New sequential design procedures for multivariable systems based on Gauss-Jordan factorisation

G.F. Bryant L.F. Ysung

Indexing terms: Gauss-Jordanjactorisatron, Multivariable systems, Sequential design

Abstract: In the paper, we have exploited the use of Gauss-Jordan factorisation to simplify the design of a multivariable system. It shows that the effects of closing a multivariable feedback system in sequential order can also be obtained by per- forming successive Gauss-Jordan eliminations on its return difference matrix. This simple elimi- nation procedure enables us to transform a multi- variable design into a series of multiinput single-output designs and to develop new sequen- tial design procedures with which the well known Nyquist and root loci techniques can be applied. The design of the precompensator K(s) can then be decomposed into n stages such that each column of K(s ) can be designed sequentially.

List of main symbols

ai. C(s) CnX"(s) = class of matrices with rational function

elements D N = Nyquist contour [Rl] GE = Gauss eliminations

i{s) loops closed

MIMO = multiinput multioutput MISO = multiinput single output N [ f ( s ) , x] = winding number of a complex function

about x, which maps the D , contour onto the complex N plane

= return difference matrix of a system with the first i loops closed

= class of matrices with real elements = single input single output = sequential return difference function

= ith row vector of the matrix A = field of rational functions

= Gauss-Jordan eliminations = transfer function of a system with the first i

R'(s) R""" SISO SRDF

1 Introduction

For maintenance and implementation reasons, a large MIMO system such as those found in process industries is usually commissioned and tuned one loop at a time in sequential order. The benefit of such a practice is that a

0 IEE, 1994 Paper 1226D (C8), received 6th August 1993 G.F. Bryant is with the Industrial Automation Group, Department ol Electrical Engineering, Imperial College, South Kensington, London, United Kingdom L.F. Yeung is with the Department of Electronic Engineering, City Polytechnic of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

I E E Pro<.-Control Theory Appl., Vol . 141, No. 6 , November 1994

multivariable design problem can be decomposed into a series of single-input single-output designs. Over the last decade, a number of sequential design methods have been proposed. For example, Rosenbrock introduced a stabil- ity condition, for a sequentially closed system [l , p. 1381. Mayne proposed a sequential design procedure [2], and Shaked and MacFarlane studied the robustness problem [3]. Recently Bryant showed that the sequential design method can be formalised by applying Gauss elimi- nations [4]. He showed that the process of closing a system in sequential order can be obtained by applying Gauss elimination (GE) operations on the system return difference matrix. Also, the GE operation can be used to transform a multivariable design into a series of simple multiinput single-output (MISO) design procedures.

In this paper, an improved sequential design pro- cedure based on the Gauss-Jordan (GJ) elimination is studied. The GJ design technique shares the same pro- perty as the Gauss-elimination-based procedure in that it is also a sequential design method, but the GJ-based method provides additional information; for instance, crosscoupling and control error can be displayed on the same Nyquist array simultaneously. A new feature of these improved techniques is the use of the sequential transformaton; a partially closed system is transformed into a new transfer function which is linear in K(s), where K(s ) is the precompensator, and the transformations are performed by applying a sequence of GE or GJ oper- ations on the return difference matrix of the system. Here, the GJ factorisation is exploited to formalise and simplify a design procedure; a design is decomposed into n stages, each stage corresponding to a design of a single column of K(s).

2 Gausdordan operation and sequential stability

Let us consider a feedback system as shown in Fig. 1. Let Q(s) be defined by

Qb) = G(s)K(s) (1)

Fig. 1 Feedback system with thefirst i loops closed

427

where G(s) is the system matrix, K(s) is the precompensa- tor and

y = GU ( 2 4

U = K e (2b)

e = v - y ( 2 4

The return difference matrix R(s) E Cnx"(s) and the sensi- tivity matrices S(s) E Cnx"(s ) are defined as follows:

R(s) A I + FQ(s) S(s) A R-'(s)

The real feedback matrix F and F' are defined as follows:

(44

(4b)

Note that the feedback matrix F is used to represent the condition of the feedback loops here, where all its diago- nal elementsfi are either equal to one or zero. The case whenf, = 0 means that there is no feedback path for the ith loop, i.e. the ith loop is open.

