472:
83:
not necessarily represent the best controller in terms of the usual performance measures used to evaluate controllers such as settling time, energy expended, etc. Also, non-linear constraints such as saturation are generally not well-handled. These methods were introduced into control theory in the late 1970s-early 1980s by
164:
techniques can be used to minimize the closed loop impact of a perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance. Simultaneously optimizing robust performance and robust stabilization is difficult. One method that comes
82:
techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled. It is important to keep in mind that the resulting controller is only optimal with respect to the prescribed cost function and does
1119:
260:
758:
175:, which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes the response near the system bandwidth to achieve good robust stabilization.
467:{\displaystyle {\begin{bmatrix}z\\v\end{bmatrix}}=\mathbf {P} (s)\,{\begin{bmatrix}w\\u\end{bmatrix}}={\begin{bmatrix}P_{11}(s)&P_{12}(s)\\P_{21}(s)&P_{22}(s)\end{bmatrix}}\,{\begin{bmatrix}w\\u\end{bmatrix}}}
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of the matrix over that space. In the case of a scalar-valued function, the elements of the Hardy space that extend continuously to the boundary and are continuous at infinity is the
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814:
623:
1114:{\displaystyle ||F_{\ell }(\mathbf {P} ,\mathbf {K} )||_{\infty }=\sup _{\omega }{\bar {\sigma }}(F_{\ell }(\mathbf {P} ,\mathbf {K} )(j\omega ))}
1384:
Zames, George (1981). "Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses".
1160:
75:
techniques are readily applicable to problems involving multivariate systems with cross-coupling between channels; disadvantages of
1411:
Helton, J. William (1978). "Orbit structure of the Mobius transformation semigroup action on H-infinity (broadband matching)".
535:
1595:
1572:
1540:
930:
819:
151:. For a matrix-valued function, the norm can be interpreted as a maximum gain in any direction and at any frequency; for
1474:
753:{\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )=P_{11}+P_{12}\,\mathbf {K} \,(I-P_{22}\,\mathbf {K} )^{-1}\,P_{21}}
1358:
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17:
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1615:
1433:(1980). "Feedback stabilization of linear dynamical plants with uncertainty in the gain factor".
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601:
1620:
193:
First, the process has to be represented according to the following standard configuration:
248:
118:
8:
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50:
to synthesize controllers to achieve stabilization with guaranteed performance. To use
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comes from the name of the mathematical space over which the optimization takes place:
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systems, this is effectively the maximum magnitude of the frequency response.
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87:(sensitivity minimization), J. William Helton (broadband matching), and
68:
techniques have the advantage over classical control techniques in that
1542:
Optimal State
Estimation: Kalman, H-infinity, and Nonlinear Approaches
61:
problem and then finds the controller that solves this optimization.
1321:
to find the controller, but require several simplifying assumptions.
1348:
141:
1225:
norm of the closed loop system is mainly given through the matrix
1324:
An optimization-based reformulation of the
Riccati equation uses
57:
methods, a control designer expresses the control problem as a
1208:{\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )(j\omega )}
1311:
of the closed loop often leads to very high-order controller.
581:{\displaystyle z=F_{\ell }(\mathbf {P} ,\mathbf {K} )\,w}
1515:
Doyle, John; Francis, Bruce; Tannenbaum, Allen (1992),
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1514:
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It is therefore possible to express the dependency of
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that we want to minimize, and the measured variables
966:{\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )}
855:{\displaystyle F_{\ell }(\mathbf {P} ,\mathbf {K} )}
1587:
Multivariable
Feedback Control: Analysis and Design
1565:
Multivariable
Feedback Control: Analysis and Design
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1584:Skogestad, Sigurd; Postlethwaite, Ian (2005),
1563:Skogestad, Sigurd; Postlethwaite, Ian (1996),
1464:
1526:
178:Commercial software is available to support
215:. There are two outputs, the error signals
1465:Barbu, V.; Sritharan, Sivaguru S. (1998),
1429:
125:and bounded in the open right-half of the
1487:
1423:
1296:)). There are several ways to come to an
924:control design. The infinity norm of the
739:
720:
700:
694:
574:
504:
437:
307:
886:{\displaystyle {\mathcal {H}}_{\infty }}
787:{\displaystyle {\mathcal {H}}_{\infty }}
794:control design is to find a controller
231:to calculate the manipulated variables
14:
1608:
1467:"H-infinity Control of Fluid Dynamics"
1410:
1386:IEEE Transactions on Automatic Control
235:. Notice that all these are generally
188:
1538:
1404:
1383:
893:norm. The same definition applies to
625:is defined (the subscript comes from
223:, that we use to control the system.
1377:
511:{\displaystyle u=\mathbf {K} (s)\,v}
207:has two inputs, the exogenous input
24:
1475:Proceedings of the Royal Society A
1034:
917:{\displaystyle {\mathcal {H}}_{2}}
903:
878:
872:
779:
773:
25:
1632:
1359:Linear-quadratic-Gaussian control
1527:Green, M.; Limebeer, D. (1995),
1435:International Journal of Control
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1146:{\displaystyle {\bar {\sigma }}}
1089:
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595:linear fractional transformation
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1328:and requires fewer assumptions.
