Parameter Estimation of Systems Having a Class of Nonstationary Stochastic Disturbances

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PARAMETER ESTIMATION OF SYSTEMS HAVING A CLASS OF NONSTATIONARY STOCHASTIC DISTURBANCES Tung Sang-ng and Moneeb A. Magdy Depal"t/III'/lt 0/ E/l'l'tl"im/ allt! COII/Pllt"I" t .' lIgi/II'nillg. Till' [ ·lIh'I' nit\· of \\ 'o/lOIlI{OIlI{ Awtm/ia 25(){)

Abstract. An indirect method of parameter estimation for linear systems subject to a class of non stationar) dist urbances is proposed. A closed loop experiment using an adaptive controller is des igned to suppress the nonstationary disturbance so that the resulting system has a zero-mean stationary di sturban ce. A >pecial estimation model is then constructed for the controller. The open loop system parameters are then retrie\ed recursively from the estimated controller parameters. Keywords. Identification; adaptive control; stochastic systems; nonstationary di sturbance s.

INTRODUCTION The study of recursive parameter estimation techniques is of interest in many applications where a process model is required on-line for control, monitoring, or forecasting purposes. The literature on recursive parameter estimation methods is extensive, see, e.g., the surveys of Astrom and Eykhoff (1971); Saridis (1974); Soderstrom el. al. (1978). A lot of effort has been devoted to the analysis of recursive parameter estimation algorithm s for linear systems having stationary stochastic disturbances, Ljung (1977a , 1977b, 1981); Solo (1980); Soderstrom and Stoica (1981); Young and lakeman (1979); Dugard and Landau (1980); Goodwin and Sin (1984), to name just a few . Moreover, rigorous theoretical proofs of convergence have also been reported for some algorithms, Ljung (1977a, 1981); Solo (1980); Goodwin and Sin (1984). The offered proofs, however, are often based on assumptions which are difficult to check and may be violated in practice. This is particularly the case when the system under consideration is subject to nonstationary disturbances. To implement a model reference adaptive controller or a selftuning controller using a LQG or a multi-step ahead design criterion, the parameters of the open loop systen are required (Ljung and Trulsson 1981 , Astrom and Wittenmark 1984, Grimble 1984). For systems disturbed by zero-mean stationary noi se, the open loop system parameters can be obtained by direct identification methods. However, for systems subject to nonstationary disturbances, many of the usual mathematical tool s used for recursive identification analysis are not applicable . For example, it is not possible to reduce the analysis to the study of Lyapunov-like function s (Solo 1979) or to relate the behav iour of th e algorithm to the solution of an ordinary differential equation (Ljung, 1977a, 1981; Solo 1980). Plant s subj ect to non statio nary disturbances are common in industrial processes. Typical exa mples are paper mills, steel rolling mills and man y industrial chemical processes (Box and lenkin s 1970).

In this paper, we propose an indirect method of parameter estimation for linear systems subject to a class of non stationary disturbances. We consider the class of nonstationary disturbances that can be modelled by passi ng white noise through an Autoregressivc Integrated Moving Average (ARIMA) filter with seasonal and oscillatory behaviour. By a proper design of a closed loop identification experiment, the nonstationary di sturbances can be reduced to a zero-mean stationary stochastic disturbance in clo sed loop. The system model can then be retrieved from the closed loop system identified by the applica tion of parameter estimation techniques with prO\'en converge nce property.

SYSTEM DESCRIPTION In this paper , the system will be described by a polynomial representation. The following notation will be used:

A(q - t) = ao +' atq - I + ... + anaq - na where q - I is the backward shift operator q - Iy(t) = y(1 - I). If Go = I, the polynomial is said to be monic and the degree of the polynomial A (q - 1lis written as no or degA. The ~rgument of the polynomial is dropped if there is no ambiguity. Now consider a linear system described by the following difference equation:

where A (q - I) and B (q - I) are co-prime and stable polynomials, k 2: I is a positive integer , {u(!)} , {yU)} are the control and output sequences respectively, {w (I)} is a nonstationary stochastic disturbance described by: D ' (q -I)w(l) = C(q - I)e(l)

