Elementary process dynamics and control: Part two

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part two

/ .../

A.Johnson

W. de Koning

ELEMENTARY

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part two

ELEMENTARY

PROCESS DYNAMICS AND CONTROL

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Copyright @ 1975 Vereniging voor Studie-en StudStudie-entStudie-enbelangStudie-en te Delft

Ezelsveldlaan 40 Delft. Holland Set in Press Roman

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-

-, . _ - -- - ~ - - - -

-PREFACE

This second,and final,volume of elementary process dynamics and con trol is an introduetion to some of the more modern topics and approaches to problem solving. A knowledge of the material of Part I is assumed and constant referenceis made to it. As previously,there are a number of worked examples, questions and bibliographical not es concerning the references in each chapter.

The contents of Part IJrange from the state space representation of dynamical systems to the control of chemical reactors. Some navel, as far as is known, proofs of elementary relationships appear, for example concerning multivariable systems in Chapter 13. Furthermore, half of Chapter 14 is devoted to the study of the structure of mathematical models of systems -a topic which isnot treated elsewhere at this elementary level. It has been. included in the firm belief that it will attain a significant importance in the future.

Once again , it is a pleasure to record the encouragement, help and toleranee sho wn by so many people,and to expressgratitude to the VSSDfor their su ppor t and effort.The fact that this second part represents one of the prod uc ts of a most stimulating and rewardingcollaboration between two enginee rs originally from different disciplines and countries merits also, we feel, note.

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.--

- - -

-18.3. 18.3.!. 18.3.2. 18.4. 18.5. 18.6.

The dynamics of an idea1 pipe reactor lsotherma1 reaction

Non-isothermal reaction Non-idea1 reactors

The con trol of chemica1 reactors References and bibliographica1 notes

270 270 272 275 279 281 APPENDIX C.

THE EXTENSIONS OF THE EULER-LAGRANGE METHOD OF DYNAMIC OPTIMIZATION TO MULTIDIMENSIONAL CONTROL PROCESSES 283 APPENDIX D.

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ADmTIONAL NOMENCLATURE PART 2.

B bottom product molar flowrate (molesjs) c heat capacity (JtCm3)

C number of components

D top product molar flowrare (molesjs) f frequency (radjs, Hz)

G molar feedrate (molesjs) L molar liquid flowrate (rnoles/s)

Lr molar liquid flowrate in rectifying section (molesjs) L

s molar liquid flowrate in stripping section (molesjs) Mmolar liquid accumulation (moles)

n (subscript) tray nurnber N number of trays in column r reaction rate (rnoles/rrr'.s) s Laplace parameter S separation factor

V molar vapour flowrate (rnoles/s) w noise or disturbance

À thermal conductivity (J/msoC) ~ system varia bie

I/> cross sectional area (rn")

ADDITIONAL SYMBOLS PART 2

X(jw)Fourier transform of xït) x * sampled form of x

I I

determinant of matrix A Z[x] z-transform of x * x(z) z-transform of x* {f(t)} number series

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"What[or doyou want new ideas?" ask ed Mr. Schultz, "Cheaper fuel, cheaper wages , harder work, that is all the new ideas I want." Evelyn waugh - The Loved One

12 THE STATE SPACE APPROACH

12.1. introduetion

The great advantage that working in the Laplace domain offers, it will be recalled, is that manipulations with transfer functions occur in an algebraic fashion. Any a1ternative approach in the time domain c1early can expect some tough competition from the "classical" techniques described in previous chapters. Some of the points that make the so-called state spa ce approach a serious con tender for the description of dynamic systems are the following: (i) The extension from s.i.s.o. (Ch. 2, p. 11) to multivariable, m.i.rn.o.

(mu1ti input - multi output) systems is straightforward,

(ij) Systems having parameters which change with time (time varying systems) can be accomodated,

(iii) controllers, which make use of the powerful mathematics of optimal control theory, can easily be designed for the dynamic process, (iv) Liapunov methods, for the analysis of the stability of the system, are

applicable.

Other very desirabIe features could be added to the above list.

In essence, the state space description ofany dynamic system consists of one or more sets ofdifferential equations. We have already (Ch. 5, p. 59) come across another time domain representation where integral equations- convolut-ion integrals - were used, and the two must not be confused. Nor is the differential operator form of model [I] , also in the time domain, meant here. In the following section we'll specify precisely what we understand by a state space model;so that comparison with the transfer function techniques already described is facilitated we will treat here the s.i.s.o, case, defering our comments on m.i. m.o . systems until the next chapter.

12.2. the state and output equations

1f the dynamic relationship between the input, u, and the output, y, of a system can be expressed in the form

i

=

A(th+ b(t)u y = c(t)!C.,

(12-1 ) (12-2)

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then the system is said to be in the (strictly proper) state space form.Equations (12-1) and (12-2) are frequently referred to as the state and output equations respectively. Here, the (nxl ) column vector ~ contains the n states (orstate variables)of the systern;A(t) is a matrix of time varying parameters, a..(t),

IJ called the (nxn) plant orstate matrix; b is a vector of time varying parameters, bi' called the (nx1) input ordrivingvector and cis the (1 xn) output vector. To complete the description of the dynamics a third equation is necessary, wherein the initial conditions (i.e. the values at t = 0+) of the states are set, for example:

(12-3) A significant simplification occurs whenever the system parameters (a ..,b.,c.) IJ 1 1 are time invariant. For the most simple types of problems this can be assumed to be the case.We then have the following continuous, time-invariant, Iinear (strictly proper) state space form:

~

=

Ax +bu

-

-y = c~ ~(O+) = ~o· (12-4 ) (12-5) (12-3) These equations, defining a systemS) and sometimes written S) (A,b,c) for the sake of brevity, will be found time and again in the Iiterature, and in the rest of this book.

It is important not to understimate the difficulties [2] in transforming our model equations (EPDC p. 30) into the state space forrn. In section 12,4 we give some suggestions for special cases, although no attempt is made to outline the systematic procedures which do exist [2] to cope with these difficulties. Note also that the state,~, of the system is not uniq ue, in other words it is often just as possible to base a state space representation upon a number of, say, pressures, as it is to base it upon a number of temperatures. Wiberg [3] discusses some factors affecting the choice of ~.

Theorderof the system described by eqs (12-1),(12-2) and (12-3) is equal to n. This sterns from the fact (Ch. 2,p. 11) that an n-th order o.d.e. can be written as n simultaneous first order o.d.e.'s. Equivalently , we could say that we are working in an n dimensional spa ce, since n = dim Ü~J.

Example 12.1.

Derive a state space model for a pneumatic cantral valve,whose transfer function is given [eh.3.p. 35] by

D.X(s) k -)

2

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Defin ing xI = ÁX deviation instem travel

o 0

x₂=ÁX =XI U

=

tiP allowsus to write thisas

o XI =x₂ o _

Is

k2 1 x₂-- k ,x₂- /C, x_I +/C ' u 4 4 4 y _{= X I'} which in matrix notationis:

y=[1 Ol:!.

Q57:Suppose that the system under consideration containsa puretime delay. Can this be accomodatedin a state spacemodel? Suggest possible approximate statespace models [seeCh.3,p. "34].1nChapt er16 a discrete-time version of eqs(12-3) to (12-5) will be presentedwhich could easily handle time delay (delayor) systems.

