A new approach to the decoupling problem of linear time-invariant systems

A Neu Approach to the Decoupling Problem of Linear Time-invariant Systems hy

P.

N.

Nutional

and

PARASKEVOPOULOS

Technical

Department

University

of Electrical

F. N.

KOUMBOULIS

of Athens,

Engineering,

Division

Science, Athens, Greece

of Computer

15773 Zographou,

ABSTRACT : A new approach to the input-output decoupling problem of linear time-invariant systems via proportional state and output feedback is presented. A major feature of the proposed approach is that it reduces the solution of the decoupling problem to that of solving a linear algebraic system of equations. This system of equations greatly facilitates the solution of the ,following four major aspects of the decoupling problem: necessary and sujficient conditions, general analytical expressions for the controller matrices, general analytical expressions for the diagonal elements of the closed-loop system, and the structural properties of the closed-loop system.

I. Introduction

Input-output decoupling is one of the central control design problems, since it aims at reducing a multi-input, multi-output system to a set of single-input, singleoutput systems, thus greatly facilitating the control strategy. The decoupling problem was originally considered in Refs (1) and (2). The first major results of the problem were reported in (3) and (4) for linear time-invariant systems described by jr(t) = Ax(t)+Bu(t),

y(t)

= Cx(t)

(1.1)

where x E KY’,u, y E Iw”. In (3) and (4), the following major aspects of the decoupling problem were successfully studied: necessary and sufficient conditions for the problem to have a solution (3), the general family of the decoupling matrices (4), and the structure of the decoupled closed-loop system in connection with the pole assignment problem together with decoupling (4). Subsequently, many papers have been published on the subject (567) which may be grouped as follows : decoupling via state feedback (5-S), decoupling via output feedback (9920), decoupling and pole placement (4, 6, 8, 64), decoupling via dynamic compensators (12-22), group or block decoupling (2327), triangular decoupling (28,29), optimal decoupling (30-32), decoupling of singular systems (3337), decoupling of time-delay systems (38-42), of time-varying systems (4346), of nonlinear systems (477Sl), of two-dimensional systems (52, 53). Sensitivity reduction in decoupling (54), decoupling with simultaneous disturbance rejection (5557), adaptive decoupling (5%60), geometric approach to decoupling (6, 7, 21, 28, 66), transfer function approach to decoupling (12,29,6163,65,67).

t The Frankhn lnst~tute 001&0032!Y? U.OOf0.00

347

P. N. Paraskevopoulos

and F. N. Koumboulis

In this paper, a new approach is proposed for the input-output decoupling of linear time-invariant systems via proportional state and output feedback. This new approach appears to be superior to known techniques and may be described briefly as follows : let H(s, F, G) be the m x m transfer function matrix of the closed-loop system using the state feedback u = Fx + Go. The decoupling problem is to find F and G such that H(s, F, G) = diag {hi(s)}, with hi(s) + 0. This equation is the starting point of our approach. Application of some algebraic manipulations to this equation leads to a new equation, linear essentially in F and G. Making use of the expansion of rational functions in a series of negative powers of s in both sides of this equation, we arrive at an identity of infinite polynomials of sP ‘. This identity holds if and only if a finite number of the first coefficients of the same powers of s- ’ of both sides of the identity are equal. Following this approach, a finite number of equations is produced. This set of equations is a linear homogeneous system of equations of the form BiIIi = 0, i = 1,2,. . , m, where the unknown vector 8; involves not only the parameters of the controller matrices, but also the parameters of the closed-loop system, and the known matrix TI, involves the Markov parameters of the state and of the output of the open-loop system. On the basis of this system of equations, the following four major aspects of the decoupling problem are resolved : (1) Necessary and sufficient conditions for decoupling. (2) General analytical expressions for the controller matrices. (3) General analytical expressions for the diagonal elements of the decoupled closed-loop transfer function. (4) Structural properties of the decoupled closed-loop system. Following a similar approach, the problem of decoupling via the output feedback law u = Ky+Go reduces to a linear algebraic system of equations. This set of equations is then used to resolve, for the case of output feedback, the four major aspects of the decoupling problem mentioned above. Most of the results derived here have already been reported in the literature, wherein several different approaches have been applied and the results derived are in various forms. The approach presented in this paper has the following distinct characteristics : -It -It

unifies the solution of the state and output feedback decoupling problems. unifies the solution of many design problems (input-output decoupling, triangular decoupling, block decoupling, exact model matching, disturbance rejection, observer design, etc.), and for different types of systems (singular, timedelay, time-varying, etc.) (68). -For certain categories of systems, e.g. of singular systems, it solves several design problems [decoupling (3%37), and exact model matching (69) via pure proportional state and output feedback, minimum observers (70), etc.], for which all known techniques appear to fail. In addition, the present approach has produced the following results, first in the field of decoupling : the cancelled out poles appearing in the closed-loop transfer 348

Journalofthe

Franklin Pergamon

Institute Press plc

Linear Time-invariant System Decoupling function matrix via state and output feedback ; the general analytical expressions of the output feedback controller matrices ; structure of the decoupled closed-loop system via output feedback. The known matrix TI, in the equation BiTTi= 0 involves, as mentioned earlier, the Markov parameters of the state and output of the open-loop system. Thus TI, can be constructed experimentally on the basis of the measured state and output vector, and hence it can be determined without resorting to a state space model (A, B, C) which requires the difficult task of identification. This fact has a benefactory influence on the computational aspect on the problem of solving 0,TI, = 0, since the measurement errors appear linear in TI,. We remark that a basic point in the overall philosophy of our approach is to transform the decoupling problem from a problem in the field of rational functions, to one in the vector space of the coefficients of the power series expansion of these functions. This way, the problem is reduced to a simple algebraic equation. The inverse transformation is used to determine the diagonal elements (rational functions) of the closed-loop system, for their corresponding power series coefficients. It is mentioned that the present results are part of the material reported in (68).

