IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
A Necessary Algebraic Condition for Controllability and Observability of Linear Time-Varying Systems Hugo Leiva and Stefan Siegmund Abstract—In this note, we give an algebraic condition which is necessary ( ) ( ) + ( ) ( ), ( ) = ( ) ( ), for the system ( ) = either to be totally controllable or to be totally observable, where , , , and the matrix functions , and are ( 2), ( 1) and ( 1) times continuously differentiable, respectively. All conditions presented here are in terms of known quantities and therefore easily verified. Our conditions can be used to rule out large classes of time-varying systems which cannot be controlled and/or observed no matter what the nonzero time-varying coefficients are. This work is motivated by the deep result of Silverman and Meadows.
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ii) Totally controllable on (0, T ) if it is completely controllable on every subinterval of (0, T ). Definition 1.2 (Completely/Totally Observable): Let T > 0. Then, (1.1) is said to be the following. i) Completely observable on (0, T ) if any initial state x0 at t = 0 can be determined from the knowledge of the system output y (t) and the control u(1) on [0, T ]. ii) Totally observable on (0, T ) if it is completely observable on every subinterval of (0, T ). It is well known (see, e.g., [2]) that controllability and observability properties of (1.1) in [0; T ], T > 0, can be fully determined by analyzing the following Gramian matrices:
( )=
W T
Index Terms—Algebraic condition, linear time-varying control systems, noncontrollability, nonobservability.
( )=
T 0
In this note, we give a necessary algebraic condition for the controllability and observability associated with the time-varying control system x
0 (t) = A(t)x(t) + B (t)u(t);
( )= ( ) ( )
y t
where x(t) 2 , y (t) 2 and the control hypothesis in this note is that: (H) The matrix functions d
q
C t x t
()2
p
u t
t
A t
t
C t
d
;
d
t
B t
d
u
u
L
;
Under this hypothesis, the unique solution of the differential equation (1.1) with the initial condition x0 at time t = 0, is given by t
( ) = 8( ) 0 +
x t
t x
0
8(t)8(s)01 B (s)u(s) ds;
t
2
where 8(t) is the fundamental matrix of the linear system x0 i.e.,
80 ( ) = ( )8( ) t
A t
t ;
8(0) =
x
. The main
;
(1.2)
= ()
A t x
t
t
dt;
8( )3 3 ( ) ( )8( ) t
0
B t B
C
t C t
t dt
respectively. However, the fundamental matrix 8(t) is rarely known in closed form and computing W (t), Z (t) is not a happy prospect, e.g., in [7] a sufficient differential-algebraic condition for controllability and observability of the particular system
(1.1)
3 7! ( ) 2 d2d 3 7! ( ) 2 d2p and 3 7! ( ) 2 q2d are ( 0 2), ( 0 1) and ( 0 1) times continuously differentiable, respectively, and the control function (1) is a locally integrable function p ). on , i.e., 2 1loc (
t
T
Z T
I. INTRODUCTION
8( )01 ( ) 3 ( )[8( )3 ]01
0 (t) = A(t)x(t) + Bu(t); y(t) = C x(t); A(t) =
l i=1
i (t)Ai
a
is given under the hypothesis that i) the scalar functions ai (1) are C 1 and ii) the Lie-algebra, generated by the constant matrices A1 ; . . . ; Al under the commutator product [Ai ; Aj ] = Ai Aj 0 Aj Ai , has dimension d, which implies (see, e.g., [8]) that the fundamental matrix 8(t), at least locally, can be represented as a product of exponential matrices
8( ) = g (t)A g (t)A 1 1 1 g (t)A where = ( 1 2 . . . l ) is the solution of a nonlinear differential equation 0 ( ) = ( ( )), (0) = 0. Moreover, it is assumed that g
g
t
e
g ;g ;
;g
t
f t; g t
e
e
g
is differential-algebraically independent, i.e., for each k exists no nontrivial polynomial p with
g
(
2
there
0 00 ; . . . ; g(k) ) = 0:
p g; g ; g
,
I:
Definition 1.1 (Completely/Totally Controllable): Let T > 0. Then system (1.1) is said to be the following. i) Completely controllable on (0, T ) if for any initial state x0 at t = 0, and any desired final state x1 at t = T , there exists a 1 locally integrable control u 2 Lloc ((0; T ); p ) defined on the interval [0, T ] such that the corresponding solution x(1) of (1.1) satisfies x(T ) = x1 .
