Computer-aided multivariable control system design for processes with time delays

Computers and Chemical Engineering Vol. 6, No. 4. pp. 31 l-326, 1982 Printed in Great Britain.

W%-1354/82/04031 I-16$03.00/0 0 1982 Pergamon Press Ltd.

COMPUTER-AIDED MULTIVARIABLE CONTROL SYSTEM DESIGN FOR PROCESSES WITH TIME DELAYS BABATUNDE A. OGuNNAIKEtand W. HARMON RAY* Departmentof ChemicalEngineering,Universityof Wisconsin,Madison,WI 53706,U.S.A.

(Received20 March 1981;receivedfor publication 8 Nooember1981) Ah&act-A computer-aided control system design package is described which handles processes with multiple time delays and allows subsequent design of interaction compensators. The package synthesizes control system designs for complex problems in a short time while also providing dynamic simulation for the purpose of evaluating controller performance. Example problems are presented along with a case study on sensitivity to errors in the delays. Scope-Computer-aided design methods are in general available for systems described by rational functions. For time delay systems, the approach has been to approximate the irrational e-” with rational Pad6 approximations. However, the resulting control systems designs then fail to deal explicitly with the presence of time delays. The present CAD package approaches this problem through the use of a time delay compensator developed previously by Ogunnaike and Ray ill]. The design procedure is carried out in two steps, (i) design of a time delay compensator followed by (ii) design for a compensated (or delay free) system. The package accepts models both in the state-space and in the frequency domain and calculates the equations necessary for the implementation of the time delay compensator. In addition G*(s), the delay compensated version of the system model is provided. Any desired standard multivariable design for C*(s) can then be carried out. Simulation testing of the designed controller is also readily available. Conclusions and Siicance-The technique of using a time delay compensator (to reduce controller design for time delay systems to the simpler task of designing for processes without delays) has been incorporated into an interactive computer program. The program furnishes the necessary equations for the implementation of the time delay compensator, and provides G*(s) (the delay compensated process model) to be used with any standard multivariable (MV) design of the user’s choice. Resulting controller structures are much simpler and any multivariable controller design technique which exists for systems with no delays can be applied directly to time delay systems with no additional modification. Chemical reactor, and distillation column examples have been used for demonstration. A case study on sensitivity to errors in the delays suggests that the techniques used are fairly robust.

INTRODUCTION

In designing linear multivariable control systems, one may choose to do the analysis in one of two ways: (i) in the time domain, dealing directly with differential equations, or (ii) in the Laplace transform domain, dealing with input/output transfer function matrices, and frequency rather than time. For each of these design approaches, a number of design procedures suitable for interactive use with the computer have been developed. These are often referred to as computer aided design (CAD) techniques. Computer aided control systems design methods have been made popular by the early work of Rosenbrock, MacFarlane and their coworkers at UMIST (see, e.g. Rosenbrock[l3,14]; MacFarlaneM) and have since found many applications in process control. For example,

*Authorto whomcorrespondenceshouldbe addressed. tB. A. Ogunnaikeis on leavefrom the Departmentof Chemical

Engineering,University of Lagos,Lagos,Nigeria.

Schwanke et a1.[16] and Bilec and Wood[l] have used these techniques for developing controller structures for distillation columns, while Shah and Fisher1171 have developed schemes to achieve feedforward disturbance minimization using these frequency domain methods. Today computer packages exist which offer a wide variety of programs that can be used for design, analysis and simulation of control systems [see [121for a partial list]. The general theoretical concepts of computer aided designs in the frequency domain have their foundation in the classical, single variable frequency domain results of Nyquist[91, Bode[Zl and Evans[3]. As most industrial systems are multivariable in nature, there was the need to extend the single variable results to cover multivariable systems. This extension has its origin in the pioneering work of Rosenbrock, MacFarlane and co-workers (see MacFarlane[% 61,Munro andIbrahim[8],Rosenbrock[lf 151, MacFarlane and Belletrutti[7]). All the theorems involved in the development of $ese multivariable methods are based on having rational functions (of the complex variable s) as elements of the transfer function matrices which represent the systems.