Itis vitallyimportant that we gleen as much information possible concerning

the general behaviour of the state space model to help in co ming to quick

prognoses in actual problem situations. Suppose, then, that SI (A,b,c) is at

rest (i.e .

t

ti

and

y

==

0) at some equilibrium value for t

<

O. Following

accepted practice,and our previous remarks concerningperturbation varia bles

(Ch. 2, p. 22),this equilibrium is taken as ~

=

0,u

=

O. Now at t

=

0 a

distu rban ce is assumed to affect the system;in principle this disturbance could

comeeitherthrough u,or through a sudden change in the initial state of the

system K(O) = .1f_o' or through both. To keep the picture simple yet

meaning-ful we'll take it that at time t

=

0 ~(O+)

=

K_o

*"

0 is the only disturbing

factor. The one dimensional (first order) case where 2Ç = [x I] is sketched

in Fig. 12.\.

We see that what eq. (12-3) means is that a sudden step-like change occurs

in the state at time t = O. Notice that an instantaneous change also manifests

itself in the velo city of the state

(Ji.

₁),a fact easily induced from eq. (12-4).

Now ifthe system SI is asymptotically stabie or b.i.b.o. stabie (Ch. 11, p.

115)-stability being determined by the poles of SI' which are in turn solely positioned

by the coefficients a .. of the plant matrix,A -, then the states wil! tend to

IJ

return of their own volition to the equilibrium state (XI = 0). The output

variabie,y, can be seen to play only a secondary role in the story - it can be

thought of as being on a lower hierarchical level than that of the state varia bles.

To complete the picture we have shown in Fig. 12.1 the effect of an (extern

-al to the system SI) induced step change in u.This produces in the state

variables a response simil~rto that caused by the non-zero initial conditions ~o'

However, the ultimate rest state (~= 0) is now non-zero ,in accordance with

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- l - l - l - l 0-

_o·

I 0-I I I

1

_o

i

_I -Xli VI

1

I I I

Fig. 12.1. State space model dynamics.

(12-6) We have deliberately adopted a rather pedantic style to firmly est ablish some general characteristics of the dynamic behaviour of om state space model. Discussion of the analytical solution of SI (A,b,c) is deferred until Section 12.6.

With these understood, we turn our attention to two variations on .t he theme of SI(A,b ,c).

Consider firstly the homogeneous form of eq.(12-1), SI (A,c):

~

=

A(th

together with eq. (12-2) and (12-3). In the time-invariant form

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we find many applications where batclï chemical reactions take place.

Exa mple 12.2.

Theequationsdescribingabatch reactorwherein two firstorde r con secutivereactions take placehavealreadybeenderived (Ch.4. p. 52). Casttheseint o a statespace model.

Definin gtheth reestate variables xI,x₂,x₃bythe relations

XI =CA x₂=C_B

x₃=C_c

wetransfarmeqs (4-391 to (4-41) [Ch. 4,pp.52-3) into:

that is,

:!

=

AJ:.

To com pletethestatespace representation weneed

Y

=c

J:

wherethe nature ofcwould depend upon which states were available for measurem ent,

and

Discussion of the solution of eq. (12-6) is left until section 12.5.

Q58. What isthe matrix state equation of the Hamiltonian system [3]: H=! qTVq+!pT Tp,

wherethe equatio ns ofmotion are

o

aH

.

0

aH

q.

=

a-

ana p.

=-a-

?

1 Pi 1 qj

To close this sectionwe studybriefly instantaneous (memoryless) systerns. Consider ,for example, a subsystem Sj which comprisesof a gain element only. This cannot be represented by an SI (A,b,c) or SI (A,c) model; in fact, a more general form, called the improper

**,

or memoryless, state space model must be used:

SI (A,b,c,d)

x

=

A(t)X+ b(t)u y

=

c(t)~

+

d(t)u

t

(12-8) (12-9)

*

The characterT den ote s the transpo se.

*

The term improper isused [2] sin ce when d(t)"* 0 the transfer func tio n G(s) correspo nding toSI(Ab.c.d) doesnot tend to zero as s+~.

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The control signal,u, can now act directly upon the output y.

12.3. disturbances

Sooner or later in process control or dynamics, we encounter thes.i.s.o. system which cannot be adequately described by models of the type S(A,b ,c ,d ). What is found to be lacking in eqs (12-8) and (12-9) is the possibility of introducing a comprehensive range of disturbances. It would be untrue to say that the state space models given so far are totally unable to cope with disturbances; as we saw in the last section, any disturbance causing an instant-aneous, once-and-f'or-all jump in the state variables (at t =0) can be

accomodated. However, not all disturbances are of this type. A possibility would be to allow disturbances to enter into the syste m through thecontrol variabie, u,but then we would no longer be able to talk ofu as the manipulated variabie, since part of it would not be within our power to manipulate. An-other possibility is to create new states (thus increasing the dimension of the state space) - "dummy" states which simulate the effect of the disturbances. of the disturbances.

Example 12.3.

Transfarm the system whose states are described by

:I

=A,!+bu+!!'.

where!!:isa constan t (for 0';;;;t .;;;;~)disturbance vector acting upon the system

(dim[w} ';;;;dim(x) ).

Definea set of dummy state variables.~.according to !. =!:':: and notethat

I

=O. Nowaugmentthe original states with the dummy states, giving

or

~

--

'""-' ~ =A~+ bu,

where Iisthe identity or unit matrix.and

:!

iscalled the augmentedstate vector.

We can expect problems here, however, whenever our disturbance cannot be transformed into a dummy state model SI (A*), eq.(12-6).

To overcome all these difficulties it is sometimes necessary to resort to the use of the following "disturbance-state space" model:

{

X

= A(t)x

+

b(t)u

+

e(t)w S (A,b,c,d,e,f) -

-I Y= c(th+d(t)u +f(t)w

(12-10) (12-11) This form of model is ideally suited to our own (i.e. process control and dynamics) needs. Consider the state space description of a heat-exchanger: Example 12.4.

In Fig.12.2issketched a constant temperature bath. We can control either or bath of the [luidflows,F_wandFk' Neglecting any heat loss to the environment itispossible to write the energy balances on the Ma fluids as follows (see for example [4]):

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Fk.Ski c91d rlUid in

warm

rluid in ~

F

_w

Swi 11

J

F

_w

,

"

'<

ew

>(

-'t ""'- ,....

---

-

jo »; ~

eb

~ ..-.-A .-....,

---

;. ,...,

----

--..; 1-Je ~ >I 1-

,.

eb eb )l. ;I- IJ-

₊

Fk Sk ~ Fig. 12.2.

Castingthese equations in perturbation variablesand linearising gives:

where we have assumed that the parametersaare time invariant.Regarding the two input temperatures as possible disturbancesourees gives the model:

whichisin the form:

K

=

AK+BM.+Ell'..

15.(0)='!o

Note that thisisa multi-input system.

Frequently we shall opt to let our disturbance variabie w be a stochastic (Ch. 7, p. 70) quantity, having a zero mean value. In Chapter 15 the reason behind this move - the remarkably simple structure of the "optimum" controller - wil! become .apparent. For disturbance with a deterministic character, a useful disturbance model - with which we can generate steps, ramps etc. etc.- is given by

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J:Y = Hl..

1

= Gz.+k G(t)

(12-12) (12-13) The use of this type of disturbance generator is described elsewhere [5].

Q. 59: Derivea state space model of a water purification plant consisting of a numberof (what must be assumedleak ing] reservoirs, conneered in series.

