Control Eng. Practice, Vol. 5, No. 4, pp. 477-483, 1997 Copyright © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/97 $17.00 + 0.00

Pergamon PII:S0967-0661(97)00027-0

DISTURBANCE REJECTION USING OUTPUT AND INPUT ESTIMATION. APPLICATION TO THE FRICTION COMPENSATION OF A DC MOTOR W. Nuninger, B. Balaud and F. Kratz Centre de Recherche en Automatique de Nancy - CNRS URA 821, InstitutNational Polytechniquede Lorraine, 2, Avenue de la For~tde Haye, 54 516 Vandoeuvre-lts-Nancy Cedex, France ([email protected])

(Received May 1996; in final form February 1997)

Abstract: In this paper, a method of rejecting disturbances in controlled systems is developed. Friction compensation for electromechanical processes, based on the friction estimation using an observer, is more specially studied. This procedure yields to a simultaneous state and input estimation through a fixed-size sliding observation window. The method is then tested and its performance is verified on a real electromechanical process, i.e. a DC motor. Copyright © 1997 Elsevier Science Ltd

Keywords: Friction, compensation, state estimation, input estimation, sliding window, unknown input.

1. INTRODUCTION

In the literature, friction compensation has been studied by several authors such as Gilbart and Winston (1974), who developed an adaptive compensation for an optical tracking telescope. For his part, Walrath (1984) proposed a pre-compensation of friction torque using a dynamic model, whereas Kubo, et al. (1986), tested the adaptive compensation for a robot arm control. Brandenburg, et al. (1987), Canudas de Wit (1988) and Schafer and Brandenburg (1989), have also studied friction compensation using an adaptive control based on the friction estimation. In the case of periodic disturbance rejection, one can consult (Hillerstr6m, 1994). The historical study given above was the starting point to solve the compensation problem for the benchmark of the authors' laboratory: a DC motor.

Real electromechanical processes usually present disturbances like Coulomb and viscous frictions. These frictions always play an important part in the mechanical systems, and therefore intervene in the dynamics. Unfortunately, the frictions are not known a priori because they depend on environmental factors, such as temperature and lubrication for instance. That is why frictions can be considered as system disturbances. Nevertheless, it is possible to evaluate them from some measured process variables. Moreover, friction is usually considerable, so it is necessary to take it into account in the control law. This means that it is necessary to find a control law so that the system output no longer relies on the disturbance. A solution is to use the estimation of the friction in order to control the system that is making a friction compensation.

In this paper, an on-line estimation of the disturbance is also used in order to implement a control law that can solve the disturbance-rejection problem. But, 477

478

W. Nuninger et al.

unlike the previous works, the method described here is based on a simultaneous output and input estimation using a fixed-size sliding observation window. As the system model presents uncertainties, the infinite dynamic observer memory generates a phenomenon referred to as "divergence". The relevance of the finite memory observer is to overcome this drawback. Indeed, its structure has an intrinsically finite memory process, which naturally limits the uncertainties accumulation. This kind of observer has been studied by Janyene (1987) and by Kratz, e t al. (1993, 1995). Both approaches restrict the dimension of the on-line reconstruction of measured variables taking the given mathematical model. These methods usually use the following calculation aspects: -the estimation problem is formulated on a limited horizon, - a sliding horizon is used. The present work shows that such a method is useful to estimate the friction in electromechanical processes in order, further, to make the friction compensation using the estimation. The first part of this communication presents the generalised state estimator that makes a simultaneous estimation of the state and the input of the system. The second part extends the results on a fixed-size sliding horizon. Then, a DC motor is presented and an explanation is given of how the friction compensation is implemented, using the disturbance estimation. Finally, some experimental results are given.

of known constant covariance matrices, Vy and V,. It should be stressed that the two last equations in (1) respectively represent the state and control measurements. Note that these two equations can be lumped together using the measurement vector [yT ZT ]r SO that the conventional state-space representation is obtained. In addition, the formulation (1) includes disturbances within the input u. Consider the time horizon of size N, i.e. the measurements for k from 1 to N. One can then define extended state and output vectors such as: x I

Yi

vl 1

u 1

z~

W L

Y:

x~

X ~: =

Z N

V: EN

(2)

z I

UN~ I

"N -~

XN

Y~

VN

J

Then, the model structure (1) can be contracted into the following model:

{

MX~ = 0 Z~ ---'HX N + E N

(3)

with: 'A

B

M=

-I n A

B

-I,,

\.

