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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
also like to gratefully acknowledge the helpful comments of an anonymous reviewer who suggested Corollary 4.1 and a simplification of the
proof of Theorem 2.1.
Finite-Time Control of Discrete-Time Linear Systems
REFERENCES
Abstract—In this note, we consider the finite-time stabilization of
discrete-time linear systems subject to disturbances generated by an
exosystem. Finite-time stability can be used in all those applications
where large values of the state should not be attained, for instance in
the presence of saturations. The main result provided in the note is a
sufficient condition for finite-time stabilization via state feedback. This
result is then used to find some sufficient conditions for the existence of
an output feedback controller guaranteeing finite-time stability. All the
conditions are then reduced to feasibility problems involving linear matrix
inequalities (LMIs). Some numerical examples are presented to illustrate
the proposed methodology.
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[3] R. Bhatia, Matrix Analysis. New York: Springer-Verlag, 1997.
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[5] M. K. Çamlibel, J. C. Willems, and M. N. Belur, “On the dissipativity
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[6] J.
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“On
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[7] P. Faurre, M. Clerget, and F. Germain, Opérateurs Rationnels Positifs. Paris, France: Dunod, 1979.
[8] A. Ferrante and L. Pandolfi, “On the solvability of the positive real
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[9] T. Iwasaki, S. Hara, and H. Yamauchi, “Structure/control design integration with finite frequency positive real property,” in Proc. Amer. Control
Conf., Chicago, IL, 2000, pp. 549–553.
[10] T. Kailath, Linear System Theory. Upper Saddle River, NJ: PrenticeHall, 1980.
[11] R. E. Kalman, “Lyapunov functions for the problem of Lur’e in automatic control,” in Proc. Nat. Acad. Sci., vol. 49, 1963, pp. 201–205.
[12] P. Lancaster and R. Rodman, Algebraic Riccati Equation. Oxford,
U.K.: Oxford Science Public, 1995.
[13] A. Lindquist and G. Picci, “A geometric approach to modeling and estimation of linear stochastic systems,” J. Math. Syst., Estimat., Control,
vol. 1, pp. 241–333, 1991.
[14] G. Meinsma, Y. Shrivastava, and M. Fu, “A dual formulation of mixed
and on the losslessness of (
) scaling,” IEEE Trans. Autom.
Control, vol. 42, no. 7, pp. 1032–1036, Jul. 1997.
[15] A. A. Nudel’man and N. A. Schwartzman, “On the existence of the solutions to certain operatorial inequalities” (in Russian), Sib. Math. Z., vol.
16, pp. 562–571, 1975.
[16] Y. Oono and K. Yasuura, “Synthesis of finite passive 2 -terminal networks with prescribed scattering matrices,” Mem. Eng. Faculty, Kyushu
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[17] L. Pandolfi, “An observation on the positive real Lemma,” J. Math. Anal.
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[18] V. M. Popov, “Absolute stability of nonlinear systems of automatic control” (in Russian), Avt. i Telemekh., vol. 22, pp. 961–979, 1961.
[19] A. Rantzer, “On the Kalman–Yakubovich–Popov Lemma,” Syst. Control
Lett., vol. 28, pp. 7–10, 1996.
[20] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Autom. Control, vol. AC-16, no. 6,
pp. 621–634, Dec. 1971.
[21] V. A. Yakubovich, “The frequency theorem in control theory,” Siberian
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[22] D. C. Youla, “On the factorization of rational matrices,” IRE Trans. Inf.
Theory, vol. IT-7, pp. 172–189, 1961.
Francesco Amato and Marco Ariola
Index Terms—Finite-time stability, linear systems, output feedback, state
feedback.
I. INTRODUCTION
When dealing with the stability of a system, a distinction should
be made between classical Lyapunov stability and finite-time stability
(FTS) (or short-time stability). The concept of Lyapunov asymptotic
stability is largely known to the control community; conversely a
system is said to be finite-time stable if, once we fix a time-interval, its
state does not exceeds some bounds during this time-interval. Often
asymptotic stability is enough for practical applications, but there are
some cases where large values of the state are not acceptable, for
instance in the presence of saturations. In these cases, we need to
check that these unacceptable values are not attained by the state; for
these purposes FTS could be used.
