Structured adaptive control, or how to solve LMIs with Simulink - - PowerPoint PPT Presentation

Structured adaptive control,

• r how to solve LMIs with Simulink

Alexandru - Razvan LUZI Dimitri PEAUCELLE IEIIT-CNR Torino, october 2012

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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, not gradient approximation ‘MIT rule’ ■ Framework initiated by V.A. Yakubovich in the late 1960’s

• Contributions: new adaptive control law with asymptotic structure

+ may solve LMIs

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Plan

Plan

1

Passivity-based adaptive control

2

LMIs are strict-passifiable systems

3

Structured adaptive control

4

Numerical Example

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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) = Ax(t) + Bu(t) , y(t) = Cx(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 u(t) = K(t)y(t) + w(t) , ˙ K(t) = −y(t)yT(t)Γ makes the closed-loop globally strictly-passive.

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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

∃Q ≻ 0 : (A + BFC)TQ + Q(A + BFC) ≺ 0 , QB = CT It happens to be an LMI constraint! ∃Q ≻ 0 : ATQ + QA + CT(F T + F)C ≺ 0 , QB = CT ■ Finding F solution to the LMI is equivalent to simulating the system with the adaptive control law and taking F = K(∞).

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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

ATP + PA + C TC PB + C TD BTP + DTC −γ21 + DTD

• ≺ 0 ,

P = PT ≻ 0.

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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

ATP + PA + C TC PB + C TD BTP + DTC −γ21 + DTD

• ≺ 0 ,

P = PT ≻ 0. ▲ All LMI constraints can be gathered in one   ATP + PA + C TC PB + C TD BTP + DTC −γ21 + DTD −P   ≺ 0 , P = PT

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LMIs are strict-passifiable systems

▲ All LMI constraints can be gathered in one   ATP + PA + C TC PB + C TD BTP + DTC −γ21 + DTD −P   ≺ 0 , P = PT

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LMIs are strict-passifiable systems

▲ All LMI constraints can be gathered in one   ATP + PA + C TC PB + C TD BTP + DTC −γ21 + DTD −P   ≺ 0 , P = PT ▲ Can be decomposed in a sum with elementary matrix variables A + BT

P

  P P −P   BP + BT

γ (−γ21)Bγ ≺ 0 ,

P = PT A =   C TC C TD DTC DTD   , BP =   A B 1 1   , Bγ =

• 1
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LMIs are strict-passifiable systems

▲ Can be decomposed in a sum with elementary matrix variables A + BT

P

  P P −P   BP + BT

γ (−γ21)Bγ ≺ 0 ,

P = PT

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LMIs are strict-passifiable systems

▲ Can be decomposed in a sum with elementary matrix variables A + BT

P

  P P −P   BP + BT

γ (−γ21)Bγ ≺ 0 ,

P = PT ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + BTFB ≺ 0 , B =

• BP

Bγ

• with the structural equality constraints

F = FP Fγ

• ,

FP =   P P −P   , P = PT , Fγ = −γ21

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LMIs are strict-passifiable systems

▲ Can be decomposed in a sum with elementary matrix variables A + BT

P

  P P −P   BP + BT

γ (−γ21)Bγ ≺ 0 ,

P = PT ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + BTFB ≺ 0 , B =

• BP

Bγ

• with the structural equality constraints

F = FP Fγ

• ,

FP =   P P −P   , P = PT , Fγ = −γ21

• The constraint A + BTFB ≺ 0 holds iff

(A, B, C = BT) is strictly-passifiable by F.

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LMIs are strict-passifiable systems

▲ Can be decomposed in a sum with elementary matrix variables A + BT

P

  P P −P   BP + BT

γ (−γ21)Bγ ≺ 0 ,

P = PT ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + BTFB ≺ 0 , B =

• BP

Bγ

• with the structural equality constraints

F = FP Fγ

• ,

FP =   P P −P   , P = PT , Fγ = −γ21

• The constraint A + BTFB ≺ 0 holds iff

(A, B, C = BT) is strictly-passifiable by F. ■ LMI converted to strict-passification problem, with equality constraints.

