Structured adaptive control, or how to solve LMIs with Simulink Alexandru - Razvan LUZI Dimitri PEAUCELLE IEIIT-CNR Torino, october 2012 1/21 Introduction Introduction n Direct adaptive control: Adaptation of control gains done directly based on measurements. s 6= Indirect adaptive control: Estimator of model parameters + scheduled control gain n Feedback-loop stabilizing gains, MRAC not considered n Lyapunov based stability proofs, not gradient approximation ‘MIT rule’ n Framework initiated by V.A. Yakubovich in the late 1960’s l Contributions: new adaptive control law with asymptotic structure + may solve LMIs 2/21 Plan Plan 1 Passivity-based adaptive control 2 LMIs are strict-passifiable systems 3 Structured adaptive control 4 Numerical Example 3/21 Passivity-based adaptive control Passivity-based adaptive control of LTI systems Theorem The following two conditions are equivalent: Ê There exists a static control u(t) = F y (t) + w (t) for the system x(t) ˙ = Ax(t) + Bu(t) , y (t) = Cx(t) , z(t) = y (t) that makes the closed-loop strictly passive (with respect to w /z). Ë For all Γ 0 the following adaptive control u(t) = K (t)y (t) + w (t) , K˙ (t) = −y (t)y T (t)Γ makes the closed-loop globally strictly-passive. 4/21 Passivity-based adaptive control l Strict-passivity includes asymptotic stability of x = 0 l Adaptive control converges to K (∞): strictly-passifying static gain s Theorem for square systems - extensions exist for non-square systems s Not all stabilizable systems are strictly-passifiable - modified adaptive laws exist for stabilizable systems l Condition Ê also reads in terms of matrix inequalities as ∃Q 0 : (A + BF C)T Q + Q(A + BF C) ≺ 0 , QB = CT It happens to be an LMI constraint! ∃Q 0 : AT Q + QA + CT (F T + F )C ≺ 0 , QB = CT n Finding F solution to the LMI is equivalent to simulating the system with the adaptive control law and taking F = K (∞). 5/21 LMIs are strict-passifiable systems All LMIs define strict-passifiable systems l Let us consider an example: LMIs for an upper bound on the H∞ norm T A P + PA + C T C PB + C T D ≺ 0 , P = P T 0. BT P + DT C −γ 2 1 + D T D 6/21 LMIs are strict-passifiable systems All LMIs define strict-passifiable systems l Let us consider an example: LMIs for an upper bound on the H∞ norm T A P + PA + C T C PB + C T D ≺ 0 , P = P T 0. BT P + DT C −γ 2 1 + D T D s All LMI constraints can be gathered in one T A P + PA + C T C PB + C T D 0 BT P + DT C −γ 2 1 + D T D 0 ≺ 0 , P = P T 0 0 −P 6/21 LMIs are strict-passifiable systems s All LMI constraints can be gathered in one T A P + PA + C T C PB + C T D 0 BT P + DT C −γ 2 1 + D T D 0 ≺ 0 , P = P T 0 0 −P 7/21 LMIs are strict-passifiable systems s All LMI constraints can be gathered in one T A P + PA + C T C PB + C T D 0 BT P + DT C −γ 2 1 + D T D 0 ≺ 0 , P = P T 0 0 −P s Can be decomposed in a sum with elementary matrix variables 0 P 0 2 T P 0 0 BP + BT A + BT γ (−γ 1)Bγ ≺ 0 , P = P P 0 0 −P CTC A = DT C 0 CTD 0 A B 0 D T D 0 , BP = 1 0 0 , Bγ = 0 1 0 0 0 1 0 0 7/21 LMIs are strict-passifiable systems s Can be decomposed in a sum with elementary matrix variables 0 P 0 2 T P 0 0 BP + BT A + BT γ (−γ 1)Bγ ≺ 0 , P = P P 0 0 −P 8/21 LMIs are strict-passifiable systems s Can be decomposed in a sum with elementary matrix variables 0 P 0 2 T P 0 0 BP + BT A + BT γ (−γ 1)Bγ ≺ 0 , P = P P 0 0 −P s Or equivalently when gathering all variables in a block-diagonal matrix A + BT F B ≺ 0 , B = BP Bγ F = FP 0 with the structural equality constraints 0 P 0 0 , FP = P 0 0 , P = P T , Fγ = −γ 2 1 Fγ 0 0 −P 8/21 LMIs are strict-passifiable systems s Can be decomposed in a sum with elementary matrix variables 0 P 0 2 T P 0 0 BP + BT A + BT γ (−γ 1)Bγ ≺ 0 , P = P P 0 0 −P s Or equivalently when gathering all variables in a block-diagonal matrix A + BT F B ≺ 0 , B = BP Bγ F = FP 0 with the structural equality constraints 0 P 0 0 , FP = P 0 0 , P = P T , Fγ = −γ 2 1 Fγ 0 0 −P l The constraint A + BT F B ≺ 0 holds iff (A, B, C = BT ) is strictly-passifiable by F . 