1

2.4. relation

ship

of th

e sta

te

s

pace d

escrip tion

wit

h other model

description

s

12.4.1. differential operatorform of model

We have alreadyseen in Exarnples(12-1) and (12-2) how,given a sirnple syste min the forrnof a differential equation (or in differential operator [I] form) it is on ly a questio n of a few simp lesubstitutions before we arrive at the state space mod el. This isnot true when we must handle systerns where term s suc h as

3, 0J

etc. occur.Such syste rnsare frequently encountered _ an y processmodelhaving one or more zeroes,any non-minimum phase system etc. The trans for ma tio ns [61 to be applied in these cases are iIIustr at ed below.

Example12.5.

Derive a state space mod el of arealJ-term controller.

The dynam icsof a com me rcialP-I-Dcontrollercan be described bythedifferentlal equation (cf eh.9,p,96):

(12-14)

whereqisthe inpu t signaland cisthe controlleroutput. We willnot concern ourselves at present with the initia! condit ions.

Defin e the followingstate variables:

XI =c - 'Ylq, "1constantthroughnur;

o 0

x 2 =c - 'Ylq- _'Y2q.

o

It follows that x₂=xI- 'Y2 q

and on differentia tingthat (12-15)

o 00 00 0

x2= C - "11q - 'Y2q

Substituting fromeq. (12-14)forCOandbyrep eateduse ofthe definedstatevariables in

place of~andc,~efindeventually that

~

2

=-':;x2 - a3xI+{{31_ 'YI }"!l+{!!.2.- a2'Yl - 'Y

2

}

q+

{

P..J _

a2'Y _ a3'YI}q.

al al al al al al al 2 al

Onchoosing "11

=

~

and "1₂

=

{32-

a2

~1

theabove eq uatio nreduces to

I al al

(12-16)

Equations(12-15)and (12- 16)tagether farmthe P-I-D controller state space model:

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-12) -13)

1

c

=

[1

where1'3 isthebracketed term in eq.(12·16).

lt is interesting to notethat in the above example, a model of the theoretical P-I-D controller would have yielded higher order derivatives in q than in c. This is in conflict with om previous (Ch. II,p .115) statement that for physical-Iy meaningful systems,the number of zeroes must be less than the number of poles (following this definition is eq. (12-14) also unrealizable).

The answer to this paradox is that a realizable system must be so for all frequencies; if we restriet our range of interest to frequencies within a certain band, then it may weil be possible to realise systems with more zeroes than poles.

12.4.2. transfer function model

It is an easy matter to derive the relationship between the transfer function and the (strictly proper) time invariant state space form of a s.i.s.o system. Laplace transforming eqs. (I 2-4) and (I2-5) we have that

l) (si - A)~

=

bii +~o

y

=

c~

Assuming that ~o = 0 (e.g. perturbation variables) we find

y

= c(sl - A)-l bii

so that the transfer function G (s) is given by G(s) = c(sl - A)-l b

(I2-17) (I 2-18)

(12-19)

(I 2-20) Q.60: Derive the relationship between input and output for an improper time invariant s.i.s.o system. Compute this quantity [rom the state model developed in Example12-5for thecommercial P-1-D controllerand check your answer with that previouslystated (Ch. 9. p.96).

It might be as weil for the reader to check at this point that hisj her knowledge of elementary matrix operations is up to scratch; nearly af! other texts on

control at this introductory level contain a section or appendix on matrices.

12.5. canonical forms

It is natural to ask at a certain point whether all 112 elements of the state

matrix are really necessary if we are concerned only with specifying the dyn-arnic behaviour of11simultaneous differential equations. Do "simpler" or

computationally better conditioned forms of the state space models presented so far exist? It has already been noted that the state space forms given are not unique- any non-singular linear transformation matrix P can be used to

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construct a new set of similar equations. For example, let K =

pi

define a

new set of state variables,

'i

If the original system was described by eq. (I2-3),

eq. C12-8) and eq.C12-9), then substitution for the new state gives:

o

pi

= APK

+

bu y

=

cP~+du

xCo+) =

p-I X

-

-a

Rearranging (12-2I) in state space format:

o

~= p-IAP~+p-Ibu

(12-21) (12-22) (12-23)

(12-24) Eqs. (12-22) to (12-24) are easily seen to be the required proper state space model:

12.5.1. Jordan canonical form

Let us suppose that the state matrix A has no repeated roots. Under this

assumption - usually the system is said to have distinct eigen valu es - it ean

be proved (see [8) for referenees) that a transformation matrix P exists sueh

that the new eanonieal systern SI CA,b,e,'d) has the form of eqs.(12-25) to

(12-27) where

It turns out that there are a number of unique canonical forms of the matrices

A, band

c.

By canonical is meant that a number of unit veetors are present

in the coefficient or state matrix, either as row veetors or column vectors.

We shall deal with only two canonical forms here, although many others exist

[7). As we have said, these canonical forms can be most advantageously

employed in real engineering applieations; sinee they have fewer non-zero elements than the general case, the eomputational burden is almost always redueed. o ~

=

A~

+

bu y =

ex

+du

ico+) =

ia

' V -I

fl~fAr

c.

=-

cP

1:

==

-

1b

(12-25) C12-26) (12-27)

_s

_:

ti .1 o c s-a

,

[

\

1...₂ A= 1...3

La

and

a

b=

Here the \ ,1...₂' . . . ,À_{n are the n distinet eigenvalues, i.e. solutions of the} equation

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pI.!

-

AI

=

0

t

( 12-28) and PI'P2'P₃, .. . 'Pn are constants called residues, for which formulas exist [ 3). The transformation matrix P, rather ironically, can only be given numerical valuesonce the n eigenvalues are known, and anyway it is unlikely that an a priori knowledge wiJl be available as to whether or not these eigenvalues are distinct. For multiple roots the structure of

A

and

b

changes slightly -

A

becoming composed of a number of so-called J ordan blocks

1 \

1 À.

I

1

A. .

= I IJ

_I

i

0

L À. I I À. I

- and efficient numerical techniques exist [3], [8] to transform any general system into the J ordan form. Should the system have complex eigenvalues, then the state vector can be augrnented (as shown in [3]) to give the "real" Jordan form.

Not only is the Jordan form unique but it also expresses the time behaviour of a s.i.s.o system having distinct eigenvalues in at most (3n

+

I) non-zero coefficients. We shall see later that for stability studies or when we come to solve the state space equation this particular form of

Ais a naturally attractive

one,

Q.61: Prove that the eigenvalues of the original matrix A are the same as those of

A

[6].

12.5.2 .the phase variabie canonical form

Ifthe system of interest can be expressed as a rational (Ch. 11, p. 115) transfer function *

G(s) -

13

_asn +

13

₁sn-I + .. .+13_n-₁s+13_n

sn+a_Isn 1

+

_. _{. .}+a_n-Is+a_n ( 12-29)

then the phase variabie canonical form can be written down from inspeetion of eq. (12-29) as SI (A *,b*,c*.d ), where

t

In accordance with [10] we will deno te the determinant of a matrix with bars and that of a scalar by "dct" to avoid confusion in the latter case with the absolute value of a scalar.

* Kwakemaak [8], p.83 gives necessary and sufficient conditions for a transformation to phase variabie canonical form to be possible.

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0 1

0

0 1 0 0 0 A*

=

b*

=

0

-a

_n

-a

_n_-I _-a_l c*

=

W

_n - an130

_~

_n

_-

_I

- a_n__l13_o ₁₃ 1

-

a\13o

I

and d

=

₁₃₀ (uncha nged).

Here all the information conta ining the system has again been conde nsed into at most (3n+ I) non-zero coe ff icien ts.

