Hybrid multivariable control systems

Hybrid Multivariable Control Systems R. C. AMARA Introduction

The associated inversion formula is

The class of systems treated in this paper is that characterized by a multiplicity of inputs and outputs. In particular, an analysis is made of linear, hybrid, multivariable control systems and a practical synthesis procedure is described. Although the paper deals only with a consideration of the stochastic signal case, an extension of the method to the deterministic (steps, ramps, etc.) signal case can be readily made. In recent years, interest in multivariable systems operating with both continuous and sampled signals has increased greatly. One principal reason for this growth in attention is that digital components are being considered for use more frequently as compensating elements in control systems operating with continuous signals. Such devices accept data in sampled form, operate on it in a discrete fashion, and yield sampled output signals. In their simplest forms these elements may be realized by configurations of resistance--capacitance networks and electronic switches; in their most complex form such sampleddata controllers may be digital computers in which the discrete transfer-function matrix to be realized is represented by the computer programme. Since even the simplest multivariable control systems require rather complex controllers, the potential use of digital computers in such systems becomes extremely attractive. Single-variable hybrid and sampled-data systems have been extensively investigated both from the standpoint of analysis and synthesis. In most of this work the description of the sampling process in terms of pulse amplitude modulation has been used. From this point of view the output of a sampler with continuous input is assumed to be composed of a train of equally spaced impulses with a spacing T (sampling period) and an amplitude equal to the value of the input signal at the sampling instant. Thus, if r(/) is a continuous input, the sampled output r*(/) may be expressed as r(mT)r)(t -

r*(1) = n/ = -

II/T)

I r(mT) = -

h;

-

r(mT) e -

III T •

(I)

(2)

:t:

oc

L n/ = -

r(II/T)z- IIt

(4)

-

(5)

'i:

where b(z) is the z transform of the weighting sequence b(I/T). It has been shown that a necessary and sufficient condition for the physical realizability of b(z) is that it be expressible in the form

where r*(s) is the transform of the sampled signal r*(/). Now defining z = e T s and representing r*(s) by r(z), where r(z) is z{r(s)}, equation 2 may be written in the more common z transform notation r(z) =

r(z)z ", -1 d z

b(I/T)z- " Il =

co iiI =

~.

b(z) =

OO

")

~

where r is the unit circle in the z plane . Similarly, it can be shown that a pulsed transfer function exists which expresses the ratio of sampled output to sampled input in a system as

where b is the delta function, T is the uniform sampling period, and m takes on integer values only. As in continuous systems, transformation methods prove to be extremely useful in sampled-data analysis. The Fourier transform of equation I yields r*(s) =

.

.

(3)

OO

.If "'- l' 111 L Ji/=O

b(z) =

7 -

lI t

(6)

.Y

I

+ L

w"z- n

Il= O

For such a function, stability requires that its poles be located inside the unit circle. In addition to these fundamental z transform concepts, other useful sampled system relations may be considered; these will be introduced in the course of the analysis as required.

Basic System Equations The principal properties of the systems to be considered and the criterion which is to be used as a measure of performance are enumerated below. (I) The input signals are members of stationary, random, continuous processes and have power spectra expressible as rational functions of z. (2) The fixed or plant elements are stable, continuous, linear, and time invariant, and are also expressible as rational functions of z. (3) The discrete controllers to be realized are linear and time invariant. (4) The criterion of system performance is the minimization of the weighted sum of the mean square errors between the set of actual outputs and a set of desired outputs. Errors are evaluated at all instants of time. The foregoing conditions or assumptions are essentially direct analogues of those normally applying to the equivalent single-variable case. The minimization criterion does not include constraints on the mean square values of any signals; however, as in the case of single- variable systems, such constraints may be subsequently included within this framework. Figure 1 (a) shows the basic hybrid multivariable control system; as can be seen, direct negative feedback from output to input is considered to be an integral part of the system. In the illustration, the plant matrix G is q x p and the controller matrix B' is p x q; also the signals R, M ' , C, D and E are

349

359

R. C. AMARA

B'(vT)

(I)

TT-

~~ + _........

b dvI ) 11·

•

b(vl)

,q

o

0

.

0

T-

f----T -

.

. .