Most of the results in the literature are focused on Lyapunov stability. Some early results on FTS can be found in [1], [2] and [3]; more
recently the concept of FTS has been revisited in the light of recent results coming from linear matrix inequalities (LMIs) theory, which has
made it possible to find less conservative conditions for guaranteeing
FTS and finite time stabilization of uncertain, linear continuous-time
systems (see [4]).
Conversely, in this note, we deal with discrete-time systems. The
first result is a sufficient condition for the existence of a state feedback
controller which guarantees the finite-time stabilization of a discretetime linear system subject to disturbances generated by an exosystem.
This result is then used to find a possible solution to the static output
feedback finite-time stabilization problem. It is finally shown how these
conditions can be turned into LMIs based feasibility problems.
The note is organized as follows. In Section II, the definition of finite-time stability is recalled and specialized to the discrete-time case,
and the problem we want to solve is formally stated. In Section III,
the first main result of this note, a sufficient condition for the existence
of a state feedback controller guaranteeing finite time stabilization of
the closed-loop system, is provided. We also show that our sufficient
condition for finite-time stabilization recovers, under stronger assumptions, asymptotic stability. In Section IV, the static output feedback case
is discussed. The conditions for finite time stabilization found in the
previous sections are then turned into optimization problems involving
Manuscript received April 11, 2003; revised May 6, 2004 and December 21,
2004. Recommended by Associate Editor D. E. Miller.
F. Amato is with the School of Computer and Biomedical Engineering, Università degli Studi Magna Græcia di Catanzaro, 88100 Catanzaro, Italy.
M. Ariola is with the Dipartimento di Informatica e Sistemistica, Università
degli Studi di Napoli Federico II, 80125 Napoli, Italy (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAC.2005.847042
0018-9286/$20.00 © 2005 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
LMIs, and some numerical examples are provided in Section V. Our
conclusions are drawn in Section VI.
II. PROBLEM STATEMENT AND PRELIMINARIES
In this note, we consider the following discrete-time linear system:
x(k + 1) = Ax(k) + Bu(k) + Gw(k)
w(k + 1) = F w(k)
725
Problem 1: Given (1), find a state feedback controller (2) such
that the closed-loop system is finite-time bounded with respect to
(x ; w ; ; R; N ).
This problem will be solved in Section III. In Section IV, we will
make use of these results to tackle the more challenging output-feedback problem.
III. STATE FEEDBACK STABILIZATION
(1a)
(1b)
Let us first consider the following discrete-time system:
where A 2 n2n , B 2 n2m , G 2 n2r , and F 2 r2r .
Given system (1), we consider the static state feedback controller
u(k) = Kx(k)
(2)
where K 2 m2n . The main aim of this note is to find some sufficient
conditions which guarantee that the system given by the interconnection of (1) with the controller (2) is bounded over a finite-time interval.
The general idea of finite-time stability concerns the boundedness of
the state of a system over a finite time interval for given initial conditions; this concept can be formalized through the following definition,
which is an extension to discrete-time systems of the one given in [1].
Definition 1 (Finite-Time Stability): The discrete-time linear system
x(k + 1) = Ax(k);
k2
0
is said to be finite-time stable with respect to (x ; ; R; N ), where R is
a positive–definite matrix, 0 < x < , and N 2 0 , if
xT (0)Rx(0) x2
) xT (k)Rx(k) < 2
k2
AT P1 A 0 P1
GT P1 A
AT P1 G
<0
G P1 G + F T P2 F 0 P2
2 min (P~1 )
2
max (P~1 )x2 + max (P2 )w
<
N
V
((x(k + 1); w (k + 1))
) xT (k)Rx(k) < 2
8 k 2 f 1; . . . ; N g:
In Definition 2, we allow the possibility that x be 0 (no free response), and that w be 0 (no forced response). In this latter case, we
recover Definition 1.
