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
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.
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.
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
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
H loopshaping: S scheme Disturbance attenuation w 6 - - - K P + +? - W 1 - z 1 W 1 S γ S(jω) γ, for all ω IR. W 1 (jω) 10 1 S: Amplitude 10 0 10 1 10 2 10 2 10 3 10 4 Frequency [Hz]
H loopshaping: S/KS scheme Disturbance attenuation with bounded control 6-6- - 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. 10 1 10 2 S: Amplitude 10 0 10 1 KS: Amplitude 10 1 10 0 10 1 10 2 10 2 10 3 10 4 Frequency [Hz] 10 2 10 2 10 3 10 4 Frequency [Hz]
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
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
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
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]
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
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
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
The CD player system
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
Plant characteristics Amplitude of P at a fixed track position 10 2 10 1 10 0 10 1 10 2 10 2 10 3 10 4 Frequency [Hz] The gain of P varies with track position: Relative gain 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.025 0.03 0.035 0.04 0.045 0.05 0.055 Radial displacement [m]
Disturbance characteristics Spectrum of the disturbance at a fixed track 10 8 10 6 Disturbance spectrum 10 4 10 2 10 0 10 2 10 4 10 6 10 0 10 1 10 2 10 3 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.
LPV model of the CD z 2 A w 2 pi 2 H w 1 pi 6 H W 2 + + 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
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 10 1 10 0 10 1 10 1 10 2 10 3 10 4 Frequency [Hz] 10 1 Sensitivity amplitude 10 0 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 Frequency [Hz]
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.
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)
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] 0.3 0.25 0.2 0.15 0.1 0.05 0 20 25 30 35 Rotational frequency [Hz] LPV controller LTI controller Internal controller
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)
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
Experimental results: fast parameter transition Tracking error [V] 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 Parameter transition LPV tracking error: Time [s]
Experimental results: fast parameter transition Tracking error [V] 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 Time [s] LTI tracking error
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.