Example 12. 6

In Example12.5 we deriveda statespace model ofa real3-term controller.Whatis the phase variabie cananicalfarm of the model?

From eq. (12-14) or(9-l1) ane hasth at

Thephasevariabiecananicalspacemodel is therefare

Thereaderisadvisedto satisfyhimselfastothe similarityof this mod el andthatp revious-ly obtained.

Note that the coe fficients of A* in the phase variabIe for rncan only be

real numbers, unlike the corresponding elements of the

A

matrix of the lordan

form. The transformation from the phase variabIe form to the lordan form

and vice versa, i.e.

and

is interesting and often useful. lt turns out [6) the P isthe so-called V

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12.6. the dynamic response of the linear homogeneous state model

We now explore the dynamic response which is to be expected from various st ate space modeis. The aim will be to present as non-mathematical analysis as possible - sin ce numeroustexts exist [3,6,8 ,9] where the contrary is true

-and we shall start with the simplest possible case,a system described by the homogeneous time invariant model already discussed, viz.

1 1 1 " I "2 "n p=

,,2 ,,2

_{" 2}

1 2 n ~n-I ~n-I _" _n_-I I 2 n

where " are the n distinct eigenvalues,and

A =

p-I A*P

Q. 62: Prov ethat theeigenvalues ofA

*

are the same asthose oftheoriginal state

matrixA. y = c~ ~(O+) =~o' (12-30) (12-31 ) (12-7) (12-5) (12-3)

given. The question which is posed is the following: given the initial conditions of the state variables, eq. (12-3),what is the dynamic response of the system SI (A,c ) or in other words what is thesolution (if it exists) to eq. (l2-7)? Two methods leading to the answer to this question are next presented.

Areasonabie assumption is that the response of x will be approximated by a Taylors series vector expansion:

(12-32)

(12-33)

(12-34) where Ek (k

=

0,I,2, . . . )are constant column vectors. At t

=

0, ~(O+)

=

~o

=

Eo' By repeatedly differentiating eq.(12-32) we find that

~= EI + 2E₂t + 3E₃t2 + . ..

:. ~(O)

=

EI

=

A~o o~

=

2E

2 + 6E3t + .. .

:. 01(0)_-

=

2E₂

=

A~(O)_-

=

A2x_-0

etc. Using these relationships we can write eq. (12-32) as

A2 A3

x

=

(I + At +- t2 _{+ - t}3 ₊ _)x _- _e_{At x}

- 2! 3! . . . -0 - -0 (12-35)

and so y = ceAt~o' (12.36)

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method of analysis proceeds as follows.

Laplace tranformation of eq.(12-7) gives, in analogy to the scalar case: s~(s) - ~o = A~(s)

:. ~(s)

=

(sI - A)-I~o Taking the inverse transform:

x

=

L-I _{(sI _- _A)_-I_}_.x - -0 (12-37) (12-38) (12-39 ) and so y = cCI

{(sI - A)-Iho' (I 2-40)

Cornparing equat ions (I2-36) and (12-40) it is apparent that the structure of the solution is

y= cD.(t)~o'

whe re D.(t )

=

eAt

=

Cl {(sI - A)-I}X . i,O

Thequantity D.(t) is called thematrizan t orstate transition matrix.Properties of D.(t) arefully discus sed in [3,6 ,10]. We now investigate how the response is calculat ed in practi ce.

Example 12.7.

Co nsiderthe statespace modelof a batch reactor (Example12.2). What is itsd

yn-amicresponse?

From Examp le 12. 2it follows that

[

s

0 0J [

-kl

sI- A= 0 s 0 - k_l

o

0 S 0

The inversematrix, (sI- Arl,called thereso lven tofA [8J.is foundin theusual way:

(sI- Arl adj (sI- A) IsI-A

I

1 s+k 2

o

s(s+kI) k 2(s+kl)

:

-

1

I s -l

From anytable of Laplace transform s we can no w find,term for term, the inverse trans -formsof theresol vent:

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7)

8)

9) 0)

Comparing the work which was necessary to find the response with that

by conventional techniques (Ch. 4, p. 53),the reader might well be doubt

-ful of the practicality of the newly presented state space approach.The

apparent clumsiness is removed,however ,when the Jordan cano nical form is

used.

Q.63: Put themodel of Example12.2 in Jordan farm and rework thereactor response.

12.7. the dynamic response of the linear non-homogeneous state

model

The next step is to consider the inclusion of a cont ro l signal, u(t) and a disturbance w in the model, i.e. S/A,b,c ,d,e ,f) ,eqs.(12-3), (12-10) and (12-11).

Laplace transforming eq. (12-10) :

sx(s) - ~o = AX(s)

+

bg(s)

+

e~ (s)

:. "R(s) =L{eAt h o

+

L{eAt( b!I( s)

+

e~ (s» }.

By convolution the second term can be transformed back into the time domain. We find that

t

~

=

eAt~o

+

J

eA(t-T){bu

+

ewjdr o

The response of SI (A,b ,c,d,e,f) is therefore

t Y= cét~o +cJ eA(t-T)[bu+ew}dT+du+fw.

o

(12-41) (12-42) (12-43)

On comparing this equation with that of the homogeneoussystern (eq. 12-36)

the influence of u and w upon the response iseasy to see. Perhaps not so

obvious is the fact that the scalar, disturbancefree version of eq. (12-43) has

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Example12.8.

Supposethat a systemisnot controlled, but nevertheless acted upon bya step

disturb-ance at t=O.Whatisthe system response?

Dur system becomes(eh. 2.p. /9):

l=A~+!..U(t=O) (!!...is(n x/)vector)

and[or simplicity we assume y

=

c~. From eq.(12-43)wehave that

I I

Y=ceAI~o+cf eA(I-r).i!.Udr=ceAlx

o+ce A t f e-Ard r.!l.. =

o 0

=ceAI{io- A-I(e-AI- I)!..}.

We recom me n dthe reader to try the following simple calculation.

Q. 64: Sho w that theresponseof the system

toa unitstep changein u at t=0 isgivenby

12,8.

eigenvalues, time constants and stabilit

y

It is very impor tan t to under st an dtherelat ion sh ips bet ween the eigen

-value s of the system (t he roots of eq.(I 2-28» andits time constan ts and poles

- terrns which were freq ue ntly encou nte red in Part I of this book. The easiest

way to do this is to begin with the transfer function of the system SI (Ajb,c),

whic h can be writte n (eq. (12-20» as

G(s) :: c(sI - A)-I b or equivalent ly si ti s~ v; sc b p I~ p E fi 1 G(s) :: cadj (sI - A)b IsI - AI (12-44 )

Nowgoing back (Ch. 11,p. I 15)tothe splitting of a ratio nal transfer function

int o the ratio of two polynomialsin s, we see that the den o minat or of eq.

(12-44 ) occur s in the syst e m characteristic equatio n,i.e.

Is! - AI=

o

.

orin ot he r words that the eigenvalue softhe system are the same as the poles

t

of the s))tem. It follo ws also (see Q. 55) that for simple processes described

.by time constant s, the rule of th umb

t

FOTan equivalentstate space represcn tat ion of the zerosof the systern, sec [12]. 144

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1

"eigenvalue

=

-"""'t ',--- -=-- -

-irne constan t should be committed to memory.

Of course, since the words pole and eigenvalue are interchangeable,every

-that has already been said concerning the dependenee of the stability of a system upon the poles of the system will be eq ually applicable to the eigen

-values of the system. However, just as during our earlier discussion (Ch. 11,p.