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LMIs are strict-passifiable systems

■ Procedure applies to any LMI: concludes with search of passifying F =    F1 ... FN    for a (symmetric) system (A, B, C = BT) with additional structural equality constraints that can be compacted in vec(Fi) = Sixi ⇔ Uivec(Fi) = 0 where vec(Fi) is the vector composed of stacked columns of Fi, xi are vectors of independent scalar decision variables and Ui = S⊥

i .

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LMIs are strict-passifiable systems

■ Procedure applies to any LMI: concludes with search of passifying F =    F1 ... FN    for a (symmetric) system (A, B, C = BT) with additional structural equality constraints that can be compacted in vec(Fi) = Sixi ⇔ Uivec(Fi) = 0 where vec(Fi) is the vector composed of stacked columns of Fi, xi are vectors of independent scalar decision variables and Ui = S⊥

i .

• When starting from the canonical representation L0 +

j ˆ

xjLj ≺ 0, then the structural constraints are all of the type Fj = ˆ xj1rj1 −ˆ xj1rj2

• and rj1 + rj2 can be very large and Uj is huge.
• F and Uis expected to be smaller when matrix representation.

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Structured adaptive control

Block-diagonal adaptive control with asymptotic structure

Theorem The following two conditions are equivalent: ➊ There exists a decentralized static control ui(t) = Fiyi(t) + wi(t) satisfying the structural constraints Uivec(Fi) = 0 for the system ˙ x(t) = Ax(t) +

• Biui(t) ,

yi(t) = Cix(t) , z(t) = y(t) that makes the closed-loop strictly passive (with respect to w/z). ➋ For all Γi ≻ 0, αi > 0 the following adaptive control ui(t) = Ki(t)yi(t) + wi(t) , ˙ Ki(t) = −yi(t)yT

i (t)Γi − αi · mat

• UT

i Ui · vec(Ki(t))

• Γi

makes the closed-loop globally strictly-passive.

‘mat’ is the function such that mat(vec(F)) = F.

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Structured adaptive control

Proof of ➊ ⇒ ➋

• ➊ reads as

∃F, ∃Q ≻ 0 : (A + BFC)TQ + Q(A + BFC) < 0, QB = CT, F = diag

• · · ·

Fi · · ·

• ,

Ui · vec(Fi) = 0 (1)

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Structured adaptive control

Proof of ➊ ⇒ ➋

• ➊ reads as

∃F, ∃Q ≻ 0 : (A + BFC)TQ + Q(A + BFC) < 0, QB = CT, F = diag

• · · ·

Fi · · ·

• ,

Ui · vec(Fi) = 0 (1)

• Let the Lyapunov function for the non-linear system (with adaptive law)

V (x, K) = 1 2

• xTQx +
• i

Tr

• (Ki − Fi)Γ−1(Ki − Fi)T

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Structured adaptive control

Proof of ➊ ⇒ ➋

• ➊ reads as

∃F, ∃Q ≻ 0 : (A + BFC)TQ + Q(A + BFC) < 0, QB = CT, F = diag

• · · ·

Fi · · ·

• ,

Ui · vec(Fi) = 0 (1)

• Let the Lyapunov function for the non-linear system (with adaptive law)

V (x, K) = 1 2

• xTQx +
• i

Tr

• (Ki − Fi)Γ−1(Ki − Fi)T
• After manipulations and using QB = CT, Ui · vec(Fi) = 0, we get:

˙ V (x, K) = xT(A+BFC)TQx +wTz −

• i

αi(Ui ·vec(Ki))T(Ui ·vec(Ki)).

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Structured adaptive control

Proof of ➊ ⇒ ➋ (continued)

˙ V (x, K) = xT(A+BFC)TQx +wTz −

• i

αi(Ui ·vec(Ki))T(Ui ·vec(Ki)).

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Structured adaptive control

Proof of ➊ ⇒ ➋ (continued)

˙ V (x, K) = xT(A+BFC)TQx +wTz −

• i

αi(Ui ·vec(Ki))T(Ui ·vec(Ki)). ▲ First term is strictly negative due to (1), until x = 0, ▲ Last term is strictly negative, until Ui · vec(Ki) = 0 ■ If no perturbations (w = 0) the system converges to the attractor A = {(x, K) : x = 0 , Ui · vec(Ki) = 0} ■ On the attractor ˙ Ki = 0: the gains Ki(∞) are constant.