(2 .2)

where:

D. = (I _ q - l) D.p = (I - q - P),

P > I is a positive integer and - 2:$ '" :$ O. The above noise model is a generalization of the ARIMA model considered by Box and 1 enkins (1970). The term s ~ and ~p represent linear and seasonal trends respecti vely. The quadratic term represents a sustained oscillatory disturbance to the system where the frequency of oscillation is determined by "'. Oscillatory di sturbances are often pre sent in many industrial plant s. The cause ma y be due to nonlinear elements. In a digital control system, it may be caused by quantization error in the analogue to digital converters (Isermann 1981, pp . 464). A special case of the abo\'e noise mod el is to replace the white noise sequence {e(1l1 by the Dirac delta fun.:tion cS'o (l) where b,o = I for' = ' 0, otherwi se b'o(l ) = O. In thi s case, the model represe nt s a purely det erministic disturbance. For further discussion on systems ha\'ing purely determini stic disturbances, sec Goodwin and Chan (1983) and Ng and ;\Iagdy (1983) .

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In Ihe fo ll o win g , ec tiom , Wc shall pr esent a method to identify the polynomial; 1I ('I I), B(q - I), C(q - I) and D(q - I) . A number of ass umption s will be mad e in regard to the plant and di sturban ces. Th ese ar e: 1.

The time delay k is known.

2.

The pol yno mial degrees n a , Ilh, known .

3.

The numb er of poles (both real and compl ex ) on the unit circlc of the noi se model are known .

"l'

and lid are

PARA~'.tETER

ESTI\IA TIO N

Clearly we are co nce rned with systems ha lin g unkn ow n para met ers, In ord er to achi el'e th e control obje ({i,,~s di s(ll ssed in the la st section , adaptil'e co ntro l tec hniqu e, ca n be u, ed . \\'e apply th e well known tec hniqu e of deril'ing a n estimation model for the controll er (referred to as direct co ntro l or ImpliCIt identifi cation meth od in ada pti l'c co ntro l termin o logl' ) a nd then appll' pro\'en recursive parameter est im at ion tc,:hniqu es to th e estimation model. If the pa ra met er, 01 t he e, tlmall o n model CO Il\'CTg c. th e po lyno mial s A , B. C and 0 ca n th en be ret ri c\ed fro m th e estim ated param etcrs. Using th e ;ystem mod el (2 . 1) , (2.2) and th e co ntroll er equa tion s (3.1) to (3.5) a nd a fter so me al ge braic ma nipul a ti ons, we obtain:

CLOSED LOOP EXPERIMENT In order to appl y identification method s that have proven con vergence pro pert y, Ihe trend and the oscillator y behal'iour caused by poles on the unit circl e of the noi se ,ransfer function must be elimin ated. We propose to accompli sh thi s by a closed loop ex periment.

H.I) where t (I) is the control error defined by ((I) =

r(l) -

11,. (1 -

k). E4ua tio n (4.1) ca n be rewritt en in the form :

Consid er a gen eral controller: (4.2) 11 (r)

(3.1 )

Where {urU)} is a persistently exciting set point pe rturbation sequence. The control sequence {lI(1)} consists of a feedback term from the output and a feedforward term from the set point. This type of controller has been widely used in selftuning controllers (Clarke and Gawthrop 1975; Astrom and Wittenmark 1980) and in model reference adaptive control (Egardt 1979). Particular choices of polynomials R, Sand T result in several well known designs of adaptive controllers (Egardt 1979; Astrom and Wittenmark 1980; Magdy and Ng 1983). Clearly, a particular closed loop behaviour may be achieved by an appropriate choice of polynomials R, Sand T. For example, in order to suppress the nonstationary disturbances such that the resulting control error converges to a stationary, zero-mean, moving average sequence of order k - I, then (for details, see Magdy and Ng 1983): 1.

k) + Rte(r) .

