From Convex Optimization to Linear Matrix Inequalities

Dep. of Information Engineering University of Pisa (Italy) From Convex Optimization to Linear Matrix Inequalities eng. Sergio Grammatico grammatico.sergio@gmail.com Class of Identification of Uncertain Systems 2012/ 13 held by prof. Andrea Caiti S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 1 / 30

Outline 1 Introduction 2 Semi-Definite Programming From Convex Optimization to Semi-Definite Programming 3 Linear Matrix Inequalities Definition Brief Historical Perspective First simple problems 4 Software SeDuMi and CVX Examples 5 Conclusion S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 2 / 30

Convex Optimization and Linear Matrix Inequalities (LMIs) Thanks to Stephen Boyd, Carsten Scherer et al. This presentation is mainly based on S. Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2004. S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), 1994. S. Boyd, Lecture notes in Convex Optimization, Standford University. S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 4 / 30

Convex Optimization General problem Convex Optimization Problem min x R n sub. to f(x) g i (x) 0, i = 1, 2,..., m Ax = b (1) f, g 1,..., g m convex functions, A R p n, b R p. Global Optimality Any local optimal point of a convex problem is globally optimal S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 6 / 30

Special Case Linear Programming (LP) Linear Programming min x R n c x sub. to Gx h, Ax = b (2) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 7 / 30

Special Case Quadratic Programming (QP) Quadratic Programming min x R n x P x + q x sub. to Gx h, Ax = b (3) P positive semi-definite P 0 (P = P, eig(p ) 0). S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 8 / 30

Another Special Case Semi-Definite Programming (SDP) Semi-Definite Programming min x R n c x sub. to F 0 + x 1 F 1 + x 2 F 2 +... + x n F n 0, Ax = b (4) F i = F i R n n S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 9 / 30

Linear Matrix Inequalities Definition Standard LMI F (x) := F 0 + n x i F i 0 (5) i=1 The inequality constraint of an SDP is an LMI. Multiple LMIs as a unique LMI: [ ] [ ] [ ˆF0 0 ˆF1 0 ˆFn 0 + x 1 +... + x n 0 F0 0 F1 0 Fn ] 0 (6) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 11 / 30

Linear Matrix Inequalities Feasibility and Optimization Feasibility LMI Problem Find x R n such that F (x) 0 Strict Feasibility LMI Problem Find x R n and α R >0 such that F (x) αi n LMI Optimization Problem Find x R n such that F (x) 0 c x is minimized S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 12 / 30

Linear Matrix Inequalities A trick for NonLinear MIs Schur Complement Given Q(x) = Q(x), R(x) 0, S(x) affinely dependent on x, the NonLinear MI Q(x) S(x)R(x) 1 S(x) 0 (7) is equivalent to the LMI [ Q(x) S(x) S(x) R(x) ] 0. (8) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 13 / 30

Brief Historical Perspective Back to Lyapunov 1 First LMI in Control Theory (A.M. Lyapunov) Lyapunov Inequality (1890) ẋ = Ax A.S. P 0 : A P + P A 0 (9) Lyapunov equality: A P + P A = Q, P, Q 0 Matrices as variables 2 Further contributions Lur e (Stability under Saturations via LMIs); Kalman, Yakubovich, Popov (Positive Real Lemma);... S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 14 / 30

Brief Historical Perspective Riccati revisited 3 On the LQ Optimal Control (J.C. Willems) Algebraic Riccati (LM)Inequality (1971) A P + P A + Q ( P B + C ) R 1 ( P B + C ) 0 (10) is equivalent (by Schur Complement) to [ A P + P A + Q P B + C ( P B + C ) R ] 0. (11) 4 Interior-point algorithms for LMIs (Y.E. Nesterov and A. Nemirovski) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 15 / 30

Brief Historical Perspective Today LMIs are used for: 1 Stability analysis of uncertain systems 2 Robust control design for uncertain systems 3 Robust optimization 4 System identification 5... S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 16 / 30

Simple LMI problems Stability Analysis of Uncertain Linear Systems Linear Differential Inclusions (Time-varying Uncertain Linear Systems) ẋ conv {A i x i [1, s]} (12) Quadratic Stability (Sufficient Condition) P 0 : A i P + P A i 0 i = (12) A.S. (13) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 17 / 30

Simple LMI problems Robust Linear State-Feedback Control design (1) Controlled Linear Differential Inclusions ẋ conv {A i x + B i u i [1, s]}, (14) 1 u(x) = Kx 2 ẋ conv {(A i + B i K)x} 3 NonLinear MI in P, K: (A i + B i K) P + P (A i + B i K) 0 i (15) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 18 / 30