To keep the argument simple, the overall system Q(s) is assumed to have no uncontrollable and no unob- servable modes on the right-half s plane. The closed-loop transfer function of the system becomes

F A diag {fi; i = 1, ..., n} = I

F' 6 diag {fl, . . . , fi, 0, . . . , 0) respectively

H(s) 6 H ( F , Q , KXs) = Q(s)R-'(s) (5)

It can be shown that R(s) being invertible over the Nyquist contour D, is a necessary condition for the system to be closed-loop stable. Therefore, the sensitivity matrix S(s) exists and can be obtained by successive Gauss-Jordan operations on R(s).

First, let us introduce some basic properties of GJ operations. The GJ eliminations on a matrix A are a series of row operations applied to A as follows:

( 6 4

where Ni(s) represents row operations of the ith stage of GJ elimination, and it is constructed from an identity matrix by filling the ith column with di:

N"Nn-1 . . . N2N1A = I

1 0 . . . ni,i . . . 0 0 0 1 . . . .;' . . . 0 0

Ni=[O . . 0 " ' n:i . . 'I' 'i o] . . . . . . . . . .

0 1 0 0 .. . n i , . ..

Let all partitioned matrices be divided into ( i , n - i ) x (i, n - i) blocks. The first stage of GJ operation on R(s)

can be represented as follows:

Generally, R("(s) denotes the resultant matrix after i stages of successive GJ eliminations on R(s), and hence

428

we have

N'(s) . . N2(s)N1(s)R(s) = R"'(s)

where ~ y \ E ci x (n - 9(s), $\ E c ( n - i) x ( n - i ) (s) and I,, E R' '. The ith column of N' can be computed by

(9)

The above eliminations can be carried out either alge- braically or numerically for a range of complex fre- quencies s =jwi ; i = 1, .. ., r. The algebraic approach should be used if rational transfer functions are required, and it is more suitable for hand calculations or by a CAD package capable of manipulating symbolic mathe- matics (for example MAXSYMA). The second approach is more suitable for Bode- or Nyquist-type techniques.

Obviously, R(O)(s) = R(s) and R(")(s) = I are the special cases, and also R'" = N'R('- '). Therefore, by eqn. (Sa), the sensitivity matrix S(s) is simply equal to

1

S(S) a R-'(s) = ,n N'(s) (10)

Note that rii- ')(s) are the pivot elements obtained at each stage of the GJ eliminations, and they have the following special properties.

Property I: Let R(s) E C n X n ( s ) be invertible over D,, then :

I = "

1 rjj- 1)(s)

A1 : det "(s) = nii(s) = -

A2: det R(s) = flr$-')(s) i = 1

A4: N[det R(s), 01 = 1 N[ri!-l)(s), 01 s E D, (14) i = 1

A5: S(S) = fl N'(s) (15) i = 1

Proof: Condition A1 can be derived from eqns. 6b and 9. Condition A2 is results from eqn. 10 and condition Al. Condition A4 is the result obtained by applying the prin- ciple of argument to A2. Condition A3 is proved in Refer- ence 5.

We can immediately see that, by condition A4, the GJ procedure can be used to transform the original multi- variable Nyquist stability test into a series of encircle- ment tests based on the transfer functions r$-')(s) as follows.

Theorem I : GJ sequential Nyquist stability: Let R(s) a I + G(s)K(s) be the return difference matrix of the compensated feedback system as shown in Fig. 1. If

(i) The matrix R(s) is invertible over D, 1

the closed-loop system is asymptotically stable, where po is the number of unstable modes of the system.

I E E Proc.-Control Theory Appl., Vol. 141, No. 6, November 1994

Proof: The generalised Nyquist stability has been proved elsewhere, [6, 71. The first part of the Nyquist stability criteria being satisfied by R(s) is invertible over D,, which is equivalent to det R(s) # 0 over D, . The second condition can be established easily using eqns. 11 to 15.