(4.4)

The estimation model (4.1) ha s been wid ely di scu ssed in the literature on self-tuning control and model referenc e adaptive control (Egardt 1979; Goodwin and Sin 1984; Magd y and Ng \983). A number of well known recursive identification algorithms with proven convergence propert y can be used to estimate 0, consistently under certain assumptions. The most important of which is that the polynomial C satisfies a strict > O. For detailed proof positive real condition, i.e., and other assumptions, see Goodwin and Sin (1984).

&- !

(3.5)

Where deg R = k - I, lis = max[lIc - k ,na + nr - k). When the system param eter s are known , straightforward algebrai c manipulations of (2.1), (3.1) to (3.5) show that the above design forces the Closed loop system to sati sfy : y (l) = ur{l -

= [u(l), ... ,1/(1 - "r) , y(l) , .. . , y (l - II s ) , lI r (t),· .. ,lIr U - 11,. )]

(4.3)

(4.5)

(3.2)

T = C

J

(3.4)

Bearing the abo ve in mind, we now choose:

R'

l'

<1> ,

"Cl/c

(3.3)

The polynomial R must include all the poles of the noise transfer function.

R = R'B

0; ;:::; [ro.· · . , r" r'SO. · · .. 5"s ' I ,l'\ . ..

We note that the controller used for the closed loop experiment contains poles on the unit circle . During the transient period, it is likely that the estimated R may have zeroes located in the unstable region, thus causing large fluctuations in the control signal. In order to avoid the zeroes of R wandering into the unstable region, one can compute the zeroes of the polynomial R, where R denotes the estimates of R, and then constrain s the zeroes of R to lie on or outside the unit circle at each it eration. Clearly thi s is not a satisfactory solution as polynomial rooting is both time consuming and sensitive to parameter variation s. Another method is to rearrange the estimation model in such a way that the zeroes of R on the unit circle are in fact restricted to lie on th e unit circle. Let:

The polynomial T must include the polynomial

C. 2.

wh ere

(3.6)

We remark that the controller we have chosen for closed loop experiments is similar to the deadbeat controller oft en used in determini stic system s. Such de sign is allracti ve becau se of it s simplicity and ' the ease with which an estimation model can be constructed. However, it has a number of drawback s such as large control effort may be required and it is restri cted to minimum phase plant s. In secti on 6, we shall show that ext ension to a genel al model reference adaptil'e controll er requires onl y minor modification to the method being proposed. Discu ss ion on the po ssibility of extension to non-minimum phase plants will al so be given.

(4.6)

then R

(4.7)

and (4. 8)

where

e;

0 ;(1) = [(1 + 0 (1 -

= [h o, ,, l1 o ,h l . . . . h"h

q 2 )u(l), U (l - I), I)U(I - 1) , .. . , 0 (1

-

(-l.9)

l

I)U(I

(4,10) II h ))

I) = I + cx (l - I)q - 1 + q 2, 0< (1 - I) is obtained from the first two te rm s of 0(I - I), th e est im ates of 0 at time P - 3. Substitutin g (4 .9) and (4 . 10) int o t-I and n h = (4.1), it follow s tha t (4.2) can be rew ritt en as :

a (t -

"r -

- k

f(1) = q

c

[eT 0 (1)1 + Rt e U )

(4 . 11 )

:\ollstatiollar\, SlOchastic Distll rbanees where:

TA ·1 /

(0 . 1)

(4.12) [~;(I),y(T), ... ,y(t - n s ),u,(I) , ... ,u,(I - ne)] (4.13) We note that without re-arrangemenr, the estimation model (4.8) conralns bilinear terms ahj,i = I, .. . ,nh ' The proposed re-arrangemenr treats the first two terms differently to obtain an estimate of a recursively . The most up-to-date estim ates of a is then used to prefilter the input data in the next iteration. The proposed method has some similarity to the generalised least squares method. The explicit identificaiton of a also allows the condition - 2:s a :s 0 to be monitored continuousl y. We also note that the unstable modes of the system have been shifted into the data vector (4.6) and (4.10). This ensures that the controller does not contain critical poles to be stimated . This reduces the possibility of the controller wandering into the unstable region. Furthermore, the proosed estimation model also reduces the number of parameters to be estimated by p +

and E ' (I)

respecti vel y, wherc

if the zeroes of A ·H are placed at \\ ell damped position s, conrrol effo rt s ca n be redu ced in so me cases ;