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13:
1:
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1317:-based approaches solve two
1309:Youla-Kucera parametrization
809:{\displaystyle \mathbf {K} }
763:Therefore, the objective of
254:In formulae, the system is:
91:(gain margin optimization).
7:
1332:
165:close to achieving this is
121:-valued functions that are
10:
1637:
1326:linear matrix inequalities
1447:10.1080/00207178008922838
618:{\displaystyle F_{\ell }}
59:mathematical optimization
1398:10.1109/tac.1981.1102603
1365:Rosenbrock system matrix
926:transfer function matrix
133:) > 0; the
1590:(2nd ed.), Wiley,
1545:, Wiley, archived from
1517:Feedback Control Theory
1354:H-infinity loop-shaping
1498:10.1098/rspa.1998.0289
1413:Adv. Math. Suppl. Stud
1236:is given in the form (
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967:
918:
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754:
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185:controller synthesis.
1529:Linear Robust Control
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1539:Simon, Dan (2006),
1482:(1979): 3009–3033,
189:Problem formulation
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1597:978-0-470-01167-6
1574:978-0-471-94277-1
1431:Tannenbaum, Allen
1319:Riccati equations
1232:(when the system
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1061:
1042:
16:(Redirected from
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1339:Blaschke product
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89:Allen Tannenbaum
21:
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1531:, Prentice Hall
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1218:The achievable
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1616:Control theory
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1392:(2): 301–320.
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1155:singular value
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145:singular value
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129:defined by Re(
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1549:on 2010-12-30
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1621:Hardy spaces
1586:
1564:
1551:, retrieved
1547:the original
1541:
1528:
1516:
1479:
1473:
1458:Bibliography
1438:
1434:
1425:
1416:
1412:
1406:
1389:
1385:
1379:
1303:controller:
1297:
1290:
1283:
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1241:
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173:loop-shaping
167:
158:
157:
149:disk algebra
140:norm is the
134:
130:
112:
105:
101:
95:
93:
85:George Zames
76:
69:
62:
51:
46:are used in
43:
37:
36:
28:
27:
26:
1519:, MacMillan
1441:(1): 1–16.
1344:Hardy space
591:Called the
227:is used in
114:Hardy space
94:The phrase
1610:Categories
1553:2006-07-05
1419:: 129–197.
1371:References
816:such that
239:, whereas
203:The plant
18:H infinity
1567:, Wiley,
1506:121983192
1484:CiteSeerX
1200:ω
1170:ℓ
1138:¯
1135:σ
1103:ω
1073:ℓ
1059:¯
1056:σ
1048:ω
1035:∞
999:ℓ
940:ℓ
879:∞
829:ℓ
780:∞
732:−
708:−
645:ℓ
611:ℓ
551:ℓ
40:-infinity
1349:H square
1333:See also
249:matrices
142:supremum
123:analytic
1315:Riccati
237:vectors
111:is the
102:control
44:methods
35:(i.e. "
1594:
1571:
1504:
1486:
1124:where
593:lower
119:matrix
1502:S2CID
1470:(PDF)
1361:(LQG)
627:lower
1592:ISBN
1569:ISBN
529:as:
247:are
243:and
153:SISO
1494:doi
1480:545
1443:doi
1394:doi
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629:):
598:,
525:on
117:of
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1277:D
1270:D
1266:2
1263:C
1259:1
1256:C
1252:2
1249:B
1245:1
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1238:A
1234:P
1227:D
1223:∞
1220:H
1203:)
1197:j
1194:(
1191:)
1187:K
1183:,
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1175:(
1166:F
1109:)
1106:)
1100:j
1097:(
1094:)
1090:K
1086:,
1082:P
1078:(
1069:F
1065:(
1040:=
1030:|
1024:|
1020:)
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1012:,
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1004:(
995:F
990:|
985:|
961:)
957:K
953:,
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936:F
910:2
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850:)
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834:(
825:F
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723:K
712:P
705:I
702:(
697:K
686:P
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669:=
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662:K
658:,
654:P
650:(
641:F
607:F
576:w
572:)
568:K
564:,
560:P
556:(
547:F
543:=
540:z
527:w
523:z
506:v
502:)
499:s
496:(
492:K
488:=
485:u
460:]
454:u
447:w
441:[
433:]
427:)
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412:P
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359:s
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335:=
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295:P
291:=
286:]
280:v
273:z
267:[
245:K
241:P
233:u
229:K
225:v
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217:z
213:u
209:w
205:P
183:∞
180:H
171:∞
168:H
162:∞
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138:∞
135:H
131:s
109:∞
106:H
99:∞
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80:∞
77:H
73:∞
70:H
66:∞
63:H
55:∞
52:H
38:H
32:∞
29:H
20:)
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