8/21 LMIs are strict-passifiable systems s Can be decomposed in a sum with elementary matrix variables 0 P 0 2 T P 0 0 BP + BT A + BT γ (−γ 1)Bγ ≺ 0 , P = P P 0 0 −P s Or equivalently when gathering all variables in a block-diagonal matrix A + BT F B ≺ 0 , B = BP Bγ F = FP 0 with the structural equality constraints 0 P 0 0 , FP = P 0 0 , P = P T , Fγ = −γ 2 1 Fγ 0 0 −P l The constraint A + BT F B ≺ 0 holds iff (A, B, C = BT ) is strictly-passifiable by F . n LMI converted to strict-passification problem, with equality constraints. 8/21 LMIs are strict-passifiable systems n Procedure applies to any LMI: concludes with search of passifying F1 0 .. F = . 0 FN for a (symmetric) system (A, B, C = BT ) with additional structural equality constraints that can be compacted in vec(Fi ) = Si xi ⇔ Ui vec(Fi ) = 0 where vec(Fi ) is the vector composed of stacked columns of Fi , xi are vectors of independent scalar decision variables and Ui = Si⊥ . 9/21 LMIs are strict-passifiable systems n Procedure applies to any LMI: concludes with search of passifying F1 0 .. F = . 0 FN for a (symmetric) system (A, B, C = BT ) with additional structural equality constraints that can be compacted in vec(Fi ) = Si xi ⇔ Ui vec(Fi ) = 0 where vec(Fi ) is the vector composed of stacked columns of Fi , xi are vectors of independent scalar decision variables and Ui = Si⊥ . l When starting from the canonical representation L0 + then the structural constraints are all of the type xˆj 1rj1 0 Fj = 0 −ˆ xj 1rj2 P j xˆj Lj ≺ 0, and rj1 + rj2 can be very large and Uj is huge. l F and Ui s expected to be smaller when matrix representation. 9/21 Structured adaptive control Block-diagonal adaptive control with asymptotic structure Theorem The following two conditions are equivalent: Ê There exists a decentralized static control ui (t) = Fi yi (t) + wi (t) satisfying the structural constraints Ui vec(Fi ) = 0 for the system X x(t) ˙ = Ax(t) + Bi ui (t) , yi (t) = Ci x(t) , z(t) = y (t) that makes the closed-loop strictly passive (with respect to w /z). Ë For all Γi 0, αi > 0 the following adaptive control ui (t) = Ki (t)yi (t) + wi (t) , K˙ i (t) = −yi (t)yiT (t)Γi − αi · mat UiT Ui · vec(Ki (t)) Γi makes the closed-loop globally strictly-passive. ‘mat’ is the function such that mat(vec(F )) = F . 10/21 Structured adaptive control Proof of Ê ⇒ Ë l Ê reads as ∃F , ∃Q 0 : T (A + BF C)T Q + Q(A + BF C) < 0, QB = C , F = diag · · · Fi · · · , Ui · vec(Fi ) = 0 (1) 11/21 Structured adaptive control Proof of Ê ⇒ Ë l Ê reads as ∃F , ∃Q 0 : T (A + BF C)T Q + Q(A + BF C) < 0, QB = C , F = diag · · · Fi · · · , Ui · vec(Fi ) = 0 (1) l Let the Lyapunov function for the non-linear system (with adaptive law) ! X 1 V (x, K ) = x T Qx + Tr (Ki − Fi )Γ−1 (Ki − Fi )T 2 i 11/21 Structured adaptive control Proof of Ê ⇒ Ë l Ê reads as ∃F , ∃Q 0 : T (A + BF C)T Q + Q(A + BF C) < 0, QB = C , F = diag · · · Fi · · · , Ui · vec(Fi ) = 0 (1) l Let the Lyapunov function for the non-linear system (with adaptive law) ! X 1 V (x, K ) = x T Qx + Tr (Ki − Fi )Γ−1 (Ki − Fi )T 2 i l After manipulations and using QB = CT , Ui · vec(Fi ) = 0, we get: X V˙ (x, K ) = x T (A + BF C)T Qx + w T z − αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i 11/21 Structured adaptive control Proof of Ê ⇒ Ë (continued) V˙ (x, K ) = x T (A + BF C)T Qx + w T z − X αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i 12/21 Structured adaptive control Proof of Ê ⇒ Ë (continued) V˙ (x, K ) = x T (A + BF C)T Qx + w T z − X αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i s First term is strictly negative due to (1), until x = 0, s Last term is strictly negative, until Ui · vec(Ki ) = 0 n If no perturbations (w = 0) the system converges to the attractor A = {(x, K ) : x = 0 , Ui · vec(Ki ) = 0} n On the attractor K˙ i = 0: the gains Ki (∞) are constant. 