The first part of this note will address the following problem.
(4a)
(4b)
< V
((x(k ); w (k ))
(5)
where x(k + 1) and w(k + 1) are evaluated along the solutions of (3).
Applying iteratively (5), we obtain
V
((x(k ); w (k ))
< kV
k = 1; . . . ; N:
(6)
Now, letting P~1 = R01=2 P1 R01=2 and using the fact that we have
1,
kV
((x(0); w (0))=
((x(0); w (0)) ;
k
k
xT (0)P1 x(0) + wT (0)P2 w(0)
max (P~1 )xT (0)Rx(0)
+
N
w(k + 1) = F w(k)
xT (0)Rx(0) x2
2
wT (0)w(0) w
T
where P~1 = R01=2 P1 R01=2 .
Proof: Let us assume that xT (0)Rx(0) x2 and wT (0)w(0) 2
w . We want to prove that if conditions (4) hold, then xT (k)Rx(k) <
2 for all k = 1; . . . ; N .
Let V ((x(k); w(k)) = xT (k)P1 x(k)+ wT (k)P2 w(k). Simple calculations show that (4a) implies
0
is said to be finite-time bounded with respect to (x ; w ; ; R; N ),
where R is a positive definite matrix, 0 x < , w 0 and
N 2 0 , if
(3a)
(3b)
Lemma 1 (Sufficient Conditions for Finite-Time Boundedness):
System (3) is finite-time bounded with respect to (x ; w ; ; R; N ) if
there exist positive–definite matrices P1 2 n2n and P2 2 r2r and
a scalar 1 such that the following conditions hold:
8k 2 f1; . . . ; N g:
Remark 1: Lyapunov asymptotic stability (LAS) and FTS are independent concepts: A system which is FTS may be not LAS; conversely
a LAS system could be not FTS if, during the transients, its state exceeds the prescribed bounds. If we restrict our attention to what happens on a finite-time interval, we can consider Lyapunov stability as an
“additional” requirement. The case of a system being both finite-time
stable and Lyapunov stable is recovered as a particular case by the conditions found in this note (see Remark 3).
Next, let us consider the case when the state is subject also to some
external signals. This leads us to the next definition, which recovers
Definition 1 as a particular case.
Definition 2 (Finite-Time Boundedness): The discrete-time linear
system
x(k + 1) = Ax(k) + Gw(k);
x(k + 1) = Ax(k) + Gw(k)
w(k + 1) = F w(k):
max (P2 )wT (0)w(0)
2
max (P~1 )x2 + max (P2 )w
(7)
and
V
T
T
((x(k ); w (k )) = x (k )P1 x(k ) + w (k )P2 w (k )
x T (k )P 1 x (k )
min (P~1 )xT (k)Rx(k):
(8)
Putting together (6)–(8), we obtain
xT (k)Rx(k) <
1
min (P~1 )
2
max (P~1 )x2 + max (P2 )w
N : (9)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
= 1 2 ...
From (9), it follows that (4b) implies that, for all k
; ; ; N,
xT k Rx k < 2 . Then, system (3) is finite-time bounded with respect to x ; w ; ; R; N .
Conditions similar to those of Lemma 1 can be obtained for the
simpler case of finite-time stability. Let us consider the following discrete-time system:
() ()
(
)
Now, let us go back to our original problem, that is to find sufficient
conditions which guarantee that the interconnection of (1) with the controller (2)
xk
( + 1) = (A + BK )x(k) + Gw(k)
(13a)
( + 1) = F w(k)
(13b)
is finite-time bounded with respect to (x ; w ; ; R; N ). The solution
( + 1) = Ax(k):
(10)
Corollary 1 (Sufficient Conditions for Finite-Time Stability):
System (10) is finite-time stable with respect to x ; ; R; N if there
exist a positive–definite matrix P 2 n2n and a scalar such
that the following conditions hold:
(
AT P A 0 P <
)
0
( ~)
( ~)
where P
R01=2 P R01=2 and
P
max P =min P .
