[PPT] - From passivity-based adaptive control to LMI tuned adaptive control PowerPoint Presentation

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From passivity-based adaptive control to LMI tuned adaptive control

Dimitri Peaucelle FRontiers of ADaptive and nonlinear CONtrol (FRADCON 2018) ECC Workshop, Limassol, June 12, 2018

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Saint Petersburg, Russia

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Robust | Direct Adaptive

[Peaucelle 2000]

L(P) ≺ 0 ⇓ ˙ x = A(∆)x is robustly stable

[Fradkov 1974]

   ˙ x = Ax + Bu y = Cx    is Hyper Minimum Phase ⇓

Closed-loop x-strictly passive with

u = Ky, ˙ K = −yyT Γ, ∀Γ ≻ 0

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LMIs for passivity

■ ˙ x = Acx + Bu, y = Cx strictly passive iff ∃P ≻ 0 : AT

∃ P ≻ 0 ǫ > 0 F : (A + BFC)T P + P(A + BFC) = AT P + CT F T C + PA + CT FC ≺ −ǫI , PB = CT ▲ SOF design for strict passivity is an LMI problem ▲ ˙ x = Ax + Bu, y = Cx is Hyper Minimum Phase

iff it is "almost passive" (ie ∃F s.t. closed-loop strictly passive)

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LMIs for passivity based adaptive control

■ ˙ x = Ax + Bu, y = Cx HMP

, then it is x-strictly passive with adaptive control :

u = Ky, ˙ K = −yyT Γ, ∀Γ ≻ 0

▲ LMI implies along trajectories with u = Ky d dt(xT Px) + 2yT (F − K)y < −ǫx2 ▲ and d

▲ Hence V (x, K) < −ǫx2, i.e. asymptotic convergence of x to 0

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Robust & Direct Adaptive?

L(P, F) ≺ 0 ⇓    ˙ x = A(∆)x + B(∆)u y = C(∆)x    is robustly HMP ⇓

Closed-loop robustly stable with u = Ky, ˙

K = −yyT Γ, ∀Γ ≻ 0 ▲ LMI-based results to prove robustness of adaptive control.

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Robust & Direct Adaptive?

Closed-loop robustly stable with u = Fy

⇑ L(P, F) ≺ 0 ⇓    ˙ x = A(∆)x + B(∆)u y = C(∆)x    is robustly HMP ⇓

Closed-loop robustly stable with u = Ky, ˙

K = −yyT Γ, ∀Γ ≻ 0 ▼ Purpose of non-linear adaptive control if static output feedback (SOF) does the job? ▼ K is unbounded and shall diverge due to noise on measurements ▲ Special case where robust SOF design is LMI!

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Robust & Direct Adaptive?

Closed-loop robustly stable with u = Fy

⇑ B(P, F, G) ≺ 0 ⇓          ˙ x = A(∆)x + B(∆)u y = C(∆)x z = Gy         

is robustly HMP w.r.t. (u, z)

⇓

Closed-loop robustly stable with u = Ky, ˙

K = −zyT Γ, ∀Γ ≻ 0 ▼ BMI conditions (but with heuristic to solve it locally ▲ ) ▼ K is unbounded and shall diverge due to noise on measurements ▼ Purpose of non-linear adaptive control if SOF does the job?

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Robust & Direct Adaptive?

Closed-loop robustly stable with u = Fys

⇑ B(P, F, G, D) ≺ 0 ⇓          ˙ x = A(∆)x + B(∆)u ys = C(∆)x + Du z = Gys         

is robustly HMP w.r.t. (u, z)

⇓

Closed-loop robustly stable with u = Kys, ˙

K = −zyT

▼ BMI conditions (but with heuristic to solve it locally ▲ ) ▼ K is unbounded and shall diverge due to noise on measurements ▼ Purpose of non-linear adaptive control if SOF does the job? ▼ Purpose of controlling ys ?

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Robust & Direct Adaptive?

D K Sys

⇔

D K Sys

The shunt is equivalent to a feedback on the control gain

▲ Even thought K is unbounded, the actual gain seen by the plant K ⋆ D is bounded ■ Justifies the search for adaptive control with gains forced to be bounded

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Robust & Direct Adaptive?

