Simple Adaptive Control without Passivity Assumptions and Experiments on Satellite Attitude Control DEMETER Benchmark Dimitri Peaucelle Adrien Drouot Christelle Pittet Jean Mignot IFAC World Congress / Milano / August 28 - September 2, 2011
Introduction ■ Considered problem: Σ u y ● Stabilization with simple adaptive control ˙ K K = − Gyy T Γ + φ , u = Ky ● For MIMO LTI systems Σ ■ Assumptions: u y ● There exists a (given) stabilizing static output feedback F u = Fy θ Σ ■ Why simple adaptive control? u y ● Expected to be more robust K ● No need for estimation K � = F (ˆ θ ) 1 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Outline ■ Design of the adaptive law ● Virtual feedthrough D & barrier function φ for bounding K ● LMI based results ■ Preliminary tests on a satellite Benchmark ● Attitude control ● Adaptive PD gains 2 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ Passivity-Based Adaptive Control [Fradkov 1974, 2003] & Simple Adaptive Control [Kaufman, Barkana, Sobel 94] ● Let Σ ∼ ( A, B, C, D ) be a MIMO system with m inputs / p ≥ m outputs. ● If ∃ ( G, F ) ∈ ( R p × m ) 2 such that the following system is passive + G Σ v u y z F ● then the following adaptive law stabilizes the system for all Γ > 0 ˙ K = − Gyy T Γ , u = Ky 3 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ Underlying properties ● Passivity implies that for all ∆ + ∆ ∗ ≥ 0 the following system is stable + G Σ v u y z F −∆ ● i.e. all gains ( F − ∆ G ) stabilize the system, for ∆+∆ ∗ ≥ 0 , possibly large ● ˙ K = − Gyy T Γ “pushes” the gains in that direction until stability is reached ▲ In practice: Need to limit growth of K . Modifications of adaptive law ˙ K = − Gyy T Γ + φ ( K ) ( eg. φ ( K ) = − σK ) 4 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ What if Σ is not passifiable by ( G, F ) ? ● ∃ S a feedthrough (or Shunt) such that the following system is passive S + + G Σ v u y z F ● then the adaptive law stabilizes the system Σ + S . ▲ In practice: S should be small for tracking issues ( u = K ( y + Su ) ) Σ u y ▲ The actual gain is bounded + K ˆ K = K ( 1 − KS ) − 1 S K 5 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ Proposed result ● Stability proof based on a modified feedthrough scheme: D + + G Σ v u y z F ● K bounded thanks to a modification of the adaptive law: ˙ K = − Gyy T Γ − ψ D ( K ) · ( K − F ) , u = Ky ▲ ψ D is a deadzone: no modification when K is close to F if || K − F || 2 ψ D ( K ) = 0 • ≤ ν ▲ ψ D is a barrier: goes to infinity when K reaches border of accepted region if || K − F || 2 ψ D ( K ) → + ∞ • → νβ ( β > 1) 6 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ LMI-based design of ( G, D = µ 2 1 , ν ) assuming a given stabilizing F ● Step 1 (LMI): minimize µ such that following system is passive D � � A T ( F ) P + PA ( F ) PB − C T G T + + Σ G < 0 v u y z B T P − GC − µ 1 F ● Step 2 (LMI): maximize ν , the size of admissible adaptive gains ( ˆ F − F ) T T ≥ 0 , Tr ( T ) ≤ νµ, ( ˆ µ − 1 1 F − F ) A T ( F ) Q + QA ( F ) + νµβC T C < 0 . Q > 0 , + R + C T ( G T ( ˆ F − F ) T G ) C F − F ) + ( ˆ QB − C T G T R ≥ 0 , B T Q − GC µ 1 7 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Design of the adaptive law ■ LMI-based design of ( G, D = µ 2 1 , ν ) assuming a given stabilizing F ▲ Procedure guaranteed to succeed if F stabilizes the system ▲ K remains in a convex set around F (appreciated by engineers) ▲ ν may be small, i.e. small admissible adaptation ( K ≃ F ) ▲ LMI results can be easily extended to uncertain systems ⇒ proof of robustness of adaptive control for given uncertainty set ▲ In the robust case, step 2 is based on the existence of a PDSOF ˆ F ( θ ) Compared to ˆ F ( θ ) , the adaptive gain K needs not the estimation of θ ▲ Lyapunov function for global stability of PBAC V ( x, K ) = x T Qx + Tr (( K − ˆ F )Γ − 1 ( K − ˆ F ) T ) 8 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ■ DEMETER satellite attitude control Disturbances T T d + c + Flexible Reaction + Satellite wheels Flight software Filter Speed PD Velocity Star Controller Bias Estimator Tracker ! !# + + !" ! + + " # r r Ground Guidance 9 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ■ First experiment: adaptive tuning of PD gains of second axis with given filter 1 Σ sat , 2 ( s )Σ filter , 2 ( s ) ● Corresponds to OF for 2 × 1 system Σ estim ( s ) s Σ estim ( s ) = s +0 . 5 Σ sat , 2 ( s ) = 0 . 04736 s 2 +0 . 0006546 s +0 . 2991 s 2 ( s 2 +0 . 01387 s +6 . 338) 0 . 5411 s 4 − 3 . 678 s 3 − 4 . 99 s 2 − 1 . 747 s − 0 . 1241 Σ filter , 2 ( s ) = s (0 . 25 s 5 +1 . 961 s 4 +5 . 094 s 3 +5 . 722 s 2 +3 . 068 s +0 . 5784) � � ● Given stabilizing SOF F = 0 . 1 2 . 10 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ■ First experiment: adaptive tuning of PD gains of second axis with given filter ● Comparison of given SOF (dotted) and obtained PBAC 15 10 δθ [deg] 5 0 − 5 0 10 20 30 40 50 60 Time [s] 2 δω [deg/s] 1 0 − 1 − 2 0 10 20 30 40 50 60 Time [s] ▲ Engineers have well chosen the PD and filter gains 11 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ■ Second experiment: same model but modified initial stabilizing F � � ● Modified SOF F = 0 . 3 2 ● PBAC design gives � � G = , µ = 196 . 49 , ν = 0 . 0244 28 . 13 − 165 . 56 10 5 δθ [deg] 0 − 5 0 50 100 150 Time [s] 2 1 δω [deg/s] 0 − 1 − 2 0 50 100 150 Time [s] 12 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ● Evolution of K during the simulation 2.25 0.35 2.2 0.3 2.15 0.25 K p 0.2 2.1 0.15 2.05 0 50 100 150 K d 2 Time [s] 1.95 2.2 1.9 2.15 K d 1.85 2.1 2.05 1.8 2 1.75 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0 50 100 150 K p Time [s] ▲ Remains in largest circle (region such that � K − F � • ≤ νβ ) ▲ When the system settles it converges in the smaller circle ( � K − F � • ≤ ν ) 13 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
DEMETER satellite attitude control ■ Trird experiment: Same adaptive law but applied to first axis of satellite (robustness test) 5 δθ [deg] 0 − 5 0 50 100 150 Time [s] 1 δω [deg/s] 0 − 1 − 2 0 50 100 150 Time [s] ▲ PBAC not affected by the uncontrollable oscillating mode 14 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
Conclusions ■ LMI-based method that guarantees stability of PBAC ● Applies to any stabilizable LTI MIMO system ● Adaptive gains remain bounded ● Adaptive gains remain close to initial given values ■ Prospectives ● Enlarge admissible region for K : ellipsoids rather than discs [IFAC 2011] ● Structured control (decentralized etc.) ● Guaranteed robustness for time-varying uncertainties ● Take advantage of flexibilities on G for engineering issues (saturations...) ● ... 15 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011
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