Structured adaptive control for solving LMIs - LAAS
Structured adaptive control for solving LMIs Alexandru-Razvan Luzi, Alexander L. Fradkov, Jean-Marc Biannic, Dimitri Peaucelle
IFAC-ALCOSP Caen, july 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 ? n LMIs: Linear Matrix Inequalities X X max bi y i : F 0 + yi Fi ≺ 0 l LMIs are SDP: Semi-Definite Programming min c T x
: Ax = b , mat(x) 0
l Primal-dual, convex, solvers in polynomial-time [Nesterov, ...] l Nice parser: YALMIP l Many control problems have LMI formulations, mainly in robust control P 0 , AT P + PA ≺ 0 l New results for: combinatorial optimization, robust optimization, algebraic geometry, cryptography, optimal control...
2/20
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
3/20
Outline
Outline
1
Passivity-based adaptive control
2
LMIs are strict-passifiable systems
3
Structured adaptive control
4
Numerical Example
4/20
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.
5/20
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 (∞).
6/20
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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
7/20
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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 l Converted with simple manipulations into one simple LMI A + BT F B ≺ 0 s with structural equality constraints on F 0 P 0 FP 0 F = , FP = P 0 0 , P = P T , Fγ = −γ 2 1 0 Fγ 0 0 −P
A=
CT C DT C 0
CT D DT D 0
0 0 0
, B=
A 1 0 0
B 0 0 1
0 0 1 0
7/20
LMIs are strict-passifiable systems
n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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 l Converted with simple manipulations into one simple LMI A + BT F B ≺ 0 s with structural equality constraints on F 0 P 0 FP 0 F = , FP = P 0 0 , P = P T , Fγ = −γ 2 1 0 Fγ 0 0 −P n The constraint A + BT F B ≺ 0 holds iff (A, B, C = BT ) is strictly-passifiable by F (condition Ê). s LMI converted to strict-passification problem, with equality constraints.
8/20
LMIs are strict-passifiable systems
n Procedure applies to any LMI:
l Concludes with search of passifying gain F =
F1
0 ..
.
0 FN l for a (symmetric) system (A, B, C = BT ) l with additional structural equality constraints that can be compacted in Ui vec(Fi ) = 0 Where vec(Fi ) is the vector composed of stacked columns of Fi .
s All constraints Ui vec(Fi ) = 0 include the constraint Fi = Fi T .
9/20
Structured adaptive control
Block-diagonal adaptive control with asymptotic structure Theorem Assume A = AT and C = BT , then the following two are equivalent: Ê There exists a symmetric decentralized static control ui (t) = Fi yi (t) satisfying structural constraints Ui vec(Fi ) = 0 that stabilizes asymptotically X x(t) ˙ = Ax(t) + Bi ui (t) , yi (t) = Ci x(t). 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 asymptotically stable and the adaptive gains converge to constant values Fi = Ki (∞) solution to condition Ê. (‘mat’ is the function such that mat(vec(F )) = F )
10/20
Structured adaptive control
Proof of Ê ⇒ Ë l Stability of a symmetric matrix A + BF C proved by V (x) = 12 x T x, i.e. Ê implies ∃F
:
(A+ BF C)T + (A+ BF C) < 0, F = diag · · · Fi · · · , Ui · vec(Fi ) = 0
(1)
l Let the Lyapunov function for the non-linear system (with adaptive law) ! X 1 −1 T T V (x, K ) = x x+ Tr (Ki − Fi )Γ (Ki − Fi ) 2 i
l After manipulations, using B = CT , Ui · vec(Fi ) = 0, we get: X V˙ (x, K ) = x T (A + BF C)T x − αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i
11/20
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V˙ (x, K ) = x T (A + BF C)T x −
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 The system converges to the attractor A = {(x, K ) : x = 0 , Ui · vec(Ki ) = 0} n Reasoning in [Ioannou&Sun 96] allows to conclude that Ki (t) converges to a constant gain Ki (∞).
12/20
Structured adaptive control
Proof of Ë ⇒ Ê
l The system with adaptive control is globally asymptotically stable, it converges to an asymptotically stable equilibrium: Fi = Ki (∞) are stabilizing gains
13/20
Structured adaptive control
Summary n All LMI problems are equivalent to static output-feedback strict-passification problems with structure constraints: - A = AT - 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 (∞) = Fi are solutions of the LMIs. s If does not converges the LMIs are infeasible.
14/20
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
15/20
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
15/20
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
16/20
Numerical Example
s Convergence to structured values of the adapted gains Ki
0 0 4.6330 K1 (∞) = 1.0671 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
17/20
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
18/20
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 .
19/20
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 ?
