Sliding mode control

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features of the proposed adaptive sliding mode controller are as follows: (i) The structured (or parametric) uncertainties and unstructured uncertainties (un-modeled dynamics, unknown external disturbances) are synthesized into a single type uncertainty term called lumped uncertainty. Therefore, a linearly parameterized dynamic model of the system is not required, and the simple structure and computationally efficient properties of this approach make it suitable for the real-time control applications. (ii) The adaptive sliding mode control scheme design relies on the online estimated uncertainty vector rather than relying on the worst-case scenario (i.e., bounds of uncertainties). Therefore, a-priory knowledge of the bounds of uncertainties is not required, and at each time instant, the control input compensates for the uncertainty that exists. (iii) The developed continuous control law using fundamentals of the sliding mode control theory eliminates the chattering phenomena without trade-off between performance and robustness, which is prevalent in boundary-layer approach.

16481: 15886: 9037: 16476:{\displaystyle {\begin{aligned}{\dot {\hat {\mathbf {x} }}}&=A{\hat {\mathbf {x} }}+B\mathbf {u} +LM\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-1\\L_{2}\end{bmatrix}}M\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\\&=A{\hat {\mathbf {x} }}+{\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}}{\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}\\&=A_{\text{obs}}{\hat {\mathbf {x} }}+B_{\text{obs}}\mathbf {u} _{\text{obs}}\end{aligned}}} 8585: 6967: 306:, and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the theoretical description of sliding modes, the system stays confined to the sliding surface and need only be viewed as sliding along the surface. However, real implementations of sliding mode control approximate this theoretical behavior with a high-frequency and generally non-deterministic switching control signal that causes the system to "chatter" in a tight neighborhood of the sliding surface. Chattering can be reduced through the use of 9032:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\overbrace {{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}{\dot {\mathbf {x} }}} ^{{\dot {\sigma }}(\mathbf {x} )}={\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}=\overbrace {} ^{\frac {\partial {\sigma (\mathbf {x} )}}{\partial {\mathbf {x} }}}\underbrace {\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u\right)} ^{\dot {\mathbf {x} }}} _{{\text{( i.e., an }}n\times 1{\text{ vector )}}}} 15467: 97: 6587: 2897: 15037: 6962:{\displaystyle {\dot {\mathbf {x} }}=\overbrace {f(\mathbf {x} ,t)-B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)} ^{f(\mathbf {x} ,t)+B(\mathbf {x} ,t)u}=f(\mathbf {x} ,t)\left(\mathbf {I} -B(\mathbf {x} ,t)\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}\right)} 3526: 13376: 4364: 5055: 2059:) exists and is reachable along system trajectories. The principle of sliding mode control is to forcibly constrain the system, by suitable control strategy, to stay on the sliding surface on which the system will exhibit desirable features. When the system is constrained by the sliding control to stay on the sliding surface, the system dynamics are governed by reduced-order system obtained from Equation ( 2707: 15462:{\displaystyle {\mathord {\overbrace {\begin{bmatrix}{\dot {e}}_{2}\\{\dot {e}}_{3}\\\vdots \\{\dot {e}}_{n}\end{bmatrix}} ^{{\dot {\mathbf {e} }}_{2}}}}=A_{2}{\mathord {\overbrace {\begin{bmatrix}e_{2}\\e_{3}\\\vdots \\e_{n}\end{bmatrix}} ^{\mathbf {e} _{2}}}}+L_{2}v(e_{1})=A_{2}\mathbf {e} _{2}+L_{2}v_{\text{eq}}=A_{2}\mathbf {e} _{2}+L_{2}A_{12}\mathbf {e} _{2}=(A_{2}+L_{2}A_{12})\mathbf {e} _{2}.} 3320: 13059: 4158: 4856: 8352: 14913: 11658:. In these sliding mode observers, the order of the observer dynamics are reduced by one when the system enters the sliding mode. In this particular example, the estimator error for a single estimated state is brought to zero in finite time, and after that time the other estimator errors decay exponentially to zero. However, as first described by Drakunov, a 11226: 14744: 2892:{\displaystyle \underbrace {\overbrace {\sigma ^{\intercal }} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0\qquad {\text{(i.e., }}{\tfrac {\operatorname {d} V}{\operatorname {d} t}}<0{\text{)}}} 9240: 3521:{\displaystyle {\dot {\sigma }}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\dot {\mathbf {x} }} ^{\tfrac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}={\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}} 6504: 13371:{\displaystyle {\begin{aligned}{\dot {\mathbf {e} }}&={\dot {\hat {\mathbf {x} }}}-{\dot {\mathbf {x} }}\\&=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})-A\mathbf {x} -B\mathbf {u} \\&=A({\hat {\mathbf {x} }}-\mathbf {x} )+Lv({\hat {x}}_{1}-x_{1})\\&=A\mathbf {e} +Lv(e_{1})\end{aligned}}} 4359:{\displaystyle {\mathord {\underbrace {D^{+}{\Bigl (}{\mathord {\underbrace {2{\mathord {\overbrace {\sqrt {V}} ^{{}\propto \|\sigma \|_{2}}}}} _{W}}}{\Bigr )}} _{D^{+}W\,\triangleq \,{\mathord {{\text{Upper right-hand }}{\dot {W}}}}}}}={\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu } 5050:{\displaystyle \underbrace {\overbrace {\sigma ^{\intercal }} ^{\tfrac {\partial V}{\partial \sigma }}\overbrace {\dot {\sigma }} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} _{\tfrac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\mathord {\overbrace {\|\sigma \|_{2}} ^{\sqrt {V}}}})^{\alpha }} 6366: 11797: 11404: 8187: 394:

or a similar technique of applying a continuous signal to an output that can only take discrete states. Sliding mode control has many applications in robotics. In particular, this control algorithm has been used for tracking control of unmanned surface vessels in simulated rough seas with high degree

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One application of sliding mode controller is the control of electric drives operated by switching power converters. Because of the discontinuous operating mode of those converters, a discontinuous sliding mode controller is a natural implementation choice over continuous controllers that may need to

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that have more moderate control action. In particular, because actuators have delays and other imperfections, the hard sliding-mode-control action can lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics. Continuous control design methods are not as susceptible to these

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origin. One of the compensatory methods is the adaptive sliding mode control method proposed in which uses estimated uncertainty to construct continuous control law. In this method chattering is eliminated while preserving accuracy (for more details see references and ). The three distinguished

10294: 5368: 8528: 10994: 895: 712: 14035: 1444:, it has the ability to drive trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better than asymptotic). However, once the trajectories reach the sliding surface, the system takes on the character of the sliding mode (e.g., the origin 366:. Because the control can be as simple as a switching between two states (e.g., "on"/"off" or "forward"/"reverse"), it need not be precise and will not be sensitive to parameter variations that enter into the control channel. Additionally, because the control law is not a 16721: 14562: 5764: 3957: 16617: 17161:

having finite frequency and amplitude. Chattering is a harmful phenomenon because it leads to low control accuracy, high wear of moving mechanical parts, and high heat losses in power circuits. For more details, see Utkin, Vadim; Lee, Jason Hoon (July 2006),

17116: – A topology that combines four switches forming the four legs of an "H". Can be used to drive a motor (or other electrical device) forward or backward when only a single supply is available. Often used in actuator in sliding-mode controlled systems. 12527: 9096: 8078: 6377: 63:

method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will

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Although various theories exist for sliding mode control system design, there is a lack of a highly effective design methodology due to practical difficulties encountered in analytical and numerical methods. A reusable computing paradigm such as a

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is uniformly bounded away from zero, the sliding mode will be maintained as if there was no disturbance. The invariance property of sliding mode control to certain disturbances and model uncertainties is its most attractive feature; it is strongly

7731: 6243: 11671: 8347:{\displaystyle {\dot {V}}(\sigma (\mathbf {x} ))=\overbrace {\sigma (\mathbf {x} )^{\text{T}}} ^{\tfrac {\partial V}{\partial \mathbf {x} }}\overbrace {{\dot {\sigma }}(\mathbf {x} )} ^{\tfrac {\operatorname {d} \sigma }{\operatorname {d} t}}} 4068: 3756: 11550: 11241: 9831: 17062: 14145: 12092: 11646:. These non-linear high-gain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time. Additionally, switched-mode observers have attractive measurement noise resilience that is similar to a 523: 7142:

It follows from these theorems that the sliding motion is invariant (i.e., insensitive) to sufficiently small disturbances entering the system through the control channel. That is, as long as the control is large enough to ensure that

1998: 14908:{\displaystyle {\mathord {\underbrace {v_{\text{eq}}} _{\text{scalar}}}}={\mathord {\underbrace {A_{12}} _{1\times (n-1) \atop {\text{ vector}}}}}{\mathord {\underbrace {\mathbf {e} _{2}} _{(n-1)\times 1 \atop {\text{ vector}}}}}.} 13732: 5665: 11976: 11888: 2422: 5988: 1735:(i.e., it switches from one state to another as trajectories move across this surface), the surface is reached in finite time. Once a trajectory reaches the surface, it will slide along it and may, for example, move toward the 10825: 1031: 310:

or boundary layers around the sliding surface, or other compensatory methods. Although the system is nonlinear in general, the idealized (i.e., non-chattering) behavior of the system in Figure 1 when confined to the

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can, however, be utilized to transform a 'unsolvable problem' of optimal design into a practically solvable 'non-deterministic polynomial problem'. This results in computer-automated designs for sliding model control.

10114: 5273: 967: 11221:{\displaystyle u(x,{\dot {x}})={\begin{cases}|{\dot {x}}|+k+1&{\text{if }}\underbrace {x+{\dot {x}}} <0,\\-\left(|{\dot {x}}|+k+1\right)&{\text{if }}\overbrace {x+{\dot {x}}} ^{\sigma }>0\end{cases}}} 8429: 5262: 10983: 5176: 734: 551: 14332: 357:

to slide along the restricted sliding mode subspace. Trajectories from this reduced-order sliding mode have desirable properties (e.g., the system naturally slides along it until it comes to rest at a desired

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must be more strongly bounded away from zero. That is, if it vanishes too quickly, the attraction to the sliding mode will only be asymptotic. To ensure that the sliding mode is entered in finite time,

16868: 16807: 12282: 15836: 14236: 12859: 7186: 4117: 3680: 3019: 16624: 12344: 14739:{\displaystyle 0={\dot {\sigma }}=a_{11}{\mathord {\overbrace {e_{1}} ^{{}=0}}}+A_{12}\mathbf {e} _{2}-{\mathord {\overbrace {v_{\text{eq}}} ^{v(\sigma )}}}=A_{12}\mathbf {e} _{2}-v_{\text{eq}}.} 10669: 15891: 13064: 10481: 7137: 7063: 6572: 6038: 5861: 5518: 3638: 2687: 2322: 2251: 2212: 2170: 2127: 2057: 5673: 7906: 6199: 3871: 17759:

Continuous Nested Algorithms: The Fifth Generation of Sliding Mode Controllers. In: Recent Advances in Sliding Modes: From Control to Intelligent Mechatronics, X. Yu, O. Efe (Eds.), pp. 5–35

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equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control

9612: 7344: 4638: 12689: 12208: 3277: 3173: 16529: 12916: 6109: 2648: 1136: 14183: 13822: 10107: 8570: 16523: 4820: 1613: 16941: 13441: 6235: 3562: 1269: 9889: 9235:{\displaystyle u(\mathbf {x} )={\begin{cases}u^{+}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )>0\\u^{-}(\mathbf {x} )&{\text{if }}\sigma (\mathbf {x} )<0\end{cases}}} 8170: 1763: 1472: 4150: 215: 11445: 9527: 9486: 7858: 5088: 2941: 2600: 1698: 1653: 13503: 12408: 16976: 12793: 10585: 10400: 6499:{\displaystyle \mathbf {u} =-\left({\frac {\partial \sigma }{\partial \mathbf {x} }}B(\mathbf {x} ,t)\right)^{-1}{\frac {\partial \sigma }{\partial \mathbf {x} }}f(\mathbf {x} ,t)} 6069: 3308: 3204: 270: 14429: 13618: 11617: 7917: 7527: 17128: – Another (feedback) method of encoding a continuous range of values in a signal that rapidly switches between two states (i.e., a kind of specialized sliding-mode control) 15670: 9384: 9284: 8142:-dimensional sliding mode simplex). This surface may have favorable properties (e.g., when the plant dynamics are forced to slide along this surface, they move toward the origin 1422:; however, the sliding mode is enforced by creasing the vector field with high-gain feedback. Like a marble rolling along a crack, trajectories are confined to the sliding mode. 7098: 5437: 3817: 15614: 12539: 10524: 10339: 4672: 1410:. Sliding mode control forces the system trajectories into this subspace and then holds them there so that they slide along it. This reduced-order subspace is referred to as a 15711: 15499: 14498: 13764: 9695: 9419: 9319: 7215: 5547: 4538: 4500: 3591: 3099: 10069: 9089: 8422: 8391: 6361:{\displaystyle {\frac {\partial \sigma }{\partial \mathbf {x} }}\overbrace {\left(f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\mathbf {u} \right)} ^{\dot {\mathbf {x} }}=\mathbf {0} } 3237: 3133: 3050: 1729: 1167: 1074: 13542: 12725: 11792:{\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=A\mathbf {x} +B\mathbf {u} \\y={\begin{bmatrix}1&0&0&\cdots &\end{bmatrix}}\mathbf {x} =x_{1}\end{cases}}} 7595: 7461: 7267: 15871: 15773: 15537: 14943: 14554: 7817: 7417: 4726: 2386:

produces outputs that only assume two states, but the effective output swings through a continuous range of motion. These complications can be avoided by using a different

1880: 16998: 16746: 13848: 12370: 12002: 11399:{\displaystyle u(x,{\dot {x}})=-(|{\dot {x}}|+k+1)\underbrace {\operatorname {sgn} (\overbrace {{\dot {x}}+x} ^{\sigma })} _{{\text{(i.e., tests }}\sigma >0{\text{)}}}} 10550: 10365: 9445: 9345: 6529: 6135: 5822: 5479: 4425: 2088: 1848: 1822: 1576: 1550: 1340: 1318: 1193: 4780: 4696: 14527: 7785: 5890: 5796: 4580: 4006: 3998: 3689: 11452: 7376: 5916: 5573: 5398: 5206: 3785: 9702: 4445: 16908: 14380: 13784: 12758: 12403: 9653: 5457: 5112: 4840: 3070: 1400: 17003: 15742: 15029: 12125: 15564: 15002: 14975: 14456: 14043: 13569: 12152: 12010: 8140: 4394: 3863: 1912: 1367: 1296: 427: 11556:

So once the system reaches the sliding mode, the system's 2-dimensional dynamics behave like this 1-dimensional system, which has a globally exponentially stable

10859: 9919: 8108: 7576: 7065:, and the trajectory conditions from the reaching phase now reduce to the above derived simpler condition. Hence, the system can be assumed to follow the simpler 4756: 2974: 1796: 335: 304: 154: 128: 14341:

must be greater than a scaled version of the maximum possible estimator errors for the system (i.e., the initial errors, which are assumed to be bounded so that

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condition after some initial transient during the period while the system finds the sliding mode. The same motion is approximately maintained when the equality

13626: 2545:{\displaystyle V(\sigma (\mathbf {x} ))={\frac {1}{2}}\sigma ^{\intercal }(\mathbf {x} )\sigma (\mathbf {x} )={\frac {1}{2}}\|\sigma (\mathbf {x} )\|_{2}^{2}} 47:

control signal (or more rigorously, a set-valued control signal) that forces the system to "slide" along a cross-section of the system's normal behavior. The

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of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a

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design method that produces a continuous controller. In some cases, sliding-mode control designs can be approximated by other continuous control designs.

17203: 10684: 1402:, then the output will track the desired sinusoid. In sliding-mode control, the designer knows that the system behaves desirably (e.g., it has a stable 972: 10289:{\displaystyle {\dot {\sigma }}={\dot {x}}_{1}+{\dot {x}}_{2}={\dot {x}}+{\ddot {x}}={\dot {x}}\,+\,\overbrace {a(t,x,{\dot {x}})+u} ^{\ddot {x}}} 5363:{\displaystyle \operatorname {sgn} (\sigma )\neq \operatorname {sgn} ({\dot {\sigma }})\qquad {\text{and}}\qquad |{\dot {\sigma }}|\geq \mu >0} 104:

trajectory of a system being stabilized by a sliding mode controller. After the initial reaching phase, the system states "slides" along the line

16910:). However, a similar procedure can be used to design a sliding mode observer for a vector of weighted combinations of states (i.e., when output 12921: 8523:{\displaystyle {\begin{cases}{\dot {\sigma }}<0&{\text{if }}\sigma >0\\{\dot {\sigma }}>0&{\text{if }}\sigma <0\end{cases}}} 914: 15838:

forces the estimate of the measured state to have zero error in finite time. Additionally, white zero-mean symmetric measurement noise (e.g.,

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First-Order Continuous Adaptive Sliding Mode Control for Robot Manipulators with Finite-Time Convergence of Trajectories to Real sliding Mode

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that ensures that there is always a control that can move a trajectory to move closer to the sliding mode. Then, once the sliding mode where

5214: 890:{\displaystyle \mathbf {u} (t)\triangleq {\begin{bmatrix}u_{1}(t)\\u_{2}(t)\\\vdots \\u_{m-1}(t)\\u_{m}(t)\end{bmatrix}}\in \mathbb {R} ^{m}} 707:{\displaystyle \mathbf {x} (t)\triangleq {\begin{bmatrix}x_{1}(t)\\x_{2}(t)\\\vdots \\x_{n-1}(t)\\x_{n}(t)\end{bmatrix}}\in \mathbb {R} ^{n}} 10866: 5120: 14030:{\displaystyle {\dot {\sigma }}={\dot {e}}_{1}=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(e_{1})=a_{11}e_{1}+A_{12}\mathbf {e} _{2}-v(\sigma )} 14247: 1271:. That is, under the control law, whenever the system is started away from the origin, it will return to it. For example, the component 13009: 1433:

or a manifold (i.e., the sliding surface) such that the system trajectory exhibits desirable behavior when confined to this manifold.

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of this distance. So the sliding-mode control acts like a stiff pressure always pushing in the direction of the sliding mode where

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For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output

16812: 16751: 16716:{\displaystyle u_{\text{obs}}\triangleq {\begin{bmatrix}\mathbf {u} \\\operatorname {sgn} ({\hat {x}}_{1}-x_{1})\end{bmatrix}}.} 12215: 15778: 14191: 12805: 7146: 4080: 3643: 2979: 1418:

of the closed-loop system. Trajectories along this subspace can be likened to trajectories along eigenvectors (i.e., modes) of

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with respect to this differential equation, then the system will slide along the sliding mode surface toward the equilibrium.

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must always be bounded firmly away from zero. To ensure this condition, sliding mode controllers are discontinuous across the

17858: 17793: 17774: 17747: 17709: 17690: 17671: 17629: 17604: 17564: 17499: 17433: 17408: 17383: 17264:. 15th International Workshop on Variable Structure Systems and Sliding Mode Control. Graz University of Technology, Austria. 17215: 12289: 17728: 17542: 1320:

may represent the difference some output is away from a known signal (e.g., a desirable sinusoidal signal); if the control

10590: 5759:{\displaystyle \{\mathbf {x} \in \mathbb {R} ^{n}:\sigma ^{\intercal }(\mathbf {x} ){\dot {\sigma }}(\mathbf {x} )<0\}} 68:

along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a

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can be built that brings the estimation error for all estimated states to zero in a finite (and arbitrarily small) time.

10405: 3952:{\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }\leq -\mu {\sqrt {V}}.} 7103: 7029: 6538: 6004: 5827: 5484: 3604: 2653: 2288: 2217: 2178: 2136: 2093: 2023: 17823: 17312:"Autonomous Navigation and Obstacle Avoidance of Unmanned Vessels in Simulated Rough Sea States - Villanova University" 17106: – Sliding mode control is often implemented as a bang–bang control. In some cases, such control is necessary for 16612:{\displaystyle B_{\text{obs}}\triangleq {\begin{bmatrix}B&{\begin{bmatrix}-M\\L_{2}M\end{bmatrix}}\end{bmatrix}},} 7865: 6158: 5863:. Moreover, if the reachability conditions from Theorem 1 are satisfied, the sliding mode will enter the region where 280:

Figure 1 shows an example trajectory of a system under sliding mode control. The sliding surface is described by

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as the system both flows through a continuous state space but also moves through different discrete control modes.

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Zeinali M.; Notash L. (2010). "Adaptive sliding mode control with uncertainty estimator for robot manipulators".

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Mahini; et al. (2013). "An experimental setup for autonomous operation of surface vessels in rough seas".

12522:{\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},\dots ,{\hat {x}}_{n})\in \mathbb {R} ^{n}} 9840: 8145: 1738: 1447: 4122: 2278: 2020:

The vital part of SMC design is to choose a control law so that the sliding mode (i.e., this surface given by

1042: 159: 11413: 9495: 9454: 8073:{\displaystyle s_{1}{\dot {x}}_{1}+s_{2}{\dot {x}}_{2}+\cdots +s_{n-1}{\dot {x}}_{n-1}+s_{n}{\dot {x}}_{n}=0} 7826: 5063: 4448: 2909: 2575: 2270: 1666: 1621: 1407: 13453: 156:

surface is chosen because it has desirable reduced-order dynamics when constrained to it. In this case, the

16950: 12763: 10555: 10370: 6043: 4842:, this result implies that the rate of approach to the sliding mode must be firmly bounded away from zero. 3282: 3178: 223: 14385: 13574: 12645:{\displaystyle {\dot {\hat {\mathbf {x} }}}=A{\hat {\mathbf {x} }}+B\mathbf {u} +Lv({\hat {x}}_{1}-x_{1})} 11563: 7473: 15619: 9351: 9251: 4397: 2903: 2282: 7068: 5403: 3790: 17915: 17073: 15569: 10488: 10303: 7726:{\displaystyle \sigma (\mathbf {x} )\triangleq s_{1}x_{1}+s_{2}x_{2}+\cdots +s_{n-1}x_{n-1}+s_{n}x_{n}} 4647: 60: 17275:

Utkin, Vadim I. (1993). "Sliding Mode Control Design Principles and Applications to Electric Drives".

15687: 15475: 14461: 13740: 9658: 9395: 9295: 7191: 5523: 4505: 4454: 3567: 3075: 17078: 10989:

So the system can be feedback stabilized (to return to the sliding mode) by means of the control law

10030: 9063: 8396: 8361: 3211: 3107: 3024: 1703: 1141: 1048: 81: 13511: 12694: 11680: 11033: 9711: 9122: 8438: 7425: 7231: 17467: 17131: 16809:, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal 15849: 15751: 15504: 14921: 14532: 7790: 7384: 4701: 4063:{\displaystyle {\frac {1}{\sqrt {V}}}{\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ,} 3751:{\displaystyle {\frac {\operatorname {d} V}{\operatorname {d} t}}\leq -\mu ({\sqrt {V}})^{\alpha }} 1853: 17289: 17140: – Another modulation scheme that produces continuous motion through discontinuous switching. 16981: 16729: 13831: 12353: 11985: 11659: 11545:{\displaystyle {\dot {x}}=-x\qquad {\text{(i.e., }}\sigma (x,{\dot {x}})=x+{\dot {x}}=0{\text{)}}} 10529: 10344: 9428: 9328: 6512: 6118: 5805: 5462: 4403: 2071: 1831: 1805: 1559: 1533: 1323: 1301: 1176: 17548: 17538: 9826:{\displaystyle {\begin{cases}{\dot {x}}_{1}=x_{2}\\{\dot {x}}_{2}=a(t,x_{1},x_{2})+u\end{cases}}} 4761: 4677: 14503: 7757: 5866: 5772: 5769:

That is, when initial conditions come entirely from this space, the Lyapunov function candidate

4543: 3965: 17462: 17284: 17176: 17137: 17125: 17057:{\displaystyle \sigma (\mathbf {x} )\triangleq {\hat {\mathbf {y} }}-\mathbf {y} =\mathbf {0} } 11233: 9619: 7539: 7352: 5994: 5895: 5552: 5377: 5185: 3764: 2383: 2379: 909: 722: 391: 48: 14140:{\displaystyle \mathbf {e} _{2}\triangleq (e_{2},e_{3},\ldots ,e_{n})\in \mathbb {R} ^{(n-1)}} 12087:{\displaystyle A\triangleq {\begin{bmatrix}a_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}}} 9386:

is some control (e.g., possibly extreme, like "off" or "reverse") that ensures Equation (

4430: 1414:, and when closed-loop feedback forces trajectories to slide along it, it is referred to as a 17591:. Lecture Notes in Control and Information Sciences. Vol. 193. London: Springer-Verlag. 17103: 16880: 14352: 13769: 12730: 12375: 9625: 9492:

back and forth along this sliding surface (i.e., the true trajectory may not smoothly follow

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is some control (e.g., possibly extreme, like "on" or "forward") that ensures Equation (

5442: 5097: 4825: 3055: 1372: 518:{\displaystyle {\dot {\mathbf {x} }}(t)=f(\mathbf {x} ,t)+B(\mathbf {x} ,t)\,\mathbf {u} (t)} 379: 15720: 15007: 12103: 15542: 14980: 14948: 14434: 14147:

is the collection of the estimator errors for all of the unmeasured states. To ensure that

13547: 12130: 8113: 4372: 3830: 2947: 2266: 1993:{\displaystyle \left\{\mathbf {x} \in \mathbb {R} ^{n}:\sigma _{k}(\mathbf {x} )=0\right\}} 1885: 1345: 1274: 1210: 1087: 342: 273: 17757:

Fridman, L.; Moreno, J.; Bandyopadhyay, B.; Kamal Asif Chalanga, S.; Chalanga, S. (2015).

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Finding feedback gains so that the system trajectory intersects and stays on the manifold.

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is constant, and so sliding mode trajectories are described by the differential equation

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is achieved, the system will stay on that sliding mode. Along sliding mode trajectories,

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Drakunov, S.V. (1983). "An adaptive quasioptimal filter with discontinuous parameters".

13727:{\displaystyle \sigma ({\hat {x}}_{1},{\hat {x}})\triangleq e_{1}={\hat {x}}_{1}-x_{1}.} 17829: 17786:

Advances in Variable Structure Systems and Sliding Mode Control—Theory and Applications

17570: 17340: 6142: 5660:{\displaystyle \{\mathbf {x} \in \mathbb {R} ^{n}:\sigma (\mathbf {x} )=\mathbf {0} \}} 1196: 350: 12210:

is a row vector corresponding to the influence of the first state on the other states,

17854: 17833: 17819: 17803: 17789: 17770: 17743: 17724: 17705: 17686: 17667: 17648: 17625: 17600: 17574: 17560: 17495: 17450: 17429: 17404: 17379: 17211: 17088: 15846:, and hence the noise will have little effect on the equivalent sliding mode control 13825: 12346:

is a column vector representing the influence of the other states on the first state.

11971:{\displaystyle \mathbf {u} \triangleq (u_{1},u_{2},\dots ,u_{r})\in \mathbb {R} ^{r}} 11883:{\displaystyle \mathbf {x} \triangleq (x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}} 11650:. For simplicity, the example here uses a traditional sliding mode modification of a 11630: 8173: 7225:

As discussed in an example below, a sliding mode control law can keep the constraint

5799: 2407: 2387: 399: 32: 17845:. Studies in Systems, Decision and Control. Vol. 271. London: Springer-Verlag. 17451:"Genetic algorithm automated approach to the design of sliding mode control systems" 17344: 5983:{\displaystyle {\frac {\partial \sigma }{\partial {\mathbf {x} }}}B(\mathbf {x} ,t)} 17892: 17884: 17846: 17811: 17788:. Studies in Systems, Decision and Control. Vol. 24. London: Springer-Verlag. 17762: 17761:. Studies in Systems, Decision and Control. Vol. 24. London: Springer-Verlag. 17592: 17552: 17472: 17369: 17365: 17332: 17294: 17240: 11557: 10820:{\displaystyle |{\dot {x}}|+|a(t,x,{\dot {x}})|\geq |{\dot {x}}+a(t,x,{\dot {x}})|} 6138: 1731:

trajectories will approach the sliding surface, and because the control law is not

1403: 1200: 412: 359: 96: 40: 16947:). In each case, the sliding mode will be the manifold where the estimated output 12795:

is an observer gain vector that serves a similar purpose as in the typical linear

17107: 17098: 12350:

The goal is to design a high-gain state observer that estimates the state vector

5998: 5439:

toward the switching surface should have a non-zero lower bound. So, even though

2378:

however, this smooth behavior may not be truly realizable. Similarly, high-speed

1766: 1094: 1038: 1026:{\displaystyle B:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n\times m}} 383: 375: 17766: 5892:

is more strongly bounded away from zero in finite time. Hence, the sliding mode

17640: 17617: 17093: 15714: 15673: 13443:

is the estimator error for the first state estimate. The nonlinear control law

12796: 11643: 9535: 7464: 7219: 5091: 4783: 2603: 2325: 1769:

with a contour of constant height along which trajectories are forced to move.