3 Sequential transformation

A major advantage of these new GE/GJ-based approaches is in the way that the precompensator is being designed. The usual approaches were either to limit the precompensator to be a diagonal or a triangular rational matrix. Another common approach was to implement the precompensator in multistages as follows M :

(16) where K'(s) is constructed from an identity matrix by filling the Ith column with a column of transfer function. With such a specially structured precompensator, the ith column of the return difference matrix of the system becomes a function of the Ith column of K'(s) only. Hence, the task of designing the precompensator is sim- plified to designing one K'(s) at a time. But such imple- mentation leads to unnecessarily high-order controllers. For instance, let

K(s ) = K"(s) . . K2(s )K1(s )

K ( S ) = K ~ ( s ) K + )

where k : , and k:, are selected in the first design stage, whereas k i 2 and k i 2 are selected in the second stage. The resultant K(s ) becomes

k I l ( 4 = k:l(s) + k:2(S)k:I(S) (184

k12(4 = k : 2 ( ~ ) ( 184

k 2 1 M = k:2(S)k:I(S) ( W

k 2 2 M = k : 2 ( ~ ) (184 We can see that k, , (s) and k,,(s) have more terms than the remaining elements of K(s), and therefore their order can be expected to be higher. For example, if each stage of these precompensators takes a simple PI (proportional plus integral) form

& k',(s) = a t +

where a:j and bZ are real parameters. The combined effect of kll(s) becomes

From eqn. 20, we can see that the order of the integral part in k, , (s) is doubled and it can lead to controller saturation. As the number of controller stages increases in proportion to the size of the system, which is typical of such a multistage approach, the complexity and the order of the individual transfer functions of K(s) becomes very large, and more seriously, the designer has no notion of the final form of K(s).

However, in this paper, it is shown that the above problems can be removed by using the sequential trans- formation which consists of a series of simple GJ elimi- nations. This new design procedure does not have the structural restrictions mentioned above, and also such

I E E Proc.-Control Theory Appl., Vol. 141, No. 6, November 1994

restrictions are unnecessary; each column of the precom- pensator can be designed in sequential order. Also, in contrast to, for instance, the characteristic frequency/gain approaches [7] (which is based on eigenvalues of the system matrix), the GJ/GE procedures involve rational polynomial operations only and hence the Nyquist stabil- ity criterion is based on the rational polynominal rli- ')(s) which is well defined and easy to compute.

The strength of the GJ- and GE-based sequential design methods is that the original problem can be trans- formed into a series of simpler problems which involve the separate design of each column of the precompensa- tor K(s) . In this Section, we show that the Gauss-Jordan operations can be used to transform a partially closed feedback system into a new system which is dependent on and linear in K(s). We also show that the successive GJ operations on R(s) are intimately related to the operation of closing the feedback loops of a system in sequential order.

Let S'(s) denote the resultant effects of the first i stages of GJ operations and be partitioned into (i, n - i) x ( i , n - i) blocks:

S'(s) N'(s) .. . N'(s)N'(s) (214

Then, eqn. 8 can be rewritten in partitioned form as follows:

S'(s)R(s) = R'"(s) (224

From the above equations, we can deduce the following identities:

Sl,(s) = RT,'(s) E C'"'(S)

S';,(s) = -R21(S)R;,'(S) E C'"-""'(s) (234

(23N

(234 R'" -

(4 (234

S'(S) = N'(.S)S''-')(S) (234

R(0 12 - - ~ - 1 11 ( )RI2(S) E CiX'"-')(s)

2 2 - -R21(~)R;,'(S)RIZ(S)

+ R (S)E C ( n - i ) x ( n - i ) 22

By substituting R(s) = I + G(s)K(s) into eqn. 23a, we have

S'(s)(I + G(s)K(s)) = R"'(s)

Si@) + S'(s)G(s)K(s) = R'"(s)

S'(s)G(s)K(s) = R"'(s) - S'(S) (24)

G'(s) S'(s)G(s) (25)

Now, let us define a new term G'(s):

(For the case of i = 0, we have S o = I and Go@) 4 C(s).) Then eqn. 24 become

G'(s)K(s) = R"'(s)S'(s) (26)

(27)

Eqn. 27 is the result of the above sequential transform- ation via GJ operations. We can see that the left-hand

= [ -S \ , (S) R:'\(s) - I , ,

429

side is linear in K(s) . The right-hand side has a physical interpretation as follows.

