92:
The LQR algorithm reduces the amount of work done by the control systems engineer to optimize the controller. However, the engineer still needs to specify the cost function parameters, and compare the results with the specified design goals. Often this means that controller construction will be an
88:
with weighting factors supplied by a human (engineer). The cost function is often defined as a sum of the deviations of key measurements, like altitude or process temperature, from their desired values. The algorithm thus finds those controller settings that minimize undesired deviations. The
2338:
3724:
525:
3995:
Model predictive control and linear-quadratic regulators are two types of optimal control methods that have distinct approaches for setting the optimization costs. In particular, when the LQR is run repeatedly with a receding horizon, it becomes a form of
3236:
2789:
104:, also known as pole placement, in which there is a clearer relationship between controller parameters and controller behavior. Difficulty in finding the right weighting factors limits the application of the LQR based controller synthesis.
1927:
2036:
954:
3832:
3575:
1540:
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then the problem is known as the polynomial-quadratic regulator (PQR). Again, the Al'Brekht algorithm can be applied to reduce this problem to a large linear one which can be solved with a generalization of the
2562:
2124:
3934:
3587:
1822:
1160:
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93:
iterative process in which the engineer judges the "optimal" controllers produced through simulation and then adjusts the parameters to produce a controller more consistent with design goals.
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3006:
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2113:
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255:
84:
The settings of a (regulating) controller governing either a machine or process (like an airplane or chemical reactor) are found by using a mathematical algorithm that minimizes a
3094:
2846:
2426:
1021:
1423:
1083:
217:
3289:
2600:
1286:
3884:
2934:
2880:
4237:
Borggaard, Jeff; Zietsman, Lizette (July 2020). "The
Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems".
1579:
645:
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2365:
167:
3961:
3415:
2900:
1697:
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772:
668:
609:
589:
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295:
275:
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1938:
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3427:
3838:
Note that one way to solve the algebraic
Riccati equation is by iterating the dynamic Riccati equation of the finite-horizon case until it converges.
4149:
1434:
2333:{\displaystyle J=x_{H_{p}}^{T}Q_{H_{p}}x_{H_{p}}+\sum \limits _{k=0}^{H_{p}-1}\left(x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k}+2x_{k}^{T}Nu_{k}\right)}
4056:
2437:
17:
3719:{\displaystyle P={\mathcal {A}}^{T}P{\mathcal {A}}-{\mathcal {A}}^{T}PB\left(R+B^{T}PB\right)^{-1}B^{T}P{\mathcal {A}}+{\mathcal {Q}}}
3892:
1709:
4266:
1099:
4122:
Shaiju, AJ, Petersen, Ian R. (2008). "Formulas for discrete time LQR, LQG, LEQG and minimax LQG optimal control problems".
3300:
1294:
4377:
4355:
4106:
611:
is the cross-term (control and state) cost matrix, the feedback control law that minimizes the value of the cost is:
69:
520:{\displaystyle J=x^{T}(t_{1})F(t_{1})x(t_{1})+\int \limits _{t_{0}}^{t_{1}}\left(x^{T}Qx+u^{T}Ru+2x^{T}Nu\right)dt}
676:
3990:
3964:
can be applied to reduce this problem to one that can be solved efficiently using tensor based linear solvers.
2939:
1173:
1610:
3960:
If the state equation is quadratic then the problem is known as the quadratic-quadratic regulator (QQR). The
3022:
2052:
775:
3231:{\displaystyle J=\sum \limits _{k=0}^{\infty }\left(x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k}+2x_{k}^{T}Nu_{k}\right)}
300:
225:
39:
2797:
2377:
49:
is called the LQ problem. One of the main results in the theory is that the solution is provided by the
965:
3418:
2784:{\displaystyle P_{k-1}=A^{T}P_{k}A-(A^{T}P_{k}B+N)\left(R+B^{T}P_{k}B\right)^{-1}(B^{T}P_{k}A+N^{T})+Q}
1700:
1381:
1041:
175:
4398:
4189:
3978:
3247:
4000:(MPC). In general, however, MPC does not rely on any assumptions regarding linearity of the system.