II. Preliminaries In the frequency

domain,

system (1.1) takes the form

sX(s) = AX(s)+BU(s), To system (2.1) apply the proportional

Y(s) = CX(s).

state feedback

U(s) = FX(s) + GSZ(s), The problem

of decoupling

consists

law

Q(s) E R”.

of finding matrices

H(s) = C(sI-A-BF)-

(2.1)

(2.2)

G and F such that

‘BG = diag {hi(s)} ;

hi(s) f 0.

Clearly, for (2.3) to hold, it is necessary that G must be invertible. matrix manipulations, (2.3) may be written as follows : P(s)J(s)C(sI-A)-‘B

(2.3) After certain

= ~-Q(sI-A)~‘B,

(2.4)

where I. = G-‘,

Q = GP ‘F,

P(s) = diag {p,(s)},

J(s) = diag (&+ ‘}

min {j : c,AjB # 0, P,(S)

=~-~,-‘[h(~)l-’

and 4 = n_l

if cAjB = o I

1

j=O,l,...,n-1) >

yj

with c, denoting the ith row of C. Equation (2.4) is equivalent following condition holds, det I- # 0.

i

to (2.3) if the (2.6)

At this point, we introduce our method for solving (2.4). This method considers as unknowns, not only r and a’, but also p,(s) and reduces the decoupling problem Vol. 329, No. 2, pp. 3477369. Prmted in Great Bntain

1992

349

P. N. Paraskevopoulos

and F. N. Koumboulis

to that of solving a linear algebraic system of equations. Note that once r and Q, and pi(S) are determined, one may readily determine G, F and hi(s) by using (2.5) to yield G = r-‘,

F = T-‘@

and

hi(s) = sPdrP’[pi(s)lP’.

(2.7)

Hence, our method determines not only the controller matrices G and F, but also the form of the diagonal elements hi(s) of the closed-loop system. To apply this method we start by considering the ith row of (2.4), i.e. by considering the relation p;(s)sdg+ ’ c,(sI-A))‘B

= y,--i(~I-A))‘B,

(2.8)

where yi and cp’ are the ith rows of r and @‘, respectively. sides of (2.8) in a series of negative powers of s to yield [(pi)o~O+(pi)‘~~‘+

. ..][c~A~B~~+c~A~~“Bs~‘+

Next, we expand

both

’ - cp,ABs- 2 - . . . ,

(2.9)

. ..I

= yiso -cp,Bs-

where we have substituted p,(s) = (P~)~s’+ (pJlsp ‘+ . . It is mentioned that according to (2.8),p,(s) cannot have a polynomial part. Equation (2.9) is an identity of two proper rational functions where the denominator of pi(s) has degree at the most n. In order that (2.9) holds, it suffices that only the first 2n-t 1 coefficients of the expansion of both sides of (2.9) are equal (71), thus yielding the following set of 2n + 1 algebraic vector equations : (2.10a)

(Pi)oCPB = Y[ i

(pi)&4+B

k=

= -rp,Ak-~‘B,

1,2 ,...,

2~2,

(2.1 Ob)

l/=0

where c,+ denotes

the ith row of C*, where CIA”1 c*=

: .

(2.11)

.

[ c,Adm I Equations

(2.10a, b) may be expressed

as follows :

more compactly

e,rI, = 0,

(2.12)

where . . (Pi>Znl

0, = k&‘L i (PJO(Pi),

n, =

0

B

...

I -1

0

..

c:B

1 350

c*AB

A2”--

0

..

c,*A2”B c:AZnP ‘B

0

c:B

... ..

0

0

...

(2.13a)

,B

(2.13b)

c:B Journal

of the Franklin Pergamon

Institute Press plc

Linear

Time-invariant

System Decoupling

Equation (2.12) is the linear homogeneous system of equations sought with 2nm + m equations and 3n + m + 1 unknowns. Clearly, the solution of the decoupling problem is now reduced to that of solving (2.12) for rp,, y, and (pJj, subject to the condition (2.6). Equation (2.12) plays a fundamental role in our approach since on the basis of this equation we will derive the necessary and sufficient conditions for the problem to have a solution, the general analytical expressions of the controller matrices G and F, and the general form of the diagonal elements hi(s) of the closed-loop system. For this reason, (2.12) may be called the state feedback decoupling

design equation.

III. Necessary

and Suflcient

Conditions

Define B* = C*B. We establish Theorem

the following

(3.1)

theorem.

3.1

The necessary the proportional

and sufficient conditions for the decoupling state feedback law (2.2), are

of system

det B* # 0. Proof : (Necessity).

From

(3.2)

(2.12), we have ~8 = (Pi)ocl*B.

Substituting

this expression

(2.1) via

(3.3)

in (2.6), we readily derive condition

(3.2).

Proof: (Sz@ciency). Using (3.2) a-id letting (poO = 1, (p,), = . . . = (P!)~~ = 0, yi = c,*B, and cp, = -c,*A. Equation (2.12) is solved, and condition (2.6) is satisfied. n The proof of the above theorem constitutes an alternate (but simpler) proof compared to that first reported in (3). On the basis of the proof of Theorem 3.1, a special solution (G*, F*) may be constructed as in the following Corollary. Corollary

3.1

If system (2.1) satisfies Theorem 3.1 then a special controller matrices G and F is given by

solution

(G*, F*) for the

G* = (I$*)-’

(3.4a)

F* = - (B*)- ‘C*A.

(3.4b)

Proof : As shown in the sufficiency part of Theorem 3.1, the solution of (2.12) : yi = c:B, cppi= -@A, (pJ,, = 1, (pi), = . . = (pJZn = 0 satisfies also condition (2.6). Substituting these expressions in (2.7), we derive the special pair (G*, F*) given in (3.4). Vol. 329, No. 2, pp. 347-369. 1992 Printed in Great Bntam

351

P. N. Paraskevopoulos Example

and F. N. Kournboulis

3.1

Consider

The transfer

the system of the form (2.1), where (3)

function

matrix of the open-loop

C(sI-A)Application

of the present B*=[;

The resulting

A],

closed-loop

~ l/(s- 1) -/ _ 0

l/(s-I)2 -li<,l)

iB =

technique

system is

readily yields

G*=[‘:

;I,

system transfer

C(sI-A-BF*))‘BG*

F*=[_;

function

1:

Solution for

;I-

matrix is

= diag

.

f,f i

IV. General

1

I

G and F

In order to derive the general analytical expression of the controller matrices G and F, we first apply to the open-loop system (2.1) the special feedback pair (G*, F*) derived in Corollary 3.1. This is a key point in our approach, facilitating the study of the structural properties of the closed-loop system. To this end, define f; = (G*))‘(F-F*),

G = (G*))‘G

(4.1)

V(s) = RX(s) + M(s).