Property 2 : After i stages of GJ operations on R(s), we obtain the following transfer functions which are equal to the transfer functions of the system as shown in Fig. 2 with the first i loops closed:

(28) H'(s) 4 H(F', Q(s), K(s ) ) = G'(s)K(s)

Fig. 2 Transferfunction "(s) after the sequential transformation

For proof, see the Appendix. We have just shown that H'(s) is the transfer function

with the first i loops being closed, and it is linear in K(s) . By examining eqns. 26 and 27, we can draw an analogy between G(s)K(s) and G'(s)K(s), and therefore can inter- pret C'(s) as the new 'open-loop' system matrix with the first i loops closed. From eqn. 28, we can further deduce the following identities:

Hi,(s) = Ill - S;I(s) (294

(296)

( 2 9 ~ )

H ; , ( s ) = G\,(s)K,I(s) + Gi,(s)K,,(s) = -~Z,(S)

H i , ( s ) = G ~ , ( s ) K , , ( s ) + G i 2 ( ~ ) K Z 2 ( ~ ) = R(&(s)

H;,(S) = G\I(S)KIZ(S) + GiZ(S)K22(S)

= R$\(s) - lZ2 (294 From the above equations, we can make the following observations. In simple terms, the ideal compensated closed-loop transfer function is when H(s) + I . But actually this ideal case can hardly be achieved owing to reasons such as finite system bandwidth, actuator limits and system robustness requirements, etc. However, the basic design goals are to minimise the error term S; and the crosscoupling terms S i , and RyL.

4 Effects of precompensators

Let A , denote the ith column vector of a matrix A. By eqn. 1, the ith column of Q and R are dependent on K l , i.e. Qi = G K , and Ri = ei + G K , . Therefore, by eqns. 7 to 9, it can be shown that the first GJ operator NI is a function of the first column of K . Also, the second column of R"'(s) becomes

By eqn. 9, the computation of the second GJ operator N 2 is based on the second column of R"'. Together with the previous observations that R , is a function of the second column of K , the second GJ operator is a function of the first two columns of K . Therefore, by induction, the above GJ properties can be summarised into the following.

430

Property 3 : A6: The ith columns of Q(s) and R(s) are dependent

on the ith column of K(s) . A7: The GJ operator N' the sensitivity matrix s' and

G' are dependent on the first i columns of K only, i.e. K , , and K,,.

AS: After i stages of GJ operations on R, then RY\ and R# are dependent on K , , and K , , only (see eqns. 29c and d).

In practice, as K , , and K , , do not affect the stability of the first i loops and they will be determined by the suc- ceeding steps of the GJ procedure, we can initially assume that K , , = +,, and K , , = I , , for the system with the first i loops being closed. Therefore we denote such a special form of K by K' as follows:

With K = K', we can read Gi, and Gi, directly from H'(F', Q, K'), i.e.

c:,(K') = H i , ( K ' ) (324

G;,(K') = H ; , ( K ' ) (32b) By examining eqn. 29d, we can obtain the following important equation:

r{f-"(s) = 1 + Cqt; ' (s)k, , (s) (33)

The term rj:-lJ(s) is a transfer function obtained by ele- mentary transformation on the return difference R(s). We name r:j- ' ) ( s ) the sequential return difference function (SRDF) of variable s. It is easy to see from the eqn. 33 that the ith SRD function is linear in k&), and the ith column of K ( s ) determines the stability of the ith loop with the first (i - 1) loops being closed). By property 3, the row vector gj;'(s) is known at the ith design stage and can be regarded as the new multiinput single-output (MISO) system function between input i and output i with the first (i ~ 1) loops closed. For example, gy.(s) is identical to gl.(s).

As the GJ operation cannot prevent pole-zero cancel- lation, the set of poles and zeroes formed from all the SRD functions ( r $ - l J ( s ) ; i = I , . . , , n) is the subset of the closed-loop system poles and zeroes. The missing zeroes are e(s)p,(s), where e(s) is the least common denominator of all non-principal minors of R(s) with all factors common to d(s) removed, d(s) is the least common denominator of all nonzero principal minors of all orders of R(s), and pd(s) is the decoupling zeroes corresponding to the uncontrollable and unobservable modes of the system (see Reference 7). Here we assume that Q(s) has no modes corresponding to uncontrollable and unob- servable modes is the right-half s plane.

In the following Sections, two design procedures are developed. One is based on the Nyquist approach and the other on the sequential root loci approach.