0

+

G(t)

~-

0 0

r~~.

M'*(I)

,(vT)

bp"

,(vI)

~ __

-

+

M-

.

((I) D(t) E, (t) diet)

C2(t) +,£<~~'f2(1) d 2 (t) 0

0

°b pq

C(I) c,(t)

. 0

0

.°

Cq(t~lq(iJ

°

0

0

-

+

I

. . 0

(a)

dt)

~ ~ S(," ~M'(O

C (I)

G(tJ

(b) F(!{lIre 1. Feedback and cascade representalions of rhe hybrid colllrol probleflls

and ~!lk" TYI'; (.) and ~' ;,II.'I";

p

X I or q X I column matrices as required . Instead of working directly with the feedback configuration shown, it is more convenient to replace it by an equivalent cascade combination of matrices as indicated in Figure I (h). The relationship between the matrices B' and B is readily derived to be

=

B'(z)

B(z)[1 -

G(z)B(Z)] - l

~!h" T",,; (:x)

r

(7)

are defined by

dl lI""gk" T(I)gh i(1 x

~(id"",, ; (- Il) =

In the event that B '(z) is not physically realizable because all the zeros of the determinant [I - G(z)B(z)] do not lie within the unit circle in the z plane, a discrete q X q matrix F(z) may be inserted in the feedback loop. This can be chosen to introduce sufficient freedom to satisfy the condition that all the zeros of the determinant F(z)[I -

= ( a:

(.)

+ 'l.);

. ",

H"" ~,,d,,(t + /1 )gh i(t)

J _ -,."dl

(11)

Now a minimization of equation 9 may be performed with respect to the h ij or h,. 1 by a standard variational procedure. The result is that 00

h,.irT)

~'i( /uT - rT

')-

r =- oc

.f =

-

+ xT)

Cfj

~"",7"1,, (xT) - ~(;,II' '''' ;( -uT) = 0

G(z)B(z)]

u

lie within the unit circle. From Figure I (a) the sum of the weighted mean square errors for a plant with p inputs and q outputs is

~

0;

i

=

=

I, 2. . . p ; j

I, 2 . . . q

(12)

are the qp minimizing equations which must be satisfied. When the h,.1 satisfy equation 12, the minimum realizable error

(8) where the ell and d" are the actual and desired outputs, respectivel y, and 11"11 is the weighting coefficient for the h output. A direct expansion of equation 8 in terms of the elements of matrix B, the elements of matrix G, and suitably defined correlation function yields 'I

£,,,2(1)

="<

1

It ..... = 1

fI

(I

jJ

- '"

IT t i =-

jJ

",,' 1 j

-=

,,-

1 k

-=

1/

". -

1 I = 1 I~

-:1_

T ,, =~ b,)uT)~r}',.'",,< -uT)]

T.,

"

=- ~

h,',(uT) :f.

,,l' =

-

h,I,(rT)

J.

~( /I (uT - rT - xT) x cb"l.I' TUI., (xT) 'Z

(9)

where the correlation functions J' /I ( ) and ~(;,II. (.) are defined by I Y ( 10) ~Xy(lp) = 2N+ I ", =~_ sx(/IIT)r(/IIT+ y') s~ X)

(\ 3)

X

:to " .1'= -

is achieved . An expression for evaluating it may be obtained by noting that equation 9 reduces to

when equation 12 is satisfied. The frequency domain versions of equations 12 and 13 are most useful in obtaining numerical solutions. Equation 12 may be converted by a z transformation to

f;'(z)

=

fl

jJ

//

~

~

~

h= I k= l ,~ '

b,.,(rT)r ' l: --=: -

Z

(\4) .T =

-

'J.,

1/ =

-

-:I")

where fj,(z) is analytic within the unit circle in the z plane.

