Structured adaptive control for solving LMIs Alexandru-Razvan Luzi, Alexander L. Fradkov, Jean-Marc Biannic, Dimitri Peaucelle CNES, CCT SCA, 12 february 2014 Published at IFAC-ALCOSP 2013 By-product of research work on adaptive satellite attitude control: ”Structured adaptive attitude control of a satellite” , A.R. Luzi, D. Peaucelle, J.-M. Biannic, Ch. Pittet, J. Mignot, International Journal of Adaptive Control and Signal Processing 2013 1/20
What are LMIs ? What are LMIs ? ■ LMIs: Linear Matrix Inequalities � � max b i y i : F 0 + y i F i ≺ 0 ● LMIs are SDP: Semi-Definite Programming min c T x : Ax = b , mat( x ) ≻ 0 ● Primal-dual, convex, solvers in polynomial-time [Nesterov, ...] ● Nice parser: YALMIP ● Many control problems have LMI formulations, mainly in robust control A T P + PA ≺ 0 P ≻ 0 , ● New results for: combinatorial optimization, robust optimization, algebraic geometry, cryptography, optimal control... 2/20
Introduction Introduction ■ Direct adaptive control: Adaptation of control gains done directly based on measurements. ▲ � = Indirect adaptive control: Estimator of model parameters + scheduled control gain ■ Feedback-loop stabilizing gains, MRAC not considered ■ Lyapunov based stability proofs ■ Framework initiated by V.A. Yakubovich in the late 1960’s ● Contributions: new adaptive control law with asymptotic structure + may solve LMIs 3/20
Outline Outline Passivity-based adaptive control 1 LMIs are strict-passifiable systems 2 Structured adaptive control 3 Numerical Example 4 4/20
Passivity-based adaptive control Passivity-based adaptive control of LTI systems Theorem The following two conditions are equivalent: ➊ There exists a static control u ( t ) = Fy ( t ) + w ( t ) for the system x ( t ) = A x ( t ) + B u ( t ) , ˙ y ( t ) = C x ( t ) , z ( t ) = y ( t ) that makes the closed-loop strictly passive (with respect to w / z). ➋ For all Γ ≻ 0 the following adaptive control ˙ K ( t ) = − y ( t ) y T ( t )Γ u ( t ) = K ( t ) y ( t ) + w ( t ) , makes the closed-loop globally strictly-passive. 5/20
Passivity-based adaptive control ● Strict-passivity includes asymptotic stability of x = 0 ● Adaptive control converges to K ( ∞ ): strictly-passifying static gain ▲ Theorem for square systems - extensions exist for non-square systems ▲ Not all stabilizable systems are strictly-passifiable - modified adaptive laws exist for stabilizable systems ● Condition ➊ also reads in terms of matrix inequalities as ( A + B F C ) T Q + Q ( A + B F C ) ≺ 0 , Q B = C T ∃ Q ≻ 0 : It happens to be an LMI constraint! A T Q + Q A + C T ( F T + F ) C ≺ 0 , Q B = C T ∃ Q ≻ 0 : ■ Finding F solution to the LMI is equivalent to simulating the system with the adaptive control law and taking F = K ( ∞ ). 6/20
LMIs are strict-passifiable systems All LMIs define strict-passifiable systems ■ Let us consider an example: ● LMIs for an upper bound on the H ∞ norm of G ( s ) ∼ ( A , B , C , D ) � A T P + PA + C T C PB + C T D � P = P T ≻ 0 . ≺ 0 , B T P + D T C − γ 2 1 + D T D 7/20
LMIs are strict-passifiable systems All LMIs define strict-passifiable systems ■ Let us consider an example: ● LMIs for an upper bound on the H ∞ norm of G ( s ) ∼ ( A , B , C , D ) � A T P + PA + C T C PB + C T D � P = P T ≻ 0 . ≺ 0 , B T P + D T C − γ 2 1 + D T D ● Converted with simple manipulations into one simple LMI A + B T F B ≺ 0 ▲ with structural equality constraints on F � F P P 0 0 � 0 , F γ = − γ 2 1 P = P T , F = , F P = P 0 0 0 F γ − P 0 0 0 � � � A B � C T C C T D 0 1 0 0 A = , B = D T C D T D 0 0 0 1 7/20 0 0 0 0 1 0
LMIs are strict-passifiable systems ■ Let us consider an example: ● LMIs for an upper bound on the H ∞ norm of G ( s ) ∼ ( A , B , C , D ) � A T P + PA + C T C PB + C T D � P = P T ≻ 0 . ≺ 0 , − γ 2 1 + D T D B T P + D T C ● Converted with simple manipulations into one simple LMI A + B T F B ≺ 0 ▲ with structural equality constraints on F � F P 0 P 0 � 0 , P = P T , F γ = − γ 2 1 F = , F P = P 0 0 F γ 0 0 0 − P ■ The constraint A + B T F B ≺ 0 holds iff ( A , B , C = B T ) is strictly-passifiable by F (condition ➊ ). ▲ LMI converted to strict-passification problem, with equality constraints. 8/20
LMIs are strict-passifiable systems ■ Procedure applies to any LMI: F 1 0 ... ● Concludes with search of passifying gain F = F N 0 ● for a (symmetric) system ( A , B , C = B T ) ● with additional structural equality constraints that can be compacted in U i vec( F i ) = 0 � � Where vec( F i ) is the vector composed of stacked columns of F i . ▲ All constraints U i vec( F i ) = 0 include the constraint F i = F i T . 9/20