374:

time (i.e., better than asymptotic behavior). Under certain common conditions,

363: 354: 36: 20: 17897: 17888: 17875:

Drakunov S.V., Utkin V.I.. (1992). "Sliding mode control in dynamic systems".

17544:[1992] Proceedings of the 31st IEEE Conference on Decision and Control 17476: 17336: 17311: 7463:). That is, when the system is constrained this way, it behaves like a simple 17909: 17756: 17375: 17083: 15874: 15681: 12284:

is a matrix representing the influence of the other states on themselves, and

11647: 7467: 1659:

The sliding-mode-control law switches from one state to another based on the

725: 85: 44: 17815: 17556: 12999:{\displaystyle \mathbf {e} =(e_{1},e_{2},\dots ,e_{n})\in \mathbb {R} ^{n}} 1430: 3596: 2398:

The following theorems form the foundation of variable structure control.

17158: 5374:

That is, the system should always be moving toward the switching surface

962:{\displaystyle f:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}} 101: 17259: 8572:

because it is the product of a negative and a positive number. Note that

398:

Sliding mode control must be applied with more care than other forms of

17850: 17596: 15677: 11655: 5579:

from one non-zero value to another as trajectories cross the manifold.

5257:{\displaystyle \operatorname {sgn} (\sigma ){\dot {\sigma }}\leq -\mu } 4074: 2285:

sense. Roughly speaking, the resulting closed-loop system moving along

1419: 338: 218: 17683:

Modern Sliding Mode Control Theory - New perspectives and applications

17298: 9451:

The resulting trajectory should move toward the sliding surface where

2701:), a sufficient condition for the existence of a sliding mode is that 15713:

system can be stabilized in exactly the same way as a typical linear

3827:

This condition ensures that for the neighborhood of the sliding mode

1083: 10978:{\displaystyle |{\dot {x}}|+k+1>|{\dot {x}}|+|a(t,x,{\dot {x}})|} 6972:

The resulting system matches the sliding mode differential equation

5824:

trajectories are sure to move toward the sliding mode surface where

5171:{\displaystyle \sigma {\dot {\sigma }}\leq -\mu |\sigma |^{\alpha }} 17113: 9488:. Because real systems have delay, sliding mode trajectories often 7470:, and so it has a globally exponentially stable equilibrium at the 6581:

Likewise, the system trajectories on the sliding mode behave as if

4758:(i.e., the system enters the sliding mode) in finite time. Because 905: 307: 52: 14349:

is sufficiently large, it can be assumed that the system achieves

14327:{\displaystyle M>\max\{|a_{11}e_{1}+A_{12}\mathbf {e} _{2}|\}.} 9529:, but it will always return to the sliding mode after leaving it). 8084:

which is a reduced-order system (i.e., the new system is of order

7378:

has a finite upper bound. In this case, the sliding mode is where

4850:

In the context of sliding mode control, this condition means that

11410:

Assuming that the system trajectories are forced to move so that

10027:

By the previous example, we must choose the feedback control law

7820: 84:, like a system under SMC, may be viewed as a special case of a 17122: – Uses switching-mode control to drive continuous outputs 13046:{\displaystyle \mathbf {e} ={\hat {\mathbf {x} }}-\mathbf {x} } 9925:

that is known). For this system, choose the switching function

1772:

The sliding (hyper)surface/manifold is typically of dimension

17645:

Variable Structure Systems: From Principles to Implementation

12691:

is a nonlinear function of the error between estimated state

10016:{\displaystyle \sigma (x_{1},x_{2})=x_{1}+x_{2}=x+{\dot {x}}} 2214:, the control action is capable of maintaining the system at 17718: 7860:. When trajectories are forced to slide along this surface, 7272:

in order to asymptotically stabilize any system of the form

2371:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\mathbf {0} ;} 1523:{\displaystyle \sigma :\mathbb {R} ^{n}\to \mathbb {R} ^{m}} 403:

problems and can be made to mimic sliding-mode controllers.

349:

Intuitively, sliding mode control uses practically infinite

17719:

Shtessel, Y.; Edwards, C.; Fridman, L.; Levant, A. (2014).

17539:"Sliding-mode observers based on equivalent control method" 17157:'Chatter' or 'chattering' is the undesirable phenomenon of 11785: 11214: 9819: 9228: 8516: 7016:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=\mathbf {0} } 17874: 17802: 17680: 17661: 17401:

Differential Equations with Discontinuous Right-hand Sides

16863:{\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})} 16802:{\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})} 12277:{\displaystyle A_{22}\in \mathbb {R} ^{(n-1)\times (n-1)}} 12127:

is a scalar representing the influence of the first state

1828:

is the number of input signals (i.e., control signals) in

17740:

Sliding Mode Control. The Delta-Sigma Modulation Approach

17704:. London: The Institution of Engineering and Technology. 17681:

Bartolini, G.; Fridman, L.; Pisano, A.; Usai, E. (2008).

15831:{\displaystyle v=M\operatorname {sgn} ({\hat {x}}_{1}-x)} 14231:{\displaystyle v(\sigma )=M\operatorname {sgn} (\sigma )} 12854:{\displaystyle L={\begin{bmatrix}-1\\L_{2}\end{bmatrix}}} 7181:{\displaystyle \sigma ^{\intercal }{\dot {\sigma }}<0} 4112:{\displaystyle \operatorname {d} W/{\operatorname {d} t}} 3675:{\displaystyle \operatorname {d} V/{\operatorname {d} t}} 3014:{\displaystyle \sigma ^{\intercal }{\dot {\sigma }}<0} 17638: 4845: 17784:

Li, S.; Yu, X.; Fridman, L.; Man, Z.; Wang, X. (2017).

17664:

Advances in Variable Structure and Sliding Mode Control

14529:

may be replaced with the equivalent continuous control

3597:

Reachability: Attaining sliding manifold in finite time

2401: 1406:) provided that it is constrained to a subspace of its 17840: 16646: 16564: 16551: 16346: 16299: 16286: 16173: 16049: 15842:) only affects the switching frequency of the control 15185: 15050: 13620:). Hence, the sliding mode control switching function 12820: 12339:{\displaystyle A_{12}\in \mathbb {R} ^{1\times (n-1)}} 12025: 11734: 8316: 8258: 4959: 4924: 4883: 3382: 2851: 2810: 2775: 2734: 2269:, and so existence and uniqueness of solutions to the 904:-dimensional input vector that will be used for state 760: 577: 17841:

Steinberger, M.; Horn, M.; Fridman, L., eds. (2020).

17233:

International Journal of Mechanism and Machine Theory

17134: – A generalized form of delta-sigma modulation. 17006: 16984: 16953: 16916: 16883: 16815: 16754: 16732: 16627: 16532: 16495: 15889: 15852: 15781: 15754: 15723: 15690: 15622: 15572: 15545: 15507: 15478: 15040: 15010: 14983: 14951: 14924: 14758: 14565: 14535: 14506: 14464: 14437: 14388: 14355: 14250: 14194: 14153: 14046: 13859: 13834: 13792: 13772: 13743: 13629: 13577: 13550: 13514: 13456: 13387: 13062: 13012: 12924: 12870: 12808: 12766: 12733: 12697: 12661: 12542: 12411: 12378: 12356: 12292: 12218: 12162: 12133: 12106: 12013: 11988: 11896: 11808: 11674: 11566: 11455: 11416: 11244: 10997: 10869: 10837: 10687: 10593: 10558: 10532: 10491: 10408: 10373: 10347: 10306: 10117: 10077: 10033: 9933: 9897: 9843: 9705: 9661: 9628: 9549: 9498: 9457: 9431: 9398: 9354: 9331: 9298: 9254: 9099: 9066: 8588: 8540: 8432: 8399: 8364: 8190: 8148: 8116: 8090: 7920: 7868: 7829: 7793: 7760: 7598: 7558: 7476: 7428: 7387: 7355: 7281: 7234: 7194: 7149: 7106: 7071: 7032: 6981: 6590: 6541: 6515: 6380: 6246: 6210: 6161: 6121: 6080: 6046: 6007: 5935: 5898: 5869: 5830: 5808: 5775: 5676: 5605: 5555: 5526: 5487: 5465: 5445: 5406: 5380: 5276: 5217: 5188: 5123: 5100: 5066: 4859: 4828: 4791: 4764: 4738: 4704: 4680: 4650: 4588: 4546: 4508: 4457: 4433: 4406: 4375: 4161: 4125: 4083: 4009: 3968: 3874: 3833: 3793: 3767: 3692: 3646: 3607: 3570: 3537: 3323: 3285: 3245: 3214: 3181: 3141: 3110: 3078: 3058: 3027: 2982: 2956: 2912: 2710: 2656: 2650:

is the distance away from the sliding manifold where

2612: 2578: 2425: 2333: 2291: 2256: 2220: 2181: 2139: 2096: 2074: 2026: 1929: 1888: 1856: 1834: 1808: 1778: 1741: 1706: 1669: 1624: 1584: 1562: 1536: 1530:

that represents a kind of "distance" that the states

1486: 1474:

may only have asymptotic stability on this surface).

1450: 1375: 1348: 1326: 1304: 1277: 1218: 1179: 1144: 1102: 1051: 975: 917: 737: 554: 430: 317: 286: 226: 162: 136: 110: 17808:

Advanced and Optimization Based Sliding Mode Control

17699: 17490:

Utkin, Vadim; Guldner, Jürgen; Shi, Jingxin (1999).

10664:{\displaystyle u<-|{\dot {x}}+a(t,x,{\dot {x}})|} 6535:, the rapid switching across the sliding mode where 362:). The main strength of sliding mode control is its 17843:

Variable-Structure Systems and Sliding-Mode Control

17662:Edwards, C.; Fossas Colet, E.; Fridman, L. (2006). 17230: 15031:acts like an output vector for the error subsystem 10476:{\displaystyle u>|{\dot {x}}+a(t,x,{\dot {x}})|} 5114:is scalar valued, the sufficient condition becomes 17164:Chattering Problem in Sliding Mode Control Systems 17056: 16992: 16970: 16935: 16902: 16862: 16801: 16740: 16715: 16611: 16517: 16475: 15865: 15830: 15767: 15736: 15705: 15664: 15608: 15558: 15531: 15493: 15461: 15023: 14996: 14969: 14937: 14907: 14738: 14548: 14521: 14492: 14450: 14423: 14374: 14326: 14230: 14177: 14139: 14029: 13842: 13816: 13778: 13758: 13726: 13612: 13563: 13536: 13497: 13435: 13370: 13045: 12998: 12910: 12853: 12787: 12752: 12719: 12683: 12644: 12521: 12397: 12364: 12338: 12276: 12202: 12146: 12119: 12086: 11996: 11970: 11882: 11791: 11642:Sliding mode control can be used in the design of 11611: 11544: 11439: 11398: 11220: 10977: 10853: 10819: 10663: 10579: 10544: 10518: 10475: 10394: 10359: 10333: 10288: 10101: 10063: 10015: 9913: 9883: 9825: 9689: 9647: 9606: 9521: 9480: 9439: 9413: 9378: 9339: 9313: 9278: 9234: 9083: 9031: 8564: 8522: 8416: 8385: 8346: 8164: 8134: 8102: 8072: 7900: 7852: 7811: 7779: 7725: 7570: 7521: 7455: 7411: 7370: 7338: 7261: 7209: 7180: 7132:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 7131: 7092: 7058:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 7057: 7015: 6961: 6567:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 6566: 6523: 6498: 6360: 6229: 6193: 6129: 6103: 6063: 6033:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 6032: 5982: 5910: 5884: 5856:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 5855: 5816: 5790: 5758: 5659: 5582: 5567: 5541: 5513:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 5512: 5473: 5451: 5431: 5392: 5362: 5256: 5200: 5170: 5106: 5082: 5049: 4834: 4814: 4774: 4750: 4720: 4690: 4666: 4632: 4574: 4532: 4494: 4439: 4419: 4388: 4358: 4144: 4111: 4062: 3992: 3951: 3857: 3822: 3811: 3779: 3750: 3674: 3633:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 3632: 3585: 3556: 3520: 3302: 3271: 3231: 3198: 3167: 3127: 3093: 3064: 3044: 3013: 2968: 2935: 2891: 2695:) and the sliding surface given by Equation ( 2682:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2681: 2642: 2594: 2544: 2370: 2317:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2316: 2246:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2245: 2207:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2206: 2165:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2164: 2122:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2121: 2082: 2052:{\displaystyle \sigma (\mathbf {x} )=\mathbf {0} } 2051: 1992: 1906: 1874: 1842: 1816: 1790: 1757: 1723: 1692: 1647: 1607: 1570: 1544: 1522: 1466: 1394: 1361: 1334: 1312: 1290: 1263: 1187: 1161: 1130: 1068: 1025: 961: 889: 706: 517: 329: 298: 264: 209: 148: 122: 17492:Sliding Mode Control in Electromechanical Systems 17423: 15501:for the unmeasured states converges to zero, the 6578:as if it were driven by this continuous control. 4251: 4180: 2281:. Thus the solutions are to be understood in the 17907: 17700:Fridman, L.; Barbot, J.-P.; Plestan, F. (2016). 17647:. London: The Insitite of Electrical Engineers. 14257: 13447:can be designed to enforce the sliding manifold 9844: 7901:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=0} 6194:{\displaystyle {\dot {\sigma }}(\mathbf {x} )=0} 17615: 17494:. Philadelphia, PA: Taylor & Francis, Inc. 17489: 17175:Other pulse-type modulation techniques include 9607:{\displaystyle {\ddot {x}}=a(t,x,{\dot {x}})+u} 7339:{\displaystyle {\ddot {x}}=a(t,x,{\dot {x}})+u} 4633:{\displaystyle 2{\sqrt {V(t)}}\leq V_{0}-\mu t} 80:. In the context of modern control theory, any 17783: 15680:must be negative). Hence, provided that it is 12684:{\displaystyle v:\mathbb {R} \to \mathbb {R} } 12203:{\displaystyle A_{21}\in \mathbb {R} ^{(n-1)}} 5593:) and sliding surface given by Equation ( 4502:which is represented by differential equation 3272:{\displaystyle {\dot {\sigma }}(\mathbf {x} )} 3168:{\displaystyle {\dot {\sigma }}(\mathbf {x} )} 2133:Ensure that the system is capable of reaching 17737: 17622:Sliding Mode Control. Theory and Applications 17513: 17511: 17198: 17196: 17166:, vol. 10.1109/VSS.2006.1644542., pp. 346–350 14977:states to the trajectory of the output state 12911:{\displaystyle L_{2}\in \mathbb {R} ^{(n-1)}} 11623: 6104:{\displaystyle {\dot {\sigma }}=\mathbf {0} } 2643:{\displaystyle \|\sigma (\mathbf {x} )\|_{2}} 1131:{\displaystyle \mathbf {u} (\mathbf {x} (t))} 14318: 14260: 14178:{\displaystyle \sigma {\dot {\sigma }}<0} 13817:{\displaystyle \sigma {\dot {\sigma }}<0} 12372:using only information from the measurement 11982:is a scalar equal to the first state of the 11660:sliding mode observer for non-linear systems 10102:{\displaystyle \sigma {\dot {\sigma }}<0} 9872: 9847: 8565:{\displaystyle \sigma {\dot {\sigma }}<0} 6152:on the sliding mode can be found by solving 5921: 5753: 5677: 5654: 5606: 5077: 5067: 5012: 5005: 4803: 4792: 4222: 4215: 2631: 2613: 2589: 2579: 2528: 2510: 1765:origin. So the switching function is like a 1618:A state that is on this sliding surface has 1578:that is outside of this sliding surface has 1093:A common task is to design a state-feedback 382:; hence, sliding mode control describes the 17277:IEEE Transactions on Industrial Electronics 16518:{\displaystyle A_{\text{obs}}\triangleq A,} 14945:represents the contribution from the other 14458:is constant (i.e., 0) along this manifold, 10587:, the control law should be picked so that 10402:, the control law should be picked so that 4815:{\displaystyle \|{\mathord {\cdot }}\|_{2}} 1608:{\displaystyle \sigma (\mathbf {x} )\neq 0} 76:consisting of the boundaries is called the 17806:; Incremona, G.P.; Cucuzzella, M. (2019). 17530: 17508: 17392: 17360: 17358: 17356: 17354: 17251: 17224: 17208:Deterministic control of uncertain systems 17193: 17000:with zero error (i.e., the manifold where 16936:{\displaystyle \mathbf {y} =C\mathbf {x} } 16726:That is, by augmenting the control vector 15880:The final version of the observer is thus 14500:as well. Hence, the discontinuous control 13436:{\displaystyle e_{1}={\hat {x}}_{1}-x_{1}} 9618:which can be expressed in a 2-dimensional 8110:because the system is constrained to this 7532: 6230:{\displaystyle \mathbf {u} (\mathbf {x} )} 5267:which is equivalent to the condition that 5208:, the scalar sufficient condition becomes 3557:{\displaystyle \mathbf {u} (\mathbf {x} )} 2689:). For the system given by Equation ( 1440:Because sliding mode control laws are not 1264:{\displaystyle \mathbf {x} =^{\intercal }} 17896: 17483: 17466: 17316: 17288: 14115: 12986: 12886: 12775: 12677: 12669: 12509: 12308: 12234: 12178: 11958: 11870: 10223: 10219: 9884:{\displaystyle \sup\{|a(\cdot )|\}\leq k} 8165:{\displaystyle \mathbf {x} =\mathbf {0} } 5690: 5619: 4280: 4276: 2393: 2261:Note that because the control law is not 1945: 1758:{\displaystyle \mathbf {x} =\mathbf {0} } 1510: 1495: 1467:{\displaystyle \mathbf {x} =\mathbf {0} } 1425:The sliding-mode control scheme involves 1007: 998: 984: 949: 940: 926: 877: 694: 500: 17536: 17517: 17442: 17398: 17374:(3rd ed.). Upper Saddle River, NJ: 11637: 6509:That is, even though the actual control 4145:{\displaystyle W\triangleq 2{\sqrt {V}}} 210:{\displaystyle s=x_{1}+{\dot {x}}_{1}=0} 95: 17589:Variable Structure and Lyapunov Control 17586: 17351: 17257: 17202: 15873:. Hence, the sliding mode observer has 15744:is viewed as the output matrix (i.e., " 13786:must always have opposite signs (i.e., 13006:be the state estimator error. That is, 11440:{\displaystyle \sigma (\mathbf {x} )=0} 9522:{\displaystyle \sigma (\mathbf {x} )=0} 9481:{\displaystyle \sigma (\mathbf {x} )=0} 8575: 7853:{\displaystyle \sigma (\mathbf {x} )=0} 7585: 5587:For the system given by Equation ( 5094:. For the case when switching function 5083:{\displaystyle \|{\mathord {\cdot }}\|} 2936:{\displaystyle \sigma (\mathbf {x} )=0} 2595:{\displaystyle \|{\mathord {\cdot }}\|} 2412: 1916: 1693:{\displaystyle \sigma (\mathbf {x} )=0} 1648:{\displaystyle \sigma (\mathbf {x} )=0} 1045:can be used to guarantee that solution 417: 217:surface corresponds to the first-order 17908: 17424:Perruquetti, W.; Barbot, J.P. (2002). 17364: 17322: 17268: 13498:{\displaystyle 0={\hat {x}}_{1}-x_{1}} 12918:is a column vector. Additionally, let 1914:-dimensional sliding surface given by 17702:Recent Trends in Sliding Mode Control 17274: 16971:{\displaystyle {\hat {\mathbf {y} }}} 12788:{\displaystyle L\in \mathbb {R} ^{n}} 10580:{\displaystyle {\dot {\sigma }}<0} 10395:{\displaystyle {\dot {\sigma }}>0} 6064:{\displaystyle \sigma (\mathbf {x} )} 4846:Consequences for sliding mode control 3303:{\displaystyle \sigma (\mathbf {x} )} 3199:{\displaystyle \sigma (\mathbf {x} )} 370:, the sliding mode can be reached in 265:{\displaystyle {\dot {x}}_{1}=-x_{1}} 17721:Sliding Mode Control and Observation 17245:10.1016/j.mechmachtheory.2009.08.003 15676:(i.e., the real part of each of its 14424:{\displaystyle {\hat {x}}_{1}=x_{1}} 14345:can be picked large enough; al). If 13613:{\displaystyle {\hat {x}}_{1}=x_{1}} 12533:states. The observer takes the form 11612:{\displaystyle (x,{\dot {x}})=(0,0)} 8579: 7589: 7522:{\displaystyle (x,{\dot {x}})=(0,0)} 5997:. That is, the system has a kind of 2416: 2402:Theorem 1: Existence of sliding mode 1920: 1138:(i.e., a mapping from current state 421: 386:for a broad set of dynamic systems. 17426:Sliding Mode Control in Engineering 15665:{\displaystyle (A_{2}+L_{2}A_{12})} 9379:{\displaystyle u^{-}(\mathbf {x} )} 9279:{\displaystyle u^{+}(\mathbf {x} )} 13: 17868: 17448: 17210:. London: Peter Peregrinus Press. 15472:So, to ensure the estimator error 14868: 14811: 11978:is a vector of inputs, and output 8906: 8885: 8729: 8708: 8643: 8622: 8330: 8319: 8269: 8261: 7093:{\displaystyle {\dot {\sigma }}=0} 6943: 6935: 6887: 6879: 6728: 6720: 6672: 6664: 6465: 6457: 6409: 6401: 6258: 6250: 5947: 5939: 5432:{\displaystyle |{\dot {\sigma }}|} 4973: 4962: 4938: 4927: 4894: 4886: 4335: 4324: 4099: 4084: 4036: 4025: 3889: 3878: 3812:{\displaystyle 0<\alpha \leq 1} 3707: 3696: 3662: 3647: 3426: 3418: 3398: 3385: 3350: 3342: 2865: 2854: 2824: 2813: 2789: 2778: 2745: 2737: 2257:Existence of closed-loop solutions 1477:The sliding-mode designer picks a 16:Method in nonlinear control theory 14: 17927: 15609:{\displaystyle (n-1)\times (n-1)} 10519:{\displaystyle x+{\dot {x}}>0} 10334:{\displaystyle x+{\dot {x}}<0} 5918:will be attained in finite time. 5667:surface is reachable is given by 4728:in finite time, which means that 4667:{\displaystyle {\sqrt {V}}\geq 0} 4451:. So, by comparison to the curve 1552:are away from a sliding surface. 406: 17877:International Journal of Control 17587:Zinober, Alan S.I., ed. (1994). 17455:International Journal of Control 17050: 17042: 17028: 17014: 16986: 16958: 16929: 16918: 16734: 16650: 16459: 16434: 16350: 16268: 16161: 16144: 16037: 16020: 15946: 15929: 15900: 15706:{\displaystyle \mathbf {e} _{2}} 15693: 15494:{\displaystyle \mathbf {e} _{2}} 15481: 15446: 15392: 15357: 15309: 15248: 15143: 14852: 14710: 14645: 14493:{\displaystyle {\dot {e}}_{1}=0} 14303: 14049: 14002: 13932: 13836: 13759:{\displaystyle {\dot {\sigma }}} 13737:To attain the sliding manifold, 13335: 13267: 13253: 13229: 13218: 13160: 13143: 13116: 13094: 13071: 13039: 13025: 13014: 12926: 12591: 12574: 12549: 12416: 12358: 11990: 11898: 11810: 11765: 11715: 11704: 11687: 11424: 9690:{\displaystyle x_{2}={\dot {x}}} 9506: 9465: 9433: 9414:{\displaystyle {\dot {\sigma }}} 9369: 9333: 9314:{\displaystyle {\dot {\sigma }}} 9269: 9212: 9191: 9160: 9139: 9107: 9074: 8991: 8961: 8938: 8911: 8896: 8810: 8780: 8757: 8734: 8719: 8693: 8660: 8648: 8633: 8605: 8407: 8302: 8273: 8237: 8213: 8172:). Taking the derivative of the 8158: 8150: 7885: 7837: 7606: 7210:{\displaystyle {\dot {\sigma }}} 7125: 7114: 7051: 7040: 7009: 6998: 6947: 6905: 6891: 6857: 6843: 6824: 6796: 6773: 6746: 6732: 6690: 6676: 6642: 6619: 6595: 6560: 6549: 6517: 6483: 6469: 6427: 6413: 6382: 6354: 6339: 6321: 6307: 6284: 6262: 6220: 6212: 6178: 6123: 6097: 6054: 6026: 6015: 5967: 5952: 5849: 5838: 5810: 5740: 5717: 5681: 5650: 5639: 5610: 5542:{\displaystyle {\dot {\sigma }}} 5506: 5495: 5467: 5459:may become vanishingly small as 4533:{\displaystyle {\dot {z}}=-\mu } 4495:{\displaystyle z(t)=z_{0}-\mu t} 3626: 3615: 3601:To ensure that the sliding mode 3586:{\displaystyle {\dot {\sigma }}} 3547: 3539: 3531:and so the feedback control law 3507: 3489: 3475: 3452: 3430: 3392: 3367: 3354: 3293: 3262: 3222: 3189: 3158: 3118: 3094:{\displaystyle {\dot {\sigma }}} 3035: 2946:Roughly speaking (i.e., for the 2920: 2675: 2664: 2623: 2520: 2490: 2476: 2439: 2361: 2350: 2310: 2299: 2239: 2228: 2200: 2189: 2158: 2147: 2115: 2104: 2076: 2045: 2034: 1972: 1936: 1836: 1810: 1751: 1743: 1708: 1677: 1632: 1592: 1564: 1538: 1460: 1452: 1328: 1306: 1220: 1181: 1146: 1112: 1104: 1053: 739: 556: 502: 487: 464: 435: 17738:Sira- Ramirez, Hebertt (2015). 17417: 11477: 10064:{\displaystyle u(x,{\dot {x}})} 9084:{\displaystyle u(\mathbf {x} )} 8417:{\displaystyle u(\mathbf {x} )} 8386:{\displaystyle {\dot {V}}<0} 6204:for the equivalent control law 5583:Theorem 2: Region of attraction 5325: 5319: 3823:Explanation by comparison lemma 3232:{\displaystyle u(\mathbf {x} )} 3128:{\displaystyle u(\mathbf {x} )} 3045:{\displaystyle u(\mathbf {x} )} 2844: 1724:{\displaystyle \mathbf {x} (t)} 1162:{\displaystyle \mathbf {x} (t)} 1069:{\displaystyle \mathbf {x} (t)} 353:to force the trajectories of a 91: 17624:. London: Taylor and Francis. 17305: 17169: 17151: 17032: 17018: 17010: 16962: 16857: 16832: 16822: 16796: 16771: 16761: 16699: 16674: 16664: 16438: 16399: 16374: 16364: 16272: 16248: 16223: 16213: 16148: 16124: 16099: 16089: 16024: 16000: 15975: 15965: 15933: 15904: 15825: 15807: 15797: 15659: 15623: 15603: 15591: 15585: 15573: 15520: 15508: 15441: 15405: 15291: 15278: 14964: 14952: 14882: 14870: 14831: 14819: 14688: 14682: 14516: 14510: 14396: 14314: 14264: 14225: 14219: 14204: 14198: 14132: 14120: 14107: 14062: 14024: 14018: 13961: 13948: 13696: 13670: 13664: 13643: 13633: 13585: 13571:after some finite time (i.e., 13537:{\displaystyle {\hat {x}}_{1}} 13522: 13470: 13408: 13361: 13348: 13318: 13293: 13283: 13271: 13257: 13246: 13208: 13183: 13173: 13147: 13098: 13053:. The error dynamics are then 13029: 12978: 12933: 12903: 12891: 12720:{\displaystyle {\hat {x}}_{1}} 12705: 12673: 12639: 12614: 12604: 12578: 12553: 12501: 12489: 12461: 12439: 12429: 12420: 12331: 12319: 12269: 12257: 12251: 12239: 12195: 12183: 11950: 11905: 11862: 11817: 11665:Here, consider the LTI system 11606: 11594: 11588: 11567: 11507: 11486: 11428: 11420: 11365: 11328: 11315: 11299: 11282: 11278: 11269: 11248: 11143: 11126: 11054: 11037: 11022: 11001: 10971: 10967: 10940: 10933: 10925: 10908: 10888: 10871: 10847: 10839: 10813: 10809: 10782: 10760: 10752: 10748: 10721: 10714: 10706: 10689: 10657: 10653: 10626: 10604: 10469: 10465: 10438: 10416: 10258: 10231: 10058: 10037: 9963: 9937: 9907: 9899: 9868: 9864: 9858: 9851: 9807: 9775: 9595: 9568: 9510: 9502: 9469: 9461: 9373: 9365: 9273: 9265: 9216: 9208: 9195: 9187: 9164: 9156: 9143: 9135: 9111: 9103: 9078: 9070: 8971: 8957: 8948: 8934: 8900: 8892: 8873: 8828: 8790: 8776: 8767: 8753: 8723: 8715: 8697: 8689: 8637: 8629: 8609: 8601: 8411: 8403: 8306: 8298: 8242: 8233: 8220: 8217: 8209: 8203: 8129: 8117: 7889: 7881: 7841: 7833: 7610: 7602: 7516: 7504: 7498: 7477: 7456:{\displaystyle {\dot {x}}+x=0} 7365: 7359: 7327: 7300: 7262:{\displaystyle {\dot {x}}+x=0} 7118: 7110: 7044: 7036: 7002: 6994: 6915: 6901: 6867: 6853: 6834: 6820: 6806: 6792: 6783: 6769: 6756: 6742: 6700: 6686: 6652: 6638: 6629: 6615: 6553: 6545: 6493: 6479: 6437: 6423: 6371:and so the equivalent control 6317: 6303: 6294: 6280: 6224: 6216: 6182: 6174: 6058: 6050: 6019: 6011: 5977: 5963: 5842: 5834: 5785: 5779: 5744: 5736: 5721: 5713: 5643: 5635: 5599:), the subspace for which the 5499: 5491: 5425: 5408: 5344: 5327: 5316: 5301: 5289: 5283: 5230: 5224: 5158: 5149: 5038: 4996: 4603: 4597: 4556: 4550: 4467: 4461: 3987: 3975: 3921: 3910: 3852: 3840: 3739: 3728: 3619: 3611: 3551: 3543: 3485: 3471: 3462: 3448: 3297: 3289: 3266: 3258: 3226: 3218: 3193: 3185: 3162: 3154: 3122: 3114: 3101:have opposite signs. That is, 3039: 3031: 2924: 2916: 2668: 2660: 2627: 2619: 2524: 2516: 2494: 2486: 2480: 2472: 2446: 2443: 2435: 2429: 2354: 2346: 2324:is approximated by the smooth 2303: 2295: 2265:, it is certainly not locally 2232: 2224: 2193: 2185: 2151: 2143: 2108: 2100: 2038: 2030: 1976: 1968: 1901: 1889: 1718: 1712: 1681: 1673: 1636: 1628: 1596: 1588: 1505: 1252: 1227: 1156: 1150: 1125: 1122: 1116: 1108: 1063: 1057: 1002: 944: 861: 855: 838: 832: 802: 796: 779: 773: 749: 743: 678: 672: 655: 649: 619: 613: 596: 590: 566: 560: 512: 506: 497: 483: 474: 460: 451: 445: 1: 17520:Automation and Remote Control 17449:Li, Yun; et al. (1996). 17186: 15866:{\displaystyle v_{\text{eq}}} 15768:{\displaystyle v_{\text{eq}}} 15532:{\displaystyle (n-1)\times 1} 14938:{\displaystyle v_{\text{eq}}} 14549:{\displaystyle v_{\text{eq}}} 7819:. The sliding surface is the 7812:{\displaystyle 1\leq i\leq n} 7578:). The switching function is 7412:{\displaystyle {\dot {x}}=-x} 4721:{\displaystyle {\sqrt {V}}=0} 1875:{\displaystyle 1\leq k\leq m} 16993:{\displaystyle \mathbf {y} } 16978:follows the measured output 16748:with the switching function 16741:{\displaystyle \mathbf {u} } 13843:{\displaystyle \mathbf {x} } 12365:{\displaystyle \mathbf {x} } 11997:{\displaystyle \mathbf {x} } 10831:and by the assumption about 10545:{\displaystyle \sigma >0} 10360:{\displaystyle \sigma <0} 9440:{\displaystyle \mathbf {x} } 9340:{\displaystyle \mathbf {x} } 7582:to be the linear combination 7542:described by Equation ( 6524:{\displaystyle \mathbf {u} } 6130:{\displaystyle \mathbf {x} } 5817:{\displaystyle \mathbf {x} } 5474:{\displaystyle \mathbf {x} } 4420:{\displaystyle 2{\sqrt {V}}} 3640:is attained in finite time, 2083:{\displaystyle \mathbf {x} } 1843:{\displaystyle \mathbf {u} } 1817:{\displaystyle \mathbf {x} } 1571:{\displaystyle \mathbf {x} } 1545:{\displaystyle \mathbf {x} } 1335:{\displaystyle \mathbf {u} } 1313:{\displaystyle \mathbf {x} } 1188:{\displaystyle \mathbf {u} } 7: 17767:10.1007/978-3-319-18290-2_2 17685:. Berlin: Springer Verlag. 17666:. Berlin: Springer Verlag. 17428:. Marcel Dekker Hardcover. 17067: 16874:to the 2-input LTI system. 15566:must be chosen so that the 14337:That is, positive constant 9388: 9288: 9045: 8393:, the feedback control law 8178: 7739: 7544: 7026:, the sliding mode surface 5595: 5589: 4775:{\displaystyle {\sqrt {V}}} 4691:{\displaystyle {\sqrt {V}}} 4582:, it must be the case that 4398:upper right-hand derivative 3021:, the feedback control law 2697: 2691: 2558: 2068:To force the system states 2061: 2006: 1802:is the number of states in 1205: 1078: 531: 10: 17932: 17643:; Spurgeon, Sarah (2004). 17074:Variable structure control 14522:{\displaystyle v(\sigma )} 11624:Automated design solutions 11232:which can be expressed in 8578: 7780:{\displaystyle s_{i}>0} 7588: 7139:only approximately holds. 5885:{\displaystyle {\dot {V}}} 5791:{\displaystyle V(\sigma )} 4822:of the switching function 4575:{\displaystyle z(0)=z_{0}} 3993:{\displaystyle V\in (0,1]} 2415: 2172:from any initial condition 1919: 420: 413:nonlinear dynamical system 61:variable structure control 17889:10.1080/00207179208934270 17477:10.1080/00207179608921865 17337:10.1017/s0263574712000720 17079:Variable structure system 15004:. In particular, the row 9921:has a finite upper bound 7371:{\displaystyle a(\cdot )} 5922:Theorem 3: Sliding motion 5911:{\displaystyle \sigma =0} 5568:{\displaystyle \sigma =0} 5393:{\displaystyle \sigma =0} 5201:{\displaystyle \alpha =1} 3780:{\displaystyle \mu >0} 1850:. For each control index 82:variable structure system 17258:Zeinali, Meysar (2018). 17144: 17132:Pulse-density modulation 14918:This equivalent control 12529:be the estimates of the 12405:. Hence, let the vector 4440:{\displaystyle \propto } 2906:of the surface given by 17816:10.1137/1.9781611975840 17557:10.1109/CDC.1992.371368 17537:Drakunov, S.V. (1992). 17399:Filippov, A.F. (1988). 16903:{\displaystyle y=x_{1}} 16870:is viewed as a control 14375:{\displaystyle e_{1}=0} 13779:{\displaystyle \sigma } 12753:{\displaystyle y=x_{1}} 12398:{\displaystyle y=x_{1}} 9648:{\displaystyle x_{1}=x} 8424:must be chosen so that 7533:Control design examples 5452:{\displaystyle \sigma } 5107:{\displaystyle \sigma } 4835:{\displaystyle \sigma } 4782:is proportional to the 4540:with initial condition 3564:has a direct impact on 3065:{\displaystyle \sigma } 2279:Picard–Lindelöf theorem 1395:{\displaystyle x_{1}=0} 1043:Picard–Lindelöf theorem 390:be applied by means of 86:hybrid dynamical system 35:method that alters the 17177:delta-sigma modulation 17138:Pulse-width modulation 17126:Delta-sigma modulation 17058: 16994: 16972: 16943:uses a generic matrix 16937: 16904: 16864: 16803: 16742: 16717: 16613: 16519: 16477: 15877:–like features. 