- Email: [email protected]
Uniform exact model matching for a class of linear time-varying analytic systems
Systems & Control Letters 19 (1992) 313-323 North-Holland
313
Uniform exact model matching for a class of linear time-varying analytic systems K.G. Arvanitis and P.N. Paraskevopoulos National Technical Universityof Athens, Division of Computer Science, Department of Electrical Engineering, 157 73 Zographou, Athens, Greece Received 27 January 1991 Revised 22 December 1991 and 25 April 1992
Abstract: The problem of uniform exact model matching is studied via state and output feedback, for a class of linear time-varying analytic systems. The following two major issues are resolved: The nessecary and sufficient conditions for the problem to have a solution and the general analytical expressions for the controller matrices. A major feature of the proposed approach, is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation. Keywords: Exact model matching; linear time-varying analytic systems; state feedback; output feedback; 'generalized' Hankel matrix.
I. Introduction This p a p e r is devoted to the problem of uniform exact model matching of linear time-varying analytic (1.t.v.a.) systems, i.e. for systems described by
ic(t)=A(t)x(t) +B(t)u(t),
y(t)=C(t)x(t),
(1.1)
where A(t) ~ ~nxn, B(t) ~ ~ × " , C(t) ~ R pxn are time-varying matrices with entries in D(T) (the commutative ring of analytic functions defined on the time interval T = (tt, t 2) with values in R). The problem of exact model matching is of both theoretical and practical importance [4, 6-10, 12-17]. The first basic results in the field were reported in [6,14-17], for linear time-invariant systems. Subsequently, a large n u m b e r of papers have been published on the subject, covering several caregories of systems such as linear time-varying, two dimensional, generalized state space, nonlinear, etc. With regard to the exact model matching problem of linear time-varying systems, the respective published results are limited [8]. The approach presented in designing static or dynamic controllers for exact model matching of linear time-varying systems in [8] is based on the algebraic equivalence between the closed-loop system and the model and it reduces the problem to that of solving a linear system of first-order differential equations. In this paper, the concept of uniform exact model matching for linear time-varying analytic systems is introduced and a 'generalized' Hankel matrix approach is presented to solve the problem using state and output static feedback. A brief description of the proposed technique is as follows: Starting with the definition of uniform exact model matching and upon using some algebraic manipulations on this definition, the problem is reduced to that of solving a linear nonhomogeneous algebraic matrix equation. This equation plays a central role in our approach and it is called the uniform exact model matching design equation. On the basis of this equation, the following two aspects of the uniform exact model matching problem are investigated: The neessecary and sufficient conditions for the problem to have a solution and the general analytical expressions of the controller matrices.
Correspondence to: Prof. P.N. Paraskevopoulos, National Technical University of Athens, Division of Computer Science, Department of Electrical Engineering, 157 73 Zographou, Athens, Greece. 0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
314
The motivation for using the above technique is due to the technique proposed in [10] for solving the exact model matching problem for linear time-invariant generalized state space systems. The technique in [10] is based on the idea of appropriately manipulating the basic exact model matching equation so as to reduce the solution of the problem to that of solving an algebraic system of equations involving the Markov parameters of the state and of the output of the open loop system. The results reported in this paper on exact model matching of time-varying systems are superior over known results [8], mainly because here, an algebraic matrix equation, rather than a differential matrix equation, is needed to be solved. Furthermore, no restrictions are imposed on the model or on the orders of the open-loop system and of the model, as compared to [8]. It is mentioned that the present result are part of the material reported in [1].
2. Preliminaries The weighting pattern matrix of system (1.1), under the assumption that the initial state vector is zero, is well known and has the form
W(t, r) = C(t)q~(t, "r)B(r), Vt > r and Vt, r ~ T, where ~ ( t , ~-) is the state transition matrix associated with put description of system (1.1). The following definitions will be useful in the sequel.
(2.1)
A(t). Clearly, (2.1) constitutes the input-out-
Definition 2.1. The sequences of the matrices, {Fo(t)} and {Qo(t)} are defined as
Fp+,(t) =Fp(t)A(t) +Fp~l)(t),
Fo(t ) : = C ( t ) ,
(2.2a)
and
Q p + , ( t ) = - A ( t ) Q p ( t ) + Q p o) ( )t , respectively, where
Qo(t):=B(t),
(2.2b)
P(~)(t) indicates the K-th time-derivative of P(t).
Definition 2.2. The 'generalized' Hankel matrix of the system (1.1) is defined as [3]
H(t)={np,K(t)},
p=l,2,...,
K=l, 2,...,
(2.3)
where
Ho,x( t ) = l'p_l( t )Qo( t ), p > l ,
(2.4a)
Hp,~(t)=-Hp+L~_l(t )+H¢,~_,( (~) t ), V p > l a n d V K > 2 .
(2.4b)
and
Definition 2.3. The 'characteristic numbers' di(t) of the system (1.1) are defined as
di(t) where
=
minp:
~n-1
(.yi)p(t)Qo(t) 4:0, for t ~ T and p = 0 , 1 if(~,i)o(t)Qo(t)=O, f o r t ~ T a n d f o r a l l p ,
n - 1,
Vi~Jp
(2.5)
(~,i)p(t) is the i-th row of the matrix Fo(t) and Jp --- {1, 2 . . . . . p}.
Note that in Definition 2.2, the di's are in general time-varying quantities. However, on the basis of the analyticity assumptions for the matrices A(t), B(t) and C(t) we can divide the interval T into subintervals over which the di's are constant. For simplicity, we assume in this paper that the di's a r e constant Vt ~ T.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
315
3. Uniform exact model matching via state feedback
3.1. Formulation of the problem Consider the 1.t.v.a. system (1.1) with weighting pattern matrix given by (2.1). To this system we apply the following proportional state feedback law
u(t) = F ( t ) x ( t ) +G(t)to(t)
(3.1)
where to(t) ~ ~ " is the new input vector. In order to insure linear independence of the m external inputs [17], G(t) is assumed invertible Vt ~ T. It is further assumed that the entries of G(t) and F(t) belong in D(T). Then, the closed-loop system is
ic(t)=Ac(t)x(t ) +Bc(t)u(t), where A~(t)=A(t)+B(t)F(t) described in state-space as
y(t)=C(t)x(t),
(3.2)
and Bc(t)=B(t)G(t). Let the desired model be a 1.t.v.a. system M,
,rm(t) = A m ( t ) X m ( t ) + B m ( t ) U m ( t ) ,
Ym(t) = C m ( t ) X m ( t )
(3.3)
~ ~ nm>(nm, Bm(t) ~ ~ nm>(m and Cm(t) ~ R p×'m are time-varying matrices with entries in D(T). Let Wc(t, z) and Wm(t, r) be the weighting pattern matrices of the closed-loop system (3.2) and of
where Am(t)
the desired model (3.3), respectively. Then, the problem of uniform exact model matching studied can be stated as follows: Find the matrices G(t) and F(t) such that Wc(t, ~-) and Wm(t, r) are equal for all values of t and ~- which belong to the time interval T, i.e. such that W~(t, ~-) = l'gm(t , "r),
Vt>~" and Vt, ~ ' ~ T .
(3.4)
Note that if relation (3.4) holds only for a single point t~ ~ T, then one may refer to this special problem as the complete exact model matching problem. In this paper we study only the uniform exact model matching problem, since this is, obviously, the general case. Next, an equivalent definition to the above problem will be given. To this end, let H~(t) and Hm(t) be the 'generalized' Hankel matrices of the closed-loop system (3.2) and of the given model (3.3), respectively. The matrices Hc(t) and Hm(t) are defined similarly to H(t) for the open-loop system (1.1) in relation (2.3). It is clear that relation (3.4) is satisfied if and only if the following relation holds [3]:
Hc(t)=Hm(t),
Vt~T.
(3.5)
Also, let (yi,~)p(t)Qc,o(t) and (Yi,m)p(t)Qm,o(t) be the i-th elementary rows of any p-th block row of the first block column of the matrices He(t) and Hm(t), respectively, where (,ri,c)p(t) and ('gi,m)p(t) are the i-th rows of the matrices Fc,p(t) and Fm,p(t), respectively. Note that the matrices Fc,p(t) and Fm,p(t) are defined similarly to the matrices Fp(t) for the open-loop system in (2.2a). Furthermore, the matrices Qc,o(t) and Qm,0(t) are defined similarly to the matrix Qo(t) for the open-loop system in (2.2b). Using the above, the definition of uniform exact model matching (3.5) may be equivalently written as
('ri,c)p( t )Qc.o( t ) = ( ]ti,m)p( t )Qm,o( t ), Vi ~J,, Vp > 0 and Vt ~ T.