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3997:
3944:
46:
3943:
problem, and so the problem is often solved repeatedly with a receding horizon. This is a form of
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97:
4393:
4143:
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120:
43:
72:. Like the LQR problem itself, the LQG problem is one of the most fundamental problems in
8:
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101:
100:. As such, it is not uncommon for control engineers to prefer alternative methods, like
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4079:
4037:
4018:"Robustness results in linear-quadratic Gaussian based multivariable control designs"
1922:{\displaystyle {\mathcal {A}}^{T}P+P{\mathcal {A}}-PBR^{-1}B^{T}P+{\mathcal {Q}}=0\,}
1090:
4320:
4258:
4254:
4212:
4204:
4131:
4071:
4029:
2031:{\displaystyle {\mathcal {A}}=A-BR^{-1}N^{T}\qquad {\mathcal {Q}}=Q-NR^{-1}N^{T}\,}
949:{\displaystyle A^{T}P(t)+P(t)A-(P(t)B+N)R^{-1}(B^{T}P(t)+N^{T})+Q=-{\dot {P}}(t)\,}
332:
is the control input. Given a quadratic cost function for the system, defined as:
89:
magnitude of the control action itself may also be included in the cost function.
4325:
4298:
3827:{\displaystyle {\mathcal {A}}=A-BR^{-1}N^{T}\qquad {\mathcal {Q}}=Q-NR^{-1}N^{T}}
3570:{\displaystyle P=A^{T}PA-(A^{T}PB+N)\left(R+B^{T}PB\right)^{-1}(B^{T}PA+N^{T})+Q}
31:
4238:
4017:
3981:; this is feasible provided that the degree of the polynomial is not too high.
73:
38:
at minimum cost. The case where the system dynamics are described by a set of
35:
4407:
4083:
4075:
4041:
4033:
85:
1535:{\displaystyle J=\int _{0}^{\infty }\left(x^{T}Qx+u^{T}Ru+2x^{T}Nu\right)dt}
96:
The LQR algorithm is essentially an automated way of finding an appropriate
3241:
the optimal control sequence minimizing the performance index is given by:
2371:
the optimal control sequence minimizing the performance index is given by:
65:
61:
4163:
2594:
is found iteratively backwards in time by the dynamic
Riccati equation:
3973:
4216:
4370:
Mathematical
Control Theory: Deterministic Finite Dimensional Systems
4208:
4315:
4249:
2557:{\displaystyle F_{k}=(R+B^{T}P_{k+1}B)^{-1}(B^{T}P_{k+1}A+N^{T})\,}
1545:
the feedback control law that minimizes the value of the cost is:
297:-dimensional real-valued vector) is the state of the system and
3417:
is the unique positive definite solution to the discrete time
3929:{\displaystyle C\mathbf {x} +D\mathbf {u} \leq \mathbf {e} .}
60:
LQR controllers possess inherent robustness with guaranteed
57:), a feedback controller whose equations are given below.
4399:
Mathematica function for Linear
Quadratic Regulator design
4299:"On Approximating Polynomial-Quadratic Regulator Problems"
4188:
Scokaert, Pierre O. M.; Rawlings, James B. (August 1998).
1817:{\displaystyle A^{T}P+PA-(PB+N)R^{-1}(B^{T}P+N^{T})+Q=0\,}
3886:
may be allowed. One common constraint is the linear one:
4015:
1155:{\displaystyle -{\dot {\lambda }}=Qx+Nu+A^{T}\lambda }
4394:
MATLAB function for Linear
Quadratic Regulator design
4297:
Borggaard, Jeff; Zietsman, Lizette (1 January 2021).
3895:
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341:
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3387:{\displaystyle F=(R+B^{T}PB)^{-1}(B^{T}PA+N^{T})\,}
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1375:For a continuous-time linear system described by:
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4099:Analysis and Control of Dynamic Economic Systems
3955:
3016:For a discrete-time linear system described by:
2046:For a discrete-time linear system described by:
2041:
1361:{\displaystyle \lambda (t_{1})=F(t_{1})x(t_{1})}
117:For a continuous-time linear system, defined on
68:, and they also are part of the solution to the
4345:
4016:Lehtomaki, N.; Sandell, N.; Athans, M. (1981).
4346:Kwakernaak, Huibert; Sivan, Raphael (1972).
4148:: CS1 maint: multiple names: authors list (
3991:Model predictive control § MPC vs. LQR
3984:
744:{\displaystyle K=R^{-1}(B^{T}P(t)+N^{T})\,}
4121:
4324:
4314:
4248:
4190:"Constrained Linear Quadratic Regulation"
3383:
3280:
3074:
3001:{\displaystyle Ax_{H_{p}-1}+Bu_{H_{p}-1}}
2553:
2417:
2104:
2027:
1918:
1813:
1665:
1570:
1224:{\displaystyle 0=Ru+N^{T}x+B^{T}\lambda }
945:
740:
636:
312:
237:
3939:The finite horizon version of this is a
1699:is found by solving the continuous time
1669:{\displaystyle K=R^{-1}(B^{T}P+N^{T})\,}
774:is found by solving the continuous time
4057:"Guaranteed Margins for LQG Regulators"
3078:{\displaystyle x_{k+1}=Ax_{k}+Bu_{k}\,}
2108:{\displaystyle x_{k+1}=Ax_{k}+Bu_{k}\,}
70:LQG (linear–quadratic–Gaussian) problem
14:
4406:
4364:
4240:2020 American Control Conference (ACC)
4197:IEEE Transactions on Automatic Control
4064:IEEE Transactions on Automatic Control
4022:IEEE Transactions on Automatic Control
3950:
79:
4164:"Ch. 8 - Linear Quadratic Regulators"
4054:
3088:with a performance index defined as:
2118:with a performance index defined as:
325:{\displaystyle u\in \mathbb {R} ^{m}}
250:{\displaystyle x\in \mathbb {R} ^{n}}
4350:(1st ed.). Wiley-Interscience.