116) we were careful to distinguish between open-and c/osed-Ioopsysterns,

so we must be equally cau tio us when dealing with system eigenvalues. It will be recalled, for example, that we proved (Ch.11,p. 117) that the closed lo o p poles of a s.i.s.osystern depend upon the open loop polesand zeros. Freq ue

nt-Iy new interpretations can be found for state space representations if one is prepared to manipulate the equations a little, as the following example shows. Example12.9.

Thelinear s.i.s.osystem

~=A:'5.+bu, ~(O)=~o y =c~

is controlled via a proportionalfeedback controllerhaving a setpoint.v.Derive expressions for the closedloo p transferfunction, T(s) relatingthe outpu ty to the setpointv[11] .

Theclosedloop system is

~ =A~+bu y

=

c:'5.

u

-ts

+v.

r-:o.

Themost obvious approach to [ind T(s) is as follows:

J=A~+b[;!+bv :.~=(sl - A - bfj-I b v

:.T(s)=c(s1- A - bfj-lb.

Not so obvious is:

y =c~=c(sl -Ar1bu. Letting z=f~ gives y =c(s1- Arlb(z +v) Z=[~= [(s1-Arlb(z+v)= (1- [Is I- Ar 1br1.[(sl-Arlbv :.y

=

c(s1- Arl

Ir[

f(~

- Ar 1

~

J

+)

v ~1- (s1-A) :..J . c(s1- Ar1b ..T(s)

=

1_f(sl -Ar1b

NotleeI/Owfromthe first expression for T(s) itfollows that the closedloo p poles[eigen

-values)are givenby the solutionof IsI-A- b fl=O.

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12.9. references and bibliographicaI notes

A comparison of state space, tran sferfunction and differential operator modelsofa process to be controlled can befound in

[I): WOLOVlCH, W.A. IEEE Trans. Autom. Con trol, Ac-19,April 1974,pp. 127 - 30. The difficulties in formulating a systern's dynamic behaviour in the state space have led Rosenbrock to propose an alternative representation. See,for example,

[2): ROSENBROCK, H.H. Computer-Aided Con trol Systern Design,Academie Press. 1974. The rneaning of "state" is discussed in

[3]: WIBERG, O.M. State Space and LinearSystems.Schuurn'sOutline Series, McGraw-Hili,1971.

Our heat-exchanger exampleinSection 12.3 is treated in detail in [4): MOZLEY, J.M. Ind. Eng.Chern.,48, p.1035 (1956).

The accomodation of external disturbances in statespace models is covered in

[5): JOHNSON, C.D.IEEE Trans. Autom. Control, Ac-16, 6, (Dec. 1971), pp.635 - 644, while Ogata in his book

[6): OGATA, K. Modern Control Engineering,Prentice - Hall, 1970

gives severalexamples of the transformations from one model rcpresenta tion to another.

A recent review of canonical forms for linear multivariable systems is that of [7): SINHA, N.K.and ROZSA, P. Int.1. Control.23, 6,(1976), pp.865-883.

Natural reading to followon this short introduetion to the state space approach would be Chapter 1 of the book by

[8]: KWAKERNAAK, H. and SIVAN,R. Linear Optimal ControlSysterns, JohnWiley, 1972

and/orthe mathematically orientated

[9): FRIEDLY, J.C.DynarnicBehaviour of Processes, Prentice-Hall, 1972. Likewise the book by

[10]: ROSENBROCK, H.H. State-space and MultivariableTheory,Nelson, 1970 is a good advancedtext.Some manipulations of s.i.s.ostate space forms are to be found in

[11): JOHNSON,A. Disturbance Localization,PO ResearchGroup Technical Report No. 9, TH Delft,1976.

A treatment of the state space equivalent representation of zeros is to be found in [12) KOUVARIT AKIS, Band MACFARLANE, A. G.J. Int. 1. of Con trol, 23, 2,pp.

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We progress in this chapter to the treatment of m.i.m .o. systems with all

their problems: non-square matrices, interaction and so forth, which have not

up till now been encountered. In doing so, we try to compare two approaches;

the state space representation (Sect io n 13.2) and the aIternative transfer function

formulation (Section 13.3), which has been extensiveIy deveIoped for multi

-variabie systems over the last decade by H.H. Rosenbrock and A.G.J.

Mac-FarIane .

My father. af ter long refle ction,

pre-pared a theory of his own, which, as he fondly hoped , would take the wind out of Lyell's sails,and justify geology

to godly readers of Genesis . . . God

hid the fossils in the rocks in order to tempt geologists into infidelity . Edmund Gosse - Father and Son

13 .1.

introduetion

13 MUL

TI VARIA BLE S

YSTE MS

13.2 . mu

lt ivariabIe st

ate

spac

e rep

resenta tions

The effect of additionai inputs and outputs of a system upon the state

space representations given in the preceding chapter are only, of course,

notat-io nal, That is to say,our system model retainsits s.i.s.o ,structure, but the

input and output coefficient veetors become replaced by matrices. In

short-hand form,

model equations s.i .s.o , rn.i.m .o. nonhomogeneous (12-1) 10 (12-3) SI (A,b,c) S2(A,B,C) homo geneous (12-2), (12-3), (12-6) SI(A,c) S2(A,C)

improper (12-3),(12-8), (12-9) SI(Aib.c.d) S2(A,B,C,D) disturbance (12-3),(12-10),(12-11) SI (A,b,c ,d,e ,f) S2(A,B,C,D,E,F)

where, if we assume that dim (y)

=

r, dim (y)

=

m and dim (w)

=

i,the n

the above matrices are: A(nxn); B(nxr): C(mxn):D{mxr}; E(nxl) and

F(mxi ).

Just as withs.i.s.o. systems,a number of very usefuI canonicaiforrns exist; unfortunateIy in the rn.i.m.o, case these farms are not unique. Suppose the model has the form

R=A~+ B!! y

=

C~ ~(O)= ~o (13-I)

cr

3-2)

cr

3-3) or SI (A,B,C) for short. Then just as with the s.i.s.o .systern,a (n xn) trans -formation matrix Poften exists such that

(30)

where

A

=

p- 1AP, B

=

P-1B an d

C

=

CP. Sixdiffer e nt m.i.m. o . cano nical forrns are at present known [1), two of which have particularly simple structures - theoutp u t identifiable cano nicalfarm, where theB-matrixis arbitrary while

A

and

C

have the fo rms :

A

=

[

~-+-lj

and

C

=

[1

i

0]

and the input identifiable canonical [orm wher e

C

is arbit ra ry but

Methods for obtaining the appro priat e transformation mat ri ces are presented in [1).

Following the development in the previous chapt er it isalso interest ing to extend the concept of a transfer function to the rn.i.m.o,system. The term

transfer matrix has been coined to relate the multiple output to the multiple input:

yes) = G(s)~(s)

where G is an (m xr) matrix-valuedrational function having the form

G(s) =

(13-4 )

the element C ..(s) being the transfer function re lating the i-t h output to the

IJ

j-th input,provided allother inputs are zero. What is meant here can beseen

ifeq. (13-4) is written out in full for the i-th output: y.(s) = G.) (s)u)(s) +G'

2u2(s) +...+G..(s)u.(s) +

I I I IJ J

+.. .+Gj/s)u/s). (13-5)

In th e mult ivar iab le system there can, in general, be no question of invert -ing G(s),since in general G(s) is not square (m

*

r).Thus, contrary to the s.i.s.o. case where

u(s) = G- 1(s)y(s) (13-6 )

except in certain trivial cases the inverse transfer matrix does not exist, a l-though each of its elements may weil be invertible.