(3.6)
In what follows, the uniform exact model matching problem will be solved on the basis of relation (3.6), rather than relations (3.4) or (3.5). The motivation for using (3.6) is that it greatly facilitates the solution of the problem.
3.2. Derivation of the uniform exact model matching desing equation Let dc,i(t) and dm,i(t) be the 'characteristic numbers' of the closed-loop system (3.2) and of the model (3.3), respectively, which are defined similar to the d / s for the open-loop system in (2.5) and which are assumed to be constant Vt ~ T. Clearly, since the 'characteristic numbers' remain invariant under state
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
316
feedback and a non-singular transformation of the inputs [5], a necessary condition for the uniform exact model matching problem via state feedback to have a solution is
di = dc,i = dm,i Vt ~ T, Vi ~ Jp.
(3.7)
Now asume that (3.7) holds. Using (2.5) and (3.7), relation (3.6) may further be written as
( %.c)o( t)Qc.o( t) = (,Yi,m)o( t)Qm,o( t),
Vi ~Jp, Vp > d i and Vt ~ T.
(3.8)
One may readily derive the following relationship [5]:
(ri,c)di(t) = (Yi)d,(t),
Vi ~Jp, Vt E T.
(3.9)
We next study (3.8) for p = d i, d i + 1. . . . . For p = d i, relation (3.8) becomes
(yi,c)d,(t)Qc,o(t) = (Yi,m)d,(t)Qm.o(t),
Vi ~Jp, Vt ~ T.
(3.10)
We introduce the following definitions:
I,t~,di(t ) = (Yi.m)d,(t)Qm,o(t)
and
G(t) =G-l(t).
(3.11)
Introducing (3.9) in (3.10) and upon using definition (3.11), relation (3.10) reduces to
( ~'i)a,( t )Qo( t ) = [~ti,di( t )(J( t ).
(3.12)
For p = d~ + 1, relation (3.8) takes on the form
('Yi.c)a,+~(t)Qc,o(t) = (],i,m)a,+l(t)Qm,o(t),
Vi EJp, Vt ~ T.
(3.13)
=d(t)F(t),
(3.14)
We introduce the following definitions:
iXi,a~+x(t ) = (ri,m)a~+l(t)Om,o(t)
and
q~(t)
Introducing the definition of rc,o(t) and relations (3.9) and (3.14) in (3.13), relation (3.13) becomes
[( ~/i)a,( t )Ac( t ) + ( ~,i)~al~)(t ) ]ao( t ) = I~i,di+ l( t )G( t ).
(3.15)
Further, introducing in (3.15) the definition of Ac(t), relation (3.15) becomes [(r~)a,+l(t) + [(ri)a,(t)Qo(t)]F(t)]Qo(t)
=lzi,d,+l(t)G(t).
(3.16)
Finally, introducing (3.12) and (3.14) in (3.16), the original relation (3.13) reduces to
( ]/i)di+ l( t )ao( t ) = ~i,di+ l( t )G( t ) -- #.i,di( t )clJ( t )Qo( t ).
(3.17)
For any p > d~ + 2, by repeatidly using the above procedure, relation (3.8) reduces to
(r~)d,+j(t)Qo(t)=[l~i,a,+j(t)]od(t)-
lJ;0
I"I~i,di+j--K--I(I)]K+I(CI'))K(t)
]
Qo(t),
Vj > l ,
(3.18)
where
[la.i,ai+j( t)]o
:=
('Yi,m)d,+j( t)Om,o( t),
[ll"i,di+j(t)]K
=
)IK, [~i,di+j(t)]K_l+ [l~i,d~+j-l( t'I(D
[ ~i,a,( t ) ]~ = I~i,ai( t )
:=
(3.19a)
Vj >_ 1 ,
('Yi,m)cli( t )Qm,o( t ),
Vj 2 1 and VK > 1,
(3.19b) (3.19c)
K.G.Arvanitis,P.N.Paraskevopoulos/ Uniformexactmodelmatching
317
and where the sequence of the time-varying analytic matrices {(~)~(t)} is defined as (q))~+l(t) = ( ~ ) ~ ( t ) A ( . t ) + (q~)~)(t),
(q))0(t):=q)(t).
(3.20)
Relation (3.18), together with relation (3.12), constitute an infinite differential system of equations with respect to the unknowns t~(t) and qD(t) (and hence, with respect to the unknown controller matrices G(t) and F(t)). Furthermore, it is pointed out that relations (3.12) and (3.18), hold for all i ~Jp. Grouping appropriately (3.12) and (3.18) for i ~Jt,, we arrive at the following differential system of equations: n~,o(t) = Mo*o( t ) ( ~ ( t ) ,
(3.21a) r
BPo(t) =MJ,*o(t)~(t) - I £ Mj*~.~(t)(@)~_l(t) t K=I
Qo(t),
V j > 1,
(3.21b)
where
-~l'd'+j(t) ]
T1)d'+j(I)QO(t) ] ( ~,2)~+j( t)Qo( t) Bj*o(t)
Vj > O,
M?,o( t ) =
=
( rp) d,+j( t )Qo( t )
(3.22a)
J
M~,,(t) =Mo*o(t), Vp _> 1 and
M~*p(t) =Mj*p_,(t) +M*d)j-l,p~'t), Vj>_ 1 and Vp >_ 1.
(3.22b)
Now, define fl = rank B~,o(t). Note that/3 is in general a time-dependent quantity. However, because of the analyticity assumptions on the matrices used, we can always partition T into subintervals over which /3 is constant. For simplicity in the presentation of our approach, we assume that /3 is constant over T. Clearly, for the system (3.21a) to have a solution for G(t) for all t ~ T, it is necessary and sufficient that rank B~.o( t ) = rank[ B~,o( t ) l Mo*o(t ) ] =/3,
Vt ~ T.
(3.23a)
Furthermore, in order to guarantee that the solution of (3.21a) is in the desired form, i.e. is a nonsingular matrix for all t E T, the following relationship must hold: rank B~,o( t ) = rank Mo*o(t ),
Vt ~ T.
(3.23b)
Clearly, conditions (3.23a) and (3.23b), may be stated as one, as follows: rank B~,o( t ) = rank Mo*,o(t ) = rank[ B~,o( t ) l Mo*o(t ) ] =/3,
Vt ~ T.
(3.24)
The class of l.t.v.a, systems studied in this paper is restricted to the case where/3 = m. Note that for this particular class of time-varying analytic systems and for T = [t0,~) , there exists a certain type of inverse. To make this clear, in what follows the invertibility result due to Silverman for scalar systems [11] will be generalized to the multivariable case. To this end, using the definition of d i, one may readily establish that
y~P)(t) = ( r i ) p ( t ) x ( t )
and
y~di+l)(t) = ('Yi)d,+l(t)x(t) + ('gi)d,(t)Qo(t)U(t)
for i ~Jp and Vt ~ [t0,oo). Defining
y(2d2+1)/ y*(t) =
.
,
A*(t) =
(l'2)d2+l(t) (~v)d.+l(t)
('Yl)d,( t )Qo( t ) (~2)d2( t)Qo( t) B~,o( t ) = ('Yp) dp( t )Qo( t )
K.G. Arvanitis, P.N. Paraskevopoulos / Uniformexactmodelmatching
318
it is obvious that
y*( t) =A*( t)x( t) + B~,o( t)u( t ). Since rank B~,o(t) = m, the matrix /~(t) = B~,T(t)B~,o(t) is nonsingular Vt ~ [t0,oo). Solving the previous equation with respect to u(t) we have
u( t) = B - l ( t)B~T( t ) y * ( t) - B - l ( t)B~,T( t)A*( t)x( t). Introducing the previous equation in (1.1) yields J?(t) = [ A ( t ) - B ( t ) B - l ( t ) B ~ , T ( t ) A * ( t ) ] x ( t )
+B(t)B-l(t)B~,T(t)y*(t),
u( t) = B - l ( t)B~,T( t)y*( t) - B - l ( t)B~,T( t)A*( t)x( t). The above set of equations represent an inverse of system (1.1), in the sense that if y(t) is the output of (1.1) to the initial state x 0 at t o and the input u(t) on [to,~), then the output of the above system to the initial state x 0 at t 0 and input y*(t), is u(t) on [t0,oo). For the class of system where /3 =rn, let j,, for r = 1, 2 . . . . . m, be the indices of the linearly independent rows of the matrices B~,o(t) and M~,o(t). Furthermore, let E be the m ×/3 matrix defined as
E=
. I , where ej~=[0 . . . . . 0 , 1 , 0 . . . . . . 0] ( l o n j~-thposition), V z = I , 2 . . . . . m. ej~
(3.25)
[e;.,/ Multiplying both sides of relations (3.21a) and (3.21b) from the left by E we have B~,0(t) =/ll0*0(t) ¢~(t),
(3.26a)
Bj*o(t) =ll;lj*,o(t)G(t ) -
Y'~ ~14y*_~,~(t)(~)K_l(t ) Qo(t),
Vj>_ 1,
(3.26b)
[K=I
where
Bj,p(t) =EBTR(t ) and
Mj,p( t) = EM~*o(t ) , Vj > 0 and p _> 0.
