318:
31:
210:
173:
1007:
221:
In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature
217:
In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable"). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the
1095:
The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers. The PID controller is probably the most-used
222:
set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.
1306:
218:
building. The control action is the switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature.
1338:, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in
229:
is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."
1342:). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if
743:
119:, by controlling the power output of the vehicle's engine. Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent.
1476:
1582:
1684:
2089:
1925:
2224:
852:
236:
is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."
1840:
1761:
1988:
1165:
565:
974:
921:
490:
433:
2242:
However, in practice, a pure differentiator is neither physically realizable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a
111:
on a road vehicle; where external influences such as hills would cause speed changes, and the driver has the ability to alter the desired set speed. The
592:
1359:
1482:
1593:
1996:
298:(i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through
213:
An electromechanical timer, normally used for open-loop control based purely on a timing sequence, with no feedback from the process
1857:
2144:
751:
156:
In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed
88:
and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.
17:
137:
uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
2412:
317:
2334:"Feedback and control systems" - JJ Di Steffano, AR Stubberud, IJ Williams. Schaums outline series, McGraw-Hill 1967
1785:
1706:
1301:{\displaystyle u(t)=K_{P}e(t)+K_{I}\int ^{t}e(\tau ){\text{d}}\tau +K_{D}{\frac {{\text{d}}e(t)}{{\text{d}}t}}.}
1946:
307:
246:
226:
287:. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
866:, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If
115:
in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or
496:
1065:
934:
869:
187:
104:
84:) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with
34:
Example of a single industrial control loop; showing continuously modulated control of process flow.
2243:
439:
382:
311:
2435:
158:
69:
2239:. With this tuning in this example, the system output follows the reference input exactly.
1073:
303:
120:
56:
1092:, referring to the three terms operating on the error signal to produce a control signal.
103:, control algorithms, and actuators is arranged in an attempt to regulate a variable at a
8:
202:
2404:
2307:
1044:
924:
134:
116:
2295:
2264:
1350:
373:
346:
299:
196:
2396:
2317:
2276:
1069:
73:
65:
2408:
738:{\displaystyle Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).}
126:
Closed-loop controllers have the following advantages over open-loop controllers:
1339:
76:. Its name comes from the information path in the system: process inputs (e.g.,
2321:
1311:
The desired closed loop dynamics is obtained by adjusting the three parameters
1001:
342:
166:
108:
81:
2280:
2429:
2400:
1011:
112:
2381:
2246:
type approach or a differentiator with low-pass roll-off are used instead.
1471:{\displaystyle u(s)=K_{P}\,e(s)+K_{I}\,{\frac {1}{s}}\,e(s)+K_{D}\,s\,e(s)}
1041:
96:
47:
1577:{\displaystyle u(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right)e(s)}
30:
209:
140:
130:
disturbance rejection (such as hills in the cruise control example above)
1990:, we can express the PID controller transfer function in series form as
1679:{\displaystyle C(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right).}
1081:
279:(difference) between the reference and the output to change the inputs
172:
372:) do not depend on time), the systems above can be analysed using the
2362:
2344:
1040:
A proportional–integral–derivative controller (PID controller) is a
1689:
As an example of tuning a PID controller in the closed-loop system
1077:
92:
51:
2312:
2084:{\displaystyle C(s)=K\left(1+{\frac {1}{sT_{I}}}\right)(1+sT_{D})}
858:
of the system. The numerator is the forward (open-loop) gain from
77:
225:
The definition of a closed loop control system according to the
123:
do not make use of feedback, and run only in pre-arranged ways.
338:
100:
85:
162:
and serves to further improve reference tracking performance.
1159:
is the tracking error, a PID controller has the general form
1920:{\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}
1343:
1779:
are some constants. The plant output is fed back through
1006:
2219:{\displaystyle K={\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}
847:{\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}
2265:"Feedback for physicists: A tutorial essay on control"
376:
on the variables. This gives the following relations:
2382:"PID control system analysis, design, and technology"
2147:
1999:
1949:
1860:
1788:
1709:
1596:
1485:
1362:
1168:
992:) and the output closely tracks the reference input.