15867: 15832: 15769: 15738: 15737:{\displaystyle A_{12}} 15707: 15666: 15610: 15560: 15533: 15495: 15463: 15025: 15024:{\displaystyle A_{12}} 14998: 14971: 14939: 14909: 14740: 14550: 14523: 14494: 14452: 14425: 14376: 14328: 14232: 14179: 14141: 14031: 13844: 13818: 13780: 13760: 13728: 13614: 13565: 13544:tracks the real state 13538: 13499: 13437: 13372: 13047: 13000: 12912: 12855: 12789: 12754: 12721: 12685: 12646: 12523: 12399: 12366: 12340: 12278: 12204: 12148: 12121: 12120:{\displaystyle a_{11}} 12088: 11998: 11972: 11884: 11793: 11613: 11546: 11441: 11400: 11222: 10979: 10855: 10821: 10665: 10581: 10546: 10520: 10477: 10396: 10361: 10335: 10290: 10103: 10065: 10017: 9915: 9885: 9827: 9691: 9649: 9608: 9523: 9482: 9441: 9415: 9380: 9341: 9315: 9280: 9236: 9085: 9033: 8566: 8524: 8418: 8387: 8348: 8166: 8136: 8104: 8074: 7902: 7854: 7813: 7781: 7727: 7572: 7523: 7457: 7413: 7372: 7340: 7263: 7211: 7182: 7133: 7094: 7059: 7017: 6963: 6568: 6525: 6500: 6362: 6231: 6195: 6150:equivalent control law 6131: 6105: 6065: 6034: 5984: 5912: 5886: 5857: 5818: 5792: 5760: 5661: 5569: 5543: 5514: 5475: 5453: 5433: 5394: 5364: 5258: 5202: 5172: 5108: 5084: 5051: 4836: 4816: 4776: 4752: 4722: 4692: 4668: 4634: 4576: 4534: 4496: 4441: 4421: 4390: 4360: 4285:Upper right-hand 4146: 4113: 4064: 3994: 3953: 3859: 3813: 3781: 3752: 3676: 3634: 3587: 3558: 3522: 3304: 3273: 3233: 3200: 3169: 3129: 3095: 3066: 3046: 3015: 2970: 2937: 2893: 2683: 2644: 2596: 2546: 2394:Theoretical foundation 2384:delta-sigma modulation 2380:pulse-width modulation 2372: 2318: 2247: 2208: 2166: 2123: 2084: 2053: 1994: 1908: 1876: 1844: 1818: 1792: 1759: 1725: 1694: 1649: 1609: 1572: 1546: 1524: 1468: 1412:sliding (hyper)surface 1396: 1363: 1336: 1314: 1292: 1265: 1189: 1163: 1132: 1070: 1027: 963: 891: 708: 519: 392:pulse-width modulation 331: 300: 277: 266: 211: 150: 124: 78:sliding (hyper)surface 17742:. Basel: Birkhauser. 17723:. Basel: Birkhauser. 17059: 16995: 16973: 16938: 16905: 16865: 16804: 16743: 16718: 16614: 16520: 16478: 15868: 15833: 15770: 15739: 15708: 15667: 15611: 15561: 15559:{\displaystyle L_{2}} 15534: 15496: 15464: 15026: 14999: 14997:{\displaystyle x_{1}} 14972: 14970:{\displaystyle (n-1)} 14940: 14910: 14741: 14551: 14524: 14495: 14453: 14451:{\displaystyle e_{1}} 14426: 14377: 14329: 14233: 14180: 14142: 14032: 13845: 13819: 13781: 13761: 13729: 13615: 13566: 13564:{\displaystyle x_{1}} 13539: 13500: 13438: 13373: 13048: 13001: 12913: 12856: 12790: 12755: 12722: 12686: 12647: 12524: 12400: 12367: 12341: 12279: 12205: 12149: 12147:{\displaystyle x_{1}} 12122: 12089: 11999: 11973: 11885: 11794: 11638:Sliding mode observer 11614: 11547: 11442: 11401: 11223: 10980: 10856: 10822: 10666: 10582: 10547: 10521: 10478: 10397: 10362: 10336: 10291: 10104: 10066: 10018: 9916: 9886: 9828: 9692: 9650: 9609: 9524: 9483: 9442: 9416: 9381: 9342: 9316: 9281: 9237: 9086: 9034: 8567: 8525: 8419: 8388: 8349: 8167: 8137: 8135:{\displaystyle (n-1)} 8105: 8075: 7903: 7855: 7814: 7782: 7728: 7573: 7524: 7458: 7414: 7373: 7341: 7264: 7212: 7183: 7134: 7095: 7060: 7018: 6964: 6574:forces the system to 6569: 6526: 6501: 6363: 6232: 6196: 6132: 6106: 6066: 6035: 5985: 5913: 5887: 5858: 5819: 5793: 5761: 5662: 5570: 5544: 5515: 5476: 5454: 5434: 5395: 5365: 5259: 5203: 5173: 5109: 5085: 5052: 4837: 4817: 4777: 4753: 4723: 4693: 4669: 4635: 4577: 4535: 4497: 4442: 4422: 4391: 4389:{\displaystyle D^{+}} 4361: 4147: 4114: 4065: 3995: 3954: 3860: 3858:{\displaystyle V\in } 3814: 3782: 3753: 3677: 3635: 3588: 3559: 3523: 3305: 3274: 3234: 3201: 3170: 3130: 3096: 3067: 3047: 3016: 2971: 2938: 2894: 2684: 2645: 2597: 2547: 2373: 2319: 2248: 2209: 2167: 2124: 2085: 2054: 1995: 1909: 1907:{\displaystyle (n-1)} 1877: 1845: 1819: 1793: 1760: 1726: 1695: 1650: 1610: 1573: 1547: 1525: 1469: 1397: 1364: 1362:{\displaystyle x_{1}} 1337: 1315: 1293: 1291:{\displaystyle x_{1}} 1266: 1190: 1164: 1133: 1071: 1028: 964: 892: 709: 520: 332: 301: 267: 212: 151: 125: 99: 55:control law is not a 17004: 16982: 16951: 16914: 16881: 16813: 16752: 16730: 16625: 16530: 16493: 15887: 15850: 15779: 15752: 15721: 15688: 15620: 15570: 15543: 15505: 15476: 15038: 15008: 14981: 14949: 14922: 14756: 14563: 14533: 14504: 14462: 14435: 14386: 14353: 14248: 14192: 14151: 14044: 13857: 13832: 13790: 13770: 13741: 13627: 13575: 13548: 13512: 13454: 13385: 13060: 13010: 12922: 12868: 12806: 12764: 12731: 12695: 12659: 12540: 12409: 12376: 12354: 12290: 12216: 12160: 12131: 12104: 12011: 11986: 11894: 11806: 11672: 11564: 11453: 11414: 11242: 10995: 10867: 10835: 10685: 10591: 10556: 10530: 10489: 10406: 10371: 10345: 10304: 10115: 10075: 10031: 9931: 9895: 9841: 9703: 9659: 9626: 9547: 9496: 9455: 9429: 9396: 9352: 9329: 9296: 9252: 9097: 9064: 8586: 8538: 8430: 8397: 8362: 8188: 8146: 8114: 8088: 7918: 7866: 7827: 7791: 7758: 7596: 7556: 7548:) with single input 7474: 7426: 7385: 7353: 7279: 7232: 7192: 7147: 7104: 7069: 7030: 6979: 6588: 6539: 6513: 6378: 6244: 6208: 6159: 6119: 6078: 6044: 6005: 5933: 5896: 5867: 5828: 5806: 5773: 5674: 5603: 5553: 5524: 5485: 5463: 5443: 5404: 5378: 5274: 5215: 5186: 5121: 5098: 5064: 4857: 4826: 4789: 4762: 4736: 4702: 4678: 4648: 4644:. Moreover, because 4586: 4544: 4506: 4455: 4431: 4404: 4373: 4159: 4123: 4081: 4007: 3966: 3872: 3831: 3791: 3765: 3690: 3644: 3605: 3568: 3535: 3321: 3283: 3243: 3212: 3179: 3139: 3108: 3076: 3056: 3025: 2980: 2954: 2910: 2708: 2654: 2610: 2576: 2423: 2331: 2289: 2267:Lipschitz continuous 2218: 2179: 2137: 2094: 2072: 2024: 1927: 1886: 1854: 1832: 1806: 1776: 1739: 1704: 1667: 1622: 1582: 1560: 1534: 1484: 1448: 1373: 1346: 1324: 1302: 1298:of the state vector 1275: 1216: 1177: 1142: 1100: 1049: 973: 915: 735: 552: 428: 378:requires the use of 343:exponentially stable 315: 284: 274:exponentially stable 224: 160: 134: 108: 72:and the geometrical 25:sliding mode control 17120:Switching amplifier 12797:Luenberger observer 11802:where state vector 11652:Luenberger observer 10854:{\displaystyle |a|} 10677:triangle inequality 9914:{\displaystyle |a|} 8534:Hence, the product 8103:{\displaystyle n-1} 7571:{\displaystyle m=1} 4751:{\displaystyle V=0} 2969:{\displaystyle m=1} 2541: 1791:{\displaystyle n-m} 1408:configuration space 1369:quickly returns to 368:continuous function 330:{\displaystyle s=0} 299:{\displaystyle s=0} 149:{\displaystyle s=0} 123:{\displaystyle s=0} 57:continuous function 17898:10338.dmlcz/135339 17851:10.1007/BFb0033675 17597:10.1007/BFb0033675 17054: 16990: 16968: 16933: 16900: 16860: 16799: 16738: 16713: 16704: 16609: 16600: 16592: 16515: 16473: 16471: 16404: 16335: 16327: 16201: 16074: 15863: 15828: 15765: 15734: 15703: 15662: 15606: 15556: 15529: 15491: 15459: 15235: 15127: 15021: 14994: 14967: 14935: 14905: 14899: 14865: 14842: 14808: 14784: 14777: 14736: 14546: 14519: 14490: 14448: 14421: 14372: 14324: 14228: 14175: 14137: 14027: 13840: 13814: 13776: 13756: 13724: 13610: 13561: 13534: 13495: 13433: 13368: 13366: 13043: 12996: 12908: 12851: 12845: 12785: 12750: 12717: 12681: 12642: 12519: 12395: 12362: 12336: 12274: 12200: 12144: 12117: 12084: 12078: 12004:state vector. Let 11994: 11968: 11880: 11789: 11784: 11758: 11609: 11542: 11437: 11396: 11395: 11377:(i.e., tests 11372: 11218: 11213: 11102: 10975: 10851: 10817: 10661: 10577: 10542: 10516: 10473: 10392: 10357: 10331: 10286: 10099: 10061: 10013: 9911: 9881: 9823: 9818: 9687: 9645: 9604: 9519: 9478: 9437: 9411: 9376: 9337: 9311: 9276: 9232: 9227: 9091:is chosen so that 9081: 9029: 9028: 9005: 8562: 8520: 8515: 8414: 8383: 8344: 8341: 8279: 8176:in Equation ( 8162: 8132: 8100: 8070: 7898: 7850: 7809: 7777: 7723: 7568: 7519: 7453: 7409: 7368: 7336: 7259: 7207: 7178: 7129: 7090: 7055: 7013: 6959: 6564: 6521: 6496: 6358: 6227: 6191: 6127: 6101: 6061: 6030: 5980: 5908: 5882: 5853: 5814: 5788: 5756: 5657: 5565: 5539: 5510: 5471: 5449: 5429: 5390: 5360: 5254: 5198: 5168: 5104: 5080: 5047: 4986: 4984: 4956: 4949: 4902: 4832: 4812: 4772: 4748: 4718: 4688: 4664: 4630: 4572: 4530: 4492: 4437: 4417: 4386: 4356: 4303: 4260: 4246: 4239: 4142: 4109: 4060: 3990: 3949: 3855: 3809: 3777: 3748: 3672: 3630: 3583: 3554: 3518: 3409: 3300: 3269: 3229: 3196: 3165: 3125: 3091: 3062: 3052:is picked so that 3042: 3011: 2966: 2950:control case when 2933: 2889: 2876: 2837: 2835: 2807: 2800: 2753: 2679: 2640: 2592: 2542: 2527: 2368: 2314: 2277:guaranteed by the 2271:closed-loop system 2243: 2204: 2162: 2119: 2080: 2049: 1990: 1904: 1872: 1840: 1814: 1788: 1755: 1721: 1690: 1645: 1605: 1568: 1542: 1520: 1479:switching function 1464: 1392: 1359: 1332: 1310: 1288: 1261: 1203:in Equation ( 1185: 1159: 1128: 1076:to Equation ( 1066: 1033:are assumed to be 1023: 959: 887: 866: 704: 683: 515: 384:optimal controller 327: 296: 278: 262: 207: 146: 120: 17916:Nonlinear control 17860:978-3-030-36620-9 17795:978-3-319-62895-0 17776:978-3-319-18289-6 17749:978-3-319-17256-9 17711:978-1-78561-076-9 17692:978-3-540-79016-7 17673:978-3-540-32800-1 17639:Sabanovic, Asif; 17631:978-0-7484-0601-2 17606:978-3-540-19869-7 17566:978-0-7803-0872-5 17501:978-0-7484-0116-1 17435:978-0-8247-0671-5 17410:978-90-277-2699-5 17385:978-0-13-067389-3 17371:Nonlinear Systems 17299:10.1109/41.184818 17217:978-0-86341-170-0 17104:Bang–bang control 17089:Nonlinear control 17035: 16965: 16835: 16774: 16677: 16635: 16540: 16503: 16466: 16454: 16441: 16427: 16377: 16275: 16226: 16151: 16102: 16027: 15978: 15936: 15912: 15907: 15860: 15810: 15762: 15748:"). That is, the 15339: 15259: 15243: 15160: 15150: 15135: 15116: 15086: 15063: 14932: 14897: 14895: 14849: 14847: 14840: 14838: 14794: 14792: 14782: 14771: 14763: 14761: 14730: 14692: 14676: 14670: 14627: 14615: 14581: 14543: 14475: 14399: 14166: 13885: 13869: 13805: 13753: 13699: 13667: 13646: 13588: 13525: 13508:so that estimate 13473: 13411: 13296: 13260: 13186: 13150: 13123: 13106: 13101: 13078: 13032: 12708: 12617: 12581: 12561: 12556: 12492: 12464: 12442: 12423: 11694: 11631:genetic algorithm 11585: 11540: 11528: 11504: 11481: 11465: 11392: 11378: 11364: 11357: 11344: 11320: 11318: 11295: 11266: 11204: 11197: 11190: 11169: 11139: 11095: 11078: 11075: 11050: 11019: 10964: 10921: 10884: 10806: 10773: 10745: 10702: 10650: 10617: 10568: 10507: 10462: 10429: 10383: 10322: 10285: 10282: 10271: 10255: 10216: 10201: 10186: 10165: 10143: 10127: 10090: 10055: 10010: 9837:Also assume that 9760: 9724: 9684: 9592: 9559: 9408: 9308: 9203: 9151: 9057: 9056: 9053: 9052: 9025: 9011: 9001: 8998: 8985: 8922: 8920: 8919: 8917: 8880: 8820: 8817: 8804: 8740: 8701: 8686: 8674: 8667: 8654: 8598: 8553: 8502: 8488: 8464: 8450: 8374: 8343: 8340: 8313: 8295: 8281: 8278: 8255: 8248: 8200: 8174:Lyapunov function 8055: 8017: 7973: 7941: 7878: 7754:where the weight 7751: 7750: 7747: 7746: 7495: 7438: 7397: 7324: 7291: 7244: 7204: 7169: 7081: 6991: 6952: 6896: 6813: 6763: 6737: 6681: 6602: 6474: 6418: 6349: 6346: 6333: 6267: 6171: 6090: 5958: 5879: 5800:Lyapunov function 5733: 5536: 5421: 5340: 5323: 5313: 5242: 5136: 5034: 5032: 5025: 4983: 4951: 4948: 4921: 4916: 4904: 4901: 4880: 4862: 4860: 4770: 4710: 4686: 4656: 4606: 4518: 4415: 4345: 4319: 4318: 4297: 4286: 4232: 4207: 4203: 4189: 4187: 4166: 4164: 4140: 4046: 4020: 4019: 3944: 3918: 3899: 3736: 3717: 3580: 3517: 3514: 3501: 3435: 3411: 3408: 3379: 3374: 3359: 3333: 3255: 3151: 3088: 3002: 2887: 2875: 2848: 2834: 2802: 2799: 2772: 2767: 2755: 2752: 2731: 2713: 2711: 2570: 2569: 2566: 2565: 2508: 2460: 2408:Lyapunov function 2388:nonlinear control 2343: 2018: 2017: 2014: 2013: 1037:and sufficiently 543: 542: 539: 538: 442: 400:nonlinear control 380:bang–bang control 237: 192: 130:. The particular 33:nonlinear control 17923: 17902: 17900: 17883:(4): 1029–1037. 17864: 17837: 17799: 17780: 17753: 17734: 17730:978-0-81764-8923 17715: 17696: 17677: 17658: 17635: 17610: 17579: 17578: 17534: 17528: 17527: 17515: 17506: 17505: 17487: 17481: 17480: 17470: 17446: 17440: 17439: 17421: 17415: 17414: 17396: 17390: 17389: 17362: 17349: 17348: 17320: 17314: 17309: 17303: 17302: 17292: 17272: 17266: 17265: 17255: 17249: 17248: 17228: 17222: 17221: 17200: 17180: 17173: 17167: 17155: 17063: 17061: 17060: 17055: 17053: 17045: 17037: 17036: 17031: 17026: 17017: 16999: 16997: 16996: 16991: 16989: 16977: 16975: 16974: 16969: 16967: 16966: 16961: 16956: 16946: 16942: 16940: 16939: 16934: 16932: 16921: 16909: 16907: 16906: 16901: 16899: 16898: 16869: 16867: 16866: 16861: 16856: 16855: 16843: 16842: 16837: 16836: 16828: 16808: 16806: 16805: 16800: 16795: 16794: 16782: 16781: 16776: 16775: 16767: 16747: 16745: 16744: 16739: 16737: 16722: 16720: 16719: 16714: 16709: 16708: 16698: 16697: 16685: 16684: 16679: 16678: 16670: 16653: 16637: 16636: 16633: 16618: 16616: 16615: 16610: 16605: 16604: 16597: 16596: 16586: 16585: 16542: 16541: 16538: 16524: 16522: 16521: 16516: 16505: 16504: 16501: 16482: 16480: 16479: 16474: 16472: 16468: 16467: 16464: 16462: 16456: 16455: 16452: 16443: 16442: 16437: 16432: 16429: 16428: 16425: 16413: 16409: 16408: 16398: 16397: 16385: 16384: 16379: 16378: 16370: 16353: 16340: 16339: 16332: 16331: 16321: 16320: 16277: 16276: 16271: 16266: 16254: 16247: 16246: 16234: 16233: 16228: 16227: 16219: 16206: 16205: 16195: 16194: 16164: 16153: 16152: 16147: 16142: 16130: 16123: 16122: 16110: 16109: 16104: 16103: 16095: 16079: 16078: 16071: 16070: 16040: 16029: 16028: 16023: 16018: 16006: 15999: 15998: 15986: 15985: 15980: 15979: 15971: 15949: 15938: 15937: 15932: 15927: 15914: 15913: 15908: 15903: 15898: 15896: 15872: 15870: 15869: 15864: 15862: 15861: 15858: 15845: 15837: 15835: 15834: 15829: 15818: 15817: 15812: 15811: 15803: 15774: 15772: 15771: 15766: 15764: 15763: 15760: 15747: 15743: 15741: 15740: 15735: 15733: 15732: 15712: 15710: 15709: 15704: 15702: 15701: 15696: 15671: 15669: 15668: 15663: 15658: 15657: 15648: 15647: 15635: 15634: 15615: 15613: 15612: 15607: 15565: 15563: 15562: 15557: 15555: 15554: 15538: 15536: 15535: 15530: 15500: 15498: 15497: 15492: 15490: 15489: 15484: 15468: 15466: 15465: 15460: 15455: 15454: 15449: 15440: 15439: 15430: 15429: 15417: 15416: 15401: 15400: 15395: 15389: 15388: 15379: 15378: 15366: 15365: 15360: 15354: 15353: 15341: 15340: 15337: 15331: 15330: 15318: 15317: 15312: 15306: 15305: 15290: 15289: 15274: 15273: 15261: 15260: 15258: 15257: 15256: 15251: 15244: 15239: 15232: 15231: 15211: 15210: 15197: 15196: 15180: 15178: 15175: 15174: 15162: 15161: 15159: 15158: 15157: 15152: 15151: 15146: 15141: 15136: 15131: 15124: 15123: 15118: 15117: 15109: 15094: 15093: 15088: 15087: 15079: 15071: 15070: 15065: 15064: 15056: 15045: 15043: 15030: 15028: 15027: 15022: 15020: 15019: 15003: 15001: 15000: 14995: 14993: 14992: 14976: 14974: 14973: 14968: 14944: 14942: 14941: 14936: 14934: 14933: 14930: 14914: 14912: 14911: 14906: 14901: 14900: 14898: 14896: 14893: 14891: 14866: 14861: 14860: 14855: 14844: 14843: 14841: 14839: 14836: 14834: 14809: 14804: 14803: 14786: 14785: 14783: 14780: 14778: 14773: 14772: 14769: 14745: 14743: 14742: 14737: 14732: 14731: 14728: 14719: 14718: 14713: 14707: 14706: 14694: 14693: 14691: 14677: 14672: 14671: 14668: 14662: 14660: 14654: 14653: 14648: 14642: 14641: 14629: 14628: 14626: 14619: 14616: 14611: 14610: 14601: 14599: 14596: 14595: 14583: 14582: 14574: 14555: 14553: 14552: 14547: 14545: 14544: 14541: 14528: 14526: 14525: 14520: 14499: 14497: 14496: 14491: 14483: 14482: 14477: 14476: 14468: 14457: 14455: 14454: 14449: 14447: 14446: 14430: 14428: 14427: 14422: 14420: 14419: 14407: 14406: 14401: 14400: 14392: 14381: 14379: 14378: 14373: 14365: 14364: 14348: 14344: 14340: 14333: 14331: 14330: 14325: 14317: 14312: 14311: 14306: 14300: 14299: 14287: 14286: 14277: 14276: 14267: 14237: 14235: 14234: 14229: 14184: 14182: 14181: 14176: 14168: 14167: 14159: 14146: 14144: 14143: 14138: 14136: 14135: 14118: 14106: 14105: 14087: 14086: 14074: 14073: 14058: 14057: 14052: 14036: 14034: 14033: 14028: 14011: 14010: 14005: 13999: 13998: 13986: 13985: 13976: 13975: 13960: 13959: 13941: 13940: 13935: 13929: 13928: 13916: 13915: 13906: 13905: 13893: 13892: 13887: 13886: 13878: 13871: 13870: 13862: 13849: 13847: 13846: 13841: 13839: 13823: 13821: 13820: 13815: 13807: 13806: 13798: 13785: 13783: 13782: 13777: 13765: 13763: 13762: 13757: 13755: 13754: 13746: 13733: 13731: 13730: 13725: 13720: 13719: 13707: 13706: 13701: 13700: 13692: 13685: 13684: 13669: 13668: 13660: 13654: 13653: 13648: 13647: 13639: 13619: 13617: 13616: 13611: 13609: 13608: 13596: 13595: 13590: 13589: 13581: 13570: 13568: 13567: 13562: 13560: 13559: 13543: 13541: 13540: 13535: 13533: 13532: 13527: 13526: 13518: 13504: 13502: 13501: 13496: 13494: 13493: 13481: 13480: 13475: 13474: 13466: 13446: 13442: 13440: 13439: 13434: 13432: 13431: 13419: 13418: 13413: 13412: 13404: 13397: 13396: 13377: 13375: 13374: 13369: 13367: 13360: 13359: 13338: 13324: 13317: 13316: 13304: 13303: 13298: 13297: 13289: 13270: 13262: 13261: 13256: 13251: 13236: 13232: 13221: 13207: 13206: 13194: 13193: 13188: 13187: 13179: 13163: 13152: 13151: 13146: 13141: 13129: 13125: 13124: 13119: 13114: 13108: 13107: 13102: 13097: 13092: 13090: 13080: 13079: 13074: 13069: 13052: 13050: 13049: 13044: 13042: 13034: 13033: 13028: 13023: 13017: 13005: 13003: 13002: 12997: 12995: 12994: 12989: 12977: 12976: 12958: 12957: 12945: 12944: 12929: 12917: 12915: 12914: 12909: 12907: 12906: 12889: 12880: 12879: 12860: 12858: 12857: 12852: 12850: 12849: 12842: 12841: 12799:. Likewise, let 12794: 12792: 12791: 12786: 12784: 12783: 12778: 12759: 12757: 12756: 12751: 12749: 12748: 12726: 12724: 12723: 12718: 12716: 12715: 12710: 12709: 12701: 12690: 12688: 12687: 12682: 12680: 12672: 12651: 12649: 12648: 12643: 12638: 12637: 12625: 12624: 12619: 12618: 12610: 12594: 12583: 12582: 12577: 12572: 12563: 12562: 12557: 12552: 12547: 12545: 12532: 12528: 12526: 12525: 12520: 12518: 12517: 12512: 12500: 12499: 12494: 12493: 12485: 12472: 12471: 12466: 12465: 12457: 12450: 12449: 12444: 12443: 12435: 12425: 12424: 12419: 12414: 12404: 12402: 12401: 12396: 12394: 12393: 12371: 12369: 12368: 12363: 12361: 12345: 12343: 12342: 12337: 12335: 12334: 12311: 12302: 12301: 12283: 12281: 12280: 12275: 12273: 12272: 12237: 12228: 12227: 12209: 12207: 12206: 12201: 12199: 12198: 12181: 12172: 12171: 12153: 12151: 12150: 12145: 12143: 12142: 12126: 12124: 12123: 12118: 12116: 12115: 12093: 12091: 12090: 12085: 12083: 12082: 12075: 12074: 12063: 12062: 12049: 12048: 12037: 12036: 12003: 12001: 12000: 11995: 11993: 11981: 11977: 11975: 11974: 11969: 11967: 11966: 11961: 11949: 11948: 11930: 11929: 11917: 11916: 11901: 11889: 11887: 11886: 11881: 11879: 11878: 11873: 11861: 11860: 11842: 11841: 11829: 11828: 11813: 11798: 11796: 11795: 11790: 11788: 11787: 11781: 11780: 11768: 11763: 11762: 11756: 11718: 11707: 11696: 11695: 11690: 11685: 11618: 11616: 11615: 11610: 11587: 11586: 11578: 11551: 11549: 11548: 11543: 11541: 11538: 11530: 11529: 11521: 11506: 11505: 11497: 11482: 11479: 11467: 11466: 11458: 11446: 11444: 11443: 11438: 11427: 11405: 11403: 11402: 11397: 11394: 11393: 11390: 11379: 11376: 11373: 11368: 11363: 11358: 11353: 11346: 11345: 11337: 11333: 11331: 11302: 11297: 11296: 11288: 11285: 11268: 11267: 11259: 11227: 11225: 11224: 11219: 11217: 11216: 11203: 11198: 11193: 11192: 11191: 11183: 11173: 11171: 11170: 11167: 11163: 11159: 11146: 11141: 11140: 11132: 11129: 11103: 11098: 11097: 11096: 11088: 11076: 11073: 11057: 11052: 11051: 11043: 11040: 11021: 11020: 11012: 10984: 10982: 10981: 10976: 10974: 10966: 10965: 10957: 10936: 10928: 10923: 10922: 10914: 10911: 10891: 10886: 10885: 10877: 10874: 10860: 10858: 10857: 10852: 10850: 10842: 10826: 10824: 10823: 10818: 10816: 10808: 10807: 10799: 10775: 10774: 10766: 10763: 10755: 10747: 10746: 10738: 10717: 10709: 10704: 10703: 10695: 10692: 10675:However, by the 10670: 10668: 10667: 10662: 10660: 10652: 10651: 10643: 10619: 10618: 10610: 10607: 10586: 10584: 10583: 10578: 10570: 10569: 10561: 10551: 10549: 10548: 10543: 10525: 10523: 10522: 10517: 10509: 10508: 10500: 10482: 10480: 10479: 10474: 10472: 10464: 10463: 10455: 10431: 10430: 10422: 10419: 10401: 10399: 10398: 10393: 10385: 10384: 10376: 10366: 10364: 10363: 10358: 10340: 10338: 10337: 10332: 10324: 10323: 10315: 10295: 10293: 10292: 10287: 10284: 10283: 10275: 10272: 10267: 10257: 10256: 10248: 10226: 10224: 10218: 10217: 10209: 10203: 10202: 10194: 10188: 10187: 10179: 10173: 10172: 10167: 10166: 10158: 10151: 10150: 10145: 10144: 10136: 10129: 10128: 10120: 10108: 10106: 10105: 10100: 10092: 10091: 10083: 10070: 10068: 10067: 10062: 10057: 10056: 10048: 10022: 10020: 10019: 10014: 10012: 10011: 10003: 9991: 9990: 9978: 9977: 9962: 9961: 9949: 9948: 9924: 9920: 9918: 9917: 9912: 9910: 9902: 9890: 9888: 9887: 9882: 9871: 9854: 9832: 