Semidefinite Programming Duality and Linear Time-invariant Systems

Semidefinite Programming Duality and Linear Time-invariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAAS-CNRS, Toulouse, France

SDP DUALITY AND LTI SYSTEMS 1 Basic ideas Many control constraints yield LMIs, many control problems are SDPs

SDP DUALITY AND LTI SYSTEMS 1 Basic ideas Many control constraints yield LMIs, many control problems are SDPs LMIs are convex constraints, SDPs are convex optimization problems From duality theory in convex optimization: Theorem of alternatives for LMIs SDP duality Explore implication of convex duality theory on underlying control problem: New (often simpler) proofs for classical results Some new results

SDP DUALITY AND LTI SYSTEMS 2 LMIs and Semidefinite Programming V is a finite-dimensional Hilbert space, S is a subspace of Hermitian matrices, F : V S is a linear mapping, F 0 S Inequality F(x) + F 0 0 is an LMI SDP is an optimization of the form: minimize: c, x V subject to: F(x) + F 0 0

SDP DUALITY AND LTI SYSTEMS 3 A theorem of alternatives for LMIs Exactly one of the following statements is true 1. F(x) + F 0 > 0 is feasible 2. There exists Z S s.t. Z 0, F adj (Z) = 0, F 0, Z S 0 (F adj ( ) denotes adjoint map, i.e., x V, Z S, F(x), Z S = x, F adj (Z) V ) Variants available for nonstrict inequalities such as F(x) + F 0 0 and F(x) + F 0 0, and with additional linear equality constraints F(x) = 0 Typically get weak alternatives, need additional conditions (constraint qualifications) to make them strong

SDP DUALITY AND LTI SYSTEMS 4 Proof of theorem of alternatives LMI F(x) + F 0 > 0 infeasible iff F 0 C = {C F(x) + C > 0 for some x V}

SDP DUALITY AND LTI SYSTEMS 4 Proof of theorem of alternatives LMI F(x) + F 0 > 0 infeasible iff F 0 C = {C F(x) + C > 0 for some x V} C is open, nonempty and convex, so there exists hyperplane strictly separting F 0 and C: Z 0 s.t. F 0, Z S < C, Z S for all C C Z 0 s.t. F 0, Z S < F(x) + X, Z S for all x V, X > 0

SDP DUALITY AND LTI SYSTEMS 5 Application: A Lyapunov inequality LMI A P + P A < 0 is feasible, or There exists Z s.t. Z 0, AZ + ZA = 0 Factoring Z = UU, can show AU = US, S has pure imaginary eigenvalues Thus: LMI A P + P A < 0 is infeasible if and only if A has a pure imaginary eigenvalue

SDP DUALITY AND LTI SYSTEMS 6 Other results P > 0, A P + P A < 0 is infeasible iff λ i (A) 0 for some i A P + P A 0 is infeasible iff A is similar to a purely imaginary diagonal matrix A P + P A 0, P 0 is infeasible iff λ i (A) 0 for all i A P + P A 0, P B = 0 is infeasible iff all uncontrollable modes of (A, B) are nondefective and correspond to imaginary eigenvalues P 0, A P + P A 0, P B = 0 is infeasible iff all uncontrollable modes of (A, B) correspond to eigenvalues with positive real part P 0, A P + P A 0, P B = 0 is infeasible iff (A, B) is controllable

SDP DUALITY AND LTI SYSTEMS 7 Frequency-domain inequalities: The KYP Lemma Inequalities of the form [ (jωi A) 1 B I ] M [ (jωi A) 1 B I ] > 0 are commonly encountered in systems and control: Linear system analysis and design Digital filter design Robust control analysis Examples of constraints: H(jω) < 1 (small gain), RH(jω) > 0 (passivity), H(jω) + H(jω) + H(jω) H(jω) < 1 (mixed constraints)

SDP DUALITY AND LTI SYSTEMS 8 The Kalman-Yakubovich-Popov Lemma FDI holds for all ω iff LMI is feasible [ (jωi A) 1 B I ] M [ A P + P A P B B P 0 [ (jωi A) 1 B ] I M < 0 ] > 0 Infinite-dimensional constraint reduced to finite-dimensional constraint No sampling in frequency required

