Dynamic Feedback and Disturbance Rejection in Large Scale Composite Systems*

DYNAMIC FEEDBACK AND DISTURBANCE REJECTION IN LARGE SCALE COMPOSITE SYSTEMS*

.... t U. Ozguner Coordinated Science Laboratory and Department of Electrical Engineering University of Illinois Urbana, Illinois 61801 USA

W. R. Perkins Coordinated Science Laboratory and Department of Electrical Engineering University of Illinois Urbana, Illinois 61801 USA

ABSTRACT

THE CONTROLLABILITY OF TANDEM CONNECTED SYSTEMS

A system is modelled in Large Scale Composite (LSC) form. Local dynamic feedback is implemented, with step-disturbance rejection by proportional-integral control being studied in detail. Controllability questions are treated in the context of tandem-connected subsystems. An economic application illustrates the approach.

The following lemma, due to Hautus [8], will be used throughout this paper.

INTRODUCTION

Note than when A~cr(A) AI-A is of full rank and the condition is satisfied. Hence (1) has to be checked only at AEa (A).

Lemma 1:

The system (A,B) is controllable if P{AI-A

B}

=n

(l)t

for all A. :1=

There exist many systems that can be mode led as an interconnection of smaller subsystems. There has been a recent interest in the properties of such complex structures of interconnected subsystems, and results on the existence question, stability and controllability have been reported [1-7]. In a recent work [7J we have considered the controllability question for a Large Scale Composite (LSC) system composed of linear, time-invariant, dynamic subsystems, and linear time-invariant interconnections. That work is extended here so that dynamic feedback is also considered. It is assumed that each subsystem has a local input and a locallycontrollable [7] state model. Local dynamic feedback is to be implemented for a variety of reasons. It is shown first that numerous problems in linear control theory can, as far as the controllability question is concerned, be treated in a tandem connected two-subsystem framework. This viewpoint is then applied to the LSC system model and dynamic feedback in general, and step-disturbance rejection with proportional-integral (P.I.) control in particular is treated. The results are applied to a simple model of a multisector economy.

l Consider two tandem connected systems {Al,Bl,C }, 2 2 2 . 2 1 {A ,B ,C } w1th u = y. Clearly the Hautus controllability test is P rI-AI

(2 )

LB c

2 l

1 2 to be checked at AEcr(A )Ucr(A ). theorem can then be stated:*

The following

Theorem 1: Two tandem connected controllable subsystems SI and s2 are controllable if and only if a)

2 l For every AEa(A )ncr(A )

pt:::: b)

*This work was supported in part by the Joint Services Electronics Program (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract DAAB-07-72-C0259, and in part by the National Science Foundation under Grant ENG-74-2009l.

,,:.2:

2 For every AEcr(A ), p[A!-A

2:

B::} "1'"2'

A~cr(Al)ncr(A2)

2

. B GCA)] = n . 2

t P{X} denotes "rank of X." :1=

tpresent address: IBM T.J. Watson Research Center, Yorktown Heights, New York, 10598, USA.

cr(x) denotes the set of eigenvalues of x.

*For closely related results see [6] and [18].

345

(3 )

(4)

2

Proof: The general case is given in (2). For 2 AEcr(A ) and A~cr(Al)ncr(A2), AI-AI is of full rank, thus for the case

p

0

nl

P~B2cl

r ,I

AI-A 2

0

nl

Corollary 1.3:

(U-Al)-lBll

B2Dl (AI_Al)-lB

l

P :

o

2

Corollary 1.2: If all eigenvalues of A are equal 2 and A is diagonalizable, the tandem connected systems are controllable if and only if G(A) is of full raw rank.

H:.2: B:~

l:::: II

is of full row rank for all AEcr(A ).

U_A2

~

J

If s2 is a system of integrators,

x = Bu, the tandem connected systems are controllable, if and only if G(O) has full raw rank.