9830: 9829: 9824: 9822: 9821: 9806: 9805: 9793: 9792: 9768: 9767: 9762: 9761: 9753: 9745: 9744: 9732: 9731: 9726: 9725: 9717: 9696: 9694: 9693: 9688: 9686: 9685: 9677: 9671: 9670: 9654: 9652: 9651: 9646: 9638: 9637: 9613: 9611: 9610: 9605: 9594: 9593: 9585: 9561: 9560: 9552: 9528: 9526: 9525: 9520: 9509: 9487: 9485: 9484: 9479: 9468: 9446: 9444: 9443: 9438: 9436: 9420: 9418: 9417: 9412: 9410: 9409: 9401: 9385: 9383: 9382: 9377: 9372: 9364: 9363: 9346: 9344: 9343: 9338: 9336: 9320: 9318: 9317: 9312: 9310: 9309: 9301: 9285: 9283: 9282: 9277: 9272: 9264: 9263: 9241: 9239: 9238: 9233: 9231: 9230: 9215: 9204: 9201: 9194: 9186: 9185: 9163: 9152: 9149: 9142: 9134: 9133: 9110: 9090: 9088: 9087: 9082: 9077: 9060:The control law 9047: 9038: 9036: 9035: 9030: 9027: 9026: 9023: 9012: 9010:( i.e., an 9009: 9006: 9000: 8999: 8994: 8989: 8986: 8981: 8977: 8964: 8941: 8925: 8923: 8918: 8916: 8915: 8914: 8904: 8903: 8899: 8883: 8881: 8876: 8872: 8871: 8853: 8852: 8840: 8839: 8826: 8824: 8819: 8818: 8813: 8808: 8805: 8800: 8796: 8783: 8760: 8744: 8742: 8741: 8739: 8738: 8737: 8727: 8726: 8722: 8706: 8700: 8696: 8688: 8687: 8679: 8675: 8670: 8669: 8668: 8663: 8658: 8655: 8653: 8652: 8651: 8641: 8640: 8636: 8620: 8617: 8615: 8608: 8600: 8599: 8591: 8580: 8576: 8571: 8569: 8568: 8563: 8555: 8554: 8546: 8529: 8527: 8526: 8521: 8519: 8518: 8503: 8500: 8490: 8489: 8481: 8465: 8462: 8452: 8451: 8443: 8423: 8421: 8420: 8415: 8410: 8392: 8390: 8389: 8384: 8376: 8375: 8367: 8353: 8351: 8350: 8345: 8342: 8339: 8328: 8317: 8314: 8309: 8305: 8297: 8296: 8288: 8284: 8282: 8280: 8277: 8276: 8267: 8259: 8256: 8251: 8250: 8249: 8246: 8240: 8228: 8226: 8216: 8202: 8201: 8193: 8171: 8169: 8168: 8163: 8161: 8153: 8141: 8139: 8138: 8133: 8109: 8107: 8106: 8101: 8079: 8077: 8076: 8071: 8063: 8062: 8057: 8056: 8048: 8044: 8043: 8031: 8030: 8019: 8018: 8010: 8006: 8005: 7981: 7980: 7975: 7974: 7966: 7962: 7961: 7949: 7948: 7943: 7942: 7934: 7930: 7929: 7907: 7905: 7904: 7899: 7888: 7880: 7879: 7871: 7859: 7857: 7856: 7851: 7840: 7818: 7816: 7815: 7810: 7786: 7784: 7783: 7778: 7770: 7769: 7741: 7732: 7730: 7729: 7724: 7722: 7721: 7712: 7711: 7699: 7698: 7683: 7682: 7658: 7657: 7648: 7647: 7635: 7634: 7625: 7624: 7609: 7590: 7586: 7577: 7575: 7574: 7569: 7551: 7528: 7526: 7525: 7520: 7497: 7496: 7488: 7462: 7460: 7459: 7454: 7440: 7439: 7431: 7418: 7416: 7415: 7410: 7399: 7398: 7390: 7377: 7375: 7374: 7369: 7345: 7343: 7342: 7337: 7326: 7325: 7317: 7293: 7292: 7284: 7268: 7266: 7265: 7260: 7246: 7245: 7237: 7216: 7214: 7213: 7208: 7206: 7205: 7197: 7187: 7185: 7184: 7179: 7171: 7170: 7162: 7159: 7158: 7138: 7136: 7135: 7130: 7128: 7117: 7099: 7097: 7096: 7091: 7083: 7082: 7074: 7064: 7062: 7061: 7056: 7054: 7043: 7022: 7020: 7019: 7014: 7012: 7001: 6993: 6992: 6984: 6968: 6966: 6965: 6960: 6958: 6954: 6953: 6951: 6950: 6941: 6933: 6931: 6930: 6922: 6918: 6908: 6897: 6895: 6894: 6885: 6877: 6860: 6846: 6827: 6812: 6799: 6776: 6764: 6759: 6749: 6738: 6736: 6735: 6726: 6718: 6716: 6715: 6707: 6703: 6693: 6682: 6680: 6679: 6670: 6662: 6645: 6622: 6610: 6608: 6604: 6603: 6598: 6593: 6573: 6571: 6570: 6565: 6563: 6552: 6530: 6528: 6527: 6522: 6520: 6505: 6503: 6502: 6497: 6486: 6475: 6473: 6472: 6463: 6455: 6453: 6452: 6444: 6440: 6430: 6419: 6417: 6416: 6407: 6399: 6385: 6367: 6365: 6364: 6359: 6357: 6348: 6347: 6342: 6337: 6334: 6329: 6325: 6324: 6310: 6287: 6271: 6269: 6268: 6266: 6265: 6256: 6248: 6236: 6234: 6233: 6228: 6223: 6215: 6200: 6198: 6197: 6192: 6181: 6173: 6172: 6164: 6136: 6134: 6133: 6128: 6126: 6110: 6108: 6107: 6102: 6100: 6092: 6091: 6083: 6070: 6068: 6067: 6062: 6057: 6039: 6037: 6036: 6031: 6029: 6018: 5989: 5987: 5986: 5981: 5970: 5959: 5957: 5956: 5955: 5945: 5937: 5917: 5915: 5914: 5909: 5891: 5889: 5888: 5883: 5881: 5880: 5872: 5862: 5860: 5859: 5854: 5852: 5841: 5823: 5821: 5820: 5815: 5813: 5797: 5795: 5794: 5789: 5765: 5763: 5762: 5757: 5743: 5735: 5734: 5726: 5720: 5712: 5711: 5699: 5698: 5693: 5684: 5666: 5664: 5663: 5658: 5653: 5642: 5628: 5627: 5622: 5613: 5574: 5572: 5571: 5566: 5548: 5546: 5545: 5540: 5538: 5537: 5529: 5519: 5517: 5516: 5511: 5509: 5498: 5480: 5478: 5477: 5472: 5470: 5458: 5456: 5455: 5450: 5438: 5436: 5435: 5430: 5428: 5423: 5422: 5414: 5411: 5400:, and its speed 5399: 5397: 5396: 5391: 5369: 5367: 5366: 5361: 5347: 5342: 5341: 5333: 5330: 5324: 5321: 5315: 5314: 5306: 5263: 5261: 5260: 5255: 5244: 5243: 5235: 5207: 5205: 5204: 5199: 5177: 5175: 5174: 5169: 5167: 5166: 5161: 5152: 5138: 5137: 5129: 5113: 5111: 5110: 5105: 5089: 5087: 5086: 5081: 5076: 5075: 5056: 5054: 5053: 5048: 5046: 5045: 5036: 5035: 5033: 5028: 5026: 5021: 5020: 5019: 5003: 5001: 4985: 4982: 4971: 4960: 4957: 4952: 4950: 4947: 4936: 4925: 4922: 4917: 4909: 4907: 4905: 4903: 4900: 4892: 4884: 4881: 4876: 4875: 4866: 4864: 4841: 4839: 4838: 4833: 4821: 4819: 4818: 4813: 4811: 4810: 4801: 4800: 4781: 4779: 4778: 4773: 4771: 4766: 4757: 4755: 4754: 4749: 4731: 4727: 4725: 4724: 4719: 4711: 4706: 4697: 4695: 4694: 4689: 4687: 4682: 4673: 4671: 4670: 4665: 4657: 4652: 4643: 4639: 4637: 4636: 4631: 4620: 4619: 4607: 4593: 4581: 4579: 4578: 4573: 4571: 4570: 4539: 4537: 4536: 4531: 4520: 4519: 4511: 4501: 4499: 4498: 4493: 4482: 4481: 4446: 4444: 4443: 4438: 4426: 4424: 4423: 4418: 4416: 4411: 4395: 4393: 4392: 4387: 4385: 4384: 4365: 4363: 4362: 4357: 4346: 4344: 4333: 4322: 4320: 4314: 4310: 4305: 4304: 4302: 4301: 4300: 4299: 4298: 4290: 4287: 4284: 4272: 4271: 4261: 4256: 4255: 4254: 4248: 4247: 4245: 4240: 4235: 4234: 4233: 4231: 4230: 4229: 4211: 4208: 4199: 4198: 4196: 4184: 4183: 4177: 4176: 4151: 4149: 4148: 4143: 4141: 4136: 4118: 4116: 4115: 4110: 4108: 4097: 4069: 4067: 4066: 4061: 4047: 4045: 4034: 4023: 4021: 4015: 4011: 3999: 3997: 3996: 3991: 3958: 3956: 3955: 3950: 3945: 3940: 3929: 3928: 3919: 3914: 3900: 3898: 3887: 3876: 3864: 3862: 3861: 3856: 3818: 3816: 3815: 3810: 3786: 3784: 3783: 3778: 3757: 3755: 3754: 3749: 3747: 3746: 3737: 3732: 3718: 3716: 3705: 3694: 3681: 3679: 3678: 3673: 3671: 3660: 3639: 3637: 3636: 3631: 3629: 3618: 3592: 3590: 3589: 3584: 3582: 3581: 3573: 3563: 3561: 3560: 3555: 3550: 3542: 3527: 3525: 3524: 3519: 3516: 3515: 3510: 3505: 3502: 3497: 3493: 3492: 3478: 3455: 3439: 3437: 3436: 3434: 3433: 3424: 3416: 3410: 3407: 3396: 3395: 3383: 3380: 3375: 3370: 3365: 3363: 3361: 3360: 3358: 3357: 3348: 3340: 3335: 3334: 3326: 3309: 3307: 3306: 3301: 3296: 3278: 3276: 3275: 3270: 3265: 3257: 3256: 3248: 3238: 3236: 3235: 3230: 3225: 3205: 3203: 3202: 3197: 3192: 3174: 3172: 3171: 3166: 3161: 3153: 3152: 3144: 3134: 3132: 3131: 3126: 3121: 3100: 3098: 3097: 3092: 3090: 3089: 3081: 3071: 3069: 3068: 3063: 3051: 3049: 3048: 3043: 3038: 3020: 3018: 3017: 3012: 3004: 3003: 2995: 2992: 2991: 2975: 2973: 2972: 2967: 2942: 2940: 2939: 2934: 2923: 2898: 2896: 2895: 2890: 2888: 2885: 2877: 2874: 2863: 2852: 2849: 2846: 2836: 2833: 2822: 2811: 2808: 2803: 2801: 2798: 2787: 2776: 2773: 2768: 2760: 2758: 2756: 2754: 2751: 2743: 2735: 2732: 2727: 2726: 2717: 2715: 2688: 2686: 2685: 2680: 2678: 2667: 2649: 2647: 2646: 2641: 2639: 2638: 2626: 2601: 2599: 2598: 2593: 2588: 2587: 2560: 2551: 2549: 2548: 2543: 2540: 2535: 2523: 2509: 2501: 2493: 2479: 2471: 2470: 2461: 2453: 2442: 2417: 2413: 2377: 2375: 2374: 2369: 2364: 2353: 2345: 2344: 2336: 2323: 2321: 2320: 2315: 2313: 2302: 2252: 2250: 2249: 2244: 2242: 2231: 2213: 2211: 2210: 2205: 2203: 2192: 2171: 2169: 2168: 2163: 2161: 2150: 2128: 2126: 2125: 2120: 2118: 2107: 2089: 2087: 2086: 2081: 2079: 2058: 2056: 2055: 2050: 2048: 2037: 2008: 1999: 1997: 1996: 1991: 1989: 1985: 1975: 1967: 1966: 1954: 1953: 1948: 1939: 1921: 1917: 1913: 1911: 1910: 1905: 1881: 1879: 1878: 1873: 1849: 1847: 1846: 1841: 1839: 1827: 1823: 1821: 1820: 1815: 1813: 1801: 1797: 1795: 1794: 1789: 1764: 1762: 1761: 1756: 1754: 1746: 1730: 1728: 1727: 1722: 1711: 1699: 1697: 1696: 1691: 1680: 1654: 1652: 1651: 1646: 1635: 1614: 1612: 1611: 1606: 1595: 1577: 1575: 1574: 1569: 1567: 1551: 1549: 1548: 1543: 1541: 1529: 1527: 1526: 1521: 1519: 1518: 1513: 1504: 1503: 1498: 1473: 1471: 1470: 1465: 1463: 1455: 1401: 1399: 1398: 1393: 1385: 1384: 1368: 1366: 1365: 1360: 1358: 1357: 1342:can ensure that 1341: 1339: 1338: 1333: 1331: 1319: 1317: 1316: 1311: 1309: 1297: 1295: 1294: 1289: 1287: 1286: 1270: 1268: 1267: 1262: 1260: 1259: 1223: 1201:dynamical system 1194: 1192: 1191: 1186: 1184: 1172: 1168: 1166: 1165: 1160: 1149: 1137: 1135: 1134: 1129: 1115: 1107: 1075: 1073: 1072: 1067: 1056: 1032: 1030: 1029: 1024: 1022: 1021: 1010: 1001: 993: 992: 987: 968: 966: 965: 960: 958: 957: 952: 943: 935: 934: 929: 903: 896: 894: 893: 888: 886: 885: 880: 871: 870: 854: 853: 831: 830: 795: 794: 772: 771: 742: 720: 713: 711: 710: 705: 703: 702: 697: 688: 687: 671: 670: 648: 647: 612: 611: 589: 588: 559: 533: 524: 522: 521: 516: 505: 490: 467: 444: 443: 438: 433: 422: 418: 336: 334: 333: 328: 305: 303: 302: 297: 271: 269: 268: 263: 261: 260: 245: 244: 239: 238: 230: 216: 214: 213: 208: 200: 199: 194: 193: 185: 178: 177: 155: 153: 152: 147: 129: 127: 126: 121: 41:nonlinear system 17931: 17930: 17926: 17925: 17924: 17922: 17921: 17920: 17906: 17905: 17871: 17869:Further reading 17861: 17826: 17796: 17777: 17750: 17731: 17712: 17693: 17674: 17655: 17641:Fridman, Leonid 17632: 17607: 17583: 17582: 17567: 17535: 17531: 17526:(9): 1167–1175. 17516: 17509: 17502: 17488: 17484: 17447: 17443: 17436: 17422: 17418: 17411: 17397: 17393: 17386: 17363: 17352: 17321: 17317: 17310: 17306: 17273: 17269: 17256: 17252: 17229: 17225: 17218: 17204:Zinober, A.S.I. 17201: 17194: 17189: 17184: 17183: 17174: 17170: 17156: 17152: 17147: 17099:Optimal control 17070: 17049: 17041: 17027: 17025: 17024: 17013: 17005: 17002: 17001: 16985: 16983: 16980: 16979: 16957: 16955: 16954: 16952: 16949: 16948: 16944: 16928: 16917: 16915: 16912: 16911: 16894: 16890: 16882: 16879: 16878: 16851: 16847: 16838: 16827: 16826: 16825: 16814: 16811: 16810: 16790: 16786: 16777: 16766: 16765: 16764: 16753: 16750: 16749: 16733: 16731: 16728: 16727: 16703: 16702: 16693: 16689: 16680: 16669: 16668: 16667: 16655: 16654: 16649: 16642: 16641: 16632: 16628: 16626: 16623: 16622: 16599: 16598: 16591: 16590: 16581: 16577: 16574: 16573: 16560: 16559: 16557: 16547: 16546: 16537: 16533: 16531: 16528: 16527: 16500: 16496: 16494: 16491: 16490: 16470: 16469: 16463: 16458: 16457: 16451: 16447: 16433: 16431: 16430: 16424: 16420: 16411: 16410: 16403: 16402: 16393: 16389: 16380: 16369: 16368: 16367: 16355: 16354: 16349: 16342: 16341: 16334: 16333: 16326: 16325: 16316: 16312: 16309: 16308: 16295: 16294: 16292: 16282: 16281: 16267: 16265: 16264: 16252: 16251: 16242: 16238: 16229: 16218: 16217: 16216: 16200: 16199: 16190: 16186: 16183: 16182: 16169: 16168: 16160: 16143: 16141: 16140: 16128: 16127: 16118: 16114: 16105: 16094: 16093: 16092: 16073: 16072: 16066: 16062: 16059: 16058: 16045: 16044: 16036: 16019: 16017: 16016: 16004: 16003: 15994: 15990: 15981: 15970: 15969: 15968: 15945: 15928: 15926: 15925: 15915: 15899: 15897: 15895: 15894: 15890: 15888: 15885: 15884: 15857: 15853: 15851: 15848: 15847: 15843: 15813: 15802: 15801: 15800: 15780: 15777: 15776: 15759: 15755: 15753: 15750: 15749: 15745: 15728: 15724: 15722: 15719: 15718: 15697: 15692: 15691: 15689: 15686: 15685: 15653: 15649: 15643: 15639: 15630: 15626: 15621: 15618: 15617: 15571: 15568: 15567: 15550: 15546: 15544: 15541: 15540: 15506: 15503: 15502: 15485: 15480: 15479: 15477: 15474: 15473: 15450: 15445: 15444: 15435: 15431: 15425: 15421: 15412: 15408: 15396: 15391: 15390: 15384: 15380: 15374: 15370: 15361: 15356: 15355: 15349: 15345: 15336: 15332: 15326: 15322: 15313: 15308: 15307: 15301: 15297: 15285: 15281: 15269: 15265: 15252: 15247: 15246: 15245: 15234: 15233: 15227: 15223: 15220: 15219: 15213: 15212: 15206: 15202: 15199: 15198: 15192: 15188: 15181: 15179: 15177: 15176: 15170: 15166: 15153: 15142: 15140: 15139: 15138: 15137: 15126: 15125: 15119: 15108: 15107: 15106: 15103: 15102: 15096: 15095: 15089: 15078: 15077: 15076: 15073: 15072: 15066: 15055: 15054: 15053: 15046: 15044: 15042: 15041: 15039: 15036: 15035: 15015: 15011: 15009: 15006: 15005: 14988: 14984: 14982: 14979: 14978: 14950: 14947: 14946: 14929: 14925: 14923: 14920: 14919: 14892: 14869: 14867: 14856: 14851: 14850: 14848: 14846: 14845: 14835: 14812: 14810: 14799: 14795: 14793: 14791: 14790: 14779: 14768: 14764: 14762: 14760: 14759: 14757: 14754: 14753: 14727: 14723: 14714: 14709: 14708: 14702: 14698: 14678: 14667: 14663: 14661: 14659: 14658: 14649: 14644: 14643: 14637: 14633: 14618: 14617: 14606: 14602: 14600: 14598: 14597: 14591: 14587: 14573: 14572: 14564: 14561: 14560: 14540: 14536: 14534: 14531: 14530: 14505: 14502: 14501: 14478: 14467: 14466: 14465: 14463: 14460: 14459: 14442: 14438: 14436: 14433: 14432: 14415: 14411: 14402: 14391: 14390: 14389: 14387: 14384: 14383: 14360: 14356: 14354: 14351: 14350: 14346: 14342: 14338: 14313: 14307: 14302: 14301: 14295: 14291: 14282: 14278: 14272: 14268: 14263: 14249: 14246: 14245: 14193: 14190: 14189: 14158: 14157: 14152: 14149: 14148: 14119: 14114: 14113: 14101: 14097: 14082: 14078: 14069: 14065: 14053: 14048: 14047: 14045: 14042: 14041: 14006: 14001: 14000: 13994: 13990: 13981: 13977: 13971: 13967: 13955: 13951: 13936: 13931: 13930: 13924: 13920: 13911: 13907: 13901: 13897: 13888: 13877: 13876: 13875: 13861: 13860: 13858: 13855: 13854: 13835: 13833: 13830: 13829: 13797: 13796: 13791: 13788: 13787: 13771: 13768: 13767: 13745: 13744: 13742: 13739: 13738: 13715: 13711: 13702: 13691: 13690: 13689: 13680: 13676: 13659: 13658: 13649: 13638: 13637: 13636: 13628: 13625: 13624: 13604: 13600: 13591: 13580: 13579: 13578: 13576: 13573: 13572: 13555: 13551: 13549: 13546: 13545: 13528: 13517: 13516: 13515: 13513: 13510: 13509: 13489: 13485: 13476: 13465: 13464: 13463: 13455: 13452: 13451: 13444: 13427: 13423: 13414: 13403: 13402: 13401: 13392: 13388: 13386: 13383: 13382: 13365: 13364: 13355: 13351: 13334: 13322: 13321: 13312: 13308: 13299: 13288: 13287: 13286: 13266: 13252: 13250: 13249: 13234: 13233: 13228: 13217: 13202: 13198: 13189: 13178: 13177: 13176: 13159: 13142: 13140: 13139: 13127: 13126: 13115: 13113: 13112: 13093: 13091: 13089: 13088: 13081: 13070: 13068: 13067: 13063: 13061: 13058: 13057: 13038: 13024: 13022: 13021: 13013: 13011: 13008: 13007: 12990: 12985: 12984: 12972: 12968: 12953: 12949: 12940: 12936: 12925: 12923: 12920: 12919: 12890: 12885: 12884: 12875: 12871: 12869: 12866: 12865: 12844: 12843: 12837: 12833: 12830: 12829: 12816: 12815: 12807: 12804: 12803: 12779: 12774: 12773: 12765: 12762: 12761: 12744: 12740: 12732: 12729: 12728: 12727:and the output 12711: 12700: 12699: 12698: 12696: 12693: 12692: 12676: 12668: 12660: 12657: 12656: 12633: 12629: 12620: 12609: 12608: 12607: 12590: 12573: 12571: 12570: 12548: 12546: 12544: 12543: 12541: 12538: 12537: 12530: 12513: 12508: 12507: 12495: 12484: 12483: 12482: 12467: 12456: 12455: 12454: 12445: 12434: 12433: 12432: 12415: 12413: 12412: 12410: 12407: 12406: 12389: 12385: 12377: 12374: 12373: 12357: 12355: 12352: 12351: 12312: 12307: 12306: 12297: 12293: 12291: 12288: 12287: 12238: 12233: 12232: 12223: 12219: 12217: 12214: 12213: 12182: 12177: 12176: 12167: 12163: 12161: 12158: 12157: 12138: 12134: 12132: 12129: 12128: 12111: 12107: 12105: 12102: 12101: 12077: 12076: 12070: 12066: 12064: 12058: 12054: 12051: 12050: 12044: 12040: 12038: 12032: 12028: 12021: 12020: 12012: 12009: 12008: 11989: 11987: 11984: 11983: 11979: 11962: 11957: 11956: 11944: 11940: 11925: 11921: 11912: 11908: 11897: 11895: 11892: 11891: 11874: 11869: 11868: 11856: 11852: 11837: 11833: 11824: 11820: 11809: 11807: 11804: 11803: 11783: 11782: 11776: 11772: 11764: 11757: 11755: 11750: 11745: 11740: 11730: 11729: 11720: 11719: 11714: 11703: 11686: 11684: 11683: 11676: 11675: 11673: 11670: 11669: 11644:state observers 11640: 11626: 11577: 11576: 11565: 11562: 11561: 11537: 11520: 11519: 11496: 11495: 11478: 11457: 11456: 11454: 11451: 11450: 11423: 11415: 11412: 11411: 11389: 11375: 11374: 11359: 11336: 11335: 11334: 11332: 11321: 11319: 11298: 11287: 11286: 11281: 11258: 11257: 11243: 11240: 11239: 11212: 11211: 11199: 11182: 11181: 11174: 11172: 11166: 11164: 11142: 11131: 11130: 11125: 11124: 11120: 11114: 11113: 11087: 11086: 11079: 11077: 11072: 11070: 11053: 11042: 11041: 11036: 11029: 11028: 11011: 11010: 10996: 10993: 10992: 10970: 10956: 10955: 10932: 10924: 10913: 10912: 10907: 10887: 10876: 10875: 10870: 10868: 10865: 10864: 10846: 10838: 10836: 10833: 10832: 10812: 10798: 10797: 10765: 10764: 10759: 10751: 10737: 10736: 10713: 10705: 10694: 10693: 10688: 10686: 10683: 10682: 10656: 10642: 10641: 10609: 10608: 10603: 10592: 10589: 10588: 10560: 10559: 10557: 10554: 10553: 10531: 10528: 10527: 10499: 10498: 10490: 10487: 10486: 10468: 10454: 10453: 10421: 10420: 10415: 10407: 10404: 10403: 10375: 10374: 10372: 10369: 10368: 10346: 10343: 10342: 10314: 10313: 10305: 10302: 10301: 10274: 10273: 10247: 10246: 10227: 10225: 10208: 10207: 10193: 10192: 10178: 10177: 10168: 10157: 10156: 10155: 10146: 10135: 10134: 10133: 10119: 10118: 10116: 10113: 10112: 10082: 10081: 10076: 10073: 10072: 10047: 10046: 10032: 10029: 10028: 10002: 10001: 9986: 9982: 9973: 9969: 9957: 9953: 9944: 9940: 9932: 9929: 9928: 9922: 9906: 9898: 9896: 9893: 9892: 9867: 9850: 9842: 9839: 9838: 9817: 9816: 9801: 9797: 9788: 9784: 9763: 9752: 9751: 9750: 9747: 9746: 9740: 9736: 9727: 9716: 9715: 9714: 9707: 9706: 9704: 9701: 9700: 9676: 9675: 9666: 9662: 9660: 9657: 9656: 9633: 9629: 9627: 9624: 9623: 9584: 9583: 9551: 9550: 9548: 9545: 9544: 9505: 9497: 9494: 9493: 9464: 9456: 9453: 9452: 9432: 9430: 9427: 9426: 9400: 9399: 9397: 9394: 9393: 9368: 9359: 9355: 9353: 9350: 9349: 9332: 9330: 9327: 9326: 9300: 9299: 9297: 9294: 9293: 9268: 9259: 9255: 9253: 9250: 9249: 9226: 9225: 9211: 9200: 9198: 9190: 9181: 9177: 9174: 9173: 9159: 9148: 9146: 9138: 9129: 9125: 9118: 9117: 9106: 9098: 9095: 9094: 9073: 9065: 9062: 9061: 9022: 9008: 9007: 8990: 8988: 8987: 8960: 8937: 8930: 8926: 8924: 8921: 8910: 8909: 8905: 8895: 8888: 8884: 8882: 8867: 8863: 8848: 8844: 8835: 8831: 8827: 8825: 8809: 8807: 8806: 8779: 8756: 8749: 8745: 8743: 8733: 8732: 8728: 8718: 8711: 8707: 8705: 8692: 8678: 8677: 8676: 8659: 8657: 8656: 8647: 8646: 8642: 8632: 8625: 8621: 8619: 8618: 8616: 8604: 8590: 8589: 8587: 8584: 8583: 8545: 8544: 8539: 8536: 8535: 8514: 8513: 8499: 8497: 8480: 8479: 8476: 8475: 8461: 8459: 8442: 8441: 8434: 8433: 8431: 8428: 8427: 8406: 8398: 8395: 8394: 8366: 8365: 8363: 8360: 8359: 8329: 8318: 8315: 8301: 8287: 8286: 8285: 8283: 8272: 8268: 8260: 8257: 8245: 8241: 8236: 8229: 8227: 8212: 8192: 8191: 8189: 8186: 8185: 8157: 8149: 8147: 8144: 8143: 8115: 8112: 8111: 8089: 8086: 8085: 8058: 8047: 8046: 8045: 8039: 8035: 8020: 8009: 8008: 8007: 7995: 7991: 7976: 7965: 7964: 7963: 7957: 7953: 7944: 7933: 7932: 7931: 7925: 7921: 7919: 7916: 7915: 7884: 7870: 7869: 7867: 7864: 7863: 7836: 7828: 7825: 7824: 7792: 7789: 7788: 7765: 7761: 7759: 7756: 7755: 7717: 7713: 7707: 7703: 7688: 7684: 7672: 7668: 7653: 7649: 7643: 7639: 7630: 7626: 7620: 7616: 7605: 7597: 7594: 7593: 7557: 7554: 7553: 7549: 7535: 7487: 7486: 7475: 7472: 7471: 7430: 7429: 7427: 7424: 7423: 7389: 7388: 7386: 7383: 7382: 7354: 7351: 7350: 7316: 7315: 7283: 7282: 7280: 7277: 7276: 7236: 7235: 7233: 7230: 7229: 7196: 7195: 7193: 7190: 7189: 7161: 7160: 7154: 7150: 7148: 7145: 7144: 7124: 7113: 7105: 7102: 7101: 7073: 7072: 7070: 7067: 7066: 7050: 7039: 7031: 7028: 7027: 7008: 6997: 6983: 6982: 6980: 6977: 6976: 6946: 6942: 6934: 6932: 6923: 6904: 6890: 6886: 6878: 6876: 6875: 6871: 6870: 6856: 6842: 6841: 6837: 6823: 6795: 6772: 6765: 6745: 6731: 6727: 6719: 6717: 6708: 6689: 6675: 6671: 6663: 6661: 6660: 6656: 6655: 6641: 6618: 6611: 6609: 6594: 6592: 6591: 6589: 6586: 6585: 6559: 6548: 6540: 6537: 6536: 6516: 6514: 6511: 6510: 6482: 6468: 6464: 6456: 6454: 6445: 6426: 6412: 6408: 6400: 6398: 6397: 6393: 6392: 6381: 6379: 6376: 6375: 6353: 6338: 6336: 6335: 6320: 6306: 6283: 6276: 6272: 6270: 6261: 6257: 6249: 6247: 6245: 6242: 6241: 6219: 6211: 6209: 6206: 6205: 6177: 6163: 6162: 6160: 6157: 6156: 6122: 6120: 6117: 6116: 6096: 6082: 6081: 6079: 6076: 6075: 6053: 6045: 6042: 6041: 6025: 6014: 6006: 6003: 6002: 5999:controllability 5966: 5951: 5950: 5946: 5938: 5936: 5934: 5931: 5930: 5924: 5897: 5894: 5893: 5871: 5870: 5868: 5865: 5864: 5848: 5837: 5829: 5826: 5825: 5809: 5807: 5804: 5803: 5774: 5771: 5770: 5739: 5725: 5724: 5716: 5707: 5703: 5694: 5689: 5688: 5680: 5675: 5672: 5671: 5649: 5638: 5623: 5618: 5617: 5609: 5604: 5601: 5600: 5585: 5575:manifold; they 5554: 5551: 5550: 5528: 5527: 5525: 5522: 5521: 5505: 5494: 5486: 5483: 5482: 5481:approaches the 5466: 5464: 5461: 5460: 5444: 5441: 5440: 5424: 5413: 5412: 5407: 5405: 5402: 5401: 5379: 5376: 5375: 5343: 5332: 5331: 5326: 5320: 5305: 5304: 5275: 5272: 5271: 5234: 5233: 5216: 5213: 5212: 5187: 5184: 5183: 5162: 5157: 5156: 5148: 5128: 5127: 5122: 5119: 5118: 5099: 5096: 5095: 5071: 5070: 5065: 5062: 5061: 5041: 5037: 5027: 5015: 5011: 5004: 5002: 5000: 4999: 4972: 4961: 4958: 4937: 4926: 4923: 4908: 4906: 4893: 4885: 4882: 4871: 4867: 4865: 4863: 4861: 4858: 4855: 4854: 4848: 4827: 4824: 4823: 4806: 4802: 4796: 4795: 4790: 4787: 4786: 4765: 4763: 4760: 4759: 4737: 4734: 4733: 4729: 4705: 4703: 4700: 4699: 4681: 4679: 4676: 4675: 4651: 4649: 4646: 4645: 4641: 4615: 4611: 4592: 4587: 4584: 4583: 4566: 4562: 4545: 4542: 4541: 4510: 4509: 4507: 4504: 4503: 4477: 4473: 4456: 4453: 4452: 4449:proportionality 4432: 4429: 4428: 4427:and the symbol 4410: 4405: 4402: 4401: 4380: 4376: 4374: 4371: 4370: 4334: 4323: 4321: 4309: 4289: 4288: 4283: 4282: 4281: 4267: 4263: 4262: 4250: 4249: 4241: 4225: 4221: 4210: 4209: 4197: 4195: 4194: 4190: 4188: 4186: 4185: 4179: 4178: 4172: 4168: 4167: 4165: 4163: 4162: 4160: 4157: 4156: 4135: 4124: 4121: 4120: 4098: 4093: 4082: 4079: 4078: 4035: 4024: 4022: 4010: 4008: 4005: 4004: 3967: 3964: 3963: 3939: 3924: 3920: 3913: 3888: 3877: 3875: 3873: 3870: 3869: 3832: 3829: 3828: 3825: 3819:are constants. 