HETTHÉSSY, J.

KEVICZKY, L.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

RELATIONS BETWEEN THE MINIMUM VARIANCE CONTROL AND THE OPTIMUM TRANSFER FUNCTION IN THE WIENER SENSE By L. KEVICZKY and J. HETTHESSY* Department of Automation, Technical University, Budapest Presented by Prof. Dr. F. CS . .\.KI Nowadays a huge progress of the stochastic control theory is s~en in the field of control engineering. An enormous amount of books and papers deal with the problem to control a process influenced by stochastic environment. Not only theoretical results [1, 2], but a lot of successful industrial applica-tions [3,4,5,6] areknown from the last few years. One of the most applicable and powerful methods from the stochastic control strategies is the minimum variance control [3,4,6,7,8], where the purpose of the control is to minimize the variance of the controlled signal about its desired value. For the solution of that sort of problems, the optimum transfer function in the WIENER sense is available. The purpose of this paper is to show that the minimum variance control strategy can be derived by means of the optimum transfer function in the WIENER sense. While in the general case the determination of the optimum transfer function in the W IENER sense has difficulties in connection with the proper factorizations of the power spectra, in the case where the system is described by an ASTRi:bI model, this factorization is easy to perform through a simple polynomial separation. 1. Statement of the problem Consider a single input-single output discrete time (sampled) linear system 'with constant and known parameters, which can be described by the difference equation: y(t) ., '" + bmu(t - d - m) , (m;S: n; t = 0, ±l, ±2, ... ) "Institute for Electrical Power Research. (1) 28 L. KEVICZKY and J. HEITHESSY where y is the controlled signal, uis the control signal, and d is the time delay of the process. Introducing the j!olynomials and in accordance 'with the interpretation of the backward shift operator Z-1 the system equation (1) can be written in the following form: A(Z-l) y(t) = B(Z-1) u(t - d) . (2) By choosing U o = 1 the minimal number of the necessary parameters is ensured. The stochastic environment and the noisy measurement situation are taken into account by an additive stochastic process reduced to the output of the system. This stochastic process is driven by a white noise e(t), charac-terized by zero mean value and variance 1. By means of polynomial R( -1) - 1 ...L -1 I I -R Z - I T1Z T' .• T TRz a moving average (xm) or an autoregressive (xa) stochastic process of e(t) can be generated in the following way [3]: 1 Xa(t) = R(Z-1) e(t) = e(t) - T1xa(t - 1) -Xm(t)= R(Z-l) e(t) = e(t) T1e(t - 1) + ... In the general case the process can be disturbed by an autoregressive moving average stochastic process of the driving noise e(t) with variance 1: where (k < n). Taking into consideration the additive disturbance Xn in Eq. (2) we get B(Z-1) C(Z-1) y(t) = u(t - d) + }. e(t). A(Z-1) A(Z-1) (3) RELATIONS BETWEEN THE MINIMUl"f VARIANCE C01VTROL AND OPTIMU1',f TRANSFER 29 In order to have a stahle system and to avoid numerical instahility the polyno-mials ZnA{Z-l) and ZkC(Z-l) are assumed to have all their zeros inside the unit circle [2,9]. Notice that the coincidence of the nominators of the process B(z-l)/A(z-l) to he controlled and that of the C(z-l)/A(Z-l) model of disturb-ances mean no special condition, because a system can always be arranged to have the form of Eq. (3), where A(Z-l) = A1(Z-1) A 2(z":'1) B(Z-l) __ B1(2;-1) A 2(Z-1) C(Z-l) = C1(Z-1) A1(z-1) . l e(tJ C(Z-I) ./l~ I ' ~ Xn (I) _U_f<;...!lJ_-l' ,~g~:j Z-dt-I---'0f------- y (t} Fig. 1 The flow chart of the so-called ASTRoM-model by Eq. (3) is shown in Fig. 1. The problem is to determine the control signal u(t) from the knowledge of the system parameters and the ohservations {y(t), y(t - 1), ... , u(t - 1), u(t - 2), ... } in such a way that the loss function will be as small as possible. E{ .