5

5.1

1 = 1

Sequential design procedure via Nyquist approach

Basic Gauss-Jordan Nyquist design procedure

Data: R(s) = I + G(s)K(s) and R(s) E CnXn(s)

po = the number of unstable modes of the system C(s)

Step 0: Let ~ ' ( s ) = I ; ~ ' ( s ) = ~ ( s ) ; R"J(s) = R(s); N0(s) = I ; So(s) = I ; Ho(s) = R(s) - I ; Yo(s) = G(s); i = 1

I E E Proc.-Control Theory App l . , Vol . 141, No. 6 , November I994

Step I : Prepare the Nyquist array Y ' - ' ( s ) formed by S i - ' , Gi- ' and Hi-' as follows:

where Y ' - ' ( s ) is partitioned into the following blocks: Y i ; l ( s ) E Cys) , , - ' Y"(S) 1 2 E c'"'"-" (4, y ; ; ' ( 4 E

(4. c(n - 0 x i(s), and y ; , (s) E c'" - i) x (n - i)

Step 2: (Nyquist encirclement criterion) Examine the Nyquist array Y'- '(s), and choose a col-

umn of controller k&); I = 1, , . . , n. Then: 2.1 Compute rlj-')(s) = 1 + 2.2 Plot the Nyquist loci r!j- ')(s) for all s E D,, and

count the number of anticlockwise encirclements about the origin, i.e. let ni = N[rl;-", 01.

2.3 The total number of encirclements must be

2.4 If the gain and the stability margin requirements are satisfied, proceed to the next stage, or repeat Steps 2.1 to 2.6 with a new set of controllers k,As).

2.5 Update the controller Ki(s) by filling in the ith column with k,,(s):

K'(s) = [K . , ( s ) I ' . I K.,(s) I ' . 1 e i + I ' ' I e,] 2.6 With the newly found column vector K.,(s), we can

update the current return difference matrix R"-"(s) by setting

gj;'(s)k,As)

I:=' fl; = -Po.

R"-" .i = G'-'(s)K&) + ei

(See eqns. 29c and d. ) Note that the following columns R!f-", I = i + 1,. .., n are not affected by K' .

Step 3: Compute the eliminator N'(s), and perform one stage of the Gauss-Jordan elimination on R("(s), then update the following matrices

3.1 R ( ~ ) = ~ i ( ~ ) ~ ( i - l ) ( ~ )

3.2 S'(S) = N'(s)S'-'(s) 3.3 GAS) = Si(s)G(s) = N'(s)G'-'(s)

Step 4 : , Compute the partial closed-loop transfer function H'(s)

Now examine each of the submatrices of H'(s). H i l should be as close to I , , as possible, and H i z and H i , should be as small as possible. The last term H;, requires cautious attention as it is the next open-loop transfer function (i.e. between Y, and V,). We would like H i Z to be 'well behaved's0 that the successive loops are not too difficult to design. The example shown later highlights this problem; tightening the first i loops may mean that the following loops are difficult to compensate with simple controllers. At this stage, a compromising solution is still an art largely dependent on the experience and the skill of the designer. Current researches are involved in developing procedures to automate this part via non- linear programming techniques.

If the requirements on the transfer function H'(s) are satisfied, proceed to the next step, or redesign the con- troller k , ' ( ~ ) ; I = 1, . . . , n from Step 1.

Step 5 : Set i = i + 1, If i < n + 1, then G O T 0 Step 1, else STOP

It is usual, but not necessary, to ensure each ri : - ' ) is stable, for if each stage of a design is stable, it will mean better system integrity. In this way, the system will at least remain stable if loops are broken in the reverse order as that being closed.

5.2 Example of the sequential Nyquist approach A 3-by-3 simplified jet engine S ( A , B, C, D) was chosen for this example. All input and output variables are scaled by the appropriate nominal values. The maximum order of the plant is nine states, and some transfer func- tions of the plant are nonminimum phase. A detailed description on the modelling of this jet engine can be . -

1 found-in Reference 8.

A =

B =

- 10.000 0 0 0

0.1432 2.871

-0.2469e - 1 100.0 - 1.31 1

0 0

- 36.000 0

- 12791.00 26230.00 - 3708.0

0 1 1862.00

16300 0 0 0 0 0 0 6.400 0 0 0 50.00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 .00 - 6720

0 0 0 0 0 0

0 0 0 0 0 0

-2.460 0 - 243.69 - 3.245 61.844 1.642 1.5579 0.1685e - 1

0 0 -61.50 -2.163

0.2604e - 3 1.4242 0.2363e - 1 0.3324e - 3 -0.3536e - 3 0.2898e - 7 0.2444 0.3349e - 1 0.2766e - 3 0.6284e - 7 0.7417e-2 71.343 0.1578 0.4235e-2 -0.2822e-2