SDP DUALITY AND LTI SYSTEMS 9 Control-theoretic proof of KYP Lemma Suppose LMI [ A P + P A P B B P 0 is feasible ] M < 0 Then [ ] (jωi A) 0 < 1 ( [ B A M P + P A P B I B P 0 = [ B ( jωi A ) 1 I ] [ ] (jωi A) M 1 B I ]) [ (jωi A) 1 B I ]

SDP DUALITY AND LTI SYSTEMS 10 New KYP lemma proof More general version of the KYP Lemma: Suppose M 22 > 0 [ A P + P A P B B P 0 is feasible iff (jωi A)u = Bv, (u, v) 0 = [ u ] M < 0, v ] M [ u v ] > 0 A can have imaginary eigenvalues If A has no imaginary eigenvalues, recover classical version

SDP DUALITY AND LTI SYSTEMS 11 Duality-based KYP Lemma proof Infeasibility of equivalent to existence of Z s.t. [ A P + P A P B B P 0 ] M < 0 Z = [ ] Z11 Z 12 Z12 Z 22 0, Z 11 A + AZ 11 + Z 12 B + BZ 12 = 0, TrZM 0 Must have Z 11 0. Hence, factor Z as [ ] Z11 Z 12 Z12 Z 22 = [ U 0 V ˆV ] [ U V 0 ˆV ], where U has full rank

SDP DUALITY AND LTI SYSTEMS 12 Can show ( [ US AU = BV, Tr U V ] [ ]) U M 0, V with S + S = 0 Take Schur decomposition of S: S = m i=1 jω iq i q i, with i q iq i = I Then q k [ U V ] M [ U V ] q k 0 for some k Define u = Uq k, v = V q k. Then [ u v ] [ ] u M 0 v and (jωi A)u = Bv

SDP DUALITY AND LTI SYSTEMS 13 Outline Theorem of alternatives for LMIs, and their applications SDP duality, and its application

SDP DUALITY AND LTI SYSTEMS 14 Primal and dual SDPs Primal SDP: minimize: c, x V subject to: F(x) + F 0 0 Dual SDP maximize F 0, Z S subject to F adj (Z) = c, Z 0 If Z is dual feasible, then TrF 0 Z p If x is primal feasible, then c T x d Under mild conditions, p = d At optimum, (F(x opt ) + F 0 ) Z opt = 0

SDP DUALITY AND LTI SYSTEMS 15 Application of duality: Bounds on H norm Stable LTI system ẋ = Ax + Bu, x(0) = 0, y = Cx Transfer function H(s) = C(sI A) 1 B H norm of H defined as H = sup σ max (H(s)) Rs>0 H 2 equals maximum energy gain H 2 = max u y T y ut u

SDP DUALITY AND LTI SYSTEMS 16 H computation as an SDP minimize: subject to: β[ A P + P A + C C P B B P βi ] 0 ( H 2 = β opt ) Dual problem maximize: TrCZ 11 C subject to: Z 11 A + AZ 11 + Z 12 B + BZ12 = 0 [ Z11 Z12 ] Z 12 0, Z 22 TrZ 22 = 1

SDP DUALITY AND LTI SYSTEMS 17 Control-theoretic interpretation of dual problem Suppose u(t) any input that steers state from x(t 1 ) = 0 to x(t 2 ) = 0, for some T 1, T 2. Let y(t) be the corresponding output Define Z 11 = T2 T 1 x(t)x(t) dt, Z 12 = T2 T 1 x(t)u(t) dt, Z 22 = T2 T 1 u(t)u(t) dt Can show Z 11, Z 12 and Z 22 are dual feasible TrZ 22 = T 2 T 1 u(t) u(t) dt = 1 normalizes input energy Dual objective is corresponding output energy, gives lower bound: TrCZ 11 C = T2 T 1 y(t) y(t) dt

SDP DUALITY AND LTI SYSTEMS 18 Recall primal problem: New upper bounds on H minimize: subject to: β[ A P + P A + C C P B B P βi ] 0 A primal feasible point is P = 2W o, β = 4λ max (W o BB W o, C C) where W o is observability Gramian, obtained by solving W o A + A W o + C C = 0 Thus, new upper bound on H is given by 2 λ max (W o BB W o, C C)