(4.2)

Substituting definitions (4.1) and (4.2) in the feedback law U(s) = FX(s) + Gfi(s) we have U(s) = F*X(s) + G*V(s). This expression for U(s) reveals that the application of the feedback law U(s) = FX(s) + Gn(s) to system (2.1) is equivalent to the application of the feedback law (4.2) to the following system sX(s) = (A+BF*)X(s)

+BG*V(s),

Y(s) = CX(s).

(4.3)

It is remarked that system (4.3) is the decoupled closed-loop system derived after the application of the special feedback pair (G*, F*) given in Corollary 3.1 to the original system (2.1). Note that system (4.3) is control law equivalent to system (2.1), as first pointed out in (4). From (4.1)-(4.3), it follows that in order to determine the general forms of the matrices G and F, it suffices to determine the general forms of the matrices G and E, since from (4.1) we have Journal

352

of the Franklin Peraamon

Institute Press plc

Linear Time-invariant G=G*c

and

System

F=F*+G*@.

Decoupling (4.4)

Applying the results of Section II to the present decoupling problem of system (4.3) via the feedback law (4.2), the problem now reduces to that of solving the following new decoupling design equation e,l=l; = 0,

(4.5)

where ei = [@I! ?i : (P,> O(Pi>I . . . (Pi> *nl 0

A

...

A,2”-‘A

e,

0

...

0

0 .. .

ej .. .

..,

0

0

...

e,

-IO...

0, =

-0

(4.6a)

0 (4.6b)

under the constraint

detF#O;

The matrices

and the vectors appearing

A, = A+BF*

f=

(4.7)

in (4.5) are as follows :

= A-B(C*B))‘C*A,

A = BG* = B(C*B))’

(4.8b)

and ei is a 1 x m row unity vector having the unity in its ith position. As indicated in the proof of Corollary 3.1, the derivation of the special feedback pair (G*, F*) is based on letting (pJO = 1, (pJ , = . . . = (pJz, = 0, or equivalently on letting p,(s) = 1. Substituting p,(s) = 1 in (2.7) we readily conclude that the transfer function matrix of system (4.3) is C(sI-A,))‘A

= diag(s-4-l).

(4.9)

The simplicity of the form of r^l, in (4.6b) is due to (4.9). Finally, it is remarked that to the condition according to definition (4.4), the constraint (4.7) is equivalent (2.6). In what follows, on the basis of Eq. (4.5), we derive the general expressions for the feedback matrices G and F. Hence we define Vol

329, No. 2, pp. 347-369,

Prmted

,n Great

Rntam

,992

353

P. N. Paraskevopoulos

and F. N. Koumboulis

L=

[A;A,A;...$-‘A]

(4.10)

Li = [Ai; . . ..A.-‘Ai]

(4.11)

Ai = [a, ;...$_,;O$,+,;.

..$,,,]

(4.12)

where M, =

L,L: +NTN+ [

c

=

6,- I 1 A’,S,Sr(A:)’ j=O

max cri ;

CT,= G-rankL,,

1

(4.14a)

e = rankL

(4.14b)

and where N is the (n-e) x n full row rank matrix which is orthogonal to the matrix L. It is mentioned that from (4.14b), (4.9) (4.10), (4.11) and (3.2) one may readily prove that cr, > 1. We now establish the following theorem. Theorem

4.1

Assume that system (2.1) satisfies the conditions general analytical expression of the feedback matrices G = (B*))‘P;‘; F=

-@*)-I

i j=

where the only A,, the arbitrary arbitrary matrix Proof : From relations

PO = diag{(p,),}

AjSi+TN I

of Theorem 3.1. Then G and F are

(4.15)

= 1 I; A/ +C*A

the

diag {(ii)/},

(4.16)

free elements are the elements of the arbitrary diagonal matrices invertible diagonal matrix PO, and the elements of the m x (n -l) T. definitions (4.11) and (4.14b), we derive the following dependence

A,“c+‘6, = -A{[6,.

. A~~~-‘GJal-A{[Aj~.

. .~A~~~‘A,],,&,

j = 0, 1,. . . (4.17)

where

4 = [(do. . . (aJr,- 11 a: = [(EJ, . . (&),

(4.18a)

!. . . ! (07J,+ Il+c,.. . (o?Jmn].

(4.18b)

Clearly, the coefficients (uJy may be uniquely determined from (4.17) for j = 0. It is remarked that, from (4.17) and (4.9) one may readily show that (E,)~ = . . . = (ai)<,, = 0. Also substituting (4.17) in (4.5), we derive the following recursive relation

354

Jaurnalofthe

Franklin Institute Pergamon Press plc

Linear Time-invariant 0,-l 1

1&A:+j6, = -

Decoupling

j = 0, 1, . . .

(a,)yi$iA~fjGi;

q=d,+

System

(4.19)

I

Using the above recursive equations, and after some algebraic breaks down to the following three equations :

manipulations,

(4.5)

i, = (pJoei

(4.20)

i3JI, = 0

(4.21)

0,-I (Pt)oz+j = - ,=F+, (C(i)q(Pi)q+,,

j = 1729.. .

(4.22)

where in (4.21) (4.23)

ai = [@iiPtl; Pi = [(Pl) 1, . . .2 (Pi),]

(4.24) Relation

(4.20) may be written

more compactly f = diag {(pJo}

as (4.25)

= PO.