350

360

HYBRID MUL T1VARIABLE CONTROL SYSTEMS

The various z transforms are now recognizable and since all factors are rational in z, equation 14 reduces to

If the b"l' b"2' .. b"q have identical poles, and in the most general case none of these corresponds to the poles within the unit circle of the cPr;r,(z) or cP'jd,(Z) in the z plane, then it follows for n such poles :X,,(ll = I, 2 . . . n)

(15)

(1 -

:x"z-l)bkl(Z)cP",,(z)lz~~,,

+ ... (I

[
(I -

Ir
1'1) T

TG

(-

Since
where
When the plant matrix G becomes the identity matrix, equation 16 reduces to the form in which the plant matrix imposes no restrictions on the ET2(t)mill which may be obtained. For this, equation 16 reduces to (18)

[
It can be shown that the method of solution which is applied to this simpler special case can also be used in the most general control case. Accordingly, equation 18 will be used as the basis for developing the method. For the most general form of the matrix
+ ... [b"iz)cP"'Q(z)]~ =

(22) [cP"d,(Z)]+

Both
(19) [b,.2(Z)cP'Q',(Z)]+

+

:x"rl)b,.iz)cPrqr,(z)lz~""

(21) (17)

+

-

log z

B(rl)
[bkl(z)cP",'(z)]-'-

+ (l

-:x"rl)b,.q(z)cP,"",(z)lz~"" = 0

matrix
notation, equation 13 becomes

11{ Yr

:x"rl)b"(z)cP,.,r,(z)lz~",,

It is at once apparent from the form of the equations that the

(16)

where B1'(z) is the transpose of B(z) and
ET2(t)mill = 21Tj

:x"z-l)bkl(Z)cP,.,/t(z)lz~""

+ ... (I


-

-

(20)

where [.]T stands for a partial-fraction expansion in terms of poles within the unit circle. Now let
+ (l

- :x"z~l)b"'I(Z)cPrl",
[b"2(Z)cP'q''(z)]+

+ ... [b,.q(z)cP,.,r.,(z)]+

= [cP,"d,(Z)]-r

351

361

R. C. AMARA

operating on B(z) simultaneously produce a composite coefficient matrix which operates on both the rows and columns of B(z).

A detailed expansion of equation 22 shows that it may always be reduced to the form (23) where K(z) is the pq x pq effective coefficient matrix, and where B(z) and rdy(r 1) are the pq x I matrices formed by arranging sequentially by rows the elements of B(z) and <1>r Tdirl), respectively. The steps by which K(z) may be generated from the basic system matrices is shown in Figure 2, where the

.",,(,) ...•",,('j] ;

.pY'Y'(Z)

r


:

. .pYPY/z )

.p(lPY'(z) .p"r,(z) <1>'T(Z)

=

Z{<1>r,.(s)}

=

r

.pr,r,(z) .

:

.p(rQ"z) q

""

_

.

.

.

(luo.(z)

. -r"",('1

$o,(I,(z) .

.

$y,I;.(z)

l·/,j =

:

<1> r,.(z)

= {.D gg

<>

rr}

.p

.p

:'~'('l . cpypy/z)

r:

,-

K

11=1,2. v = 1,2 .

. .p(fuY.(z)

<1>gg -

.Drr

q

.

<1>rr(z) .

,

oM] }p . <1>,.,.(z)

=

[~,.'M~,,('j} {$,•.,M~ .. (,j} ... ($"'"M~"('j:] {cpy,y/z) <> <1>,,.(z)}

bkb)·

(4) The residues for each pole on both sides of the equations are equated and the resulting set of linear algebraic equations is solved. This completely specifies all bkz(z) . The foregoing series of steps may be applied in a straightforward manner to the soluticm of hybrid multivariable problems satisfying the conditions enumerated earlier. The multi variable solutions always show improvement in minimum realizable error over the corresponding single-variable solutions which are included as special cases. The cost of this improvement is in the form of more complex transfer functions. Considerations in the Determination of the Optimum Sampling Period

'[(fU~X;;---;P~~J~' .pyu~v(z)]')

1>u uuv(z) -

into its component equations automatically yields the coefficient matrix K(z). In recapitulation, the principal steps of the implicit method of solution may be listed below. (I) The determinant of the effective coefficient matrix of BT(z) is set equal to zero; the roots of this equation which lie within the unit circle in the z plane determine the natural poles of all bkZ(z). (2) The poles within the unit circle in the z plane of the .prh(Z) not appearing in the .p'jT,(Z) are added to the poles determined above. (3) A partial fraction expansion is made and the number of independent equations is counted. This determines the total number of polynomial coefficients for the numerators of the