4096:
3105:
2841:{\displaystyle P_{H_{p}}=Q_{H_{p}}}
2421:{\displaystyle u_{k}=-F_{k}x_{k}\,}
2194:
24:
4290:
4230:
3784:
3741:
3711:
3701:
3627:
3616:
3600:
3120:
1987:
1944:
1907:
1862:
1840:
1454:
1016:{\displaystyle P(t_{1})=F(t_{1}).}
25:
4425:
4387:
1428:with a cost function defined as:
1371:Infinite-horizon, continuous-time
4136:10.3182/20080706-5-KR-1001.01483
3919:
3911:
3900:
1418:{\displaystyle {\dot {x}}=Ax+Bu}
1078:{\displaystyle {\dot {x}}=Ax+Bu}
1026:The first order conditions for J
591:is the control cost matrix, and
212:{\displaystyle {\dot {x}}=Ax+Bu}
27:Linear optimal control technique
3846:In practice, not all values of
3781:
3284:{\displaystyle u_{k}=-Fx_{k}\,}
3012:Infinite-horizon, discrete-time
1984:
113:Finite-horizon, continuous-time
42:and the cost is described by a
4348:Linear Optimal Control Systems
4259:10.23919/ACC45564.2020.9147286
4181:
4115:
4090:
4048:
4009:
3968:Polynomial-quadratic regulator
3841:
3558:
3526:
3481:
3456:
3380:
3348:
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3310:
2772:
2733:
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2454:
1798:
1769:
1753:
1738:
1662:
1633:
1355:
1342:
1336:
1323:
1314:
1301:
1281:{\displaystyle x(t_{0})=x_{0}}
1262:
1249:
1007:
994:
985:
972:
942:
936:
909:
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409:
396:
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377:
371:
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130:
34:is concerned with operating a
13:
1:
4003:
3956:Quadratic-quadratic regulator
3581:This can be also written as:
2902:is driven to its final state
2042:Finite-horizon, discrete-time
1827:This can be also written as:
959:with the boundary condition:
776:Riccati differential equation
40:linear differential equations
4326:10.1016/j.ifacol.2021.06.090
551:is the initial cost matrix,
7:
3879:{\displaystyle x_{k},u_{k}}
107:
10:
4430:
4372:(2nd ed.). Springer.
4130:(2). Elsevier: 8773–8778.
3988:
3419:algebraic Riccati equation
1701:algebraic Riccati equation
571:is the state cost matrix,
51:linear–quadratic regulator
18:Linear-quadratic regulator
4097:Chow, Gregory C. (1986).
3979:Bartels-Stewart algorithm
3972:If the state equation is
2929:{\displaystyle x_{H_{p}}}
2875:{\displaystyle u_{H_{p}}}
98:state-feedback controller
4124:IFAC Proceedings Volumes
4076:10.1109/TAC.1978.1101812
4034:10.1109/TAC.1981.1102565
3998:model predictive control
3985:Model-predictive control
3945:model predictive control
2794:from terminal condition
4055:Doyle, John C. (1978).
1574:{\displaystyle u=-Kx\,}
640:{\displaystyle u=-Kx\,}
3930:
3880:
3828:
3720:
3571:
3411:
3388:
3285:
3232:
3124:
3079:
3002:
2930:
2896:
2882:is not defined, since
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2334:
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2109:
2032:
1923:
1818:
1693:
1670:
1598:
1575:
1536:
1419:
1362:
1282:
1235:4) Boundary conditions
1225:
1166:3) Stationary equation
1156:
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768:
745:
664:
641:
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4168:underactuated.mit.edu
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3389:
3286:
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3080:
3003:
2931:
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2843:
2786:
2589:
2587:{\displaystyle P_{k}}
2559:
2423:
2362:
2360:{\displaystyle H_{p}}
2335:
2193:
2110:
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164:
162:{\displaystyle t\in }
4243:. pp. 818–823.