Substituting (4.25) in (4.8) and upon substituting the resulting relation in (4.4), we derive the general solution for G as given in (4.15). In order to facilitate the solution of (4.21) for & we observe that from the definition of ci the following rank equality holds rank [Li: 6;. . . A,“1‘S,] = rank I?, = rank L.

(4.26)

Hence, the C, last rows of l=Ij are linearly dependent upon its first n rows. Thus the parameters (p,), , . . , (pJ,,, are arbitrary. Then, Eq. (4.21) may be considered as the following non-homogeneous equation &,[Li$i. . . A~~‘d,] = -p,[O&,]. Equation (4.27) is always solvable for &. Its general solution A and is given by 4; = i,Nwhere ?, is an 1 x (n-d) arbitrary more compactly, as follows :

= -

(4.27) is derived in Appendix

5 (pi)q6T(AT)q-‘M,~‘, y= I row vector.

$ P,S,+TN; I

‘I+=

Expression

,

(4.28) (4.28) may be written

P, = diag{(pJj},

(4.29)

j=

where use was made of the definitions (4.13) and (4.14b). Finally, using the definitions A, = Pi ‘P, and T = -Pi ‘T, and substituting (4.29) in (4.8a) and upon Vol.

329, No. 2, pp. 347-369,

Printed

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1992

355

P. N. Paraskevopoulos

and F. N. Koumboulis

substituting the resulting relation in (4.4), we derive the general expression for F given in (4.16). n The form of the general expressions for G and F given in (4.15) and (4.16), respectively, is more convenient for implementation compared to other results (4). On the basis of the proof of Theorem 4.1 one may readily prove the following corollary. Corollary 4.1

The special pair (G*, F*) given in Corollary 3.1 may be derived from the general expressions for G and F given in (4.15) and (4.16), if we let T=O, It is remarked Example

(pJO=l,

(Q,=O,

that the expressions

j=1,2

,...,

c.

(4.30)

in (4.30) satisfy the constraint

(4.7).

4.1

Consider the system of Example 3.1. Application of the results of the present section yields the following general expressions for G and F G =

1 [ ,

F =

’-(/I)

_ -0 _ _ m;p -(h),-1,

-t

p2p1p -t,,-2

-1’ t -(I ) ?‘_ _ f _*! p-5 ! t,,+l I ’

where the parameters (P,)~, (p2),,, (A,), , (i2), , t, , and t2, are arbitrary satisfying the inequality (P,)~ (p2),, # 0.

V. General Form of the Closed-loop

parameters

System

In order to derive the general form of diagonal elements of the closed-loop system, we first determine the general form of p,(s) - (pJO. It is well known that there is a unique bilateral correspondence between a strictly proper rational function with denominator polynomial having degree less than or equal to y1 and the vector involving as its elements the first 2n coefficients of the negative power series expansion of the rational function (71). We use this property to construct the rational function p,(s) - (pl)” from the vector [(p,) ,, . . . , (pJzn]. According to (4.22), all coefficients (P,)~,+,, j = 1,2,. are linearly dependent as mentioned in the proof of upon (P,)~,+,~ l,. . , (PA,,+ I+,. Furthermore, These observations Theorem 4.1, the coefficients (pi), , . . . , (p,),, are arbitrary. lead to the conclusion that p,(s)-(p,), may be realised as a transfer function of minimum order Go, having the form (71) Pi(S) - (Pi)0

Now if we substitute (5.1) in (2.7), we derive the following the ith diagonal element of the closed-loop system

general expression

Journal

356

of the

Franklm Pergamon

for

lnst~tutc Press plc

Linear Time-invariant ,

hi(s) = (Pi)0

a,(s)

Bl(s>

s”~+- ’ + (aJo,_ ,s-+ 2+ . . . +

1

= (PJO’

System Decoupling

so’+ (/?&_ ,s”~~~-j

(!&+

,s”

(/?Jos”

1’

(5.2)

where

(PI>,

-I

=

tAi)j+

(a,>o,-j

1 (~r)y(Cli)o,-l+y,

+

y=1

(5.3)

where CAi)j

=

(Pi)0 ‘(pi).,,

(az)d, =

'.

=

(al>0

=

0.

(5.4)

Commenting on the structure of (5.2), we remark that the numerator polynomial LX~(S) is fixed and depends entirely upon the open-loop system. On the contrary, the denominator depends on the completely free parameters (A;), , . , (A.,),,. These degrees of freedom may be very useful to satisfy other design requirements such as pole assignment, sensitivity reduction, etc.

Example 5.1 Consider the system of Example 3.1. Using the results of the present section we derive the general form of the transfer function of the closed-loop system (P*)O’ s+(&>,

H(s) = diag

VI. Structural Properties

of the Closed-loop

.

System

The morphological characteristics of the general form of the ith diagonal element h,(s) of the closed-loop system given in (5.2) facilitates the study of the simultaneous problem of input-output decoupling and pole assignment. Thus relation (5.3) is rewritten more compactly as /IL = $A, +a;,

(6.1)

where P, = [(Pz)b,-. 13

(BI>,

23 . 9 (Pi>01

a, = [(aJli,~I, . . . , (ah,+I, 4 = [(&>I> Voi.32'). No.2.p~ 317-369.IYY? Prmicd I" Greal Br,la,n

(U2>.

. .>

0, . , 01

m,1

(6.2a)

(6.2b) (6.2~)

357

P. N. Paraskevopoulos

l

and F. N. Koumboulis

Ccli)r7-

1

I

..

O

(%)d,+I

.

(%LP

I

. .

Cai)d,+

1

. . .O

0

...O

0

, (6.2d)

0

1

(a-

1

1 where the matrix &has dimensions oLx cri. On the basis of Eq. (6.1) we establish the following Theorem

theorem.