3.

it is possible to ext end th e propo sed meth od to non -minimum phase plant s . In thi s case . we factori ze the polynomial B into S + B - , wh ere

In section 4 , we rearranged the estimation model to a void the critical poles of the conrroller wanderin g inro un stable regi o ns in order to avoid large fluctuations of control signals . Another cause of large control signals may be due to the controller having integrators in the feedback loop. In practical implementation , precautions similar to that discussed in Isermann (1981) should be taken to prevent the problem of "inregrator wind up" . One plausible method of limiting the control signal u(t) (after the transient period) is to make use of the fact that equation (3.2) has infinite number of solutions . Consider the following modifications the design:

Step I : Estimate the (k-I) parameters of the monic polynomial

• R le(t)

= dt)

-k· T q

- -.-[8
R=

R' - q-kF(q-l)

S=

(5.1)

Step 2 : Compute the polynomial BD from (3.3) and (3 .4) using (part of 0) and R I obtained in Step I,

(6.3)

S + AF(q - l)

(6.4)

where F(q - I) is an arbitrary polynomial. It is clear that Rand S still satisfy (3.2) and the control input is now given by:

0,

u(l) =

(5.2)

Step 3: Compute the polynomial AD from (3.2), (3.4) and (3.5) using R I obtained in Step I and a in 0,

T_u,(I) -

BR

.5_ y (l)

(6,5)

BR

Appropriate choice of F(q - I) can thu s be used ID limit Ihe conrrol signal. With these chan ges in the co ntroll er, th e estima· tion model is now replaced by:

(5 .3)

Step 4 : Compute the common factor and P2 .

D of the polynomials PI

(6 .6)

where q - k

;(1)=«1) -

Step 5 : Compute polynomials D obtained in Step 4.

A and iJ

using (5 .2), (5.3) and

Thus at each update of 0, an estimate of the system and noise parameters can be obtained if required. We remark that the polynomial R I in (5. I) can be estimated using standard least squares or instrumentarvariable methods. DISCUSSION AND EXTENSION In the closed loop experiment discussed in section 3, the design can easily be replaced by a model reference adaptive controller . In this case, (3.2)and (4 . 1) are replaced by:

IS VOL 2- E

.-1 .1/.

B + and B - represenr the stable and un stabl e factors of B respectively . The pol yno mial B - is then included in the reference mod el BI/ . For detailed d iscussion of the meth od, see Astrom and Wittenmark (1980).

In the sYstem we are considering, it may be desirable to overestimate ho so that when a is computed from the second term on the right hand side of (4.9), the condition - 2:s a :s 0 is more likely to be satisfied. This is simply to avoid the zeroes of the quadratic term wandering into the unstable region .

Rio

" SIl Il, (I).

2.

h .

Assuming that the parameter vector of the controller can be estimated consistently, we are now in a position to retrieve the polynomials A , iJ, C and D, where A denotes an estimate of A etc., from the estimates 0 at each update of O. The following presents one of several possibilities .

'I

better conrrol o\·er the dosed loop ,ystem be· ha viour ;

0.5:S~:SOO.

RETRIEVAL OF OPEN LOOP SYSTEM

,.1 .11 .1' (1) -

I.

3. It has been shown (Astrom and Wittenmark 1973) that the estimate h 0 can be fixed a priori and if an estimation algorithm converges for hO , it will still converge (Ljung 1977a) if

=

. ' (1)

S ·I/ are the reference mod el pol ynomiab. With appropri ate modifications to equation s in secti o ns .. a nd 5. th e meth od proposed in thi s pa per can be used to re trie\·c th e o pen loo p parameter s. The advantages o f using a ge nera l refere nce model include:

A F(q C•

q - ~k

I)

.1'(1)

+

iJ F(q C

_ I) 11(1)

(6.7)

