Minimum Information for Arbitrary Regulation

Copyright © IFAC Automatic Control ill Petroleum . Pelroche mica l and Desalinatioll Industries. Kuwait. 1986

OPTIMIZATION AND APPLICATION TO DRILLING

MINIMUM INFORMATION FOR ARBITRARY REGULATION A. K. El-Sakkary Deparlmfllt uf Systems Ellgilleerillg, Ullil 'ersity of Pelm/eum alld MilleraLI, Dhahmll J1261, Sa/ldi A.rabia Abstract. ~ objective of this stwy is to draw conclusions concerning the requireITEnts on a non-minimum phase process to perform as close to a reference m::xiel as we desire. The stwy is corrlucted in the fr~rk of minimax optimal sensitivity theory pioneered by Zarres [1 J and developed by Zarres and Francis [ 2 J. The design approach developed therein is adapted in this report. Keywords. M:XIel Reference 5cherre. Sensitivity.

Perfect tracking.

Optimal

Minimax

The problem is to find a stabilizing controller C to minimize the If' norm Ilxlloo of X, where X is

In many industrial processes, it is desirable to design a feedback controller so that the output of the plant arbitrarily tracks each one of several reference trajectories. A design which is established on Wiener-Hopf-Kalman quadratic theory depends on the desired reference signal and the tracking error increases when signals which vary significantly are employed. The approach considered in this report is founded on tracking a reference m::xiel, and therefore works for a class of reference trajectories. The objective is to detennine the minimum infonration on the process to achieve an optimal design which can bring the tracking error over all concerned trajectories to zero. The design method suggested here is interpreted in the fr~rk of optimal minimax sensitivity theory introduced by Zames [1] and developed by Zarres and Francis.

the difference between W and the closed loop feedback system (PC/l+PC)W as pointed out in Figure 1 and 2. Al ternatively, find a stabilizing controller C to minimize Ilx l loo and bring it to zero i f

possible, where C is proper. 2.

By perfect tracking we rrean that the tracking

error can be made as SllI3.11 as we desire. 'Ib reach the main conclusions of this paper we need to state the follONing theorem fron [3] . 'll-lEDREr1 1

a) I f P has no zeros in Re (s)

sup iG(jw) 1<00

=

0 is

exists, which is unique nonzero function. The function Xo is expressed in the form Xo

= PCoI(l+PCo)W

- W = W/(l+Pc )

O

(1)

Define J

O = inf {llxlloo : C is a stabilizing controller }

This infinmum mayor may not be achieved. I f it is, the corresponding C will be called an optimal (minimax) controller CO. In [2,3J conditions are

The Hardy space HP , 1:;;p$ is the Banach space of QT'1) lex valued functions G which a re ana 1yt i c in t~~ half plane Re(s»O and which satisfy

II Glloo

0, then Xo

b) I f P has neither poles nor zeros on the imaginary axis but at least one zero in Re (5) > 0, then an optimal weighted sensitivity function Xo

NCYI'ATION AND TER-1INOLCGY

[o/": G(j wIPdw 1 l / p :;oo , 1:>p:;;oo

~

the unique optimal weighted sensitivity function.

The paper is organized as follows: A brief account of the terminology is given. The optimal minimax sensitivity technique is briefly outlined and the pertinent results are sumarized. The tracking problem is then formulated and its association with minimax optimal sensitivity is highlighted. Concluding remarks concerning necessary infonration for arbitrary tracking are stated. Sorre solved exanple to derronstrate the use of the obtained results are provided. The relation between optimal tracking and the celebrated IVHK quadratic criterion is outlined.

II Gll p

Perfect Tracking:

derived for the existence of an optimal controller. ~ver, it turns out that only in trivial cases is an optimal controller proper. Al ternatively, a sequence C of proper stabilizing n controllers are sought while I lw/ (l+PCn) ) 1100 con-

RI (s) will dencte the set of rational functions

verges to J

rr = H~IlRI (s).

O

as n-->

00 .

The problem of perfect tracking considered in this paper is solvable in the fr~rk of optimal minimax sensitivity theory with J = o. An optimizO ing sequence C of proper stabilizing controln lers satisfies the relationship

1. AssU!Tptions: W is the reference m::xiel and P is the plant required to track I'" to diminish the tracking error to zero. W and P are strictly proper and stable and W is minimum phase. A function C in RI (s) is a stabilizing controller if the feedback system is stable.

3.