Proof: The proof can be obtained along the guidelines of Lemma
x k TPx k .
1 once we let V x k
Remark 2 (Conservativeness of Corollary 1): The conditions of
Corollary 1 give us the opportunity of making some considerations
regarding the conservativeness of our approach. Indeed it is easy to
check that a necessary and sufficient condition for finite-time stability
of system (10) with respect to x ; ; R; N is given by
(
max R Ak R0
)
<
;
x
1 . . . ; Ng
k2f ;
(12)
where max denotes the maximum singular value.
In order to compare the necessary and sufficient condition (12) with
the sufficient conditions (11) we have randomly generated 1,000 discrete-time linear systems. For each sample we have computed the minimum such that the given system is FTS wrt x ; ; R; N with some
fixed values for x , N and R and we have then evaluated the following
quantity:
(
)
0 true
err% = 100 1 sutrue
err
)
=1
1
1
(
)
=1
0Q1
AQ1 BL
+
0
1
(AQ1 + BL)T
)
0
0 Q1
<
G
GT
F T Q2 F 0 Q2
2
2<
max Q2 w
N
max Q1
x2
min Q1
0
(14a)
(14b)
(~ )+ ( )
(~ )
~ 1 = R1=2 Q1 R1=2 . In this case the controller K is given by
where Q
K = LQ101 .
Proof: Let us consider Lemma 1 with Q1 = P101 and Q2 = P2 .
Condition (4b) can be rewritten as in (14b) recalling that for a positive–definite matrix Q
( ) = min (1Q01 ) :
max Q
^ = + BK and let us consider condition (4a), which in
Now, let A
A
this case reads
^
^
1
AT Q101 A 0 Q0
1
0
1
T
G Q1 A
^
^
+
AT Q101 G
F T Q2 F
GT Q01 G
1
0 Q2
0
< : (15)
Pre- and postmultiplying (15) by the symmetric matrix
Q1
0
0
I
the following equivalent condition is obtained:
where true denotes the exact value of computed using (12) and su
its estimated value obtained applying (11). The average value of
%
over these 1,000 samples has evaluated to 19%.
Conditions like (12) have not been pursued here since, if used for the
finite-time boundedness case or for the stabilization problem, they lead
to conditions which cannot be solved efficiently from a numerical point
of view. On the other hand, Corollary 1 can be extended to both these
problems, as shown in the next sections, leading to conditions which
can be turned into LMIs.
Remark 3 (Asymptotic Stability From Corollary 1): If conditions
, then (10) is finite-time
(11) in Corollary 1 are satisfied with stable with respect to x ; ; R; N for all N 2 0 and it is also asymp, it is straightforward to see that the
totically stable [5]. Indeed, if left-hand side of (11a) reduces to the first difference V of V along the
solutions of (10). Then, since V is negative definite, (10) is asymptotically stable. The key idea in our proofs is to relax the condition of
negative definiteness of V : indeed, according to (11a), V does not
need to be negative definite but just not greater than 0 V . Simi, then (3) is finite-time bounded
larly, if conditions (4) hold for with respect to x ; w ; ; R; N for all N 2 0 and it is also asymptotically stable.
(
(
(11b)
x
cond( ~) =
( ( )) = ( ) ( )
of this problem is given by the following theorem.