311

B. A. OWNNAIKE

312

Especially in process control, one frequently encounters transfer function matrices with elements that are irrational-usually in the form of a time delay e-“. To date, the usual approach has been to look at the dilIiculties associated with irrational functions as being mathematical rather than physical (Rosenbrock[lS], p. 94). AS a result, these irrational functions have been approximated with rational functions. A popular type is the Pad6 approximation which for a 2nd order approximation takes the form e

--Ts_ 1 - 742 + T2S2/12 - 1+ 42 + ?s2/12

The accuracy of the approximation deteriorates as s = jo increases. The degree of accuracy increases with increase in the order of the Pad6 approximation but so does the complexity. Often, this leads to transfer function matrices of considerable complexity and naturally, the analysis and resulting controllers will be correspondingly complex. In addition, note that the resulting control systems design fails to deal explicitly with the presence of time delays. This paper is concerned with computer aided design for this class of systems based on an alternative design technique, which circumvents the need to use rational approximations and therefore considerably reduces the complexity of the resulting controller structure. The CAD algorithm employed here is based on the multivariable time delay compensator developed previously (Ogunnaike and Ray[ll]). The control system design task is divided into two parts: (i) the design and implementation of a time delay compensator and (ii) standard multivariable controller design for a compensated system. In the next section we discuss how our algorithm simplifies the design for time delay systems. Following this we discuss the structure of the computer program, and provide examples to illustrate the application of the method. MULTIvARIABLE SYSl’EMS WlTBTIMEDELAYS Rather than tackle the design problem for multivariable systems directly, Rosenbrock[H-151 introduced the concept of diagonal dominance which immediately allowed Single Input, Single Output (SISO) results to be directly applicable. In a similar vein we propose that the design for time delay systems be approached via the preliminary design of a compensator for the time delays followed by application of one of the standard multivariable design methods to the compensated system. Consider the block diagram shown in Fig. 1 where G(s) is the transfer function matrix of a time delay system. If K is chosen to be a set of conventional PID controllers represented by G, then the characteristic equation of the closed loop system is

and W. H. RAY

It has been shown [ 111that by including a compensator GK = G* - G as shown in Fig. 2 (where G* is G without the delays) the system characteristic equation becomes Det(Z t G,G*) = 0

(2)

which contains no time delays; i.e. introducing the compensator GK as shown eliminates the time delays from the characteristic equation. It is easy to see that Fig. 2 could eduivalently be drawn as in Fig. 3. This makes clear the fact that the overall system to be controlled by the controller G, is the process G*(s) which is G compensated for delays. Thus the controller structure can be designed for G* using any of the available MV design techniques and the overall control system will be implemented along with the compensator, GK. From the preceding discussion, the design is thus seen to be in two stages. (i) Compensate for the delays in G. (ii) Design for G*. Let us discuss each aspect in more detail. Delay compensation This step involves obtaining G, from the given information and carrying out implementation in real time. Since GK = G* - G will consist of functions of the Laplace variable s, real time implementation raises the problem of realization. There are two possible forms the process model may take. The first is a time domain model given by

Y =

x Cidt i

7,).

We note that the real time implementation of the compensator G, requires producing from it the signal y*(r) -y(t) based on inputs u(e). It is clear that in-

Fii. 2. Block diagramof system with time delay compensation.

Det(Z + CC,) = 0

/*y&y;

which contains time delays.

r------

-piJ-

Y

Fig. 1. Conventional multivariable feedback control system.

___-__-

(i’(s)

______

~

L)++lgLq Fig. 3. An alternative form of Fig. 3.

Computer-aided multivariable control system design for processes with time delays

tegration of (3) and (4) will yield y(t). y*(t) is obtained from integrating the following set of equations.

313

Note that if we define 2=x*-x

i*=Ax*tBu

(5)

y* = cx*

(6)

where

w =y*-y then subtracting (11) from (13) and (12) from (17) yields

A=x

t

Ai, B=x

B,, C=x

j

I

C.

(7) i=A~t~Bj[u(t)-u(t-Bj)] i

In practice, analytical solutions requiring only quadratures may be used to solve for y* - y. Note that with this form of model, if the control system is to be designed in the state-space, no additional work need be done, since the compensator equations are implicit in the model. [Equations (5) and (6) are obtained directly from (3), (4).] However, if it is desired to carry out the design in the frequency domain, we need to obtain G(s) [and hence G*(s)] from the given model. To obtain the required frequency domain representations, Laplace transformation of (3) and (4) yields the transfer function G(~)=[~C~e-‘i”][s~-~A~e-“]-‘[~Bje-P”] (8) and by definition, setting yi = pi = & = 0 for all i, j in (8) yields G*(S)=?

