V ( x ( t )) < V ( x (0)) (s) G i.e. is passive t 0 [ - - PowerPoint PPT Presentation

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 i.e.

u(t) Σ (s) G y(t) F

is passive    

V (x(t)) < V (x(0)) +

t

• 0 [u(θ)∗Gy(θ)] dθ

    then whatever positive Γi > 0, Σ is G-passified by the PBAC

u(t) = K(t)y(t) , ˙ Ki(t) = −y∗

i (t)ΓiGy(t) .

Technical remark

➞ One can always take F = −kG with k sufficiently large ➞ F exists if GΣ is hyper-minimum-phase

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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

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① Bounded PBAC

Bounded Passification Based Adaptive Algorithm

➞ Let B a bounded set of Cm and let a penalty function φ : Cm → Cm: 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 Γi > 0, Σ(∆) is robustly G-passified by the BPBAC

u(t) = K(t)y(t) , ˙ Ki(t) = −y∗

i (t)ΓiGy(t) − Γiφ(Ki(t)) .

➞ Convergence of the BPBAC is such that x(∞) = 0 , K(∞) ∈ B , Σ(∆) ⋆ K(∞) is G-passive

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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

V (x, K, ∆) = 1 2x∗H(∆)x + 1 2

l

• i=1

(Ki − Fi(∆))∗Γ−1

i (Ki − Fi(∆)) & 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.

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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 ¯ h1 > H > 0 , HB = C∗G∗ , t > −1 HA(0) + A∗(0)H − kC∗G∗GC < t1 ➞ Minimize t using PenBMI, if it returns t < 0, the procedure succeeded.

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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

HB∆ B∗

∆H

−1

   + ρ    C∗

∆

D∗

∆

      C∗

∆

D∗

∆

  

∗

< 0

then Σ(∆) ⋆ F is G-passive for all uncertainties ∆∗∆ ≤ ρ1 where Σ(∆):

˙ x = [A(0) + B∆∆(1 − D∆∆)−1C∆]x + Bu , y = Cx .

Remarks

➞ Maximization of ρ is LMI ➞ If ∆ ⊂ ∆ρ = {∆∗∆ ≤ ρ1} the problem is solved with F(∆) = F ➞ Replacing in the LMIs A(0) by A(∆i) for some ∆i, gives a couple (Fi, ρi)

such that Σ(∆) ⋆ Fi is robustly G-passive w.r.t. {∆i + ∆ : ∆ ∈ ∆ρi}.

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③ LMI design of F(∆)

Solving the LMI conditions for some finite sequence {∆i} gives sequences {Fi}, {ρi} that defines an F(∆) for {∆i + ∆ : ∆ ∈ ∆ρi}.

ρ

1 2 3 4

∆

ρ ρ ρ ∆

1 2 4

∆ ∆ ∆

3

If ∆ ⊂ {∆i + ∆ : ∆ ∈ ∆ρi} the robustness BPABC G-passification conditions are fulfilled.

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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 = 5km.

For two choices of high gain k

G1PenBMI = 10−2

• 4.5404

2.8436 1.7107

• G2PenBMI = 10−3
• 8.4000

5.4505 3.0961

• and after simple analysis step:

G1 =

• 4

3 2

• , G2 =
• 8

5 3

• &

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④ Example : autonomous aircraft

➞ Choice of LMI representable B (bounded set of control gains) F =

• f1

f2 f3

• ,

− 10 ≤ fi ≤ 10

and φ dead-zone penalty function of the BPBAC:

φ(fi) = fi − sat10(fi)

10 −10

φ f

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④ Example : autonomous aircraft

➞ LMI design of admissible h intervals

−8 −6 −4 −2 2 4 6 −1.5 −1 −0.5 0.5 1 1.5 Real axis Imaginary axis −8 −6 −4 −2 2 4 6 −2 −1.5 −1 −0.5 0.5 1 1.5 2 Real axis Imaginary axis

Admissible ∆ ∈ C for G1 Admissible ∆ ∈ C for G2 Result is h ∈ [0 9.2976] Result is h ∈ [0 9.6213] Simulations are done with the choice of G = G2.

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④ Example : autonomous aircraft

Simulations with Γi = 1

h = 5km h = 9.6km

1 2 3 4 5 6 7 8 −40 −35 −30 −25 −20 −15 −10 −5 5 10 1 2 3 4 5 6 7 8 −10 −5 5

y(t) k(t)

10 20 30 40 50 60 70 80 90 100 −60 −40 −20 20 40 60 10 20 30 40 50 60 70 80 90 100 −10 −8 −6 −4 −2 2 4

y(t) k(t) h = 8km with noisy measurements and saturated input (±20)

1 2 3 4 5 6 7 8 9 10 −20 −15 −10 −5 5 10 15 20 1 2 3 4 5 6 7 8 9 10 −40 −30 −20 −10 10 20 30

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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 Γi design results : convergence speed, disturbance rejections ...

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