From passivity-based adaptive control to LMI tuned adaptive control or how Alexander Fradkov convinced me that direct adaptive control can be robust Dimitri Peaucelle FRontiers of ADaptive and nonlinear CONtrol (FRADCON 2018) ECC Workshop, Limassol, June 12, 2018
Saint Petersburg, Russia FRADCON’2018 α 70 ! D. Peaucelle 1 June 12, 2018
Robust | Direct Adaptive [Fradkov 1974] [Peaucelle 2000] x = Ax + Bu ˙ is Hyper Minimum Phase L ( P ) ≺ 0 y = Cx ⇓ ⇓ x = A (∆) x is robustly stable ˙ Closed-loop x -strictly passive with K = − yy T Γ , ∀ Γ ≻ 0 u = Ky, ˙ FRADCON’2018 α 70 ! D. Peaucelle 2 June 12, 2018
LMIs for passivity ■ ˙ x = A c x + Bu , y = Cx strictly passive iff A T c P + PA c ≺ − ǫI, PB = C T ∃ P ≻ 0 : ● F static output feecback ( A c = A + BFC ), closed loop strictly passive iff P ≻ 0 ( A + BFC ) T P + P ( A + BFC ) , PB = C T ∃ : ǫ > 0 = A T P + C T F T C + PA + C T FC ≺ − ǫI F ▲ 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) FRADCON’2018 α 70 ! D. Peaucelle 3 June 12, 2018
LMIs for passivity based adaptive control ■ ˙ x = Ax + Bu , y = Cx HMP , then it is x -strictly passive with adaptive control : K = − yy T Γ , ∀ Γ ≻ 0 u = Ky, ˙ ● Proof with Lyapunov function V ( x, K ) = x T Px + 2 Tr ( K − F ) T Γ − 1 ( K − F ) ▲ LMI implies along trajectories with u = Ky d dt ( x T Px ) + 2 y T ( F − K ) y < − ǫ � x � 2 dt (2 Tr ( K − F ) T Γ − 1 ( K − F )) = − 2 y T ( F − K ) y ▲ and d ▲ Hence V ( x, K ) < − ǫ � x � 2 , i.e. asymptotic convergence of x to 0 FRADCON’2018 α 70 ! D. Peaucelle 4 June 12, 2018
Robust & Direct Adaptive? L ( P, F ) ≺ 0 ⇓ x = A (∆) x + B (∆) u ˙ is robustly HMP y = C (∆) x ⇓ K = − yy T Γ , ∀ Γ ≻ 0 Closed-loop robustly stable with u = Ky, ˙ ▲ LMI-based results to prove robustness of adaptive control. FRADCON’2018 α 70 ! D. Peaucelle 5 June 12, 2018
Robust & Direct Adaptive? Closed-loop robustly stable with u = Fy ⇑ L ( P, F ) ≺ 0 ⇓ x = A (∆) x + B (∆) u ˙ is robustly HMP y = C (∆) x ⇓ K = − yy T Γ , ∀ Γ ≻ 0 Closed-loop robustly stable with u = Ky, ˙ ▼ 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! FRADCON’2018 α 70 ! D. Peaucelle 6 June 12, 2018
Robust & Direct Adaptive? Closed-loop robustly stable with u = Fy ⇑ B ( P, F, G ) ≺ 0 ⇓ x = A (∆) x + B (∆) u ˙ is robustly HMP w.r.t. ( u, z ) y = C (∆) x z = Gy ⇓ K = − zy T Γ , ∀ Γ ≻ 0 Closed-loop robustly stable with u = Ky, ˙ ▼ 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? FRADCON’2018 α 70 ! D. Peaucelle 7 June 12, 2018
Robust & Direct Adaptive? Closed-loop robustly stable with u = Fy s ⇑ B ( P, F, G, D ) ≺ 0 ⇓ x = A (∆) x + B (∆) u ˙ is robustly HMP w.r.t. ( u, z ) y s = C (∆) x + Du z = Gy s ⇓ Closed-loop robustly stable with u = Ky s , ˙ K = − zy T s 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? ▼ Purpose of controlling y s ? FRADCON’2018 α 70 ! D. Peaucelle 8 June 12, 2018
Robust & Direct Adaptive? D Sys ⇔ Sys K K D 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 FRADCON’2018 α 70 ! D. Peaucelle 9 June 12, 2018