6.1

All poles of the denominator of the general form of the decoupled closed-loop system may be arbitrarily assigned using the free parameters (A,), of the feedback matrix F given in Theorem 4.1. Proof: From (6.2d), we observe that & is invertible. Hence, if we select the vector pi = p*, and since (a,)[ are known, then from (6.1) we have that the parameter vector Iz, takes on the value

Thus, the poles of a polynomial with order gi are arbitrarily allocated. W On the basis of the proof of Theorem 6.1, one may readily establish the following corollary. Corollary 6.1

The maximum number v of the poles which may be arbitrarily decoupled closed-loop transfer function is

assigned

in the

(6.3) Using Corollary Theorem

6.1, we establish

the following

theorem.

6.2

All poles of the characteristic polynomial arbitrarily assigned together with decoupling rank[Ai...~A~~‘A]

=;+i,$

of the closed-loop if and only if

I- 1

system

may be

rank [Ai: . ..&‘AJ.

Proof : Clearly, from Corollary 6.1 all the poles of the characteristic polynomial of the closed-loop system may be arbitrarily assigned together with decoupling if and only if v = n. Now if we substitute the definition of ci, given in (4.14b), in the relation (6.3) and upon substituting the resulting relation in v = n we derive relation (6.4). n

358

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Linear

Time-invariant

System Decoupling

In the rest of this section we will determine the cancelled out poles of the closedloop system. We start by observing that the poles of the denominator of the general form of the decoupled closed-loop system are the roots of the product polynomial b(s) = II:= ,fli(s). This is true since pi(s) are prime among themselves in the sense that the coefficients of every pi(s) may, in general, be chosen as arbitrary. Clearly, the poles which are cancelled out in every choice of poles of the denominator of the transfer function of the decoupled closed-loop system are the roots of the polynomial p”(s) given by the following relation det (sl-A-BF) P”(S) = -P(s)

;

/j(s) = lfi p(J)

where det [sI -A - BF] is the characteristic polynomial of the closed-loop On the basis of (6.5). we prove the following proposition. Proposition

system.

6.1

The poles which are cancelled out in the general form of the decoupled closedloop transfer function are feedback invariant, i.e. they depend entirely upon the open-loop system and are the roots of the polynomial det [C(sI - A) B”(S) = Proof:

‘B] det @I- A)

;

4s)

Substitute

cc(s) = fJ cc;(s). l= I

(6.6)

for p,(s) in (2.5) to yield

h,(s), given by (5.2), in the expression

(6.7) Introducing (6.7) in Eq. (2.4) and upon applying the determinant sides of the resulting equation we derive the following expression det [I-F(sI-A)

‘B] = [det P,][det G] det [C(sI-A)-

On the basis of (6.8) and the well-known det(sI-A-BF) and (4.15) relation

determinant

operator

‘B] :I:;.

to both

(6.8)

property

= det (I-F(sI-A))‘B)(det

(sI-A)},

(6.5) reduces to

p”(s) = det (B*)) ’ det ~~~ [‘(‘I

-“;(;P

det (“I -A)

= (det (B*)m l)/Qs).

(6.9)

From (6.9), we observe that since det (B*) ’ # 0 the roots of pu(s) and p”(s) are identical. Hence Proposition 2.1 has been proven. W On the basis of Proposition 6.1 and Theorem 6.1 one may prove the following corollary. Corollary

6.2

The problem if the condition Vol. 329. No. 2, pp. 347-369. Prmted I” Great Br~lam

of simultaneous decoupling and stabilization (3.2) holds and the roots of the polynomial 1992

is solvable if and only pU(s) are stable. 359

P. N. Paraskevopoulos

and F, N. Koumhoulis

Example 6.1 Consider the system of Example 3.1. The cancelled system are the roots of the polynomial

out poles of the closed-loop

P”(S) = -(s-l). Hence the closed-loop

VII. Input-output

system cannot

be stabilized.

Decoupling via Pvopovtional

Output Feedback

7.1. Necessary and sufJicient conditions Consider applying to system (2.1) the proportional

output

feedback

law

U(s) = KY(s) + GQ(s). The decoupling

problem

(7.1)

here is to find K and G such that

H(s) = C(sI-A-BKC))‘BG

= diag {hi(s)} ;

h,(s) + 0.

(7.2)

Clearly, this problem is equivalent to the problem of decoupling using the proportional state feedback control law (2.2), where now the feedback matrix F has the special form F = KC. Hence, following the approach presented in Section II, we may readily derive the following output,feedback decoupling design equation

e;fi,= 0,

(7.3)

where ei

=

[+i

i Yi

! (Pz)O(Pi) I . . . (P~IZnl

0

CB

...

-1

0

...

::B

c*AB

0

c:B

...

0

0

. . .

(7.4a)

CA’“-‘B 0 cFA2”B

(7.4b)

$A*“+ ‘B

Cf%

where $i

=

(7.5)

Y,K

and r, d,, (pi), and CT have been defined in (2.5) and (2.11) respectively, r must satisfy condition (2.6). From the m first equations in (7.3), we have

and where

r = P,)C*B = POB*. Substituting (7.6) in (2.6) we derive the following decoupling via output feedback of system (2.1) 360

(7.6) necessary

conditions

Journal

for the

of the Frankhn Pcrgamon

,nst,tute Press plc

Linear detB*

Time-invariant

System Decoupling

(pJO # 0.

# 0,

Now, using (7.6) and (7.7), Eq. (7.3) may be broken follows :

(7.7)

down into two equations

as

yi = (~zhcl*B

(7.8)

T,Q,= a>

(7.9)

where 5, = [3i

Q1

4i

: (4)

1

. . . CAi)2nl

CB

CAB

.

c?B

c,*AB

...

c,+A2”-‘B

(7.10a)

CA2”-‘B

0

c:B

. .. .

q?AZnp *B

0

0

...

c:B

(7.1Ob)

-[c*AB$:A2B;...;c~A2”-‘B],

(7.1Oc)

where $i = (Pi)0 (AJj = (pJj(pi)O’,

‘llri

(7.10d)

j = 1,. . . ,2n.

(7.10e)

Equation (7.9) is a linear non-homogeneous algebraic system of equations. On the basis of this equation, all subsequent results will be derived. Hence the following theorem. Theorem

7.1

The necessary the proportional

and sufficient conditions for the decoupling output feedback law u = Ky + Go, are (i) det B* # 0 (ii) rank

of system

(2.1) via

(7.11) = rank Qi,

V i = 1,. . . , m.

