Conditions LMI de synth` ese de commandes adaptatives directes pour les syst` emes lin´ eaires ‘presque stables’ Dimitri PEAUCELLE LAAS-CNRS - Universit´ e de Toulouse - FRANCE Cooperation program between CNRS, RAS and RFBR: A. Fradkov, B. Andrievsky Application to Demeter satellite with CNES: A. Drouot, Ch. Pittet, J. Mignot Application to ‘helicopter’ benchmark: B. Andrievsky, V. Mahout
Introduction Simple adaptive control (SAC) For a system y ( t ) = [Σ u ]( t ) to follow reference y r ˙ K ( t ) = − Gy ( t ) e T ( t )Γ , e ( t ) = y ( t ) − y r ( t ) u ( t ) = K ( t ) e ( t ) , ■ K is driven to minimize the square of the error J ( t ) = e T ( t ) e ( t ) ● In the scalar case k = − γ ∂ ( y − y r ) ˙ ( y − y r ) ≃ − γgye ∂k ● Gy : approximation of the gradient of J with respect to K (for the closed-loop) ● Γ > O : weight on the adaptation speed ● [Fradkov, Kaufman et al, Ioannou, Barkana] G is chosen with respect to closed-loop passivity conditions ▲ Choice of G depends of the systems model ▲ Adaptive control intended for uncertain systems: robust 1 Seminar at LAGIS, Lille, October 2010
Outline ❶ Passivity conditions for Simple Adaptive Control (SAC) ● LMI formulas for SAC - robustness issues ❷ Robustness to noise on measurements & to parametric uncertainty ● Barrier type corrective term ● Passivity with respect to an output with feedthrough gain ❸ Examples and some other features of SAC ● L 2 performance ● Stable neighborhood of the origin in case of time-varying systems 2 Seminar at LAGIS, Lille, October 2010
❶ Passivity conditions for SAC SAC for LTI systems ( u ∈ R m , y ∈ R p , m ≤ p ) ● Let a linear system ˙ x = Ax + Bu , y = Cx ˙ K = − Gyy T Γ u = Ky , ● and SAC ■ Closed-loop stability is guaranteed if ∃ F : ˙ x = ( A + BFC ) x + Bw, z = GCx strictly passive or equivalenty if ∃ F, P > O : ( A + BFC ) T P + P ( A + BFC ) < O , PB = C T G T � �� � Υ ● Proof using Lyapunov function V ( x, K ) = x T Px + Tr (( K − F )Γ − 1 ( K − F ) T ) . V = x T Υ x + 2 x T PB ( K − F ) y + 2 Tr ( ˙ ˙ K Γ − 1 ( K − F ) T ) = x T Υ x 3 Seminar at LAGIS, Lille, October 2010
❶ Passivity conditions for SAC SAC versus SOF ■ Closed-loop stability with SAC is guaranteed if system is ‘almost passive’ ∃ F, P : ( A + BFC ) T P + P ( A + BFC ) < O , PB = C T G T � �� � Υ ▲ Stability with SAC proved by existence of some stabilizing SOF ( u = Fy ) - Why complicating the control ? ● The condition happens to be LMI+E (for given G ): ∃ F, P : A T P + PA + C T ( G T F + F T G ) C < O , PB = C T G T ● Any F = − kG with k large enough stabilizes the system (high gain) ▲ Not all SOF stabilizable systems will satisfy such constraints ▲ The SAC design problem is to find G : non convex problem. 4 Seminar at LAGIS, Lille, October 2010
❶ Passivity conditions for SAC Robustness of SAC ● Let an uncertain LTI system ˙ x = A (∆) x + B (∆) u , y = C (∆) x ˙ K = − Gyy T Γ u = Ky , ● and SAC ■ Closed-loop robust stability with SAC is guaranteed if ∃ F (∆) , P (∆) : A T (∆) P (∆) + P (∆) A (∆) + C T (∆)( G T F (∆) + F T (∆) G ) C (∆) < O P (∆) B (∆) = C T (∆) G T ● Robustness techniques may be applied to the LMI (given G ) ▲ Equality constraint almost impossible to guarantee robustly P (∆) B (∆) = C T (∆) G T , ∀ ∆ ∈ ∆ ∆ !!! 5 Seminar at LAGIS, Lille, October 2010
❶ Passivity conditions for SAC Divergence of SAC due to noise ● Assume noisy measurements y ( t ) = Cx ( t ) + d ( t ) K = − Gyy T Γ = − G ( Cx + d )( x T C T + d T )Γ ˙ ▲ K ( t ) will diverge even if x → 0 (if d does not go to zero). ▲ Not acceptable in practice ● Most often corrective terms are added such as ˙ K = − Gyy T Γ − µ ( K − F 0 ) ▲ But then K ( t ) → F 0 : the closed-loop characteristics tend to those with SOF u = F 0 y - Why complicating the control ? 6 Seminar at LAGIS, Lille, October 2010
