An LMI Approach to the Control of a Compact Disc Player. Marco Dettori SC Solutions Inc. Santa Clara, California

Transcription

1 An LMI Approach to the Control of a Compact Disc Player Marco Dettori SC Solutions Inc. Santa Clara, California IEEE SCV Control Systems Society Santa Clara University March 15, 2001

2 Overview of my Ph.D. research project Joint project between: - Systems and Control Group, Delft University of Technology, The Netherlands. - Philips Research Laboratories. Theoretical side: Development of LMI algorithms for analysis and synthesis of control systems. Practical side: Application to a Compact Disc Player system: - Multi-objective design. - Gain-scheduling design.

3 Outline Theoretical part LMIs in control theory. Gain-scheduling for LPV systems. Application Gain-scheduling design for CD player. Experimental set-up. Implementation results. Conclusions and discussion.

4 What are LMIs? A generic Linear Matrix Inequality is: F (x) < 0 x R m, F (x) is a real symmetric matrix F (.) is an affine mapping < means negative definite. Form with matrix variables: [ x X := 1 x 2 x 3 x 4 ] A XB + B X A + C<0 Solving an LMI is a convex optimization problem: min t λ max (A XB + B X A + C) < t

5 Relevance in Control Theory: examples Lyapunov stability criterion. x= Ax is asymptotically stable iff X > 0, A X + XA <0 Bounded Real Lemma The system. x = Ax + Bu y = Cx has H norm smaller than 1 iff X > 0, A X + XA + XBB X + C C<0 iff X > 0, [ A X + XA + C C XB B X I ] < 0

6 H loopshaping: S scheme Disturbance attenuation w K P + +? - W 1 - z 1 W 1 S γ S(jω) γ, for all ω IR. W 1 (jω) 10 1 S: Amplitude Frequency [Hz]

7 H loopshaping: S/KS scheme Disturbance attenuation with bounded control K z 2 6 W 2 P + + w? - W 1 - z1 ( W 1 S W 2 KS ) γ W1 (jω)s(jω) 2 + W 2 (jω)k(jω)s(jω) 2 γ, ω IR S: Amplitude KS: Amplitude Frequency [Hz] Frequency [Hz]

8 Benefits of LMI formulation Recent development of powerful interior point methods for convex optimization (1994). Available software packages: LMI Toolbox for Matlab: Gahinet, Nemirovskij, Laub, Chilali SP: Boyd, Vandenberghe LMITOOL: El Ghaoui, Delebecque, Nikoukhah SDPpack: Overton, Alizadeh et al. Through LMIs it is possible to numerically solve problems otherwise unsolvable. Multi-objective Control Linear Parametrically Varying Control

9 Motivation Gain scheduling: design controller for nonlinear systems using linear design tools. Classical approach: - linearize the system at various operating points - design linear controller at each point - interpolate to get a global controller Drawback: no systematic way to perform interpolation. LPV approach: - systematic design method - no need for interpolation step - based on LMI techniques

10 From nonlinear to LPV systems Given the nonlinear system. x = a(x, q 1 )+b 1 (x, q 1 )w + b 2 (x, q 1 )u z = c 1 (x, q 1 )+d 1 (x, q 1 )w + d 2 (x, q 1 )u y = c(x, q 1 )+d(x, q 1 )w Suppose x =0is an equilibrium for all q 1. Rewrite. x = A(x,q 1 )x + B 1 (x,q 1 )w + B 2 (x,q 1 )u z = C 1 (x,q 1 )x + D 1 (x,q 1 )w + D 2 (x,q 1 )u y = C(x,q 1 )x + D(x,q 1 )w Arrive at LPV system: - Replace x by q ( 2 - Define p = q 1 q 2 ) Π

11 From nonlinear to LPV systems LPV system llpv ẋ = A(p(t))x + B 1(p(t))w + B 2 (p(t))u z = C 1 (p(t))x + D 1 (p(t))w + D 2 (p(t))u y = C(p(t))x + D(p(t))w. where p(t) Π for all t. Example:. x= x sin(x) can be transformed into. x= p x, p [ 1, 1]