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Structured adaptive control

Proof of ➊ ⇒ ➋ (continued)

˙ V (x, K) = xT(A+BFC)TQx +wTz −

• i

αi(Ui ·vec(Ki))T(Ui ·vec(Ki)).

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Structured adaptive control

Proof of ➊ ⇒ ➋ (continued)

˙ V (x, K) = xT(A+BFC)TQx +wTz −

• i

αi(Ui ·vec(Ki))T(Ui ·vec(Ki)). ■ For initial conditions at equilibrium and nonzero perturbations 0 ≤ V (x(t), K(t)) = t ˙ V (x, K)dτ < t wTz dτ ⇒ the system is strictly passive.

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Structured adaptive control

Proof of ➋ ⇒ ➊

• The system with adaptive control is globally asymptotically stable,

it converges to a asymptotically stable equilibrium: F i = Ki(∞) are stabilizing gains.

• Same reasoning holds for passivity.

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Structured adaptive control

Summary

■ All LMI problems are equivalent to static output-feedback strict-passification problems with structure constraints:

• gain F is block-diagonal
• sub-blocks should satisfy Uivec(Fi) = 0.

■ If a structured strict-passification problem admits solutions, the block-diagonal adaptive law with asymptotic structure will converge to

• ne of these.
• The LMIs can be solved by simulating the adaptive controlled systems.

▲ If the system converges Ki(∞) contain solutions of the LMIs. ▲ If does not converges the LMIs are infeasible.

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Numerical Example

Numerical example

• Consider the transfer function:

G(s) = s2 + s + 1 s2 + s + 2

• Problem: compute the H∞ norm (or at least an upper bound).

▲ In Matlab: norm(G, Inf, 1e-4) = 1.3251

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Numerical Example

Numerical example

• Consider the transfer function:

G(s) = s2 + s + 1 s2 + 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 ˙ Ki = −yiyT

i Γi − αi · mat

• UT

i Ui · vec(Ki)

• Γi ,

y1 ∈ R6 , y2 ∈ R with structural asymptotic constraints : F1 =   P PT −P   , P = PT ∈ R2×2 , F2 = −γ21 = −γ2.

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Numerical Example

• Parameters for simulating the adaptive law (simulation in Simulink)

▲ Initial conditions x = (1 . . . 1)T and Ki = 0 ▲ Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1

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Numerical Example

• Parameters for simulating the adaptive law (simulation in Simulink)

▲ Initial conditions x = (1 . . . 1)T and Ki = 0 ▲ Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1

• Convergence to zero of the ‘outputs’ yi

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Numerical Example

▲ Convergence to structured values of the adapted gains Ki

K1(∞) = 2 6 6 6 6 6 6 4 4.6330 1.0671 1.0671 10.7960 4.6330 1.0671 1.0671 10.7960 −4.6330 −1.0671 −1.0671 −10.7960 3 7 7 7 7 7 7 5 K2(∞) = −7.1307

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Numerical Example

▲ Evolution of the (1 : 2, 3 : 4) elements of K1 that converge to P ▲ Solution of the LMIs P = 4.6330 1.0671 1.0671 10.7960

• ,

γ = 2.6703 ≥ 1.3251 = γopt

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Numerical Example

• Test for feasible / unfeasible cases

▲ Only K1 is adapated, γ is slowly linearly modified ▲ Unstable behavior when γ < 1.3251 = γopt.

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Conclusions

Conclusions et perspectives

■ LMI feasibility problems can be solved by simulating systems ▲ Need for a parser to convert LMIs to adaptive control problem ▲ Simulation time is large - what is the best implementation ? ▲ Is simulation time polynomial w.r.t. size of problem ? ■ What about LMI optimization problems ? ▲ Decreasing parameters until system becomes unstable ? ▲ Minimizing gap with dual LMI problem (it works). ▲ Other ? ■ Solving time-varying LMI problems ?

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