0 0 0 0

-2.158 - 5.941 -0.2554

0 6.862

0 0 0 0 0 0 0 0

-915.50 0.5731 -281.60 0.1897 - 5.03 0.7794e - 2

0 - 5.00 740.50 1.195

0 0 0 0

134.20 57.05

0.5807 0

- 171.50

0.1599e - 1 -0.378Oe - 4 0.7542e - 2 -0.212Oe - 3 0.2578e - 5 -0.3678e - 8

-0,1240 0.1847e-2 -0.4588 J I E E Proc.-Control Theory Appl., Vol. 141, No . 6 , November 1994 43 1

Fig. 3 is the Nyquist array of the open-loop transfer func- tions of the plant, i.e. Yo(s). The open-loop plant is not dominant; for example, g3&) is about 30 times larger than gll(s) at low frequency. Thus, it is sensible to reduce crosscoupling at the start. Let us proceed to Stage 1 with this in mind.

before, but q3,(s) is still too high, and a small amount of phase lag is also introduced into qI1(s). It is obvious that proper cancellation can not be achieved by a simple K(s). The next step is to find a column vector k.,(s) which can approximately cancel some of the offdiagonal elements. From Fig. 3a, g3,(s) is the largest component of the first

i..' a

; ( 1

t b

C

Fig. 3 B Column I with K!, = (1, O,O,O)' b Column I with decoupling c Column 1 with decoupling and lead-lag compensation d Columns 2 and 3 unchanged by K '

Nyquist plots ofYo(K'(s))for different K'. Note that Yo($ = G(s)

Stage 1: Let us attempt to make an initial design by using a simple controller on C(s), (note that Co(s) = G(s)). The aim of this step is to let the designer become familiar with the system in order to progress to a more advanced design. From Fig. 3 we have the following observations:

(i) gI1(s) and gI2(s) are approximately out of phase. (ii) g2,(s) and gZ2(s) are roughly equal when o is less

than 1 rad/s. (iii) g3,(s) and g&) are roughly similar when o is less

than 1 rad/s. (iv) g2,(s) and g3,(s) is relatively very large with respect

to g,,(s); therefore, the crosscoupling from loop 2 and loop 3 to loop 1 is high and may cause problems at a later stage.

We observed that gI1(s) has infinite gain and approx- imately 30" phase margin. This means that the simplest controller for the first loop can be just a scalar gain, i.e. the first column of K(s) can be (10, 0, O)T. But, unfor- tunately, we found that the subsequent designs of loop 2 and loop 3 are unsatisfactory in terms of gain and phase margins. We concluded that this is mainly due to the effect of the crosscoupling terms gZ1(s) and g3,(s).

Stage 2: From the above facts (i) to (iv), g2,(s) and g3!(s) can be reduced by subtracting from column 1 a portion of column 2 of G(s). The first column of K(s) is therefore chosen to be K.,(s) = (1, -0.7, O ) T . Fig. 4 is the Nyquist plot of Y'(G, K'). Now, q2 , (s ) is much smaller than

432

- 1 I .

32

-1 0 32

d

column of G(s) and therefore is the prime target to be reduced by compensation.

Stage 3: The approximated transfer functions of g3,(s), g32(s) and g33(s) are computed from Bode plots as follows:

143 p 3 1 ( S ) = (S + 4.5Xs + 17)

832(S) = (s2 + 6s + 18.5) 860

8.57(~ - 3) Q 3 3 ( 4 =

After a few tries, we found that the following K'(s) yields the best decoupling result

(s + 4.5) (s2 + 6s + 18.5) I K'(s) =

Note that, although g3,(s) is better approximated by a second-order transfer function, a better decoupling effect can be achieved with g3,(s) being approximated by a second-order transfer function p3,(s) = 2420/(s + 4.5).