3792: 3789: 3788: 3766: 3763: 3762: 3742: 3738: 3731: 3706: 3695: 3693: 3691: 3688: 3687: 3661: 3656: 3645: 3642: 3641: 3625: 3614: 3606: 3603: 3602: 3599: 3572: 3571: 3569: 3566: 3565: 3546: 3538: 3536: 3533: 3532: 3506: 3504: 3503: 3488: 3474: 3451: 3444: 3440: 3438: 3429: 3425: 3417: 3415: 3397: 3391: 3384: 3381: 3366: 3364: 3362: 3353: 3349: 3341: 3339: 3325: 3324: 3322: 3319: 3318: 3292: 3284: 3281: 3280: 3261: 3247: 3246: 3244: 3241: 3240: 3221: 3213: 3210: 3209: 3188: 3180: 3177: 3176: 3157: 3143: 3142: 3140: 3137: 3136: 3117: 3109: 3106: 3105: 3080: 3079: 3077: 3074: 3073: 3057: 3054: 3053: 3034: 3026: 3023: 3022: 2994: 2993: 2987: 2983: 2981: 2978: 2977: 2955: 2952: 2951: 2919: 2911: 2908: 2907: 2884: 2864: 2853: 2850: 2845: 2823: 2812: 2809: 2788: 2777: 2774: 2759: 2757: 2744: 2736: 2733: 2722: 2718: 2716: 2714: 2712: 2709: 2706: 2705: 2674: 2663: 2655: 2652: 2651: 2634: 2630: 2622: 2611: 2608: 2607: 2583: 2582: 2577: 2574: 2573: 2536: 2531: 2519: 2500: 2489: 2475: 2466: 2462: 2452: 2438: 2424: 2421: 2420: 2404: 2396: 2360: 2349: 2335: 2334: 2332: 2329: 2328: 2309: 2298: 2290: 2287: 2286: 2259: 2238: 2227: 2219: 2216: 2215: 2199: 2188: 2180: 2177: 2176: 2175:Having reached 2157: 2146: 2138: 2135: 2134: 2114: 2103: 2095: 2092: 2091: 2075: 2073: 2070: 2069: 2044: 2033: 2025: 2022: 2021: 1971: 1962: 1958: 1949: 1944: 1943: 1935: 1934: 1930: 1928: 1925: 1924: 1887: 1884: 1883: 1855: 1852: 1851: 1835: 1833: 1830: 1829: 1825: 1809: 1807: 1804: 1803: 1799: 1777: 1774: 1773: 1767:topographic map 1750: 1742: 1740: 1737: 1736: 1707: 1705: 1702: 1701: 1676: 1668: 1665: 1664: 1631: 1623: 1620: 1619: 1591: 1583: 1580: 1579: 1563: 1561: 1558: 1557: 1537: 1535: 1532: 1531: 1514: 1509: 1508: 1499: 1494: 1493: 1485: 1482: 1481: 1459: 1451: 1449: 1446: 1445: 1429:Selection of a 1380: 1376: 1374: 1371: 1370: 1353: 1349: 1347: 1344: 1343: 1327: 1325: 1322: 1321: 1305: 1303: 1300: 1299: 1282: 1278: 1276: 1273: 1272: 1255: 1251: 1219: 1217: 1214: 1213: 1180: 1178: 1175: 1174: 1170: 1145: 1143: 1140: 1139: 1111: 1103: 1101: 1098: 1097: 1052: 1050: 1047: 1046: 1011: 1006: 1005: 997: 988: 983: 982: 974: 971: 970: 953: 948: 947: 939: 930: 925: 924: 916: 913: 912: 901: 881: 876: 875: 865: 864: 849: 845: 842: 841: 820: 816: 813: 812: 806: 805: 790: 786: 783: 782: 767: 763: 756: 755: 738: 736: 733: 732: 718: 698: 693: 692: 682: 681: 666: 662: 659: 658: 637: 633: 630: 629: 623: 622: 607: 603: 600: 599: 584: 580: 573: 572: 555: 553: 550: 549: 501: 486: 463: 434: 432: 431: 429: 426: 425: 409: 316: 313: 312: 285: 282: 281: 272:, which has an 256: 252: 240: 229: 228: 227: 225: 222: 221: 195: 184: 183: 182: 173: 169: 161: 158: 157: 135: 132: 131: 109: 106: 105: 100:Figure 1: 94: 21:control systems 17: 12: 11: 5: 17929: 17919: 17918: 17904: 17903: 17870: 17867: 17866: 17865: 17859: 17838: 17825:978-1611975833 17824: 17800: 17794: 17781: 17775: 17754: 17748: 17735: 17729: 17716: 17710: 17697: 17691: 17678: 17672: 17659: 17653: 17636: 17630: 17612: 17611: 17605: 17581: 17580: 17565: 17529: 17507: 17500: 17482: 17468:10.1.1.43.1654 17461:(3): 721–739. 17441: 17434: 17416: 17409: 17391: 17384: 17350: 17331:(5): 703–715. 17315: 17304: 17267: 17250: 17223: 17216: 17206:, ed. (1990). 17191: 17190: 17188: 17185: 17182: 17181: 17168: 17149: 17148: 17146: 17143: 17142: 17141: 17135: 17129: 17123: 17117: 17111: 17101: 17096: 17094:Robust control 17091: 17086: 17081: 17076: 17069: 17066: 17052: 17048: 17044: 17040: 17034: 17030: 17023: 17020: 17016: 17012: 17009: 16988: 16964: 16960: 16931: 16927: 16924: 16920: 16897: 16893: 16889: 16886: 16859: 16854: 16850: 16846: 16841: 16834: 16831: 16824: 16821: 16818: 16798: 16793: 16789: 16785: 16780: 16773: 16770: 16763: 16760: 16757: 16736: 16724: 16723: 16712: 16707: 16701: 16696: 16692: 16688: 16683: 16676: 16673: 16666: 16663: 16660: 16657: 16656: 16652: 16648: 16647: 16645: 16640: 16631: 16620: 16608: 16603: 16595: 16589: 16584: 16580: 16576: 16575: 16572: 16569: 16566: 16565: 16563: 16558: 16556: 16553: 16552: 16550: 16545: 16536: 16525: 16514: 16511: 16508: 16499: 16484: 16483: 16461: 16450: 16446: 16440: 16436: 16423: 16419: 16416: 16414: 16412: 16407: 16401: 16396: 16392: 16388: 16383: 16376: 16373: 16366: 16363: 16360: 16357: 16356: 16352: 16348: 16347: 16345: 16338: 16330: 16324: 16319: 16315: 16311: 16310: 16307: 16304: 16301: 16300: 16298: 16293: 16291: 16288: 16287: 16285: 16280: 16274: 16270: 16263: 16260: 16257: 16255: 16253: 16250: 16245: 16241: 16237: 16232: 16225: 16222: 16215: 16212: 16209: 16204: 16198: 16193: 16189: 16185: 16184: 16181: 16178: 16175: 16174: 16172: 16167: 16163: 16159: 16156: 16150: 16146: 16139: 16136: 16133: 16131: 16129: 16126: 16121: 16117: 16113: 16108: 16101: 16098: 16091: 16088: 16085: 16082: 16077: 16069: 16065: 16061: 16060: 16057: 16054: 16051: 16050: 16048: 16043: 16039: 16035: 16032: 16026: 16022: 16015: 16012: 16009: 16007: 16005: 16002: 15997: 15993: 15989: 15984: 15977: 15974: 15967: 15964: 15961: 15958: 15955: 15952: 15948: 15944: 15941: 15935: 15931: 15924: 15921: 15918: 15916: 15911: 15906: 15902: 15893: 15892: 15856: 15840:Gaussian noise 15827: 15824: 15821: 15816: 15809: 15806: 15799: 15796: 15793: 15790: 15787: 15784: 15758: 15731: 15727: 15715:state observer 15700: 15695: 15661: 15656: 15652: 15646: 15642: 15638: 15633: 15629: 15625: 15605: 15602: 15599: 15596: 15593: 15590: 15587: 15584: 15581: 15578: 15575: 15553: 15549: 15528: 15525: 15522: 15519: 15516: 15513: 15510: 15488: 15483: 15470: 15469: 15458: 15453: 15448: 15443: 15438: 15434: 15428: 15424: 15420: 15415: 15411: 15407: 15404: 15399: 15394: 15387: 15383: 15377: 15373: 15369: 15364: 15359: 15352: 15348: 15344: 15335: 15329: 15325: 15321: 15316: 15311: 15304: 15300: 15296: 15293: 15288: 15284: 15280: 15277: 15272: 15268: 15264: 15255: 15250: 15242: 15238: 15230: 15226: 15222: 15221: 15218: 15215: 15214: 15209: 15205: 15201: 15200: 15195: 15191: 15187: 15186: 15184: 15173: 15169: 15165: 15156: 15149: 15145: 15134: 15130: 15122: 15115: 15112: 15105: 15104: 15101: 15098: 15097: 15092: 15085: 15082: 15075: 15074: 15069: 15062: 15059: 15052: 15051: 15049: 15018: 15014: 14991: 14987: 14966: 14963: 14960: 14957: 14954: 14928: 14916: 14915: 14904: 14890: 14887: 14884: 14881: 14878: 14875: 14872: 14864: 14859: 14854: 14833: 14830: 14827: 14824: 14821: 14818: 14815: 14807: 14802: 14798: 14789: 14776: 14767: 14747: 14746: 14735: 14726: 14722: 14717: 14712: 14705: 14701: 14697: 14690: 14687: 14684: 14681: 14675: 14666: 14657: 14652: 14647: 14640: 14636: 14632: 14625: 14622: 14614: 14609: 14605: 14594: 14590: 14586: 14580: 14577: 14571: 14568: 14539: 14518: 14515: 14512: 14509: 14489: 14486: 14481: 14474: 14471: 14445: 14441: 14418: 14414: 14410: 14405: 14398: 14395: 14371: 14368: 14363: 14359: 14335: 14334: 14323: 14320: 14316: 14310: 14305: 14298: 14294: 14290: 14285: 14281: 14275: 14271: 14266: 14262: 14259: 14256: 14253: 14239: 14238: 14227: 14224: 14221: 14218: 14215: 14212: 14209: 14206: 14203: 14200: 14197: 14174: 14171: 14165: 14162: 14156: 14134: 14131: 14128: 14125: 14122: 14117: 14112: 14109: 14104: 14100: 14096: 14093: 14090: 14085: 14081: 14077: 14072: 14068: 14064: 14061: 14056: 14051: 14038: 14037: 14026: 14023: 14020: 14017: 14014: 14009: 14004: 13997: 13993: 13989: 13984: 13980: 13974: 13970: 13966: 13963: 13958: 13954: 13950: 13947: 13944: 13939: 13934: 13927: 13923: 13919: 13914: 13910: 13904: 13900: 13896: 13891: 13884: 13881: 13874: 13868: 13865: 13838: 13813: 13810: 13804: 13801: 13795: 13775: 13752: 13749: 13735: 13734: 13723: 13718: 13714: 13710: 13705: 13698: 13695: 13688: 13683: 13679: 13675: 13672: 13666: 13663: 13657: 13652: 13645: 13642: 13635: 13632: 13607: 13603: 13599: 13594: 13587: 13584: 13558: 13554: 13531: 13524: 13521: 13506: 13505: 13492: 13488: 13484: 13479: 13472: 13469: 13462: 13459: 13430: 13426: 13422: 13417: 13410: 13407: 13400: 13395: 13391: 13379: 13378: 13363: 13358: 13354: 13350: 13347: 13344: 13341: 13337: 13333: 13330: 13327: 13325: 13323: 13320: 13315: 13311: 13307: 13302: 13295: 13292: 13285: 13282: 13279: 13276: 13273: 13269: 13265: 13259: 13255: 13248: 13245: 13242: 13239: 13237: 13235: 13231: 13227: 13224: 13220: 13216: 13213: 13210: 13205: 13201: 13197: 13192: 13185: 13182: 13175: 13172: 13169: 13166: 13162: 13158: 13155: 13149: 13145: 13138: 13135: 13132: 13130: 13128: 13122: 13118: 13111: 13105: 13100: 13096: 13087: 13084: 13082: 13077: 13073: 13066: 13065: 13041: 13037: 13031: 13027: 13020: 13016: 12993: 12988: 12983: 12980: 12975: 12971: 12967: 12964: 12961: 12956: 12952: 12948: 12943: 12939: 12935: 12932: 12928: 12905: 12902: 12899: 12896: 12893: 12888: 12883: 12878: 12874: 12862: 12861: 12848: 12840: 12836: 12832: 12831: 12828: 12825: 12822: 12821: 12819: 12814: 12811: 12782: 12777: 12772: 12769: 12747: 12743: 12739: 12736: 12714: 12707: 12704: 12679: 12675: 12671: 12667: 12664: 12653: 12652: 12641: 12636: 12632: 12628: 12623: 12616: 12613: 12606: 12603: 12600: 12597: 12593: 12589: 12586: 12580: 12576: 12569: 12566: 12560: 12555: 12551: 12516: 12511: 12506: 12503: 12498: 12491: 12488: 12481: 12478: 12475: 12470: 12463: 12460: 12453: 12448: 12441: 12438: 12431: 12428: 12422: 12418: 12392: 12388: 12384: 12381: 12360: 12348: 12347: 12333: 12330: 12327: 12324: 12321: 12318: 12315: 12310: 12305: 12300: 12296: 12285: 12271: 12268: 12265: 12262: 12259: 12256: 12253: 12250: 12247: 12244: 12241: 12236: 12231: 12226: 12222: 12211: 12197: 12194: 12191: 12188: 12185: 12180: 12175: 12170: 12166: 12155: 12141: 12137: 12114: 12110: 12095: 12094: 12081: 12073: 12069: 12065: 12061: 12057: 12053: 12052: 12047: 12043: 12039: 12035: 12031: 12027: 12026: 12024: 12019: 12016: 11992: 11965: 11960: 11955: 11952: 11947: 11943: 11939: 11936: 11933: 11928: 11924: 11920: 11915: 11911: 11907: 11904: 11900: 11877: 11872: 11867: 11864: 11859: 11855: 11851: 11848: 11845: 11840: 11836: 11832: 11827: 11823: 11819: 11816: 11812: 11800: 11799: 11786: 11779: 11775: 11771: 11767: 11761: 11754: 11751: 11749: 11746: 11744: 11741: 11739: 11736: 11735: 11733: 11728: 11725: 11722: 11721: 11717: 11713: 11710: 11706: 11702: 11699: 11693: 11689: 11682: 11681: 11679: 11639: 11636: 11625: 11622: 11621: 11620: 11608: 11605: 11602: 11599: 11596: 11593: 11590: 11584: 11581: 11575: 11572: 11569: 11554: 11553: 11552: 11536: 11533: 11527: 11524: 11518: 11515: 11512: 11509: 11503: 11500: 11494: 11491: 11488: 11485: 11476: 11473: 11470: 11464: 11461: 11436: 11433: 11430: 11426: 11422: 11419: 11408: 11407: 11406: 11388: 11385: 11382: 11371: 11367: 11362: 11356: 11352: 11349: 11343: 11340: 11330: 11327: 11324: 11317: 11314: 11311: 11308: 11305: 11301: 11294: 11291: 11284: 11280: 11277: 11274: 11271: 11265: 11262: 11256: 11253: 11250: 11247: 11230: 11229: 11228: 11215: 11210: 11207: 11202: 11196: 11189: 11186: 11180: 11177: 11165: 11162: 11158: 11155: 11152: 11149: 11145: 11138: 11135: 11128: 11123: 11119: 11116: 11115: 11112: 11109: 11106: 11101: 11094: 11091: 11085: 11082: 11071: 11069: 11066: 11063: 11060: 11056: 11049: 11046: 11039: 11035: 11034: 11032: 11027: 11024: 11018: 11015: 11009: 11006: 11003: 11000: 10987: 10986: 10985: 10973: 10969: 10963: 10960: 10954: 10951: 10948: 10945: 10942: 10939: 10935: 10931: 10927: 10920: 10917: 10910: 10906: 10903: 10900: 10897: 10894: 10890: 10883: 10880: 10873: 10849: 10845: 10841: 10829: 10828: 10827: 10815: 10811: 10805: 10802: 10796: 10793: 10790: 10787: 10784: 10781: 10778: 10772: 10769: 10762: 10758: 10754: 10750: 10744: 10741: 10735: 10732: 10729: 10726: 10723: 10720: 10716: 10712: 10708: 10701: 10698: 10691: 10673: 10672: 10671: 10659: 10655: 10649: 10646: 10640: 10637: 10634: 10631: 10628: 10625: 10622: 10616: 10613: 10606: 10602: 10599: 10596: 10576: 10573: 10567: 10564: 10541: 10538: 10535: 10515: 10512: 10506: 10503: 10497: 10494: 10483: 10471: 10467: 10461: 10458: 10452: 10449: 10446: 10443: 10440: 10437: 10434: 10428: 10425: 10418: 10414: 10411: 10391: 10388: 10382: 10379: 10356: 10353: 10350: 10330: 10327: 10321: 10318: 10312: 10309: 10297: 10296: 10281: 10278: 10270: 10266: 10263: 10260: 10254: 10251: 10245: 10242: 10239: 10236: 10233: 10230: 10222: 10215: 10212: 10206: 10200: 10197: 10191: 10185: 10182: 10176: 10171: 10164: 10161: 10154: 10149: 10142: 10139: 10132: 10126: 10123: 10098: 10095: 10089: 10086: 10080: 10060: 10054: 10051: 10045: 10042: 10039: 10036: 10025: 10024: 10023: 10009: 10006: 10000: 9997: 9994: 9989: 9985: 9981: 9976: 9972: 9968: 9965: 9960: 9956: 9952: 9947: 9943: 9939: 9936: 9909: 9905: 9901: 9880: 9877: 9874: 9870: 9866: 9863: 9860: 9857: 9853: 9849: 9846: 9835: 9834: 9833: 9820: 9815: 9812: 9809: 9804: 9800: 9796: 9791: 9787: 9783: 9780: 9777: 9774: 9771: 9766: 9759: 9756: 9749: 9748: 9743: 9739: 9735: 9730: 9723: 9720: 9713: 9712: 9710: 9683: 9680: 9674: 9669: 9665: 9644: 9641: 9636: 9632: 9616: 9615: 9614: 9603: 9600: 9597: 9591: 9588: 9582: 9579: 9576: 9573: 9570: 9567: 9564: 9558: 9555: 9539: 9538: 9536:dynamic system 9531: 9530: 9518: 9515: 9512: 9508: 9504: 9501: 9477: 9474: 9471: 9467: 9463: 9460: 9449: 9448: 9447: 9435: 9407: 9404: 9375: 9371: 9367: 9362: 9358: 9347: 9335: 9307: 9304: 9275: 9271: 9267: 9262: 9258: 9244: 9243: 9242: 9229: 9224: 9221: 9218: 9214: 9210: 9207: 9199: 9197: 9193: 9189: 9184: 9180: 9176: 9175: 9172: 9169: 9166: 9162: 9158: 9155: 9147: 9145: 9141: 9137: 9132: 9128: 9124: 9123: 9121: 9116: 9113: 9109: 9105: 9102: 9080: 9076: 9072: 9069: 9055: 9054: 9051: 9050: 9041: 9039: 9024: vector ) 9021: 9018: 9015: 9004: 8997: 8993: 8984: 8980: 8976: 8973: 8970: 8967: 8963: 8959: 8956: 8953: 8950: 8947: 8944: 8940: 8936: 8933: 8929: 8913: 8908: 8902: 8898: 8894: 8891: 8887: 8879: 8875: 8870: 8866: 8862: 8859: 8856: 8851: 8847: 8843: 8838: 8834: 8830: 8823: 8816: 8812: 8803: 8799: 8795: 8792: 8789: 8786: 8782: 8778: 8775: 8772: 8769: 8766: 8763: 8759: 8755: 8752: 8748: 8736: 8731: 8725: 8721: 8717: 8714: 8710: 8704: 8699: 8695: 8691: 8685: 8682: 8673: 8666: 8662: 8650: 8645: 8639: 8635: 8631: 8628: 8624: 8614: 8611: 8607: 8603: 8597: 8594: 8574: 8573: 8561: 8558: 8552: 8549: 8543: 8532: 8531: 8530: 8517: 8512: 8509: 8506: 8498: 8496: 8493: 8487: 8484: 8478: 8477: 8474: 8471: 8468: 8460: 8458: 8455: 8449: 8446: 8440: 8439: 8437: 8413: 8409: 8405: 8402: 8382: 8379: 8373: 8370: 8356: 8355: 8354: 8338: 8335: 8332: 8327: 8324: 8321: 8312: 8308: 8304: 8300: 8294: 8291: 8275: 8271: 8266: 8263: 8254: 8244: 8239: 8235: 8232: 8225: 8222: 8219: 8215: 8211: 8208: 8205: 8199: 8196: 8160: 8156: 8152: 8131: 8128: 8125: 8122: 8119: 8099: 8096: 8093: 8082: 8081: 8080: 8069: 8066: 8061: 8054: 8051: 8042: 8038: 8034: 8029: 8026: 8023: 8016: 8013: 8004: 8001: 7998: 7994: 7990: 7987: 7984: 7979: 7972: 7969: 7960: 7956: 7952: 7947: 7940: 7937: 7928: 7924: 7910: 7909: 7908: 7897: 7894: 7891: 7887: 7883: 7877: 7874: 7849: 7846: 7843: 7839: 7835: 7832: 7808: 7805: 7802: 7799: 7796: 7776: 7773: 7768: 7764: 7749: 7748: 7745: 7744: 7735: 7733: 7720: 7716: 7710: 7706: 7702: 7697: 7694: 7691: 7687: 7681: 7678: 7675: 7671: 7667: 7664: 7661: 7656: 7652: 7646: 7642: 7638: 7633: 7629: 7623: 7619: 7615: 7612: 7608: 7604: 7601: 7584: 7583: 7567: 7564: 7561: 7534: 7531: 7518: 7515: 7512: 7509: 7506: 7503: 7500: 7494: 7491: 7485: 7482: 7479: 7452: 7449: 7446: 7443: 7437: 7434: 7420: 7419: 7408: 7405: 7402: 7396: 7393: 7367: 7364: 7361: 7358: 7347: 7346: 7335: 7332: 7329: 7323: 7320: 7314: 7311: 7308: 7305: 7302: 7299: 7296: 7290: 7287: 7270: 7269: 7258: 7255: 7252: 7249: 7243: 7240: 7203: 7200: 7177: 7174: 7168: 7165: 7157: 7153: 7127: 7123: 7120: 7116: 7112: 7109: 7089: 7086: 7080: 7077: 7053: 7049: 7046: 7042: 7038: 7035: 7024: 7023: 7011: 7007: 7004: 7000: 6996: 6990: 6987: 6970: 6969: 6957: 6949: 6945: 6940: 6937: 6929: 6926: 6921: 6917: 6914: 6911: 6907: 6903: 6900: 6893: 6889: 6884: 6881: 6874: 6869: 6866: 6863: 6859: 6855: 6852: 6849: 6845: 6840: 6836: 6833: 6830: 6826: 6822: 6819: 6816: 6811: 6808: 6805: 6802: 6798: 6794: 6791: 6788: 6785: 6782: 6779: 6775: 6771: 6768: 6762: 6758: 6755: 6752: 6748: 6744: 6741: 6734: 6730: 6725: 6722: 6714: 6711: 6706: 6702: 6699: 6696: 6692: 6688: 6685: 6678: 6674: 6669: 6666: 6659: 6654: 6651: 6648: 6644: 6640: 6637: 6634: 6631: 6628: 6625: 6621: 6617: 6614: 6607: 6601: 6597: 6562: 6558: 6555: 6551: 6547: 6544: 6519: 6507: 6506: 6495: 6492: 6489: 6485: 6481: 6478: 6471: 6467: 6462: 6459: 6451: 6448: 6443: 6439: 6436: 6433: 6429: 6425: 6422: 6415: 6411: 6406: 6403: 6396: 6391: 6388: 6384: 6369: 6368: 6356: 6352: 6345: 6341: 6332: 6328: 6323: 6319: 6316: 6313: 6309: 6305: 6302: 6299: 6296: 6293: 6290: 6286: 6282: 6279: 6275: 6264: 6260: 6255: 6252: 6226: 6222: 6218: 6214: 6202: 6201: 6190: 6187: 6184: 6180: 6176: 6170: 6167: 6125: 6113: 6112: 6099: 6095: 6089: 6086: 6060: 6056: 6052: 6049: 6028: 6024: 6021: 6017: 6013: 6010: 5991: 5990: 5979: 5976: 5973: 5969: 5965: 5962: 5954: 5949: 5944: 5941: 5923: 5920: 5907: 5904: 5901: 5878: 5875: 5851: 5847: 5844: 5840: 5836: 5833: 5812: 5787: 5784: 5781: 5778: 5767: 5766: 5755: 5752: 5749: 5746: 5742: 5738: 5732: 5729: 5723: 5719: 5715: 5710: 5706: 5702: 5697: 5692: 5687: 5683: 5679: 5656: 5652: 5648: 5645: 5641: 5637: 5634: 5631: 5626: 5621: 5616: 5612: 5608: 5584: 5581: 5564: 5561: 5558: 5535: 5532: 5508: 5504: 5501: 5497: 5493: 5490: 5469: 5448: 5427: 5420: 5417: 5410: 5389: 5386: 5383: 5372: 5371: 5359: 5356: 5353: 5350: 5346: 5339: 5336: 5329: 5318: 5312: 5309: 5303: 5300: 5297: 5294: 5291: 5288: 5285: 5282: 5279: 5265: 5264: 5253: 5250: 5247: 5241: 5238: 5232: 5229: 5226: 5223: 5220: 5197: 5194: 5191: 5180: 5179: 5165: 5160: 5155: 5151: 5147: 5144: 5141: 5135: 5132: 5126: 5103: 5092:Euclidean norm 5079: 5074: 5069: 5058: 5057: 5044: 5040: 5031: 5024: 5018: 5014: 5010: 5007: 4998: 4995: 4992: 4989: 4981: 4978: 4975: 4970: 4967: 4964: 4955: 4946: 4943: 4940: 4935: 4932: 4929: 4920: 4915: 4912: 4899: 4896: 4891: 4888: 4879: 4874: 4870: 4847: 4844: 4831: 4809: 4805: 4799: 4794: 4784:Euclidean norm 4769: 4747: 4744: 4741: 4717: 4714: 4709: 4685: 4663: 4660: 4655: 4629: 4626: 4623: 4618: 4614: 4610: 4605: 4602: 4599: 4596: 4591: 4569: 4565: 4561: 4558: 4555: 4552: 4549: 4529: 4526: 4523: 4517: 4514: 4491: 4488: 4485: 4480: 4476: 4472: 4469: 4466: 4463: 4460: 4436: 4414: 4409: 4383: 4379: 4367: 4366: 4355: 4352: 4349: 4343: 4340: 4337: 4332: 4329: 4326: 4317: 4313: 4308: 4296: 4293: 4279: 4275: 4270: 4266: 4259: 4253: 4244: 4238: 4228: 4224: 4220: 4217: 4214: 4206: 4202: 4193: 4182: 4175: 4171: 4139: 4134: 4131: 4128: 4107: 4104: 4101: 4096: 4092: 4089: 4086: 4073:which, by the 4071: 4070: 4059: 4056: 4053: 4050: 4044: 4041: 4038: 4033: 4030: 4027: 4018: 4014: 3989: 3986: 3983: 3980: 3977: 3974: 3971: 3960: 3959: 3948: 3943: 3938: 3935: 3932: 3927: 3923: 3917: 3912: 3909: 3906: 3903: 3897: 3894: 3891: 3886: 3883: 3880: 3854: 3851: 3848: 3845: 3842: 3839: 3836: 3824: 3821: 3808: 3805: 3802: 3799: 3796: 3776: 3773: 3770: 3759: 3758: 3745: 3741: 3735: 3730: 3727: 3724: 3721: 3715: 3712: 3709: 3704: 3701: 3698: 3670: 3667: 3664: 3659: 3655: 3652: 3649: 3628: 3624: 3621: 3617: 3613: 3610: 3598: 3595: 3579: 3576: 3553: 3549: 3545: 3541: 3529: 3528: 3513: 3509: 3500: 3496: 3491: 3487: 3484: 3481: 3477: 3473: 3470: 3467: 3464: 3461: 3458: 3454: 3450: 3447: 3443: 3432: 3428: 3423: 3420: 3414: 3406: 3403: 3400: 3394: 3390: 3387: 3378: 3373: 3369: 3356: 3352: 3347: 3344: 3338: 3332: 3329: 3312: 3311: 3299: 3295: 3291: 3288: 3279:positive when 3268: 3264: 3260: 3254: 3251: 3228: 3224: 3220: 3217: 3207: 3195: 3191: 3187: 3184: 3175:negative when 3164: 3160: 3156: 3150: 3147: 3124: 3120: 3116: 3113: 3087: 3084: 3061: 3041: 3037: 3033: 3030: 3010: 3007: 3001: 2998: 2990: 2986: 2976:), to achieve 2965: 2962: 2959: 2932: 2929: 2926: 2922: 2918: 2915: 2900: 2899: 2883: 2880: 2873: 2870: 2867: 2862: 2859: 2856: 2843: 2840: 2832: 2829: 2826: 2821: 2818: 2815: 2806: 2797: 2794: 2791: 2786: 2783: 2780: 2771: 2766: 2763: 2750: 2747: 2742: 2739: 2730: 2725: 2721: 2677: 2673: 2670: 2666: 2662: 2659: 2637: 2633: 2629: 2625: 2621: 2618: 2615: 