Ci(sI-FAi)-‘FBj

(9)

which is the transfer function for the system described in Eqs. (S)-(7). The second form the process model may take is a multivariable transfer function matrix G(s). Types often encountered in process control have elements of the form gii(s) = Kl e-“““Pij(s) %(S)

.

(10)

In order to implement the time delay compensator, this transfer function, G(s), must be converted to the time domain as an equivalent set of differential equations. It can be shown[lO] that a minimal realization for G(s) of the form of Eq. (10) is given by f = Ax t 2 Bju(t - &) i

(11)

y=cx

(12)

where A is a diagonal matrix consisting of poles of G(s) as its elements, while the rows of Bj and columns of C are obtained from the individual matrix of residues (for each pole of G(s)) associated with the time delay flj (see[lO] for details of procedure used to obtainminimal realizations). Then by eliminating the delays, y* is obtained from

cz.

(16)

Equations (15) and (16) must be integrated to yield w(t) = y*(t) - y(t) needed to implement the compensator GK. The undelayed transfer function G* is found by setting the delays in G equal to zero. From (lo), the elements of G* will have the form (17) Multivariable design for process G* After implementing the compensator, GK, we must now carry out multivariable controller design for the time delay compensated process G*. To accomplish this, we are free to use any of the standard multivariable control system design procedures; e.g.: Inverse Nyquist Array, characteristic locus, noninteracting control, etc. The usefulness of the compensator in designing for systems with time delays is not limited to the frequency domain alone. Time domain techniques such as optimal control and modal control[l2] can be adapted with ease for systems with time delays. The linear-quadratic optimal feedback control law for time delay systems (Eq. 3) has been obtained by Soliman and Ray [ 181as u(t) = - w-l (B:E,,(t) t E:(t, O))x(t) 0 t (B,TE,(t, s) + E:(t, s, O))x(t + s) ds I -Pn 0 t (B,TE,(t, s)tE.,(f,O, s))u(t +s)ds I -Bm

I (18)

where pn, /3,,, are respectively the largest state and input delays. The other parameters are determined from the solution of a series of ordinary and partial difIerentia1 Riccati equations shown in detail in the above cited reference. Although such results as these exist for time delay systems, observe that on incorporating the time delay compensator, we need only implement the much simpler feedback control law for an undelayed system which reads1121 u(t) = - E-‘B=S(t)x(t)

(19)

(13)

where S(t) is obtained from the matrix Riccati equation

y* = cx*

(14)

dS X = - SA - A=S t SBE-‘B=S - F.

B=xB,. i Vol. 6, No. 4-D

w =

(15)

i*=Ax*tBu

where

CACE

and

(20)

Unlike the situation for optimal control, no known

B. A. OGLINNAIKE and W. H. RAY

314

results exist on modal control for time delay systems, [apparently because of the dticulties involved in calculating the system eigenvalues in Eq. (3)3.Nevertheless, we again note that implementing the compensator permits one to apply modal control techniques to the undelayed version of Eq. (3); the resulting feedback control law will be[12] u(t) = - RKLx(t)

(21)

where R, L are respectively matrices of the normalized right and left eigenvectors for the matrix A = X Ai, and K is a diagonal matrix of proportional controller gains. We should mention that the problem usually encountered with these techniques occurs with output feedback, when measurements y = Cx are available rather than the state vector x. [Note that these feedback laws require x(t).] However these problems are overcome by using state-estimators (e.g. a Luenberger observer or a Kalman-Bucy filter) to provide the state vector x from the available measurements. CAD programs In order to carry out this design procedure, a suite of general purpose computer-aided design (CAD) programs have been developed and implemented on the PDP 1l/55 minicomputer in the process control laboratory at the University of Wisconsin. The basic structure of the CAD suite is shown in Fig. 4. Note that one may begin with either a state space or transfer function model. When starting from the state-space, and control system design is NOT desired in the frequency domain, then the program is needed only for simulation. (The system equations are already available in the time domain; hence no realization is necessary, and no Laplace transformation to obtain G*(s) is required.) When the design is to be done in the frequency domain, and the model is in the time domain, A, B, C as given by (7) are required by the program and not the AI’s, B,‘s or Ci’s since G*(s) and not G(s) is the information required for the control

system design. When starting from a transfer function model, time domain realizations are required both for the design (& implementation) of the compensator as well as for simulations. Appendix A describes the techniques used in various parts of the program. The program provides intermediate output of the time delay compensator equations and the transfer function G*. The matrix G* may then be used together with one of a number of standard MV control programs available in our library in order to produce a multivariable controller K(s). The control system design then may be simulated by combining the equations for the state space model, the time delay compensator, and the MV feedback controller with appropriate ODE integration routines. Appendix B shows how the simulations are carried out with this program using analytical solutions, thereby minimizing computing requirements. SOME-