(7.12)

Proof : (Necessity). Condition (7.11) is the same with the first necessary condition appearing in (7.7). Clearly, (7.12) is a necessary condition for Eq. (7.9) to be solvable with regard to &. Proof: (SufJiency). If we let (pJO = 1, then from (7.8) we have yi = c?B. Thus from (7.12) and (7.11) we readily observe that (7.9) is solvable and condition (2.6) has been satisfied. The present results are new and the decoupling conditions derived have the advantage over known results (10, 11, 13) in that they are simple algebraic criteria Vol. 329, No. 2. pp. 347 369, 1992 Printed in Great Britain

361

P. N. Paraskevopoulos

and F. N. Koumboulis

involving the Markov parameters c,B, c,AB, . . . of the open-loop system. Furthermore, the present results facilitate the derivation of the general analytical solution, derived below, for G and K. 7.2. General solution for G and K The general expression for G is derived from (7.6) and (2.5) to be G = (B*)-‘P,‘. The general expression

(7.13)

for K may be derived by solving (7.9). To this end, define c

C*B

C*AB

...

C*A’“-

'B1

. 0 The matrix Q* is invertible.

0

...

C*B

Its inverse has the following

[Q(O)

QU>

...

(7.14)

J form

:

QW-1)l (7.15)

where Q(0) = (C*B)

‘,

Q(j)=

-(C*B)-‘[C*AAi,‘A];

j=

1,2,...,2n-1 (7.16)

and A, = A-B(C*B)

‘C*A

and

A = B(C*B)

‘.

(7.17)

Using (7.17) and postmultiplying (7.9) by (Q*)-‘, Eq. (7.9) reduces following compact form ^?., : ‘Azn] = -C*A[A;A,A;...;A,2”-‘A] ‘I’J+[A,;A2:...;

to the

(7.18)

where use was made of the relation [B;AB;.

..;A*“-‘B](Q*)-’

= [A;A,A;...;Ap-‘A]

as well as of the observation that all rows of [CB i . . . : CA*“- ‘B] are also rows of Q*. The matrices Y, J and A, appearing in (7.18) are defined as

9-c~t].

q?],

Aj =diag{(Lj)j>;

j=

l,...,

2n.

(7.19)

m If we divide (7.18) in columns,

we readily derive the following

set of equations Journal

362

of the Frankhn Pergamon

Institute Press plc

Linear Time-invariant System Decoupling (7.2Oa)

qey = - (JJ4+ ,ey - C*AA$Si

q#di,

C,*AA$S,=O, where the product

q = 0,. . .,2n-1

q # di,

(A&+ 1 = -c*AA&

@eT is the ith column

q=O ,...,

2n-1

(7.20b) (7.20~)

of q’, and where

(7.21)

Equation

(7.20a) may be written

more compactly q =

as

-A-C*AR,

(7.22)

where R = [A$6, !A$&; We now establish

the following

. . . ! A$&],

A = diag {(AJd,+ ,}.

(7.23)

theorem

Theorem 7.2 Assume that system (2.1) satisfies the conditions of Theorem 7.1. Then general analytical expressions of the controller matrices G and K are

the

(7.24)

G = (C*B)-‘P,’ K = - (C*B) - ‘C*AR - (C*B) ~ ‘A. The elements of the diagonal matrix A and of the invertible the only free parameters in (7.24) and (7.25).

(7.25) diagonal

matrix P, are

Proof : Using (7.13) and substituting (7.22) in (7.10d), and upon substituting the resulting relation in (7.5), we derive the general expressions for G and K appearing in (7.24) and (7.25), respectively. H From (7.2Oc), we then derive an alternative expression [first established in (9)] for condition (7.12) given in the following corollary. Corollary 7.1 An alternative

criterion

to (7.12) is

CFAAzSj = 0,

q # d,,

i = l,..

.,m.

(7.26)

7.3. Structure of the closed-loop system Using relation (7.20a) and (7.20b), as well as the procedure in Section V for the derivation of the general analytical expression of the decoupled closed-loop system Vol. 329, No. 2, pp. 347-369, Printed in Great Brmain

1992

363

P. N. Paraskevopoulos and F. N. Koumboulis for the state feedback case, we derive, for the present case, the following for h,(s) hi(s) = (PJO’

1

1 cAijdt+, +

expression

s~,+‘[~-c:A(sI-A~)~‘~~]+c:AA$G~

.

(7.27)

while all other The parameters (pi),, and (,$)d,+, are the only free parameters quantities appearing in (7.27) depend entirely upon the open-loop system. In fact (Ai)d,+ I is the only one arbitrary parameter available to affect the coefficients of the denominator of h,(s) and consequently the poles of the diagonal element of the decoupled closed-loop system (using, for example, the well-known technique of root-locus). Thus the stabilizability of the transfer function of the closed-loop system together with decoupling may readily be checked by applying any of the well-known stability criteria to the denominator of hi(s). With regard to the cancelled out poles in the general form of the closed-loop system, using the same reasoning as in Section VI, we derive the following expression : B”(S) =

det [C(sI - A)

‘B] det (~1 -A)

;

a(s)

E(S) = fi Ei(S) i= I

(7.28)

‘S,],

(7.29)

and E,(S) is defined by the relation

6 6) = &+‘[I -cTA(sI-A,)~

6 6)

where i,(s) and ai are polynomials prime between themselves. For the derivation of (7.28) use was also made of the property that the denominator polynomials of hi(s) are prime among themselves in the sense that they have no common output feedback invariant poles. Example 7.1 Consider the system of Example 2.1. Application of the technique of the present section yields that system (2.1) may be decoupled via output feedback, and the general form of the controller matrices G and K is

G=

I (P2)O’

T - -0~

’

1 [ 3 K=

The general form of the closed-loop H(s) = diag

-[1+&),1 _--~--_~_--~l___ -[1+@,),1 1

.

system is (Pl)O’ ~ s-t@,),

The cancelled-out poles of the closed-loop P”(S) = -(s1).