2604:Euclidean norm 2591: 2586: 2581: 2568: 2567: 2564: 2563: 2554: 2552: 2539: 2534: 2530: 2526: 2522: 2518: 2515: 2512: 2507: 2504: 2499: 2496: 2492: 2488: 2485: 2482: 2478: 2474: 2469: 2465: 2459: 2456: 2451: 2448: 2445: 2441: 2437: 2434: 2431: 2428: 2403: 2400: 2395: 2392: 2367: 2363: 2359: 2356: 2352: 2348: 2342: 2339: 2312: 2308: 2305: 2301: 2297: 2294: 2258: 2255: 2254: 2253: 2241: 2237: 2234: 2230: 2226: 2223: 2202: 2198: 2195: 2191: 2187: 2184: 2173: 2160: 2156: 2153: 2149: 2145: 2142: 2117: 2113: 2110: 2106: 2102: 2099: 2078: 2047: 2043: 2040: 2036: 2032: 2029: 2016: 2015: 2012: 2011: 2002: 2000: 1988: 1984: 1981: 1978: 1974: 1970: 1965: 1961: 1957: 1952: 1947: 1942: 1938: 1933: 1903: 1900: 1897: 1894: 1891: 1882:, there is an 1871: 1868: 1865: 1862: 1859: 1838: 1812: 1787: 1784: 1781: 1753: 1749: 1745: 1720: 1717: 1714: 1710: 1689: 1686: 1683: 1679: 1675: 1672: 1657: 1656: 1644: 1641: 1638: 1634: 1630: 1627: 1616: 1604: 1601: 1598: 1594: 1590: 1587: 1566: 1540: 1517: 1512: 1507: 1502: 1497: 1492: 1489: 1462: 1458: 1454: 1438: 1437: 1434: 1391: 1388: 1383: 1379: 1356: 1352: 1330: 1308: 1285: 1281: 1258: 1254: 1250: 1247: 1244: 1241: 1238: 1235: 1232: 1229: 1226: 1222: 1183: 1158: 1155: 1152: 1148: 1127: 1124: 1121: 1118: 1114: 1110: 1106: 1065: 1062: 1059: 1055: 1020: 1017: 1014: 1009: 1004: 1000: 996: 991: 986: 981: 978: 956: 951: 946: 942: 938: 933: 928: 923: 920: 898: 897: 884: 879: 874: 869: 863: 860: 857: 852: 848: 844: 843: 840: 837: 834: 829: 826: 823: 819: 815: 814: 811: 808: 807: 804: 801: 798: 793: 789: 785: 784: 781: 778: 775: 770: 766: 762: 761: 759: 754: 751: 748: 745: 741: 715: 714: 701: 696: 691: 686: 680: 677: 674: 669: 665: 661: 660: 657: 654: 651: 646: 643: 640: 636: 632: 631: 628: 625: 624: 621: 618: 615: 610: 606: 602: 601: 598: 595: 592: 587: 583: 579: 578: 576: 571: 568: 565: 562: 558: 541: 540: 537: 536: 527: 525: 514: 511: 508: 504: 499: 496: 493: 489: 485: 482: 479: 476: 473: 470: 466: 462: 459: 456: 453: 450: 447: 441: 437: 408: 407:Control scheme 405: 355:dynamic system 337:surface is an 326: 323: 320: 295: 292: 289: 259: 255: 251: 248: 243: 236: 233: 206: 203: 198: 191: 188: 181: 176: 172: 168: 165: 145: 142: 139: 119: 116: 113: 93: 90: 43:by applying a 15: 9: 6: 4: 3: 2: 17928: 17917: 17914: 17913: 17911: 17899: 17894: 17890: 17886: 17882: 17878: 17873: 17872: 17862: 17856: 17852: 17848: 17844: 17839: 17835: 17831: 17827: 17821: 17817: 17813: 17809: 17805: 17801: 17797: 17791: 17787: 17782: 17778: 17772: 17768: 17764: 17760: 17755: 17751: 17745: 17741: 17736: 17732: 17726: 17722: 17717: 17713: 17707: 17703: 17698: 17694: 17688: 17684: 17679: 17675: 17669: 17665: 17660: 17656: 17654:0-86341-350-1 17650: 17646: 17642: 17637: 17633: 17627: 17623: 17619: 17616:Edwards, C.; 17614: 17613: 17608: 17602: 17598: 17594: 17590: 17585: 17584: 17576: 17572: 17568: 17562: 17558: 17554: 17550: 17546: 17545: 17540: 17533: 17525: 17521: 17514: 17512: 17503: 17497: 17493: 17486: 17478: 17474: 17469: 17464: 17460: 17456: 17452: 17445: 17437: 17431: 17427: 17420: 17412: 17406: 17402: 17395: 17387: 17381: 17377: 17376:Prentice Hall 17373: 17372: 17367: 17361: 17359: 17357: 17355: 17346: 17342: 17338: 17334: 17330: 17326: 17319: 17313: 17308: 17300: 17296: 17291: 17290:10.1.1.477.77 17286: 17282: 17278: 17271: 17263: 17262: 17254: 17246: 17242: 17238: 17234: 17227: 17219: 17213: 17209: 17205: 17199: 17197: 17192: 17178: 17172: 17165: 17160: 17154: 17150: 17139: 17136: 17133: 17130: 17127: 17124: 17121: 17118: 17115: 17112: 17109: 17105: 17102: 17100: 17097: 17095: 17092: 17090: 17087: 17085: 17084:Hybrid system 17082: 17080: 17077: 17075: 17072: 17071: 17065: 17046: 17038: 17021: 17007: 16925: 16922: 16895: 16891: 16887: 16884: 16875: 16873: 16852: 16848: 16844: 16839: 16829: 16819: 16816: 16791: 16787: 16783: 16778: 16768: 16758: 16755: 16710: 16705: 16694: 16690: 16686: 16681: 16671: 16661: 16658: 16643: 16638: 16629: 16621: 16606: 16601: 16593: 16587: 16582: 16578: 16570: 16567: 16561: 16554: 16548: 16543: 16534: 16526: 16512: 16509: 16506: 16497: 16489: 16488: 16487: 16448: 16444: 16421: 16417: 16415: 16405: 16394: 16390: 16386: 16381: 16371: 16361: 16358: 16343: 16336: 16328: 16322: 16317: 16313: 16305: 16302: 16296: 16289: 16283: 16278: 16261: 16258: 16256: 16243: 16239: 16235: 16230: 16220: 16210: 16207: 16202: 16196: 16191: 16187: 16179: 16176: 16170: 16165: 16157: 16154: 16137: 16134: 16132: 16119: 16115: 16111: 16106: 16096: 16086: 16083: 16080: 16075: 16067: 16063: 16055: 16052: 16046: 16041: 16033: 16030: 16013: 16010: 16008: 15995: 15991: 15987: 15982: 15972: 15962: 15959: 15956: 15953: 15950: 15942: 15939: 15922: 15919: 15917: 15909: 15883: 15882: 15881: 15878: 15876: 15875:Kalman filter 15854: 15841: 15822: 15819: 15814: 15804: 15794: 15791: 15788: 15785: 15782: 15756: 15729: 15725: 15716: 15698: 15683: 15679: 15675: 15654: 15650: 15644: 15640: 15636: 15631: 15627: 15600: 15597: 15594: 15588: 15582: 15579: 15576: 15551: 15547: 15526: 15523: 15517: 15514: 15511: 15486: 15456: 15451: 15436: 15432: 15426: 15422: 15418: 15413: 15409: 15402: 15397: 15385: 15381: 15375: 15371: 15367: 15362: 15350: 15346: 15342: 15333: 15327: 15323: 15319: 15314: 15302: 15298: 15294: 15286: 15282: 15275: 15270: 15266: 15262: 15253: 15240: 15236: 15228: 15224: 15216: 15207: 15203: 15193: 15189: 15182: 15171: 15167: 15163: 15154: 15147: 15132: 15128: 15120: 15113: 15110: 15099: 15090: 15083: 15080: 15067: 15060: 15057: 15047: 15034: 15033: 15032: 15016: 15012: 14989: 14985: 14961: 14958: 14955: 14926: 14902: 14888: 14885: 14879: 14876: 14873: 14862: 14857: 14828: 14825: 14822: 14816: 14813: 14805: 14800: 14796: 14787: 14774: 14765: 14752: 14751: 14750: 14733: 14724: 14720: 14715: 14703: 14699: 14695: 14685: 14679: 14673: 14664: 14655: 14650: 14638: 14634: 14630: 14623: 14620: 14612: 14607: 14603: 14592: 14588: 14584: 14578: 14575: 14569: 14566: 14559: 14558: 14557: 14537: 14513: 14507: 14487: 14484: 14479: 14472: 14469: 14443: 14439: 14416: 14412: 14408: 14403: 14393: 14369: 14366: 14361: 14357: 14321: 14308: 14296: 14292: 14288: 14283: 14279: 14273: 14269: 14254: 14251: 14244: 14243: 14242: 14222: 14216: 14213: 14210: 14207: 14201: 14195: 14188: 14187: 14186: 14172: 14169: 14163: 14160: 14154: 14129: 14126: 14123: 14110: 14102: 14098: 14094: 14091: 14088: 14083: 14079: 14075: 14070: 14066: 14059: 14054: 14021: 14015: 14012: 14007: 13995: 13991: 13987: 13982: 13978: 13972: 13968: 13964: 13956: 13952: 13945: 13942: 13937: 13925: 13921: 13917: 13912: 13908: 13902: 13898: 13894: 13889: 13882: 13879: 13872: 13866: 13863: 13853: 13852: 13851: 13827: 13811: 13808: 13802: 13799: 13793: 13773: 13750: 13747: 13721: 13716: 13712: 13708: 13703: 13693: 13686: 13681: 13677: 13673: 13661: 13655: 13650: 13640: 13630: 13623: 13622: 13621: 13605: 13601: 13597: 13592: 13582: 13556: 13552: 13529: 13519: 13490: 13486: 13482: 13477: 13467: 13460: 13457: 13450: 13449: 13448: 13428: 13424: 13420: 13415: 13405: 13398: 13393: 13389: 13356: 13352: 13345: 13342: 13339: 13331: 13328: 13326: 13313: 13309: 13305: 13300: 13290: 13280: 13277: 13274: 13263: 13243: 13240: 13238: 13225: 13222: 13214: 13211: 13203: 13199: 13195: 13190: 13180: 13170: 13167: 13164: 13156: 13153: 13136: 13133: 13131: 13120: 13109: 13103: 13085: 13083: 13075: 13056: 13055: 13054: 13035: 13018: 12991: 12981: 12973: 12969: 12965: 12962: 12959: 12954: 12950: 12946: 12941: 12937: 12930: 12900: 12897: 12894: 12881: 12876: 12872: 12846: 12838: 12834: 12826: 12823: 12817: 12812: 12809: 12802: 12801: 12800: 12798: 12780: 12770: 12767: 12745: 12741: 12737: 12734: 12712: 12702: 12665: 12662: 12634: 12630: 12626: 12621: 12611: 12601: 12598: 12595: 12587: 12584: 12567: 12564: 12558: 12536: 12535: 12534: 12514: 12504: 12496: 12486: 12479: 12476: 12473: 12468: 12458: 12451: 12446: 12436: 12426: 12390: 12386: 12382: 12379: 12328: 12325: 12322: 12316: 12313: 12303: 12298: 12294: 12286: 12266: 12263: 12260: 12254: 12248: 12245: 12242: 12229: 12224: 12220: 12212: 12192: 12189: 12186: 12173: 12168: 12164: 12156: 12139: 12135: 12112: 12108: 12100: 12099: 12098: 12079: 12071: 12067: 12059: 12055: 12045: 12041: 12033: 12029: 12022: 12017: 12014: 12007: 12006: 12005: 11963: 11953: 11945: 11941: 11937: 11934: 11931: 11926: 11922: 11918: 11913: 11909: 11902: 11875: 11865: 11857: 11853: 11849: 11846: 11843: 11838: 11834: 11830: 11825: 11821: 11814: 11777: 11773: 11769: 11759: 11752: 11747: 11742: 11737: 11731: 11726: 11723: 11711: 11708: 11700: 11697: 11691: 11677: 11668: 11667: 11666: 11663: 11661: 11657: 11653: 11649: 11648:Kalman filter 11645: 11635: 11632: 11603: 11600: 11597: 11591: 11582: 11579: 11573: 11570: 11559: 11555: 11534: 11531: 11525: 11522: 11516: 11513: 11510: 11501: 11498: 11492: 11489: 11483: 11474: 11471: 11468: 11462: 11459: 11449: 11448: 11434: 11431: 11417: 11409: 11386: 11383: 11380: 11369: 11360: 11354: 11350: 11347: 11341: 11338: 11325: 11322: 11312: 11309: 11306: 11303: 11292: 11289: 11275: 11272: 11263: 11260: 11254: 11251: 11245: 11238: 11237: 11235: 11231: 11208: 11205: 11200: 11194: 11187: 11184: 11178: 11175: 11160: 11156: 11153: 11150: 11147: 11136: 11133: 11121: 11117: 11110: 11107: 11104: 11099: 11092: 11089: 11083: 11080: 11067: 11064: 11061: 11058: 11047: 11044: 11030: 11025: 11016: 11013: 11007: 11004: 10998: 10991: 10990: 10988: 10961: 10958: 10952: 10949: 10946: 10943: 10937: 10929: 10918: 10915: 10904: 10901: 10898: 10895: 10892: 10881: 10878: 10863: 10862: 10843: 10830: 10803: 10800: 10794: 10791: 10788: 10785: 10779: 10776: 10770: 10767: 10756: 10742: 10739: 10733: 10730: 10727: 10724: 10718: 10710: 10699: 10696: 10681: 10680: 10678: 10674: 10647: 10644: 10638: 10635: 10632: 10629: 10623: 10620: 10614: 10611: 10600: 10597: 10594: 10574: 10571: 10565: 10562: 10539: 10536: 10533: 10513: 10510: 10504: 10501: 10495: 10492: 10484: 10459: 10456: 10450: 10447: 10444: 10441: 10435: 10432: 10426: 10423: 10412: 10409: 10389: 10386: 10380: 10377: 10354: 10351: 10348: 10328: 10325: 10319: 10316: 10310: 10307: 10299: 10298: 10279: 10276: 10268: 10264: 10261: 10252: 10249: 10243: 10240: 10237: 10234: 10228: 10220: 10213: 10210: 10204: 10198: 10195: 10189: 10183: 10180: 10174: 10169: 10162: 10159: 10152: 10147: 10140: 10137: 10130: 10124: 10121: 10111: 10110: 10096: 10093: 10087: 10084: 10078: 10052: 10049: 10043: 10040: 10034: 10026: 10007: 10004: 9998: 9995: 9992: 9987: 9983: 9979: 9974: 9970: 9966: 9958: 9954: 9950: 9945: 9941: 9934: 9927: 9926: 9903: 9878: 9875: 9861: 9855: 9836: 9813: 9810: 9802: 9798: 9794: 9789: 9785: 9781: 9778: 9772: 9769: 9764: 9757: 9754: 9741: 9737: 9733: 9728: 9721: 9718: 9708: 9699: 9698: 9681: 9678: 9672: 9667: 9663: 9642: 9639: 9634: 9630: 9621: 9617: 9601: 9598: 9589: 9586: 9580: 9577: 9574: 9571: 9565: 9562: 9556: 9553: 9543: 9542: 9541: 9540: 9537: 9534:Consider the 9533: 9532: 9516: 9513: 9499: 9491: 9475: 9472: 9458: 9450: 9424: 9405: 9402: 9391: 9390: 9360: 9356: 9348: 9324: 9305: 9302: 9291: 9290: 9260: 9256: 9248: 9247: 9245: 9222: 9219: 9205: 9182: 9178: 9170: 9167: 9153: 9130: 9126: 9119: 9114: 9100: 9093: 9092: 9067: 9059: 9058: 9049: 9042: 9040: 9019: 9016: 9013: 9002: 8995: 8982: 8978: 8974: 8968: 8965: 8954: 8951: 8945: 8942: 8931: 8927: 8889: 8877: 8868: 8864: 8860: 8857: 8854: 8849: 8845: 8841: 8836: 8832: 8821: 8814: 8801: 8797: 8793: 8787: 8784: 8773: 8770: 8764: 8761: 8750: 8746: 8712: 8702: 8683: 8680: 8671: 8664: 8626: 8612: 8595: 8592: 8582: 8581: 8577: 8559: 8556: 8550: 8547: 8541: 8533: 8510: 8507: 8504: 8494: 8491: 8485: 8482: 8472: 8469: 8466: 8456: 8453: 8447: 8444: 8435: 8426: 8425: 8400: 8380: 8377: 8371: 8368: 8357: 8336: 8333: 8325: 8322: 8310: 8292: 8289: 8264: 8252: 8230: 8223: 8206: 8197: 8194: 8184: 8183: 8181: 8180: 8175: 8154: 8126: 8123: 8120: 8097: 8094: 8091: 8083: 8067: 8064: 8059: 8052: 8049: 8040: 8036: 8032: 8027: 8024: 8021: 8014: 8011: 8002: 7999: 7996: 7992: 7988: 7985: 7982: 7977: 7970: 7967: 7958: 7954: 7950: 7945: 7938: 7935: 7926: 7922: 7914: 7913: 7911: 7895: 7892: 7875: 7872: 7862: 7861: 7847: 7844: 7830: 7822: 7806: 7803: 7800: 7797: 7794: 7774: 7771: 7766: 7762: 7753: 7752: 7743: 7736: 7734: 7718: 7714: 7708: 7704: 7700: 7695: 7692: 7689: 7685: 7679: 7676: 7673: 7669: 7665: 7662: 7659: 7654: 7650: 7644: 7640: 7636: 7631: 7627: 7621: 7617: 7613: 7599: 7592: 7591: 7587: 7581: 7565: 7562: 7559: 7547: 7546: 7541: 7537: 7536: 7530: 7513: 7510: 7507: 7501: 7492: 7489: 7483: 7480: 7469: 7468:linear system 7466: 7450: 7447: 7444: 7441: 7435: 7432: 7422:(i.e., where 7406: 7403: 7400: 7394: 7391: 7381: 7380: 7379: 7362: 7356: 7333: 7330: 7321: 7318: 7312: 7309: 7306: 7303: 7297: 7294: 7288: 7285: 7275: 7274: 7273: 7256: 7253: 7250: 7247: 7241: 7238: 7228: 7227: 7226: 7223: 7221: 7201: 7198: 7175: 7172: 7166: 7163: 7155: 7151: 7140: 7121: 7107: 7087: 7084: 7078: 7075: 7047: 7033: 7005: 6988: 6985: 6975: 6974: 6973: 6955: 6938: 6927: 6924: 6919: 6912: 6909: 6898: 6882: 6872: 6864: 6861: 6850: 6847: 6838: 6831: 6828: 6817: 6814: 6809: 6803: 6800: 6789: 6786: 6780: 6777: 6766: 6760: 6753: 6750: 6739: 6723: 6712: 6709: 6704: 6697: 6694: 6683: 6667: 6657: 6649: 6646: 6635: 6632: 6626: 6623: 6612: 6605: 6599: 6584: 6583: 6582: 6579: 6577: 6556: 6542: 6534: 6490: 6487: 6476: 6460: 6449: 6446: 6441: 6434: 6431: 6420: 6404: 6394: 6389: 6386: 6374: 6373: 6372: 6350: 6343: 6330: 6326: 6314: 6311: 6300: 6297: 6291: 6288: 6277: 6273: 6253: 6240: 6239: 6238: 6188: 6185: 6168: 6165: 6155: 6154: 6153: 6151: 6146: 6144: 6140: 6093: 6087: 6084: 6074: 6073: 6072: 6047: 6022: 6008: 6000: 5996: 5974: 5971: 5960: 5942: 5929: 5928: 5927: 5919: 5905: 5902: 5899: 5876: 5873: 5845: 5831: 5801: 5782: 5776: 5750: 5747: 5730: 5727: 5708: 5704: 5700: 5695: 5685: 5670: 5669: 5668: 5646: 5632: 5629: 5624: 5614: 5598: 5597: 5592: 5591: 5580: 5578: 5562: 5559: 5556: 5533: 5530: 5502: 5488: 5446: 5418: 5415: 5387: 5384: 5381: 5357: 5354: 5351: 5348: 5337: 5334: 5310: 5307: 5298: 5295: 5292: 5286: 5280: 5277: 5270: 5269: 5268: 5251: 5248: 5245: 5239: 5236: 5227: 5221: 5218: 5211: 5210: 5209: 5195: 5192: 5189: 5163: 5153: 5145: 5142: 5139: 5133: 5130: 5124: 5117: 5116: 5115: 5101: 5093: 5072: 5042: 5029: 5022: 5016: 5008: 4993: 4990: 4987: 4979: 4976: 4968: 4965: 4953: 4944: 4941: 4933: 4930: 4918: 4913: 4910: 4897: 4889: 4877: 4872: 4868: 4853: 4852: 4851: 4843: 4829: 4807: 4797: 4785: 4767: 4745: 4742: 4739: 4715: 4712: 4707: 4683: 4661: 4658: 4653: 4627: 4624: 4621: 4616: 4612: 4608: 4600: 4594: 4589: 4567: 4563: 4559: 4553: 4547: 4527: 4524: 4521: 4515: 4512: 4489: 4486: 4483: 4478: 4474: 4470: 4464: 4458: 4450: 4434: 4412: 4407: 4399: 4381: 4377: 4353: 4350: 4347: 4341: 4338: 4330: 4327: 4315: 4311: 4306: 4294: 4291: 4277: 4273: 4268: 4264: 4257: 4242: 4236: 4226: 4218: 4212: 4204: 4200: 4191: 4173: 4169: 4155: 4154: 4153: 4137: 4132: 4129: 4126: 4105: 4102: 4094: 4090: 4087: 4076: 4057: 4054: 4051: 4048: 4042: 4039: 4031: 4028: 4016: 4012: 4003: 4002: 4001: 3984: 3981: 3978: 3972: 3969: 3946: 3941: 3936: 3933: 3930: 3925: 3915: 3907: 3904: 3901: 3895: 3892: 3884: 3881: 3868: 3867: 3866: 3849: 3846: 3843: 3837: 3834: 3820: 3806: 3803: 3800: 3797: 3794: 3774: 3771: 3768: 3743: 3733: 3725: 3722: 3719: 3713: 3710: 3702: 3699: 3686: 3685: 3684: 3668: 3665: 3657: 3653: 3650: 3622: 3608: 3594: 3577: 3574: 3511: 3498: 3494: 3482: 3479: 3468: 3465: 3459: 3456: 3445: 3441: 3421: 3412: 3404: 3401: 3388: 3376: 3371: 3345: 3336: 3330: 3327: 3317: 3316: 3315: 3286: 3252: 3249: 3215: 3208: 3182: 3148: 3145: 3111: 3104: 3103: 3102: 3085: 3082: 3059: 3028: 3008: 3005: 2999: 2996: 2988: 2984: 2963: 2960: 2957: 2949: 2944: 2930: 2927: 2913: 2905: 2881: 2878: 2871: 2868: 2860: 2857: 2841: 2838: 2830: 2827: 2819: 2816: 2804: 2795: 2792: 2784: 2781: 2769: 2764: 2761: 2748: 2740: 2728: 2723: 2719: 2704: 2703: 2702: 2700: 2699: 2694: 2693: 2671: 2657: 2635: 2616: 2605: 2584: 2562: 2555: 2553: 2537: 2532: 2513: 2505: 2502: 2497: 2483: 2467: 2463: 2457: 2454: 2449: 2432: 2426: 2419: 2418: 2414: 2411: 2409: 2399: 2391: 2389: 2385: 2381: 2365: 2357: 2340: 2337: 2327: 2306: 2292: 2284: 2280: 2276: 2272: 2268: 2264: 2235: 2221: 2196: 2182: 2174: 2154: 2140: 2132: 2131: 2130: 2111: 2097: 2066: 2064: 2063: 2041: 2027: 2010: 2003: 2001: 1986: 1982: 1979: 1963: 1959: 1955: 1950: 1940: 1931: 1923: 1922: 1918: 1915: 1898: 1895: 1892: 1869: 1866: 1863: 1860: 1857: 1785: 1782: 1779: 1770: 1768: 1747: 1734: 1715: 1687: 1684: 1670: 1662: 1642: 1639: 1625: 1617: 1602: 1599: 1585: 1555: 1554: 1553: 1515: 1500: 1490: 1487: 1480: 1475: 1456: 1443: 1435: 1432: 1428: 1427: 1426: 1423: 1421: 1417: 1413: 1409: 1405: 1389: 1386: 1381: 1377: 1354: 1350: 1283: 1279: 1256: 1248: 1245: 1242: 1239: 1236: 1233: 1230: 1224: 1212: 1209:) around the 1208: 1207: 1202: 1198: 1173:to the input 1153: 1119: 1096: 1091: 1089: 1085: 1081: 1080: 1060: 1044: 1040: 1036: 1018: 1015: 1012: 994: 989: 979: 976: 954: 936: 931: 921: 918: 911: 907: 882: 872: 867: 858: 850: 846: 835: 827: 824: 821: 817: 809: 799: 791: 787: 776: 768: 764: 757: 752: 746: 731: 730: 729: 727: 724: 721:-dimensional 699: 689: 684: 675: 667: 663: 652: 644: 641: 638: 634: 626: 616: 608: 604: 593: 585: 581: 574: 569: 563: 548: 547: 546: 535: 528: 526: 509: 494: 491: 480: 477: 471: 468: 457: 454: 448: 439: 424: 423: 419: 416: 415:described by 414: 404: 401: 396: 393: 387: 385: 381: 377: 373: 369: 365: 361: 356: 352: 347: 344: 340: 324: 321: 318: 309: 293: 290: 287: 275: 257: 253: 249: 246: 241: 234: 231: 220: 204: 201: 196: 189: 186: 179: 174: 170: 166: 163: 143: 140: 137: 117: 114: 111: 103: 98: 89: 87: 83: 79: 75: 71: 67: 62: 58: 54: 50: 46: 45:discontinuous 42: 38: 34: 30: 26: 22: 17880: 17876: 17842: 17807: 17785: 17758: 17739: 17720: 17701: 17682: 17663: 17644: 17621: 17618:Spurgeon, S. 17588: 17543: 17532: 17523: 17519: 17491: 17485: 17458: 17454: 17444: 17425: 17419: 17400: 17394: 17370: 17366:Khalil, H.K. 17328: 17324: 17318: 17307: 17283:(1): 23–36. 17280: 17276: 17270: 17260: 17253: 17239:(1): 80–90. 17236: 17232: 17226: 17207: 17171: 17163: 17159:oscillations 17153: 16876: 16871: 16725: 16485: 15879: 15471: 14917: 14894: vector 14837: vector 14748: 14336: 14240: 14039: 13850:). However, 13736: 13507: 13380: 12863: 12654: 12349: 12096: 11801: 11664: 11641: 11627: 11480:(i.e., 10526:(i.e., when 10341:(i.e., when 9489: 9422: 9387: 9322: 9287: 9043: 8177: 7737: 7579: 7543: 7421: 7348: 7271: 7224: 7141: 7025: 6971: 6580: 6575: 6508: 6370: 6203: 6149: 6147: 6114: 5992: 5925: 5768: 5594: 5588: 5586: 5576: 5373: 5266: 5181: 5059: 4849: 4368: 4072: 3961: 3826: 3760: 3600: 3530: 3313: 3310:is negative. 3206:is positive. 2945: 2904:neighborhood 2901: 2847:(i.e., 2696: 2690: 2571: 2556: 2405: 2397: 2274: 2260: 2129:, one must: 2067: 2060: 2019: 2004: 1771: 1700:. Desirable 1660: 1658: 1478: 1476: 1439: 1431:hypersurface 1424: 1416:sliding mode 1415: 1411: 1204: 1092: 1077: 1041:so that the 899: 716: 544: 529: 410: 397: 395:of success. 388: 371: 348: 279: 92:Introduction 77: 70:sliding mode 69: 65: 28: 24: 18: 17804:Ferrara, A. 17547:. pp. 15678:eigenvalues 14431:). Because 13826:essentially 11558:equilibrium 11234:closed form 10552:), to make 10367:), to make 9620:state space 8182:), we have 7538:Consider a 6237:. That is, 6139:equilibrium 5995:nonsingular 4732:must reach 4698:must reach 2406:Consider a 2090:to satisfy 1420:LTI systems 1404:equilibrium 1095:control law 411:Consider a 360:equilibrium 102:Phase plane 17403:. Kluwer. 