937:
872:
754:
595:
499:
442:
385:
194:
Fundamentally, there are two types of control loop:
165:
A common closed-loop controller architecture is the
64:. A closed-loop controller uses feedback to control
1353:results in the transformed PID controller equation
1047:control technique widely used in control systems.
2218:
2083:
1982:
1919:
1834:
1755:
1678:
1576:
1470:
1300:
968:
915:
846:
737:
559:
484:
427:
1025:is the desired process value or "set point", and
2427:
2328:
240:
2389:IEEE Transactions on Control Systems Technology
2296:"Thermodynamics of feedback controlled systems"
290:This is called a single-input-single-output (
152:improved rectification of random fluctuations
1050:A PID controller continuously calculates an
180:
259:) is fed back through a sensor measurement
146:reduced sensitivity to parameter variations
2379:
2369:. Clinton, MA US: The Colonial Press, Inc.
2351:. Clinton, MA US: The Colonial Press, Inc.
2262:
1835:{\displaystyle F(s)={\frac {1}{1+sT_{F}}}}
1756:{\displaystyle P(s)={\frac {A}{1+sT_{P}}}}
1587:with the PID controller transfer function
1110:is the control signal sent to the system,
2380:Ang, K.H.; Chong, G.C.Y.; Li, Y. (2005).
2311:
1664:
1640:
1553:
1529:
1455:
1451:
1425:
1414:
1388:
263:to a comparison with the reference value
2293:
1983:{\displaystyle K_{I}={\frac {K}{T_{I}}}}
1014:of a PID controller in a feedback loop,
1005:
208:
188:Control loop § Open-loop and closed-loop
171:
29:
2127:into the closed-loop transfer function
995:
149:improved reference tracking performance
14:
2428:
1700:, consider a 1st order plant given by
2361:
2343:
2294:Cao, F. J.; Feito, M. (2009-04-10).
1064:as the difference between a desired
560:{\displaystyle E(s)=R(s)-F(s)Y(s).}
24:
1854:is also a constant. Now if we set
1072:and applies a correction based on
316:
107:(SP). An everyday example is the
25:
2447:
133:guaranteed performance even with
2418:from the original on 2013-12-13.
1090:Proportional-Integral-Derivative
186:This section is an excerpt from
2367:The Origins of Feedback Control
2349:The Origins of Feedback Control
2263:Bechhoefer, John (2005-08-31).
969:{\displaystyle |F(s)|\approx 1}
916:{\displaystyle |P(s)C(s)|\gg 1}
2373:
2355:
2337:
2287:
2256:
2078:
2056:
2009:
2003:
1798:
1792:
1719:
1713:
1606:
1600:
1571:
1565:
1495:
1489:
1465:
1459:
1435:
1429:
1398:
1392:
1372:
1366:
1279:
1273:
1238:
1232:
1203:
1197:
1178:
1172:
1036:is the measured process value.
956:
952:
946:
939:
903:
899:
893:
887:
881:
874:
838:
832:
826:
820:
814:
808:
794:
788:
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776:
764:
758:
729:
723:
717:
711:
702:
696:
683:
677:
671:
665:
659:
653:
639:
633:
627:
621:
605:
599:
551:
545:
539:
533:
524:
518:
509:
503:
479:
473:
467:
461:
452:
446:
422:
416:
410:
404:
395:
389:
321:A simple feedback control loop
13:
1:
2249:
856:closed-loop transfer function
485:{\displaystyle U(s)=C(s)E(s)}
428:{\displaystyle Y(s)=P(s)U(s)}
308:distributed parameter systems
247:Closed-loop transfer function
241:Closed-loop transfer function
227:British Standards Institution
984:) is approximately equal to
325:If we assume the controller
310:the vectors may be infinite-
283:to the system under control
7:
1132:is the desired output, and
1121:is the measured output and
143:processes can be stabilized
10:
2452:
2322:10.1103/PhysRevE.79.041118
2138:, we find that by setting
999:
244:
185:
2281:10.1103/RevModPhys.77.783
2269:Reviews of Modern Physics
1096:feedback control design.