(P&’ ~ ’ s+(h),

I .

system are the roots of the polynomial

Journal

364

1

of the Franklin Pergamon

Institute Press plc

Linear Time-invariant

System

Decoupling

VZZZ.Conclusions In this paper, a new approach is presented for the decoupling of linear timeinvariant systems via proportional state and output feedback. This approach appears to be very powerful having the following characteristics (68) :

(1) It reduces the solution

of the decoupling problem to that of solving a linear algebraic system of equations, the form of which offers itself for practical applications as well as for analytical and computational manipulations. It unifies the solution of the decoupling problem via state and output feedback. (2) It may be used to solve the decoupling problem for other types of systems, (3) such as singular (3%37), time-delay, time-varying, etc. and (4) It may be extended to solve other design problems, such as triangular block decoupling, exact model matching (69), disturbance rejection, observer design (70), etc. Furthermore, the proposed approach has facilitated lowing results first in the field of decoupling :

the derivation

of the fol-

(1) The determination of the cancelled out poles appearing in the transfer function matrix of the decoupled closed-loop system via state (Proposition 6.2) and output (Section 7.3) feedback. (2) The derivation of the general analytical expressions for the output controller matrices K and G (Theorem 7.2), as well as the structure of the closed-loop system (Section 7.3). These last results may also be derived as a special case, when treating generalized state space systems (37). Overall, the proposed approach gives new insight and unifies the solution of many design problems for different types of systems, while for certain categories of systems (such as the singular systems), it solves several design problems (3537, 69, 70), for which all known techniques appear to fail.

Acknowledgements This work was partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research and Technology, and by the “Herakles” General Cement Company of Greece.

References (1) B. S. Morgan,

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Vol. 329, No. 2, pp. 347-369, Prmted in Great Britain

1992

365

P. N. Paraskevopoulos

and F. N. Koumboulis

(4) E. G. Gilbert, “The decoupling of linear systems by state feedback”, SIAM J. Control, Vol. 7, pp. 50-63, 1969. (5) W. A. Wolovich and P. L. Falb, “On the structure of multivariable systems”, SI,4M J. Control, Vol. 7, pp. 437451, 1969. (6) W. M. Wonham and A. S. Morse, “Decoupling and pole assignment in linear multivariable systems : a geometric approach”, SIAM J. Control, Vol. 8, pp. l-18, 1970. (7) J. Descuse, J. F. Lafay and V. Kucera, “Decoupling by restricted static state feedback : the general case”, IEEE Trans. Aut. Control, Vol. 24, pp. 79-81, 1984. J. Descuse, J. F. Lafay and M. Malabre, “Solution to Morgan’s problem”, IEEE (8) Trans. Aut. Control, Vol. 33, pp. 732-739, 1988. (9) J. W. Howze, “Necessary and sufficient conditions for decoupling using output feedback”, IEEE Trans. Aut. Control, Vol. 18, pp. 4446, 1973. (10) M. J. Denham, “A necessary and sufficient condition for decoupling by output feedback”, IEEE Trans. Aut. Control, Vol. 18, pp. 5355536, 1973. IEEE Trans. Aut. Control, Vol. 20, (11) W. A. Wolovich, “Output feedback decoupling”, pp. 1488149, 1975. (12) M. Bahey Argoun and J. Van de Vegte, “Output feedback decoupling in the frequency domain”. ht. J. Control, Vol. 31, pp. 6655675, 1980. (13) J. Descuse, “A necessary and sufficient condition for decoupling using output feedback”, ht. J. Control, Vol. 31, pp. 833-840, 1980. (14) M. M. Bayoumi and T. L. Duffield, “Output feedback decoupling and pole placement in linear time-invariant systems”, IEEE Trans. Aut. Control, Vol. 22, pp. 142-143, 1977. “Decoupling of linear systems by dynamical (15) J. Hammer and P. P. Khargonekar, output feedback”. Math. Syst. Theory, Vol. 17, pp. 135-l 57, 1984. by dynamic output feedback”, (16) A. B. Ozguler and V. Eldem, “On diagonalization Kybernetica, Vol. 25, pp. 35-40, 1989. problem by constant (17) V. Eldem and A. B. Ozguler “A solution to the diagonalization precompensator and dynamic output feedback”, IEEE Trans. Aut. Control, Vol. 34, pp. 1061-1067, 1989. A. I. G. Vardulakis, “Internal stabilization and decoupling in linear multivariable (18) systems by unity output feedback compensation”, IEEE Trans. Aut. Control, Vol. 32, pp. 7355739, 1987. of linear multivariable systems by unity output (19) A. I. G. Vardulakis, “Decoupling feedback compensation”, ht. J. Control, Vol. 50, pp. 107991088, 1989. stability margin optimization with (20) M. G. Safonov and B. S. Chen, “Multivariable Proc. IEE, Vol. 129, part D, No. 6, pp. 276 decoupling and output regulation”, 282, 1982. and pole assignment by dynamic (21) W. M. Wonham and A. S. Morse, “Decoupling compensation”, SIAM J. Control, Vol. 8, pp. 317-337, 1970. for decoupling and (22) H. R. Sirisena and S. S. Choi, “Minimal order compensators arbitrary pole placement in linear multivariable systems”, ht. J. Control, Vol. 25, pp. 7555767, 1977. (231 H. Hikita, “Block decoupling and arbitrary pole assignment for a linear right-invertible ht. J. Control, Vol. 45, pp. 1641-1653, 1987. system by dynamic compensation”, of state feedback decoupling (24) S. M. Sato and P. V. Lopresti, “On the generalization theory”, IEEE Trans. Aut. Control, Vol. 16, pp. 1333139, 1971. decoupling theory”, (25) S. M. Sato and P. V. Lopresti, “New results in multivariable Automatica, Vol. 7, pp. 4999508, 1971. (26) T. W. C. Williams and P. 1. Antsaklis, “A unifying approach to the decoupling of linear multivariable systems”, ht. J. Control, Vol. 44, pp. 181-201, 1986.