} denotes mathematical expectation. 2. Determination of the optimal control by means of the system equations To determine the optimal u(t) let us write the earliest value of y influenced by u(t) on the basis of Eq. (3): 30 L. KEVICZKY and J. HETTHESSY The second term in the right side is a linear function of eft + d), eft + d .- 1), ... • •. , eft + 1), eft), eft - 1), ... Since eft), eft - 1), ... can he exactly computed from the observations {y(t),y(t ~ 1), ... u(t - d), u(t - d - 1),.;.J byEq. (3), but eft + d), eft + d - 1), ... eft + 2), eft + 1) are independent of these observations, this separation is easy by means of the polynomial equation (4) where F(Z-l) = 1 + f1z-1 + ... + fd_lZ1-d G(Z-l) = go + glZ-l + ... + gn_1i-n. Straightforward algebraic manipulations then give the optimal control law [3,10,11]: G( -1) UO(t) - - Z y(t) - B(Z-l) F(Z-l) . (5) By such a control strategy the controlled signal has the following form: y(t + d) = ).F(Z-l) eft + d) • (6) From Eq. (6) y(t) is seen to be described by a moving average stochastic process of e( t). Thus the expectable value of y is zero, while the variance of the output, that is, the minimal value of the loss function is Vmin = ).2(1 + j; + ... f:-1) • In connection with Eq. (6) the minimum variance output has to be evidently a moving average stochastic process of eft) at least of the order (d - 1). It is obvious, because the independent noises eft + d), eft + d - 1), ... • . . , eft + 2), eft + 1) appear after generating the control signal u(t). The flow chart of the optimally controlled system is shown in Fig. 2. ! eft) ~---------------------~ IN" D I 1 __ OISY proce .A C(z-J} I I I I A (z-ij , i I Juri B (Z-I) -d .<'A yfl I A(z-I) Z '<>' I I I L _____________________ ~r-------------, I I ! u(t} - G(Z-I) I i B(Z-I} F(z 1) I I I I Controfler : L ____________ _ Fig. 2 RELATIONS BETWEEN THE MINIMUM VARIANCE CONTROL AND OPTIMUM TRANSFER 31 3. Determination of the minimum variance control by PleaDS ·of the optimum transfer function in the WIENER sense Let us transform the system equation (3), the polynomial separation equation (4), as well as the equation of the optimal control hy Eq. (5) in such a way that the argument of the polynomials will he z instead of Z-l. Define A *(z) = zn A(Z-l) = zn + a1zn- 1 + ... + an B*(z) = ZnB(Z-l) = lioZn + li1zn- 1 + ... + limzn- m C*(z) = znC(z-l) = zn + c1zn- 1 + ... + CkZn- k G*(z) = znG(Z-l) =gozn + glzn-l+ ... + gn-lZ F*(z) = Zd F(Z-l) = zd + f1zn- 1 + ... + h-lZ hence, the system equation is B*(z) C*(z) y(t + d) = --u(t) + )'--e(t+ d), A*(z) A*(z) (7) the polynomial separation equation is zdC*(z) = A *(z) F*(z) + G*(z) , (8) and the optimal control is UO(t) = - zdG*(z) y(t) YO(z)y(t) B*(z) F*(z) =" '. (9) The pulse transfer function Y~(z) will he shown to equal the optimum transfer function in the WIENER sense. On the hasis of Fig. 3 the pulse transfer function hetween e(t) and y(t) is written as: W(z) = }'C*(z) zd A*(z)zd - Yv(z) B*(z) (10) .!e(t) c~ (z) .A Niz) ):!.!!L. 8* (z) -d "- , y{t) An (z). Z :>' I Yv (z,) I,. I Fig. 3 32 L. KEVICZKY and J. HE1THESSY The optimaIYv(z) is sought for, by which y(t)fQllow&" x;(t) = },F(Z-l) e(t) = }.F*(z) z -de(t) as an ideal signal. Fig. 4 shows the optimal transfer function 1 Wc(z) = ---'-" ------A*(z) zd - Yv(z) B*(z) to be determined in a semi-free configuration with" Xb(t) = e(t) as input signal as ideal signal (11 ) x;(t) = }.F*(z) z -de(t) Yj{z) = },C*(z) zd as the