Simple Adaptive Control without Passivity Assumptions and - - PowerPoint PPT Presentation

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:

• Stabilization with simple adaptive control

˙ K = −GyyTΓ + φ , u = Ky

• For MIMO LTI systems

Σ K y u

■ Assumptions:

• There exists a (given) stabilizing static output feedback

u = Fy

Σ y u F

■ Why simple adaptive control?

• Expected to be more robust
• No need for estimation K = F(ˆ

θ)

y u Σ θ K

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) ∈ (Rp×m)2 such that the following system is passive

u Σ y + z v F G

• then the following adaptive law stabilizes the system for all Γ > 0

˙ K = −GyyTΓ , 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

z Σ y v + u −∆ F G

• i.e. all gains (F −∆G) stabilize the system, for ∆+∆∗ ≥ 0, possibly large
• ˙

K = −GyyTΓ “pushes” the gains in that direction until stability is reached ▲ In practice: Need to limit growth of K. Modifications of adaptive law ˙ K = −GyyTΓ + 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

Σ z + y v + u F S G

• then the adaptive law stabilizes the system Σ + S.

▲ In practice: S should be small for tracking issues (u = K(y + Su)) ▲ The actual gain is bounded ˆ K = K(1 − KS)−1 Σ K + u K y S

5 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ Proposed result

• Stability proof based on a modified feedthrough scheme:

Σ y z + v + u F G D

• K bounded thanks to a modification of the adaptive law:

˙ K = −GyyTΓ − ψD(K) · (K − F) , u = Ky ▲ ψD is a deadzone: no modification when K is close to F ψD(K) = 0

if ||K − F||2

• ≤ ν

▲ ψD is a barrier: goes to infinity when K reaches border of accepted region ψD(K) → +∞

if ||K − F||2

• → νβ

(β > 1)

6 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ LMI-based design of (G, D = µ

21, ν) assuming a given stabilizing F

• Step 1 (LMI): minimize µ such that following system is passive

Σ y z + v + u F G D

• AT (F)P + PA(F)

PB − CT GT BT P − GC −µ1

• < 0
• Step 2 (LMI): maximize ν, the size of admissible adaptive gains

  T ( ˆ F − F)T ( ˆ F − F) µ−11   ≥ 0,

Tr(T) ≤ νµ,

Q > 0,   AT (F)Q + QA(F) + νµβCT C +R + CT (GT ( ˆ F − F) + ( ˆ F − F)T G)C   < 0.   R QB − CT GT BT Q − GC µ1   ≥ 0,

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Design of the adaptive law ■ LMI-based design of (G, D = µ

21, 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) = xTQx + 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

PD + + + ! Flexible Satellite ! + + Estimator Velocity Tracker Star Disturbances Filter Reaction wheels + + Ground Guidance Flight software !" !# " # T T

c d r r

Controller Bias Speed

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

• Corresponds to OF for 2 × 1 system

  1 Σestim(s)   Σsat,2(s)Σfilter,2(s) Σestim(s) =

s s+0.5

Σsat,2(s) = 0.04736s2+0.0006546s+0.2991

s2(s2+0.01387s+6.338)

Σfilter,2(s) =

0.5411s4−3.678s3−4.99s2−1.747s−0.1241 s(0.25s5+1.961s4+5.094s3+5.722s2+3.068s+0.5784)

• Given stabilizing SOF F =
• 0.1

2

• .

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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

10 20 30 40 50 60 −5 5 10 15 Time [s] δθ [deg] 10 20 30 40 50 60 −2 −1 1 2 Time [s] δω [deg/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 =

• 28.13

−165.56

• , µ = 196.49 , ν = 0.0244

50 100 150 −5 5 10 Time [s] δθ [deg] 50 100 150 −2 −1 1 2 Time [s] δω [deg/s]

12 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

DEMETER satellite attitude control

• Evolution of K during the simulation

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 Kp Kd 50 100 150 0.15 0.2 0.25 0.3 0.35 Time [s] Kp 50 100 150 2 2.05 2.1 2.15 2.2 Time [s] Kd

▲ 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)

50 100 150 −5 5 Time [s] δθ [deg] 50 100 150 −2 −1 1 Time [s] δω [deg/s]

▲ PBAC not affected by the uncontrollable oscillating mode

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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