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Proceedings of the Am erican Control Conferen ceChicago , fllinois June 2000Discrete-Time Robust Tracking ControlUsing A State Space Disturbance Observer

Seung-Hi Lee Young-Hoon Kim Sang-Eun BaekElectro-mechanics Lab. Samsung Advanced Institute of TechnologyP.O.Box 111, Suwon 440-600, KOREA

s h l , younghoon [email protected]

Abstract for high frequency sensor noise rejection. Thus, the Qfilter can be interpreted as a complementary sensitivityfunction of the feedback loop. For further information,the interested readers are referred to [3][4]and refer-ences therein.

This paper considers desigll of a discrete-time robusttracking control system which consists of a sta te spacedisturbance observer, a state estimator, and a statefeedback controller. A new discrete-time state spacedisturbance observer is proposed not requiring model-ing of disturbances. Through the results of applica-tions, i t is shown that the proposed method is veryeffective to compensate variety of disturbances and t oimprove the performance of a tracking control systemin the existence of external disturbances. It is also ob-served that the proposed state space disturbance ob-server allows more accurate st ate estimation in the ex-istence of modeling error and disturbances.

1 IntroductionExternal disturbances have significant impact on theperformance of a tracking controller. Accordingly,there has been much work on the problem of reject-ing the disturbances effectively. Modeling errors inth design of control systems can also cause trackingerror. Modeling errors include unmodeled dynamics,which represent high order flexible modes, as well asplant parameter uncertainties/variations during oper-ation. The difference between the out put of the plantand the output of the nominal model is regarded as anequivalent disturbance applied to the nominal model.Disturbance observers are used to estimate the equiva-lent disturbances. Thus, a disturbance observer can beused to make the plant behave like the nominal modelin the existence of the equivalent disturbances.In [3], the idea of a disturbance observer was pro-posed. In [4], the disturbance observer theory was re-fined based on the design of two degrees of freedom sevocontrollers and th e factorization approach. The designof a disturbance observer is dependent on the design ofso called the Q f i l ter , which determines robustness anddisturbance rejection performance. In the design of theQ filter, unit low frequency gain is required for distur-bance rejection, while high frequency roll-off is required0-7803-5519-9/00 $10 00 AACC 41 94

Conventional approach in the design of st at e space dis-turbance observers is to model disturbances and t o aug-ment it into the estimator model (e.g. bias and period-ical disturbance [ l ] . owever, it is in general impossi-ble to build models for arbi trary disturbances. There-fore, state space disturbance observers are hardly usedexcept for simple disturbances which can be modeledeasily.This paper considers design of a discrete-time robusttracking control system which consists of a sta te spacedisturbance observer, a state estimator, and a statefeedback controller. In this paper, we propose a newdiscrete-time state space disturbance observer not re-quiring modeling of disturbances. An application ex-ample is presented to show applicability and effective-ness of the proposed disturbance observation method.Throughout the paper, a transfer matrix in terms ofstate-space is denoted by

p M ) denotes the spec tral radius of a matrix M . It isassumed that all the vectors and matrices have appro-priate dimensions.

2 Problem StatementTo begin with, consider a continuous-time model of aplant described by

1 )x A,x+B,u+wy C , x nwhere w is the disturbance and n is the measurementnoise, A , , B, and C, are matrices with appropriate