Robust & Direct Adaptive? Closed-loop robustly stable with u = Fy ⇑ B ( P, F, G, D ) ≺ 0 ⇓ x = A (∆) x + B (∆) u ˙ is robustly HMP w.r.t. ( u, z ) y = C (∆) x z = Gy + Du ⇓ K = P E ( D ) [ − Gyy T ]Γ , ∀ Γ ≻ 0 Closed-loop robustly stable with u = Ky, ˙ where P E ( 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 FRADCON’2018 α 70 ! D. Peaucelle 10 June 12, 2018
Robust & Direct Adaptive? Closed-loop robustly stable with u = F (∆) y ⇑ B [ v ] ( P [ v ] , F [ v ] , S, G, D, K c ) ≺ 0 ⇓ K = P E ( D,K c ) [ − Gyy T ]Γ , ∀ Γ ≻ 0 Closed-loop robustly stable with u = Ky, ˙ where P E ( D,K c ) 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 FRADCON’2018 α 70 ! D. Peaucelle 11 June 12, 2018
Sketch of the proof ■ Robustness is for linear systems with uncertainties x = A ζ x + B ζ u, ˙ y = Cx where affine polytopic uncertainty is assumed for simplicity v ¯ v ¯ � � � � � � A [ v ] B [ v ] = ζ v ≥ 0 , ζ v = 1 . ζ v , A ζ B ζ v =1 v =1 Other types of uncertainties easily handled with appropriate LMI-based robustness tools ■ Matrix inequalities B [ v ] ( P [ v ] , F [ v ] , S, G, D, K c ) ≺ 0 assumed to hold for all vertices v = 1 . . . ¯ v ▲ Thanks to S-variable type result, � ¯ v inequalities hold true for all ζ ∈ { ζ v ≥ 0 , v =1 ζ v = 1 } . B ζ ( X ζ , F ζ , S, G, D, K c ) ≺ 0 FRADCON’2018 α 70 ! D. Peaucelle 12 June 12, 2018
Sketch of the proof B ζ ( P ζ , F ζ , S, G, D, K c ) ≺ 0 Implies the following inequalities along closed-loop trajectories x T P ζ x + ǫx T x + 2 y T ( D − ( K c − K ) T ( K c − K ) − G T ( K − F ζ )) y ≤ 0 2 ˙ ( K c − F ζ ) T ( K c − F ζ ) � D K = P E ( D,K c ) [ − Gyy T ]Γ is conceived in order to satisfy Adaptive law ˙ ( K c − K ) T ( K c − K ) � D K Γ − 1 + Gyy T )( K − F ) ≤ 0 Tr ( ˙ whatever F such that ( K c − F ) T ( K c − F ) � D . With these properties one gets x T P ζ x + ǫx T x + 2 Tr ˙ K T Γ − 1 ( F ζ − K ) ≤ 0 2 ˙ FRADCON’2018 α 70 ! D. Peaucelle 13 June 12, 2018
Sketch of the proof ■ Stability is proved with the parameter-dependent Lyapunov function V ζ ( x, K ) = x T P ζ x + Tr ( F ζ − K ) T Γ − 1 ( F ζ − K ) , V ζ ( x, K ) ≤ − ǫx T x ˙ ▲ Both P ζ and F ζ are parameter-dependent ▲ F ζ 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 FRADCON’2018 α 70 ! D. Peaucelle 14 June 12, 2018
Heuristic method for adaptive control design ● Design a controller C ( s ) for the nominal plant H ζ =0 ( s ) ● Convert the loop H ζ ( s ) ⋆ C ( s ) into G ζ ( s ) ⋆ K o where K o is diagonal and contains the gains to be adapted G ζ ( s ) has a descriptor state-space representation, affine in the uncertainties ζ ● Starting from K c = K o and a small set of uncertainties ζ ∈ Z o 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 H ζ ( s ) ⋆ C ( s ) is unstable ▼ Does not mean that no robustly stabilizing LTI control exists FRADCON’2018 α 70 ! D. Peaucelle 15 June 12, 2018
Applied to a satellite attitude control example [Leduc 2017] Uncertainties on the inertia (large variations during deployment of appendices) FRADCON’2018 α 70 ! D. Peaucelle 16 June 12, 2018
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! FRADCON’2018 α 70 ! D. Peaucelle 17 June 12, 2018
To be continued... Thank you! and happy α 70 ! FRADCON’2018 α 70 ! D. Peaucelle 18 June 12, 2018
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