SDP DUALITY AND LTI SYSTEMS 19 Recall dual problem New lower bounds on H maximize: TrCZ 11 C subject to: Z 11 A + AZ 11 + Z 12 B + BZ12 = 0 [ ] Z11 Z 12 Z12 0, TrZ Z 22 = 1 22 A dual feasible point is Z 11 = W c /α, Z 12 = B/(2α), Z 22 = B Wc 1 B/(4α), where α = Tr(B Wc 1 B/4) Thus new lower bound is 2 TrCW c C /(TrB W 1 c B)

SDP DUALITY AND LTI SYSTEMS 20 Primal Application of duality: LQR problem minimize: TrQZ 11 + TrZ 22 subject to: AZ [ 11 + BZ12 ] + Z 11 A + Z 12 B + x 0 x 0 0, Z11 Z 12 Z12 0 Z 22 Dual maximize: x[ 0P x 0 A subject to: P + P A + Q P B B P I ] 0, P 0

SDP DUALITY AND LTI SYSTEMS 21 The Linear-Quadratic Regulator problem ẋ = Ax + Bu, x(0) = x 0, s.t. lim t x(t) = 0 find u that minimizes J = 0 (x(t) Qx(t) + u(t) u(t)) dt, Well-known solution: Solve Riccati equation A T P + P A + Q P BB T P = 0 such that P > 0. Then, u opt (t) = B T P x(t) (Proof using quadratic optimal control theory)

SDP DUALITY AND LTI SYSTEMS 22 Duality-based proof: Basic ideas Primal problem gives upper bound on LQR objective Dual problem gives lower bound on LQR objective Optimality condition gives LQR Riccati equation

SDP DUALITY AND LTI SYSTEMS 23 Primal problem interpretation Assume u = Kx, s.t. x(t) 0 as t Then LQR objective reduces to J K = 0 x(t) (Q + K K) x(t) dt and is an upper bound on the optimum LQR objective Condition x(t) 0 as t equivalent to (A + BK) Z + Z(A + BK) + x 0 x 0 = 0, Z 0 LQR objective is Tr Z(Q + K K)

SDP DUALITY AND LTI SYSTEMS 24 Best upper bound using state-feedback: minimize: Tr Z(Q + K K) subject to: Z 0 (A + BK) Z + Z(A + BK) + x 0 x 0 = 0 With Z 11 = Z, Z 12 = ZK, Z 22 = K ZK : minimize: TrQZ 11 + TrZ 22 subject to: AZ [ 11 + BZ12 ] + Z 11 A + Z 12 B + x 0 x 0 0, Z11 Z 12 Z12 0 Z 22 (Same as primal problem)

SDP DUALITY AND LTI SYSTEMS 25 Dual problem interpretation Suppose for P 0, d dt x(t) P x(t) (x(t) Qx(t) + u(t) u(t)), for all t 0, and for all x and u satisfying ẋ = Ax + Bu, x(t ) = 0. Then, x 0P x 0 T 0 (x(t) Qx(t) + u(t) u(t)) dt, So J opt x 0P x 0 Derivative condition equivalent to LMI [ A P + P A + Q P B B P I ] 0 So lower bound to LQR objective given by dual problem

SDP DUALITY AND LTI SYSTEMS 26 Optimality conditions Stabilizability of (A, B) guarantees strict primal feasibility Detectability of (Q, A) guarantees strict dual feasibility Recall, at optimality (F(x opt ) + F 0 ) Z opt = 0. This becomes [ ] [ Z11 Z 12 A P + P A + Q P B Z12 Z 22 B P I Reduces to [ ] [ ] I K A P + P A + Q P B B = 0, P I or K = B P, with all the eigenvalues of A + BK having negative real part, and A P + P A + Q P BB P = 0 (Classical LQR result, derived using duality) ] = 0

SDP DUALITY AND LTI SYSTEMS 27 Conclusions SDP duality theory has interesting implications systems and control Implications for numerical computation: Dual problems sometimes have fewer variables Most efficient algorithms solve primal and dual together; control-theoretic interpretation can help increase efficiency