(5 )

Corollary 1.4:

If both sI and s2 are single-input,

single-output subsystems, the tandem connected composite system is controllable if and only if sI 2

has no zeroes in cr(A ).

B2cl(AI_Al)-lBl+B2D

All the above can be obtained from Theorem 1 [9]. Corollary 1.4 is the well known case of pole-zero cancellation in transfer functions.

'---

(6)

which is equivalent to

THE DISTURBANCE REJECTION PROBLEM The tandem connected system problem is crucial in several linear systems control problems. For example, the controllability conditions appear as necessary conditions in all dynamic feedback, observer theory, compensator theory and tracking, PI control and disturbance rejection problems.

(7)

which is condition b. l 2 For AEcr(A ) and A~cr(Al)ncr(A2), AI-A is of full rank, hence the general condition (2) simplifies to

which is satisfied by sI being controllable. only the case a remains.

In this section a discussion of the disturbance rejection problem will be given for a general linear time invariant system with step disturbances. This problem has been studied by Davison, Johnson, Wonham, and Bhattacharyya [10-13] among others. The difference of approach in the treatment below will be in showing the relation with the problem of controllability of a tandem connected system. The results will then be extended to a "restricted input" case, i.e. the case when only a subset of the inputs can be used.

Thus

Q.E.D.

l An important point is that, if D 0, the tandem connected subsystems controllability problem can always be reduced to checking, for AEcr(A2),

=

Consider the system P[U-A

2

2 : B C(A)] = n

(9)

2

where l 1 1 C(A) = C (AI-A - BlKl)-lB

(10)

cr(A )ncr(A

l

l

+ BlK ) = rp,

C x(t) + D u(t) + FW

(13 )

x-

(11)

1 1 1 where a (nonunique) K exists since (A ,B ) is a

Theorem 2:

controllable pair and poles of Al can be arbitrarily placed by linear state feedback. Furthermore this does not alter the controllability properties of sI' since

{Al~l} = {Al+BlKl~l}.

A x(t) + B u(t) + Ew

Y (t)

where A is an nxn, B is an nxp, E is an nxq and C is rxn. Controls are sought for asymptotic stability and output regulation, i.e. 0 and y - 0 as t - 00, where w is an arbitrary constant vector and A is nonsingular.

l and K is any matrix such that 2

X(t)

The feedback control (14)

where z(t) is the state of the system connected in tandem to the output,

(12) ~(t)

Various special cases may now be stated. For example, assume that cr(Al)ncr(A2) = rp. Then

= O·z(t)

+ Iy(t)

(15 )

solves the problem defined above if and only 1f a) b)

Corollary 1.1: For the controllability of the tandem connected systems it is sufficient that G(A)

346

(A,B) is stabilizable the system {O,I,I,O} is controllable through the inputs of the original system.

Proof: A full proof of the above will not be given here, but it will be shown that (b) is equivalent to the condition

n+r

(ii) When the injected disturbance does have unstable components only if a(A ) = ~ and

22 a(A44 )=C-, where a(A ii ) is the set of eigenvalues of A and C is the left half complex plane. ii

(16)

Here, (ii) is equivalent to saying C~ = ~. given by Davison and Smith [10]. singular,

Since A is non-

Note thatr both conditions imply the system to be stabilizable. The extension to consideration of outputs is similar to the above. If internal stabilization (i.e. X - 0 as t - 00) were not required, depending on (C), the null-space C, some unstable modes in a(A )Ua(A ) may be omitted. 22 44 Hence the output-controllable subspace of the disturbance must be within the output-controllable subspace of the control, i.e.

(17)

(20) which implies that p[-CA

-1

B+D]

= r.

where ~ and J are the range spaces of D and F respectively. Or, if there is a portion outside the output-controllable subspace, that portion must be stable.

(18)

This is equivalent to requiring that G(O) be of full row rank, where G(A) is the trans fer function matrix of the system. Referring to Corollary 1.3 this is seen to be the controllability condition for the system to be connected in tandem. Q.E.D.