4101:. Krieger Publ. Co.
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3962:Al'Brekht algorithm
3951:Related controllers
3941:convex optimization
3209:
3175:
3144:
2367:is the time horizon
2311:
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2155:
1458:
102:full state feedback
80:General description
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4303:IFAC-PapersOnLine
4268:978-1-5386-8266-1
3410:{\displaystyle P}
2895:{\displaystyle x}
1692:{\displaystyle P}
1597:{\displaystyle K}
1394:
1115:
1091:Co-state equation
1054:
1034:1) State equation
933:
767:{\displaystyle P}
663:{\displaystyle K}
604:{\displaystyle N}
584:{\displaystyle R}
564:{\displaystyle Q}
544:{\displaystyle F}
290:{\displaystyle n}
270:{\displaystyle x}
188:
16:(Redirected from
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4203:(8): 1163–1169.
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2102:
2087:
2086:
2071:
2070:
2037:
2035:
2034:
2029:
2026:
2025:
2016:
2015:
1991:
1990:
1983:
1982:
1973:
1972:
1948:
1947:
1928:
1926:
1925:
1920:
1911:
1910:
1898:
1897:
1888:
1887:
1866:
1865:
1850:
1849:
1844:
1843:
1823:
1821:
1820:
1815:
1797:
1796:
1781:
1780:
1768:
1767:
1722:
1721:
1698:
1696:
1695:
1690:
1675:
1673:
1672:
1667:
1661:
1660:
1645:
1644:
1632:
1631:
1603:
1601:
1600:
1595:
1580:
1578:
1577:
1572:
1541:
1539:
1538:
1533:
1525:
1521:
1514:
1513:
1492:
1491:
1473:
1472:
1457:
1452:
1424:
1422:
1421:
1416:
1396:
1395:
1387:
1367:
1365:
1364:
1359:
1354:
1353:
1335:
1334:
1313:
1312:
1287:
1285:
1284:
1279:
1277:
1276:
1261:
1260:
1230:
1228:
1227:
1222:
1217:
1216:
1201:
1200:
1161:
1159:
1158:
1153:
1148:
1147:
1117:
1116:
1108:
1084:
1082:
1081:
1076:
1056:
1055:
1047:
1022:
1020:
1019:
1014:
1006:
1005:
984:
983:
955:
953:
952:
947:
935:
934:
926:
908:
907:
883:
882:
870:
869:
797:
796:
773:
771:
770:
765:
750:
748:
747:
742:
736:
735:
711:
710:
698:
697:
669:
667:
666:
661:
646:
644:
643:
638:
610:
608:
607:
602:
590:
588:
587:
582:
570:
568:
567:
562:
550:
548:
547:
542:
526:
524:
523:
518:
510:
506:
499:
498:
477:
476:
458:
457:
442:
441:
440:
430:
429:
428:
408:
407:
389:
388:
370:
369:
357:
356:
331:
329:
328:
323:
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315:
296:
294:
293:
288:
276:
274:
273:
268:
256:
254:
253:
248:
246:
245:
240:
218:
216:
215:
210:
190:
189:
181:
169:, described by:
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155:
154:
142:
141:
21:
4429:
4428:
4424:
4423:
4422:
4420:
4419:
4418:
4414:Optimal control
4404:
4403:
4390:
4380:
4366:Sontag, Eduardo
4358:
4342:
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4281:
4279:
4269:
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2975:
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2903:
2887:
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2799:
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2762:
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2689:
2685:
2684:
2666:
2662:
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2623:
2608:
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2599:
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2574:
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2512:
2508:
2496:
2492:
2477:
2473:
2467:
2463:
2445:
2441:
2439:
2436:
2435:
2411:
2407:
2401:
2397:
2385:
2381:
2379:
2376:
2375:
2351:
2347:
2345:
2342:
2341:
2319:
2315:
2306:
2301:
2285:
2281:
2272:
2267:
2254:
2250:
2241:
2236:
2231:
2227:
2213:
2209:
2208:
2197:
2182:
2178:
2177:
2173:
2165:
2161:
2160:
2156:
2150:
2143:
2139:
2138:
2126:
2123:
2122:
2098:
2094:
2082:
2078:
2060:
2056:
2054:
2051:
2050:
2044:
2021:
2017:
2008:
2004:
1986:
1985:
1978:
1974:
1965:
1961:
1943:
1942:
1940:
1937:
1936:
1906:
1905:
1893:
1889:
1880:
1876:
1861:
1860:
1845:
1839:
1838:
1837:
1835:
1832:
1831:
1792:
1788:
1776:
1772:
1760:
1756:
1717:
1713:
1711:
1708:
1707:
1684:
1681:
1680:
1656:
1652:
1640:
1636:
1624:
1620:
1612:
1609:
1608:
1589:
1586:
1585:
1553:
1550:
1549:
1509:
1505:
1487:
1483:
1468:
1464:
1463:
1459:
1453:
1448:
1436:
1433:
1432:
1386:
1385:
1383:
1380:
1379:
1373:
1349:
1345:
1330:
1326:
1308:
1304:
1296:
1293:
1292:
1272:
1268:
1256:
1252:
1244:
1241:
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1212:
1208:
1196:
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1175:
1172:
1171:
1143:
1139:
1107:
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1101:
1098:
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1046:
1045:
1043:
1040:
1039:
1029:
1001:
997:
979:
975:
967:
964:
963:
925:
924:
903:
899:
878:
874:
862:
858:
792:
788:
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731:
727:
706:
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690:
686:
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384:
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180:
179:
177:
174:
173:
150:
146:
137:
133:
122:
119:
118:
115:
110:
82:
32:optimal control
28:
23:
22:
15:
12:
11:
5:
4427:
4417:
4416:
4402:
4401:
4396:
4389:
4388:External links
4386:
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4362:
4356:
4341:
4340:
4309:(9): 329–334.