A

(4a) B

-I n

2. G E N E R A L I S E D 'STATE ESTIMATOR iC This paper considers the following state-space model of the system in discrete-time representation:

D H ~

(4b)

"''

D : A x k + Bu k Yk = Cxk + Vk Z k = Du k + w k

Xk÷ I

(1)

(

C

where ' ' stands for the zero matrix and I, lor the identity matrix of rank n.

where: XR is the state vector of dimension n, Uk is the input vector of dimension m, Yk is the output measurement vector of dimension p, ZR is the input measurement vector of dimension q, v k and w k are respectively the output and input measurement noise vectors of appropriate dimensions. A, B, C, D are real matrices of appropriate dimensions. It is assumed that both error vectors VR and w k are respectively zero-mean Gaussian independent vectors

The previous representation (3) of the system is more convenient in order to estimate the system state and input, taking account of N measurements of the output. So the estimation problem can be expressed as the minimisation of the criterion (5a) with respect to the estimation of Xs submitted to the constraint (5b): JN = 1

HXN-Z~

MX~ = 0 with:

:

(5a)

(5b)

Disturbance Rejection Using Output and Input Estimation

V-

.

.

•

V

L ....

.

.

(6)

v,J

Indeed, the problem is reduced to the minimisation of the criterion (10a) with respect to the estimation of XkL, subject to the constraint (10b): J~L =

This problem is solved using the Lagrange method. The following estimation is obtained: XN = PR-tHrV-tZ~

(7)

where: R = HrV-LH + MrM

(8a)

P = I - R-'Mr(MrR-'M)-'M.

(8b)

This solution requires the inversion of a large matrix. So, for an on-line treatment, it is necessary to have an appropriate formalism. Several techniques have been developed (such as recursive ones or techniques using a gliding observation horizon). In the following section, this last method is presented (KratzBousghiri, 1994; Bousghiri, et al., 1994).

479

HX~r - Z ,

-,

MX~, = 0

(lOa) (lOb)

There is no difficulty in solving this problem, as it is identical to problem (5). So, using the Lagrange method, as for equation (7), the following solution is obtained: ~(kL = pR-IHrV-IZu-

(11)

with P and R expressed by (8a) and (8b) respectively. Matrices M, H and V are identical to the matrices appearing in (4a), (4b) and (6) with the appropriate dimensions, taking into account the width of the observation window. Note that it is only necessary to compute fik and ~-k÷,which are the two latest component values of the vector ~(u..

3. GENERALISED STATE ESTIMATOR ON A FIXED-SIZE SLIDING WINDOW

So, if the matrix PR-'HrV -' is partitioned as:

To allow an on-line treatment, the computation of both XR and UR estimations is based on dynamic observation in the past and present time. It has been shown by Darouach and Zasadzinski (1992) and Ragot, et al. (1992), that the filter memory is of finite size, and the estimates can be calculated using only a given number of measurements. Moreover, they show that the inclusion of additional measurements does not provide more information for the state and input estimation. So it is possible to limit the calculation size. Indeed, the term in the block that forms the lower limit of matrix P (8b) tends towards zero. As a consequence, the inclusion of the corresponding measurements provides no additional information about fit and ~k÷,. Therefore, it is possible to limit the size of the matrix P by keeping only terms that differ significantly from zero. Therefore, one can again write the estimation problem based upon a sliding observation window Z~ of fixed size L, defining the following vectors:

(12)

where the block PL is a (n+m,(p+q)L+p) matrix, following estimates of fik and x~÷t are obtained: /fi~ ) = PLZu.

Xk+l

(13)

There is no difficulty in applying eq. (13) if it is assumed that the width of the data window is correctly chosen. Indeed, it can be shown (Darouach and Zasadzinski, 1992) that if there is any significant change in the new estimate or covariance, the estimate can be calculated only on the fixed number L of measurements. Note that the initial conditions are taken with respect to eq. (11) using the L first measurements of the output, that is why the initial conditions are obtained as:

Xk- L

X~ =

I

Uk

Xk+l

Vk- L

Yk-L

z,..._L

uk.-t

ZkL

I

Yk+,

~(oL = PR-IHTV-IZoL •

Wk-!

(9)

E N -~

Vk W k Vk+l

(14)

Thus, the estimation algorithm is summarised in the following statements, where the four first steps, (a) to (d), are computed off-line and stored:

480

(a)

W. Nuninger et Determination of the width L of the sliding window.

(b) Computation of matrix PR