Disturbance rejection with simultaneous linear exact model matching for a class of nonlinear systems

J. Franklin hilt. Vol. 335B. No. 7, pp. 1299 1325, 1998

Pergamon

Pll: S0016-0032(97)00074-4

~ 1998The Franklin Institute Published by Elscvicr Sci. . . . Ltd Printed in Grcat Britain 0016 003298 $19.00 + 0.00

Disturbance Rejection with Simultaneous Linear Exact Model Matchingfor a Class of Nonlinear Systems b V A. S. T S I R I K O S *

and

P. N. P A R A S K E V O P O U L O S

Department o f Electrical and Computer Engineering, Division oJ'Computer Science, National Technical Unirersitv o [ A thens, 15773 Zographou, Athens, Greece (Received in f i n a l j b r m 6 M a y 1997; accepted 23 July 1997)

ABSTRACT : hi this paper, the comhhwd desiyn problem of disturbance r@,ction and ollhwar exact model matching ria static state or static state and measurement .l~'edhack is considered. A new approach is presented which solves the problem jor a class of q[fine nonsquare nonlinear systems. The proposed approach reduces the probh,m ofdetermining the leedhack control law, with or without disturbance measurements, to that Ofsolrmg a system o[Jirst-order partial d(,(lk'rential equations. Based on th is s3'stem of equa t ions, algebraic (i.e. easi0 ' checkable) necessap3, and sl({'ficient conditions ./or the ~k',s'i.qn prohh, m to hare a solution are established. Furthermore, sobbing this system (71 equations, the general anah'tical expression.lbr the fi, edhack control law is explicitly determined.

!'; 1998 The Franklin Institute. Published by Elsevier Science Ltd Keywords: nonlinear control systems: disturbance rejection: model matching: linearization.

L Introduction

This paper deals with the problem of disturbance rejection with simultaneous linear exact model matching via static state ( D R L E M M ) or via static state and measurement feedback ( D R L E M M M F) for a class of nonlinear systems. Hence, the contribution of the paper is two-fold: it solves the problem of disturbance rejection (DR) and it produces a closed-loop system with linear input/output (i/o) description. Each one of these problems is of great theoretical and practical importance and has been extensively but separately studied in the literature. The motivation for considering the combined problem is that it is desirable for the closed-loop system to be disturbance decoupled and furthermore to have a linear i/o description, thus facilitating the application of known linear techniques to improve further the performance of the closed-loop system. The D R problem of nonlinear systems via static state feedback is studied in (1-8),

*Tel: + 30-1-7722501; fax: + 30-1-7722459: e-mail: tsirikos(a contro ece ntua gr and parask(a control.ece.ntua.gr 1299

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A. S. Tsirikos and P. N. Parasket~opoulos

where differential geometric tools have been used. For the case of affine nonlinear systems, necessary and sufficient conditions in terms of controlled invariant distributions are derived (I) and (2). In the case where the disturbances enter in a nonlinear manner in state space equations, necessary and sufficient conditions in a differential geometric context are reported in (3). For affine nonlinear systems, the construction of the control law is reduced to that of determining certain controlled invariant distributions. The determination of the appropriate distributions (for details, see (4)) is usually a very difficult task, except for the class of square systems (in which the number of inputs equals the number of outputs), as is pointed out in (8) and (9). Furthermore, explicit expression of the appropriate state feedback, called 'standard noninteracting feedback' (1, 8), is available only for square and i/o decouplable systems. The DR problem with stability has been studied for S|SO in (7) and for square and i/o decouplable systems in (5, 6). It is remarked that the standard noninteracting feedback leads to a closed-loop system with linear i/o description (i, 5, 8). The main disadvantage of these approaches is that, in general, they do not lead to a closed-loop system with linear i/o description of the largest dimension. Dynamic versions of the DR problem are reported in (9, 10). In (9), the case of square systems is considered. The approach is based on Singh's inversion algorithm yielding left-invertible (regular) controllers. In (10) nonsquare systems are considered and nonregular controllers are allowed. The construction of the feedback is given in terms of a constructive algorithm based on a modified version of Singh's inversion algorithm and of zero dynamics algorithm (8). The linear exact model matching (LEM M) problem is part of a more general problem, known in the literature as the linearization problem and it was first treated in (11, 12). The linearization problem may be separated into two major problems. First there is the state linearization problem where the objective is to determine a control law (static or dynamic) and a state transformation which, when applied to a nonlinear system, modify its input-state representation to the input-state representation of a linear system (8, 13--18). Secondly, there is the input-output linearization problem where the objective is to determine a control law which, when applied to a nonlinear system, results in a closed-loop system with a linear i/o description (8, 19-25). The closed-loop system may either be any arbitrary linear system (19-24) or have the same i/o description with a pre-specified linear model (25). This last case constitutes the L E M M problem. The LEMM problem via dynamic state and static state or output feedback is studied in (25). This paper is first devoted to the D R L E M M problem using static state feedback. The D R L E M M problem is solved for a class of nonsquare nonlinear systems. Many practical systems belong to the class considered in the paper and therefore it is a class of nonlinear systems often studied in the literature. The proposed approach has the advantage that it reduces the problem of determining the desired control law to that of solving a nonhomogeneous system of first-order partial differential equations. This system of equations is called the D R L E M M design equations and plays an important role in our approach. Based on the D R L E M M design equations, algebraic (i.e. easily checkable) necessary and sufficient conditions for the problem to have a solution are established. Furthermore, solving the D R L E M M design equations, the general form

Simultaneous Linear Exact Model Matchin,q

1301

of the desired control law is explicitly determined. In addition, the necessary and sufficient conditions derived may be used to give guidelines for the selection of the admissible linear models which satisfy the D R L E M M problem. Using the proposed approach one can, in general, produce an i/o linear subsystem of larger dimension than the dimension of the i/o linear subsystem produced by the standard noninteracting feedback (1, 5, 8). Furthermore, the standard noninteracting feedback is derived here as a special case. The paper also considers the D R L E M M M F problem, for the case where disturbance measurements are available. The proposed approach is successfully extended to solve this problem using the so-called measurement feedback (feedback of the disturbance measurements) (8). It is noted that the use of measurement feedback may facilitate the solution of the D R L E M M problem, since the necessary and sufficient conditions for the solvability of D R L E M M problem are now weakened. The respective results for the cases of linear time-invariant (26) and time-varying systems (27) are derived as special cases. Similar results for simultaneous disturbance rejection and decoupling of linear time-varying analytic systems and of nonlinear systems may be found in (28, 29) and (30), respectively. The present results form part of the material reported in (31).

II. Preliminaries Consider the nonlinear system described in state space as = g0(x) + G(x)u + D(x)~, x(0) = x0, y = h(x),

where the input u e ~'" and the disturbance ~ e R ~ are piecewise analytic vector valued functions, the output y e Rp, the state x evolves on an open subset Z of R", go and each column of G and D, denoted by gi and di, respectively, are analytic vector valued functions on Z and h is an analytic mapping from ~ to ~P. This paper is concerned only with this kind of system. The nonlinear system under consideration may be equivalently written as (S): = Eo(x) + E(x

, x(0) = x0,

y = h(x),

where E0(x): = g0(x)

and

E(x): = [G(x)

D(x)].

