Structured adaptive control for solving LMIs Alexandru-Razvan Luzi, - - PowerPoint PPT Presentation

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Structured adaptive control for solving LMIs Alexandru-Razvan Luzi, - - PowerPoint PPT Presentation

Sep 25, 2023 249 likes •480 views

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

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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

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What are LMIs ?

What are LMIs ?

■ LMIs: Linear Matrix Inequalities max

  • biyi

: F0 +

  • yiFi ≺ 0
  • LMIs are SDP: Semi-Definite Programming

min cTx : 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

P ≻ 0 , ATP + PA ≺ 0

  • New results for: combinatorial optimization, robust optimization,

algebraic geometry, cryptography, optimal control...

2/20

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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 ■ Framework initiated by V.A. Yakubovich in the late 1960’s

  • Contributions: new adaptive control law with asymptotic structure

+ may solve LMIs

3/20

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Outline

Outline

1

Passivity-based adaptive control

2

LMIs are strict-passifiable systems

3

Structured adaptive control

4

Numerical Example

4/20

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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.

5/20

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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(∞).

6/20

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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 of G(s) ∼ (A, B, C, D)

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

  • ≺ 0 ,

P = PT ≻ 0.

7/20

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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 of G(s) ∼ (A, B, C, D)

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

  • ≺ 0 ,

P = PT ≻ 0.

  • Converted with simple manipulations into one simple LMI

A + BTFB ≺ 0 ▲ with structural equality constraints on F F = FP Fγ

  • ,

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

  • CT C

CT D DT C DT D

  • , B =
  • A

B 1 1 1

  • 7/20
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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)

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

  • ≺ 0 ,

P = PT ≻ 0.

  • Converted with simple manipulations into one simple LMI

A + BTFB ≺ 0 ▲ with structural equality constraints on F 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 (condition ➊). ▲ LMI converted to strict-passification problem, with equality constraints.

8/20

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LMIs are strict-passifiable systems

■ Procedure applies to any LMI:

  • Concludes with search of passifying gain F =

   F1 ... FN   

  • for a (symmetric) system (A, B, C = BT)
  • with additional structural equality constraints that can be compacted in

Uivec(Fi) = 0

  • Where vec(Fi) is the vector composed of stacked columns of Fi.
  • ▲ All constraints Uivec(Fi) = 0 include the constraint Fi = Fi T.

9/20

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Structured adaptive control

Block-diagonal adaptive control with asymptotic structure

Theorem Assume A = AT and C = BT, then the following two are equivalent: ➊ There exists a symmetric decentralized static control ui(t) = Fiyi(t) satisfying structural constraints Uivec(Fi) = 0 that stabilizes asymptotically ˙ x(t) = Ax(t) +

  • Biui(t) ,

yi(t) = Cix(t). 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 asymptotically stable and the adaptive gains converge to constant values Fi = Ki(∞) solution to condition ➊.

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

10/20

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Structured adaptive control

Proof of ➊ ⇒ ➋

  • Stability of a symmetric matrix A + BFC proved by V (x) = 1

2xTx,

i.e. ➊ implies ∃F : (A + BFC)T + (A + BFC) < 0, 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

  • xTx +
  • i

Tr

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

˙ V (x, K) = xT(A + BFC)Tx −

  • 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)Tx −

  • 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. ■ The system converges to the attractor A = {(x, K) : x = 0 , Ui · vec(Ki) = 0} ■ Reasoning in [Ioannou&Sun 96] allows to conclude that Ki(t) converges to a constant gain Ki(∞).

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Structured adaptive control

Proof of ➋ ⇒ ➊

  • The system with adaptive control is globally asymptotically stable,

it converges to an asymptotically stable equilibrium: Fi = Ki(∞) are stabilizing gains

13/20

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Structured adaptive control

Summary

■ 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 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(∞) = Fi are solutions of the LMIs. ▲ If does not converges the LMIs are infeasible.

14/20

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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

15/20

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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.

15/20

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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

16/20

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Numerical Example

▲ Convergence to structured values of the adapted gains Ki

K1(∞) =         4.6330 1.0671 1.0671 10.7960 4.6330 1.0671 1.0671 10.7960 −4.6330 −1.0671 −1.0671 −10.7960         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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