12/21 Structured adaptive control Proof of Ê ⇒ Ë (continued) V˙ (x, K ) = x T (A + BF C)T Qx + w T z − X αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i 13/21 Structured adaptive control Proof of Ê ⇒ Ë (continued) V˙ (x, K ) = x T (A + BF C)T Qx + w T z − X αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i n For initial conditions at equilibrium and nonzero perturbations Z t Z t V˙ (x, K )dτ < w T z dτ 0 ≤ V (x(t), K (t)) = 0 0 ⇒ the system is strictly passive. 13/21 Structured adaptive control Proof of Ë ⇒ Ê l The system with adaptive control is globally asymptotically stable, it converges to a asymptotically stable equilibrium: F i = Ki (∞) are stabilizing gains. l Same reasoning holds for passivity. 14/21 Structured adaptive control Summary n All LMI problems are equivalent to static output-feedback strict-passification problems with structure constraints: - gain F is block-diagonal - sub-blocks should satisfy Ui vec(Fi ) = 0. n If a structured strict-passification problem admits solutions, the block-diagonal adaptive law with asymptotic structure will converge to one of these. l The LMIs can be solved by simulating the adaptive controlled systems. s If the system converges Ki (∞) contain solutions of the LMIs. s If does not converges the LMIs are infeasible. 15/21 Numerical Example Numerical example l Consider the transfer function: s2 + s + 1 s2 + s + 2 norm (or at least an upper bound). G (s) = l Problem: compute the H∞ s In Matlab: norm(G, Inf, 1e-4) = 1.3251 16/21 Numerical Example Numerical example l Consider the transfer function: s2 + s + 1 s2 + s + 2 norm (or at least an upper bound). G (s) = l Problem: compute the H∞ s In Matlab: norm(G, Inf, 1e-4) = 1.3251 s LMI problem converted to adaptive passification K˙ i = −yi yiT Γi − αi · mat UiT Ui · vec(Ki ) Γi , y1 ∈ R6 , y2 ∈ R with structural asymptotic constraints : 0 P 0 F1 = P T 0 0 , P = P T ∈ R2×2 , F2 = −γ 2 1 = −γ 2 . 0 0 −P 16/21 Numerical Example l Parameters for simulating the adaptive law (simulation in Simulink) s Initial conditions x = (1 . . . 1)T and Ki = 0 s Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1 17/21 Numerical Example l Parameters for simulating the adaptive law (simulation in Simulink) s Initial conditions x = (1 . . . 1)T and Ki = 0 s Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1 l Convergence to zero of the ‘outputs’ yi 17/21 Numerical Example s Convergence to structured values of the adapted gains Ki 2 0 6 0 6 6 4.6330 K1 (∞) = 6 6 1.0671 6 4 0 0 K2 (∞) = −7.1307 0 0 1.0671 10.7960 0 0 4.6330 1.0671 0 0 0 0 1.0671 10.7960 0 0 0 0 0 0 0 0 −4.6330 −1.0671 0 0 0 0 −1.0671 −10.7960 3 7 7 7 7 7 7 5 18/21 Numerical Example s Evolution of the (1 : 2, 3 : 4) elements of K1 that converge to P s Solution of the LMIs 4.6330 1.0671 P= , γ = 2.6703 ≥ 1.3251 = γopt 1.0671 10.7960 19/21 Numerical Example l Test for feasible / unfeasible cases s Only K1 is adapated, γ is slowly linearly modified s Unstable behavior when γ < 1.3251 = γopt . 20/21 Conclusions Conclusions et perspectives n LMI feasibility problems can be solved by simulating systems s Need for a parser to convert LMIs to adaptive control problem s Simulation time is large - what is the best implementation ? s Is simulation time polynomial w.r.t. size of problem ? n What about LMI optimization problems ? s Decreasing parameters until system becomes unstable ? s Minimizing gap with dual LMI problem (it works). s Other ? n Solving time-varying LMI problems ? 21/21

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