345:(i.e., elements of their
251:The output of the system
181:Open-loop and closed-loop
121:Open-loop control systems
2401:10.1109/TCST.2005.847331
1346:systems are considered.
923:, i.e., it has a large
314:(typically functions).
234:Feedback Control System
62:non-feedback controller
2244:phase-lead compensator
2220:
2085:
1984:
1921:
1836:
1757:
1680:
1578:
1472:
1351:Laplace transformation
1302:
1037:
970:
917:
854:is referred to as the
848:
739:
561:
486:
429:
322:
214:
177:
91:In the case of linear
40:closed-loop controller
35:
2221:
2086:
1985:
1922:
1837:
1758:
1681:
1579:
1473:
1303:
1088:is an initialism for
1009:
971:
918:
849:
740:
562:
487:
430:
320:
275:then takes the error
212:
176:A basic feedback loop
175:
33:
2145:
1997:
1947:
1858:
1786:
1707:
1594:
1483:
1360:
1166:
996:PID feedback control
935:
870:
752:
593:
497:
440:
383:
57:open-loop controller
54:, in contrast to an
927:with each value of
203:closed-loop control
200:(feedforward), and
50:which incorporates
44:feedback controller
27:Feedback controller
18:Closed-loop control
2216:
2081:
1980:
1917:
1832:
1753:
1676:
1574:
1468:
1298:
1045:feedback mechanism
1038:
966:
913:
844:
735:
557:
482:
425:
323:
302:instead of simple
294:) control system;
271:). The controller
215:
178:
36:
2300:Physical Review E
2162:
2049:
1978:
1910:
1830:
1751:
1649:
1538:
1423:
1293:
1287:
1268:
1244:
842:
687:
374:Laplace transform
347:transfer function
333:, and the sensor
306:values. For some
197:open-loop control
16:(Redirected from
2443:
2420:
2419:
2417:
2386:
2377:
2371:
2370:
2359:
2353:
2352:
2341:
2335:
2332:
2326:
2325:
2315:
2291:
2285:
2284:
2260:
2238:
2225:
2223:
2222:
2217:
2215:
2214:
2202:
2201:
2189:
2188:
2176:
2175:
2163:
2155:
2137:
2126:
2115:
2104:
2090:
2088:
2087:
2082:
2077:
2076:
2055:
2051:
2050:
2048:
2047:
2046:
2030:
1989:
1987:
1986:
1981:
1979:
1977:
1976:
1964:
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1958:
1942:
1926:
1924:
1923:
1918:
1916:
1912:
1911:
1909:
1908:
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1898:
1889:
1870:
1869:
1853:
1841:
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1838:
1833:
1831:
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1827:
1805:
1778:
1769:
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1754:
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1749:
1748:
1726:
1699:
1685:
1683:
1682:
1677:
1672:
1668:
1663:
1662:
1650:
1642:
1639:
1638:
1626:
1625:
1583:
1581:
1580:
1575:
1561:
1557:
1552:
1551:
1539:
1531:
1528:
1527:
1515:
1514:
1477:
1475:
1474:
1469:
1450:
1449:
1424:
1416:
1413:
1412:
1387:
1386:
1337:
1328:
1319:
1307:
1305:
1304:
1299:
1294:
1292:
1288:
1285:
1282:
1269:
1266:
1263:
1261:
1260:
1245:
1242:
1228:
1227:
1218:
1217:
1193:
1192:
1158:
1131:
1120:
1109:
1070:process variable
1063:
1035:
1024:
975:
973:
972:
967:
959:
942:
922:
920:
919:
914:
906:
877:
853:
851:
850:
845:
843:
841:
797:
771:
744:
742:
741:
736:
692:
688:
686:
642:
616:
566:
564:
563:
558:
491:
489:
488:
483:
434:
432:
431:
426:
74:dynamical system
21:
2451:
2450:
2446:
2445:
2444:
2442:
2441:
2440:
2426:
2425:
2424:
2423:
2415:
2384:
2378:
2374:
2360:
2356:
2342:
2338:
2333:
2329:
2292:
2288:
2261:
2257:
2252:
2229:
2210:
2206:
2197:
2193:
2184:
2180:
2171:
2167:
2154:
2146:
2143:
2142:
2128:
2117:
2106:
2095:
2072:
2068:
2042:
2038:
2034:
2029:
2022:
2018:
1998:
1995:
1994:
1972:
1968:
1963:
1954:
1950:
1948:
1945:
1944:
1940:
1933:
1928:
1904:
1900:
1894:
1890:
1888:
1881:
1877:
1865:
1861:
1859:
1856:
1855:
1851:
1846:
1823:
1819:
1809:
1804:
1787:
1784:
1783:
1776:
1771:
1767:
1744:
1740:
1730:
1725:
1708:
1705:
1704:
1690:
1658:
1654:
1641:
1634:
1630:
1621:
1617:
1616:
1612:
1595:
1592:
1591:
1547:
1543:
1530:
1523:
1519:
1510:
1506:
1505:
1501:
1484:
1481:
1480:
1445:
1441:
1415:
1408:
1404:
1382:
1378:
1361:
1358:
1357:
1340:process control
1335:
1330:
1326:
1321:
1317:
1312:
1284:
1283:
1265:
1264:
1262:
1256:
1252:
1241:
1223:
1219:
1213:
1209:
1188:
1184:
1167:
1164:
1163:
1133:
1122:
1111:
1100:
1068:and a measured
1054:
1026:
1015:
1004:
998:
955:
938:
936:
933:
932:
902:
873:
871:
868:
867:
798:
772:
770:
753:
750:
749:
748:The expression
643:
617:
615:
611:
594:
591:
590:
498:
495:
494:
441:
438:
437:
384:
381:
380:
249:
243:
238:
237:
191:
183:
28:
23:
22:
15:
12:
11:
5:
2449:
2439:
2438:
2436:Control theory
2422:
2421:
2395:(4): 559–576.
2372:
2354:
2336:
2327:
2286:
2275:(3): 783–836.
2254:
2253:
2251:
2248:
2227:
2226:
2213:
2209:
2205:
2200:
2196:
2192:
2187:
2183:
2179:
2174:
2170:
2166:
2161:
2158:
2153:
2150:
2092:
2091:
2080:
2075:
2071:
2067:
2064:
2061:
2058:
2054:
2045:
2041:
2037:
2033:
2028:
2025:
2021:
2017:
2014:
2011:
2008:
2005:
2002:
1975:
1971:
1967:
1962:
1957:
1953:
1938:
1931:
1915:
1907:
1903:
1897:
1893:
1887:
1884:
1880:
1876:
1873:
1868:
1864:
1849:
1843:
1842:
1826:
1822:
1818:
1815:
1812:
1808:
1803:
1800:
1797:
1794:
1791:
1774:
1764:
1763:
1747:
1743:
1739:
1736:
1733:
1729:
1724:
1721:
1718:
1715:
1712:
1687:
1686:
1675:
1671:
1667:
1661:
1657:
1653:
1648:
1645:
1637:
1633:
1629:
1624:
1620:
1615:
1611:
1608:
1605:
1602:
1599:
1585:
1584:
1573:
1570:
1567:
1564:
1560:
1556:
1550:
1546:
1542:
1537:
1534:
1526:
1522:
1518:
1513:
1509:
1504:
1500:
1497:
1494:
1491:
1488:
1478:
1467:
1464:
1461:
1458:
1454:
1448:
1444:
1440:
1437:
1434:
1431:
1428:
1422:
1419:
1411:
1407:
1403:
1400:
1397:
1394:
1391:
1385:
1381:
1377:
1374:
1371:
1368:
1365:
1333:
1324:
1315:
1309:
1308:
1297:
1291:
1281:
1278:
1275:
1272:
1259:
1255:
1251:
1248:
1240:
1237:
1234:
1231:
1226:
1222:
1216:
1212:
1208:
1205:
1202:
1199:
1196:
1191:
1187:
1183:
1180:
1177:
1174:
1171:
1002:PID controller
1000:Main article:
997:
994:
965:
962:
958:
954:
951:
948:
945:
941:
912:
909:
905:
901:
898:
895:
892:
889:
886:
883:
880:
876:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
796:
793:
790:
787:
784:
781:
778:
775:
769:
766:
763:
760:
757:
746:
745:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
691:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
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641:
638:
635:
632:
629:
626:
623:
620:
614:
610:
607:
604:
601:
598:
578:) in terms of
568:
567:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
492:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
435:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
343:time-invariant
245:Main article:
242:
239:
192:
184:
182:
179:
167:PID controller
154:
153:
150:
147:
144:
138:
131:
109:cruise control
82:electric motor
80:applied to an
26:
9:
6:
4:
3:
2:
2448:
2437:
2434:
2433:
2431:
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2383:
2376:
2368:
2364:
2358:
2350:
2346:
2340:
2331:
2323:
2319:
2314:
2309:
2306:(4): 041118.
2305:
2301:
2297:
2290:
2282:
2278:
2274:
2270:
2266:
2259:
2255:
2247:
2245:
2240:
2236:
2232:
2211:
2207:
2203:
2198:
2194:
2190:
2185:
2181:
2177:
2172:
2168:
2164:
2159:
2156:
2151:
2148:
2141:
2140:
2139:
2135:
2131:
2124:
2120:
2113:
2109:
2102:
2098:
2073:
2069:
2065:
2062:
2059:
2052:
2043:
2039:
2035:
2031:
2026:
2023:
2019:
2015:
2012:
2006:
2000:
1993:
1992:
1991:
1973:
1969:
1965:
1960:
1955:
1951:
1941:
1934:
1913:
1905:
1901:
1895:
1891:
1885:
1882:
1878:
1874:
1871:
1866:
1862:
1852:
1824:
1820:
1816:
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1810:
1806:
1801:
1795:
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1777:
1745:
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1737:
1734:
1731:
1727:
1722:
1716:
1710:
1703:
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1697:
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1673:
1669:
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1655:
1651:
1646:
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1631:
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1507:
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1498:
1492:
1486:
1479:
1462:
1456:
1452:
1446:
1442:
1438:
1432:
1426:
1420:
1417:
1409:
1405:
1401:
1395:
1389:
1383:
1379:
1375:
1369:
1363:
1356:
1355:
1354:
1352:
1347:
1345:
1341:
1336:
1327:
1318:
1295:
1289:
1276:
1270:
1257:
1253:
1249:
1246:
1235:
1229:
1224:
1220:
1214:
1210:
1206:
1200:
1194:
1189:
1185:
1181:
1175:
1169:
1162:
1161:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1129:
1125:
1118:
1114:
1107:
1103:
1097:
1093:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1061:
1057:
1053:
1048:
1046:
1043:
1033:
1029:
1022:
1018:
1013:
1012:block diagram
1008:
1003:
993:
991:
987:
983:
979:
963:
960:
949:
943:
930:
926:
910:
907:
896:
890:
884:
878:
865:
861:
857:
835:
829:
823:
817:
811:
805:
802:
799:
791:
785:
779:
773:
767:
761:
755:
732:
726:
720:
714:
708:
705:
699:
693:
689:
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674:
668:
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656:
650:
647:
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624:
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612:
608:
602:
596:
589:
588:
587:
585:
581:
577:
573:
554:
548:
542:
536:
530:
527:
521:
515:
512:
506:
500:
493:
476:
470:
464:
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419:
413:
407:
401:
398:
392:
386:
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378:
377:
375:
371:
367:
363:
359:
355:
351:
348:
344:
340:
336:
332:
328:
319:
315:
313:
309:
305:
301:
297:
293:
288:
286:
282:
278:
274:
270:
266:
262:
258:
254:
248:
235:
232:Likewise; "A
231:
228:
223:
219:
211:
207:
205:
204:
199:
198:
189:
174:
170:
168:
163:
161:
160:
151:
148:
145:
142:
139:
136:
132:
129:
128:
127:
124:
122:
118:
114:
113:PID algorithm
110:
106:
102:
98:
94:
89:
87:
83:
79:
75:
71:
67:
63:
59:
58:
53:
49:
45:
41:
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2366:
2357:
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2339:
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2272:
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2258:
2241:
2234:
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2133:
2129:
2122:
2118:
2111:
2107:
2100:
2096:
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1936:
1929:
1847:
1844:
1772:
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1150:
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1138:
1134:
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1116:
1112:
1105:
1101:
1098:
1094:
1089:
1085:
1074:proportional
1059:
1055:
1051:
1049:
1042:control loop
1039:
1031:
1027:
1020:
1016:
989:
985:
981:
977:
928:
863:
859:
855:
747:
583:
579:
575:
571:
570:Solving for
569:
369:
365:
361:
357:
353:
349:
334:
330:
329:, the plant
326:
324:
295:
291:
289:
284:
280:
276:
272:
268:
264:
260:
256:
252:
250:
233:
224:
220:
216:
206:(feedback).
201:
195:
193:
164:
157:
155:
125:
97:control loop
90:
61:
55:
48:control loop
43:
39:
37:
1052:error value
312:dimensional
159:feedforward
95:systems, a
2363:Mayr, Otto
2345:Mayr, Otto
2250:References
1082:derivative
99:including
2313:0805.4824
2094:Plugging
1349:Applying
1247:τ
1236:τ
1221:∫
961:≈
931:, and if
908:≫
528:−
117:overshoot
2430:Category
2413:Archived
2365:(1969).
2347:(1970).
1078:integral
1066:setpoint
586:) gives
141:unstable
105:setpoint
93:feedback
52:feedback
1084:terms.
976:, then
364:), and
300:vectors
101:sensors
86:sensors
78:voltage
70:outputs
2409:921620
2407:
2116:, and
1943:, and
1845:where
1766:where
1080:, and
339:linear
304:scalar
66:states
2416:(PDF)
2405:S2CID
2385:(PDF)
2308:arXiv
2237:) = 1
135:model
72:of a
46:is a
1770:and
1344:MIMO
1329:and
1149:) −
1141:) =
925:norm
341:and
337:are
296:MIMO
292:SISO
2397:doi
2318:doi
2277:doi
1099:If
1086:PID
862:to
356:),
68:or
60:or
42:or
2432::
2411:.
2403:.
2393:13
2391:.
2387:.
2316:.
2304:79
2302:.
2298:.
2273:77
2271:.
2267:.
2105:,
1937:KT
1935:=
1927:,
1320:,
1076:,
1010:A
169:.
38:A
2399::
2324:.
2320::
2310::
2283:.