Therefore, while necessary and sufficient controllability conditions for (1.1) are easily established, often these conditions are of limited practicality. The breakthrough work of Silverman and Meadows [6] shows under the Hypothesis (H) that the total controllability and total observability of (1.1) can be characterized in terms of A(t), B (t), C (t) and their appropriate derivatives. Since these matrices are a priori known, these results provide a means of verifying the controllability and observability tests that are easily attainable. For the sake of convenience, we formulate the Silverman–Meadows results. To this end, define the controllability matrix of (1.1)
c (t) = [P0 (t)jP1 (t)j . . . jPd01 (t)]
Q
Manuscript received April 28, 2003; revised July 25, 2003. Recommended by Associate Editor F. M. Callier. This work was performed while the second author was an EMMY NOETHER FELLOW supported by the German Research Council (DFG) under Grant Si801. H. Leiva is with the Department of Mathematics, Universidad de Los Andes, Merida, Venezuela (e-mail:
[email protected]). S. Siegmund is with the FB Mathematik, J.W. Goethe-Universität, 60054 Frankfurt a.M., Germany (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2003.820145
where
( )= ( ) k+1 ( )= 0 ( ) k ( )+ k0 ( )
P0 t
B t ; P
t
A t P
t
P
t ; k
= 0 . . . 02
The observability matrix is defined by
0018-9286/03$17.00 © 2003 IEEE
o (t) = [S0 (t)jS1 (t)j . . . jSd01 (t)]
Q
;
;d
:
(1.3)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
where
b) Assume that the condition
S0 (t) = C
3
(t);
Sk
+1 (t) = 0A(t)Sk (t)+ Sk0 (t);
k = 0;
...;d
02
n
(1.4) The following theorem gives sufficient conditions for complete controllability and observability of system (1.1) and a characterization of total controllability and observability. Theorem 1.3 (Silverman–Meadows [6]): On the interval (0, T ), (1.1) is a) completely controllable if rankQc (t) = d for some t 2 (0; T ); b) totally controllable if and only if rankQc (t) = d on any subinterval of (0, T ); c) completely observable if rankQo (t) = d for some t 2 (0; T ); d) totally observable if and only if rankQo (t) = d on any subinterval of (0, T ). The work of Silverman and Meadows [6] was done more than 30 years ago and has become so highly regarded that it can be found in virtually every graduate level linear system textbook. A recent approach using a module-theoretic framework can be found in [1]. One might ask, how could it be possible to improve on results which are so easy to verify? On the other hand, if one wants to show that system (1.1) is not totally controllable and/or not totally observable then one has to check that the matrices Qc (t) and Qo (t), respectively, do not have full rank for t in every subinterval in (0, T ). The goal of this note is to use Theorem 1.3 to find checkable conditions in terms of the matrices A(t), B (t) and C (t) which can easily be verified and ensure the noncontrollability and/or nonobservability of system (1.1). To achieve this aim we separate the time-dependence of the matrices A(t), B (t) and C (t) from their linear time-independent structure. Although our representation is not unique, it provides a useful tool to rule out large classes of systems (1.1) which are not totally controllable and totally observable, respectively.
j
l
i
=1
=
bi (t)Bi ; C (t)
=
i
=1
m
j
=1
l
ImAi
...;i =1
111
Ai Bj
+
i ;
i
=1
ImAi
Bj +ImBj
d
holds, where ImA denotes the image of the operator (1.1) is not totally controllable on ( 0; T ).
rank[Ai
Ai
3 3 rank[A A i
i
111 : 3 3 111 : Ai Bj
Ai Cj
k k
A
(2.2) . Then,
= 0; 1; . . . ; d
j
j
q
(2.3)
. . . ; i1 ; j );
fk (ik
01 ; . . . ; i1 ; j );
...;
1 ; j );
fk (i
fk (j )
ik ; . . . ; i1 2 f1; . . . ; lg, j 2 f1; . . . ; mg, of functions from to , which are polynomials in a1 (1); . . . ; al (1), b1 (1); . . . ; bm (1) and their derivatives up to order (d 0 2) and (d 0 1), respectively, such that the matrix functions Pk (t), k = 0; . . . ; d 0 1, given by (1.3), can be written as shown in (2.4) at the bottom of the next page, where the sum li ;...;i =1 is defined to be 0, if k = 0. b) For every k 2 f0; . . . ; d 0 1g, there exists a family
ci (t)Ci
l
1 1 1+
=1
which are the well-know necessary and sufficient Kalman conditions for noncontrollability and nonobservability, respectively (see, e.g., [2]). To prove Theorem 2.1 we need the following lemma. Lemma 2.3: a) For every k 2 f0; . . . ; d 0 1g, there exists a family
i
(2.1) with 1 l d2 , 1 m dp, 1 n qd, and where the ai (1), bi (1) and ci (1) are (d 0 2), (d 0 1) and (d 0 1) times continuously differentiable scalar functions in (0, T ), and Ai , Bi , Ci are constant matrices. Then, we have the following main result. Theorem 2.1 (Noncontrollability and Nonobservability): a) Assume that the condition
i
3 C 3 +ImC 3
ImAi
jABj 1 1 1 jAd01 B] 0 for all t 2 [0; T ]. It is possible to derive a similar condition from the proof of our Theorem 2.1, but the result in [4] is stronger. iii) If system (1.1) is time-independent then (2.1) holds with l = m = n = 1 and a1 (1) = b1 (1) = c1 (1) 1 and conditions (2.2) and (2.3) can be written, respectively, as
II. MAIN RESULTS
A(t)
l
:
gk (ik ;
. . . ; i1 ; j );
gk (ik
01 ; . . . ; i1 ; j );
...;
1 ; j );
gk (i
gk (j )
. . . ; i1 2 f1; . . . ; lg, j 2 f1; . . . ; ng, of functions from to , which are polynomials in a1 (1); . . . ; al (1), c1 (1); . . . ; cn (1) and their derivatives up to order (d 0 2) and (d 0 1), respectively, such that the matrix functions Sk (t), k = 0; . . . ; d 0 1, given by (1.4), can be written as shown in the second equation at the bottom of the next page. Proof: We prove only a), since b) is analogous. We proceed by induction on k 2 f0; . . . ; d 0 1g. For k = 0 we get m b (t)Bj , hence f0 (j )(t) = bj (t). Now assume P0 (t) = j =1 j that the representation (2.4) holds for a k 2 f0; . . . ; d 0 2g. We show that (2.4) holds for k + 1 with corresponding scalar functions fk+1 (ik+1 ; ik ; . . . ; i1 ; j ); . . . ; fk+1 (i1 ; j ); fk+1 (j ). Therefore, we ik ;
0 1; 0 1;
= 0; 1; . . . ; d
j j
= 1; . . . ; m; ir6=0 = 1; . . . ; l] = 1; . . . ; n; ir6=0 = 1; . . . ; l]