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and S. G. Tzafestas, “Group decoupling theory for a gener(27) P. N. Paraskevopoulos alized multivariable control system”, ht. J. Systems Sci., Vol. 6, pp. 239-248, 1975. decoupling of linear multivariable (28) A. S. Morse and W. M. Wonham, “Triangular systems”, IEEE Trans. Aut. Control, Vol. 15, pp. 447449, 1970. (29) C. Comault and J. M. Dion, “Transfer matrix approach to the triangular block decoupling problem”, Automatica, Vol. 19, pp. 533-542, 1983. E. Yore, “Optimal decoupling control”, Proc. JACC, pp. 327-336, 1968. (30) (31) G. Hirzinger, “Decoupling multivariable systems by optimal control techniques”, Znt. J. Control, Vol. 22, pp. 157-168, 1975. and A. J. Van der Schaft, “Direct approach to almost (32) R. Marino, W. Respondek disturbance and almost input-output decoupling”, Znt. J. Control, Vol. 48, pp. 353383, 1988. and P. N. Paraskevopoulos, “Decoupling and pole-zero place(33) M. A. Christodoulou ment in singular systems”, 9th World Congress ZFAC, Vol. 1, pp. 245-250, Budapest, 1984. of generalized (34) M. A. Shayman and Z. Zhou, “Feedback control and classification linear systems”, IEEE Trans. Aut. Control, Vol. 32, pp. 483494, 1982. “The decoupling of generalized state and F. N. Koumboulis, (35) P. N. Paraskevopoulos space systems via state feedback”, IEEE Trans. Aut. Control, to appear. “Decoupling and pole assignment in and F. N. Koumboulis, (36) P. N. Paraskevopoulos generalized state space systems”, Proc. IEE, part-D, accepted for publication. and F. N. Koumboulis, “Output feedback decoupling of (37) P. N. Paraskevopoulos generalized state space systems”, Automatica, accepted for publication. “On the decoupling of multivariable (38) S. G. Tzafestas and P. N. Paraskevopoulos, ht. J. Control, Vol. 17, pp. 407415, 1973. control systems with time-delays”, of differential-difference systems (39) R. D. Jacubow and M. M. Bayoumi, “Decoupling using state feedback”, ht. J. Systems Sci., Vol. 8, pp. 587-599, 1977. (40) Z. V. Rekasius and R. L. Milcrarek, “Decoupling without prediction of systems with delays”, JACC, pt II, pp. 147&1475, 1977. and arbitrary coefficient assignment in time delay systems”, (41) M. Kono, “Decoupling Syst. Control Lett., Vol. 3, pp. 349-354, 1983. and coefficient assignment for (A, B, C, D) time-delay (42) M. L. Liu, “Decoupling systems”, ht. J. Control, Vol. 50, pp. 1089-I 101, 1989. of and inverses for time-varying linear systems”, IEEE (43) W. A. Porter, “Decoupling Trans. Aut. Control, Vol. 14, pp. 3788380, 1969. multivariable systems by decoupling and by (44) E. Freund, “Design of time-variable inverse”, IEEE Trans. Aut. Control, Vol. 15, pp. 1833185, 1970. and S. G. Tzafestas, “On feedback decoupling of linear time(45) P. N. Paraskevopoulos varying, time-delay control systems”, ht. J. Systems Sci., Vol. 4, pp. 167-178, 1973. “On the decoupling of linear multivariable (46) A. K. Majumdar and A. K. Ghoudhoury, systems”, ht. J. Control, Vol. 50, pp. 1089-l 101, 1989. of a class of nonlinear systems”, IEEE (47) S. Nazar and Z. V. Rekasius, “Decoupling Trans. Aut. Control, Vol. 16, pp. 257-260, 1971. de systemes non lineaires series generatrices non com(48) D. Claude, “Decouplage mutatives et algebres de Lie”, SIAM J. Control, Vol. 24, pp. 5622578, 1986. and S. Monaco, “Nonlinear decoupling via (49) A. Isidori, A. J. Krener, C. Gori-Giori feedback : a differential geometric approach”, IEEE Trans. Aut. Control, Vol. 26, pp. 331-345, 1981. of nonlinear systems”, Syst. Control Lett., Vol. 1, pp. 2422 (50) D. Claude, “Decoupling 248, 1982. Voi 329, No 2. pp. 341-369, Printed in Great Entam

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368

Journal of the Franklin Institute Pergamon Press plc

Linear Time-invariant

System

Decoupling

Appendix In this Appendix we will derive the general establish the following Lemma.

solution

of Eq. (4.27). To this end we first

Lemma Al Equation ~,[L,~NT~6,~...~A~-‘6,]

= -pi[OjO;I,J

(Al)

is always solvable with regard to I& for every (p,),. Prooj’ : As is known, Eq. (A 1) is solvable with regard to 4, for every (p,), iff rank

L, ) NT I si...&-‘s, o ; -o- I _ _ i _ _ ~ =rank[Li/NT!6i...A>P’6,]. 0, [, 1

From the form of the matrix on the left-hand

(A2)

side of (A2), it may also be written as

rank[L,~NT]+~i=rank[L,~NT~6,...A>P16,]

(A3)

which is an identity since the columns of NT are orthogonal to the columns of L and the column vectors S;, . , A>- ‘6, are linearly independent from the columns of L,. W Since (Al) is solvable and rank [Li /NT ! 6, &- ‘S,] = n the matrix M, defined in (4.14a) is invertible, the unique solution of (Al) is

ijc = -p,[iST..

&TAT)“,-‘]“M, ’ = -

c (p,),6:‘(A:)“y- 1

‘M,- ‘_

644)

Clearly, since (A4) is a solution of (Al), then it is also a special solution of (4.27). Then the general solution of (4.27) is the sum of the solution for @; of the respective homogeneous equation (cji[Li i 6,. . A?- ‘S,] = 0) and the special solution (A4). This sum is given in (4.28).

Vol. 329, No. 2. pp. 347-369. 1992 Printed m Great BrCain

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