Structured adaptive control Block-diagonal adaptive control with asymptotic structure Theorem Assume A = A T and C = B T , then the following two are equivalent: ➊ There exists a symmetric decentralized static control u i ( t ) = F i y i ( t ) satisfying structural constraints U i vec( F i ) = 0 that stabilizes asymptotically � x ( t ) = A x ( t ) + ˙ B i u i ( t ) , y i ( t ) = C i x ( t ) . ➋ For all Γ i ≻ 0 , α i > 0 the following adaptive control u i ( t ) = K i ( t ) y i ( t ) + w i ( t ) , ˙ K i ( t ) = − y i ( t ) y T � U T � i ( t )Γ i − α i · mat i U i · vec( K i ( t )) Γ i makes the closed-loop globally asymptotically stable and the adaptive gains converge to constant values F i = K i ( ∞ ) solution to condition ➊ . (‘mat’ is the function such that mat(vec( F )) = F ) 10/20
Structured adaptive control Proof of ➊ ⇒ ➋ ● Stability of a symmetric matrix A + B F C proved by V ( x ) = 1 2 x T x , i.e. ➊ implies ( A + B F C ) T + ( A + B F C ) < 0 , ∃ F : (1) � � F = diag · · · F i · · · , U i · vec( F i ) = 0 ● Let the Lyapunov function for the non-linear system (with adaptive law) � ( K i − F i )Γ − 1 ( K i − F i ) T �� V ( x , K ) = 1 � � x T x + Tr 2 i ● After manipulations, using B = C T , U i · vec( F i ) = 0, we get: V ( x , K ) = x T ( A + B F C ) T x − ˙ � α i ( U i · vec( K i )) T ( U i · vec( K i )) . i 11/20
Structured adaptive control Proof of ➊ ⇒ ➋ (continued) ˙ � V ( x , K ) = x T ( A + B F C ) T x − α i ( U i · vec( K i )) T ( U i · vec( K i )) . i ▲ First term is strictly negative due to (1), until x = 0, ▲ Last term is strictly negative, until U i · vec( K i ) = 0. ■ The system converges to the attractor A = { ( x , K ) : x = 0 , U i · vec( K i ) = 0 } ■ Reasoning in [Ioannou&Sun 96] allows to conclude that K i ( t ) converges to a constant gain K i ( ∞ ). 12/20
Structured adaptive control Proof of ➋ ⇒ ➊ ● The system with adaptive control is globally asymptotically stable, it converges to an asymptotically stable equilibrium: F i = K i ( ∞ ) are stabilizing gains 13/20
Structured adaptive control Summary ■ All LMI problems are equivalent to static output-feedback strict-passification problems with structure constraints: - A = A T - gain F is block-diagonal - sub-blocks should satisfy U i vec( F i ) = 0. ■ If a structured strict-passification problem admits solutions, the block-diagonal adaptive law with asymptotic structure will converge to one of these. ● The LMIs can be solved by simulating the adaptive controlled systems. ▲ If the system converges K i ( ∞ ) = F i are solutions of the LMIs. ▲ If does not converges the LMIs are infeasible. 14/20
Numerical Example Numerical example ● Consider the transfer function: G ( s ) = s 2 + s + 1 s 2 + s + 2 ● Problem: compute the H ∞ norm (or at least an upper bound). ▲ In Matlab: norm(G, Inf, 1e-4) = 1.3251 15/20
Numerical Example Numerical example ● Consider the transfer function: G ( s ) = s 2 + s + 1 s 2 + s + 2 ● Problem: compute the H ∞ norm (or at least an upper bound). ▲ In Matlab: norm(G, Inf, 1e-4) = 1.3251 ▲ LMI problem converted to adaptive passification � � y 1 ∈ R 6 , ˙ K i = − y i y T U T i Γ i − α i · mat i U i · vec( K i ) Γ i , y 2 ∈ R with structural asymptotic constraints : P 0 0 P = P T ∈ R 2 × 2 , P T F 2 = − γ 2 1 = − γ 2 . F 1 = , 0 0 0 0 − P 15/20
Numerical Example ● Parameters for simulating the adaptive law (simulation in Simulink) ▲ Initial conditions x = (1 . . . 1) T and K i = 0 ▲ Γ 1 = 1000 · 1 , Γ 2 = 10, α 1 = α 2 = 1 ● Convergence to zero of the ‘outputs’ y i 16/20
Numerical Example ▲ Convergence to structured values of the adapted gains K i 0 0 4 . 6330 1 . 0671 0 0 0 0 1 . 0671 10 . 7960 0 0 4 . 6330 1 . 0671 0 0 0 0 K 1 ( ∞ ) = 1 . 0671 10 . 7960 0 0 0 0 0 0 0 0 − 4 . 6330 − 1 . 0671 0 0 0 0 − 1 . 0671 − 10 . 7960 K 2 ( ∞ ) = − 7 . 1307 17/20
Numerical Example ▲ Evolution of the (1 : 2 , 3 : 4) elements of K 1 that converge to P ▲ Solution of the LMIs � 4 . 6330 1 . 0671 � P = , γ = 2 . 6703 ≥ 1 . 3251 = γ opt 1 . 0671 10 . 7960 18/20
Numerical Example ● Test for feasible / unfeasible cases ▲ Only K 1 is adapated, γ is slowly linearly modified ▲ Unstable behavior when γ < 1 . 3251 = γ opt . 19/20
Recommend
More recommend