(3.27)
It is obvious that the matrices /~,o(t) and /I,lo*o(t) are nonsingular. Now, define
Lo(t ) = / ~ , o 1(t)/ll0*0(t).
(3.28)
Introducing (3.28) in (3.26a) and (3.26b) yields G(t) = Lol(t),
(3.29a)
(~)j_,(t)Qo(t)=Lj(t),
Vj>I,
(3.29b)
where L I ( t ) = )l~O*l-l[/141",o( t ) L o l(t) - J ~ o ( t ) ] ,
[
(3.30a)
l
" , Vj> 2. Lj(t) = "M0*j-I(t) Mj*,o(t)L o- l(t) - Y'~ IITIy*--K,K(t)LK(t) -Sj,o(t) "* K=I
(3.30b)
In what follows, the differential system of equations in (3.29b) will be appropriately manipulated to be reduced to an equivalent algebraic system of equations. To this end, we will first establish the following lemma.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
319
Lemma 3.1. The following relationship holds:
( @ ) J - ' ( t ) Q ° ( t ) = ~" ( - 1 ) K j - K - 1
@(t)QK(t)]
, W>_I.
(3.31)
.=0
Proof. We will establish the proof of Lemma 3.1 by using the perfect inductive method. Clearly, for j = 1,
relation (3.33) holds. For j = 2 we have
(@)l(t)Qo(t) = @o)(t)Qo(t ) + @(t)A(t)Qo(t ) = [@(t)Qo(t)](1)+ @(t)A(t)Qo(t ) - @(t)Q~ol)(t) = [@(t)Qo(t)] 0 ) - @(t)Ql(t), where use was made of relations (3.20) and (2.2b). Hence, for j = 2 relation (3.31) holds. Now, assume that relation (3.31) holds for j = p, i.e. assume that
( ~ ) o _ , ( t ) Q o ( t ) = ,=0 ~'~ ( - 1 ) "
1 - 1 [tl~(t)QK(t)](P-K-1)" p -p -K
(3.32)
We will show that relation (3.31) holds for j =p + 1, i.e., p
( , ) , ( t ) Q o ( t ) = ~_,( - 1 ) ' ( .=0
p
p --K
) [ , ( t)Q,(t)] <'-') .
(3.33)
In order to prove (3.33) we make use of relation (3.20) to yield
( ¢ro)p( t )Qo( t ) = ¢lJ(pl)_,(t )Qo( t ) + ¢rOp_l( t ) A( t )Qo( t ) = [ ~op_,( t )Qo( t ) ] (') - ¢rop_,( t )Q,( t ).
(3.34) Introducing relation (3.32) in (3.34) we have
(@),(t)Qo(t)=
(
p-1
~ (-1)" .=0
(
-- 1
pP-K-1
)
[@(t)Q~(t)](P-~-
1)
-tlJp-l(t)Ql(t)"
(3.35)
The last term in the right hand side of (3.35) may further be written as
tlJp_l( t)Ql( t ) = [tl~p_2(t)Ql( t)] (1)- ¢IJ)_2(t)Q2( t),
(3.36)
where in order to derive (3.36) use was made of relations (3.20) and (2.2b). By repeatedly using this procedure we arrive at the following relation:
tl~o_l(t)Ql(t ) = ~ ( - 1 ) " .=1
- 1 [q~(t )QK(t)] (o--) . K
(3.37)
Introducing relation (3.37) in (3.35) we arrive at the desired expression (3.33), where appropriate use was made of the following well known relation
This completes the proof of Lemma 3.1.
[]
320
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
Introduce relation (3.31) of Lemma 3.1 in (3.30b). The resulting relation, in a detailed form is as follows:
dP( t )Qo( t ) = L,( t ),
(3.38a)
[ ~ ( t )Qo( t ) ] (1) - ¢,( t )al( t ) = L2( t ) ,
(3.38b)
[q~( t)Qo (t)](2) _ ( 21)[ q~( t)Ql(/)] (1) + ~i~( t)Q2(t) = L 3 ( t ) ,
(3.38c)
The system of equations (3.38) will next be appropriately manipulated so as to eliminate the derivative operator in ~(t). To this end, substituting relation (3.38a) in (3.38b) yields
~ ( t ) Q l ( t ) = - L 2 ( t ) + L]]'(t).
(3.39)
Substituting relations (3.39) and (3.38a) in (3.38c) yields
• (t)Q2(t)=L3(t)+
(~)L~)(t) - L]2)(t).
(3.40)
By repeatedly using this procedure, the system of equations (3.38) can be transformed to the following equivalent system of equations:
q~(t)Qj_l(t ) = Z i ( t ) ,
Vj>l,
(3.41)
where
Zi(t) = ( - - 1 ) j-1 E ( - - 1 ) J-K j -- 1 L ~ _ K ) ( t )
Zl(t ) :=L,(t),
K=I
j
K
(3.42) '
and where use was made of the definition (3.20). Clearly, the system of equation (3.41) is algebraic, with
respect to its unknown matrix ~ ( t ) . Hence, the system of equations (3.29) has been reduced to the
algebraic system of equations (~(t) = L o l ( t ) ,
(3.43a)
q~(t)Qj_l(t ) = Z i ( t ) ,
Vj> 1,
(3.43b)
Relation (3.43b) may be truncated up to its n first equations, since the following relationship holds: rank[Q0(t)lQl(t)l...lQj(/)] =rank[Q0(t)lQ~(t)l...lQ,_l(t)],
Vj>n.
(3.44)
Upon using (3.44), relations (3.43a) and (3.43b) may finally be written as 6~(t) = L o l ( / ) ,
(3.45a)
q~( t)S( t) = Z( t),
(3.45b)
where
S(t)=[Qo(t)lQl(t)l'"
IQn_l(t)]
and
Z(t)=[Zl(t)lZ2(t)l
... IZn(t)].
(3.46)
Note that the matrix S(t) is the well known uniform state controllability matrix of the open loop system. Clearly, the solution of the uniform exact model matching problem, for the case where rank B~,o(t) = m, is now reduced to that of solving (3.45) for G(t) and ~(t), subject to the condition det t~(t) 4: 0. Note that once t~(t) and ~ ( t ) are determined, one may readily determine G(t) and F(t) by using (3.11) and (3.14) to yield
G(t)=G-I(t)
and
F(t)=t~-](t)~(t).
(3.47)
K.G. Arvanitis, P.N. Paraskevopoulos / Uniformexact'modelmatching
321
Equation (3.45) 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, and the general analytical expressions of the controller matrices G(t) and F(t). For this reason, equation (3.45) may be called the state feedback uniform exact model matching design equation.
3.3. Necessary and sufficient conditions and solution of the problem We first prove the following theorem. Theorem 3.1. Assume that /3 = rank B~.o(t)= m. Then, the necessary and sufficient conditions for the
uniform exact model matching problem to have a solution for G(t) and F(t) are d i = dm,i,
Vi ~Jp, Vt ~ T,
(3.48a)
rank /~,0(t) = rank h40*0(t ) = r a n k [ / ~ , 0 ( t ) I ~t0*,0(t)] = m, r
Vt ~ T,
(3.48b)
~
rank/-S3t?-/= rankS(t),
tz(t)J
Vt ~ T.