29

Necessary Conditions for Perfect Tracking:

30

A. K. El-Sakkan·

Obviously, from equ(l) we have for any real w 1I XOI I", ~ IW(j w)/( l+P(j w}C(j w) ) I· (2) If P is strictly proper (p(joo) = 0), then (2) reduces to (3)

The inequality (3) is necessary for the existence of an optimal minimax controller and can be used to determine the necessary conditions on the reference model W. It follows from eou(3) that W has to be strictly proper if we want to achieve exact tracking (X =0. implies that W(joo) =0) . O Another conclusion from (2), ( by using the maximum modulus principle and substituting the right half plane zeros of P for j w) is that if P is nonminimum phase then we cannot bring the tracking error to zero. This can happen only if W is nonminimum phase and has exactly the same right half plane zeros of P. Of course such coincidence between the structure of P and the reference model W is practically impossible. Furthermore, if we want to achieve (3) together with arbitrary small tracking error using a non-proper W, then C has to include a differentiation element. FRACTIONAL TRANSFORMATIONS As in Zames and Francis[2] we shall use a transformation which serves to parameterize the feedbacks and keep the feedback system stable. This ensures a stable design. Let Q be a new variable defined by the fractional transformation Q = C/(l+PC)

(4)

As PC ~ -1 by hypothesis, (4) is well-defined. Moreover, C can be constructed from Q by the reverse transformation C = Q/(l-PQ)

(5)

and vice-versa. Observe that as P(s) is strictly proper, (4) and (5) map any strictly proper C(s) into strictly proper Q(s), and vice versa. We shall view Q as a representation of the cascade controller C, which is useful for design purposes. The function X introduced in (1) is X = (l-PQ)W The optimal solution Q belongs to (l+s) vH'" for O some positive integer v, which is obviously nonproper and therefore cannot be realized by physical feedback. The most we can hope for is to find a sequence of strictly proper Q in H'" to give optimal tracking function whichnapproaches the minimax optimal solution. CONSTRUCTION OF STRICTLY PROPER OPTIMAL CONTROLLERS In [2] two procedures are given for constructing a sequence of strictly proper Qn.Here we will include the simpler procedure, which is valid for strictly proper reference model W. This procedure involves the modification of Q by O some high-frequency attenuation. Let Qs(s) be any strictly proper function which stabilizes P(s). Let Qn(s), n=],2,3 ... , be defined as foll ows: v Qn(s) = Qs(s) + (QO - Qs)(s)[n/(s+n)J . (6) Consequently, the corresponding sequence of controllers is expressed by the fractional

transformation (7) C (s) = Q (s)L l-P(s}Q (s)J- l n n n we now state the following result from Zames [2]:

THEOREM 2 The functions Q defined by (6) are strictly proper elementsnin H"' , stabilize P(s), and are optimal in the sense that they produce tracking error functions which approach the infimal value J, i. e. , -- >

(8)

EXAMPLES As an application of the concepts outlined in this paper we provide few examples for illustration: Example 1 Let p(s) =1/(5+1) and W(s) be strictly proper and stable. A stabilizing controller is chosen to be C(s) = l/(s+l). The optimal minimal value of J is J = 0(from[2]). Thp co r respondin0 to th~ stabili2ing. Cs is ~(s) = (1+5)[(5+1)2 +lr 1 The optimal function Q obtained from J o =0 is O QO = (1 +s), whi ch is non-proper. The sequence Qn constructed from Equ(6) is to give Q (5) = (s+1)/[(s+1}2+1]+L(s+1) -

(~+1)/[(S+1}2+1](n/s+n)2

Qn is an optimal sequence in the sense that it stabilizes P and results I n -- >J O = O. The corresponding sequence of optimal controllers C can be obtained by substituting for P and Qn n into Equation(7) to produce C = [(s+n}2 +n 2 (s+1)2 J /[s(s+1)(s+2n))]. n Obviously, as n -- > '" , Cn -- > C. Observe that the controllers C have integrator action which is in agreement w9th the classical servomechanism problem for asymptotic tracking. Example 2 Let p(s) = (2s+3) / (s+2)(s-1) and W(s) = 2/ (5+1). A strictly proper stabilizing controller is Cs(S} = 1/ (5+1). The optimal minimal value of J is J O = O. The 0 corr 7sponding to the stabilizing Cs is s Q (s) = (S2+S_2} / (S3+ 2S2+S+1). 5

The optimal function Q obtained from J O = 0 is O Q = (S2+ s- 2)/(2s+3), which is non-proper . O The sequence Q is constructed from Equation(6) n to give Q (5) = n

(s2+s_2)/s3+2s2+S+l )+[(S+2)(S-1)/2s+3)-(s2+ s-2)/ (s3+ 2s 2+ s +l )J(n/s+n)2.