Outline ❶ Passivity conditions for Simple Adaptive Control (SAC) ● LMI formulas for SAC - robustness issues ❷ Robustness to noise on measurements & to parametric uncertainty ● Barrier type corrective term ● Passivity with respect to an output with feedthrough gain ❸ Examples and some other features of SAC ● L 2 performance ● Stable neighborhood of the origin in case of time-varying systems 7 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Dead-zone + barrier corrective term ● Usual corrective term ˙ K = − Gyy T Γ − µ ( K − F 0 ) ▲ Corrective term always active even if K does not diverge ▲ Corrective term does not guarantee K to be bounded in given set ■ Proposed corrective term ˙ K = − Gyy T Γ − φ ( K − F 0 )Γ � �� � ˆ K φ ( ˆ K ) = ψ ( � ˆ K � 2 ) ˆ K dψ ψ (0 ≤ k ≤ ν ) = 0 , dk ( ν ≤ k ≤ βν ) > 0 , ψ ( νβ ) = + ∞ K � 2 = Tr ( ˆ ● Example: weighted Frobenius norm � ˆ K ˆ D ˆ K T ) and ψ ( ν ≤ k ≤ βν ) = exp ( µk − log( βν − k )) 8 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Dead-zone + barrier corrective term ■ Proposed corrective term ˙ K = − Gyy T Γ − φ ( K − F 0 )Γ � �� � ˆ K φ ( ˆ K ) = ψ ( � ˆ K � 2 ) ˆ K dψ ψ (0 ≤ k ≤ ν ) = 0 , dk ( ν ≤ k ≤ βν ) > 0 , ψ ( νβ ) = + ∞ ● Corrective term active only when � K − F 0 � > ν ● Guarantees that K ( t ) is bounded around F 0 : � K − F 0 � < νβ ▲ β > 1 can be chosen arbitrarily based on practical considerations ● ˆ D defines the geometry of the set � ˆ K � = Tr ( ˆ K ˆ D ˆ K T ) ≤ ν ● ν defines the dead-zone and barrier levels ▲ Best to maximize the set � ˆ i.e. maximize ν and minimize Tr ˆ K � ≤ ν , D 9 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Feedthrough gain for robust passivity ˙ K = − Gyy T Γ x = Ax + Bu , y = Cx ˙ u = Ky , with SAC ■ Closed-loop stability is guaranteed if ∃ F : ˙ x = ( A + BFC ) x + Bw, z = GCx strictly passive or equivalenty if ∃ F, P : ( A + BFC ) T P + P ( A + BFC ) < O , PB = C T G T ▲ Need for conditions without equality constraints ⇒ need for a feedthrough gain ( z = GCx + Dw ) 10 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Feedthrough gain for robust passivity ■ Passivity conditions without equality constraints x = ( A + BFC ) x + Bw, z = GCx + Dw strictly passive ˙ if and only if ( A + BFC ) T P + P ( A + BFC ) PB − C T G T < O ∃ P : B T P − GC − D − D T ● Feedthrough gain D always exists if system is SOF stabilizable ● If D is small, then conditions are close to original ones ▲ Choice of F = − kG with k ≫ 1 no more valid ▲ Gains should be bounded ● Gains are bounded thanks to corrective term φ 11 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Main result - part 1 ● Let F 0 be a stabilizing SOF for x = Ax + Bu , y = Cx ˙ ■ There exists ( P > O , G, ˆ D ) solution to ( A + BF 0 C ) T P + P ( A + BF 0 C ) PB − C T G T < O − ˆ B T P − GC D ● minimize Tr ˆ D and choose • G for the adaptation gain ˙ K = − Gyy T Γ − φ ( K − F 0 )Γ � �� � ˆ K K � 2 = Tr ( ˆ • � ˆ K ˆ D ˆ K T ) for the norm in the corrective term φ • F 0 as the center of the set around which the adaptation is performed 12 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Main result - part 2 ■ ( F 0 , G, ˆ D ) being chosen, there exist ( Q > O , R, F, T, ν ) solution to QB − C T G T R ≥ O ˆ B T Q − GC D ( F − F 0 ) T T ≥ O , Tr T ≤ ν ˆ D − 1 ( F − F 0 ) ( A + BF 0 C ) T Q + Q ( A + BF 0 C ) + νβC T C + R + C T ( G T ( F − F 0 ) + ( F − F 0 ) T G ) C < O ● maximize ν and take it for the levels in the corrective term φ ■ SAC defined by ( G, F 0 , ˆ D, ν ) stabilizes the system. Proof with V ( x, K ) = x T Qx + Tr (( K − F )Γ − 1 ( K − F ) T ) . 13 Seminar at LAGIS, Lille, October 2010
❷ Robustness to noise on measurements & to parametric uncertainty Characteristic of the results ● ‘Almost passive’ conditions extended to ‘almost stable’ SAC can be applied to all SOF stabilizable systems ● Stability is proved for SAC with the corrective barrier function Moreover, K ( t ) is strictly bounded, even w.r.t. perturbations and noise ● The gain K ( t ) remains ‘close’ to initial SOF guess F 0 Interesting feature for practitioners: keep close to a ‘safe’ situation Benefit of adaptation expected to be improved if domain is large [Submitted to IFAC World Congress 2011, Milano] 14 Seminar at LAGIS, Lille, October 2010
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