17187:References 17108:optimality 15682:observable 12154:on itself, 11656:LTI system 8358:To ensure 6533:continuous 4075:chain rule 3314:Note that 2410:candidate 2263:continuous 1733:continuous 1442:continuous 1035:continuous 376:optimality 364:robustness 339:LTI system 219:LTI system 17834:198313071 17575:120072463 17549:2368–2370 17463:CiteSeerX 17285:CiteSeerX 17039:− 17033:^ 17022:≜ 17008:σ 16963:^ 16845:− 16833:^ 16820:⁡ 16784:− 16772:^ 16759:⁡ 16687:− 16675:^ 16662:⁡ 16639:≜ 16568:− 16544:≜ 16507:≜ 16439:^ 16387:− 16375:^ 16362:⁡ 16303:− 16273:^ 16236:− 16224:^ 16211:⁡ 16177:− 16149:^ 16112:− 16100:^ 16087:⁡ 16053:− 16025:^ 15988:− 15976:^ 15963:⁡ 15934:^ 15910:˙ 15905:^ 15820:− 15808:^ 15795:⁡ 15598:− 15589:× 15580:− 15524:× 15515:− 15241:⏞ 15217:⋮ 15148:˙ 15133:⏞ 15114:˙ 15100:⋮ 15084:˙ 15061:˙ 14959:− 14886:× 14877:− 14863:⏟ 14826:− 14817:× 14806:⏟ 14775:⏟ 14721:− 14686:σ 14674:⏞ 14656:− 14613:⏞ 14579:˙ 14576:σ 14514:σ 14473:˙ 14397:^ 14223:σ 14217:⁡ 14202:σ 14164:˙ 14161:σ 14155:σ 14127:− 14111:∈ 14092:… 14060:≜ 14022:σ 14013:− 13943:− 13883:˙ 13867:˙ 13864:σ 13803:˙ 13800:σ 13794:σ 13774:σ 13751:˙ 13748:σ 13709:− 13697:^ 13674:≜ 13665:^ 13644:^ 13631:σ 13586:^ 13523:^ 13483:− 13471:^ 13421:− 13409:^ 13306:− 13294:^ 13264:− 13258:^ 13223:− 13212:− 13196:− 13184:^ 13148:^ 13121:˙ 13110:− 13104:˙ 13099:^ 13076:˙ 13036:− 13030:^ 12982:∈ 12963:… 12898:− 12882:∈ 12824:− 12771:∈ 12706:^ 12674:→ 12627:− 12615:^ 12579:^ 12559:˙ 12554:^ 12505:∈ 12490:^ 12477:… 12462:^ 12440:^ 12421:^ 12326:− 12317:× 12304:∈ 12264:− 12255:× 12246:− 12230:∈ 12190:− 12174:∈ 12018:≜ 11954:∈ 11935:… 11903:≜ 11866:∈ 11847:… 11815:≜ 11753:⋯ 11692:˙ 11583:˙ 11526:˙ 11502:˙ 11484:σ 11472:− 11463:˙ 11418:σ 11381:σ 11370:⏟ 11361:σ 11355:⏞ 11342:˙ 11326:⁡ 11293:˙ 11276:− 11264:˙ 11201:σ 11195:⏞ 11188:˙ 11137:˙ 11118:− 11100:⏟ 11093:˙ 11048:˙ 11017:˙ 10962:˙ 10919:˙ 10882:˙ 10804:˙ 10771:˙ 10757:≥ 10743:˙ 10700:˙ 10648:˙ 10615:˙ 10601:− 10566:˙ 10563:σ 10534:σ 10505:˙ 10460:˙ 10427:˙ 10381:˙ 10378:σ 10349:σ 10320:˙ 10280:¨ 10269:⏞ 10253:˙ 10214:˙ 10199:¨ 10184:˙ 10163:˙ 10141:˙ 10125:˙ 10122:σ 10088:˙ 10085:σ 10079:σ 10053:˙ 10008:˙ 9935:σ 9876:≤ 9862:⋅ 9758:˙ 9722:˙ 9682:˙ 9590:˙ 9557:¨ 9500:σ 9459:σ 9406:˙ 9403:σ 9392:) (i.e., 9361:− 9306:˙ 9303:σ 9292:) (i.e., 9206:σ 9183:− 9154:σ 9017:× 9003:⏟ 8996:˙ 8983:⏞ 8907:∂ 8890:σ 8886:∂ 8878:⏞ 8858:… 8815:˙ 8802:⏞ 8730:∂ 8713:σ 8709:∂ 8684:˙ 8681:σ 8672:⏞ 8665:˙ 8644:∂ 8627:σ 8623:∂ 8596:˙ 8593:σ 8551:˙ 8548:σ 8542:σ 8505:σ 8486:˙ 8483:σ 8467:σ 8448:˙ 8445:σ 8372:˙ 8334:⁡ 8326:σ 8323:⁡ 8311:⏞ 8293:˙ 8290:σ 8270:∂ 8262:∂ 8253:⏞ 8231:σ 8207:σ 8198:˙ 8124:− 8095:− 8053:˙ 8025:− 8015:˙ 8000:− 7986:⋯ 7971:˙ 7939:˙ 7876:˙ 7873:σ 7831:σ 7804:≤ 7798:≤ 7693:− 7677:− 7663:⋯ 7614:≜ 7600:σ 7493:˙ 7436:˙ 7404:− 7395:˙ 7363:⋅ 7322:˙ 7289:¨ 7242:˙ 7202:˙ 7199:σ 7167:˙ 7164:σ 7156:⊺ 7152:σ 7108:σ 7079:˙ 7076:σ 7034:σ 6989:˙ 6986:σ 6944:∂ 6939:σ 6936:∂ 6925:− 6888:∂ 6883:σ 6880:∂ 6848:− 6761:⏞ 6729:∂ 6724:σ 6721:∂ 6710:− 6673:∂ 6668:σ 6665:∂ 6633:− 6600:˙ 6543:σ 6466:∂ 6461:σ 6458:∂ 6447:− 6410:∂ 6405:σ 6402:∂ 6390:− 6344:˙ 6331:⏞ 6259:∂ 6254:σ 6251:∂ 6169:˙ 6166:σ 6088:˙ 6085:σ 6048:σ 6009:σ 5948:∂ 5943:σ 5940:∂ 5900:σ 5877:˙ 5832:σ 5783:σ 5731:˙ 5728:σ 5709:⊺ 5705:σ 5686:∈ 5633:σ 5615:∈ 5557:σ 5534:˙ 5531:σ 5520:surface, 5489:σ 5447:σ 5419:˙ 5416:σ 5382:σ 5352:μ 5349:≥ 5338:˙ 5335:σ 5311:˙ 5308:σ 5299:⁡ 5293:≠ 5287:σ 5281:⁡ 5252:μ 5249:− 5246:≤ 5240:˙ 5237:σ 5228:σ 5222:⁡ 5190:α 5164:α 5154:σ 5146:μ 5143:− 5140:≤ 5134:˙ 5131:σ 5125:σ 5102:σ 5078:‖ 5073:⋅ 5068:‖ 5043:α 5023:⏞ 5013:‖ 5009:σ 5006:‖ 4994:μ 4991:− 4988:≤ 4977:⁡ 4966:⁡ 4954:⏟ 4942:⁡ 4934:σ 4931:⁡ 4919:⏞ 4914:˙ 4911:σ 4898:σ 4895:∂ 4887:∂ 4878:⏞ 4873:⊺ 4869:σ 4830:σ 4804:‖ 4798:⋅ 4793:‖ 4659:≥ 4625:μ 4622:− 4609:≤ 4528:μ 4525:− 4516:˙ 4487:μ 4484:− 4435:∝ 4354:μ 4351:− 4348:≤ 4339:⁡ 4328:⁡ 4295:˙ 4278:≜ 4258:⏟ 4237:⏟ 4223:‖ 4219:σ 4216:‖ 4213:∝ 4205:⏞ 4152:), means 4130:≜ 4103:⁡ 4088:⁡ 4055:μ 4052:− 4049:≤ 4040:⁡ 4029:⁡ 3973:∈ 3937:μ 3934:− 3931:≤ 3926:α 3908:μ 3905:− 3902:≤ 3893:⁡ 3882:⁡ 3838:∈ 3804:≤ 3801:α 3769:μ 3744:α 3726:μ 3723:− 3720:≤ 3711:⁡ 3700:⁡ 3666:⁡ 3651:⁡ 3609:σ 3578:˙ 3575:σ 3512:˙ 3499:⏞ 3427:∂ 3422:σ 3419:∂ 3402:⁡ 3389:⁡ 3377:⏞ 3372:˙ 3351:∂ 3346:σ 3343:∂ 3331:˙ 3328:σ 3287:σ 3253:˙ 3250:σ 3183:σ 3149:˙ 3146:σ 3086:˙ 3083:σ 3060:σ 3000:˙ 2997:σ 2989:⊺ 2985:σ 2914:σ 2869:⁡ 2858:⁡ 2828:⁡ 2817:⁡ 2805:⏟ 2793:⁡ 2785:σ 2782:⁡ 2770:⏞ 2765:˙ 2762:σ 2749:σ 2746:∂ 2738:∂ 2729:⏞ 2724:⊺ 2720:σ 2658:σ 2632:‖ 2617:σ 2614:‖ 2590:‖ 2585:⋅ 2580:‖ 2529:‖ 2514:σ 2511:‖ 2484:σ 2468:⊺ 2464:σ 2433:σ 2341:˙ 2338:σ 2293:σ 2222:σ 2183:σ 2141:σ 2098:σ 2028:σ 1960:σ 1941:∈ 1896:− 1867:≤ 1861:≤ 1783:− 1671:σ 1626:σ 1600:≠ 1586:σ 1506:→ 1488:σ 1257:⊺ 1243:… 1197:stabilize 1016:× 1003:→ 995:× 945:→ 937:× 910:functions 873:∈ 825:− 810:⋮ 753:≜ 690:∈ 642:− 627:⋮ 570:≜ 440:˙ 308:deadbands 250:− 235:˙ 190:˙ 17910:Category 17810:. SIAM. 17620:(1998). 17368:(2002). 17345:31903795 17325:Robotica 17114:H-bridge 17068:See also 11168:if 11074:if 10109:. Here, 10071:so that 9423:positive 9323:negative 9202:if 9150:if 8501:if 8463:if 7787:for all 7529:origin. 4640:for all 4447:denotes 3962:So, for 2326:dynamics 2283:Filippov 1556:A state 1169:at time 906:feedback 341:with an 53:feedback 37:dynamics 15684:, this 15674:Hurwitz 15616:matrix 15539:vector 14382:(i.e., 11654:for an 11447:, then 9891:(i.e., 9490:chatter 7912:and so 7821:simplex 7552:(i.e., 6531:is not 5182:Taking 5090:is the 4396:is the 4077:(i.e., 2606:(i.e., 2602:is the 1086:and is 276:origin. 31:) is a 17857: 17832: 17822: 17792: 17773: 17746: 17727: 17708: 17689: 17670: 17651: 17628: 17603: 17573: 17563: 17498: 17465: 17432: 17407: 17382: 17343: 17287: 17214: 16486:where 14781:scalar 14556:where 14241:where 14185:, let 14040:where 13381:where 12864:where 12760:, and 12655:where 12097:where 9622:(with 9246:where 7823:where 7580:picked 7465:stable 7220:robust 6143:stable 6115:If an 5577:switch 5060:where 4369:where 3761:where 3239:makes 3135:makes 2948:scalar 2572:where 1798:where 1211:origin 1088:unique 1084:exists 1039:smooth 908:. The 900:is an 726:vector 717:is an 545:where 372:finite 17830:S2CID 17571:S2CID 17341:S2CID 17145:Notes 16872:input 15717:when 10485:When 10300:When 9697:) as 9421:) is 9321:) is 7540:plant 7349:when 5798:is a 4119:with 2902:in a 1195:) to 723:state 74:locus 66:slide 49:state 39:of a 17855:ISBN 17820:ISBN 17790:ISBN 17771:ISBN 17744:ISBN 17725:ISBN 17706:ISBN 17687:ISBN 17668:ISBN 17649:ISBN 17626:ISBN 17601:ISBN 17561:ISBN 17496:ISBN 17430:ISBN 17405:ISBN 17380:ISBN 17212:ISBN 14255:> 14170:< 13828:all 13824:for 13809:< 13766:and 11384:> 11206:> 11105:< 10905:> 10598:< 10572:< 10537:> 10511:> 10413:> 10387:> 10352:< 10326:< 10094:< 9655:and 9220:< 9168:> 8557:< 8508:< 8492:> 8470:> 8454:< 8378:< 7772:> 7188:and 7173:< 6148:The 5926:Let 5802:and 5748:< 5355:> 3798:< 3787:and 3772:> 3072:and 3006:< 2879:< 2839:< 1824:and 1661:sign 1199:the 969:and 728:and 351:gain 17893:hdl 17885:doi 17847:doi 17812:doi 17763:doi 17593:doi 17553:doi 17473:doi 17333:doi 17295:doi 17241:doi 17064:). 16817:sgn 16756:sgn 16659:sgn 16634:obs 16619:and 16539:obs 16502:obs 16465:obs 16453:obs 16426:obs 16359:sgn 16208:sgn 16084:sgn 15960:sgn 15792:sgn 15672:is 14749:So 14258:max 14214:sgn 11560:at 11323:sgn 11236:as 9845:sup 9425:at 9325:at 6576:act 6141:is 5993:be 5322:and 5296:sgn 5278:sgn 5219:sgn 4400:of 2382:or 2275:not 2273:is 2065:). 29:SMC 19:In 17912:: 17891:. 17881:55 17879:. 17853:. 17828:. 17818:. 17769:. 17599:. 17569:. 17559:. 17551:. 17541:. 17524:44 17522:. 17510:^ 17471:. 17459:64 17457:. 17453:. 17378:. 17353:^ 17339:. 17329:31 17327:. 17293:. 17281:40 17279:. 17237:45 17235:. 17195:^ 15859:eq 15761:eq 15730:12 15655:12 15437:12 15386:12 15338:eq 15017:12 14931:eq 14801:12 14770:eq 14729:eq 14704:12 14669:eq 14639:12 14593:11 14542:eq 14297:12 14274:11 13996:12 13973:11 13926:12 13903:11 12299:12 12225:22 12169:21 12113:11 12072:22 12060:21 12046:12 12034:11 11890:, 10861:, 10679:, 7222:. 4674:, 4000:, 3865:, 3593:. 2943:. 1090:. 1082:) 23:, 17901:. 17895:: 17887:: 17863:. 17849:: 17836:. 17814:: 17798:. 17779:. 17765:: 17752:. 17733:. 17714:. 17695:. 17676:. 17657:. 17634:. 17609:. 17595:: 17577:. 17555:: 17504:. 17479:. 17475:: 17438:. 17413:. 17388:. 17347:. 17335:: 17301:. 17297:: 17247:. 17243:: 17220:. 17179:. 17110:. 17051:0 17047:= 17043:y 17029:y 17019:) 17015:x 17011:( 16987:y 16959:y 16945:C 16930:x 16926:C 16923:= 16919:y 16896:1 16892:x 16888:= 16885:y 16858:) 16853:1 16849:x 16840:1 16830:x 16823:( 16797:) 16792:1 16788:x 16779:1 16769:x 16762:( 16735:u 16711:. 16706:] 16700:) 16695:1 16691:x 16682:1 16672:x 16665:( 16651:u 16644:[ 16630:u 16607:, 16602:] 16594:] 16588:M 16583:2 16579:L 16571:M 16562:[ 16555:B 16549:[ 16535:B 16513:, 16510:A 16498:A 16460:u 16449:B 16445:+ 16435:x 16422:A 16418:= 16406:] 16400:) 16395:1 16391:x 16382:1 16372:x 16365:( 16351:u 16344:[ 16337:] 16329:] 16323:M 16318:2 16314:L 16306:M 16297:[ 16290:B 16284:[ 16279:+ 16269:x 16262:A 16259:= 16249:) 16244:1 16240:x 16231:1 16221:x 16214:( 16203:] 16197:M 16192:2 16188:L 16180:M 16171:[ 16166:+ 16162:u 16158:B 16155:+ 16145:x 16138:A 16135:= 16125:) 16120:1 16116:x 16107:1 16097:x 16090:( 16081:M 16076:] 16068:2 16064:L 16056:1 16047:[ 16042:+ 16038:u 16034:B 16031:+ 16021:x 16014:A 16011:= 16001:) 15996:1 15992:x 15983:1 15973:x 15966:( 15957:M 15954:L 15951:+ 15947:u 15943:B 15940:+ 15930:x 15923:A 15920:= 15901:x 15855:v 15844:v 15826:) 15823:x 15815:1 15805:x 15798:( 15789:M 15786:= 15783:v 15757:v 15746:C 15726:A 15699:2 15694:e 15660:) 15651:A 15645:2 15641:L 15637:+ 15632:2 15628:A 15624:( 15604:) 15601:1 15595:n 15592:( 15586:) 15583:1 15577:n 15574:( 15552:2 15548:L 15527:1 15521:) 15518:1 15512:n 15509:( 15487:2 15482:e 15457:. 15452:2 15447:e 15442:) 15433:A 15427:2 15423:L 15419:+ 15414:2 15410:A 15406:( 15403:= 15398:2 15393:e 15382:A 15376:2 15372:L 15368:+ 15363:2 15358:e 15351:2 15347:A 15343:= 15334:v 15328:2 15324:L 15320:+ 15315:2 15310:e 15303:2 15299:A 15295:= 15292:) 15287:1 15283:e 15279:( 15276:v 15271:2 15267:L 15263:+ 15254:2 15249:e 15237:] 15229:n 15225:e 15208:3 15204:e 15194:2 15190:e 15183:[ 15172:2 15168:A 15164:= 15155:2 15144:e 15129:] 15121:n 15111:e 15091:3 15081:e 15068:2 15058:e 15048:[ 15013:A 14990:1 14986:x 14965:) 14962:1 14956:n 14953:( 14927:v 14903:. 14889:1 14883:) 14880:1 14874:n 14871:( 14858:2 14853:e 14832:) 14829:1 14823:n 14820:( 14814:1 14797:A 14788:= 14766:v 14734:. 14725:v 14716:2 14711:e 14700:A 14696:= 14689:) 14683:( 14680:v 14665:v 14651:2 14646:e 14635:A 14631:+ 14624:0 14621:= 14608:1 14604:e 14589:a 14585:= 14570:= 14567:0 14538:v 14517:) 14511:( 14508:v 14488:0 14485:= 14480:1 14470:e 14444:1 14440:e 14417:1 14413:x 14409:= 14404:1 14394:x 14370:0 14367:= 14362:1 14358:e 14347:M 14343:M 14339:M 14322:. 14319:} 14315:| 14309:2 14304:e 14293:A 14289:+ 14284:1 14280:e 14270:a 14265:| 14261:{ 14252:M 14226:) 14220:( 14211:M 14208:= 14205:) 14199:( 14196:v 14173:0 14133:) 14130:1 14124:n 14121:( 14116:R 14108:) 14103:n 14099:e 14095:, 14089:, 14084:3 14080:e 14076:, 14071:2 14067:e 14063:( 14055:2 14050:e 14025:) 14019:( 14016:v 14008:2 14003:e 13992:A 13988:+ 13983:1 13979:e 13969:a 13965:= 13962:) 13957:1 13953:e 13949:( 13946:v 13938:2 13933:e 13922:A 13918:+ 13913:1 13909:e 13899:a 13895:= 13890:1 13880:e 13873:= 13837:x 13812:0 13722:. 13717:1 13713:x 13704:1 13694:x 13687:= 13682:1 13678:e 13671:) 13662:x 13656:, 13651:1 13641:x 13634:( 13606:1 13602:x 13598:= 13593:1 13583:x 13557:1 13553:x 13530:1 13520:x 13491:1 13487:x 13478:1 13468:x 13461:= 13458:0 13445:v 13429:1 13425:x 13416:1 13406:x 13399:= 13394:1 13390:e 13362:) 13357:1 13353:e 13349:( 13346:v 13343:L 13340:+ 13336:e 13332:A 13329:= 13319:) 13314:1 13310:x 13301:1 13291:x 13284:( 13281:v 13278:L 13275:+ 13272:) 13268:x 13254:x 13247:( 13244:A 13241:= 13230:u 13226:B 13219:x 13215:A 13209:) 13204:1 13200:x 13191:1 13181:x 13174:( 13171:v 13168:L 13165:+ 13161:u 13157:B 13154:+ 13144:x 13137:A 13134:= 13117:x 13095:x 13086:= 13072:e 13040:x 13026:x 13019:= 13015:e 12992:n 12987:R 12979:) 12974:n 12970:e 12966:, 12960:, 12955:2 12951:e 12947:, 12942:1 12938:e 12934:( 12931:= 12927:e 12904:) 12901:1 12895:n 12892:( 12887:R 12877:2 12873:L 12847:] 12839:2 12835:L 12827:1 12818:[ 12813:= 12810:L 12781:n 12776:R 12768:L 12746:1 12742:x 12738:= 12735:y 12713:1 12703:x 12678:R 12670:R 12666:: 12663:v 12640:) 12635:1 12631:x 12622:1 12612:x 12605:( 12602:v 12599:L 12596:+ 12592:u 12588:B 12585:+ 12575:x 12568:A 12565:= 12550:x 12531:n 12515:n 12510:R 12502:) 12497:n 12487:x 12480:, 12474:, 12469:2 12459:x 12452:, 12447:1 12437:x 12430:( 12427:= 12417:x 12391:1 12387:x 12383:= 12380:y 12359:x 12332:) 12329:1 12323:n 12320:( 12314:1 12309:R 12295:A 12270:) 12267:1 12261:n 12258:( 12252:) 12249:1 12243:n 12240:( 12235:R 12221:A 12196:) 12193:1 12187:n 12184:( 12179:R 12165:A 12140:1 12136:x 12109:a 12080:] 12068:A 12056:A 12042:A 12030:a 12023:[ 12015:A 11991:x 11980:y 11964:r 11959:R 11951:) 11946:r 11942:u 11938:, 11932:, 11927:2 11923:u 11919:, 11914:1 11910:u 11906:( 11899:u 11876:n 11871:R 11863:) 11858:n 11854:x 11850:, 11844:, 11839:2 11835:x 11831:, 11826:1 11822:x 11818:( 11811:x 11778:1 11774:x 11770:= 11766:x 11760:] 11748:0 11743:0 11738:1 11732:[ 11727:= 11724:y 11716:u 11712:B 11709:+ 11705:x 11701:A 11698:= 11688:x 11678:{ 11619:. 11607:) 11604:0 11601:, 11598:0 11595:( 11592:= 11589:) 11580:x 11574:, 11571:x 11568:( 11539:) 11535:0 11532:= 11523:x 11517:+ 11514:x 11511:= 11508:) 11499:x 11493:, 11490:x 11487:( 11475:x 11469:= 11460:x 11435:0 11432:= 11429:) 11425:x 11421:( 11391:) 11387:0 11366:) 11351:x 11348:+ 11339:x 11329:( 11316:) 11313:1 11310:+ 11307:k 11304:+ 11300:| 11290:x 11283:| 11279:( 11273:= 11270:) 11261:x 11255:, 11252:x 11249:( 11246:u 11209:0 11185:x 11179:+ 11176:x 11161:) 11157:1 11154:+ 11151:k 11148:+ 11144:| 11134:x 11127:| 11122:( 11111:, 11108:0 11090:x 11084:+ 11081:x 11068:1 11065:+ 11062:k 11059:+ 11055:| 11045:x 11038:| 11031:{ 11026:= 11023:) 11014:x 11008:, 11005:x 11002:( 10999:u 10972:| 10968:) 10959:x 10953:, 10950:x 10947:, 10944:t 10941:( 10938:a 10934:| 10930:+ 10926:| 10916:x 10909:| 10902:1 10899:+ 10896:k 10893:+ 10889:| 10879:x 10872:| 10848:| 10844:a 10840:| 10814:| 10810:) 10801:x 10795:, 10792:x 10789:, 10786:t 10783:( 10780:a 10777:+ 10768:x 10761:| 10753:| 10749:) 10740:x 10734:, 10731:x 10728:, 10725:t 10722:( 10719:a 10715:| 10711:+ 10707:| 10697:x 10690:| 10658:| 10654:) 10645:x 10639:, 10636:x 10633:, 10630:t 10627:( 10624:a 10621:+ 10612:x 10605:| 10595:u 10575:0 10540:0 10514:0 10502:x 10496:+ 10493:x 10470:| 10466:) 10457:x 10451:, 10448:x 10445:, 10442:t 10439:( 10436:a 10433:+ 10424:x 10417:| 10410:u 10390:0 10355:0 10329:0 10317:x 10311:+ 10308:x 10277:x 10265:u 10262:+ 10259:) 10250:x 10244:, 10241:x 10238:, 10235:t 10232:( 10229:a 10221:+ 10211:x 10205:= 10196:x 10190:+ 10181:x 10175:= 10170:2 10160:x 10153:+ 10148:1 10138:x 10131:= 10097:0 10059:) 10050:x 10044:, 10041:x 10038:( 10035:u 10005:x 9999:+ 9996:x 9993:= 9988:2 9984:x 9980:+ 9975:1 9971:x 9967:= 9964:) 9959:2 9955:x 9951:, 9946:1 9942:x 9938:( 9923:k 9908:| 9904:a 9900:| 9879:k 9873:} 9869:| 9865:) 9859:( 9856:a 9852:| 9848:{ 9814:u 9811:+ 9808:) 9803:2 9799:x 9795:, 9790:1 9786:x 9782:, 9779:t 9776:( 9773:a 9770:= 9765:2 9755:x 9742:2 9738:x 9734:= 9729:1 9719:x 9709:{ 9679:x 9673:= 9668:2 9664:x 9643:x 9640:= 9635:1 9631:x 9602:u 9599:+ 9596:) 9587:x 9581:, 9578:x 9575:, 9572:t 9569:( 9566:a 9563:= 9554:x 9517:0 9514:= 9511:) 9507:x 9503:( 9476:0 9473:= 9470:) 9466:x 9462:( 9434:x 9389:5 9374:) 9370:x 9366:( 9357:u 9334:x 9289:5 9274:) 9270:x 9266:( 9261:+ 9257:u 9223:0 9217:) 9213:x 9209:( 9196:) 9192:x 9188:( 9179:u 9171:0 9165:) 9161:x 9157:( 9144:) 9140:x 9136:( 9131:+ 9127:u 9120:{ 9115:= 9112:) 9108:x 9104:( 9101:u 9079:) 9075:x 9071:( 9068:u 9048:) 9046:5 9044:( 9020:1 9014:n 8992:x 8979:) 8975:u 8972:) 8969:t 8966:, 8962:x 8958:( 8955:B 8952:+ 8949:) 8946:t 8943:, 8939:x 8935:( 8932:f 8928:( 8912:x 8901:) 8897:x 8893:( 8874:] 8869:n 8865:s 8861:, 8855:, 8850:2 8846:s 8842:, 8837:1 8833:s 8829:[ 8822:= 8811:x 8798:) 8794:u 8791:) 8788:t 8785:, 8781:x 8777:( 8774:B 8771:+ 8768:) 8765:t 8762:, 8758:x 8754:( 8751:f 8747:( 8735:x 8724:) 8720:x 8716:( 8703:= 8698:) 8694:x 8690:( 8661:x 8649:x 8638:) 8634:x 8630:( 8613:= 8610:) 8606:x 8602:( 8560:0 8511:0 8495:0 8473:0 8457:0 8436:{ 8412:) 8408:x 8404:( 8401:u 8381:0 8369:V 8337:t 8331:d 8320:d 8307:) 8303:x 8299:( 8274:x 8265:V 8247:T 8243:) 8238:x 8234:( 8224:= 8221:) 8218:) 8214:x 8210:( 8204:( 8195:V 8179:3 8159:0 8155:= 8151:x 8130:) 8127:1 8121:n 8118:( 8098:1 8092:n 8068:0 8065:= 8060:n 8050:x 8041:n 8037:s 8033:+ 8028:1 8022:n 8012:x 8003:1 7997:n 7993:s 7989:+ 7983:+ 7978:2 7968:x 7959:2 7955:s 7951:+ 7946:1 7936:x 7927:1 7923:s 7896:0 7893:= 7890:) 7886:x 7882:( 7848:0 7845:= 7842:) 7838:x 7834:( 7807:n 7801:i 7795:1 7775:0 7767:i 7763:s 7742:) 7740:4 7738:( 7719:n 7715:x 7709:n 7705:s 7701:+ 7696:1 7690:n 7686:x 7680:1 7674:n 7670:s 7666:+ 7660:+ 7655:2 7651:x 7645:2 7641:s 7637:+ 7632:1 7628:x 7622:1 7618:s 7611:) 7607:x 7603:( 7566:1 7563:= 7560:m 7550:u 7545:1 7517:) 7514:0 7511:, 7508:0 7505:( 7502:= 7499:) 7490:x 7484:, 7481:x 7478:( 7451:0 7448:= 7445:x 7442:+ 7433:x 7407:x 7401:= 7392:x 7366:) 7360:( 7357:a 7334:u 7331:+ 7328:) 7319:x 7313:, 7310:x 7307:, 7304:t 7301:( 7298:a 7295:= 7286:x 7257:0 7254:= 7251:x 7248:+ 7239:x 7176:0 7126:0 7122:= 7119:) 7115:x 7111:( 7088:0 7085:= 7052:0 7048:= 7045:) 7041:x 7037:( 7010:0 7006:= 7003:) 6999:x 6995:( 6956:) 6948:x 6928:1 6920:) 6916:) 6913:t 6910:, 6906:x 6902:( 6899:B 6892:x 6873:( 6868:) 6865:t 6862:, 6858:x 6854:( 6851:B 6844:I 6839:( 6835:) 6832:t 6829:, 6825:x 6821:( 6818:f 6815:= 6810:u 6807:) 6804:t 6801:, 6797:x 6793:( 6790:B 6787:+ 6784:) 6781:t 6778:, 6774:x 6770:( 6767:f 6757:) 6754:t 6751:, 6747:x 6743:( 6740:f 6733:x 6713:1 6705:) 6701:) 6698:t 6695:, 6691:x 6687:( 6684:B 6677:x 6658:( 6653:) 6650:t 6647:, 6643:x 6639:( 6636:B 6630:) 6627:t 6624:, 6620:x 6616:( 6613:f 6606:= 6596:x 6561:0 6557:= 6554:) 6550:x 6546:( 6518:u 6494:) 6491:t 6488:, 6484:x 6480:( 6477:f 6470:x 6450:1 6442:) 6438:) 6435:t 6432:, 6428:x 6424:( 6421:B 6414:x 6395:( 6387:= 6383:u 6355:0 6351:= 6340:x 6327:) 6322:u 6318:) 6315:t 6312:, 6308:x 6304:( 6301:B 6298:+ 6295:) 6292:t 6289:, 6285:x 6281:( 6278:f 6274:( 6263:x 6225:) 6221:x 6217:( 6213:u 6189:0 6186:= 6183:) 6179:x 6175:( 6137:- 6124:x 6111:. 6098:0 6094:= 6059:) 6055:x 6051:( 6027:0 6023:= 6020:) 6016:x 6012:( 5978:) 5975:t 5972:, 5968:x 5964:( 5961:B 5953:x 5906:0 5903:= 5874:V 5850:0 5846:= 5843:) 5839:x 5835:( 5811:x 5786:) 5780:( 5777:V 5754:} 5751:0 5745:) 5741:x 5737:( 5722:) 5718:x 5714:( 5701:: 5696:n 5691:R 5682:x 5678:{ 5655:} 5651:0 5647:= 5644:) 5640:x 5636:( 5630:: 5625:n 5620:R 5611:x 5607:{ 5596:2 5590:1 5563:0 5560:= 5507:0 5503:= 5500:) 5496:x 5492:( 5468:x 5426:| 5409:| 5388:0 5385:= 5370:. 5358:0 5345:| 5328:| 5317:) 5302:( 5290:) 5284:( 5231:) 5225:( 5196:1 5193:= 5178:. 5159:| 5150:| 5039:) 5030:V 5017:2 4997:( 4980:t 4974:d 4969:V 4963:d 4945:t 4939:d 4928:d 4890:V 4808:2 4768:V 4746:0 4743:= 4740:V 4730:V 4716:0 4713:= 4708:V 4684:V 4662:0 4654:V 4642:t 4628:t 4617:0 4613:V 4604:) 4601:t 4598:( 4595:V 4590:2 4568:0 4564:z 4560:= 4557:) 4554:0 4551:( 4548:z 4522:= 4513:z 4490:t 4479:0 4475:z 4471:= 4468:) 4465:t 4462:( 4459:z 4413:V 4408:2 4382:+ 4378:D 4342:t 4336:d 4331:V 4325:d 4316:V 4312:1 4307:= 4292:W 4274:W 4269:+ 4265:D 4252:) 4243:W 4227:2 4201:V 4192:2 4181:( 4174:+ 4170:D 4138:V 4133:2 4127:W 4106:t 4100:d 4095:/ 4091:W 4085:d 4058:, 4043:t 4037:d 4032:V 4026:d 4017:V 4013:1 3988:] 3985:1 3982:, 3979:0 3976:( 3970:V 3947:. 3942:V 3922:) 3916:V 3911:( 3896:t 3890:d 3885:V 3879:d 3853:] 3850:1 3847:, 3844:0 3841:[ 3835:V 3807:1 3795:0 3775:0 3740:) 3734:V 3729:( 3714:t 3708:d 3703:V 3697:d 3669:t 3663:d 3658:/ 3654:V 3648:d 3627:0 3623:= 3620:) 3616:x 3612:( 3552:) 3548:x 3544:( 3540:u 3508:x 3495:) 3490:u 3486:) 3483:t 3480:, 3476:x 3472:( 3469:B 3466:+ 3463:) 3460:t 3457:, 3453:x 3449:( 3446:f 3442:( 3431:x 3413:= 3405:t 3399:d 3393:x 3386:d 3368:x 3355:x 3337:= 3298:) 3294:x 3290:( 3267:) 3263:x 3259:( 3227:) 3223:x 3219:( 3216:u 3194:) 3190:x 3186:( 3163:) 3159:x 3155:( 3123:) 3119:x 3115:( 3112:u 3040:) 3036:x 3032:( 3029:u 3009:0 2964:1 2961:= 2958:m 2931:0 2928:= 2925:) 2921:x 2917:( 2886:) 2882:0 2872:t 2866:d 2861:V 2855:d 2842:0 2831:t 2825:d 2820:V 2814:d 2796:t 2790:d 2779:d 2741:V 2698:2 2692:1 2676:0 2672:= 2669:) 2665:x 2661:( 2636:2 2628:) 2624:x 2620:( 2561:) 2559:3 2557:( 2538:2 2533:2 2525:) 2521:x 2517:( 2506:2 2503:1 2498:= 2495:) 2491:x 2487:( 2481:) 2477:x 2473:( 2458:2 2455:1 2450:= 2447:) 2444:) 2440:x 2436:( 2430:( 2427:V 2366:; 2362:0 2358:= 2355:) 2351:x 2347:( 2311:0 2307:= 2304:) 2300:x 2296:( 2240:0 2236:= 2233:) 2229:x 2225:( 2201:0 2197:= 2194:) 2190:x 2186:( 2159:0 2155:= 2152:) 2148:x 2144:( 2116:0 2112:= 2109:) 2105:x 2101:( 2077:x 2062:2 2046:0 2042:= 2039:) 2035:x 2031:( 2009:) 2007:2 2005:( 1987:} 1983:0 1980:= 1977:) 1973:x 1969:( 1964:k 1956:: 1951:n 1946:R 1937:x 1932:{ 1902:) 1899:1 1893:n 1890:( 1870:m 1864:k 1858:1 1837:u 1826:m 1811:x 1800:n 1786:m 1780:n 1752:0 1748:= 1744:x 1719:) 1716:t 1713:( 1709:x 1688:0 1685:= 1682:) 1678:x 1674:( 1655:. 1643:0 1640:= 1637:) 1633:x 1629:( 1615:. 1603:0 1597:) 1593:x 1589:( 1565:x 1539:x 1516:m 1511:R 1501:n 1496:R 1491:: 1461:0 1457:= 1453:x 1390:0 1387:= 1382:1 1378:x 1355:1 1351:x 1329:u 1307:x 1284:1 1280:x 1253:] 1249:0 1246:, 1240:, 1237:0 1234:, 1231:0 1228:[ 1225:= 1221:x 1206:1 1182:u 1171:t 1157:) 1154:t 1151:( 1147:x 1126:) 1123:) 1120:t 1117:( 1113:x 1109:( 1105:u 1079:1 1064:) 1061:t 1058:( 1054:x 1019:m 1013:n 1008:R 999:R 990:n 985:R 980:: 977:B 955:n 950:R 941:R 932:n 927:R 922:: 919:f 902:m 883:m 878:R 868:] 862:) 859:t 856:( 851:m 847:u 839:) 836:t 833:( 828:1 822:m 818:u 803:) 800:t 797:( 792:2 788:u 780:) 777:t 774:( 769:1 765:u 758:[ 750:) 747:t 744:( 740:u 719:n 700:n 695:R 685:] 679:) 676:t 673:( 668:n 664:x 656:) 653:t 650:( 645:1 639:n 635:x 620:) 617:t 614:( 609:2 605:x 597:) 594:t 591:( 586:1 582:x 575:[ 567:) 564:t 561:( 557:x 534:) 532:1 530:( 513:) 510:t 507:( 503:u 498:) 495:t 492:, 488:x 484:( 481:B 478:+ 475:) 472:t 469:, 465:x 461:( 458:f 455:= 452:) 449:t 446:( 436:x 325:0 322:= 319:s 294:0 291:= 288:s 258:1 254:x 247:= 242:1 232:x 205:0 202:= 197:1 187:x 180:+ 175:1 171:x 167:= 164:s 144:0 141:= 138:s 118:0 115:= 112:s 51:- 27:(

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