Minimum Informatioll for Arbitran' Regulatioll

On is an optimal sequence in the sense that it stabilizes P and produces I n --> J O =0. The corresponding sequence of optimal controllers en can bp obtained bv substituting for P and On into Equation(7). Example 3 Let Pis) (1-s)/2(2+1)2 and W(s) = 2/(s+1). A strictly proper stabilizing controller is Cs(s) 1/ (s+ 1).

mal minimax sensitivity theory. It has been concluded that a non-minimum phase plant cannot track a minimum phase process arbitrarily. For arbitrary tracking the optimal controller as revealed by Example 1 has an integrating action (l/s) which is consistent with the requirements of asymptotic tracking in the classical servomechanism problem. The results of this paper can be extended for multivariable systems along the lines of [7]. REFERENCES [1] ZAMES,G. (198 1 ) , Feedback and optimal senstivity: '~odel reference transformations, Multiplicative semi norms, and Approximate inverses. IEEE Trans. Aut. Control, Vol.AC-26, No.2, April,pp.301-320.

The optimal minimal value of J is J = W(l) O (from [2]). The Os corresponding to the stabilizing Cs is

[2] Zames,G. and Francis,B.(1983), Feedback,minimax sensitivity, and optimal robustness. IEEE Trans. Aut. Control, Vol.AC-28, No. 5, June, pp.5!l5-600.

Qs(s) = 2(1+5)2/(?s3+6s2+5s+3). The optimal function q~ obtained from J 0 o = (1+s)2 , which is non-proper.

is

Q

The sequence ~n is constructed from Equatio l(6) to give 3 2 Q (s) = 2(s+1 )2/(2s +6s +5s+3)+[(s+1 )3_?(s+1)2 n 3 2 3 Its +fs +55 +1) l (n/s+ n) . On is an optimal serua~ce in the s~~se that it stabilizes P and produces I ~-> J O = 1. n The corresponding sequence of optimal cnntrollers Cn can be obtained by substitutin r for P and On into Equation(7). nbserve that in this case with the presence of rhp zeros we fail to achieve perfect tracking. COMPARISON WITH WIENER-HOPF OPTIMIZATION The well known Wiener Hopf-Kalman quadratic minimizat;on orocedure[4,5], provides the o~timal controller for inputs which have a fixed frequency response.

[3] Francis, B.(1981), Notes on optimal sensitivity theory: The single-input single-output case. Tech. report, Department of Electrical Engineerin~, University of Waterloo, 1931. [4] Youla,D..(.Jabr,H., and Bongiorno,Jr. ,J.J. (1976), Modern Wiener-Hopf design of optimal controllers. - part I and 11, IEEE Trans. Aut. Control, Vol.AC-21, Feb., pp.3-13, and June, pp. 219-338. [5] Youla, D.C . and BonCliorno, Jr., J. J. (1985), " Feedback Theory of Two Degrees of Freedom Optimal \~iener-Hopf Design, IEEE Trans. Aut. Control, Vol.AC-30, No.7,July. [6] Feintuch,A. and Francis,B. (1985), Uniformly Optimal Control of Linear Feedback Systems, Automatica, Vol. 2], No.5 , pp 5.pp 563-574. [7] Francis,B., Helton,J.W., and ZAMES,G. (1984), ~ro Optimal Feedback Controllers for Linear Multivariable Systems IEEE Trans. Aut. Control Vol.AC-29, No.10, Oct.

In Fig. 2, let u be the reference input obtained by passing unit variance white noise d throuQh the filter W(s). The mean square value of the error e is 11ell / = (1/2 ~ )_ oo lOOl e(j (J) 12dw (9) 2 and the WHK objective is to minimize the L norm of e subject to the constraint that C stabilizes the feedback system while taking the input as the unit impulse. Employing Parseval 's2theorem we obtain an optimization problem in H . To conclude, the Wiencr-Honf problem is equivalent to the nroblem of minimizinr the H2 norm of the tracking error function e and can be solved by the oo same method as the H problem. The result is that there is a unique function eO of minimal norm. lhus the WHK approach minimizes i !Xl !2' optimize for one input, namely an impulse, whereas the minimax approach minimizes 11XIJoo and ootimizes uniformly over all inputs in L as deta i 1ed in [3,6] . CONCLUOING REMARKS In this oaper, we have investigated the problem of determining an optimal controller which downgrades the tracking error to zero. The problem is formulated in parallel with its counter-part in opti-

~I

d

W

u

;.

Figure

~

u W

e C

Figure 2

~D~