Closed-loop robustly stable with u = Fy

⇑ B(P, F, G, D) ≺ 0 ⇓          ˙ x = A(∆)x + B(∆)u y = C(∆)x z = Gy + Du         

is robustly HMP w.r.t. (u, z)

⇓

Closed-loop robustly stable with u = Ky, ˙

K = PE(D)[−GyyT ]Γ, ∀Γ ≻ 0

where PE(D) is a projection forcing K in a set with size proportional to D−1

▼ Purpose of non-linear adaptive control if SOF does the job? ▼ BMI conditions (but with heuristic to solve it locally ▲ ) ▲ Adaptive control with bounded gains

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Robust & Direct Adaptive?

Closed-loop robustly stable with u = F(∆)y

⇑ B[v](P [v], F [v], S, G, D, Kc) ≺ 0 ⇓

Closed-loop robustly stable with u = Ky, ˙

K = PE(D,Kc)[−GyyT ]Γ, ∀Γ ≻ 0

where PE(D,Kc) is a projection forcing K in a set with size proportional to D−1

▼ BMI conditions (but with heuristic to solve it locally ▲ ) ▲ There might not be any SOF that does the job (only a gain scheduled one) ▲ Adaptive control with bounded gains

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Sketch of the proof

■ Robustness is for linear systems with uncertainties ˙ x = Aζx + Bζu, y = Cx

where affine polytopic uncertainty is assumed for simplicity

Bζ

ζv

B[v]

ζv ≥ 0,

ζv = 1.

Other types of uncertainties easily handled with appropriate LMI-based robustness tools

■ Matrix inequalities B[v](P [v], F [v], S, G, D, Kc) ≺ 0

assumed to hold for all vertices v = 1 . . . ¯

v ▲ Thanks to S-variable type result,

inequalities hold true for all ζ ∈ {ζv ≥ 0,

¯

Bζ(Xζ, Fζ, S, G, D, Kc) ≺ 0

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Sketch of the proof

Bζ(Pζ, Fζ, S, G, D, Kc) ≺ 0

Implies the following inequalities along closed-loop trajectories

2 ˙ xT Pζx + ǫxT x + 2yT (D − (Kc − K)T (Kc − K) − GT (K − Fζ))y ≤ 0 (Kc − Fζ)T (Kc − Fζ) D

Adaptive law ˙

K = PE(D,Kc)[−GyyT ]Γ is conceived in order to satisfy (Kc − K)T (Kc − K) D Tr( ˙ KΓ−1 + GyyT )(K − F) ≤ 0

whatever F such that (Kc − F)T (Kc − F) D. With these properties one gets

2 ˙ xT Pζx + ǫxT x + 2Tr ˙ KT Γ−1(Fζ − K) ≤ 0

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Sketch of the proof

■ Stability is proved with the parameter-dependent Lyapunov function Vζ(x, K) = xT Pζx + Tr(Fζ − K)T Γ−1(Fζ − K), ˙ Vζ(x, K) ≤ −ǫxT x ▲ Both Pζ and Fζ are parameter-dependent ▲ Fζ is a stabilyzing static output feedback

cannot be implemented without knowledge of ζ

▲ Adaptive control achieves the same robustness property without knowledge of ζ ▲ Methodology applies without difficulty for the design of structured adaptive gains

e.g. K can be constrained to be diagonal

▲ Result extend to descriptor uncertain systems

thus applicable to systems rational in the uncertainties ζ

▼ The results are in terms of BMIs : Non-convex

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Heuristic method for adaptive control design

Ko is diagonal and contains the gains to be adapted Gζ(s) has a descriptor state-space representation, affine in the uncertainties ζ

solve iteratively the BMIs while increasing the size of the set of uncertainties Z

▲ The algorithm converges on examples to sets Z

containing values ζ for which Hζ(s) ⋆ C(s) is unstable

▼ Does not mean that no robustly stabilizing LTI control exists

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Applied to a satellite attitude control example

[Leduc 2017] Uncertainties on the inertia (large variations during deployment of appendices)

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Tested on the full size simulation model at CNES

Baseline LTI control designed for the fully deployed configuration Adaptive control stable for all configurations Adaptive & Robust!