(3.48c)
Proof. Clearly, (3.48a) is imposed in order to go from (3.6) to (3.8). Relation (3.48b) is imposed for (3.26a) to hold and furthermore the invertibility of G(t) to be guaranteed, Vt ~ T. Moreover, relation (3.48c) is a necessary and sufficient condition for (3,45b) to have a solution for ~ ( t ) . [] In what follows, under the assumption that the conditions of Theorem 3.1 hold, we will give the solution of equation (3.45) to obtain the general solution for G(t) and ~ ( t ) . Clearly, G(t) is already determined in (3.45a). The general solution for ~ ( t ) may be shown to be
• (t) = A ( t ) S o ( t ) + Z ( t ) s T ( t ) [ S ( t ) s T ( t )
+ So(t)sT(t)]-1
(3.49)
where A(t) is an m x (n - n * ) arbitrary matrix with n* = rank S(t) and So(t) is an (n - n * ) x n matrix whose rows are the linearly independent vectors which are orthogonal to the matrix S(t). We will now establish the following theorem. Theorem 3.2. Under the assumption that the conditions of Theorem 3.1 are satisfied, the general analytical expressions of the exact model matching controller matrices G(t ) and F ( t ) , are
G( t ) = Lo( t ) :=/~,ol(t)i~ro*,o(t),
(3.50a)
F( t) = Lo( t ) A ( t )So( t) + Lo( t)Z( t)S~'( t)[ S( t ) s T ( t ) + So( t)ST( t)] -1
(3.50b)
Proof. Combining relations (3.28) and (3.47), we readily derive (3.50a). Introducing (3.49) in (3.47) and upon using (3.50a), we derive (3.50b). [] Remark 3.1. It is pointed out that when the original system is uniformly controllable, then n* = n. For this case the solution of (3.45) is unique and it has the form: G(t) = Lo(t), F ( t ) =
Lo(t)Z(t)S'r(t)[S(t)sT(t)]- 1. Remark 3.2. In the case where the open-loop system and the desired model are linear time-invariant, the present approach may be applied to solve the exact model matching problem, using obvious modifications. For this case our results reduce to those presented in [10] for the linear time-invariant case, wherein the technique for solving (3.45b) was first presented.
322
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
Remark 3.3. A very interesting feature of our approach is that the conditions of Theorem 3.1 allow a classification of linear time-varying systems. To make this clear, let the desired model M be a linear time-invariant system and the original system be a time-varying plant. In this case, the technique presented in this paper, may readily be applied to derive the controller matrices G(t) and F(t), which when applied to the original time-varying system, bring it to the form of a linear time-invariant system. The following two corollaries, which are direct concequences of Theorem 3.1 and of Remark 3.1, give the mathematical characterization of this classification. Corollary 3.1. A linear time-varying system for which rank B~,o(t)= m, Vt E T, is state feedback equivalent with a linear time-invariant system of T, if and only if the conditions of Theorem 3.1 hold true, where in this case, the matrix ~l,10*0(t) is a time-invariant matrix. Corollary 3.2. A linear time-varying system is state feedback equivalent with a linear controllable time-invariant system on T, if (1) rank B~,o(t) = m, Vt ~ T. (2) d i = d6m, Vt E T. (3) rank[B~.0(t) JM0*0] = rank M*0,0 = m , V t ~ T. , (4) rank S(t) = n, Vt ~ T. Note that the problem of reducing a linear time-varying system to a state feedback equivalent time-invariant system, has already been invastigated by Brunovsky [2], wherein necessary and sufficient conditions were established for this important problem to have a solution. The classification of linear time-varying systems stated by Corollary 3.2, is 'weaker' than the classification given in [2]. However, the classification stated by Corollary 3.1 appears to be new since it is not restricted only to the case of controllable time-invariant systems. Extension of the results of the present paper to the more general problem of uniform exact model matching and the corresponding system classification problems in the case where rank B~,o(t) ~ m is an open problem.
4. U n i f o r m e x a c t m o d e l m a t c h i n g v i a o u t p u t f e e d b a c k
Consider applying to system (1.1) the output feedback law u ( t ) = K ( t ) y ( t ) + G(t)to(t). To solve the present uniform exact model matching problem the procedure presented in Section 3 may readily be extended to yield and
G(t)=Lol(t)
l~(t)C(t)S(t)=Z(t)
(4.1)
where /~(t) = t~(t)K(t), and S(t), Z(t) are defined in (3.46). Clearly, (4.1) corresponds to (3.45). Hence, all the results of the previous section may readily be extended to the present case. A classification of time-varying systems under output feedback, analogous to that under state feedback (see Remark 3.3), may readily be stated as in the following corollary: Corollary 4.1. A linear time-varying system for which rank B~,o(t)= m, Vt ~ T, is output feedback equivalent with a linear time-invariant system on T, if and only if d i = di,m,
(4.2a)
Vt ~ T,
rank[B~.o(t)[/l~o*o] , = rank M"* o,o=m, rank[-C ( t ) S ( t - ) ] = r a n k C ( t ) S ( t ) ,
[ Z(t)
]
Vt~T,
(4.2b)
Vt ~ T.
(4.2c)
Such a type of classification appears to be first in the field.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
323
5. Conclusions In this paper, an new approach is presented for the solution of the uniform exact model matching problem for a class of linear time-varying analytic systems. The particular class of systems studied is restricted by the condition/3 = m, where/3 is defined in (3.24) and m is the number of inputs. Using the proposed approach, the necessary and sufficient conditions for the problem to have a solution (Theorem 3.1) and the general analytical expressions for the controller matrices (Theorem 3.2) have been established. The present technique appears to be very powerful having the following characteristics: (1) It unifies the solution of the exact model matching problem for linear time-invariant [10] and time-varying analytic systems. (2) It unifies the solution of the uniform exact model matching problem via state and output feedback. (3) It reduces the solution of the uniform exact model matching problem to that of solving a linear algebraic matrix equation, thus greatly simplifying the derivation of the analytical expressions of the controller matrices. (4) It gives the possibility to classify linear time-varying systems, as discussed in Remark 3.3.
Acknowledgement The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research & Technology, by the 'Heracles' General Cement Co. of Greece and by the Greek State Scolarship Foundation (I.K.Y.).
References [1] K.G. Arvanitis, New techniques for the design of time-varying and multirate feedback controllers for linear multivariable systems, Ph.D. Thesis, National Technical University of Athens, Depart. of Electr. Eng., under completion. [2] P. Brunovsky, A classification of linear controllable systems, Kybernetika 3 (1970) 173-187. [3] E.W. Kamen, New results in realization theory for linear time-varying analytic systems, IEEE Trans. Automat. Control 24 (1979) 866-878. [4] V. Kucera, Exact model matching, polynomial equation approach, Internat. J. Systems Sci. 12 (1981) 1477-1484. [5] A.K. Majumdar and A.K. Choudhury, On the decoupling of linear multivariable systems, Internat. J. Control 17 (1973) 225-256. [6] B.C. Moore and L.M. Silverman, Model matching by state feedback and dynamic compensation, IEEE Trans. Automat. Control 17 (1972) 491-497. [7] P.N. Paraskevopoulos, Exact transfer-function design using output feedback, Proc. lEE Part D 123 (1976) 831-834. [8] P.N. Paraskevopoulos, Exact model matching of linear time-varying systems, Proc. lEE Part D 125 (1978) 66-70. [9] P.N. Paraskevopoulos, Nonparametric model matching of linear time-invariant multivariable systems, IEEE Trans. Indust. Electr. Contr. Instr. 26 (1979) 234-237. [10] P.N. Paraskevopoulos, F.N. Koumboulis and D.F. Anastasakis, Exact model matching of generalized state space systems, Z Optim. Theory Appl. (to appear) [11] L.M. Silverman, Properties and application of inverse systems, IEEE Trans. Automat. Control 13 (1968) 436-437. [12] S.G. Tzafestas and P.N. Paraskevopoulos, On the exact model matching controller design, IEEE Trans. Automat. Control 21 (1976) 242-246. [13] A.I.G. Vardulakis and N. Karcanias, On the stable exact model matching problem, Systems Control Lett. 9 (1985) 237-242. [14] S.H. Wang and C.A. Desoer, The exact model matching of linear multivariable systems, IEEE Trans. Automat. Control 17 (1972) 347-349. [15] S.H. Wang and E.J. Davison, Solution of the exact model matching problem, IEEE Trans. Automat. Control 17 (1972) 574. [16] S.H. Wang and E.J. Davison, A minimization algorithm for the design of linear multivariable systems, IEEE Trans. Automat. Control 18 (1973) 220-225. [17] W.A. Wolovich, The use of state feedback for exact model matching, SlAM J. Control 10 (1972) 512-523.