2279::
2235:s
2233:(
2231:H
2212:P
2208:T
2204:=
2199:D
2195:T
2191:,
2186:F
2182:T
2178:=
2173:I
2169:T
2165:,
2160:A
2157:1
2152:=
2149:K
2136:)
2134:s
2132:(
2130:H
2125:)
2123:s
2121:(
2119:C
2114:)
2112:s
2110:(
2108:F
2103:)
2101:s
2099:(
2097:P
2079:)
2074:D
2070:T
2066:s
2063:+
2060:1
2057:(
2053:)
2044:I
2040:T
2036:s
2032:1
2027:+
2024:1
2020:(
2016:K
2013:=
2010:)
2007:s
2004:(
2001:C
1974:I
1970:T
1966:K
1961:=
1956:I
1952:K
1939:D
1932:D
1930:K
1914:)
1906:I
1902:T
1896:D
1892:T
1886:+
1883:1
1879:(
1875:K
1872:=
1867:P
1863:K
1850:F
1848:T
1825:F
1821:T
1817:s
1814:+
1811:1
1807:1
1802:=
1799:)
1796:s
1793:(
1790:F
1775:P
1773:T
1768:A
1746:P
1742:T
1738:s
1735:+
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1728:A
1723:=
1720:)
1717:s
1714:(
1711:P
1698:)
1696:s
1694:(
1692:H
1674:.
1670:)
1666:s
1660:D
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1652:+
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1628:+
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1619:K
1614:(
1610:=
1607:)
1604:s
1601:(
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1572:)
1569:s
1566:(
1563:e
1559:)
1555:s
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1541:+
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1503:(
1499:=
1496:)
1493:s
1490:(
1487:u
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1463:s
1460:(
1457:e
1453:s
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1396:s
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1376:=
1373:)
1370:s
1367:(
1364:u
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1332:K
1325:I
1323:K
1316:P
1314:K
1296:.
1290:t
1286:d
1280:)
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1274:(
1271:e
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1233:(
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1201:t
1198:(
1195:e
1190:P
1186:K
1182:=
1179:)
1176:t
1173:(
1170:u
1157:)
1155:t
1153:(
1151:y
1147:t
1145:(
1143:r
1139:t
1137:(
1135:e
1130:)
1128:t
1126:(
1124:r
1119:)
1117:t
1115:(
1113:y
1108:)
1106:t
1104:(
1102:u
1062:)
1060:t
1058:(
1056:e
1034:)
1032:t
1030:(
1028:y
1023:)
1021:t
1019:(
1017:r
990:s
988:(
986:R
982:s
980:(
978:Y
964:1
957:|
953:)
950:s
947:(
944:F
940:|
929:s
911:1
904:|
900:)
897:s
894:(
891:C
888:)
885:s
882:(
879:P
875:|
864:y
860:r
839:)
836:s
833:(
830:C
827:)
824:s
821:(
818:P
815:)
812:s
809:(
806:F
803:+
800:1
795:)
792:s
789:(
786:C
783:)
780:s
777:(
774:P
768:=
765:)
762:s
759:(
756:H
733:.
730:)
727:s
724:(
721:R
718:)
715:s
712:(
709:H
706:=
703:)
700:s
697:(
694:R
690:)
684:)
681:s
678:(
675:F
672:)
669:s
666:(
663:C
660:)
657:s
654:(
651:P
648:+
645:1
640:)
637:s
634:(
631:C
628:)
625:s
622:(
619:P
613:(
609:=
606:)
603:s
600:(
597:Y
584:s
582:(
580:R
576:s
574:(
572:Y
555:.
552:)
549:s
546:(
543:Y
540:)
537:s
534:(
531:F
525:)
522:s
519:(
516:R
513:=
510:)
507:s
504:(
501:E
480:)
477:s
474:(
471:E
468:)
465:s
462:(
459:C
456:=
453:)
450:s
447:(
444:U
423:)
420:s
417:(
414:U
411:)
408:s
405:(
402:P
399:=
396:)
393:s
390:(
387:Y
370:s
368:(
366:F
362:s
360:(
358:P
354:s
352:(
350:C
335:F
331:P
327:C
285:P
281:u
277:e
273:C
269:t
267:(
265:r
261:F
257:t
255:(
253:y
190:.
20:)
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