Passification-Based Adaptive Control : Robustness Issues Dimitri PEAUCELLE † & Alexander FRADKOV ‡ & Boris ANDRIEVSKY ‡ † LAAS-CNRS - Toulouse, FRANCE ‡ IPME-RAS - St Petersburg, RUSSIA CNRS-RAS research cooperation program No. 19134 Complex Program of the Presidium of RAS #22 ”Control processes”, project 1.8.
Introduction Passification-based adaptive control [Fradkov 1974] Let an LTI Σ and assume there exists F and G such that Σ ⋆ F is G -passive u(t) y(t) V ( x ( t )) < V ( x (0)) Σ (s) G i.e. is passive t 0 [ u ( θ ) ∗ Gy ( θ )] dθ + � F then whatever positive Γ i > 0 , Σ is G -passified by the PBAC ˙ K i ( t ) = − y ∗ u ( t ) = K ( t ) y ( t ) , i ( t )Γ i Gy ( t ) . Technical remark ➞ One can always take F = − kG with k sufficiently large ➞ F exists if G Σ is hyper-minimum-phase & 1 ROCOND’06, 5-7 July 2006, Toulouse
Introduction Known Advantages of the PBAC ➚ Good observed behavior w.r.t. uncertainties and non-linearities ➚ Simple to design ➚ Based on physical meaning Drawbacks ➘ Need to prove the robustness properties ➘ Need for numerical methods for choosing G ➘ Divergence of K ( t ) due to disturbances Outline ① Modified PBAC that bounds K ( t ) + robustness conditions ② BMI design of G for the nominal system ③ LMI design of Parameter-Depenent F (∆) that fulfills robustness conditions ④ Example : autonomous aircraft & 2 ROCOND’06, 5-7 July 2006, Toulouse
① Bounded PBAC Bounded Passification Based Adaptive Algorithm ➞ Let B a bounded set of C m and let a penalty function φ : C m �→ C m : φ ( K ) = 0 ∀ K ∈ B ( K − F ) ∗ φ ( K ) ≥ 0 ∀ F ∈ B ➞ If there exists F (∆) ∈ B and G such that Σ(∆) ⋆ F (∆) is G -passive for all ∆ ∈ ∆ , then whatever positive Γ i > 0 , Σ(∆) is robustly G -passified by the BPBAC ˙ K i ( t ) = − y ∗ u ( t ) = K ( t ) y ( t ) , i ( t )Γ i Gy ( t ) − Γ i φ ( K i ( t )) . ➞ Convergence of the BPBAC is such that x ( ∞ ) = 0 , K ( ∞ ) ∈ B , Σ(∆) ⋆ K ( ∞ ) is G -passive & 3 ROCOND’06, 5-7 July 2006, Toulouse
① Bounded PBAC Proof of BPBAC properties ➞ The considered LTI uncertain models Σ(∆) are such that: x = A (∆) x + Bu , y = Cx . ˙ ➞ x = 0 and K = F (∆) are stable equilibrium points. ➞ Σ(∆) ⋆ F (∆) being G -passive implies the existence of H (∆) : H (∆) = H ∗ (∆) > 0 , H (∆) B = C ∗ G ∗ H (∆) A (∆ , F (∆)) + A ∗ (∆ , F (∆)) H (∆) < 0 and the BPBAC closed-loop system has a storage function of the class l V ( x, K, ∆) = 1 2 x ∗ H (∆) x + 1 � ( K i − F i (∆)) ∗ Γ − 1 i ( K i − F i (∆)) 2 i =1 & 4 ROCOND’06, 5-7 July 2006, Toulouse
② BMI design of G for the nominal system Conditions for BPBAC passification Existence of F (∆) ∈ B and G such that Σ(∆) ⋆ F (∆) is G -passive for all ∆ ∈ ∆ . Conservative procedure for the BPBAC conditions ➞ Design G such that Σ(∆ = 0 ) is G -passifiable via static output feedback. ➞ For a given G , prove the existence of F (∆) such that Σ(∆) ⋆ F (∆) is G -passive for all uncertainties. & 5 ROCOND’06, 5-7 July 2006, Toulouse
② BMI design of G for the nominal system G -passification of Σ( 0 ) [Fradkov 1976] H = H ∗ > 0 , HB = C ∗ G ∗ H ( A ( 0 ) + BFC ) + ( A ( 0 ) + BFC ) ∗ H < 0 Along solutions, there always exists F = − kG with k sufficiently large. BMI (YALMIP+PENBMI) design ➞ Choose a (large) value of k ➞ Choose an upper bound on H (we took ¯ h = 1 ) for scaling the solutions ➞ Declare in YALMIP the following BMI problem h 1 > H > 0 , HB = C ∗ G ∗ , t > − 1 ¯ HA ( 0 ) + A ∗ ( 0 ) H − kC ∗ G ∗ GC < t 1 ➞ Minimize t using PenBMI, if it returns t < 0 , the procedure succeeded. & 6 ROCOND’06, 5-7 July 2006, Toulouse