By substituting eqs.(13-1) and (13-2) into eq.(13-4) an expression for the transfer matrixisobtained in state space form:

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Compare this equ ation with eg. (I2-20) for the s.i.s.o.system.We shall frequent -Iy refer to the transfer matrix in subsequent sections; next, however, we con -sider what at first sight seems a complesely differe n t approach to modelling dynamic systems.

13.

3. sys

tem matri

x

rep

resentation

The fact that difficuIties can arise in transforming the model equations, derived from the basic physical and chemicallaws, into the state spa ce form has caused some people to look for re present atio ns for multivariable systems which alleviate these (and other) problems. One attractive method has been extensively described in the literature [2],[3], and is very suited to the demands of process con trolwork;in what followsan outline of this approach wiIl be given.

Starting point is the equatio ns of conservation of mass, energy and momentu m (Chap te r 3)or other relations(allof which are assu med to have beenlinearised about the initial operating point ) which togethe r describ ethe multivariab le plant or pro cess of interest . Some of these diff erential or algebraic equati on s may be of the follo wing gene ra l fo rm:

da~ da-I ~ d~ db~ db-I~ _ _I+a I + +a ~+a ~ +_ _2 +a 2 + dt" 11 dta-I . . . l,a-I dt i. e I dt" 21 dtb-I dPu dP-1U /3 dU I (3 dqu d q-I_U I +R 1+ . + + u + _ _2+/3 2 + dtP '"'11 dtp-I . . I,p-1d t l,p I dtq 2I dtq-I R dU 2 R dZu d Z_-1_u +.. '+ '"'2,q-l--::J7""t_U_l +'"'2,_qU2 + +_(ïtZR +/3_R_{I dtZ} _IR+ R dUR R +...+'"'R,Z-I(it +'"'R,zuR· ( 13-8)

Here, the ~I'~2' . . . '~N are what we sha ll call system variables, and uI,u

2' . . . ,uR are the R manipulat ed inputs. Fo r the sa ke of clarity the possibili t y of additional disturbance inputs has been neglected.

Other model equations may not have thestructure of eq. (13-8). In that case, the ir fo r m may weil be given by:

+

~+

I: +

. " ïIa-I dt ' ïI,a"I

d~? I:

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In the above eq ua tio n, Y_j, is just one ofth e YI,Y2' . . . ,Ym process out -puts - which mayor may not be a measured variabie (Ch. I, p. I). The follow-ing det ailed examp leof modellinga simple process should help the reader gras p the idea ofusing eqs. (13-8) an d (13-9) and helps somewhat to justify the approach.

Examp/e13.1.

Consid er a continuo us industrialplant for produ cingpartially whippedcream,shown in Fig.3.1. lt maynot be assumedthatthe tank contents have aconstant density.

w

Fig.13.1.

(ij

(33)

(i) Overall mass balance.

whereVis the volume of thecontents of the tank.

Defining thenormal operating point by the symbol "r"and let

~P==P - P_r ~V==V- V_r

t:.Fj==Fi- Fir t:.Fo==_{Fo - F or}

Iftheoperating point is chosen to coincide with the (steady state)initial conditions,

i.e. pr==p(O-) V r==V(O-)

!<i

r

==

F/n

For

==

Fo(O-) ,

then th elinearised overall mass balancebecomesparticularlysimp Ie, since

(13- 10)

Here we have used the fact that

- - d -

-F ir - -For

=

Fj(O ) - Fo(O )

=

dtMO ) V(O ))

=

O.

(U) Density change.

It is assumed that the density of the whippedcreaminthetankis at anv instant of time uniform throughout the tank and proportionallathespeed of rotarian of the stirreror the power inputthroughthe stirrerblades.Inany case, a first approxim -ation wouldbe to set

(13- 11) (iii} Level control loop .

Supposing that theincaming[luidhasacons tantden sity,and that apneumatic control valvewith linear trim andconstant pressuredropisused , we seefromeq. (8 -25) (Ch.8, p. 91)that

wherecisthe pneu maticcontrolsignal in therange3-15psig.I],furthermore,the

controllerisa twoterm controller,then the equation describing thelevel control lo op becomes:

t

t:.Fi=k3~V+k4J ~V('T) d'T o

(34)

or in dif f erential farm: dA Fj =k dAV +k AV dt 3dt 4 (13-12) si. E; Wl

Equations(13-10) to(13-12) together withappro priateini tialconditionsfarm a simple model for the continuo us whipped cream process.Ta cast the modelin the required farm ,defin e:

input variables: out put variabie: systemvariables: UI =Aw u₂= ÄF_i YI=ÄF_o 1;1 =Ap 1;2 = A V. 0'

Then rewriteeqs.(13-10) to13.12/ as:

_ d1;1 d1;2 YI - "(1([(+ "(2([(+u2 1;1

=

131U1 di;2 _ d_U2 a2 2( j f +a2 11;2

-

dt '

( 13-lOa) (13-1 1a) ( 13-12a) p: w rr where "(I=-V_r; "(2= - P r; 13₁=k_l; a_{2 1}=k4 and a2 2=k3·

Itiseasyto seeno wthat eq.(13-lOa)isin the farmof eq.(13-9)and eqs.(13-11a), (13.12a)are sim ilartothe genericeq.(13-8).

Q.65: writeastatespacerepresenta tion[rom theequations (13-10)to (13.12).

The next step is to Laplacetransfor m thesystemequatio nsof the type w eq . (13-8) and/ or eq. (13-9). If the warn ings concerning the use of perturb- rr ationvariab leshave been heeded, then the in iti al conditio ns will be zero (Ch. 2, fe p. 22) and the transformed syst em eq ua t io ns have the following structure: cl

w T(s)~(s) = U(s)1!: (s) yes) = Y(s ) ~ ( s)+W(s) j:!(s) (13-13) (13-14) te al

Noti ce that eqs. (13- 13) an d (13- 14) completely describe the whole process, ti whereaseqs. (13-8) an d/or (13-9) are only representative of how two systern e, equ at io ns are str uc tu red. Inot her wo rd s, we now have the veetors~,!!,J:::, where

dim {~}

=

N, dim ÜÜ

=

Rand dim

hJ

=

M. The matr icesT(s), U (s), yes) an d W(s) are not similar in natureto the tran sfer matri x G (s) consid ered in the last section; their elements are each polynomials in s, an d wit h real coefficient s a,{3;y or E:.Hence th eir name:realpolynomial matrices. Of course, ifthe dynamical processbeing described had distri bute d parameters(Ch. 2,

p. 11) or for some other reason contain ed one or more pure time delays, then

T(s), U(s), Y(s) andW(s) might weil only be approx imated by rational poly-nomials.

Note fina lly that T(s) must be a square (NxN) no n-singular matrix, other-wise eq.(13-13) isindet erminan t. This conditio n canoften beregard ed as the

(35)

~-I2)

\-IOa)

1-11a) 1-12a)

signal to seek additio nal mode leq uations .

Example13.2.

Wecontinuewith the previousexample. The transf orm edsyste meq uat ions may be

written:

yl(sj =Ils"fl(sj+'Y2s12(sj+U_2(sj

l;d

sj = (3 l

u

I(s)

a_{22s f 2(sj}+a_{2 1}~2(sj=_{su 2(sj}

or in vector notation:

Thereader is advised to work through the modelling ofanother dynamica! process(for example,this time an open-loop systern) to become familiar with this new Laplace domain representation.