{$y pg /z) <> <1> ,,(z) J

Figure 2. Generation of the effective coefficient matrix Kfor the control problem

operation of matrix superposition is used. This operation is defined as follows: two matrices X and Y of the same dimensions are superposed by multiplying corresponding elements of each matrix to obtain the corresponding element of the new matrix, W = {X <> Y}. It should be emphasized that it is not necessary to perform formally the reduction to equation 23 in order to obtain a solution to the multivariable control problem. For any specific problem, a direct expansion of equation 22

Sampled signals may arise in a single-variable or multivariable system in two principal ways. The signals may occur naturally in sampled form, such as in a scanning radar system, or continuous signals may be sampled by a discrete element in the process of performing a control function. In the first instance, the sampling period is usually set by circumstances beyond the direct control of the system designer. In the second case, however, the sampling period may be a system parameter which must be determined. This can be particularly true when a digital computer must be selected on the basis of speed or programme flexibility to perform the filtering or compensating function in a multivariable system. One approach to the problem of obtaining the optimum sampling period has been suggested for the single-variable case . 'Computer-limited' solutions for this case may be obtained by an examination of a number of possible programmes, ranging from the programme containing the maximum number of terms to those containing all lesser combinations of terms. An improvement in performance is possible with the simpler programmes since the sampling period may be proportionately reduced; this enables a faster data rate to be handled with a corresponding reduction in mean square error. Thus, if the reduction in error due to a reduced sampling period is larger than the increase in error due to the use of a less complex programme, a net gain may result. In the multivariable case, the far greater complexity of the solutions indicates that an alternative approach is desirable. Using such an approach, the determination of the optimum sampling period involves a compromise between two opposing objectives. On the one hand it is desired to achieve the lowest possible mean square output error; on the other, this must be done without making such unreasonable demands on the speed of the digital element that its cost becomes prohibitive. In order to reconcile these opposing factors a common basis must be found which can be used as a measure of overall system performance; such a common basis can be total system cost.

352

362

HYBRID MULTIVARIABLE CONTROL SYSTEMS The following hypothetical example is intended to provide a qualitative description of how such a determination of sampling period might be made. As is evident from the expressions for cT 2(t)min which apply to the hybrid signal cases, the minimum realizable error decreases as the sampling period decreases. In the limit, as the sampling period T approaches zero, the cT 2(t)lllin approaches the value for the continuous case. Figure 3 (a) is a sketch of a hypothetical variation of cT 2(t)lllin as a function of T. As shown, the typical curve can reasonably be expected to be S shaped. At small values of T the value of cT 2(t)min asymptotically approaches that for the continuous case; at large values of T, the cT 2(t)min approaches a value which is equal to the weighted

- -,, 1

~

:;

0.

E

o u

(a)

r 1;;

8 :; 0. :; o

r

T

I increasing ~

r

(b)

~ , r r r

,

r

I Optimum s.ampling I p~riod To

(c)

(d)

a plausible relation between these characteristics; as can be seen, the length of sampling period varies approximately in an inverse manner with computer cost. The smooth curve is obviously an approximation; the dotted curve has been included to show how the actual relation might appear. The optimum T may now be determined; if total system cost is to be minimized, a relative output cost must be assigned to each possible cT 2(t)min. Figure 3 (c) is a plot of such an assignment where T now appears as an implicit variable along the curve of output cost versus cT 2(t)lllill" From Figure 3 (b) and (c) a composite plot may be made of total system cost L-erSUS T; this is shown in Figure 3 (d). It is evident that small values of T will not result in optimum overall performance because computer costs become excessive; similarly, at very large values of T overall system performance deteriorates because cT 2(t)min becomes large. At some intermediate values a minimum occurs which determines a reasonably optimum sampling period T. Many variations are possible by using the relations between system characteristics described above. These do not necessarily involve considerations of cost. For example, if the computer characteristics are given, it may be required to determine the resulting sampling period and accompanying cT 2(t)min, or, if a particular value of cT 2(t)min is desired, a similar approach may be used to select or design a computer with such characteristics that it can operate with the sampling period required by the specified error. Thus, the most important observation is perhaps that such significant system characteristics as minimum realizable error, sampling period, and computer speed may be related in a simple and useful way in the design of hybrid multivariable systems. References

