June Engineering Department, Stanford University. System Analysis and Synthesis. Linear Matrix Inequalities. Stephen Boyd (E.

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1 Stephen Boyd (E. Feron :::) System Analysis and Synthesis Control Linear Matrix Inequalities via Engineering Department, Stanford University Electrical June 1993 ACC, 1

2 linear matrix inequalities (LMIs) LMI problems can be solved extremely eciently Basic idea Many problems in control can be cast in terms of 2

3 Fdist F act 2 () 3 () 1 () C(s) 4 () Question: how large can RMS(z) be? Example problem ignoring nonlinearities, RMS(z) 4:1 Monte Carlo / simulation suggests RMS(z) 5:6 orso LMI-based analysis guarantees RMS(z) 7:0 3 z

4 What is an LMI? Examples from control Solving LMI problems History of LMIs Conclusions & challenges Outline 4

5 F (x) =F0+x1F1++xmFm 0 i = F T i are given; x 2 R m is the variable F a set of polynomial inequalities in x LMI has form: Linear matrix inequality multiple LMIs can be combined into one Optimization problems over LMIs: Feasibility: nd x s.t. F (x) 0, or show that none exists Linear objective: minimize c T x subject to F (x) 0 5

6 LMI problems LMI Problems appear dicult (nonlinear, nondierentiable) have no \analytic solution" but: LMI problems are readily solved by computer Why? :::convexity 6

7 What \readily solved" means There are algorithms for LMI problems that are globally convergent (good initial guess not needed) compute the global optimum within a given tolerance, or proof of infeasibility nd computational eort required is small and grows slowly with size problem :::more details later 7

8 P > 0; A T P + PA < 0 no analytic solution, but readily solved LMI problems: two examples Symmetric matrix P is the variable: Lyapunov inequality: analytic solution (1890) Two Lyapunov inequalities: P > 0; A T P + PA < 0; ~ A T P +P ~ A<0 8

9 Examples from control What is an LMI? Outline LQR synthesis {multimodel Popov analysis { Solving LMI problems History of LMIs Conclusions & challenges 9

10 Question: is it stable? Note: implies stability of nonlinear time-varying system if for all 2 Co fa 1;:::;A L g (\global linearization") A time-varying system Consider time-varying linear system _x(t) =A(t)x(t); A(t)2Co fa 1 ;:::;A L g _x = f(x(t);t) 10

11 an LMI in P ; no analytic solution but readily solved looks simple, but more powerful than many well known (multivariable circle criteria, :::) methods Quadratic Lyapunov function search there a quadratic Lyapunov function V (z) = z T Pz that Is stability? proves Equivalent to: is there P s.t. P > 0; A T i P + PA i < 0; i =1;:::;L? 11

12 performance specications, actuator eort, RMS gain, LQR cost e.g., synthesis of linear state feedback Some extensions synthesis of gain-scheduled controllers In these cases we get LMI problems that look more complicated also have no analytic solutions are just as readily solved 12

13 y 00 (0) = y 0 (0) = y(0) = 1 u(t) 2 u(t) 2 + y(t) 2 dt We'll explore linear state feedback ::: LQR cost: Nominal nom = Z 1 J Multimodel LQR example y a 1 (t)y 00 + a 2 (t)y 0 + a 3 (t)y = a 4 (t)u Worst-case LQR cost: 0 wc = max J a i (t) 2 1=2 + y(t) 2 dt with a i (t) = Z 0

14 p Jnom p Jwc lqr 1:00 1 K 1:70 4:00 Kmmlqr K mmlqr performance guaranteed with nonlinear system Two state feedback gains: LQR vs. multi-model LQR K lqr minimizes J nom solution of Riccati equation) (requires K mmlqr minimizes upper bound on J wc solution of an LMI problem) (requires K mmlqr has traded o nominal performance for improved robustness 14

15 Examples from control What is an LMI? Outline multimodel LQR synthesis { analysis {Popov Solving LMI problems History of LMIs Conclusions & challenges 15

16 0 1 (a)da + + r Z y r hence readily solved Search for Lyap fct with Popov terms Consider system with scalar nonlinearities 1 ();:::; r () function Popov Lyapunov with terms: T Z y1 Px+ 1 =x (x) V seek V that proves performance specication We LQR cost, RMS gain, :::) (stability, reduce to LMI problem in several ways can Pyatnitsky, Hall) (Skorodinsky, 16 0 r(a)da

17 F dist F act 2 () 3 () 1 () C(s) 4 () Question: how large can RMS(z) be? Popov analysis example actuator, sensor, springs are 10% nonlinear: i(a) 0:9 a 1:1 C is simple LQG controller RMS(F dist ) 1 17 z

18 Linearized vs. Popov performance analysis For nominal linear system: compute H1 norm to nd RMS(z) 4:13 For nonlinear system, solve LMI problem to nd RMS(z) 7:00 Extensive simulation only allows guess of performance degradation This analysis guarantees a maximum performance degradation 18