4289:
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4229:
4180:
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4114:
4107:
4089:
4070:(4): 756–757.
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3989:Main article:
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2185:
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2059:
2043:
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2011:
2007:
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74:control theory
36:dynamic system
30:The theory of
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4379:0-387-98489-5
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3792:
3789:
3776:
3772:
3766:
3763:
3759:
3755:
3752:
3749:
3746:
3732:
3731:
3730:
3706:
3696:
3691:
3687:
3681:
3678:
3673:
3669:
3666:
3661:
3657:
3653:
3650:
3646:
3641:
3638:
3633:
3621:
3611:
3606:
3594:
3591:
3584:
3583:
3582:
3564:
3561:
3553:
3549:
3545:
3542:
3539:
3534:
3530:
3521:
3518:
3513:
3509:
3506:
3501:
3497:
3493:
3490:
3486:
3478:
3475:
3472:
3469:
3464:
3460:
3453:
3450:
3447:
3442:
3438:
3434:
3431:
3424:
3423:
3422:
3420:
3404:
3375:
3371:
3367:
3364:
3361:
3356:
3352:
3343:
3340:
3332:
3329:
3324:
3320:
3316:
3313:
3307:
3304:
3297:
3296:
3295:
3275:
3271:
3267:
3264:
3261:
3256:
3252:
3244:
3243:
3242:
3224:
3218:
3214:
3210:
3205:
3200:
3196:
3192:
3189:
3184:
3180:
3176:
3171:
3166:
3162:
3158:
3153:
3149:
3145:
3140:
3135:
3131:
3126:
3115:
3112:
3109:
3101:
3098:
3091:
3090:
3089:
3069:
3065:
3061:
3058:
3053:
3049:
3045:
3042:
3037:
3034:
3031:
3027:
3019:
3018:
3017:
3009:
2993:
2990:
2985:
2981:
2976:
2972:
2969:
2964:
2961:
2956:
2952:
2947:
2943:
2919:
2915:
2910:
2889:
2865:
2861:
2856:
2831:
2827:
2822:
2818:
2811:
2807:
2802:
2778:
2775:
2767:
2763:
2759:
2756:
2751:
2747:
2741:
2737:
2728:
2725:
2720:
2716:
2711:
2707:
2701:
2697:
2693:
2690:
2686:
2678:
2675:
2672:
2667:
2663:
2657:
2653:
2646:
2643:
2638:
2634:
2628:
2624:
2620:
2615:
2612:
2609:
2605:
2597:
2596:
2595:
2579:
2575:
2545:
2541:
2537:
2534:
2529:
2526:
2523:
2519:
2513:
2509:
2500:
2497:
2489:
2484:
2481:
2478:
2474:
2468:
2464:
2460:
2457:
2451:
2446:
2442:
2434:
2433:
2432:
2412:
2408:
2402:
2398:
2394:
2391:
2386:
2382:
2374:
2373:
2372:
2352:
2348:
2326:
2320:
2316:
2312:
2307:
2302:
2298:
2294:
2291:
2286:
2282:
2278:
2273:
2268:
2264:
2260:
2255:
2251:
2247:
2242:
2237:
2233:
2228:
2222:
2219:
2214:
2210:
2204:
2201:
2198:
2190:
2183:
2179:
2174:
2166:
2162:
2157:
2151:
2144:
2140:
2135:
2131:
2128:
2121:
2120:
2119:
2099:
2095:
2091:
2088:
2083:
2079:
2075:
2072:
2067:
2064:
2061:
2057:
2049:
2048:
2047:
2022:
2018:
2012:
2009:
2005:
2001:
1998:
1995:
1992:
1979:
1975:
1969:
1966:
1962:
1958:
1955:
1952:
1949:
1935:
1934:
1933:
1915:
1912:
1902:
1899:
1894:
1890:
1884:
1881:
1877:
1873:
1870:
1867:
1857:
1854:
1851:
1846:
1830:
1829:
1828:
1810:
1807:
1804:
1801:
1793:
1789:
1785:
1782:
1777:
1773:
1764:
1761:
1757:
1750:
1747:
1744:
1741:
1735:
1732:
1729:
1726:
1723:
1718:
1714:
1706:
1705:
1704:
1702:
1686:
1657:
1653:
1649:
1646:
1641:
1637:
1628:
1625:
1621:
1617:
1614:
1607:
1606:
1605:
1604:is given by:
1591:
1567:
1564:
1561:
1558:
1555:
1548:
1547:
1546:
1529:
1526:
1522:
1518:
1515:
1510:
1506:
1502:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1474:
1469:
1465:
1460:
1449:
1445:
1441:
1438:
1431:
1430:
1429:
1412:
1409:
1406:
1403:
1400:
1397:
1391:
1388:
1378:
1377:
1376:
1368:
1350:
1346:
1339:
1331:
1327:
1320:
1317:
1309:
1305:
1298:
1273:
1269:
1265:
1257:
1253:
1246:
1239:
1238:
1237:
1236:
1218:
1213:
1209:
1205:
1202:
1197:
1193:
1189:
1186:
1183:
1180:
1177:
1170:
1169:
1168:
1167:
1149:
1144:
1140:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1112:
1109:
1103:
1096:
1095:
1094:
1093:
1092:
1072:
1069:
1066:
1063:
1060:
1057:
1051:
1048:
1038:
1037:
1036:
1035:
1031:
1010:
1002:
998:
991:
988:
980:
976:
969:
962:
961:
960:
939:
930:
927:
921:
918:
915:
912:
904:
900:
896:
890:
884:
879:
875:
866:
863:
859:
852:
849:
846:
840:
834:
828:
825:
819:
813:
810:
804:
798:
793:
789:
781:
780:
779:
777:
761:
732:
728:
724:
718:
712:
707:
703:
694:
691:
687:
683:
680:
673:
672:
671:
670:is given by:
657:
633:
630:
627:
624:
621:
614:
613:
612:
598:
578:
558:
538:
514:
511:
507:
503:
500:
495:
491:
487:
484:
481:
478:
473:
469:
465:
462:
459:
454:
450:
445:
437:
433:
425:
421:
416:
412:
404:
400:
393:
385:
381:
374:
366:
362:
353:
349:
345:
342:
335:
334:
333:
317:
307:
304:
284:
264:
242:
232:
229:
206:
203:
200:
197:
194:
191:
185:
182:
172:
171:
170:
151:
147:
143:
138:
134:
127:
124:
105:
103:
99:
94:
90:
87:
86:cost function
77:
75:
71:
67:
63:
58:
56:
52:
48:
45:
41:
37:
33:
19:
4369:
4347:
4306:
4302:
4292:
4280:. Retrieved
4239:
4232:
4220:. Retrieved
4200:
4196:
4183:
4171:. Retrieved
4167:
4158:
4144:cite journal
4127:
4123:
4117:
4098:
4092:
4067:
4063:
4050:
4028:(1): 75–93.
4025:
4021:
4011:
3994:
3971:
3959:
3938:
3845:
3837:
3728:
3580:
3396:
3293:
3240:
3087:
3015:
2848:. Note that
2793:
2566:
2430:
2370:
2117:
2045:
1931:
1826:
1678:
1583:
1544:
1427:
1374:
1290:
1234:
1233:
1165:
1164:
1088:
1087:
1033:
1032:
1025:
958:
753:
649:
529:
221:
116:
95:
91:
83:
66:phase margin
59:
54:
50:
29:
3842:Constraints
4316:2009.11068
4250:1910.03396
4217:1793/10888
4004:References
3974:polynomial
257:(that is,
4335:221856517
4282:20 August
4277:203904925
4222:20 August
4173:20 August
4084:0018-9286
4042:0018-9286
3916:≤
3807:−
3796:−
3764:−
3753:−
3679:−
3622:−
3519:−
3454:−
3341:−
3265:−
3121:∞
3106:∑
2991:−
2962:−
2726:−
2647:−
2613:−
2498:−
2395:−
2220:−
2195:∑
2010:−
1999:−
1967:−
1956:−
1882:−
1868:−
1762:−
1736:−
1626:−
1562:−
1455:∞
1446:∫
1392:˙
1299:λ
1219:λ
1150:λ
1113:˙
1110:λ
1104:−
1052:˙
931:˙
922:−
864:−
829:−
692:−
628:−
417:∫
308:∈
233:∈
186:˙
128:∈
44:quadratic
4408:Category
4368:(1998).