Theorem 1 (Finite-Time Boundedness via State Feedback): System
(13) is finite-time bounded with respect to x ; w ; ; R; N if there
exist positive–definite matrices Q1 2 n2n and Q2 2 r2r , a matrix
L 2 m2n and a scalar such that
(11a)
2
cond(P~) < 1N 2
~=
1
xk
w k
=1
1
1
( 1)
^
^
^
+
Q1 AT Q101 AQ1 0 Q1
Q1 AT Q101 G
0
1
0
1
T
T
G Q1 AQ1
G Q1 G F T Q2 F
^
0 Q2
0
< :
(16)
Now, we will show that (16) is equivalent to the following:
^
^
0
Q1 AT Q101 AQ1 0 Q1
^
GT Q101 AQ1
0
0Q1
GT
^
Q1 AT Q101 G
G
F T Q2 F 0 Q2
0
< : (17)
Indeed, using Schur complements, (16) is equivalent to
GT Q101 G F T Q2 F 0 Q2 <
Q1 AT Q101 AQ1 0 Q1 0 Q1 AT Q101 G
^
+
^
^
0
2 GT Q101G + F T Q2 F 0 Q2
^ 1 < 0:
2 GT Q101AQ
(18a)
01
(18b)
Again using Schur complements, (17) is equivalent to
0Q1
GT
G
F T Q2 F 0 Q2
<
0
(19a)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
1^
Q1 A^T Q0
1 AQ1 0 Q1 0
2 0GQT1
2
Q1 A^T Q101 G
0
where Ko 2 m2p . We will find some sufficient conditions which
guarantee that the system given by the interconnection of (24) with the
controller (25)
01
G
2 F 0 Q2
FTQ
0
^ 1
GT Q101 AQ
< 0:
0Q A^T Q0
0
0
I
1
1
I
0
0
x(k + 1) = (A + BKo C )x(k) + Gw(k)
w(k + 1) = F w(k)
(19b)
It is straightforward to check that, under the assumption that Q1 > 0,
(18a) is equivalent to (19a). Moreover, in the inverse that appears in
(19b) we are only interested in the (2,2) term. Using the block matrix inversion formulas, this term is equal to (GT Q101 G + F T Q2 F 0
Q2 )01 . As a consequence (18b) is equivalent to (19b). Therefore, we
have shown that (16) is equivalent to (17). Now, let us premultiply (17)
by
I
727
1
0
(20)
is finite-time bounded with respect to (x ; w ; ; R; N ).
As in [6], let us augment C adding n 0 p rows in such a way that
T
1
^ 1
AQ
0
Q1 A^T
0Q1
GT
0
FTQ
G
F
0 Q2
2
< 0:
(21)
Recalling that A^ = A + BK and letting L = KQ1 we finally obtain
that (21) is equivalent to (14a). This concludes the proof since we have
shown that conditions (4) of Lemma 1 are equivalent to (14).
The following corollary of Theorem 1 allows us to find a controller
K such that system
x(k + 1) = (A + BK )x(k)
(22)
is finite-time stable with respect to (x ; ; R; N ).
Corollary 2 (Finite-Time Stability via State Feedback): System (22)
is finite-time stable with respect to (x ; ; R; N ) if there exist a positive
definite matrix Q 2 n2n , a matrix L 2 m2n and a scalar 1
such that
0Q
AQ + BL
(AQ + BL)
0Q
T
~)
cond(Q
<0
(23a)
2
1 < N 2
x
(23b)
~ = R1=2 QR1=2 . In this case, the controller K is given by
where Q
K = LQ01 .
Proof: The proof can be obtained as in Theorem 1, applying the
results of Corollary 1 to system (22).
C
H
=
is square and full rank. It is simple to see that there are an infinite
number of matrices H satisfying this condition.
Introducing the new state variable x
~(k ) = T x(k ), system (24) becomes
x~(k + 1) = T AT 01 x~(k) + T Bu(k) + T Gw(k)
w(k + 1) = F w(k)
y (k) = CT 01 x~(k) = (Ip 0)~
x (k )
and postmultiply it by the transpose of (20). In this way, we obtain the
following equivalent condition:
0Q
~K
~ )~
~ (k )
x~(k + 1) = (A~ + B
x(k) + Gw
w(k + 1) = F w(k)
~
K
= ( Ko
T
0):
(29)
C
H
=
0Q
1
~ 1 + BLS
~
AQ
T
~ 1 + BLS
~
(AQ
)
0Q
0
~
<0
G
T
T
~
F Q2 F 0 Q2
G
2
2
+ max (Q2 )w <
~1)
N max (Q
1
where
(24a)
Q1
(24b)
(25)
(28b)
~ = T B, G
~ = T G. Then, (26)
is invertible and let A~ = T AT 01 , B
is finite-time bounded with respect to (x ; w ; ; R; N ) if there exist
positive–definite matrices Q11 2 p2p , Q12 2 (n0p)2(n0p) and
Q2 2 r2r , a matrix L 2 m2n and a scalar 1 such that
In this section, we will consider the system
u(k) = Ko y (k)
(28a)
System (28) is in the same form as (13), the only difference being
the constraint on the controller structure. However, this constraint can
be dealt with as shown in the following result which allows us to find a
controller Ko such that system (26) is finite-time bounded with respect
to (x ; w ; ; R; N ).