We note that the parameters to be estimat ed remai n unc ha nged and ;(t) is a quantity computed from measured dat a and estimated parameters. Thus no modi ficatio n to th e eSl im ation model is needed. CONCLUSION This paper has shown that system s subject to nonstati o nary dis· turbances can be identified using closed loo p e~pc rim e n t a l tec hniques. The proposed method mak es use o f so me properties of the model reference adapti\·e co nt roller to suppress the no nstationary disturban ces so that stand ard pa ramet er estimation techniques can be appli ed. !\I odificati o ns have a lso been made to the estimation model o f th e adapti \·e co ntro ll er in such a

Tung Sang-ng and \fonccb .-\. !\Iagd\' way that poles on the unit circle of the controller are in fact restricted to lie on the unit circle. This rearrangement of the estimation model guarantees the closed loop stability and also improves the transient behaviour of the closed loop system. The retrieval of the open loop system parameters is then achieved by solving simp le a lgebraic equations. ACKNOWLEDGEMENT This work was supported by The University of Wollongong Research Grants Committee. REFERENCES Astrom, K.J. and Eykhoff, P. (1971): 'System identification A survey', AUlOmatica, 7,123-162. Astrom, K.J. and Wittenmark, B. (1973): regulators', Automatica, 9, 185-199.

'On self-tun ing

Astrom, K.J. and Wittenmark, B. (1980); 'Self-tunin g controllers based on pole-zero placement', Proc. 1££, 127, Pt D, 120-130. Astrom, K.J. and Wittenmark, B. (1984): Computer Controlled Systems Theory And Design, Prentice Hall. Box, G.E.P. and Jenkins, G.M. (1970): Time Series Analysis: Forecasting and Control, Holden Day.

Isermann, R. (1980): Digital Control Systems, Springer Verlag. Ljung, L. (1977a): 'Analysis of recursive stochastic a lgorithms', 1£££ Trans., AC-22, 551-575. Ljung, L. (1977b): 'On positive real transfer functions and the convergence of some recursive schemes', 1£££ Trans., AC-22, 539-551. Ljung, L. (1981): 'Analysis of a general recursive prediction error identification algorithm', Au tomatica, 17, 89-99. Ljung, L. and Trulsson, E. (1981): 'Adaptive control based on explicit criterion minimization', IFAC World Congress, Kyoto, Japan. Magdy, M.A. and Ng, T.S. (1983) : 'A genera li zed model reference adaptive controller with nonstationary stochastic disturbances', Technical Report, The Un iversity of Wollongong, Australia. Ng, T. S. and Magdy, M. A. (1983): 'Representation and transient behaviour of model reference adaptive controller with purely deterministic disturbances', Technical Report, The University of Woliongong, Australia. Saridis, G. N. (1974): 'Comparison of six on-line identification algorithms', Automatica , 10,69-79.

Clarke, D. W. and Gawthrop, P.J. (1975): 'Self-tuning controller', Proc. 1££, 122, 929-934.

Soderstrom, T., Ljung, L. and Gustavsson, I. (1978): 'A theoretical analysis of recursive identification methods', Automatica, 14,231-244.

Dugard, L. and Landau, I.D. (1980): 'Recursive output error identification algorithms: Theory and evaluat ion' , AUlomatica, 16,443-462.

Soderstrom, T. and Stoica P. (1981): 'Comparison of some instrumental variable methods - consistency and acc uracy aspects', Automalica, 17, 101-115.

Egardt, B. (1979): Stability of Adaptive Controllers, SpringerVerlag.

Solo, V. (1979): 'The convergence of AML', I££E Trans., VoI.AC-24,958-962.

Goodwin, G.c. and Chan, S.W. (1983): 'Model reference adaptive control of system having purely deterministic disturbances', I£EE Trans., AC-28, 855-858.

Solo, V. (1980): 'Some aspects of recursive parameter estimation' ,1nl. J. Contr. , 32, 395-410.

Goodwin, G.C. and Sin, K.S. (1984): Adaptive Filtering Prediction and Control, Prentice Hall. Grimble, M.J. (1984): 'Implicit and explicit LQG self-tuning controllers' , IFA C World Congress Budapest.

Young, P. and Jakeman, A. (1979): 'Refined instrumental variable methods of recursive time-series analysis' ,1nt. J. Conlr., 29, 1-30.