12 Analysis result for LPV systems The LPV system. x(t) =A(p(t))x(t)+B(p(t))w 1 (t) z 1 (t) =C(p(t))x(t)+D(p(t))w 1 (t) is exponentially stable and has L 2 gain w 1 z 1 smaller than γ if Lyapunov matrix X > 0 s.t. p Π A(p) X + XA(p) XB 1 (p) C 1 (p) B 1 (p) X γi D 1 (p) < 0 C 1 (p) D 1 (p) γi Infinitely many LMI s in X. Two ways to proceed: Gridding techniques Introduction of scalings

13 LFT representation of LPV systems Pulling out the parameter p:. x A B 1 B 2 x z 1 = C 1 D 1 D 12 z 2 C 2 D 21 D 2 where (.) is continuous w 1 w 2, w 2 = (p)z 2 w 2 w 1 Σ z 2 z 1

14 a a!! Controller synthesis Controller with same structure of the plant. x A c B c1 B c2 x u = C c1 D c1 D c12 y, w c = c (p)z c z c C c2 D c21 D c2 w c (p) a a!! w 2 z 2 w 1 u a a!! a a!! a a!! z 1 y K a a!! z c w c a a!! c (p) Synthesis algorithms based om LMI techniques

15 The CD player system

16 Specs for the CD track spot + error K P Disturbance suppression max. track eccentricity: 100 µm max. allowable position error: 0.1 µm Factor 1000 time-domain attenuation Relatively small bandwidth avoid high power consumption no amplification of audible noise robustness

17 Plant characteristics Amplitude of P at a fixed track position Frequency [Hz] The gain of P varies with track position: Relative gain Radial displacement [m]

18 Disturbance characteristics Spectrum of the disturbance at a fixed track Disturbance spectrum Frequency [Hz] Location of harmonics vary with rotational frequency. New high-performance applications require higher rotational frequency (30 Hz) necessity of adaptive selective disturbance suppression.

19 LPV model of the CD z 2 A w 2 pi 2 H w 1 pi 6 H W A H H W 1 ; A P H H H u H z H 1 Scheduling parameter is p =2πfrot, with 25Hz frot 35Hz. P is scheduled for gain variations. Performance filter W 1 models frequency-varying notches W 1 (s, p) = s2 +2ζ z ps + p 2 s 2 +4ζ z ps +4p 2 s 2 +2ζ p ps + p 2 s 2 +4ζ p ps +4p 2 Filter W 2 for robustness issues. + y H

20 Results: Frequency domain Amplitude of controller and sensitivity for the five frozen values frot=25, 27.5, 30, 32.5, 35 Hz: 10 2 Controller amplitude Frequency [Hz] 10 1 Sensitivity amplitude Frequency [Hz]

21 Experimental set-up Old Philips audio CD Player Set-up with two dspace systems: C40 to measure rotational frequency. Multiprocessor (C40 and Alpha) to implement the controller.

22 Implementation scheme Problems: Implementation of unstable controllers Bumpless switch from internal to external controller h ; 6 b b b - G K ext K int - ; h + 6 ; b Computer b b - P G stabilizes K ext during the start-up procedure High order controllers High sampling rate (20 khz)

23 Experimental results Comparison with a high performing LTI controller. Peak of tracking error for several fixed values of the rotational frequency 0.35 Peak of the tracking error [V] Rotational frequency [Hz] LPV controller LTI controller Internal controller

24 C40 - T c =20s T s =50s - u(k) y(k) u(k+1)? A=D mw Du(k) sum 6 D=A?? Memory Alpha Cx(k;1) mw u(k;1) 6 u(k) ; Cx(k)? Ax(k)+Bu(k) Cx(k+1)

25 Multiprocessor Setup u K RAD EXT DAC #1 DAC #2 M2 D Sum D1 FOC EXT DAC #3 DAC #4 DAC #5 DAC #6 ADC #1 RAD ERROR DS2102_B1 + 5V ADC #2 RAD CD ADC #3 FOC ERROR u master:0 to alpha:0 IPC1 in out alpha:1 to master:1 IPC2 ADC #4 FOC CD alpha M3 ADC #5 DS2001_B1 + 5V

26 Experimental results: fast parameter transition Tracking error [V] Parameter transition LPV tracking error: Time [s]

27 Experimental results: fast parameter transition Tracking error [V] Time [s] LTI tracking error

28 Conclusions LPV techniques allow to design controller with stability and performance guarantees over the whole operative range of the plant. Experimental set-up based on dspace systems to implement LPV controllers. LPV controller improves performance of the CD player servosystem as required by high-demanding applications.