THE LARGE SCALE COMPOSITE SYSTEM

A discussion, using some concepts from the geometric viewpoint in control theory [12] will now be offered.

Consider a system S composed of m subsystems [Sl,S2,.,.,Sm}. the j'th subsystem is mode led as Sj:

Define the following controllability subspaces: CB C E

C

[Ala} =6 + l6 + [Ale} = e + Ae + C

EB = B

(21)

n-l

+A 6 + An-le

(22) j j where x , u J', u , yj are n., r., r. and q. vectors l z J J J J respectively; and the matrices are real and of proper dimension. Here denote the locally

n CE C*

.J.

c*E

.J.

B

Ut

available inputs, (or local inputs, for short) and

u~ the aggregate interaction inputs from other sub-

also define C as the noncontrollable subspace. N

systems. as

Theorem 3 [7]: There exists a similarity transformation T to change (13) into Xl

An

)(2

0

x3

A 3l

x

0

4

0 A 22

0 0

A A 32 33 0

0

A 14 A 24

x

A 34

x

A 44

x

2 3 4

r0 l

B l

Xl

I

0

+

u

E2

B 3

E 3

0

0

X Y

I 1

w

U

(19)

z

where x

The composite system S than can be modeled

+ Blu l + BZu Z Cx + Dlu + DZu l Z

Ax

x(O)=x

o

(Z4) (Z5)

Hy

-l~J'

(Z3 )

u u

1 l

U I

l

!'

ml

~lJ

U

z

1

z

I:__zJI.

(Z6 )

Um

are the composite state, composite local input and composite interaction input vectors respectively.

It is now obvious that the system can be stabilized, (i)

When the injected disturbance has no un-

stable components only if a(A22)Ua(A44)=C-.

347

composite system equations can be partitioned as

~ere,

Al

0

1

2 A

I

A

(27)

I

i

0

J~(22)J I

ul

u

i B (23)

LO

...•.....

ml

L_ 1_ An important problem that has to be solved when

+

= Cx +

BU

x (O)=X

l

r

(29)

o

(30)

DuI

,m. (38)

I A(ll)

o

I B(22)fHC(1)

A(22) + B(22)fHC(2)

L~(3l+B (23 )fHC (1)

A(32) + B(23)fHC(2)

o

where the dimension of x is the sum of subsystem state vector dimensions. If so, what A, B, C, ~ are in terms of A, B , B , C, D , D and H? This l 2 l 2 problem has been considered by several researchers [4,9]. An important result is the following, where the state space associated with s. is denoted by .

J=l, ...

Hence the A matrix of the composite system is

analyzing the composite system is whether we can obtain a model such as,

x = 'Xx

block diag (B~k}

B (ik)

0

(37)

block diag (Aik}. .

I

,h

(36 )

2

Lo

A(ik)

I

~

o

J

~J:

o

i B(13)

(28)

y

A(33)

where

•

I

2

A(32)

[- B(ll)j-

+1

;0

!: u m

A(22)

A(3l)

-.J

and B , B , C, D , D are similarly constructed l l 2 2 block diagonal matrices. H shows all interconnections in the LSC system, and has the partitioned form ,

[u; I.

o

o

o

Am!

0

o

o

A(ll) j

B(22)fHC (3)

A(14) A(24)+B(22)fHC(4)

A(33 )+B (23)f HC (3)

A(34 )+B (23 )fHC (4)

o

A(44)

(39)

The state space of the composite system is L:

1

Ell L:

2

EIl •••••EIl L:

m

(31)

and there exists a solution for all x and piecewise continuous u to (23)-(25) in th2s space, if l and only if

where C has been partitioned to be compatible with the decomposition. The B matrix is

(32)

l

B (11)

Under this condition a formulation such as (29)(30) exists, where x is an

B(22)fHD

m L: n. vector, and the j=l J

system is represented as,

X

(/I + B2fHC)x + (B

l + BlHDl)u l

(33 )

y

(C + D fHC)X + (D + DlHDl)u 2 l l

(34 )

J

where -1

x .