Certain definitions are introduced which will be useful later in the material that follows (8, 32).

Definition I (lnvolutive distribution) A distribution A is called involutive if, for any T,a ~ A it holds that [T,~] ~ A, where locally

A. S. Tsirikos and P. N. Paraskevopoulos

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[t,~l(x) =~ ~ (x)t(x)- ~x (xm(x). Definition II (lnvolutive closure) Consider a distribution A and the minimal element of the family of all involutive distributions containing A. This minimal element, which is the smallest involutive distribution containing A, is called the involutive closure of A. In this paper, inv (Q(x)) denotes the involutive closure of the distribution spanned by the columns of the matrix Q(x). Furthermore, it can be proven that if the distribution spanned by the columns of Q(x) is nonsingular for all x E ~ , then the involutive distribution inv (Q(x)) is spanning locally by vector fields of the form Xi...... ,,(x) = [qi~(x),X,~ ...... i~(x)], for k > 1, where Xi,(x) & qi,(x), for i~..... i k e { 1,2 ..... n}, where q,(x) ..... q,,(x) are the columns of Q(x). Let J,,& {1,2 ..... 0}, where 0 is a positive integer, and ,~., ~{0,1 ..... oo}. The inputoutput map of a system represented by (S) is given by the Fliess functional expansion

(s) y(×o,u,~,r) = h(xo)+

L~. . . . L
~ k =

0 i 0 .....

ih

=

(1

"Ill

dc~i~.., da,,,, ~ ()

where L,h(x)&0?h(x)/~x)z(x) and the iterated integral 51~dai~... dai,, is given by the recursive relation a o ( t ) = t and [" I/,ui(z) dr, o-i(/) = <{.j'~,~i ,,,(v)dz,

for i~ {1 . . . . . m}, foriE{m+l

.re+g, s

and 5g,d % . . . da,,, = 5/,d%(t)5~d% , . . . daio, for ikc {0A,,,+;} and k e , 0 , .

Definition III For system (S), there exist nonnegative integers e/i, for each it,~,, defined by

LL/L~:Jli(x) = 0, for a l l j e 5,,+;, and k < d,, L,:;L)~,IT,(x)# O, for s o m e / e ~,,,+:. for all x belonging to a neighborhood Z0 of x~, where hi(x) is the ith c o m p o n e n t of h(x). The d, are called characteristic numbers. The definition of the characteristic numbers for linear systems is similar to Definition III for nonlinear systems.

IlL Statement o f the D R L E M M Problem Consider the linear time-invariant system (M): = AMZ + BMU, z(O) = Z,,, yM = CMz,

where z ~ N",,, w e N'", YME ~P and AM, B M and CM are matrices of appropriate dimensions with real entries. Clearly, the output YM of (M) is not affected by any disturbance.

Shnultaneous Linear Exact Model Matching

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Definition I V The output y of the nonlinear system (S) matches the output YM of the linear model (M) if and only if y(xo,u,~,t)- yu(zo,u,t) is independent ofu and ~

(3.1)

m the neighborhood of [xo,z0}, for any input u and disturbance ~. Using the Fliess functional expansion, (3.1) is equivalent to the relations

LELL, Lt- ... LF hi(x) = (Ci)MA~(B/)M, for jl = j2 . . . . .

.jl, = O, je J .... (3.2a)

LELLi, LL-... L~.:hi(x) = 0, with at least one ofj~,j2 ...... j ~ {0,Sm+;}, different than z e r o j e 5 .... (3.2b)

Lc, Lt, Lt, .

. h i ( x ). L~,,

=

.0, for.jl,.j2 . . . . .

Jk e ~0,~,,,+ I , i ~ { m + 1 . . . . . m + ~ } ,

(3.2c)

for all x e ~ °, k e {0,4~} and ie,~p, where (ei)M denotes the ith row of C and (Bj) M denotes t h e j t h column of BM. Next, consider the analytic static state feedback control law of the form (CL): u = a(x) + B ( x ) w ,

where we ~'". The control law (CL) is assumed to be nonsingular, i.e. IB(x)[ # 0 , for all x ~ Z0. Applying (CL) to the nonlinear system (S) yields a closed-loop system of the form (CLS): = f;0(x) + E" ( x ) [Wl, x ( 0 ) = x 0 , y = h(x),

where I~o(x): = E0(x) + G ( x ) a ( x ) : = g0(x) + G ( x ) a ( x ) ,

(3.3a)

and ~;(x): = [G(x)

D(×)]: = [G(x)

D(x)]

.

(3.3b)

The D R L E M M problem may be stated as follows: Determine the control law of the form (CL) for which the output of(CLS) matches the output of the linear time-invariant model (M). IV. Solution of the DRLEMM Problem

4.1. Derivation o['the D R L E M M design equations Applying Definition IV to (CLS) and grouping appropriately (3.2), the D R L E M M problem reduces to the solution of the set of equations d(L~,h~(x))E : [(Ci)MA~B)M 0],

(4.1a)

A. S. Tsirikos and P. N. Paraskevopoulos

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d(L~-LL.j... LL-,Jti(x))~; = 0,

(4.1b)

for all x~ ~0, J~,.J2......i~<~~0J,,,+;} with at least one of ./~,¢0, k~ 10,J., } and i ~ , , for the pair la(x),B(x)). Since the right-hand side of (4.1a) is a matrix with real entries it is easy to see that if (4.1a) is satisfied then (4.1b) will also be satisfied. Hence, the unknown pair {a(x),B(x)} is to be determined only by (4.1a). Let (d,)~s), (di)~CLS~and (d,)~m~, ie~p, be the characteristic numbers of the original system (S), of the closed loop system (CLS) and of the model (M), respectively. From equation (4.1a) we have that a necessary condition for the D R L E M M problem to have a solution is that (di)~cLs~=(dj)(m~, for all ieJp and x e Z ~, where use was made of Definition Ill for the characteristic numbers. Since the desired control law (CL) is assumed nonsingular and the characteristic numbers are invariant under nonsingular feedback (8), we have that (d,)(s~= (d,)~cus~, for all ie,Dp. Therefore, the above necessary condition takes on the form

di=(di)~s)=(di)~M), for all ie,~/,, a n d x e X °.

(4.2)

Using (4.2), relation (4.1a) may be written in a matrix form, as is proven in the following lemma.

