Experiment design for MIMO model identification. Marco Lovera Dipartimento di Scienze e Tecnologie Aerospaziali Politecnico di Milano

Transcription

1 Experiment design for MIMO model identification Marco Lovera Dipartimento di Scienze e Tecnologie Aerospaziali Politecnico di Milano

2 Introduction Experiment design: fundamental role in the practice of model identification Problem of optimal experiment design has been studied extensively in the model identification literature The problem of defining optimal input sequences for MIMO model identification while taking into account operational constraints is considered Step wise and multi cyclic MIMO proposed methods are based on C. Jauberthie, L. Denis Vidal, P. Coton, and G. Joly Blanchard. An optimal input design procedure. Automatica, 42(5):881884, D.E. Rivera, H. Lee, H.D. Mittelmann, and M.W. Braun. Constrained multisine input signals for plant friendly identification of chemical process systems. Journal of Process Control, 19(4): , 2009.

3 Experiment design - 1 Excite the dynamic system so that the data contain sufficient information respecting the constraints Constraints Inputs/outputs amplitude, experiment duration, system behaviour, etc.

4 Experiment design - 1 Excite the dynamic system so that the data contain sufficient information respecting the constraints Constraints Inputs/outputs amplitude, experiment duration, system behaviour, etc. Non linear model Equilibrium

5 Experiment design - 1 Excite the dynamic system so that the data contain sufficient information respecting the constraints Constraints Inputs/outputs amplitude, experiment duration, system behaviour, etc. Non linear model Equilibrium Measured outputs

6 Experiment design - 2 Fisher Information Matrix Sensitivity

7 Experiment design - 3 7/22 Asymptotic variance of θ 1 : σ M 1 =. σ Optimality criteria: scalar functions of M Optimisation problem Linear SISO models Nonlinear MIMO models Closed form solutions exist Numerical algorithms A priori knowledge on θ 7

8 Experiment design - 4 Fisher Information Matrix Sensitivity Optimisation Problem Unconstrained problem is convex... but the optimal solution might not be compatible with the dynamics!

9 Experiment design - 5 Fisher Information Matrix Sensitivity Constrained Optimisation Problem

10 Experiment design - 4 Fisher Information Matrix Sensitivity Optimisation Problem Nonconvex problem (except some cases)

11 Considered input signal classes 4 Orthogonal multisines 4 Piecewise constant Automatic control system Pilot

12 Multisine signals (SISO) Single Input Input design: parameters Bandwidth Number of harmonics Output constraints

13 Multisine signals (SISO) Single Input Input design: parameters Bandwidth Number of harmonics Output constraints Optimisation variables

14 Multisine signals (SISO) Single Input Input design: parameters Bandwidth Number of harmonics Output constraints Schroeder s phases Optimisation variables Initial solution Amplitudes Uniform Phases Schroeder

15 Multisine signals (MIMO) - 1 Multiple Input Design (Rivera et al. 2009) Different harmonic frequencies Orthogonality between signals Primary Frequency Bandwidth

16 Multisine signals (MIMO) - 2 Optimisation problem SYSTEM SENSITIVITY Optimisation variables Harmonics amplitude Harmonics phase Constraints Input amplitude Output amplitude

17 Piecewise constant signals - 1 First Step Input Input Design Parameters Number of steps Duration of each step Maximum step amplitude Output constraints Experiment duration (Jauberthie et al. 2006)

18 Piecewise constant signals - 1 First Step Input Input Design Parameters Number of steps Duration of each step Maximum step amplitude Output constraints Experiment duration Optimisation Signal shape Large number of solutions (Jauberthie et al. 2006)

19 Piecewise constant signals - 1 First Step Input Input Design Parameters Number of steps Duration of each step Maximum step amplitude Output constraints Experiment duration Optimisation Signal shape Large number of solutions (Jauberthie et al. 2006) Dynamic programming

20 Piecewise constant signals - 2 Example: SISO case Input Output

21 Piecewise constant signals - 2 Example: SISO case Input Output

22 Piecewise constant signals - 2 Example: SISO case Input Output

23 Piecewise constant signals - 2 Example: SISO case Input Output

24 Piecewise constant signals - 2 Example: SISO case Input Output

25 Piecewise constant signals - 2 Example: SISO case Input Output

26 Piecewise constant signals - 2 Example: SISO case Input Output

27 Piecewise constant signals - 2 Example: SISO case Input Output

28 Piecewise constant signals - 2 Example: SISO case Input Output

29 Piecewise constant signals - 2 Example: SISO case Input Output

30 Piecewise constant signals - 2 Example: SISO case Input Output

31 Piecewise constant signals - 3 Outputs belong to the same strip α1 0 -α1 Step 1 Step 2 J1 J2 J3 α1 0 -α1 α1 0 -α1 J4 J5 J6 J7 J8 J9 Sequence -α1 α1: the output violates the constraints Step r

32 Piecewise constant signals - 4 Second step Approximation of Initial solution First step Optimal solution Optimisation variables Duration of each step Signal amplitude Multiple input Input orthogonality Time skew

33 Quadrotor UAV

34 Quadrotor UAV 4 independent rotors Attitude control Collective Roll Pitch Yaw Collective Pitch/Roll Yaw

35 Quadrotor model 4 inputs, 7 outputs Model Parameters Moments of Inertia Aerodynamic coefficients

36 Quadrotor model 4 inputs, 7 outputs Model Parameters Moments of Inertia Aerodynamic coefficients

37 Quadrotor model 3 inputs, 7 outputs Model Parameters Moments of Inertia Aerodynamic coefficients

38 Maneuver constraints Flight condition: hover Open loop Unstable equilibrium Output constraints Roll Pitch Vertical Acceleration

39 Results: optimal cost Input signal class Cost function (J=Tr[M -1 ]) Initial guess Optimal solution Multisine 1.57 x x 10-6 Piece-wise constant x 10-7

40 Results: multisine signals Inputs Experiment duration: 38 s Bandwidth: 1 6 rad/s Outputs

41 Results: piecewise constant signals Inputs Experiment duration : 34 s Steps: 17 Outputs

42 Results: identification Output Error Method Initial error of the true value: 10% Output noise 50 runs Initial value True value Multisines Piecewise constant

43 Conclusions Problem of defining optimal input sequences for MIMO model identification has been considered Proposed method builds on results available in the literature and allows the optimisation of scalar function of the Fisher information matrix Step wise and multi cyclic MIMO inputs have been taken into account Evaluation of the proposed method by designing MIMO experiments to estimate parameters of quadrotor model