pulse transfer function of the fixed elements. xj{t) ~ Xh(t) x;JtJ A"(Z}Zd - Yv(z)B"(z} Fig. 4 According to the theory of the sampled stochastic processes [3,9, 12], the physically realizable optimal pulse transfer function can be determined by [ q)bi(Z) ] ~+ q)~(z) (12) assuming that YJ(z) has no poles or zeros outside the unit circle. This condition is met in our case. In Eq. (12) q)bb(Z) is the power spectrum of Xh q)b;(Z) is the cross-power spectra of Xb and x;, q)tb(z) and q) bi,(z) are obtaine.d by spectrum factorization where the factor q)tb(z) contains all the poles and zeros of q)bb(Z) inside the unit circle, while the factor q)bi,(z) contains all the poles and zeros of q)bb(Z) outside the unit circle. Finally, the superscript + refers to the positive time func-tions [9, 12]. RELATI01YS BETWEEN THE MINIMUM VARIA,YCE CONTROL AND OPTIMUM TRANSFER 33 In our case of interest the following relationships hold: tPbb(Z) = 1 , tPbi(Z) = tPbb(Z) Yi(Z) = }.F*(z) Z -d. Taking into consideration these relationships in Eq. (12) we get F*(z) z-d W (z) - -'-'---er - C*(z)zd (13) Substituting Eq. (13) into Eq. (ll) and expressing Yv(z), the optimal pulse transfer function Y~(z) becomes: Y~(Z) = A*(z) F*(z) - zdC*(z) _ F*(z) B*(z) z-d zdG*(z) F*(z) B*(z) Comparing the above expression of Y~(z) with Eq. (9) the two ways of solution of the minimum variance control problem are seen to give the same result. Thus it has been shown that the minimum variance control strategy can be derived by means of the optimum transfer function in the WIENER sense. Snmmary In this paper a derivation of the minimum variance control law is given by means of the optimum transfer function in the WIENER sense. It is shown that the optimum transfer function has to be determined in a semi-free configuration. The ideal signal is chosen after taking into consideration the independence of the disturbances related to the output of the system. References 1. AOKI, M.: Optimization of Stochastic Systems. Academic Press, New York 1967. 2. ASTROM, K. J.: Introduction to Stochastic Control Theory. Academic Press, N ew York 1970. 3. ASTRO:M, K. J.: Computer Control of a Paper Machine. An Application of Linear Stochastic Control Theory. IBM J. Res. Development, Vol. 9, 1967. 4. BORISSON, U.-WITTEl'<"1dARK, B.: An Industrial Application of the Self-Tuning Regulator. Report 7310 of Lund Institute of Technology, Division of Automatic Control, 1973. 5. OTOMO, T.-NAK.AGAWA, T.-AK..-\'IKE, H.: Statistical Approach to Computer Control of Cement Rotary Kilns. Automatica, Vol. 8, pp. 35-48., 1972. 6. BORISSON, U.-SYDING, R.: Self-Tuning Control of an Ore Crusher. IFAC, Budapest 1974. 7. PETERK..-\., V.: Adaptive Digital Regulation of Noisy Systems. IFAC, Prague 1970. 8. WITTENlIIARK, B.: A Self-Tuning Regulator. Report of Lund Institute of Technology, Division of Automatic Control. 1973. 9. Csin, F.: Dynamics of Control L~ops (in Hungarian). Akademiai Kiad6, Budapest 1966. 10. ASTROM, K. J.-WITTENMARK, B.: Control of Constant but Unknown Systems. Lecture on the IFAC, Paris 1972. n. KEVICZKY, L. - HETTHESSY, J.: On the Minimum Variance Control Strategies. Per. Pol. El. Eng. Vol. 18, No. 3, (1974). 12. Cs.in, F: Simplified Derivation of Optimum Transfer Functions for Digital Stochastic Processes. Per. Pol. El. Eng. Vol. 9, No. 3, (1965). Dr. Laszl6 KEVICZKY } H-1521 Budapest Dr. J eno HETTHESSY 3 Periodica Polytechnica El 20/1 

RELATIONS BETWEEN THE MINIMUM VARIANCE CONTROL AND THE OPTIMUM TRANSFER FUNCTION IN THE WIENER SENSE

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RELATIONS BETWEEN THE MINIMUM VARIANCE CONTROL AND THE OPTIMUM TRANSFER FUNCTION IN THE WIENER SENSE

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