313
Uniform exact model matching for a class of linear time-varying analytic systems K.G. Arvanitis and P.N. Paraskevopoulos National Technical Universityof Athens, Division of Computer Science, Department of Electrical Engineering, 157 73 Zographou, Athens, Greece Received 27 January 1991 Revised 22 December 1991 and 25 April 1992
Abstract: The problem of uniform exact model matching is studied via state and output feedback, for a class of linear time-varying analytic systems. The following two major issues are resolved: The nessecary and sufficient conditions for the problem to have a solution and the general analytical expressions for the controller matrices. A major feature of the proposed approach, is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation. Keywords: Exact model matching; linear time-varying analytic systems; state feedback; output feedback; 'generalized' Hankel matrix.
I. Introduction This p a p e r is devoted to the problem of uniform exact model matching of linear time-varying analytic (1.t.v.a.) systems, i.e. for systems described by
ic(t)=A(t)x(t) +B(t)u(t),
y(t)=C(t)x(t),
(1.1)
where A(t) ~ ~nxn, B(t) ~ ~ × " , C(t) ~ R pxn are time-varying matrices with entries in D(T) (the commutative ring of analytic functions defined on the time interval T = (tt, t 2) with values in R). The problem of exact model matching is of both theoretical and practical importance [4, 6-10, 12-17]. The first basic results in the field were reported in [6,14-17], for linear time-invariant systems. Subsequently, a large n u m b e r of papers have been published on the subject, covering several caregories of systems such as linear time-varying, two dimensional, generalized state space, nonlinear, etc. With regard to the exact model matching problem of linear time-varying systems, the respective published results are limited [8]. The approach presented in designing static or dynamic controllers for exact model matching of linear time-varying systems in [8] is based on the algebraic equivalence between the closed-loop system and the model and it reduces the problem to that of solving a linear system of first-order differential equations. In this paper, the concept of uniform exact model matching for linear time-varying analytic systems is introduced and a 'generalized' Hankel matrix approach is presented to solve the problem using state and output static feedback. A brief description of the proposed technique is as follows: Starting with the definition of uniform exact model matching and upon using some algebraic manipulations on this definition, the problem is reduced to that of solving a linear nonhomogeneous algebraic matrix equation. This equation plays a central role in our approach and it is called the uniform exact model matching design equation. On the basis of this equation, the following two aspects of the uniform exact model matching problem are investigated: The neessecary and sufficient conditions for the problem to have a solution and the general analytical expressions of the controller matrices.
Correspondence to: Prof. P.N. Paraskevopoulos, National Technical University of Athens, Division of Computer Science, Department of Electrical Engineering, 157 73 Zographou, Athens, Greece. 0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
314
The motivation for using the above technique is due to the technique proposed in [10] for solving the exact model matching problem for linear time-invariant generalized state space systems. The technique in [10] is based on the idea of appropriately manipulating the basic exact model matching equation so as to reduce the solution of the problem to that of solving an algebraic system of equations involving the Markov parameters of the state and of the output of the open loop system. The results reported in this paper on exact model matching of time-varying systems are superior over known results [8], mainly because here, an algebraic matrix equation, rather than a differential matrix equation, is needed to be solved. Furthermore, no restrictions are imposed on the model or on the orders of the open-loop system and of the model, as compared to [8]. It is mentioned that the present result are part of the material reported in [1].
2. Preliminaries The weighting pattern matrix of system (1.1), under the assumption that the initial state vector is zero, is well known and has the form
W(t, r) = C(t)q~(t, "r)B(r), Vt > r and Vt, r ~ T, where ~ ( t , ~-) is the state transition matrix associated with put description of system (1.1). The following definitions will be useful in the sequel.
(2.1)
A(t). Clearly, (2.1) constitutes the input-out-
Definition 2.1. The sequences of the matrices, {Fo(t)} and {Qo(t)} are defined as
Fp+,(t) =Fp(t)A(t) +Fp~l)(t),
Fo(t ) : = C ( t ) ,
(2.2a)
and
Q p + , ( t ) = - A ( t ) Q p ( t ) + Q p o) ( )t , respectively, where
Qo(t):=B(t),
(2.2b)
P(~)(t) indicates the K-th time-derivative of P(t).
Definition 2.2. The 'generalized' Hankel matrix of the system (1.1) is defined as [3]
H(t)={np,K(t)},
p=l,2,...,
K=l, 2,...,
(2.3)
where
Ho,x( t ) = l'p_l( t )Qo( t ), p > l ,
(2.4a)
Hp,~(t)=-Hp+L~_l(t )+H¢,~_,( (~) t ), V p > l a n d V K > 2 .
(2.4b)
and
Definition 2.3. The 'characteristic numbers' di(t) of the system (1.1) are defined as
di(t) where
=
minp:
~n-1
(.yi)p(t)Qo(t) 4:0, for t ~ T and p = 0 , 1 if(~,i)o(t)Qo(t)=O, f o r t ~ T a n d f o r a l l p ,
n - 1,
Vi~Jp
(2.5)
(~,i)p(t) is the i-th row of the matrix Fo(t) and Jp --- {1, 2 . . . . . p}.
Note that in Definition 2.2, the di's are in general time-varying quantities. However, on the basis of the analyticity assumptions for the matrices A(t), B(t) and C(t) we can divide the interval T into subintervals over which the di's are constant. For simplicity, we assume in this paper that the di's a r e constant Vt ~ T.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
315
3. Uniform exact model matching via state feedback
3.1. Formulation of the problem Consider the 1.t.v.a. system (1.1) with weighting pattern matrix given by (2.1). To this system we apply the following proportional state feedback law
u(t) = F ( t ) x ( t ) +G(t)to(t)
(3.1)
where to(t) ~ ~ " is the new input vector. In order to insure linear independence of the m external inputs [17], G(t) is assumed invertible Vt ~ T. It is further assumed that the entries of G(t) and F(t) belong in D(T). Then, the closed-loop system is
ic(t)=Ac(t)x(t ) +Bc(t)u(t), where A~(t)=A(t)+B(t)F(t) described in state-space as
y(t)=C(t)x(t),
(3.2)
and Bc(t)=B(t)G(t). Let the desired model be a 1.t.v.a. system M,
,rm(t) = A m ( t ) X m ( t ) + B m ( t ) U m ( t ) ,
Ym(t) = C m ( t ) X m ( t )
(3.3)
~ ~ nm>(nm, Bm(t) ~ ~ nm>(m and Cm(t) ~ R p×'m are time-varying matrices with entries in D(T). Let Wc(t, z) and Wm(t, r) be the weighting pattern matrices of the closed-loop system (3.2) and of
where Am(t)
the desired model (3.3), respectively. Then, the problem of uniform exact model matching studied can be stated as follows: Find the matrices G(t) and F(t) such that Wc(t, ~-) and Wm(t, r) are equal for all values of t and ~- which belong to the time interval T, i.e. such that W~(t, ~-) = l'gm(t , "r),
Vt>~" and Vt, ~ ' ~ T .
(3.4)
Note that if relation (3.4) holds only for a single point t~ ~ T, then one may refer to this special problem as the complete exact model matching problem. In this paper we study only the uniform exact model matching problem, since this is, obviously, the general case. Next, an equivalent definition to the above problem will be given. To this end, let H~(t) and Hm(t) be the 'generalized' Hankel matrices of the closed-loop system (3.2) and of the given model (3.3), respectively. The matrices Hc(t) and Hm(t) are defined similarly to H(t) for the open-loop system (1.1) in relation (2.3). It is clear that relation (3.4) is satisfied if and only if the following relation holds [3]:
Hc(t)=Hm(t),
Vt~T.
(3.5)
Also, let (yi,~)p(t)Qc,o(t) and (Yi,m)p(t)Qm,o(t) be the i-th elementary rows of any p-th block row of the first block column of the matrices He(t) and Hm(t), respectively, where (,ri,c)p(t) and ('gi,m)p(t) are the i-th rows of the matrices Fc,p(t) and Fm,p(t), respectively. Note that the matrices Fc,p(t) and Fm,p(t) are defined similarly to the matrices Fp(t) for the open-loop system in (2.2a). Furthermore, the matrices Qc,o(t) and Qm,0(t) are defined similarly to the matrix Qo(t) for the open-loop system in (2.2b). Using the above, the definition of uniform exact model matching (3.5) may be equivalently written as
('ri,c)p( t )Qc.o( t ) = ( ]ti,m)p( t )Qm,o( t ), Vi ~J,, Vp > 0 and Vt ~ T.