Itisa fairly simple ste p now to expresseqs. (13-13) and (13-14) in the more conciseform:

-],

(13-15)

which is analogous to the well-known matrix equation Ax = b.The partitioned

irb- matrix abo veis called thesystem matrix [2), which we shal!denote

hence-h. 2, forth by P(s). Quitea lot of chemical an d physical processes,both open and

closed loop, can bedescribed by eq. (13-15 ). Co nsider, for exa m ple, a process which we assu me can be describ ed in theopen loop situatio n by eq. (13-15)

3-13) - Iet this process be denoted _{by G(Tl'U I,vl'WI), When f}eedba ck is added

3-14 ) to the process - in the form of a secon d dynamic systern H(T₂,_U2,_V2, W2)-an d when eit her a non-constant setpoint or disturbancevectorvj,acts upon

;, the closed loop systern (see Fig.13.2) thefollowing developrnent shows how

m eq,(13-15) can still be used to describe the closed loop system. here The system equations can be written down in full as:

's) n irse, ien y- er-:he G(TI'Ul'VI,W_{I ) :} TI(s)~ = UI(s)!I y = V I(s)I +WI(s)!I H(T2,U₂,V₂,W₃) : T 2(s)I = U2(S)~ I = V2(S)~+ W2(S)~ Setpoint/disturbance :

!!.

=

I

-

I (13-16) (13-17) (13-18) (13-19) (13-20)

(36)

w s~ N

.,.

·ÁÀ

y

:t

'6!

G( T,.U,.V,.W, ) ~ H(T2 ,U2 ,V2 ,W2 ) . 13.2. Fig

After some manipulation,we find fromeqs. (13-16 ), (13-17), (13-18) and

(13-20):

TI~ -UII+UIf:=O

T2X=U2y

-V

t -

W

r

+

W C= _\I

I.'>. 1 - 1 - s: '

Vi

o

whereupon by using eq. (13-19) to eliminate

g:

:

TII- UIL+UIV2X=-UIW2Y

T2X=U2y

-V_I~

t

-

W F + W V_1- _{1 2 -}

X

=-(I + W W_{1 2 " -})\1.

c

In partitioned matrix form these equations look like

~

~--->~~-t-;,]

[

I

]

=

[

i--~

-1-

I+:~;]

l

-;]

or, by defining anaugmented system variabie vector ~ = [fT,

X

T

I

T, which does not hing more than dividing up the above partitioned form along thegiven

dotted lines, we arrive at

r a v where

T==

[

:'

U

,V,]

;

T₂ -v

==

[-VI W_IV₂],

w==w

_I

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Now using the fact [4] that the inverse of a partitioned matrix of the form

we see th at so long as the matrix [1+ WIW2] is nonsingular the closed-Ioop system matrix rnay be written as

(13-21 )

-

OP-I]

p-

I

[T

+

p~,P;'

VU

-p~;'

IV]

[-ij

=

[_~]

which has exactly the same form as eq. (13-15).

It is also interesting to compare the system matrix with both the multi-variabie state space and transfer matrix representations. The latter is easily obtained from eqs. (13-13) and (13-14).

Remembering that T(s) is a non-singular matrix, the system varia bles,

t

can be eliminated from the transformed system equations, leaving

r(s) = [V(s)T-I (s)u(s) +W(s)] ges) ( 13-22)

Comparing this with eq. (13-4) it follows that

G(s) = yes)

r '

(s) U(s) +W(s). (13-23)

In general, the relationship between G(s) and P(s), the system matrix, becomes rather messy [3]. However, for the special case of a dynamical system having as many outputs as inputs (M = R), P(s) is th en a partitioned square matrix, whose determinant can be braken down [2] into:

IP(s)1 = iT(s)IIV(s)T-I (s) U(s) +W(s)1 = IT(s)IIG(s)1 (13-24 )

ren

Although other possibilities exist, the state space representation may be developed via the transfer matrix, G(s). Recalling eq. (13-7) and comparing with eq. (13-23) suggests that

T(s)

= (

sI - A); U(s)

=

B; yes)

=

C and hence that

[ sI - A P(s) =

-C

(13-25)

which has been termed the state-space system matrix [3].

13.4. closed

-loop

systems

In the last section little attention was paid to the role of disturbances affecting the process. It will be recalled, ho wever, that a disturbance vector

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'!!.can easily be incorporated into thegeneral,linear state space mod el, eqs. (12-3),(12-10) and (12-11) giving:

x

=

AK+ By+ E~ y = CK + Dg+F~ ~(O) = K_o (13-26) (13-27) (13-28) Q. on

or SI (A,B,C,D,E,F) forshort. An attractive pictorial representation ofeqs. (13-26) and (13-27) is shown in Fig.13.3a. Thetriangular syrnbo l represents integration (see Chapter 14) and double lines dep iet the vario us vect orial i n-formation streams. Likewise, the squaresare matrices (which arenot, how -ever, themselves necessarily square). Notice how A~can be seen as in te rnal feedback. 't:I.

I

F tJ 0 P

Y

lS.

:t

I t: A

,

o

Fig.i3.3a.The process SI (A,B,C,D, E,F).

Simple manipulation of the above state spaceequations shows that an equivalent transfer matrix representation exists. Assuming that ~o = 0, we have from eq. (13-26):

~ = (si - A)-l B!i:+ (si - A)-lE~. Hence

l =

{C(sl - A)-l B

+

Dm

+

{C(s l - A)- I E

+

F}~

=

= G(s)u

+

Gct(s)y{

(13-29)

(13-30 ) The two transfer matrices together define SI;see Fig. 13.3b.

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6)

7)

8)

Q. 66: Derive a victoria/ representation of a processdescribed in terms of system variab/es,

and act ed upon noton/y by manipu/ated inputs but a/so disturbances.

%(s)

:g

+

:t..

G(s) +

Fig. 13.3b.Transfer matrix represen tation ofSI.

Let us now return to the very beginning of this story (Ch. 1,p. 1); there the distinction was made bet ween measured output varia bIes,y, and unmeasured

out pu t variables, v. We have really never paid very much attention to this point up to now - the variables y

e

~ being often referred to vaguely as

"ou t put s". In fact, not all the unmeasured output variables need be considered.

It is suff icient ,as a few moments reflection will prove, to account for only two types of outputs: the measured outputs, y, and the controlled variabies,z, - being those variabies which we wish to con trol.Conceptually we have the Venn diagram of Fig. 13.4,where thetotal space isx (x =y Uv). In vector notation ,thesystem SI(A,B,C, D,E,F ) must receive the supplementary equation:

(13-29)

where L is a(kxn) dimension unitary matrix. In transfer matrix notation,

from eqs. (13-26) and (13-29)

29) 30)

z:

= L(sI - A)-I Bg+ L(sI - A)-IE~

Fig. 13.4.Conceptual model of unmeasuredv,measuredy and controlledzvariables.

(13-30) (13-31)

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By taking account of the measurement transmitter s ,the controller s and the final control elements it is now possible to "clo se the loop" for the general, multivariabie system in a fashion analogous to the s.i.s.o ,case (Ch. 8, p. 84). Denoting the transmitters', controllers' and contral ele me n ts' transfer matrices by H(s), K(s) and J (s) respectively, the general closed loop system can be drawn in block diagram form as shown in Fig.13.5 .

MIs)

1

~

"'\:I(

s)

w

Gd(S)

.1

E

_.

_'9'-

~

_Y

KIs) Jls ) GIs) ~ x '<~ !::!. ~ Hls) Fig.13.5.