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dimensions. I t is assumed that A p ,Bp ,C,) is stabiliz-able and detectable. In addition, we assume the dis-turbance w is bounded. However, no time/frequencycharacteristics of w are assumed.Performance of a control system is significantly affectedby disturbances. Thus, it is very important t o rejectthe disturbance effectively. As mentioned before, thetransfer function approach has many drawbacks. Itrequires filtering of output measurements. Thus, th eperformance of the disturbance observer is determinedby the Q filter. In addition, the transfer function ap-proach can not be used in the multi-input-multi-output(MIMO) cases, while the state space approach canbe. In the state space approach, specific disturbancemodels are augmented in the estimator model to esti-mate the disturbances. The models are developed fromthe timelfrequency characteristics of disturbances (forexample, constant bias and periodic disturbance) [l].When the time/frequency characteristics of the dis-turbance are unknown, the assumption of piece-wiseconstant bias is used with sufficiently fast eigenvalueassignment t o the state s associated the disturbance.However, fast varying disturbances can not be observedeffectively by this approach. In addition, performanceof a state estimator can be affected.It should be noted that even a well designed distur-bance observer can not compensate disturbances per-fectly. In this paper, the remaining disturbance is re-garded as a residual disturbance. This residual distur-bance is considered asan exogenous signal in the designof a controller. Thus, the disturbance observer is to bedesigned to suppress the equivalent disturbance, whilethe controller is to be designed such that the feedbackloop is less sensitive to the residual disturbance.

3 Control Syst em DesignIn this section, we design a discrete-time control systemwhich consists of a sta te space disturbance observer, acurrent stat e estimator, and a st ate feedback controller.Firstly, the concept of the new state space disturbanceobserver is introduced. Subsequently, assuming thatthe disturbance observer is working properly, we designa state estimator and a state feedback controller forthe system with the residual disturbance. Finally, wedesign a disturbance observer to compensate equivalentdisturbances and to make the closed loop system stable.

3.1 Discrete Time ModelTObegin with, consider a discrete-time model describedbY

Gm [ ]41 95

where

and T, is the sampling period. Here, (@,I ,C) s as-sumed to be stabilizable and detectable. For the sim-plicity of formulation we assume no computation timedelay. However, the time delay can be handled eas-ily by augmenting the delayed control as an additionalstate [I].3.2 New State Spac e Disturbanc e ObserverThe basic structure of the proposed state space distur-bance observer (shown in Figure 1 is expressed by

where is the st ate of the model, z is the state ofthe plant, Kdist is the disturbance observer gain to bedetermined. If all the s ta te measurements are not avail-able, we use the estimated states ?. In this case, weuse

udist(k) Kdzst(e(k) ? k)). (4)

Udist

Plant

Figure 1: Disturbance observer scheme

The disturbance observer gain Kdist determines theconvergence rate to the disturbance w and affects thestability of the feedback loop. A condition for propergain ist will be addressed in Disturbance ObserverDesign subsection. Not like the transfer function a pproach in the s domain, there is nothing like the Q fil-ter . Thus, there is no additional dynamics associatedwith the disturbance observer itself.Given the states 3 and 2, no additional dynamics com-putation is required to observe the disturbance. Thestates 3 and 2 are to be obtained from a state esti-mator which is to be addressed in the next subsection.The disturbance observation zLdist k ) is injected intothe feedback control loop (as shown in Figure 2) tocompensate disturbances. As a result, we have a ficti-tious system in which only the residual disturbance isacting on (as shown in Figure 3). We design a sta te es-timator and a state feedback controller for this systemin the following subsections.


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W n

Figure 2: Control system with disturbance observerd n

Figure 3: System for controller design

3.3 State Estimator DesignThe current state estimator [l ] s expressed by

(5.)i ( k ) f k ) + L,(y(k) C z ( k ) )q + 1) d r i l c ) + rzL(Ic)where y is the output measurement, 2 and f are the,stat e update and the sta te prediction, respectively.Although the time/frequency characteristics of theresidual disturbance is unknown, i t is reasonable to as-sume tha t the residual disturbance is bounded. Con-sidering this fact, we design a current state estimatorin the H2 point of view. The current estimator gainL , tha t minimizes the estimation error in the H2 sensewith given weightings can be obtained by solving thediscrete algebraic Riccati equation (DARE)

Y drYV drYCT DIDT +CYCT)-'CYdrT-rlrT = o (6)The estimator gain L, is computed by

L , = YCT(CYCT+ D I D T ) - (7)where Y > 0 is the solution of 6 ) . Then, from (5) weobtain the transfer function of the s tate estimator