Lemma I Under the assumption (4.2), the infinite set of equations (4. la) may be written in the matrix form

E*(x)+ 2M~d(L~. 2' '4~)F.(×)=[M~. i=o

01

"

0

'

fork,S,,,

,,

where 4~(×): =[B(×)] la(×) and

(Cl)MAM

t~l,~h+./ M/:

=

:

#~,.]/l,+/

E,*(x):=

E(I):= ,/

(/~,/+'/,,,(x))J

:=

BM,

:

d,,+/ k(ep)mA~ d i

/[6(x)

D (x)I,

L,/ (L~:;+%(x))J

for j > 0. Proq[! In (22) it has been proven that the infinite set of equations (4. l a) is equivalent to the first 2n equations. Hence, (4.1a) is equivalent to the equations

d(L~,h;)E(x)=[(c;)mA~Bm ~+I,

d(Ls, ~ h,)E(x)=[(c~)mA~+kBm -

0] 0],

(4.3a) (4.3b)

Simultaneous Linear Exact Model Matchin9

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for i ~ r and ke~_,,_~, where use was made of Definition III for the characteristic numbers di and of (4.2). Upon substituting the definitions (3.3) for F~o(x) and l£(x) in (4.3), we get /

B(x)

d(L~'.h,)E(X)I o ~,+~ k , h,)E(x)+/=~/~,.,,,+/_/(L~./o

d(L/.

Ol]=[#i.,,'

O],

'qb)E(x)=[l'"'"+~

(4.4a) [-B ' ( x ) 0 ] 0] L 0 lJ'

(4.4b)

for ie 3r and k e ~2,, ~, where use was made of the invertibility of B(x) and the definitions of M, and ¢p(x). Finally, using the definitions of E*(x), equations (4.4) may be written in the matrix form given by the present lemma. • The equations of this lemma may further be written in a compact matrix form as MA(x) = - E*(x),

(4.5)

where [Mo M&

O]

[M,

O]

...

[M2,, ,

O]

0

[Mo

O]

.-.

[M2,,_2

O]

0

0

"'"

[Mo

--ILO ~1 ILll

E*(x)~

O]

'" IL2~I]

E*(x)

E*(x)

..

E*, ,(x)

0

E*(x)

..

E*,, 2(x)

:

:

".

:

o

o

..

E*(x)

L 0 ( x ) ~ B '(x), Lk(x)&d(L~,/'~b)E(x),

(4.6)

for k e ~z,, 1, where * stands for some matrix of appropriate dimensions. Now, let ,q= rank B*(x). In what follows we assume that the nonlinear system (S) satisfies the property fl=m, for all x s X0 Note that we are looking for a solution A(x)

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A. S. Tsirikos and P. N. Paraskevopoulos

of (4.5) with A(x) an invertible matrix, for all x E Z °, because B(x) is assumed to be invertible for all x ~ Z °. If this is the case, then, the following lemma, regarding the solvability of (4.5), may be proven. Lemma II

Under the assumption that /~=m, for all x ~ Z ~j, the necessary and sufficient conditions for (4.5) to have a solution for L~(x), where i6 t~0,• 2,, ~,~ with [L0(x)[ 4:0, are D*(x)=0,

(4.7a)

2nm =- rank[M] = rank[E*(x)] = rank[E*(x)

M],

(4.7b)

for all x ~ Z °. Proq[! We first prove the necessity. Assume that there exists a solution of (4.5) for A(x) in the form defined by (4.6). Then, by construction of the matrix M and using the definition of E~*(x), it is obvious that (4.7a) will be satisfied. In order to prove that (4.7b) holds we make use of the Kronecker-Capelli Theorem from which it follows that rank[M]=rank[M

E*(x)].

(4.8)

Since A(x) is invertible, for all x c ~ ~', eqn (4.5) is equivalent to M = - E * ( x ) A Applying the Kronecke~Capelli Theorem to the above equation, we obtain

J(x).

rank[E*(x)]=rank[E*(x)

(4.9)

M],

for all x e Z °. Combining (4.8) with (4.9) we arrive at the tbllowing rank condition rank[M]= rank[E*(x)] = rank[E*(x)

M],

(4.10)

for all x c ~). By the construction of M and E*(x) we have that rank [M] < 2nm and rank [E*(x)] > 2nm, for all x ~ ~ 0 where use was made of the assumption [:t= m. Using these rank inequalities and (4.10) it is easy to see that (4.7b) is satisfied. We now prove sufficiency. By assumption we have that B*(x) is a full column rank matrix, for all x ~ ~o. Therefore, there exists an invertible matrix V0 defined by

v,, LK,,(x)5' where P~ is a real matrix and K~)(x) is a matrix valued function on ~ , which performs to the matrix E*o(x) the row permutation VoE*(x) = V o [ B * ( x ) O ] = I

P°B*(x) LK,,(x)B*(x)

O [P°B0*(x) O] =

:],

(4.11)

where PoB*(x) is an invertible m x m matrix, for all x 6 ~o. To derive (4.11) use was made of (4.7a). Assume that condition (4.7b) holds. Then, there exists a matrix N(x) which satisfies the equation

Simultaneous Linear Exact Model Matching

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M = - E*(x)N(x).

(4.12)

Substituting (4.7a) into (4.12) and using definition (4.6) for the matrices M and E*(x), we obtain that N(x) satisfies the equation [0

0

-..

[Mo

011:-[0

0

..-

[B~(x)

0]]N(x).

Premultiplying this equation by Ko and using (4.1 1) we have Ko(x)Mo = 0.

(4.13)

Therefore. premultiplying Mo by Vo and using the above relation, we have

F PoMo q VoMo = LKo(x)Mo]-- [P°0M°I.

(4.14)

But from (4.7b), we have that Mo is a full column rank matrix and therefore m = rank [VoMo]. Combining this rank condition with (4.14) we conclude that PoMo is invertible. Furthermore, from (4.13) we obtain that the matrix Ko(x) may be selected to be a real matrix• If condition (4.7a) holds, then (4.5) reduces to the equation ~lA(x) = - E*(x),

(4.15)

where

M,, M,

I L

1

0

...

By the assumption that (4.7b) holds, eqn (4.15) has a solution for A(×). Furthermore, this solution is unique, since I~ is a full column rank matrix, and it is given by -[Lo A(x)&

0

0]

LI -[L,,

•

.

0

0

L2,

""

0]

.. .

z

L2,, 2

.

'

"'

-

-

[Lo

,

0]

where Lo(x) = [PoMo]-'Po{ B*(x)], L,(x)=[PoM,,] 'Po{M,[L,,(x) Lk(x)=[PoMo] ~P,,[M~[L,,(x)

(4.17a)

0]-E*(x)}.

(4.17b)

0]--E~(x)--~=6:M/+,L,,. ,_,(x)].

(4.17c)

for 2 _~o as is desired.

•

A. S. Tsirikos and P. N. Paraskevopoulos

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The conditions (4.7) imply that rank~[B*(x)] = rank~[B*(x)], for all x ~ Z~', where the notion o f ranks[.] is the one defined in (22). N o t e also that in this case the nonlinear system is left-invertible. Next, using the solution of (4.5) determined in the process of proving the sufficient part of L e m m a II, we can derive the equations that the u n k n o w n matrices q~(x) and B(x) must satisfy. T o this end, we will first prove the following lemma.