Theorem 2 (Finite-Time Boundedness via Static Output Feedback): Consider (24), choose a matrix H 2 (n0p)2n such that
x2
~1)
min (Q
with C 2 p2n . We will assume that C is full-row rank. Given (24),
we consider a static output feedback controller in the form
(27c)
~ = T B, G
~ = T G, and
where A~ = T AT 01 , B
IV. OUTPUT FEEDBACK STABILIZATION
(24c)
(27a)
(27b)
where Ip denotes the identity matrix of order p. Now, substituting (25)
in (27) leads to
0
x(k + 1) = Ax(k) + Bu(k) + Gw(k)
w(k + 1) = F w(k)
y (k) = Cx(k)
(26a)
(26b)
S
=
Q11
0
0
Q12
2 n2n is defined as
S
=
Ip
0
0
0
~ 1=2 Q1 R
~1 = R
~ 1=2 , being R
~ = T 0T RT 01 .
and Q
(30a)
(30b)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
~ = LSQ01 ,
The controller Ko is given by the first p columns of K
1
which is in the form (29).
Proof: Applying the results of Theorem 1, conditions (30) guar~ = LSQ01 , (28) is finite-time bounded with reantee that, choosing K
1
~ N ). Now, the proof follows noting the following
spect to (x ; w ; ; R;
two facts.
i)
The chosen structures for the matrices S and Q1 guarantee
~ is in the form (29). Indeed
that K
~
K
01 = L Ip
= LSQ1
=L
ii)
0
01
Q11
0
0
0
0
Q01
11
0
0
= ( Ko
If the problem is feasible, then the controller K = LQ101 renders (13)
finite-time bounded with respect to (x ; w ; ; R; N ).
Similarly, LMI feasibility problems can be derived from Corollary 2
and Theorem 2.
LMI Feasibility Problem 2 (From Corollary 2): Given (22) and
(x ; ; R; N ), fix a 1 and find matrices Q > 0 and L satisfying
the LMIs (23a) and
2
1 R01 < Q < N 2 R01 :
x
0
Q1021
If the problem is feasible, then the controller K = LQ01 renders (22)
finite-time stable with respect to (x ; ; R; N ).
LMI Feasibility Problem 3 (From Theorem 2): Given (26) and
(n0p)2n
such that
(x ; w ; ; R; N ), choose a matrix H 2
0):
Since x
~(k ) = T x(k ), then we have
T T
0T RT 01 )(T x) = xT Rx:
~x
x~T R
~ = (x T )(T
T
Therefore, since (28) is finite-time bounded with respect to
~ N ), then system (26) is finite-time bounded
(x ; w ; ; R;
with respect to (x ; w ; ; R; N ).
Remark 4: In Theorem 2, the choice of matrix H represents a degree of freedom, which could be exploited to improve the effectiveness of the proposed output feedback stabilization method. An open
problem, which is left for future research, is the development of a procedure to find an optimal matrix H .
Remark 5 (Static and Dynamic Feedback): In this note, we have not
considered the case of dynamic output feedback but we have restricted
our attention to the case of static output feedback. In the case of dynamic output feedback, adopting the change of variables suggested in
[7], it is possible to show that condition (4a) leads to an LMI but the
additional condition (4b) imposes a constraint which cannot be treated
efficiently from a numerical point of view. For this reason, this case has
not been treated here.