G t l [ ]CL,which is used as a target loop t ransfer funct ion3.4 State Feedback DesignTo begin with, consider a system described in Figure3 Here, d w U t , the residual disturbance. Stat efeedback control law is described by

u ( k ) K C Z T k ) -W (9)

where K , is the feedback gain to be determined suchtha t b oth tracking performance and disturbance rejec-tion are satisfactory. In order to atta in sufficient ro-bustness, the loop transfer recovery (LTR) design tech-nique is applied in the design of s tat e feedback gain K,.Given the target loop transfer function Gtl, the statefeedback gain K , is determined using the LTR method-ology at the plant output such that the loop transferfunction approaches to the target loop transfer func-tion. By solving the DARE, with control weighting R2and stat e weighting R1 qCTC for recovery,

x G ~ x ~ ,aTxr 1-2~+rTxr)- rTxdr-RI = O (10)

one can determine the state feedback gain fromK , = (rTxr R2)-*rTx+ (11)

where X 0 is the solution of (10).3.5 Analysis of Feedback LoopThe sensitivity transfer function is expressed by

wherer K,L,C r ( K , K A C )-(a K,)L,C dr K , dr K,)L,C

The complementary sensitivity transfer function is

G c s = [where1r - K,L,C r K c K C)- 6, K,) L,C 6, 'K, 6, TK,) L,C,,

It should be noted tha t the complementary sensitivityfunction is the Q filter equivalent in the design of statespace disturbance observer based control systems.3.6 Disturbance Observer DesignUsing (5) we can rewrite the disturbance observermodel (4) shown in Figure 4-(a) as

U d i s t ( k ) K d i s t L c (-Y(k)-tC z ( k ) ) (14)which is depicted in Figure 4-(b). The disturbance ob-servation u d i s t ( k ) is represented in terms of the esti-mation error y ( k ) Cf(lc). So far, we have assumedthat an appropriate disturbance observer gain Kdist is

4 96


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given. To complete design we need to determine thegain Kdzst. n general, larger gain Kdzstresults in moreaccurate disturbance observation. However, it also af-fects stability of the closed loop system. Thus, theperformance of disturbance observation as well as thestability of closed loop should be considered in the de-termination of the gain Kdzst. The following theoremprovides a condition for the disturbance observationgain Kdzst.

Theorem 1 Suppose that p a - aL,C) < 1 anda FK,) < 1. Th en , there always exist some Kdistsuch that p ( ( a @L,C rKdiStL,C) < 1, i .e . thefeedback loop is stable. Moreove r, in this case, thereexist b Ts,dist)> 0 and ks Ts,dist)> 0 such that( ( W Udis t k ) l l < 6 (T~,Kdist )o r > ks (T.,Kdist).Proof: Not presented due to space limitation

3Model

Model

U d i s t

(b)Figure 4: Current state estimator and disturbance ob-server ((a) and (b) represent the disturbanceobserver models (4) and ( 1 4 ) , espectively.)The performance of a disturbance observer is highlydependent on the sampling frequency. As the sam-pling frequency becomes higher, the disturbance ob-server can reject higher frequency disturbances. In thecase of limited sampling frequency, multi-rate imple-mentation of th e proposed disturbance observer is ef-fective.