Lemma III The following equations hold

cl(L~,~,,'

k - 1 l )L~. J '{d(ek)ad~,>E(x)}, 4,)E(x)= *y' ,~,(- 1)/~k-./-

(4.1 8)

for k > 1, where ad~¢a(x) A=[v,ad~ I~r](x), k = 1,2 ..... ad°a(x)&~r(x) and use is m a d e of the binomial coefficient defined by

(;): r, s!(r - s)."

Proof! We will use the perfect induction method. T o this end, apply (4.18) for k = 1 and k = 2 to yield d(~b)E(x) = d(~b)ad°E(x), f o r k = 1,

d(L~:,e~)E(x) = - d(4))adE, E(x) + L,:. [d(c~)ad~::,E(x)}, for k = 2. Using the definition of ad~:<,(), where j = 0 , 1 ..... one can easily see that the above equations hold. We proceed by assuming that (4.18) holds for k = Q, i.e.

d(Lj~,, '~b)E(x)= ~, ( - 1) j j= c,

e . 1 L~.' '[d(qb)ad~.,E(x)l. \~ -.1- 1

(4.19)

We will prove that (4.18) holds for k = ~ + 1. To this end, observe that the relation

d(L~:Sp)E(x) = -- d(L~:,, l~b)(ddE, E ( x ) ~- t,~;,,{d(t~:, ' ~b)(./(/~:.oE(x)~ • holds, where use was m a d e of the definition of ad~:,~(). Substituting (4.19) into the above equation yields

(4.20) The first term of the right-hand side of (4.20) m a y be further written as

- d(/~; ' 4,)aJ,~,,E(,,) = d(L~,, ~4,)ad~.E(x)-- L,~,,{~/(L~- 2,~)ad~, E(x)} = d(LeE,, 3~)ad~,E(x) + 2LE,.{d(L~,~, 3~b)ad~.. E(x)} - L~.[d(L~, 3dp)adE,>E(x)}. Continuing the above procedure we finally arrive at

Simultaneous Linear Exact Model Matching -

~'

( z-,, q~)ad;;,E(x)

1309

Q-1

=

\~-.I/

Substituting the above equation into (4.20) yields 0 <;(L~<,,.)E(x)=(-l)'(~-ol \ )L~,,l<;(.)..;~,,E/x)i

+~(-~),!\ ~ - . 1 /.IL~,,'~d(+)a,;;:,,E(,,)~+ E,(-~)'( ~_1 )L~<;'I<;(O)~';~:"E(~)~ 1 =

\~-,1-

The following relations between the binomial coefficients will be useful and may be easily proven

Using these in (4.21) yields

Hence, (4.18) holds for k = e + l, which completes the proof of the lemma. • Now, using Lemma llI and the solution of (4.5) given by (4.17), we can derive the equations that the unknown pair [4~(x),B(x)} must satisfy. These equations are given by the following theorem.

Theorem I Under the assumption that [B(x)j V:0, that (d;)~s~= (d;)~M~, for i~$p, that [4=m, for all x ~ ~o and the conditions (4.7) hold, the unknown pair {~b(x),B(x)} of (CL) will be the solution of the following set of equations, which will be called the D R L E M M design equations, B

I(x) = L,,(x),

d(q$)[Q(x)] = [Z(x)]. Here Q(x):=[E(x)

adE,E(x)

."

ad~' 2E(x)],

Z2(x)

""

Z_~,, ,(x)],

and Z(x)::[Z~(x) where

ad$:. E(x): =[ad~:,El(x)

.."

ad~:.E,,,+c(x)]

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A. S. Tsirikos and P. N. Paraskevopoulos

and Z/(x), forj~,D2,, ~, are given in terms o f the first 217- 1 nonzero M a r k o v parameters Mj o f the linear model by the recursive relations Z~(x): = Lt(x) and

for 2 <_k < 2 n - 1. Pro(?/! The first equation is just the definition o f L0(x). To prove the second equation, c o m b i n e the definition in (4.6) for ~40E(x) with (4.18) and next solve the resulting equation for d(d?)ad~£ ~E(x) to yield d(gp)ad~.il ' E(x) = Z~(x), where use was made o f the definition o f Zk(x), k ~ ,~z,,-,- Finally, grouping appropriately the above equations, one can prove that the second equation of the present theorem will be satisfied. •

d(L~:~,

4.2. Necessary attd su[licient conditions In order to determine B(x) and a(x) one has to solve the D R L E M M design equations for L0(x) and 4~(x). Once Lo(x) and ~p(x) are determined, B(x) and a(x) m a y be c o m p u t e d by the relations B(x) = L~; I(X),

(4.22a)

a(x): = B(x)q~(x) = L0 ~(x)4~(x).

(4.22b)

Hence, based on the necessary and sufficient conditions for the solvability o f the D R L E M M design equations for Lo(x) and q~(x), we can derive the necessary and sufficient conditions for the solvability of the D R LEM M problem. To this end, we first establish the necessary and sufficient conditions for the solvability o f the D R L E M M design equations for ~b(x) by proving the following lemma.

Lemma I V The first-order partial differential equation d(4~)[Q(x)] = [Z(x)],

(4.23)

where q~(x) is an m-dimensional vector valued function on ~ , Q(x) is an m × r matrix valued function on ~ and Z(x) is a matrix valued function on Z of appropriate dimensions, has a solution for ~b(x), for all x e Zo, if and only if dim.{inv[Q(x)]] = d i m ~ i n v F Q ( x ) l ; , for all x e 3<°.

kZ(x)J) Pro@ We prove tirsl the necessity. Assume that there exists a solution of (4.23) for ¢p(x). Next, consider the vector lields X~,,...4(x) and -~- .. ....~(x) detined by X i. . . . . . / 1( x ) : [qia,Xi ....... i,], where

S//(X ) = qi, (X),

":,. . . . ,(x) = Lq,(~i ........ ,) -- Lx ........ (zi~), where ~,,(x) = z,,(x),

(4.24a) (4.24b)

for ij ..... ix e ~l ..... r~~ and k = 1,2 ..... where q~ ..... q, are the colurnns of Q(x) and z~..... z,.(x) are the colurnns of Z(x). It can be easily proven that locally inv[Q(x)] = span[X,,,X,.>,,, .... X,...... 4,-- J"

(4.25a)

Simultaneous Lineal" Exact Model Matchinq

1311

LZ(x)j=spankL~--,,J'L'~,>,,J "'" L=i ...... i,~ for il ..... ix ~ I 1..... r I and k = 1,2 ..... Since q~(x) is a solution o f (4.23), we obtain d(4~)[Xi,(x)] = [~i,(x)], for i, 6 [1 ..... r}.