(3.6)
In what follows, the uniform exact model matching problem will be solved on the basis of relation (3.6), rather than relations (3.4) or (3.5). The motivation for using (3.6) is that it greatly facilitates the solution of the problem.
3.2. Derivation of the uniform exact model matching desing equation Let dc,i(t) and dm,i(t) be the 'characteristic numbers' of the closed-loop system (3.2) and of the model (3.3), respectively, which are defined similar to the d / s for the open-loop system in (2.5) and which are assumed to be constant Vt ~ T. Clearly, since the 'characteristic numbers' remain invariant under state
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
316
feedback and a non-singular transformation of the inputs [5], a necessary condition for the uniform exact model matching problem via state feedback to have a solution is
di = dc,i = dm,i Vt ~ T, Vi ~ Jp.
(3.7)
Now asume that (3.7) holds. Using (2.5) and (3.7), relation (3.6) may further be written as
( %.c)o( t)Qc.o( t) = (,Yi,m)o( t)Qm,o( t),
Vi ~Jp, Vp > d i and Vt ~ T.
(3.8)
One may readily derive the following relationship [5]:
(ri,c)di(t) = (Yi)d,(t),
Vi ~Jp, Vt E T.
(3.9)
We next study (3.8) for p = d i, d i + 1. . . . . For p = d i, relation (3.8) becomes
(yi,c)d,(t)Qc,o(t) = (Yi,m)d,(t)Qm.o(t),
Vi ~Jp, Vt ~ T.
(3.10)
We introduce the following definitions:
I,t~,di(t ) = (Yi.m)d,(t)Qm,o(t)
and
G(t) =G-l(t).
(3.11)
Introducing (3.9) in (3.10) and upon using definition (3.11), relation (3.10) reduces to
( ~'i)a,( t )Qo( t ) = [~ti,di( t )(J( t ).
(3.12)
For p = d~ + 1, relation (3.8) takes on the form
('Yi.c)a,+~(t)Qc,o(t) = (],i,m)a,+l(t)Qm,o(t),
Vi EJp, Vt ~ T.
(3.13)
=d(t)F(t),
(3.14)
We introduce the following definitions:
iXi,a~+x(t ) = (ri,m)a~+l(t)Om,o(t)
and
q~(t)
Introducing the definition of rc,o(t) and relations (3.9) and (3.14) in (3.13), relation (3.13) becomes
[( ~/i)a,( t )Ac( t ) + ( ~,i)~al~)(t ) ]ao( t ) = I~i,di+ l( t )G( t ).
(3.15)
Further, introducing in (3.15) the definition of Ac(t), relation (3.15) becomes [(r~)a,+l(t) + [(ri)a,(t)Qo(t)]F(t)]Qo(t)
=lzi,d,+l(t)G(t).
(3.16)
Finally, introducing (3.12) and (3.14) in (3.16), the original relation (3.13) reduces to
( ]/i)di+ l( t )ao( t ) = ~i,di+ l( t )G( t ) -- #.i,di( t )clJ( t )Qo( t ).
(3.17)
For any p > d~ + 2, by repeatidly using the above procedure, relation (3.8) reduces to
(r~)d,+j(t)Qo(t)=[l~i,a,+j(t)]od(t)-
lJ;0
I"I~i,di+j--K--I(I)]K+I(CI'))K(t)
]
Qo(t),
Vj > l ,
(3.18)
where
[la.i,ai+j( t)]o
:=
('Yi,m)d,+j( t)Om,o( t),
[ll"i,di+j(t)]K
=
)IK, [~i,di+j(t)]K_l+ [l~i,d~+j-l( t'I(D
[ ~i,a,( t ) ]~ = I~i,ai( t )
:=
(3.19a)
Vj >_ 1 ,
('Yi,m)cli( t )Qm,o( t ),
Vj 2 1 and VK > 1,
(3.19b) (3.19c)
K.G.Arvanitis,P.N.Paraskevopoulos/ Uniformexactmodelmatching
317
and where the sequence of the time-varying analytic matrices {(~)~(t)} is defined as (q))~+l(t) = ( ~ ) ~ ( t ) A ( . t ) + (q~)~)(t),
(q))0(t):=q)(t).
(3.20)
Relation (3.18), together with relation (3.12), constitute an infinite differential system of equations with respect to the unknowns t~(t) and qD(t) (and hence, with respect to the unknown controller matrices G(t) and F(t)). Furthermore, it is pointed out that relations (3.12) and (3.18), hold for all i ~Jp. Grouping appropriately (3.12) and (3.18) for i ~Jt,, we arrive at the following differential system of equations: n~,o(t) = Mo*o( t ) ( ~ ( t ) ,
(3.21a) r
BPo(t) =MJ,*o(t)~(t) - I £ Mj*~.~(t)(@)~_l(t) t K=I
Qo(t),
V j > 1,
(3.21b)
where
-~l'd'+j(t) ]
T1)d'+j(I)QO(t) ] ( ~,2)~+j( t)Qo( t) Bj*o(t)
Vj > O,
M?,o( t ) =
=
( rp) d,+j( t )Qo( t )
(3.22a)
J
M~,,(t) =Mo*o(t), Vp _> 1 and
M~*p(t) =Mj*p_,(t) +M*d)j-l,p~'t), Vj>_ 1 and Vp >_ 1.
(3.22b)
Now, define fl = rank B~,o(t). Note that/3 is in general a time-dependent quantity. However, because of the analyticity assumptions on the matrices used, we can always partition T into subintervals over which /3 is constant. For simplicity in the presentation of our approach, we assume that /3 is constant over T. Clearly, for the system (3.21a) to have a solution for G(t) for all t ~ T, it is necessary and sufficient that rank B~.o( t ) = rank[ B~,o( t ) l Mo*o(t ) ] =/3,
Vt ~ T.
(3.23a)
Furthermore, in order to guarantee that the solution of (3.21a) is in the desired form, i.e. is a nonsingular matrix for all t E T, the following relationship must hold: rank B~,o( t ) = rank Mo*o(t ),
Vt ~ T.
(3.23b)
Clearly, conditions (3.23a) and (3.23b), may be stated as one, as follows: rank B~,o( t ) = rank Mo*,o(t ) = rank[ B~,o( t ) l Mo*o(t ) ] =/3,
Vt ~ T.
(3.24)
The class of l.t.v.a, systems studied in this paper is restricted to the case where/3 = m. Note that for this particular class of time-varying analytic systems and for T = [t0,~) , there exists a certain type of inverse. To make this clear, in what follows the invertibility result due to Silverman for scalar systems [11] will be generalized to the multivariable case. To this end, using the definition of d i, one may readily establish that
y~P)(t) = ( r i ) p ( t ) x ( t )
and
y~di+l)(t) = ('Yi)d,+l(t)x(t) + ('gi)d,(t)Qo(t)U(t)
for i ~Jp and Vt ~ [t0,oo). Defining
y(2d2+1)/ y*(t) =
.
,
A*(t) =
(l'2)d2+l(t) (~v)d.+l(t)
('Yl)d,( t )Qo( t ) (~2)d2( t)Qo( t) B~,o( t ) = ('Yp) dp( t )Qo( t )
K.G. Arvanitis, P.N. Paraskevopoulos / Uniformexactmodelmatching
318
it is obvious that
y*( t) =A*( t)x( t) + B~,o( t)u( t ). Since rank B~,o(t) = m, the matrix /~(t) = B~,T(t)B~,o(t) is nonsingular Vt ~ [t0,oo). Solving the previous equation with respect to u(t) we have
u( t) = B - l ( t)B~T( t ) y * ( t) - B - l ( t)B~,T( t)A*( t)x( t). Introducing the previous equation in (1.1) yields J?(t) = [ A ( t ) - B ( t ) B - l ( t ) B ~ , T ( t ) A * ( t ) ] x ( t )
+B(t)B-l(t)B~,T(t)y*(t),
u( t) = B - l ( t)B~,T( t)y*( t) - B - l ( t)B~,T( t)A*( t)x( t). The above set of equations represent an inverse of system (1.1), in the sense that if y(t) is the output of (1.1) to the initial state x 0 at t o and the input u(t) on [to,~), then the output of the above system to the initial state x 0 at t 0 and input y*(t), is u(t) on [t0,oo). For the class of system where /3 =rn, let j,, for r = 1, 2 . . . . . m, be the indices of the linearly independent rows of the matrices B~,o(t) and M~,o(t). Furthermore, let E be the m ×/3 matrix defined as
E=
. I , where ej~=[0 . . . . . 0 , 1 , 0 . . . . . . 0] ( l o n j~-thposition), V z = I , 2 . . . . . m. ej~
(3.25)
[e;.,/ Multiplying both sides of relations (3.21a) and (3.21b) from the left by E we have B~,0(t) =/ll0*0(t) ¢~(t),
(3.26a)
Bj*o(t) =ll;lj*,o(t)G(t ) -
Y'~ ~14y*_~,~(t)(~)K_l(t ) Qo(t),
Vj>_ 1,
(3.26b)
[K=I
where
Bj,p(t) =EBTR(t ) and
Mj,p( t) = EM~*o(t ) , Vj > 0 and p _> 0.