4 ExampleA continuous time system described by

0 1 00 -128.59 ]z-t 2.1484e4 ]= [

y = [ 1 l z f n

is considered. Here, n denotes measurement noise andw denotes unknown but bounded disturbance of thesystem.4.1 Control System DesignFor a sampling time of 300 psec, a discrete model de-scribed by

[ 1 2 . 9 4 2 9 e - 4 10 9.6216e-1 [ 9 . 5 4 4 5 e - 4 16.3224==is obtained. We first determine a state estimatorgain L , [ 0.75 0.5 T and a state feedback gainK , [ 107.42 0.3 ] such that the sensitivity andcomplementary sensitivity of the control system aresatisfactory. The st ate estimator has p a aL,C)0.25, 0 .96, and the state feedback controller has((a FK,) 0.895, 0.93. For sufficient compensation

of the equivalent disturbance we choose a disturbanceobserver gain Kdzst [ 18 3.75 1 Then, we havep ((a @L,c KdistLcC) 0.28, 0.92. As stated inTheorem 1, the closed loop is stable and the distur-bance observer is convergent. Figure 5 shows sensitiv-ity and complementary sensitivity of the feedback loop.

Figure 5: Sensitivity and complementary sensitivity4.2 Simulation ResultsDisturbances of 0.2 sin(8nt) and 0.2 sin(8nt) +O . l s i n ( 2 4 n t ) are injected into the plant input in thesimulations of step command tracking and sinusoidalcommand tracking, respectively. For more realisticsimulations, we also introduce a control saturation ofIul 5 umax 0.7 amp, the quantization effect of ADCand DAC, and measurement noises.Figure 6 shows time response of the plant for a stepposition command. Comparisons of the model out-put with the plant output are shown. It is shown thatthe plant out put is very close to t he model outp ut fol-lowing the step command with disturbance compensa-tion. Figure 7 shows that the disturbance observer ef-fectively compensates the sinusoidal disturbance. Fig-ure 8 shows time response of the plant for a sinusoidal

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position command. Comparisons of the model outputwith the plant output are shown. It is shown that theplant output is very close to the model output and tothe sinusoidal command with disturbance compensa-tion. Figure 9 shows very good compensation of thesinusoidal d isturbances by t he disturbance observer.

1 ~ ' ~0 0 1 2 3 0 4 5 06 7 8 9

2 1 I0 1 2 3 0 4 O S 8 7 8 9 1lime [SRI

Figure 6: Step position command tracking

1 2 3 0 4 5 8 0 7 8 9 1- 0 8niMiw

Figure 7: Disturbance and disturbance observation instep position command tracking

5 ConclusionsThis paper has presented a new discrete-time statespace disturbance observer not requiring modeling ofdisturbances. A discrete-time robust tracking controlsystem is proposed, which consists of the state spacedisturbance observer, a state estimator, and a statefeedback controller. Through the results of simulations,it is shown that the proposed method is very effectiveto improve th e tracking accuracy of control systems inthe existence of external disturbances. It is observedthat the proposed disturbance observer effectively esti-mate and compensate any disturbances if t he sampling

0 1 2 3 0 4 5 6 7 6 9 1T i m 5 8 ~ 1

Figure 8: Sinusoidal position command tracking

0 6 t0 4

I0 1 2 3 0 4 5 8 0 7 8 9 1l ime [ cl

- 0 8

Figure 9: Disturbance and disturbance observation in si-nusoidal position command trackingfrequency is fast enough. In addition, accurate stateestimation was attained even in the existence of mod-eling uncertainties as well as external disturbances.

References[I] G.F. Franklin, J.D. Powell, and M.L. Workman,Digital Control of Dy nam ic System s, Addison Wesley,1990.[2] T. Murakami and K. Ohnishi, Observer-basedMotion Control: Application to Robust Control andparameter Identi f ication, Proc. of the IEEE Indus-trial Electronics Society: Asia-Pacific Workshop on Ad-vances in Motion Control, July 15-16, 1993, pp. 1-6.[3] K . Ohnishi, A New Servo Method in Mechatron-zcs, Trans. of Japanese Society of Electrical Engineers,Vol. 107-D, 1987, pp. 83-86.[4] T. Umeno and Y. Hori, Robust Speed Con-trol of DC Serv omoto rs using Moder n Tw o Degrees-of-Freedom Controller Design, IEEE Trans. on IndustrialElectronics, Vol. 38, No. 5, 1990, pp. 363-368.

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