(4.26)

Taking the Lie derivative o f 4~(x) along the vector field Xi...... i,, one has d(~)X~., j,(x) =-%~..j,(x),

(4.27)

for i~..... i~ ~ I I ..... r} and k = 1,2 ..... where use was made o f definitions (4.24) and eqn (4.23). But (4.27) constitute a set o f linear equations for d(#~) and hence, it must hold (Kronecker C a p p e l i Theorem) that dim[span{X~,X, ,,, .... X,. . . . . , . . . } } = d i m ~ s p a n ~ | 2

,,..

k_

/,|

k.L~"tlJ

- |,..

L~i2,ilJ

"'

,. . . .

"" ~i~,...,i I

for i~..... ia c ~1..... r} and k = 1,2 ..... Finally, combining (4.25) with the above dimension condition, it is easy to see that the condition o f the present lemma holds. We now prove the sufficiency. Assume that

dim{inv[Q(x]} : d i m t ~ i n v [ Q ( x ) l ~ = Q , f o r a l l x e ~o.

{

LZ(x)d)

(4.28)

The p r o o f o f the sufficiency is constructive, i.e. it will be shown that if the above condition holds, then a solution for equation (4.23) can always be constructed. To this end. denote by q~ ..... q~ the linearly independent columns o f Q(x). Next, based on q~ ..... q,, construct a local basis o f inv [Q(x)], a r o u n d x0. Denote the elements o f this basis by the set o f vector fields ql ..... q ...... q,, (obviously Q > = a), i.e. inv[Q(x)] = span {q, ..... q,,}.

(4.29)

Next, consider the basis o f the distribution spanned locally a r o u n d x0 by the columns of

[Q, q Z(x)J This basis is the set o f vectors

[::] .....[::] Using the definition o f the involutive closure o f a distribution and condition (4.28), it is easy to see that a local basis o f the distribution inv[Q(x)l

LZIx)J can be obtained, so that

1312

A. S. Tsirikos and P. N. Paraskevopoulos

LZ(x)J

(LZ,j

"

71} z,,

,4,0,

"

Now, define the augmented state vector ~ by X[ =

[ =

x2

]=

[XI ......

\'H

-~',, + l . . . . . .

[1, + ,,t]

,

x2

where x,,+~ ...... %+,,, are additional states independent of.v~ ...... %. Based oll the distributions defined in (4.29) and (4.30), a solution o f (4.23) will be constructed. To this end, consider the first-order partial differential equations i)Q [q,

""

q~'] = O.

(4.31)

In order to construct the solutions o f (4.31 we augment the matrix appearing in this equation by adding linearly independent columns (33) to yield the matrix

z,

...

z,,

0

.--

0

e,,+~

...

e .......

LZ(x)

I,,,

where the vectors q,,+ ~(x)..... q,,(x) are independent o f the vector fields qt(x) ..... q~,(x), for all x e ? < ° and ei is the ith column of I,,. Obviously, the vector fields q~(x) ..... q,,(x) constitute a local basis o f JR" and the columns o f the matrix defined by (4.32) constitute a local basis o f [R''+~. Next, define the following diffeomorfism (32)

gl

,e

~,,+ I

,5,,

dn+ I

g,~ ,,,

where

. . v,,,0 are the new coordinates %: . .[x~,.. 0 . . . . . 0] w and 4~[ denotes the tlow of f(x) at time t. It is k n o w n that the functions ~,,+:~,.... ~,,+,,,~ are independent solutions o f the firstorder partial differential equations (4.31) (33). With regard to the diffeomorfism (4.33) it m a y be proven that its Jacobian has the t'orm

~,

l,,,

Solving the above first-order partial differential equation for ~, ~e = P2(~,)+ ~:. Hence, the vector ~2 = [g,, ~ ~. . . . . ~,,+,,,]T may be written as

one

has

Simultaneous Linear Exact Model Matching

1313

~2 = -,~(~,) + ~2,

(4.34a)

2(~,): =[2,(~1) ..... 2,,,(,~. )]v:--~'2 (~,)l~, ~,I,,,.

(4.34b)

where

Since ~,,+ ~..... ~,,+,,, are solutions of (4.31 ), the equation

a~ L Z ( x ) J holds. Substituting (4.34a) in the above equation yields 8~.(x)

Q(x) = Z(x),

where use was made of the definition ~ : = x . Finally, 2(x) satisfies the first-order partial differential equations (4.23). Hence, it is proven that (4.23) has a solution for ~b(x) (the vector valued function 2(x)). This completes the proof of the sufficient part of the present lemma. • Now, we may establish the necessary and sufficient conditions for the D R L E M M problem to have a solution. These conditions are given by the following theorem. Theorem H The necessary and sufficient conditions for the solvability of the D R L E M M problem, under state feedback (CL), for the class of nonlinear system (S) with rank[B*(x)] = m, for all x ~ ~ 0 are I. (4),s, = (d,)~,,, 2. D * ( x ) = 0 , 3. 2nm= rank[M] = rank[E*(x)]=rank[E*(x)

M],

4. dim{inv[Q(x)]} = dim~invI-Q(x)l~,

LZ(x)JJ

for ie,~i, and for all x ~ °. Proo/! Condition (1) is necessary for the solvability of the D R L E M M problem. This necessary condition reduces the problem of determining the control law (CL) to the solution of (4.5) for Li(x), ie{0,,~2,,_l}, and subsequently to the solution of the D R L E M M design equations. Hence, for the class of nonlinear systems considered, condition (1) together with the necessary and sufficient conditions for the solvability of (4.5) for L(x), i t {0J2,, i}, and of D R L E M M design equations for B(x) and 4~(x), constitute the necessary and sufficient conditions for the D R L E M M problem to have a solution. In fact, equation (4.5) has a solution for Li(x), ie{OA2,, ~}, with L0(x) invertible if and only if conditions (2) and (3) hold. Because of the invertibility of L0(x) the D R L E M M design equations have a solution for B(x). Clearly, B(x), thus determined, is invertible as is desired. By Lemma 4.4, we have that condition (4) is the necessary and sufficient condition for the D R L E M M design equations to have a solution for ~b(x).

1314

A. S. Tsirikos and P. N. Paraskevopoulos

4.3. General solution for the state feedback control law As has been pointed out, to determine a(x) and B(x) one has to solve first the D R L E M M design equations for L0(x) and q~(x). Then, the feedback pair {a(x), B(x)} will be given by (4.22). Hence, in order to derive the general expression for the control law (CL), it suffices to determine the general solution of the D R L E M M design equations for Lo(x) and 4)(x). To this end we will first prove the following lemma.

Lemma V The following condition holds: inv[E(x)

adE,E(x)

...

ad']~,tE(x)] =inv[E(x)

ad~,E(x)

...

ad"E,,LE(x)

ad~,E(x)

...].