~ = T B , and G
~ = T G. Fix a
is invertible and let A~ = T AT 01 , B
1 and find matrices Q11 > 0, Q12 > 0, Q2 > 0 and L, and
positive scalars 1 and 2 satisfying the LMIs (30a) and
1 T R01 T T < Q1 < T R01 T T
< Q2 < 2 I
0 2 w x > 0
x
1
0
2
where
Q1
and S
1 R01 < Q1 < R01
0 < Q2 < 2 I
2
x2
2
+ 2 w < N
1
S
0 w
2
2
x
x
1
> 0:
< Q2 < 2 I
0 2 w x > 0:
x
1
0
2
0
Q12
=
Ip
0
0
0
:
If the problem is feasible, then the controller Ko given by the first p
columns of LSQ101 renders system (26) finite-time bounded with respect to (x ; w ; ; R; N ).
Example 1: Let us consider system
(31a)
x(k + 1) = Ax(k) + Bu(k) + Gw(k)
w(k + 1) = F w(k)
(31c)
y (k) = Cx(k)
(32)
Now, starting from Theorem 1, we obtain the following result.
LMI Feasibility Problem 1 (From Theorem 1): Given (13) and
(x ; w ; ; R; N ), fix a 1 and find matrices Q1 > 0, Q2 > 0 and
L, and positive scalars 1 and 2 satisfying the LMIs (14a) and
1 R01 < Q1 < R01
0
(31b)
for some positive numbers 1 and 2 . Inequality (31c) can be converted
to an LMI using Schur complements; indeed it is equivalent to
Q11
=
2 n2n is defined as
V. COMPUTATIONAL ISSUES AND NUMERICAL EXAMPLES
In this section, we will show how, once we have fixed a value for ,
the feasibility of the conditions stated in the previous Sections can be
turned into LMI-based feasibility problems [8]. To this aim, first of all
it is easy to check that condition (14b) is guaranteed by imposing the
conditions
C
H
=
with
A=
F
=
1
3
01 01
0: 8
0 0 :6
0:6
0 :8
B
=
C
01
G=
1
= (2
1
0
0
1
1):
The chosen F matrix has complex eigenvalues of unitary modulus;
therefore, we are considering the case of a sinusoidal disturbance.
We have considered the following three cases:
1)
finite-time stability via state feedback: a) w(0) = 0; b)
u(k) = Kx(k);
2)
finite-time boundedness via state feedback: a) w(0) 6= 0; b)
u(k) = Kx(k);
3)
finite-time boundedness via output feedback: a) w(0) 6= 0;
b) u(k) = Ko y (k).
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
In this example, we decided to perform an optimization over using
the algorithm sketched below, with the aid of the Matlab LMI Control
Toolbox [9].
Step 1) We chose some given fixed values for x , w , R and N .
Step 2) We decided an initial value for .
Step 3) We solved the specific LMI feasibility problem. Since the
parameter can range from 1 to (=x )(2=N ) (this bound
is obtained from (31c) letting 1 = 1 and 2 = 0),
starting from = 1 we kept increasing until a solution
is found or the maximum value for is reached.
Step 4) If no solution is found then the initial value for needs to
be increased, otherwise can be decreased until its minimum value is found.
In this case, we chose x = 1, w = 3, R = I2 , N = 4, and an initial
value for equal to 30.
Case 1) We solved the LMI Problem 2 and we found that the controller
K
Case 2)
1:54)
solves our problem with = 2:8 and = 1. The value
of (see Remark 3) implies that the state boundedness
is guaranteed for all N 2 0 and that the closed-loop
system is also asymptotically stable.
We solved the LMI Problem 1 and found that the controller
K
Case 3)
= (1:02
= (1:00
1:50)
guarantees the desired closed-loop properties with =
16:9 and = 1:36.
The solution of the LMI Problem 3 with H = (00:1 10)
leads to the controller Ko
Ko
= 0:51
which solves our problem with = 25:5 and = 1:43.
Remark 6: Note that in Example 1, we keep x fixed and optimize
over . In a similar way one can fix and look for the maximum admissible x guaranteeing the desired closed-loop finite-time property.