19 i.e., many other problems Other problems that reduce to LMIs optimal lter / controller realization linear controller design via Q-parametrization multi-criterion LQG inverse problem of optimal control synthesis of multipliers for analysis of systems with constant parameters unknown analysis & design for stochastic systems 19

20 Some problems that (probably) won't reduce to LMIs: output feedback synthesis state feedback synthesis for Popov, multipliers 20

21 controller comes with certicate of performance repeat f The V {K iteration x V ; design K (synthesis) x K; nd good V (analysis) g handles constraints on controller structure each step is LMI problem controller may not be globally optimal, but appears to work in practice (Safonov, Skelton) 21

22 Solving LMI problems What is an LMI? Examples from control History of LMIs Conclusions & challenges Outline 22

23 new interior-point methods (1988) extremely ecient { can exploit problem structure (Lyapunov, etc) In theory: Solving LMI problems fundamental complexity is low (polynomial) In practice: classical optimization methods do not work ellipsoid algorithm (1970s) is robust but slow 23

24 Problem: minimize linear fct of P subject to: (Ai; Qi given; P 2 R nn is the variable) independent Lyapunov equations solving solving coupled Lyapunov inequalities Example: multiple Lyapunov inequalities (L. Vandenberghe) A T i P + PA i+q i 0; i =1;:::;L number of variables: n(n +1)=2=O(n 2 ) cost of solving L independent Lyapunov equations: O(Ln 3 ) new primal-dual method solves above problem in O(L 1:2 n 4 ) ratio: O(L 0:2 n) 24

25 History of LMIs What is an LMI? Examples from control Solving LMI problems Conclusions & challenges Outline 25

26 1890s: Lyapunov's theory 1940s: Lur'e, Postnikov apply to real problems Early 1960s: Yakubovich recognizes important role of LMIs Late 1960s: graphical, ARE methods for solving some LMIs LMIs in control: key events 1980s: LMIs recognized as convex programs Early Pyatnitsky) (Skorodinsky, 1980s: Interior-point algorithms for LMIs Late & Nemirovsky) (Nesterov 26

27 Conclusions & challenges What is an LMI? Examples from control Solving LMI problems History of LMIs Outline 27

28 Conclusion and useful control problems can be cast as LMI Interesting hence solved problems, robust performance analysis some robust synthesis many other problems 28

29 LMI-based analysis / design will complement, not replace highly nonlinear feedback synthesis (e.g., geometric :::) simulation, Monte Carlo analysis 29

30 What constitutes a solution? reduction to an Algebraic Riccati equation? reduction to an LMI problem? 30

31 relation between system theory duality and optimization duality? convex output feedback? Challenges: theory what control problems can be cast as LMIs? 31

32 user interface { automated translation, block diagram,! LMI { { ecient specialized codes Try LMI-based analysis / synthesis on real problems are worst-case performance bounds too conservative? { is multi-model robust synthesis useful? { { does V {K iteration work in practice? Challenges: applications Develop useful computer-aided robustness analysis / tools synthesis 32

33 > 0 ; P > 0; L = diag( 1 ;:::; 4 ) 0; T = diag( 1 ;:::; 4 )0 objective: LMI problem from Popov example,a T P, PA,C T C,P ~ B +A T ~ C T L+ ~ C T T,PB ~ T P +L ~ CA + T ~ C L ~ C ~ B + ~ B T ~ T C L +2T ~ D L ~ CB, T B B T ~ T C L I P,B variables: looks ugly (to people) impossible to solve analytically or by hand easily solved numerically =) should be hidden from user 33

34 (Some) recent work on LMIs Yakubovich, Megretsky: S-procedure Geromel, Peres & Bernussou: robust control Khargonekar, Rotea: mixed H2/H1 Barmish, Hollot: quadratic stabilizability Doyle, Packard & associates: LMIs, LFTs, Bernstein & Haddad, Hall & How: multipliers, Popov Safonov: Km-synthesis Skelton et al.: covariance control Boyd, El Ghaoui, Feron & Balakrishnan: monograph Nesterov & Nemirovsky, Vandenberghe & Overton & Haeberly, Fan: algorithms Boyd, Gahinet: LMIlab 34

35 warning: advertisement Boyd, El Ghaoui, Feron & Balakrishnan, Linear Matrix Inequalities in System and Control Theory monograph in preparation, late 1993 (?) rough draft via ftp: ACC paper: end of Proceedings vol 2 anonymous ftp into isl.stanford.edu & 35

Optimization in. Stephen Boyd. 3rd SIAM Conf. Control & Applications. and Control Theory. System. Convex

Optimization in. Stephen Boyd. 3rd SIAM Conf. Control & Applications. and Control Theory. System. Convex Optimization in Convex and Control Theory System Stephen Boyd Engineering Department Electrical University Stanford 3rd SIAM Conf. Control & Applications 1 Basic idea Many problems arising in system and