Robust Multivariable Control

Transcription

1 Lecture 1 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet

2 Addresses mobile:

3 Feedback Why do we need feedback? r u G 1 y G When cannot we do like this?

4 Feedback Why do we need feedback? r u G 1 y G When cannot we do like this? G is unstable;

5 Feedback Why do we need feedback? r u G 1 y G When cannot we do like this? G is unstable; G is uncertain;

6 Feedback Why do we need feedback? r u G 1 y G When cannot we do like this? G is unstable; G is uncertain; G is not minimum phase (zeros in RHP, G 1 unstable).

7 Some history KF ˆx u L G y LQR/LQE is optimal in some sense What about phase and amplitude margins? Abstract: There are none, Doyle, 1978, chapter in ZDG. LTR gives margins (Loop Transfer Recovery).

8 Standard form w (system disturbances) r + u K 1 G y + (measurement disturbances) + e y = Sw + T (r e) S = (I + GK ) 1 T = GK (I + GK ) 1 S + T = (I + GK )(I + GK ) 1 = I

9 Augmented system We would also like to reduce the control signals in magnitude. KSw W KS < 1 y K u G Tw + W T < 1 w Sw WS < 1

10 Augmented system w G z u y K We want to limit the gain from w to z by a suitable choice of K. Try to reduce the gain to one or below.

11 Design methods w G z u y K 1) Specify the requirements 2) Synthesis 3) Check if the requirements are satisfied 4) Modify and repeat from 1)

12 Loop shaping Find W 1 and W 2 to shape the open loop gain W1 P W2 The controller, K, can be obtained in a synthesis step K + W2 K W1 P

13 Alternative formulation z 2 w 2 W 1 1 W 1 w 1 W 1 2 z 1 W 2 + K + P +

14 Multivariable gain Consider [ z1 z 2 z = Mw ] [ = ][ w1 w 2 ] Let w 2 = w w 2 2 = 1, and maximize z 2 = z z2 2.

15 Multivariable gain Singular value decomposition: [ M = [ = where U T U = I, V T V = I, Σ = Use svd in Matlab. ] = UΣV T ][ [ σ1 Maximum gain, M = σ(m) = σ 1 = max i σ i. Different directions in w give different gains.... ][ ] σ n 0. ] T

16 Connection to eigenvalues M = UΣV T M T M = VΣU T UΣV T = VΣ 2 V T Thus, M T MV = VΣ 2 and Σ 2 are eigenvalues to M T M. [ ] 0 M Also, the eigenvalues of M T are equal to ±σ i. 0 Computing the eigvalues can be numerically difficult

17 Synthesis Consider a static system: D 12 z D 11 w D 21 y K u Choose a K such that D 11 + D 12 KD 21 is minimized.

18 Synthesis Use orthonormal transformations to obtain a similar problem [ A Find γ = min X C B X ]. This is solved by Parrott s theorem: [ ] } [ ] γ max{ A B, A C One solution is X = [ 0 C ][ γi A A T γi ] [ B 0 This problem is related to the elimination lemma. ]

19 Multivariable phenomena throttle [%] velocity [m/s, Mach] elevator [deg] Aircraft altitude [m] How can we compare degrees and %? How can we compare m, m/s and Mach number?

20 Scaling Scale the input signals, u, so that they are limited between ±1. Scale the output signals, y, so that they are limited ±1. Scale the disturbances, d, so that they are limited ±1. G d d 1 u 1 G + y 1

21 Scaling G d d 1 u 1 G + y 1 In order to compensate fully for a disturbance so that y 1: G 1 G d 1

22 Small gain theorem G G stable and G < 1. If 1 then the closed loop system is stable

23 Small gain theorem Pull out the uncertain parameters. LFT = linear fractional transformation: δ δ 1 G

24 Lectures 1 Introduction

25 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing

26 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance

27 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR

28 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis

29 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis 6 LMIs

30 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis 6 LMIs 7 Loop shaping, model reduction

31 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis 6 LMIs 7 Loop shaping, model reduction 8 Model uncertainties, small gain theorem, LFTs

32 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis 6 LMIs 7 Loop shaping, model reduction 8 Model uncertainties, small gain theorem, LFTs 9 µ analysis and synthesis

33 Lectures 1 Introduction 2 Norms, Lyapunov equations, balancing 3 Feedback, stability, performance 4 Parametrization of regulators, Riccati equations, LQR 5 H -synthesis 6 LMIs 7 Loop shaping, model reduction 8 Model uncertainties, small gain theorem, LFTs 9 µ analysis and synthesis 10 Summary