Figure 3. Determination of optimum sampling period: (a) CT 2 (t)min versus T; (b) computer cost versus T; (c) output cost versus CT 2(t)min; (d) total system cost versus T

1

2

sum of the mean square values of the desired outputs. Clearly, to obtain the smallest possible error, the smallest possible T is necessary. However, the realization of the lowest possible T must be viewed in the perspective of total system cost. It can be easily established that the number of basic operations (addition, multiplication, storage, etc.) in the programme of the digital computer does not change as Tis varied. That is, the degree of the polynomials of the numerators and denominators of the bkl (=) depends only on the degree of the elements of such matrices as rr(z), gg(z), and rd.(Z); the form of such elements is in turn clearly independent of the sampling period T. Thus, as T varies the coefficients of the polynomials in the bk1(Z) will vary but the number of terms remains constant. This establishes that the length of time which can be allowed for each elementary operation decreases linearly with decreasing T. If T is changed to Tin, only llnth of the time is available for performing each operation. It is thus now possible to translate the variation in T to a variation in computer cost through its relation to computer speed. Figure 3 (b) represents

3

4

5

6

7

8

9

10

BoKSENBOM, A. S. and HOOD, R. General algebraic method applied to control analysis of complex engine types. NACA Tech. Rept. 980, Lewis Flight Propulsion Laboratory, Cleveland, Ohio, 1950 GOLUMB, M. and USDlN, E. A theory of multidimensional servo systems. J. Franklin Inst. 253 (1952) 29 FRANKLIN, G. F. The optimum synthesis of sampled-data systems. Tech. Report T-6/B, Columbia University Electronics Research Laboratories, New York, 1955 FREEMAN, H. The synthesis of multipole control systems. Tech. Rept. T-15/B, Columbia University Electronics Research Laboratories, New York, April 1956 KAVANAGH, R. J. The application of matrix methods to multivariable control systems. J. Franklin Inst. 262 (1956) 349 WIENER, N. and MASANI, P. The theory of multivariate stochastic processes. Acta math. Stockh. 98 (1957) III PUGACHEV, V. S. The use of canonical expansions of random functions in determining an optimum linear system. Automation and Remote Control, Moscow. U.S.S.R. NEWTON, G. C. Jr., GOULD, L. A. and KAISER, J. F. Analytical Design of Linear Feedback Controls. 1957. New York; Wiley ROBINSON, A. S. The Optimum SyntheSiS of Computer-limited Sampled-Data Systems. Eclipse-Pioneer Division, Bendix Aviation Corporation, Teterboro, New Jersey, 1957 AMARA, R. C. The linear least squares synthesis of continuous and sampled data multivariable systems. Tech. Rept. No. 40, Stanford Electronics Laboratories, Stanford, Calif., 1958

Summary The class of systems treated in this paper is that characterized by a multiplicity of inputs and outputs. In particular, an analysis is made of linear, hybrid, multivariable control systems. Starting from the basic system equations, a derivation is made of the set of minimizing equations which must be satisfied by a discrete compensator 12

operating in a continuous closed loop control system. An implicit method for solving these equations is present~d. An examination is also made of the considerations in determining the optimum sampling period. The results of the analysis are expressed compactly in matrix form. 353

363

R. C. AMARA

Sommaire de commande continu a chaine fermee. On presente une methode implicite pour la resolution de ces equations. On examine egalement les considerations qui determinent la periode optimale d'echantillonnage . Les resultats de I'analyse sont exprimes de maniere compacte sous la forme matricielle.

La classe de systemes traitee dans ce rapport est celle qui est caracterisee par une multiplicite d'entn:es et de sorties. En particulier, on fait une analyse des systemes de commande lineaires a variables multiples et hybrides. A partir des equations de base du systeme, on deduit I'ensemble des equations de minimisation qui doivent etre satisfaites par un compensateur discret, fonctionnant dans un systeme

Zusammenfassung Die in diesem Beitrag behandelten Systeme sind durch eine Vielzahl von Eingangen und Ausgangen gekennzeichnet. Speziell fur ein gemischtes Regelsystem mit mehreren Veranderlichen wird eine Analyse durchgefUhrt. Ausgehend von den grundlegenden Gleichungen der Anordnung, wird aus der Gruppe der Gleichungen fUr das Minimum eine Ableitung gebildet, die von einem im kontinuier-

lichen Regelkreis angeordneten diskontinuierlichen Kompensator erfUllt werden mul3. Zur Uisung dieser Gleichungen wird ein implizites Verfahren angegeben. Ebenso werden Uberlegungen zur Bestimmung der optimalen Tastperiode durchgefUhrt. Die Ergebnisse der Analyse sind geschlossen in Form einer Matrix zusammengefa/3t.