Proof. First, note that the involutive closure of a distribution may be constructed from a basis of the distribution. Hence, in order to prove the present lemma, it suffices to prove that rank[E(x)

adEoE(x)

...

ad:~. LE(x)] =rank[E(x)

adt:,E(x)

...

ad~- IE(x)

ad~.E(x)

...],

for all x ~ Z0. To this end, consider the ith column of ad~.oE(x) denoted by ad%,Ei(x), .j~ 3~. Furthermore, assume that this column is linearly dependent upon the columns of the matrix [E(x)

adL-E(x)

...

ad~tc01E(x)],

(4.35)

~ 2,~,(x)ad~-oE,(x).

(4.36)

for any x e Zo, i.e. j

l

ad~, E~(x)= ~ p

0 k

l

Then, it will be proven that ad~+t E~(x) is also linearly dependent upon the columns of the matrix (4.35). Using the definition of ad(), we have that +l

ad~o E~(x) = d(ad~,oEi(x))Eo(x) - d(Eo(x))ad~-oE~(x). Substituting (4.36) in this equation yields i

l

ad~,~'E,(,,)= Z ~ d(;'~,,(x))Eo(")adLE*(x) (~=0 k=l j-

/

]

+Z o /

0 k

1 k--I

0 k

0=0

I

~ 2,.,_,(x)d(Eo(x))ad~.,Ek(x)

£J--O k = l I

~ LE,,(2k(,(x))ad~,E,(x)+ Z

0--0

0

[ /

=Z

l

~ ).*~,(x)d(ad'~,,Ek(x))Eo(x) - Z

~ 2a,(x)ad~+'E*(x) k

1

Simultaneous Linear Exact Model Matching

1315

where use was made of (4.36). Using the above relation and by contradiction, it may be easily proven that all the linear independent columns of [E(x)

adEoE(x)

...

ad~. 'E(x)

ad~,E(x)

...]

are columns of the matrix [E(x)

adL.

...

ad"-'E(x)].E,,

This completes the proof of the present lemma. Lemma V leads to the following corollary which characterizes q~(x).

•

Corollary I Using Lemma V it is obvious that if the necessary and sufficient conditions of Theorem II hold then the general solution of (4.23) reduces to the solution of the equation

d(4,)Q(x)= Z(x), where Q(x):=[E(x)

a(lE,E(x)

...

acid- 'E(x)]

and Z(x):=[Z,(x)

Z2(x)

...

z,,(x)].

Finally, Corollary I together with Theorems I and I1 lead to the general solution for the control law (CL), given by the following theorem. Theorem I l i If the necessary and sufficient conditions given in Theorem II hold for the class of nonlinear systems (S) with rank[B*(x)] = m, for all x ~ Z0, then the general solution for the controller (CL) which satisfies the D R L E M M problem is given by

B(x) = L~ '(x), a(x) = Lo I(X)q#(X), {#(X) = ~[~hom(X)+ ~(X), where q~hom(X)is the solution of the homogeneous equation d(q~)(~(x)= 0 and ~,(x) is a special solution of the nonhomogeneous equation d(~b)0(x)= Z(x). In the following corollary, we will prove that under the assumption that p = m and D*(x) = 0, for all x ~ Z0, the standard noninteracting feedback satisfies the disturbance rejection problem (l, 8). Corollary H Assume that for the nonlinear system (S), p = m, rank[B*(x)] = m and D*(x)= 0, for all x ~ Zo. It will be proven that the control law of the form (CL), where

a ( x ) = -- [B*(x)] ' [L~,,+ 'h~(x),...,_~,,,I'/,,,+ 'hm(x)] T,

(4.37a)

B(x)=[B*(x)] ',

(4.37b)

1316

A. S. Tsirikos and P. N. Paraskevopoulos

called the standard noninteracting feedback, leads to a closed-loop system whose output is not affected by the disturbances. Proo[! Assume that p = m, rank[B*(x)] = m and D*(x)= 0, for all x ~ Z 0 Then, the third condition of Theorem I is equivalent to rank[Mo]=m, which provides that a possible admissible linear model satisfying the disturbance rejection problem has the form M0 = I, Mj = 0, for j > I. In what follows we will prove that for the above particular model the D R L E M M problem is solvable. To this end, observe that the first of the two D R L E M M design equations is always solvable for B(x) yielding B(x)= Ld l(x)=[B*(x)] ~. What remains is to prove that the standard noninteracting feedback (4.37) satisfies (4.23). In fact, for the particular model, the parameters Z~(x), k 6 ~:,, ~, appearing in (4.23), take on the form

(k,)

Zk(x)=(--1)~(--l) k ~ ' ~=o k .,"

I

L~-{E* j(x)}.

(4.38)

On the other hand, using the definition of Lie bracket operation, it may be proven that

d(~[~)adkolE(x): ~'

(_i)/,-j

/=0

LIEo{LELkoJ l~j~},

i k

1

for keJz,, ,. Substituting the standard noninteracting feedback (4.37) in the above relation yields

)k ,(

d(dp)ad~"'lE(x)=(-1 /~o --l)k ~ i(kk~'ll)L~"'{E* /(x)},

(4.39)

where use was made of definitions of E*(x) and ~b(x). Finally, substituting (4.38) and (4.39) into (4.23), the resulting relation is clearly an identity. • It is obvious that the particular model selected is noninteracting. In fact, the particular linear model admits a minimal realization of the form H(s) = diag~,~,,,(1/s'~,+~).

Remark I As a consequence of the present results one can obtain the solution of L E M M problem in the case where no disturbances influence the nonlinear system (S), i.e. D ( x ) = 0 , reported in (26). Clearly, in this case D * ( x ) - - 0 , j ~ .... The first, third and fourth conditions, under suitable modifications (i.e. zeroing certain columns of E*(x)), are the necessary and sufficient conditions for the L E M M problem to have a solution. Finally, the general solution for the control law will be given by Theorem III.

Remark H The proposed approach may, in general, produce a closed-loop system with linear i/o description of larger dimension (24, 25) than the dimension of the linear i/o description produced by other known i/o linearization techniques (8, 21). For instance, as a consequence of this observation, it could be shown that, for the case of square and i/o decouplable systems, the admissible linear models which satisfy the DRM EMM problem belong to a more general class than the class produced by the standard noninteracting feedback, given in Corollary 1I.

Simultaneous Linear Exact Model Matching

1317

Remark III It is noted that the linear model (M) may be selected to have the more general form in state space x = A x + Bw+ D~, y = Cx under the constraint that the transfer function from the disturbances to the output is zero. Even in this case the results of this section may still be applied since our approach takes into account only the Markov parameters of the linear model. 4.4. Illustrative example The following example illustrates the approach presented in this paper. Consider the nonlinear system of the form (S) with X1X3-~-X2e v z

Eo(x)= ]

x3

~ I

, E(x)=[G(x)

D(x)]=

0

X4 --X2X3

,

0

h .,c~+ x2x4 - x~x3

L x3

- 1

h(x) = [h~ (x), h2(x)]~ = [x,e ": + x4 -.w_x3,x:] T, defined on ~4. For the given nonlinear system the characteristic numbers are (d~)~s~= 1 and (~)~s~= 0, for all x • ~4, since dh(x)E(x) = f~

:],d(Le,fl,(x))E(x)=[1

0].