I E E Proc.-Control Theory Appl., Vol. 141, No . 6, November 1994

Fig. 3b is the Nyquist array of Q(G, K') . The corsscoup- ling terms qZ1 and q3' are significantly smaller, but qI1 does not have good stability margins. A series of lead-lag compensators were added to shape q I 1 . The best result

Stage 4 : Fig. 5 is a Nyquist plot with loop 1 closed. Now we close the first loop and proceed to design the second loop. We observed that g:,(s) and gi3(s) are well behaved functions in terms of stability, but gi2(s) has excessive 0 I p g f 2 -i

Fig. 4 Open-loop response with K' = (1, -0.7,O)

Fig. 5 Nyquist plots of Y'(K'(s)) with one loop close

was achieved by a three-stage lead compensator. The resultant compensator becomes:

K1(s) =

(lOs+90)(10s+ 150)(10s+400) (s + 90)(s + 150Ns + 400) (I 0 O 0 1 O I

(s+4.5) (lOs+90)(10s+ 150)(10s+400) (s2 + 6s + 18.5) (s + 90)(s + 150)(s + 400)

-0.27~1

The Nyquist plots of Q(s) = G(s)K1 are shown in Fig. 3c; the magnified plot of (r\'?(s) - 1) is shown in Fig. 8. We found that the best loop gain for loop 1 is u1 = 8.

I E E Proc.-Control Theory Appl., Vol. 141, No. 6, November I994

phase lag which could cause problems (note that from eqn. 32, gi3 = r\'j(K1) and g:2 = r\'j(K')). Therefore, we decided to drop gi2(s), i.e. not to feed the control signal to input 2. We found that K&) should be (1, 0, 4)' and the loop gain can be tuned to 4. The magnified version of (r$'j(s) - 1) is shown in Fig. 8. The updated controller becomes: K2(s ) =

8(s+4.5) (10s+90)(10s+ 150)(10s+400) (s2 + 6s + 18.5) (s + 90)(s + 15O)(s + 400)

0 0

0 16 1

- 2.16( 10s + 90)(10~ + 150)(10s + 400) (s + 90)(s + 150)(s + 400)

433

I

Stage 5: Fig. 6 is a Nyquist plot with two loops closed. We observed that and gZ3 both encircle the point ( - 1 ,O) except sil. The function g i l takes the -jw axis

w < 5 rad/s, and the bandwidth is much wider than the open-loop one. The crosscouplings are very small; the largest one is h12(0) which is less than 0.4 in magnitude.

1 1

Fig. 6 Nyquisf plots of Y’(K*(s)) wzth two loops closed

as its asympbol at high frequency and therefore it has The controller can be improved by fine tuning. infinite gain margin and about 90” phase margins, respec- However, as a design example, we stop at this stage. The tively. Loop 3 can be controlled by injecting signal into design can be summarised as follows: input 1 alone and can be stabilised by a simple gain. We (i) The controller obtained by GJ sequential design is found that 10 is sufficient. Finally the controller with simple: three elements are simple gain factors, and two three loops closed is as follows: elements are a combination of second-order transfer

function and a series of lead-lag compensators. All these transfer functions of K(s ) are easy to implement.

(ii) The controller is a sparse matrix; four elements out of sixteen of K(s ) are zero.

K 3 ( s ) =

6 Comments and conclusions

We have presented a sequential Nyquist design pro- Fig. 7 is the Nyquist array of the plant with three loops cedure which offers design engineers an easy way to solve closed. The result is very good, i.e. H ( s ) % 1 when a multivariable control problem as if it was a SISO

1 8(s+4.5) (10s+90)(10~+150)(10~+400) ( s z + 6 s + 18.5) (s+9O) (s+ 150)(s+400)

0 0

l 6

- 2.16( 10s + 90)( 10s + 1 SO)( 10s + 400) (s + 9OXs + 150)(s + 400) I 0

I - s ; ,

+ 1 Ay 1 4 1 - 1

f 1 , +#-s;z; - , 1 ,I 1; .. a 1

1 1 1

Fig. 7

434

Nyquist plots of the closed-loop system H(s) = I - S’(s)

I E E Proc.-Control Theory Appl., Vol. 141, N o . 6 , November I994

problem. The GJ design method is based on the classical Nyquist loop-shaping technique that most control engi- neers are familiar with. Coupled with engineering insight,

Fig. 8 Sequential returnjunctions

a good controller can be produced. The GJ operation is performed on the left-hand side of the return difference matrix R(s), and is only used as a tool to make the neces- sary sequential transformation; each GJ elimination is equivalent to the operation of closing a loop. Therefore, the resultant controller is not a function of the GJ oper- ator (i.e. N'). Note that, in procedure discribed in Section 5, the Nyquist array Y' is chosen in such a way that Y ; and Y : , are the closed-loop transfer functions with K?, ( s ) and K, , ( s ) given, and Y\,(s) and Yi2(s ) are the G;,(s) and G;,(s) matrices, respectively. Such an arrange- ment enables us to examine the partially closed system responses and the open-loop transfer function on the same Nyquist array. The next sequential return difference functions as defined in eqn. 33 can be constructed from the matrix Y'(s) easily, i.e. the product of the ( i + 1)th row of Y'(s) and the (i + 1)th column of K(s).