(40)

Consider a Large Scale Composite (LSC) system [7,9], where sybsystem s. is modeled as, j

f = (I-HD ) 2

J

l

~(13)=B(23)fHDl

(35 )

= A.. x

j

JJ

m

+ L:

(41)

i=l i=j

If subsystems are given in the canonic form, the

For each subsystem, the similarity transformation of Theorem 3 can be applied. Assume that each subsystem is locally controllable [i.e. all (A .. ,B.) JJ

J

are controllable pairs]. Then the composite system can be represented as [9,14]

Ixx~f L 1

348

[A(ll) A(3l)

0

l

xl

A(33~ x 3 +

IB(ll~

l!(3l)J

u

(42)

j = 0_ _ for every j=l, ... ,m, njxn z the augmented LSC system (45) is locally ~ontroll­ able if and only if

where A(ll), B(ll), B(3l) are block diagonal. All matrices can be determined from the set (41) after applying the transofrmation of Theorem 3. The LSC system and the pairs A(ll), B(ll) are controllable [9]. Next consider local dynamic feedback:

)

Corollary 4b:

m

p[G(O)} (43)

1

z

F

A(3l) F 3

l

0J~z ["J [B(l1] ~(3l) :3

+

In both corollaries the invertibility condition is not restrictive, as explained for Theorem 1. Notice that

After considering the controllability properties of dyanmic feedback the case of disturbance rejection can be analyzed. The following theorem is given: Theorem 5: A controllable LSC system under step disturbances is asymptotically stable and output regulated by local P.I. feedback control if and only if condition (50) holds.

Block diag [ FI} Block diag [ F~} j Block diag [F } z

F 3 F

z

This is a simple extension of our previous discussions. When the P.I. control is set up, it is obvious that it is a collection of local controllers. Hence Theorem 5 follows.

(46)

j=l, ... ,m.

Theorem 4: The augmented LSC system (45) is controllable if and only if m

z:: n.

P[AI-F z ' G(A)}

j l 2

There exist some interesting special cases related to the interconnection structure within the LSC system. Indeed it can be shown that if the interconnections are loop-free [14,15], A(33) of Eq.(42) turns out to be lower block-triangular, and hence G(A) of Eq. (48) is also lower block triangular. Thus the test (47) decomposes to m separate tests as in Corollary 4a. The loop-free (also called multilevel) structure is very important in LSC systems as it arises often in physical models and causes various decompositions and simplifications of certain properties. For a further discussion of these see [14,15].

(47)

J

for every AEcr(F )' where G(A) is defined as z

G(A)

[F

l!-A(l1) l

F3 ]

~ -A (31)

o

IT(l1]

AI-A(33

~=~ means dynamic feedback related to the

space Ca*J •

where, F l

(50)

=

(45)

u

.

j If furthermore F 0 FjAj-lB has to be of full 3 1 11 row rank for every j=l, .•. ,m.

(44) means attaching a new subsystem to the outputs of each existing subsystem, and if (43) is to place all poles of the LSC system as required, the augmented system has to be controllable, i.e. the subsystems (44) have to be controllable through the existing inputs of the LSC system. Consider the augmented system

r~ x = r~(ll) A(3l) 3

= z:: nj j=l

(44)

0

If F

(48)

B(3l) AN ECONOMIC REGULATION EXAMPLE

j is a n.xn. matrix. z J J under the assumption that

and F

Theorem 4 has been given (a[A(11)]Ub[A(33)])na(Fz)~

This is not restrictive, since local feedback can always be implemented to satisfy this condition.