Hence, B*(x)= [1 1]T and the system belongs to the class of nonlinear system studied in this paper. Also consider the linear model of the form (M) with

[: 0]

E002

Calculating the characteristic numbers of the given (M) we conclude that d,= (di)(s~= (di)~M~, for all × • N4, for i = 1,2. Hence, the first condition of Theorem I holds. The matrices E*(×) and Mj are of the form

E*(x)=[B*(x)D*(x)]=[: :],E*(x)=[B*(x)D*(x)]=I: E*(x)=[B*(x)D*(x)]=I~-i],E*(x)=[B*(x)D*(x)]=[~

;],

~],

for k > 3, and

no=E:1,

LyAe,,e~)J

for j > 1, where 5.~are functions of their arguments. Since D*(x)=0, for all x • ~ 4 the second condition of Theorem | holds. Constructing the matrices E*(x) and M, it is easy to see that the third condition of Theorem I holds. The matrix V0 is chosen as

1318

A. S. Tsirikos and P. N. Paraskevopoulos

vo [P°I F' o] = kKoJ = L-

1

It may be proven that the involutive closures of the distributions locally spanned by the columns of Q(x) and [Q(x) Z(x)] T are

mv Q(x) = span O(x , inv FQ(xq; = s p a n ffO(,,)l~ ~ respectively, where span {A(x)} denotes the distribution spanned by the columns of A(x) and where Q(x) & [q,(x)

q2(x)

Z(x)=[Z,(x)

q3(x)

q4(x)] = [E(x)

Z2(x)]=[-e,

0

[Eo(x),E(x)]],

~_~ - l ] ,

where

[E0(x),E(x)] =

0 0

' X2

Hence, the fourth condition of Theorem I is satisfied. Using all the above results we conclude that for the nonlinear system under consideration the D R L E M M problem via static state feedback is solvable. To determine the diffeomorfism (4.33), we first calculate the flows

--QI~I +.V5

L

-g2

X2

)(4

~4X2 -+-X4 ]

X5

J

0

where X~A [xT,Xs] T, with initial conditions X0=[0,0,0,0,0] T. Then, the diffeomorfism (4.33) is defined by

1319

Simultaneous Linear Exact Model Matchinq e~1(~2 -- ~3)

where ~ =[~1,~2,~3,~4]T. Hence, a special solution for 4~(x) is given by (4.34b) and it has the form 4,(x) = f'=(¢,)le,

= ¢i ,,x~-- - 0 , x ~ - 0 = ( x , e

'= + x 4 - x 2 x ~ ) -

x3.

Since for the present example 0 ( x ) is invertible, for all x ¢ ~4, we obtain that the solution of the homogeneous equation d ( ~ ) 0 ( x ) = 0 is ~bhom(X)=0. Hence, the pair {a(x),B(x)} of (CL) is given by B ( x ) = 1 and a(x) =~b(x). To show that the above feedback control law satisfies the D R L E M M problem apply (CL) to the given system to yield the closed-loop system = E0(x) + E(x)

[,,] (

, y = h(x),

where Eo(x) = Ix2ex~

0

x4 - -

-7(2X3

X2

(X4 -- X2X3)]T -- G(x)[Q Ix2 d- Q2(xI e ~2+ ' ( 4

-- X2X3)]"

Next, apply to the above closed-loop system the diffeomorfism = [~ ,G,~.~,~4] ~ = F(x)

= [x, e

': + x 4 - x 2 x ~ , x : , x ~ , x 4

- x~x~] ~

to yield the system

0

00

0

0

0

0

0

0

0

-1

"

The above system is decomposed into two linear subsystems. The first subsystem has the same description as that of the model (M), while the second is an unobservable subsystem affected by the disturbance. In fact, the control law, derived by our approach, rejects the effect of the disturbances to the output of the system.

V. Solution o f the D R L E M M M F Problem

In studying the D R L E M M problem it is possible not to be able to reject the effect of certain disturbances. This happens if the second condition of Theorem II does not hold. In this case, it is possible to overcome this difficulty if we can measure the disturbances that cannot be rejected by (CL). To this end, instead of (CL) consider the analytic control law of the form (CL)I:

1320

A. S. Tsirikos and P. N. Paraskevopoulos

u = a(x) + B(x)w + F(x)~, where we [~'", {e N: and IB(x)l #0, for all x e ~0. The control law of the form (CL)~ is a static state and measurement feedback. Applying the control law (CL)~ to the original system (S) yields a nonlinear closed loop system having the form (CLS)~:

- [w], x(0)=x0, X= l~o(x)+ E(x) y=h(x), where

l~o(X):=ao(x)+G(x)a(x), I~(x):=[G(x) D(x)]IB(0x) FT) 1. The D R L E M M M F problem consists in determining a control law of the form (CL)~, such that the output of (CLSh matches the output of (M), i.e. such that

d(L~.h,(x))f~=[(c,)MA~(B)M

01,

(5.1)

for k e ~ , , where use was made of (4.1 a). Since B(x) is invertible, we note that condition (4.2) is a necessary condition for the D R L E M M M F problem to have a solution. Furthermore, as is proven in Section 4, if condition (4.2) holds, then the infinite set of equations (5.1) reduces to the finite set of equations E*(x)[B(0x)

FT)I=[Mo

/,1 E*(x)+i=~onfl(L~, l J~b)E(x)=[M~

0],

[B 0'(X) 0]L

(5.2a)

--B 1(Ix)F(x) 1 ,

(5.2b)

for ke~z,, ~, where 4~(x), Mr and E*(x), for j > 0 , are defined as in Lemma 4.1. Equations (5.2) may be rewritten in a compact matrix form as MA(x) = -- E*(x),

(5.3)

where all quantities are defined as in Section 4, except for A(x) which is defined by

__ILO --L°F1

[El]

"'"

o

_[Lo-Lo

1 ...

0

0

....