VI. CONCLUSION
In this note, we have considered the finite-time stabilization problem
for a discrete-time linear system subject to disturbances generated by
an exosystem. The first result of the note is a sufficient condition for finite-time state feedback stabilization; then the output feedback problem
has been considered and a further condition guaranteeing the existence
of a static output feedback controller has been provided. These conditions have been then turned into optimization problems involving
LMI’s. The proposed method has been illustrated through some numerical examples.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewer for suggesting the necessary and sufficient condition (12).
REFERENCES
[1] P. Dorato, “Short time stability in linear time-varying systems,” in Proc.
IRE Int. Convention Record Part 4, 1961, pp. 83–87.
[2] L. Weiss and E. F. Infante, “Finite time stability under perturbing forces
and on product spaces,” IEEE Trans. Autom. Control, vol. AC-12, pp.
54–59, Jan. 1967.
729
[3] H. D’Angelo, Linear Time-Varying Systems: Analysis and Synthesis. Boston, MA: Allyn and Bacon, 1970.
[4] F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica,
vol. 37, no. 9, pp. 1459–1463, Sep. 2001.
[5] F. M. Callier and C. A. Desoer, Linear System Theory. New York:
Springer-Verlag, 1991.
[6] G. Garcia, J. Bernussou, and D. Arzelier, “Stabilitazion of an uncertain
linear dynamic system by state and output feedback: A quadratic stabilizability approach,” Int. J. Control, vol. 64, no. 5, pp. 839–858, 1996.
synthesis,”
[7] P. Gahinet, “Explicit controller formulas for LMI-based
Automatica, vol. 32, no. 7, pp. 1007–1014, 1996.
[8] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix
Inequalities in System and Control Theory. Philadelphia, PA: SIAM,
1994.
[9] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control
Toolbox. Natick, MA: The Mathworks, Inc., 1995.
On Distributed Delay in Linear Control Laws—Part II:
Rational Implementations Inspired From the -Operator
Qing-Chang Zhong
Abstract—This note proposes rational implementations for distributed
delay in linear control laws. The main benefit of doing so is the easy implementation of rational transfer functions. The proposed approach was
inspired from the -operator. The resulting rational implementation has
an elegant structure of chained low-pass filters. The stability of each node
can be guaranteed by the choice of the number
of the nodes. The sta-norm
bility of the closed-loop system can also be guaranteed since the
of the implementation error approaches 0 when
goes to
. Moreover,
the steady-state performance of the system is retained without the need to
change the control structure.
Index Terms— -operator, dead-time compensator, distributed delay,
finite-spectrum assignment, implementation error, modified Smith predictor, rational implementation.
I. INTRODUCTION
Distributed delays, which are finite-impulse-response (FIR) blocks,
often appear as a part of dead-time compensators for processes with
dead time, in particular, as a part of the finite-spectrum-assignment
(FSA) control law [1]–[3] or in the form of a modified Smith predictor [4], [5] for unstable processes. Distributed delays also appear
in H control of (even, stable) dead-time systems [6]–[10] and continuous-time deadbeat control [11]. Due to the requirement of internal
stability, such an FIR block has to be, approximately, implemented as
a stable block without hidden unstable poles.
A common way to do this is to replace the distributed delay by the
sum of a series of discrete (often commensurate) delays [1], [4], [5]
(other interesting implementations using resetting mechanism can be
found in [12] and [13]). There have been some arguments about the
possibility of causing instability by doing this, which has attracted a lot
of attention from the delay community; see [13]–[22]. It was proposed
as an open problem in the survey paper [23]. Very recently, it has been
1
Manuscript received July 18, 2004; revised December 9, 2004. Recommended by Associate Editor Hong Wang. This work was supported by the
EPSRC under Grant EP/C005953/1.
The author is with the School of Electronics, University of Glamorgan, Pontypridd CF37 1DL, U.K. (email: [email protected]).
Digital Object Identifier 10.1109/TAC.2005.847043
0018-9286/$20.00 © 2005 IEEE