DISCUSSION G. M.

KRANC

(U.S.A,)

Is it not possible to design systems, of the type which the author tre(!.ts in his paper, on the basis of non-interaction, thus reducing the problem to the design of single-loop systems, which could be achieved by conventional techniques?

E. I.

JURY

(U.s.A.)

This fine paper by Dr. Amara has important applications, at the present time, in practical systems which require the use of multivariable control. As Dr. Kranc has shown that a multivariable controller can improve the system performance, the natural extension of the use of a singlerate controller in the paper into a multi-rate controller becomes evident. Of particular interest is Or. Amara's Figure 3, where the effect of the sampling period Tis shown on the minimum mean square, as well as on the computer cost. This curve, confirms Dr. Kranc's observation that the smaller the value of T, the less the minimum error. I would like to ask Or. Amara if it is possible to enlarge on part (d) of Figure 3, that is, how this was calculated and whether such an optimum curve would always exist for all systems. As a last observation, Dr. Amara shows in F(fjure 3(a) that in the limit when T goes to zero, a continuous compensator behaves better than a single-rate or multi-rate digital compensator. Moreover, when T is small the computer cost is very high, as indicated in F(f{ure 3(b). Therefore, for this case, one would question the advisability of using a digital computer, and probably a continuous compensator is more economical.

A. N.

POKROYSKI

(U.S.s.R.)

As is shown in the paper of Mr. Amara, the mean square error of the system increases with increase in the sampling period. This also occurs with other formulations of the problem of optimization of discrete systems, for example, in the formulation considered by V. P. Perov. It is natural to expect that this assertion is universal , since increase in the sampling period diminishes the quantity of information arriving in the system . However, I know of no demonstration of this proposition in the general case and have not encountered such a demonstration in the literature.

As a rule, a concrete problem can be solved only by using a certain definite type of computer which is available. If the form of the algorithm is fixed, then the amount of computation is determined and consequently the minimum sampling interval and the minimum mean square error are also determined. Now if we change the actual algorithm of the computation from the optimum algorithm to a simpler one which requires a smaller amount of computation, the error increases on the one hand because the method is impaired but, on the other hand, it may later diminish by reducing the sampling interval. The problem is to determine which of these two cases is the most advantageous from the standpoint of reducing the errors. I know of no published works in which this question is considered in a general form . R. C. AMARA, ill reply. Replying to Or. Kranc, in a multivariable system, non-interaction of inputs and outputs (diagonalization) is not necessarily the most satisfactory criterion of performance. It is true that such a criterion may simplify design computations. However, it can be shown that by using the information from all the inputs for each output (that is, by combining the inputs in a particular way), an improved performance will result. The application of diversity reception in communication systems is an example of such performance. The cost of the improvement is in the form of a more complex design . Replying to Professor Jury, Figure 3 is a simple illustration of a possible method of selecting the optimum sampling period. In this figure, parts (b) and (c) are combined to give part (d). The problem of obtaining these curves is mainly due to the difficulty of assigning a suitable cost for each permissible mean square error. Depending on the form of the curves in parts (b) and (c) , a minimum mayor may not exist. If it does exist, it will be found by the method outlined . Replying to Dr. Pokrovski, the selection of the optimum sampling period which has been illustrated is subject to the restriction that the number of terms in each computer programme remains the same. If freedom is to be allowed to examine all possible programmes, then an approach similar to that suggested by Robinson (reference 9) for single-variable systems may be applied . Such a method involves an examination of a number of possible programmes, ranging from the programme containing the maximum number of terms, to those containing all lesser combinations of terms . For multi variable systems such an iterative procedure appears to involve many computational difficulties.

354

364