IL2~ 11

7]

1321

Simultaneous Linear Exact Model Matching

As in Section 4 it may be proven that (4.7b) is the necessary and sufficient condition for (5.3) to have a solution for A(x) with I.g,(x)l~ 0, for all x t ;~o, and in the desired form. Furthermore, choosing Vo as in the previous section, the solution of (5.3) for Li(x), i t {0J2,, i} and L0(x)F(x) will be given by Lo(x)=[PoMo] LPoB*(x),

(5.4a)

L0(x)F(x)= -[PoM0] ~PoD*(x),

(5.4b)

L,(x) = [PoMo] 'P0{MiLo(x)[l L~(x)=[PoMo] Ip0 MkLo(x)[1

- r ( x ) ] - E*(x)},

-r(×)]-E*(x)-EMj+lLk

(5.4c) j

,(X) , (5.4d)

j = 0

for 2 <_k < 2 n - 1. In what follows, using the definition in (4.6) for L(x), i t [0,~2,, ~}, we can define design equations, associated to D R L E M M M F problem. These equations are given by the following theorem. Theorem I V Under the assumption that IB(x)l ~0, [J=m, for all x t N.°, and conditions (4.2) and (4.7b) hold, the unknown triplet {4~(x),B(x),F(x)} of (CL), will be the solution of the following set of equations, which will be called D R L E M M M F design equations: B - I ( x ) = L0(x),

L,)(x)r(x)= - [PoMo] ~PoD*(x), d(4))[Q(x)] = [Z(x)], where all the quantities are defined in a similar way as they are in Theorem I. Based on the D R L E M M M F design equations, the necessary and sufficient conditions for the D R L E M M M F problem to have a solution may be derived. These conditions are given by the following theorem. The proof is similar to the proof of Theorem II. Theorem V The necessary and sufficient conditions for the solvability of the D R L E M M M F problem, under state feedback (CL)~, for the class of nonlinear system (S) with rank[B*(x)] = m, for all x t ~'), are 1. (di),s)=(d,)(M), 2. 2nm = rank[M] = rank[E*(x)] = rank[M

E*(x)],

LZ(x)J) for i t ~ , and for all x t ~(>. Finally, the expression of the unknown triplet {4,(x),B(x),F(x)} of the control law (CL), is given by the following theorem.

1322

A. S. Tsirikos and P. N. Paraskevopoulos

Theorem VI

If the necessary and sufficient conditions of Theorem V hold, for the class of nonlinear systems (S) with rank[B*(x)] = m, for all x ~ N °, then the general solution for the triplet {~b(x),B(x),F(x)} of (CL)~ which satisfies the D R L E M M M F problem is given by B(x) : Lo

I(X), r(X)

= --

[PoB*(x)] -IPoD*(x),

a(x) = Lo l(x)~b(x), ~b(x) = ~bhom(X)+ 2(x), where 4~hom(X)is the solution of the homogeneous equation d(~b)0(x) = 0 and ~(x) is a special solution of the nonhomogeneous equation d(q~)0(x) = Z(x). With regard to the form of the control law (CL)~, we note that there is not any restriction on the matrix F(x). Hence, certain columns of F(x) may be zero vectors. These columns are obviously the columns which are associated with the disturbances that can be rejected under (CL). Clearly, from Theorem VI we obtain that if the ith column of D*(x) is zero i.e. the ith disturbance may be rejected by (CL), then the ith column of F(x) is zero and hence the ith disturbance needs not to be measured. VI. The Special Case o f Linear Systems 6.1. Linear time-invariant systems

Consider the special case where the open-loop system is a linear time-invariant system and the control law is a proportional state feedback, i.e. (S): x = A x + E F U ' ] , x(0)=x0, y = C x ,

LCJ and (CL): u=Fx+Gw,

w h e r e A c ~ .... , E : = [ B D ] ¢ ~ × ¢ " + ~ ) , C C R P × " , F ¢ ~ ..... a n d G ¢ ~ . . . . ( I G l # 0 ) . F o r this case it is easy to see that the results of Section 4 readily coincide with the results reported in [27]. 6.2. Linear time-varyiny systems Consider a linear time-varying analytic system and a time-varying analytic proportional state feedback control law, defined in the time interval [0,T], i.e. (S)~: : A(t)x + E(t) I u l , x(0) =x0, y = C(t)x, and (CL)I: u = F(t)x + G(t)w, where

A(t)~R ..... ,

E(t):=[B(t)

D ( t ) ] ~ " ×~m+~l, C ( t ) ~ P ×',

F(t)~R .....

and

G ( t ) s R ...... (IG(t)150, for all t~[0,T]). The system (S)1 may be represented as a non-

linear analytic system with an augmented state (S)2:

Simultaneous Linear Exact Model Matching

1323

dr = E0(a) + E(6) [u], ¢ a(0) =a0, Y=h(tr), and the control law (CL), as a nonlinear analytic control law (CL)2: u = a(a) + B(a)w,

where t r : = [ x ] , ro(o'): = I X ( : ) x ] , E(tr): =[E~t)], h(a): = C(t)x, a(a) = F(t)x, B(a) = G(t). The following relationship, related to (S)2, may be easily proven: d\k 1 7 E(t)],

ad~-::' E ( a ) = - A ( t ) + l d t ) 0

for kz~2,, ~. Furthermore, defining 4~(a):=~(t)x, where ~ ( t ) : = G '(t)F(t), we have that dq~(a)=[~(t) 0]. Hence, the D R L E M M for the class of system (S):, with rank[B*(a)l=m, for all te[0,T], may be written as O(t)Q(t) = Z(t),

(6.1)

where

Hence, based on the results of the present paper, we can establish the necessary and sufficient conditions for the solvability of the problem. These conditions are 1. (d,),s,= (di)~M,, 2. rank[E*(t)]=rank[M]=rank[E*(t)

M],

for ie ~p and for all t ~ [0, T]. Also, the general solution for the controller matrices F(t) and G(t) may be found by solving the matrix equation (6.1). In the case where the model is selected to be linear time-invariant and p = m, the foregoing results coincide with those reported in (27). VII. Conclusions

In this paper, a new approach to the disturbance rejection with simultaneous linear exact model matching problem via static state ( D R L E M M ) or static state and measurement feedback ( D R L E M M M F ) for a class of nonlinear systems is presented. The

1324

A. S. Tsirikos and P. N. Paraskevopoulos

particular class of nonlinear systems considered is that of nonsquare systems with defined vector relative degree. The reason for considering the combined problem of D R with L E M M is that it is desirable for the closed-loop system not only to be disturbance decoupled but also to have the simplest i/o description, thus allowing easy further improvement of the closed-loop system performance. The proposed approach consists in reducing the D R L E M M ( M F ) (i.e. the D R L E M M or the D R L E M M M F ) problem to that of solving a nonhomogeneous system of firstorder partial differential equations, called D R L E M M ( M F ) design equations. It is proven that the necessary and sufficient conditions for the solvability of the D R L E M M ( M F ) design equations are the necessary and sufficient for the solvability of the D R L E M M ( M F ) problem. Hence, algebraic conditions for the D R L E M M ( M F ) problem to have a solution, easy to be checked, are established. Furthermore, solving this system of equations, the general analytical expression for the feedback control law is explicitly determined. The standard noninteracting feedback concerning i/o decouplable systems is derived by our algorithm as a special case. It is pointed out that the necessary and sufficient conditions derived here may be used to give guidelines for the selection of the admissible linear models which satisfy the D R L E M M ( M F ) problem. Furthermore, our approach may produce a closed-loop system with linear i/o description of larger dimension than the dimension of the linear i/o subsystem produced by known algorithms. Acknowledgements

The work presented in this paper has been partially funded by the Greek State Scholarship Foundation (I.K.Y.) and by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research and Technology (EKBAN-714).

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