Q. 67: In view of what was alreadysaid in Chapters 8and 9ofPart I,whatstructure would the state space model of this general closedloop systembelikely to have? Draw

a flow diagram similar to Fig. I3.3a.

Fig. 13. 5 isby no mean sunique ; alternative forms can often be beneficial in specific applications.

Q. 68: RearrangeFig. /3.5.so as to yielda system with an inputcompensator, T(s).

where only the setpointisused as inputto the compensator.

Suppose now tha t the variab les to be controlled are the same as the

va an th Fl fo w

u

E \\ ij

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Equations (13-34) and (13-36) are two alternative forms of_r

n

_y(s). Note that We have assumed that (I + G J K H) is invertible - in fact it can be shown that if this is so (see (5) or eq. (13-56)) then (I + HGJK) is also invertible.

Sirnilarly, if no setpoint changes o ccur, then:

variables which are measured,y = ~. The closed loop transfer matrices

/:2/S)

and_w

n

_y(s) are now defined as the real rational function matrices relating the outputs,y.(s), to the setpoints I(s) and disturbances ?{(s), respectively. FrornFig. 13.5 it is possible to write down various alternative relationships for _r

n

_y and _w

n.

_y For example, writing

(13-33) (13-32) 03-36) 03-34) 03-35) ~(s)=lCs) - HGJK~(s) yes) = (I + GJKH)-IGJK."[(s) yes) = G.J.K(lCs) - Hy(s)), yes)

=

GJK~(s)

=

GJKO + HGJK)-l l(s)

:. n

(s) =GJK(I + HGJK)-l r y

:. n

(s)

=

(I +GJK H)-l GJK r y

where we have assumed that no disturbances are acting,leads to

Alternatively, ho wever, if we begin with

then

9

--=---yes) = -G J K Hy (s) + Gd.?{(s)

:. n

(s) = (I + GJKH)-l Gd w y (13-37)

Q.69: Write on alternative [orm for the closed loop transfer matrix.

wil/SJ.

Looking at eqs. (13-34) and (13-37) we see that (I + GJ KH) is an important quantity in the case of either setpoint or disturbance changes. In fact,it occurs

Sa often in more advanced studies that it has acquired its own name-- the return

difference matrix. To understand the name, consider Figs.13.6a and 13.6b. Here

We have condensed the forward transmittance GJK into one transfer matrix G(s). When feedback is present

t

.

yes) =GI(s) - G Hy'(s), (13-38)

v while without feedback:

Yes)

=

GI(s) - G H!!:(s)

:

.

y -

X

=

GHC!ï -

y)

(13-39)

t

This interpretation of the return difference matrix differs from others ([3),[5)) in not requiring a zero setpoint vector,

r

==

O.

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G(s) H(s) no al giv sqi it ' po, of th~ Q. eq. fOl aru

-

-(13-40)

E

_+""

~ G(s)

y

x

_~

r

H(s) Fig. 13.6b.

:. !! -

Y

= (I

+

G H)(!i -

yJ

ü Fig.13.6a.

It _{is seen that the return difference thus expresses the effect of feedback}

up on the difference between the input and the output signals of an open loop system. The matrix G(s)H(s) is called the return ratio (sometimes loop gain) matrix.

13 .5.

_{poles and zeros of multivariable systems}

The concepts of poles and zeros of s.i.s.o. _{systerns cannot be}

straightfor-wardly carried over to multivariabIe systems._{This should be obvious sin}_c_{e in} dealing with a transfer matrix we are handling a real rational function matrix.

For the easiest case of square transfer matrices (dim(!! )= dim (y» it turns out (for proof, _{see p. 40 of [5]) that for an open loop system ha ving a}

trans-fer matrix G(s):

anI

IG(s) I

=

~i:;,

(13-41)

i.e.

(13-42)

A number of remarks are in order. Firstly, we notice on comparing eqs.

(13-42) and eq.(11-2) _{that the determinant of the m.i}_._m.o_. _{system must be}

taken. Secondly, the polynomial 1/I(s), _{the roots of which are the z}_eros _{of the}

Be:

ele are

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-

--

- - - : : : : -

- - - -

-~ - - - -

~-l-40) m.i.m.o. system, can be shown to be of degree n - m or less. Furthermore,

the polynomial, lP(s), is theopen loop characteristic polynomial of the systern and is given by

for the system SI(A,B,C).

Q. 70: Usingthe fact that the determinantof a scalaris the same as the scalar, prove

eq. (13-43).

lP(s)= IsI- AI (13-43)

For a more detailed survey of poles and zeros in m.i.m.o.systerns with non-square transfer matrices and cancellation of poles or zeros see [6], while auseful technique for ascertaining the stability of multivariable systems is siven in [7];we consider further only the similarities and differences between

square open and closed loop systems. Turning, therefore,to closed lo o p systems, it would be reasonable to assume, it seems,that th eclosed lo o p charac te ristic

polynomialof a system su eh as shown in Fig.13.6a would be theden o min ator of the determinant of the closed loo p transfe r matrix. Tha t is, with exactly

the same reaso ning as ineq. (13-41):

(13-44) where we are interest ed primarily in the closed loop characteristic polynom ial,

\Pe( s). This rational function must be the same wherever we choose to tak e OUr closed loop transfer matrix;the stability properties of a closed loo p syste m

are,afterall, invariant of where we look in the loop. In other words,

(13-4 5)

and

I

n

(s)1

=

l/!f(S).

r f IP/s) (13-46)

Frorn Fig. 13.6a it follows that

Before taking det er minants of the abo veequations we note the followi ng two

ele ment ary properties of determina ntsof sq ua re matr ices. First ly, if A and B

are two square rational function matrices, then

(13-51) (13-47) (13-48) (13-49) (13-50) (cf. eq.(13-34)) (cf. eq. (13-36)) IABI

=

IAIIBI n (s) = [I +G(s) H(s)]-1G(s) r y = G(s)[I

+

H(s)G( s)]-1 n (s) = [I + H(s)G(s)]-I r e rnf(s) = H(s) G(s)[ I+H(s)G(s)]-1

2)

.I)

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-

--and less obviously

(13-52) This latter result ean be proved as follows:

IA-I_AI

₌

₁₁₁

₌

₁

₌

_IA-lilA!,

from (13-51).

Henee

I~I

= IA-II,i.e. eq. (13-52).

With the help of these two properties we have then that

i.e. pre me

Notiee how the return differenee matrix appears already. From eq.(13-53) it is clear to see that

IG(s)1

Vly

(s)

I

=

II+G(s)H(s) I 1 Irste(s)1

=

II

+

H(s)G(s)1 I

st

(s)]

=

IH(s)IIG(s)1 r f

IJ

+H(s)G(s)1 IG(s)1 II

+

H(s)G(s)1 (13-53) (13-54) (13-55) isI rOl Q. for Ir+G(s)H(s)1

=

IJ

+H(s)G(s)1 À(s)

=

6.(s)' say, (13-56) cIe eq Furthermore, by writing the determinants of G(s) and H(s) according to

eq. (13-41) as

IG(s)1

=

1/!G(s)

<PG(s)

IH(s)

I

=

1/!i-I (s)

<PH(s)

and substituting all these polynomials into eqs. (13.53) to (13-55) we find that wl an

I

st

(s)]

=

1/!G(s).6.(s) r y <P G(s) À(s)

I

st

(s)]

=

6.(s) r e À(s) ( 13-57) (13-58) (13-59)

13

Wl th, int sig ie

For eqs. (13-44) to (13-46) to be compatible with the above equations requires