(3.27)
It is obvious that the matrices /~,o(t) and /I,lo*o(t) are nonsingular. Now, define
Lo(t ) = / ~ , o 1(t)/ll0*0(t).
(3.28)
Introducing (3.28) in (3.26a) and (3.26b) yields G(t) = Lol(t),
(3.29a)
(~)j_,(t)Qo(t)=Lj(t),
Vj>I,
(3.29b)
where L I ( t ) = )l~O*l-l[/141",o( t ) L o l(t) - J ~ o ( t ) ] ,
[
(3.30a)
l
" , Vj> 2. Lj(t) = "M0*j-I(t) Mj*,o(t)L o- l(t) - Y'~ IITIy*--K,K(t)LK(t) -Sj,o(t) "* K=I
(3.30b)
In what follows, the differential system of equations in (3.29b) will be appropriately manipulated to be reduced to an equivalent algebraic system of equations. To this end, we will first establish the following lemma.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
319
Lemma 3.1. The following relationship holds:
( @ ) J - ' ( t ) Q ° ( t ) = ~" ( - 1 ) K j - K - 1
@(t)QK(t)]
, W>_I.
(3.31)
.=0
Proof. We will establish the proof of Lemma 3.1 by using the perfect inductive method. Clearly, for j = 1,
relation (3.33) holds. For j = 2 we have
(@)l(t)Qo(t) = @o)(t)Qo(t ) + @(t)A(t)Qo(t ) = [@(t)Qo(t)](1)+ @(t)A(t)Qo(t ) - @(t)Q~ol)(t) = [@(t)Qo(t)] 0 ) - @(t)Ql(t), where use was made of relations (3.20) and (2.2b). Hence, for j = 2 relation (3.31) holds. Now, assume that relation (3.31) holds for j = p, i.e. assume that
( ~ ) o _ , ( t ) Q o ( t ) = ,=0 ~'~ ( - 1 ) "
1 - 1 [tl~(t)QK(t)](P-K-1)" p -p -K
(3.32)
We will show that relation (3.31) holds for j =p + 1, i.e., p
( , ) , ( t ) Q o ( t ) = ~_,( - 1 ) ' ( .=0
p
p --K
) [ , ( t)Q,(t)] <'-') .
(3.33)
In order to prove (3.33) we make use of relation (3.20) to yield
( ¢ro)p( t )Qo( t ) = ¢lJ(pl)_,(t )Qo( t ) + ¢rOp_l( t ) A( t )Qo( t ) = [ ~op_,( t )Qo( t ) ] (') - ¢rop_,( t )Q,( t ).
(3.34) Introducing relation (3.32) in (3.34) we have
(@),(t)Qo(t)=
(
p-1
~ (-1)" .=0
(
-- 1
pP-K-1
)
[@(t)Q~(t)](P-~-
1)
-tlJp-l(t)Ql(t)"
(3.35)
The last term in the right hand side of (3.35) may further be written as
tlJp_l( t)Ql( t ) = [tl~p_2(t)Ql( t)] (1)- ¢IJ)_2(t)Q2( t),
(3.36)
where in order to derive (3.36) use was made of relations (3.20) and (2.2b). By repeatedly using this procedure we arrive at the following relation:
tl~o_l(t)Ql(t ) = ~ ( - 1 ) " .=1
- 1 [q~(t )QK(t)] (o--) . K
(3.37)
Introducing relation (3.37) in (3.35) we arrive at the desired expression (3.33), where appropriate use was made of the following well known relation
This completes the proof of Lemma 3.1.
[]
320
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
Introduce relation (3.31) of Lemma 3.1 in (3.30b). The resulting relation, in a detailed form is as follows:
dP( t )Qo( t ) = L,( t ),
(3.38a)
[ ~ ( t )Qo( t ) ] (1) - ¢,( t )al( t ) = L2( t ) ,
(3.38b)
[q~( t)Qo (t)](2) _ ( 21)[ q~( t)Ql(/)] (1) + ~i~( t)Q2(t) = L 3 ( t ) ,
(3.38c)
The system of equations (3.38) will next be appropriately manipulated so as to eliminate the derivative operator in ~(t). To this end, substituting relation (3.38a) in (3.38b) yields
~ ( t ) Q l ( t ) = - L 2 ( t ) + L]]'(t).
(3.39)
Substituting relations (3.39) and (3.38a) in (3.38c) yields
• (t)Q2(t)=L3(t)+
(~)L~)(t) - L]2)(t).
(3.40)
By repeatedly using this procedure, the system of equations (3.38) can be transformed to the following equivalent system of equations:
q~(t)Qj_l(t ) = Z i ( t ) ,
Vj>l,
(3.41)
where
Zi(t) = ( - - 1 ) j-1 E ( - - 1 ) J-K j -- 1 L ~ _ K ) ( t )
Zl(t ) :=L,(t),
K=I
j
K
(3.42) '
and where use was made of the definition (3.20). Clearly, the system of equation (3.41) is algebraic, with
respect to its unknown matrix ~ ( t ) . Hence, the system of equations (3.29) has been reduced to the
algebraic system of equations (~(t) = L o l ( t ) ,
(3.43a)
q~(t)Qj_l(t ) = Z i ( t ) ,
Vj> 1,
(3.43b)
Relation (3.43b) may be truncated up to its n first equations, since the following relationship holds: rank[Q0(t)lQl(t)l...lQj(/)] =rank[Q0(t)lQ~(t)l...lQ,_l(t)],
Vj>n.
(3.44)
Upon using (3.44), relations (3.43a) and (3.43b) may finally be written as 6~(t) = L o l ( / ) ,
(3.45a)
q~( t)S( t) = Z( t),
(3.45b)
where
S(t)=[Qo(t)lQl(t)l'"
IQn_l(t)]
and
Z(t)=[Zl(t)lZ2(t)l
... IZn(t)].
(3.46)
Note that the matrix S(t) is the well known uniform state controllability matrix of the open loop system. Clearly, the solution of the uniform exact model matching problem, for the case where rank B~,o(t) = m, is now reduced to that of solving (3.45) for G(t) and ~(t), subject to the condition det t~(t) 4: 0. Note that once t~(t) and ~ ( t ) are determined, one may readily determine G(t) and F(t) by using (3.11) and (3.14) to yield
G(t)=G-I(t)
and
F(t)=t~-](t)~(t).
(3.47)
K.G. Arvanitis, P.N. Paraskevopoulos / Uniformexact'modelmatching
321
Equation (3.45) 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, and the general analytical expressions of the controller matrices G(t) and F(t). For this reason, equation (3.45) may be called the state feedback uniform exact model matching design equation.
3.3. Necessary and sufficient conditions and solution of the problem We first prove the following theorem. Theorem 3.1. Assume that /3 = rank B~.o(t)= m. Then, the necessary and sufficient conditions for the
uniform exact model matching problem to have a solution for G(t) and F(t) are d i = dm,i,
Vi ~Jp, Vt ~ T,
(3.48a)
rank /~,0(t) = rank h40*0(t ) = r a n k [ / ~ , 0 ( t ) I ~t0*,0(t)] = m, r
Vt ~ T,
(3.48b)
~
rank/-S3t?-/= rankS(t),
tz(t)J
Vt ~ T.