Simple LMI problems Robust Linear State-Feedback Control design (2) Change of Variables for the LMI design of K: S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 19 / 30

Simple LMI problems Robust Linear State-Feedback Control design (2) Change of Variables for the LMI design of K: 4 Multiply (15) both on the left and on the right by P 1 S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 19 / 30

Simple LMI problems Robust Linear State-Feedback Control design (2) Change of Variables for the LMI design of K: 4 Multiply (15) both on the left and on the right by P 1 5 Q := P 1 6 Y := KQ S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 19 / 30

Simple LMI problems Robust Linear State-Feedback Control design (2) Change of Variables for the LMI design of K: 4 Multiply (15) both on the left and on the right by P 1 5 Q := P 1 6 Y := KQ 7 LMI in Y, Q: A i Q + QA i + B i Y + Y B i 0 i (16) 8 P := Q 1, K := Y Q 1 S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 19 / 30

Solver and Matlab interface SeDuMi and CVX Solver: SeDuMi (by J.F. Sturm) http://sedumi.ie.lehigh.edu/ Matlab interface: CVX (by S. Boyd) http://cvxr.com/cvx/ CVXOPT in Python CVXGEN for C-code generation S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 21 / 30

Example 1 Variable Mass-Spring System as Uncertain Linear System (1) mẍ = kx bẋ k [k, k] k kx bx m 0 ˆx ξ 1 := x, ξ 2 := ẋ, w := k S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 22 / 30

Example 1 Variable Mass-Spring System as Uncertain Linear System (2) Variable Mass-Spring System as Uncertain Linear System [ ] 0 1 ξ = ξ (17) w 1 w [0.1, 1] [ 0 1 A 1 = 0.1 1 Robust LMI Analysis ] [, A 2 = 0 1 1 1 A i P + P A i 0, i = 1, 2 ] S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 23 / 30

Example 1 Variable Mass-Spring System as Uncertain Linear System (3) Matlab code using CVX A{1} = [0 1; -0.1-1]; A{2} = [0 1; -1-1]; n = 2; s = 2; cvx begin variable P(n,n) symmetric variable a maximize(a) P == semidefinite(n); a >= 0; for i=1:s A{i} *P + P*A{i} + a*eye(n) == -semidefinite(n); end cvx end S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 24 / 30

Example 2 Quadratic Stabilizability is only Sufficient for Uncertain Linear Systems Counterexample [ ẋ = 0 1 1 0 ] [ w x + 1 ] u (18) w [w, w] [ w A 1 = A 2 = A, B 1 = 1 ] [ w, B 2 = 1 ] Conservatism of Quadratics (18) quadratic stabilizable (searching P, K) only if w [ 1, 1] S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 25 / 30

Example 2 Quadratic Stabilizability is only Sufficient for Uncertain Linear Systems Counterexample [ ẋ = 0 1 1 0 ] [ w x + 1 ] u (18) w [w, w] [ w A 1 = A 2 = A, B 1 = 1 ] [ w, B 2 = 1 ] Conservatism of Quadratics (18) quadratic stabilizable (searching P, K) only if w [ 1, 1] (18) stabilizable w R S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 25 / 30

Example 3 Nonlinear Inverted Pendulum as Uncertain Linear System (1) I θ = mgl sin(θ) + τ 2, l I mg x 1 := θ, x 2 := θ, u = τ = ẋ 2 = mgl I sin(x 1 ) x 1 + 1 x 1 I u S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 26 / 30

Example 3 Nonlinear Inverted Pendulum as Uncertain Linear System (2) Inverted Pendulum as Uncertain Linear System [ ] [ 0 1 0 ẋ = x + aw(x) 0 b w(0) = 1 x 1 [ π, π] w(x 4 4 1) [0.9, 1] [ ] [ ] 0 1 0 1 A 1 =, A 0.9 a 0 2 = 1 a 0 [ ] 0 B 1 = B 2 = b2 ] u (19) S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 27 / 30

Example 3 Nonlinear Inverted Pendulum as Uncertain Linear System (3) Robust LMI Control design from (16) A i Q + QA i + B i Y + Y B i 0, P = Q 1, K = Y Q 1 Matlab code using CVX cvx begin variable Q(n,n) symmetric variable Y(m,n) maximize(log det(q)) Q == semidefinite(n); for i=1:s A{i}*Q + Q*A{i} + B{i}*Y + Y *B{i} == -semidefinite(n); end cvx end P = Q^(-1); K = Y*P S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 28 / 30

Conclusion LMIs in Systems and Control Theory 1 LMIs Robust Analysis and Control design 2 LMIs Uncertain Linear Systems S. Grammatico (DEI UNIPI) Introduction to LMIs Identification of Uncertain Systems 30 / 30