③ LMI design of F (∆) ”Quadratic stability”, norm-bounded result For a given G , if there exists H , F and ρ such that H > 0 , HB = C ∗ G ∗ , F ∈ B ∗ � HA ( 0 ) + C ∗ G ∗ FC � C ∗ C ∗ HB ∆ ∆ ∆ + ρ < 0 B ∗ D ∗ D ∗ − 1 ∆ H ∆ ∆ then Σ(∆) ⋆ F is G -passive for all uncertainties ∆ ∗ ∆ ≤ ρ 1 where Σ(∆) : x = [ A ( 0 ) + B ∆ ∆( 1 − D ∆ ∆) − 1 C ∆ ] x + Bu , y = Cx . ˙ Remarks ➞ Maximization of ρ is LMI ➞ If ∆ ⊂ ∆ ρ = { ∆ ∗ ∆ ≤ ρ 1 } the problem is solved with F (∆) = F ➞ Replacing in the LMIs A ( 0 ) by A (∆ i ) for some ∆ i , gives a couple ( F i , ρ i ) such that Σ(∆) ⋆ F i is robustly G -passive w.r.t. { ∆ i + ∆ : ∆ ∈ ∆ ρ i } . & 7 ROCOND’06, 5-7 July 2006, Toulouse
③ LMI design of F (∆) Solving the LMI conditions for some finite sequence { ∆ i } gives sequences { F i } , { ρ i } that defines an F (∆) for � { ∆ i + ∆ : ∆ ∈ ∆ ρ i } . ρ 3 ∆ 1 ∆ 2 ρ 1 ∆ ∆ 4 ∆ 3 ρ 2 ρ 4 If ∆ ⊂ � { ∆ i + ∆ : ∆ ∈ ∆ ρ i } the robustness BPABC G -passification conditions are fulfilled. & 8 ROCOND’06, 5-7 July 2006, Toulouse
④ Example : autonomous aircraft Linearized fourth-order model of lateral dynamics for an autonomous aircraft including model of actuator dynamics. Uncertainty is the flight altitude ( ∆ = h ). System has 3 outputs, one input. ➞ BMI design of G for the nominal system h = 5 km. For two choices of high gain k � � G 1 PenBMI = 10 − 2 4 . 5404 2 . 8436 1 . 7107 � � G 2 PenBMI = 10 − 3 8 . 4000 5 . 4505 3 . 0961 and after simple analysis step: � � � � G 1 = , G 2 = 4 3 2 8 5 3 & 9 ROCOND’06, 5-7 July 2006, Toulouse
④ Example : autonomous aircraft ➞ Choice of LMI representable B (bounded set of control gains) � � F = − 10 ≤ f i ≤ 10 , f 1 f 2 f 3 and φ dead-zone penalty function of the BPBAC: φ −10 φ ( f i ) = f i − sat 10 ( f i ) 10 f & 10 ROCOND’06, 5-7 July 2006, Toulouse
④ Example : autonomous aircraft ➞ LMI design of admissible h intervals 2 1.5 1.5 1 1 0.5 0.5 Imaginary axis Imaginary axis 0 0 − 0.5 − 0.5 − 1 − 1 − 1.5 − 1.5 − 2 − 8 − 6 − 4 − 2 0 2 4 6 − 8 − 6 − 4 − 2 0 2 4 6 Real axis Real axis Admissible ∆ ∈ C for G 1 Admissible ∆ ∈ C for G 2 Result is h ∈ [0 9 . 2976] Result is h ∈ [0 9 . 6213] Simulations are done with the choice of G = G 2 . & 11 ROCOND’06, 5-7 July 2006, Toulouse
④ Example : autonomous aircraft Simulations with Γ i = 1 h = 5 km h = 9 . 6 km 4 5 60 10 5 2 40 0 0 0 − 5 20 − 10 − 2 0 − 15 − 4 − 20 − 5 − 20 − 25 − 6 − 30 − 40 − 8 − 35 − 10 − 10 − 60 − 40 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 y ( t ) k ( t ) y ( t ) k ( t ) h = 8 km with noisy measurements and saturated input ( ± 20 ) 20 30 15 20 10 10 5 0 0 − 10 − 5 − 20 − 10 − 30 − 15 − 20 − 40 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 & 12 ROCOND’06, 5-7 July 2006, Toulouse
Conclusion ➚ LMI-based proof of robustness ➚ Easy to generalize to other uncertain modeling ➚ BMI design of G ➚ K ( t ) constrained to converge to a bounded set ➘ Conservative step : design of G a prioiri before checking the LMI robustness conditions ✪ Need for Γ i design results : convergence speed, disturbance rejections ... & 13 ROCOND’06, 5-7 July 2006, Toulouse
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