(3.48c)
Proof. Clearly, (3.48a) is imposed in order to go from (3.6) to (3.8). Relation (3.48b) is imposed for (3.26a) to hold and furthermore the invertibility of G(t) to be guaranteed, Vt ~ T. Moreover, relation (3.48c) is a necessary and sufficient condition for (3,45b) to have a solution for ~ ( t ) . [] In what follows, under the assumption that the conditions of Theorem 3.1 hold, we will give the solution of equation (3.45) to obtain the general solution for G(t) and ~ ( t ) . Clearly, G(t) is already determined in (3.45a). The general solution for ~ ( t ) may be shown to be
• (t) = A ( t ) S o ( t ) + Z ( t ) s T ( t ) [ S ( t ) s T ( t )
+ So(t)sT(t)]-1
(3.49)
where A(t) is an m x (n - n * ) arbitrary matrix with n* = rank S(t) and So(t) is an (n - n * ) x n matrix whose rows are the linearly independent vectors which are orthogonal to the matrix S(t). We will now establish the following theorem. Theorem 3.2. Under the assumption that the conditions of Theorem 3.1 are satisfied, the general analytical expressions of the exact model matching controller matrices G(t ) and F ( t ) , are
G( t ) = Lo( t ) :=/~,ol(t)i~ro*,o(t),
(3.50a)
F( t) = Lo( t ) A ( t )So( t) + Lo( t)Z( t)S~'( t)[ S( t ) s T ( t ) + So( t)ST( t)] -1
(3.50b)
Proof. Combining relations (3.28) and (3.47), we readily derive (3.50a). Introducing (3.49) in (3.47) and upon using (3.50a), we derive (3.50b). [] Remark 3.1. It is pointed out that when the original system is uniformly controllable, then n* = n. For this case the solution of (3.45) is unique and it has the form: G(t) = Lo(t), F ( t ) =
Lo(t)Z(t)S'r(t)[S(t)sT(t)]- 1. Remark 3.2. In the case where the open-loop system and the desired model are linear time-invariant, the present approach may be applied to solve the exact model matching problem, using obvious modifications. For this case our results reduce to those presented in [10] for the linear time-invariant case, wherein the technique for solving (3.45b) was first presented.
322
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
Remark 3.3. A very interesting feature of our approach is that the conditions of Theorem 3.1 allow a classification of linear time-varying systems. To make this clear, let the desired model M be a linear time-invariant system and the original system be a time-varying plant. In this case, the technique presented in this paper, may readily be applied to derive the controller matrices G(t) and F(t), which when applied to the original time-varying system, bring it to the form of a linear time-invariant system. The following two corollaries, which are direct concequences of Theorem 3.1 and of Remark 3.1, give the mathematical characterization of this classification. Corollary 3.1. A linear time-varying system for which rank B~,o(t)= m, Vt E T, is state feedback equivalent with a linear time-invariant system of T, if and only if the conditions of Theorem 3.1 hold true, where in this case, the matrix ~l,10*0(t) is a time-invariant matrix. Corollary 3.2. A linear time-varying system is state feedback equivalent with a linear controllable time-invariant system on T, if (1) rank B~,o(t) = m, Vt ~ T. (2) d i = d6m, Vt E T. (3) rank[B~.0(t) JM0*0] = rank M*0,0 = m , V t ~ T. , (4) rank S(t) = n, Vt ~ T. Note that the problem of reducing a linear time-varying system to a state feedback equivalent time-invariant system, has already been invastigated by Brunovsky [2], wherein necessary and sufficient conditions were established for this important problem to have a solution. The classification of linear time-varying systems stated by Corollary 3.2, is 'weaker' than the classification given in [2]. However, the classification stated by Corollary 3.1 appears to be new since it is not restricted only to the case of controllable time-invariant systems. Extension of the results of the present paper to the more general problem of uniform exact model matching and the corresponding system classification problems in the case where rank B~,o(t) ~ m is an open problem.
4. U n i f o r m e x a c t m o d e l m a t c h i n g v i a o u t p u t f e e d b a c k
Consider applying to system (1.1) the output feedback law u ( t ) = K ( t ) y ( t ) + G(t)to(t). To solve the present uniform exact model matching problem the procedure presented in Section 3 may readily be extended to yield and
G(t)=Lol(t)
l~(t)C(t)S(t)=Z(t)
(4.1)
where /~(t) = t~(t)K(t), and S(t), Z(t) are defined in (3.46). Clearly, (4.1) corresponds to (3.45). Hence, all the results of the previous section may readily be extended to the present case. A classification of time-varying systems under output feedback, analogous to that under state feedback (see Remark 3.3), may readily be stated as in the following corollary: Corollary 4.1. A linear time-varying system for which rank B~,o(t)= m, Vt ~ T, is output feedback equivalent with a linear time-invariant system on T, if and only if d i = di,m,
(4.2a)
Vt ~ T,
rank[B~.o(t)[/l~o*o] , = rank M"* o,o=m, rank[-C ( t ) S ( t - ) ] = r a n k C ( t ) S ( t ) ,
[ Z(t)
]
Vt~T,
(4.2b)
Vt ~ T.
(4.2c)
Such a type of classification appears to be first in the field.
K.G. Arvanitis, P.N. Paraskevopoulos / Uniform exact model matching
323
5. Conclusions In this paper, an new approach is presented for the solution of the uniform exact model matching problem for a class of linear time-varying analytic systems. The particular class of systems studied is restricted by the condition/3 = m, where/3 is defined in (3.24) and m is the number of inputs. Using the proposed approach, the necessary and sufficient conditions for the problem to have a solution (Theorem 3.1) and the general analytical expressions for the controller matrices (Theorem 3.2) have been established. The present technique appears to be very powerful having the following characteristics: (1) It unifies the solution of the exact model matching problem for linear time-invariant [10] and time-varying analytic systems. (2) It unifies the solution of the uniform exact model matching problem via state and output feedback. (3) It reduces the solution of the uniform exact model matching problem to that of solving a linear algebraic matrix equation, thus greatly simplifying the derivation of the analytical expressions of the controller matrices. (4) It gives the possibility to classify linear time-varying systems, as discussed in Remark 3.3.
Acknowledgement The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research & Technology, by the 'Heracles' General Cement Co. of Greece and by the Greek State Scolarship Foundation (I.K.Y.).
References [1] K.G. Arvanitis, New techniques for the design of time-varying and multirate feedback controllers for linear multivariable systems, Ph.D. Thesis, National Technical University of Athens, Depart. of Electr. Eng., under completion. [2] P. Brunovsky, A classification of linear controllable systems, Kybernetika 3 (1970) 173-187. [3] E.W. Kamen, New results in realization theory for linear time-varying analytic systems, IEEE Trans. Automat. Control 24 (1979) 866-878. [4] V. Kucera, Exact model matching, polynomial equation approach, Internat. J. Systems Sci. 12 (1981) 1477-1484. [5] A.K. Majumdar and A.K. Choudhury, On the decoupling of linear multivariable systems, Internat. J. Control 17 (1973) 225-256. [6] B.C. Moore and L.M. Silverman, Model matching by state feedback and dynamic compensation, IEEE Trans. Automat. Control 17 (1972) 491-497. [7] P.N. Paraskevopoulos, Exact transfer-function design using output feedback, Proc. lEE Part D 123 (1976) 831-834. [8] P.N. Paraskevopoulos, Exact model matching of linear time-varying systems, Proc. lEE Part D 125 (1978) 66-70. [9] P.N. Paraskevopoulos, Nonparametric model matching of linear time-invariant multivariable systems, IEEE Trans. Indust. Electr. Contr. Instr. 26 (1979) 234-237. [10] P.N. Paraskevopoulos, F.N. Koumboulis and D.F. Anastasakis, Exact model matching of generalized state space systems, Z Optim. Theory Appl. (to appear) [11] L.M. Silverman, Properties and application of inverse systems, IEEE Trans. Automat. Control 13 (1968) 436-437. [12] S.G. Tzafestas and P.N. Paraskevopoulos, On the exact model matching controller design, IEEE Trans. Automat. Control 21 (1976) 242-246. [13] A.I.G. Vardulakis and N. Karcanias, On the stable exact model matching problem, Systems Control Lett. 9 (1985) 237-242. [14] S.H. Wang and C.A. Desoer, The exact model matching of linear multivariable systems, IEEE Trans. Automat. Control 17 (1972) 347-349. [15] S.H. Wang and E.J. Davison, Solution of the exact model matching problem, IEEE Trans. Automat. Control 17 (1972) 574. [16] S.H. Wang and E.J. Davison, A minimization algorithm for the design of linear multivariable systems, IEEE Trans. Automat. Control 18 (1973) 220-225. [17] W.A. Wolovich, The use of state feedback for exact model matching, SlAM J. Control 10 (1972) 512-523.