20/20
IFAC-ALCOSP Caen, july 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 ? n LMIs: Linear Matrix Inequalities X X max bi y i : F 0 + yi Fi ≺ 0 l LMIs are SDP: Semi-Definite Programming min c T x
: Ax = b , mat(x) 0
l Primal-dual, convex, solvers in polynomial-time [Nesterov, ...] l Nice parser: YALMIP l Many control problems have LMI formulations, mainly in robust control P 0 , AT P + PA ≺ 0 l New results for: combinatorial optimization, robust optimization, algebraic geometry, cryptography, optimal control...
2/20
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
3/20
Outline
Outline
1
Passivity-based adaptive control
2
LMIs are strict-passifiable systems
3
Structured adaptive control
4
Numerical Example
4/20
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.
5/20
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 (∞).
6/20
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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
7/20
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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 l Converted with simple manipulations into one simple LMI A + BT F B ≺ 0 s with structural equality constraints on F 0 P 0 FP 0 F = , FP = P 0 0 , P = P T , Fγ = −γ 2 1 0 Fγ 0 0 −P
A=
CT C DT C 0
CT D DT D 0
0 0 0
, B=
A 1 0 0
B 0 0 1
0 0 1 0
7/20
LMIs are strict-passifiable systems
n Let us consider an example: l LMIs for an upper bound on the H∞ norm of G (s) ∼ (A, B, C , D) 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 l Converted with simple manipulations into one simple LMI A + BT F B ≺ 0 s with structural equality constraints on F 0 P 0 FP 0 F = , FP = P 0 0 , P = P T , Fγ = −γ 2 1 0 Fγ 0 0 −P n The constraint A + BT F B ≺ 0 holds iff (A, B, C = BT ) is strictly-passifiable by F (condition Ê). s LMI converted to strict-passification problem, with equality constraints.
8/20
LMIs are strict-passifiable systems
n Procedure applies to any LMI:
l Concludes with search of passifying gain F =
F1
0 ..
.
0 FN l for a (symmetric) system (A, B, C = BT ) l with additional structural equality constraints that can be compacted in Ui vec(Fi ) = 0 Where vec(Fi ) is the vector composed of stacked columns of Fi .
s All constraints Ui vec(Fi ) = 0 include the constraint Fi = Fi T .
9/20
Structured adaptive control
Block-diagonal adaptive control with asymptotic structure Theorem Assume A = AT and C = BT , then the following two are equivalent: Ê There exists a symmetric decentralized static control ui (t) = Fi yi (t) satisfying structural constraints Ui vec(Fi ) = 0 that stabilizes asymptotically X x(t) ˙ = Ax(t) + Bi ui (t) , yi (t) = Ci x(t). 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 asymptotically stable and the adaptive gains converge to constant values Fi = Ki (∞) solution to condition Ê. (‘mat’ is the function such that mat(vec(F )) = F )
10/20
Structured adaptive control
Proof of Ê ⇒ Ë l Stability of a symmetric matrix A + BF C proved by V (x) = 12 x T x, i.e. Ê implies ∃F
:
(A+ BF C)T + (A+ BF C) < 0, F = diag · · · Fi · · · , Ui · vec(Fi ) = 0
(1)
l Let the Lyapunov function for the non-linear system (with adaptive law) ! X 1 −1 T T V (x, K ) = x x+ Tr (Ki − Fi )Γ (Ki − Fi ) 2 i
l After manipulations, using B = CT , Ui · vec(Fi ) = 0, we get: X V˙ (x, K ) = x T (A + BF C)T x − αi (Ui · vec(Ki ))T (Ui · vec(Ki )). i
11/20
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V˙ (x, K ) = x T (A + BF C)T x −
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 The system converges to the attractor A = {(x, K ) : x = 0 , Ui · vec(Ki ) = 0} n Reasoning in [Ioannou&Sun 96] allows to conclude that Ki (t) converges to a constant gain Ki (∞).
12/20
Structured adaptive control
Proof of Ë ⇒ Ê
l The system with adaptive control is globally asymptotically stable, it converges to an asymptotically stable equilibrium: Fi = Ki (∞) are stabilizing gains
13/20
Structured adaptive control
Summary n All LMI problems are equivalent to static output-feedback strict-passification problems with structure constraints: - A = AT - 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 (∞) = Fi are solutions of the LMIs. s If does not converges the LMIs are infeasible.
14/20
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
15/20
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
15/20
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
16/20
Numerical Example
s Convergence to structured values of the adapted gains Ki
0 0 4.6330 K1 (∞) = 1.0671 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
17/20
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
18/20
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 .
19/20
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 ?
20/20