3421:(DARE):
2340:, where
108:Versions
47:function
3294:where:
2431:where:
4376:
4354:
4333:
4275:
4265:
4105:
4082:
4040:
3729:with:
1584:where
650:where
531:where
277:is an
222:where
4331:S2CID
4311:arXiv
4273:S2CID
4245:arXiv
4193:(PDF)
4060:(PDF)
1932:with
1030:are:
4374:ISBN
4352:ISBN
4284:2022
4263:ISBN
4224:2022
4175:2022
4150:link
4103:ISBN
4080:ISSN
4038:ISSN
3397:and
2567:and
1679:and
1291:and
754:and
76:.
64:and
62:gain
4321:doi
4255:doi
4213:hdl
4205:doi
4132:doi
4072:doi
4030:doi
2936:by
1089:2)
1028:min
55:LQR
4410::
4329:.
4319:.
4307:54
4305:.
4301:.
4271:.
4261:.
4253:.
4211:.
4201:43
4199:.
4195:.
4166:.
4146:}}
4142:{{
4128:41
4126:.
4078:.
4068:23
4066:.
4062:.
4036:.
4026:26
4024:.
4020:.
3947:.
3008:.
1703::
778::
4382:.
4360:.
4337:.
4323::
4313::
4286:.
4257::
4247::
4226:.
4215::
4207::
4177:.
4152:)
4138:.
4134::
4111:.
4086:.
4074::
4044:.
4032::
3924:.
3920:e
3912:u
3908:D
3905:+
3901:x
3897:C
3872:k
3868:u
3864:,
3859:k
3855:x
3834:.
3820:T
3816:N
3810:1
3803:R
3799:N
3793:Q
3790:=
3785:Q
3777:T
3773:N
3767:1
3760:R
3756:B
3750:A
3747:=
3742:A
3712:Q
3707:+
3702:A
3697:P
3692:T
3688:B
3682:1
3674:)
3670:B
3667:P
3662:T
3658:B
3654:+
3651:R
3647:(
3642:B
3639:P
3634:T
3628:A
3617:A
3612:P
3607:T
3601:A
3595:=
3592:P
3577:.
3565:Q
3562:+
3559:)
3554:T
3550:N
3546:+
3543:A
3540:P
3535:T
3531:B
3527:(
3522:1
3514:)
3510:B
3507:P
3502:T
3498:B
3494:+
3491:R
3487:(
3482:)
3479:N
3476:+
3473:B
3470:P
3465:T
3461:A
3457:(
3451:A
3448:P
3443:T
3439:A
3435:=
3432:P
3405:P
3381:)
3376:T
3372:N
3368:+
3365:A
3362:P
3357:T
3353:B
3349:(
3344:1
3337:)
3333:B
3330:P
3325:T
3321:B
3317:+
3314:R
3311:(
3308:=
3305:F
3276:k
3272:x
3268:F
3262:=
3257:k
3253:u
3225:)
3219:k
3215:u
3211:N
3206:T
3201:k
3197:x
3193:2
3190:+
3185:k
3181:u
3177:R
3172:T
3167:k
3163:u
3159:+
3154:k
3150:x
3146:Q
3141:T
3136:k
3132:x
3127:(
3116:0
3113:=
3110:k
3102:=
3099:J
3070:k
3066:u
3062:B
3059:+
3054:k
3050:x
3046:A
3043:=
3038:1
3035:+
3032:k
3028:x
2994:1
2986:p
2982:H
2977:u
2973:B
2970:+
2965:1
2957:p
2953:H
2948:x
2944:A
2920:p
2916:H
2911:x
2890:x
2866:p
2862:H
2857:u
2832:p
2828:H
2823:Q
2819:=
2812:p
2808:H
2803:P
2779:Q
2776:+
2773:)
2768:T
2764:N
2760:+
2757:A
2752:k
2748:P
2742:T
2738:B
2734:(
2729:1
2721:)
2717:B
2712:k
2708:P
2702:T
2698:B
2694:+
2691:R
2687:(
2682:)
2679:N
2676:+
2673:B
2668:k
2664:P
2658:T
2654:A
2650:(
2644:A
2639:k
2635:P
2629:T
2625:A
2621:=
2616:1
2610:k
2606:P
2580:k
2576:P
2551:)
2546:T
2542:N
2538:+
2535:A
2530:1
2527:+
2524:k
2520:P
2514:T
2510:B
2506:(
2501:1
2494:)