As demonstrated in the example in Section 5.2, the sequential design procedure demands some experience from the designer who uses it. We have to choose the structure of the controller to stabilise the system and, at the same time, balance the system performance and integ- rity. The GJ procedure does not produce a 'black-box' controller like many other methods such as the H-infinity or LQG approaches. But its sequential nature and the freedom offered to the designer to choose the familiar controller structures (such as lead/lag or PID-type controllers) make it particularly suitable for process- control systems such as those found in the field of chemi- cal engineering where plants do not close at once.

I E E Proc.-Control Theory Appl., Vol. 141, No. 6, November 1994

A different type of decomposition procedure is sug- gested by Feliu and Avello [9] to aid the precompensator design problem. The elimination is applied on the system transfer function G(s) instead of the return difference matrix, and the elimination takes place on the right-hand side of G(s). Although the Feliu and Avello method does indeed provide a solution for the design of the elements of the precompensator sequentially, the main differences are:

(i) The resultant compensator is the inverse of the actual compensator. Therefore, the compensator must be invertable and thus is nonproper.

(ii) The GJ or GE procedure directly corresponds to closing loops in sequential order; particularly the GJ method also yields the partial closed-loop transfer func- tion at the same time. But the Feliu and Avello method does not have a physical interpretation, and it is purely a procedure for testing the Nyquist stability criterion.

The main problem of the sequential approach is that, after closing some loops, the crosstransmission from the earlier loops may cause the remaining loops to be diff- cult to design, and consequently they require unneces- sarily complicated compensators. The usual remedy is 'back-tracking', i.e. to redesign the previous loop and to relax some performance criteria.

7 References

1 ROSENBROCK, H.H.: 'Computer aided control system design' (Academic Press, London, 1974)

2 MAYNE, D.Q.: 'Sequential design of linear multivariable systems', Proc. IEE , 1979, 126, (6)

3 SHAKED, U,, and MACFARLANE. A.G.J.: 'Design of linear multi- variable systems for stability under large parameter uncertainty'. IFAC Multivariable Control 1977, pp. 149-156

4 BRYANT, G.F.: 'Direct methods in multivariable control I-Gauss elimination revisited. IEE Conference, Control 85, 1, pp. 83-88

5 YEUNG, L.F.: 'Dominance and direct methods in multivariable designs'. PhD dissertation, Department of Electrical Engineering, Imperial College of Science and Technology, 1990

6 ROSENBROCK, H.H: 'State space and multivariable theory' (Nelson, London)

7 MACFARLANE, A.G.J.: 'Complex variable methods for linear multivariable feedback systems' (Taylor and Frances Ltd.. 1980)

8 SAIN, M.K., PECHKOWSKI, J.C., and MELSA, 1.L: 'Alternatives in linear multivariable control' (National Engineering Consortium, Inc., Chicago, 1978)

9 FELIU, V , and JIMENEZ AVELLO, A ' Matrix factorization method of stabilize multivariable control systems', Automatira, 1987, 23, ( 5 ) pp. 647-651

8 Appendix

Proof of Property 2: Note that the feedback matrices F and F;, are assumed to be unity matrices. The partially closed system transfer function can be found as follows

H'(s) H(F', Q(s), K(s ) ) = Q(s)(I + F'Q(s))- ' (34)

[HL(s) Hi,(s)] H\i(s) H \ h )

Qi~(s)R,,'(s) Q I Z ( S ) - Qii(s)R;,'(s)QiAs) = [Q zi(s)R,~'(s) QZZ(S) - Qzi(S)R;:(s)Qiz(s)

Sll(s) = R;:(s) (35)

1 By eqn. 23a, we have

435

By eqns. 23a, 23c and 38, we have

Then, by eqn. 34, we obtain

This completes the proof.

(42)

(43)

436 I E E Proc.-Control Theory Appl., Vol. 141, No. 6, November I994