=

Under the assumption F 0, 3 G(A) = Fl[AI-A(ll)] -lB(ll), which is block diagonal. Hence

=

Corollary 4a:

If F 0 in (45) the augmented LSC 3 system is locally controllable if and only if for j=l, ... ,m

p[AI-~z for every A E

(49)

a(~). z

349

LSC systems can be used to model interconnected systems in various areas. Here we give an example from economics in which certain predetermined values of target variables have to be achieved in the steady state. This is obtained with dynamics in the feedback loop and the problem, discussed in the previous section, of whether the augmented subsystems are controllable through the original inputs, has to be solved. First the single sector equations will be analyzed. Then extensions will be made to be a multisector model where each sector is a single subsystem. For more details on the models, see (7). Demand:' Yd

(l-s)ys+g+a

(51 )

Supply: Ys

-A (Ys -Yd)

(52)

which gives

Ys

Apply the transformations =

(53)

-SAYs+Ag+Aa

where a denotes autonomous investment and consumption expenditure which is assumed to be a known constant. The lagged response of aggregate supply (ys), to aggregate demand (Yd)' is assumed to be continuous, with time constant A. (l-s) is the propensity to consume. To regulate around the desired goals, a performance index is selected,

(64 )

=J

[q(y _y*)2+r(g_g*)2]dt, s

o

J

(54 )

s

J

J J

L:

J Ji#j

J J

Then the subsystem can then be represented as j -<., .u J.· ~ {.., x '1\ J J i#j J J

S.: x.L--A. (1 -et. ) X j +z j +A.et.

The transforms

J

Ys -

x u

Sj

+A.g~ +A.a.J.

where y* is a desired target value and g* a specifi~d level of government expenditure.

=g

y:

J

- g*

x=

s

(56)

Ax

+ Iz +

J

J

Let . J J

(67) -SAX+ A!1 + z

(57) Since A. # 0 the composite system is controllable. Considet the local state-regulator problem. PI control is applied to (66). As shown previously the integrators can be considered as separate subsystems. The controllability condition for them is G(O) and thus A (as in (67)) being nonsingular.

and the cost function is

J 00

2

2

(qx +ru )dt.

(58)

o

The solution to this regulator problem is well known, and in terms of the previous variables, this is of the "PI" control form,

det A

00

m m (-1) [

(59)

g

m A. (1- L O"i)] j=l J i=l

J

= -A. (y -y ) J s. d. J J

Aj > 0

(60)

m

(61)

L O"i # 1. i=l

(62) CONCLUSION The dynamic feedback problem in large scale systems was analyzed by considering in detail the tandem connected two-subsystem problem. The controllability conditions obtained appear as necessary conditions in various problems in control theory. These were applied to local dynamic feedback and specifically local disturbance rejection with P.I. control in LSC systems. An example of a multisector economic model was given.

where C. is the final household (consumption) demand J in sector j and the second equation specifies the consumption function. Combining,

Y

Sj

=-A.(l-O".)y +A.et. L: Y -f7.. ·(gJ·+a ·)· J J J Sj J J i#j si J

m

L et. = 1 in i=l 1 which case local feedback has to be implemented to change the eigenvalues. Thus the multisector system can be regulated to a set of desired goals with a set of Phillips-type policies applied to each sector separately, only if

This is called a Phillips-type (16) stabilization policy. Extending the simple model to more than one sector, (7) we have Yd. = C.J + g.J + a.J J 1 C. =et.L:y O~ et. < J J J i si

(68)

TT

this condition is satisfied unless

Ys .

(66)

Bu

-A.(l-O".)

where, if the constant term is denoted by z, the state equation becomes,

J =

(65 )

where

-SAX + AU + (-sn*-f7..g*+Aa)

x=

J

and S, the composite system, as

(55 )

are made, so that (53) becomes

x=

= [-A. (I-et. )y* + A .0".

zj

00

j

(63)

350

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[ 2]

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[3]

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[4]

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[5]

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[6]

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[7]

U.

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]

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[18]

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