2490:B
2485:1
2482:+
2479:k
2475:P
2469:T
2465:B
2461:+
2458:R
2455:(
2452:=
2447:k
2443:F
2413:k
2409:x
2403:k
2399:F
2392:=
2387:k
2383:u
2353:p
2349:H
2327:)
2321:k
2317:u
2313:N
2308:T
2303:k
2299:x
2295:2
2292:+
2287:k
2283:u
2279:R
2274:T
2269:k
2265:u
2261:+
2256:k
2252:x
2248:Q
2243:T
2238:k
2234:x
2229:(
2223:1
2215:p
2211:H
2205:0
2202:=
2199:k
2191:+
2184:p
2180:H
2175:x
2167:p
2163:H
2158:Q
2152:T
2145:p
2141:H
2136:x
2132:=
2129:J
2100:k
2096:u
2092:B
2089:+
2084:k
2080:x
2076:A
2073:=
2068:1
2065:+
2062:k
2058:x
2023:T
2019:N
2013:1
2006:R
2002:N
1996:Q
1993:=
1988:Q
1980:T
1976:N
1970:1
1963:R
1959:B
1953:A
1950:=
1945:A
1916:0
1913:=
1908:Q
1903:+
1900:P
1895:T
1891:B
1885:1
1878:R
1874:B
1871:P
1863:A
1858:P
1855:+
1852:P
1847:T
1841:A
1811:0
1808:=
1805:Q
1802:+
1799:)
1794:T
1790:N
1786:+
1783:P
1778:T
1774:B
1770:(
1765:1
1758:R
1754:)
1751:N
1748:+
1745:B
1742:P
1739:(
1733:A
1730:P
1727:+
1724:P
1719:T
1715:A
1687:P
1663:)
1658:T
1654:N
1650:+
1647:P
1642:T
1638:B
1634:(
1629:1
1622:R
1618:=
1615:K
1592:K
1568:x
1565:K
1559:=
1556:u
1530:t
1527:d
1523:)
1519:u
1516:N
1511:T
1507:x
1503:2
1500:+
1497:u
1494:R
1489:T
1485:u
1481:+
1478:x
1475:Q
1470:T
1466:x
1461:(
1450:0
1442:=
1439:J
1413:u
1410:B
1407:+
1404:x
1401:A
1398:=
1389:x
1356:)
1351:1
1347:t
1343:(
1340:x
1337:)
1332:1
1328:t
1324:(
1321:F
1318:=
1315:)
1310:1
1306:t
1302:(
1274:0
1270:x
1266:=
1263:)
1258:0
1254:t
1250:(
1247:x
1214:T
1210:B
1206:+
1203:x
1198:T
1194:N
1190:+
1187:u
1184:R
1181:=
1178:0
1145:T
1141:A
1137:+
1134:u
1131:N
1128:+
1125:x
1122:Q
1119:=
1073:u
1070:B
1067:+
1064:x
1061:A
1058:=
1049:x
1011:.
1008:)
1003:1
999:t
995:(
992:F
989:=
986:)
981:1
977:t
973:(
970:P
943:)
940:t
937:(
928:P
919:=
916:Q
913:+
910:)
905:T
901:N
897:+
894:)
891:t
888:(
885:P
880:T
876:B
872:(
867:1
860:R
856:)
853:N
850:+
847:B
844:)
841:t
838:(
835:P
832:(
826:A
823:)
820:t
817:(
814:P
811:+
808:)
805:t
802:(
799:P
794:T
790:A
762:P
738:)
733:T
729:N
725:+
722:)
719:t
716:(
713:P
708:T
704:B
700:(
695:1
688:R
684:=
681:K
658:K
634:x
631:K
625:=
622:u
599:N
579:R
559:Q
539:F
515:t
512:d
508:)
504:u
501:N
496:T
492:x
488:2
485:+
482:u
479:R
474:T
470:u
466:+
463:x
460:Q
455:T
451:x
446:(
438:1
434:t
426:0
422:t
413:+
410:)
405:1
401:t
397:(
394:x
391:)
386:1
382:t
378:(
375:F
372:)
367:1
363:t
359:(
354:T
350:x
346:=
343:J
318:m
313:R
305:u
285:n
265:x
243:n
238:R
230:x
207:u
204:B
201:+
198:x
195:A
192:=
183:x
157:]
152:1
148:t
144:,
139:0
135:t
131:[
125:t
53:(
20:)
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