The design procedure is best illustrated by examples. Example 1. Reactor train with delayed recycle This example, first considered in [ll], consists of a two-stage reactor train with recycle in which the irreversible, 6rst order, isothermal reaction A+B is taking place (see Fig. 5). The following model was obtained for the reactor (see above reference for details). !$=A,,x(t)tA,x(t-l)+Bu(t)+Dd

(22)

where AO= [‘it

-“2]

A,= [;

Of]

B= [“o” t5] (23)

x=

[:I], D= p;].

Fig. 4. Structure of computer aided control system design programs.

315

Computer-aided multivariable control system design for processes with time delays FRESH

domain technique such as the Inverse Nyquist Array (INA) is to be used, then given A, B, and C = I the program produces G* as

FEED

FI 4f

L

‘F

Fe,

FPfCl

INTERSTAGE

FEED

CL1

.T STREAM.1

2

“2

1

RECYCLE

R. c2

Fmcz,

FRDDUCT

STREAM,

2

Fig. 5. Two stage chemical reactor train with delayed recycle.

Because of composition analysis delays, the measurements are given by Y1=x,(t-3) y* = X*(f- 2).

(24)

Compensator equations From Eqs. (3X7), we note that for this example we have i*=Ax*+ButW

(25)

y* = cx* where

A=[;;

“;I,

(26)

C=I;

with B, D as in (23). Solving (22)-(U) and (25)-(26) will yield y* -y and hence implement the compensator. Let us illustrate several types of multivariable control for the delay compensated system. (i) Modal control. Consider, for example, that d = 0 and that a modal feedback controller is to be designed. Because we will be implementing a time delay compensator, we need only carry out this design for the undelayed system f*=Ax*tBu y = x. It is easy to show that the eigenvalues of A are AI=-1.5,

A*=-2.5

i.e. A=

[

- 1.5 o

0 - 2.5I

1* (27)

’

0.25 t 1 G*(s)= s= t 4s t 3.75 0.4s0.2 t 0.8 0.5s

With the implementation of the time delay compensator, controller design will now be carried out for G*(s) in Eq. (27). Figure 6(a) shows the Inverse Nyquist Array for G*(s), and the immediate interpretation is that G*(s) is diagonally dominant. The corresponding Gershgorin row bands are shown in Fig. 6(b) and these clearly exclude the origin by a considerable margin. This means that two single loop controllers are suiiicient to give satisfactory performance since there are no significant interactions. This conclusion drawn from the INA is completely supported by the physics of the problem; Fig. 5 shows that u, affects x1 directly but its effect on x2 is only through that portion of the reactor 1 product which goes into reactor 2. Similarly, u2 affects x2 directly and affects x1 only through the recycle stream. Thus with u, and u2 not having direct in!luences on x2 and x1 respectively, we would expect to obtain satisfactory results using two single loop controllers. Further, because none of the curves in Fig. 6(a) intersect the negative real axis, we would expect the system to be unconditionally stable with high controller gains. These conclusions are supported by simulation as shown in Fig. 7. This is the response of the system to a set-point change for reactor 1 (yld = 1) only. The two proportional controllers (k,,, = k22c= 20.0) are seen to perform very satisfactorily. Note the fast response of y, and the negligible amount by which y2 is disturbed by this set-point change for y,, indicating strong diagonal dominance. The offsets could be taken care of with the addition of integral action to the controllers. Without delay compensation, however, the situation is quite different as is shown by the INA in Fig. 8(a). (The corresponding Gershgorin row bands are shown in Fig. 8(b).) It is interesting to note that although the system is still diagonally dominant (the bands in Fig. 8(b) exclude the origin) chances for system instability are greatly increased. This inherent instability of the uncompensated system was demonstrated in an earlier publication [1l] and one example is reproduced in Fig. 9. The dashed lines in Fig. 9 show the system response to set-point changes y,,, = 0.5, yM = 1.0 for two single loop proportional controllers with gains k,,, = 3.0, kuC = 3.5. In the neighborhood of kii, = 5.0 serious instabilities set in (see [ill). When using the multidelay compensator, much higher gains kll, = kuE = 20.0 may be used giving the response shown by the solid lines in Fig. 9.

