Robust Simple Adaptive Control with Relaxed Passivity and PID - PowerPoint PPT Presentation

Robust Simple Adaptive Control with Relaxed Passivity and PID control of a Helicopter Benchmark Dimitri Peaucelle Vincent Mahout Boris Andrievsky Alexander Fradkov 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 E Σ ■ Assumptions: u y ● There exists a (given) stabilizing PI control F � u = F P y + F I Ey 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 ■ PI structure of the adaptive controller ● Adaptation that stops as the output reaches the reference ■ Design of the adaptive law: ˙ K = − Gyy T Γ + φ ● Virtual feedthrough D & barrier function φ for bounding K ● LMI based results ⇒ guaranteed robustness ■ Preliminary tests on a 3D helicopter Benchmark ● Adaptive PID 2 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

PI structure of Adaptive Control −y ■ Assumptions: ref E Σ ● There exists a (given) stabilizing PI control + y u � u = F P ( y − y ref ) + F I E ( y − y ref ) F ● Integral term: precision to constant reference ■ Impossible to apply the following adaptive control: � � y − y ref ˙ K = − Gηη T Γ + φ , η = u = Kη , � E ( y − y ref ) −y ref E Σ + y u � E ( y − y ref ) does not go to 0 NO! because K 3 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

PI structure of Adaptive Control � ■ Proposed PI adaptive control law: u = K P ( y − y ref ) + K I E ( y − y ref )   y − y ref ˙  ( y − y ref ) T Γ P + φ P K P = − G P  � E ( y − y ref ) � T Γ I + φ I ˙ �� K I = − G I ( y − y ref ) E ( y − y ref ) ● When ( y − y ref ) goes to zero adaptation stops ■ Results applicable to any controller structure of the type ˙ K 1 = − G 1 z 1 y T 1 Γ 1 + φ 1 u = K 1 y 1 + K 2 y 2 , ˙ K 2 = − G 2 z 2 y T 2 Γ 2 + φ 2 as long as stability and/or tracking imply that y 1 and z 2 go to zero. ● Can also be extended to more than 2 terms ( u = K 1 y 1 + K 2 y 2 + K 3 y 3 + . . . ) 4 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 5 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ Proposed result ● If ∃ ( G 1 , G 2 , F 1 , F 2 , D 1 , D 2 ) such that the system below is passive D 1 D 2 z 1 G + 1 G Σ u z 2 2 + y y 2 1 F 1 F 2 ● then the following adaptive law stabilizes the system for all Γ 1 > 0 , Γ 2 > 0 ˙ K 1 = − G 1 z 1 y T 1 Γ 1 − φ 1 ( K 1 ) u = K 1 y 1 + K 2 y 2 , ˙ K 2 = − G 2 z 2 y T 2 Γ 2 − φ 2 ( K 2 ) where φ 1 ( K 1 ) and φ 1 ( K 2 ) are to be determined (depend of D 1 and D 2 ). 6 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ 2 step LMI design procedure ● Step 1: Given stabilizing F 1 , F 2 solve LMI problem L 1 to get ( G 1 , G 2 , D 1 , D 2 ) :   T G 1 T A T ( F 1 , F 2 ) P + PA ( F 1 , F 2 ) PB − C T 2 G 2 T PB − C 1 B T P − G 1 C 1 L 1 :  < 0 .  − 2 D 1  0  B T P − G 2 C 2 − 2 D 2 0 ● Step 2: Given ( G 1 , G 2 , F 1 , F 2 , D 1 , D 2 ) solve LMI problem L 2 (see paper) to get α 1 and α 2 that define the functions φ 1 , φ 2 : � ( K i − F i ) T D i ( K i − F i ) � dead-zone: φ i ( K i ) = 0 ≤ α i if Tr � � ( K i − F i ) T D i ( K i − F i ) barrier: φ i ( K i ) → + ∞ → α i β if Tr 7 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ Properties of the LMI design procedure ● Applicable as soon as there is a given stabilizing control F 1 , F 2 ● The TV gains K 1 , K 2 are guaranteed to be bounded in � ( K i ( t ) − F i ) T D i ( K i ( t ) − F i ) � < α i β Tr ● It is possible to maximize the domain of admissible adaptation values by minimizing Tr ( D i ) in the first LMI step maximizing α i in the second LMI step ● Proof of stability with adaptive control using Lyapunov function 2 � F i ) T � � ( K i − ˆ − 1 ( K i − ˆ V ( x, K 1 , K 2 ) = x T Qx + F i )Γ i Tr i =1 where Q , ˆ F 1 and ˆ F 2 are solutions to the LMI problem L 2 8 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Design of the adaptive law ■ Robustness properties of the LMI design procedure ● In case of polytopic uncertainties N N � � � � � � A [ j ] B [ j ] = ζ j = 1 , ζ j ≥ 0 ζ j , A ( ζ ) B ( ζ ) j =1 j =1 ● Robust LMI version of second step (based on slack variables technique) proves stability with parameter-dependent Lyapunov function 2 � F i ( ζ )) T � � ( K i − ˆ − 1 ( K i − ˆ V ( x, K 1 , K 2 , ζ ) = x T Q ( ζ ) x + F i ( ζ ))Γ i Tr i =1 j =1 ζ j Q [ j ] and ˆ j =1 ζ j ˆ F [ j ] where Q ( ζ ) = � N F i ( ζ ) = � N i . 9 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

3DOF Helicopter control ■ 3DOF helicopter from Quanser c � - LAAS configuration ● All states measured: elevation ǫ , pitch θ , travel λ and their derivatives ● Two inputs: drag forces due to the propellers ● Non linear model: ▲ linearized at operating point: allows to design an initial PID controller (state-feddback problem) ▲ non-linearities taken into account through an uncertain linear model Robust adaptive PID control design applied 10 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

3DOF Helicopter control ■ Simulation results: travel and elevation (dotted: PID - solid: Adaptive) 10 8 Elevation 6 4 10 2 Travel 5 0 0 50 100 150 200 250 300 11.5 0 11 10.5 10 0 50 100 150 200 250 300 9.5 9 8.5 255 260 265 270 275 280 285 290 295 11 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

3DOF Helicopter control ■ Simulation results: pitch and Adaptive PID gains − 200 10 − 300 0 50 100 150 200 250 300 − 8 − 10 5 − 12 0 50 100 150 200 250 300 − 900 Pitch − 1 000 0 0 50 100 150 200 250 300 − 100 − 200 − 5 0 50 100 150 200 250 300 − 60 − 100 − 10 − 140 0 50 100 150 200 250 300 0 50 100 150 200 250 300 12 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011

Conclusions ■ LMI-based method that guarantees robust stability of Adaptive control ● Applies to any stabilizable LTI MIMO system ● Allows to keep some structure such as PID ● Adaptive gains are bounded and remain close to initial given values ■ Prospectives ● More structured control (decentralized etc.) ● Guaranteed robustness for time-varying uncertainties ● Take advantage of flexibilities on G for engineering issues (saturations...) ● Apply to the actual helicopter benchmark 13 IFAC WC / Milano / Aug. 28 - Sept. 2, 2011