and that the (normalized) matrices of right & left eigenvectors R, L are given by

Example 2. Distillation column control A distillation column (used for methanol-water separation) has been studied by Wood and Berry[20], Shah and Fisher[l7] and is reported to be well modeled by

[one can verify that lU=LR =I and that RAL=A; A = LAR]. The modal feedback control law is given by

Y(S) = G(sMs) + Gc&hf(s)

(28)

where G(s) and G&) are given by ##=-B-LRgIx where K is a diagonal matrix of proportional controller gains to be chosen as desired. (ii) Inverse Nyquist Array (MA). When a frequency

12.8 e-’ 16.7s t 1 G(s)= 66e-,‘ . 10.9 st 1 I

- 18.9 ee3’ 21.0 s + 1 - 19.4 e-‘” 14.4 s + 1 1

(29)

B. A. OGUNNAIKE and W. H. RAY

316

INUERSE

-11.9

NYWIST

RRRAY

, -3.50

Fii. 6(a). Reactor problem with delay compensation(05 o 5 5). (b) Reactor problem with delay compensation (GershgorinCircles,RowDominance)(05 o 5 5).

(30)

Here, in terms of deviation variables, y1 = overhead mole fraction methanol, y2 = bottoms mole fraction ethanol, u1 = overhead reflux flow rate, u2 = steam flow rate, d = column feed flow rate. Time-delay compensator equations From (28), we know that

,

0

- 12.8

;““__“~“~-““‘~~~~____-_____--_

I 5

”

” TIM

I ” 10

“I”” IS

20

CHINS)

Fig. 7. Response to set-point change with delay compensation and proportionalcontrol.

- - 18.9

1

G*(s)= . 16.7 s + 1 21.0 s+ 1 - 19.4 L- 66 10.9 s t 1 14.4 s t 1

With the aid of the computer program, a miniial

(31)

real-

317

Computer-aided multivariable control system design for processeswith time delays

-iEEls -::ImsQ 12.0

.

12.0

1

-14.50

,

1

-4.83

Fig. 8(a) Reactor problem without delay compensation (0 I o 5 5).

-1 in

_____- +______

3

-1r.0

-14

-10

-12.0

I

1

50

I

I -4.83

4.83

14.58

I

i

I -4.33

-14.58

I 4.s3

I 14.50

3

Fig. 8(b) Reactor problem without delay compensation (Gershgorin Circles, Row Dominance) (0 5 o 5 5).

ization for (29) is obtained as

li&) = -0.0917x,(t) t ohoml(t &(f) TIME

= -0.0694x&)

- 7)

- 1.3472u,(t - 3)

(minr)

’

i,(t)=-0.06z,(t)t0.7665[u,(t)-u,(t-l)l 0

5

IO TIME

Fig.

I5

20

(minsl

9. Response to set-point changes with and without the compensator.

iz( t) = - O.O476z2(

t) - 0.90000[ u2( t) - u2( t - 3)]

i,(t)=-0.0917z,(t)t0.6055[ur(t)-ul(t-7)] i,(t)=-0.0694z4(t)-1.3472[u*(t)-u*(t-3)]

(34)

B. A. OGUNNNKEand W. H. RAY

318

w*(t)= ZdO+ Z&) Wdf)= z3(f)+ Z&l

1’

(35)

Note how simply this compensator can be implemented, requiring only 4 integrals to be evaluated. Again, let us illustrate the design of multivariable controller for the time delay compensated system: (i) Non-interacting control for G*. We can now design for G*(s) given by (31). If in particular we are

Fig. 10(a) Block diagram considered in the INA design procedure.

concerned with only eliminating steady state interactions (steady-state decoupling) the required controller G: may be given by G: = GIG,

(36)

where G, is a diagonal matrix of controller gains and Gr may be selected as the steady state inverse of G*,

1*

0.157 -0.153 ” = G*-’ = 0.053 - 0.1036

C

(37)

The CAD package allows simulation of this control system design. This is accomplished by a realization of Eqs. (28-30). In addition to (32), we have the equations &(f) = -0.0671x,(t) t 0.2550d(t - 8.1) x,(t) = -0.0758x,(t) t 0.3712d(t - 3.4) I

(38)

with y,(t) = Y&l

x,(t) + X*(f)+ xX0

= -G(t) + Xdf) + &5(f)

1

Fig. 10(b)System designed by use of the INA and rearranged for implementation.

menting the time delay compensator with Eqs. (34), (35) implies that we need only choose L in Fig. 10(a)such that LG* possesses the desired properties. The, Inverse Nyquist Array (INA) for G* is shown in Fig. 11. Even without plotting the Gershgorin circles, it is evident that G*-’ is not diagonal dominant. We commence the design of L by choosing

(39)

when disturbances are involved [see Appendix Cl. One can then use the system simulation results to tune the controller until satisfactory system response is obtained. (ii) INA controller design for G*. Let us now use the INA method? to design a controller for the distillation column represented by Eqs. (28)-(30). Again, imple-

thereby achieving diagonalixation at zero frequency. The INA for Q-’ = G*-IL-’ is shown in Fig. 12(a). We add the Gershgorin circles in Fig. 12(b) to check for row dominance. Although the bands of circles exclude the origins in both diagrams, we note that there is still room for improvement. After a single row operation followed by a scaling, the resulting design

tWe have a slight change in notation here to conform to the methodology developed by Rosenbrock[lS].

L-, = 19.0 - 18.4 13.2 -38.8 I

INVERSE PIQUIST MmY

II -0.i73

+I.&$

0 b58

0.i73

-0.k73

: -0.858

Fig. 11. Inverse Nyqnist array for C*(s) (0 zzo ~2).

0.658

0.i73

(41)

319

Computer-aided multivariable control system design for processes with time delays

_;li-;-li -1.212

-0.404

0.464

=-&II-J2 1.212

Fig. 12(a)Inverse Nyquist array for Q-’ = G*-‘L-’ with L-’ obtained from Eqn (45) (0 5 o 5 2).

’

-34.8

-1.b

I

I

-0.404

I

0.404

1

t.212

68

-7 0

Fig. 12(b)Gershgorin row bands for the diagonal elements of Fig. 16(a)(Gershgorin circles at o = 0.1,0.2,. . . .2).

was tried. The corresponding INA is shown in Fig. 13(a) and with its Gershgorin bands for row dominance in Fig. 13(b). Since none of the diagrams intersect the negative real axis, there is no concern about stability limits for the feedback gains. (Indeed, our time delay compensator has seen to that.) The final step is the rearrangement of the system as in Fig. 10(b) so that the desired value of y can be used. (Note that the resulting controller has only proportional action.) From (41),

1*

0.0785 - 0.0372 L = [ 0.0267 0.0384.

Simulated system response to a step change in the overhead composition (from %.25 to 97) is shown in Fig. 14. The continuous lines represent the response for kc,, = k,= 2.0. Note that because the controller has only proportional action, there is considerable offset in the final overhead composition. When the gains are increased (kc,, = ka2 = 5.0, k,,, = kq2 = lO.O),the response is shown in dashed lines and the offset i6 seen to be reduced as the controller gains are increased. The

profiles for the bottoms composition show that a fair degree of decoupling is achieved since the set-point change in the overhead composition does not aflect the bottoms composition after an initial brief transient. Sensitivity testing Owing to the fact that all the preceding discussion assumes that the process model is in perfect agreement with the process itself, we now briefly take up the question of sensitivity to modeling errors. Since this design procedure is one that compensates for time delays, it seems appropriate to investigate the effect of errors in estimating the system delays. We choose the distillation column example (Eqs. 29, 30) in which the system model delays represented in the matrix below are taken to be the true values.

1

1 3 Matrix of Actual delays = 7 3 . [

(42)

Fire 15 shows the system response (for feedback gains k,,, = kal = 5) to a set-point change in the overhead as in Fig. 19. When the compensator is implemented with

320

B. A. OGUNNNKE and W. H.

RAY

50

Fig. 13(a) Inverse Nyquist array for Q-’ = G*-‘L? with final J_? choice shown in Eqn (46) (OS o 12). (b) Gershgorin row bands for the diagonal elements of Fig. 17(a)(Gershgorin circles at o = 0.1,0.2,. . .2).

13

._._. -._.-.-. ---.-.-. -.-.-.-.-.-.-.. c _ ,,._____._._._ II

3:

,----~__________________~~~~~~~~~~~---------~~~

TINE