In this example, we illustrate the proposed method by Theorem 4.1 by using MATLAB. As
stated in example 1, the system is stabilizable and observable, and satisfies Assumption 3,
and is open-loop unstable.
⎡1 0 0⎤
⎡1 0⎤
Let 8.27
γ =
, R =
⎢
⎥
⎢
, Q = 0 1 0 , 3.78
ρ =
00 , and δ = 0.0010 , then we solve the
0 1⎥
⎣
⎦
⎢
⎥
⎢0 0 1⎥
⎣
⎦
parameter-dependent discrete-time Riccati equation (21) to get
⎡18.5238 3.8295
0.1664 ⎤
X
⎢
⎥
∞ =
3.8295 51.3212 23.3226 > 0
⎢
⎥
,
⎢ 0.1664 23.3226 22.7354⎥
⎣
⎦
⎡ 0.7292
0.0560
−
0.0024
−
⎤
2 T
⎡609.6441 723.0571⎤
U
γ
⎢
⎥
= −
= −
−
>
=
1
I
1
B X∞ 1
B
0.0560
0.2496
0.3410
0
⎢
⎥
, U 2 ⎢
⎥ ,
⎢
⎣723.0571 863.5683
0.0024
−
0.3410
−
⎦
⎣
0.6676 ⎥⎦
⎡14.2274
0.0723
−
0
⎤
2
T
ρ I H
⎢
⎥
−
= −
>
KU 2 H
0.0723 14.2020
0
0
K
⎢
⎥
.
⎢
0
0
14.2884⎥
⎣
⎦
176
New Trends in Technologies
Based on this, the non-fragile discrete-time state feedback mixed LQR/ H∞ controller is
⎡ 0.4453
−
0.1789
−
0.0682
−
⎤
F∞ = ⎢
0.1613
1.1458
1.0756⎥
−
−
−
⎣
⎦
6. Conclusion
In this chapter, we first study the discrete time state feedback mixed LQR/ H∞ control
problem. In order to solve this problem, we present an extension of the discrete time
bounded real lemma. In terms of the stabilizing solution to a discrete time Riccati equation,
we derive the simple approach to discrete time state feedback mixed LQR/ H∞ control
problem by combining the Lyapunov method for proving the discrete time optimal LQR
control problem with the above extension of the discrete time bounded real lemma, the
argument of completion of squares of Furuta & Phoojaruenchanachi (1990) and standard
inverse matrix manipulation of Souza & Xie (1992).A related problem is the standard H∞
control problem (Doyle et al., 1989a; Iglesias & Glover, 1991; Furuta & Phoojaruenchanachai,
1990; Souza & Xie, 1992; Zhou et al. 1996), another related problem is the H∞ optimal
control problem arisen from Basar & Bernhard (1991). The relations among the two related
problem and mixed LQR/ H∞ control problem can be clearly explained by based on the
discrete time reference system (9)(3). The standard H∞ control problem is to find an
admissible controller K such that the H∞ -norm of closed-loop transfer matrix from
disturbance input w to the controlled output z is less than a given number γ > 0 while the
H∞ optimal control roblem arisen from Basar & Bernhard (1991) is to find an admissible
controller such that the H∞ -norm of closed-loop transfer matrix from disturbance input w
to the controlled output
γ >
0
z is less than a given number
0 for the discre time reference
system (9)(3). Since the latter is equivalent to the problem that is to find an admissible
controller K such that
ˆ
sup ∈ inf { }
w
, we may recognize that the mixed LQR/ H∞ control
2
L
K J
+
problem is a combination of the standard H∞ control problem and H∞ optimal control
problem arisen from Basar & Bernhard (1991). The second problem considered by this
chapter is the non-fragile discrete-time state feedback mixed LQR/ H∞ control problem
with controller uncertainty. This problem is to extend the results of discrete-time state
feedback mixed LQR/ H∞ control problem to the system (2)(4) with controller uncertainty.
In terms of the stabilizing solution to a parameter-dependent discrete time Riccati equation,
we give a design method of non-fragile discrete-time state feedback mixed LQR/ H∞
controller, and derive necessary and sufficient conditions for the existence of this non-
fragile controller.
7. References
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a dynamic game approach. Boston, MA: Birkhauser.
D. S. Bernstein, and Haddad W. M. (1989). LQG control with an H∞ performance bound: A
Riccati equation approach, IEEE Trans. Aut. Control. 34(3), pp. 293- 305.
J. C. Doyle, Glover K., Khargonekar P. P. and Francis B. A. (1989a) . State-space solutions to
standard H 2 and H∞ control problems . IEEE Trans. Aut. Control, 34(8), pp. 831-
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J. C. Doyle, Zhou K., Glover K. and Bodenheimer B. (1994). Mixed H 2 and H∞ perfor-
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D. Famularo, Dorato P., Abdallah C. T., Haddad W. M. and Jadbabaie A. (2000). Robust non-
fragile LQ controllers: the static state case, INT. J. Control, 73 (2),pp.159-165.
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W. M. Haddad, and Corrado J. R. (2000). Robust resilient dynamic controllers for systems
with parametric uncertainty and controller gain variations, INT. J. Control, 73(15),
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P. A. Iglesias, and Glover K. (1991). State-space approach to discrete-time H∞ control, INT.
J. Control, 54(5), pp. 1031- 1073.
L. H. Keel, and Bhattacharyya S. P. (1997). Robust, fragile, or optimal ? IEEE Trans. Aut.
Control, 42(8), pp. 1098-1105
L. H. Keel, and Bhattacharyya S. P. (1998). Authors’ Reply. IEEE Trans. Aut. Control, 43(9),
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approach, IEEE Trans. Aut. Control, 36(7), pp. 824-837.
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D. J. N. Limebeer, Anderson B. D. O., Khargonekar P. P. and Green M. (1992). A game
theoretic approach to H∞ control for time-varying systems. SIAM J. Control and
Optimization, 30(2), pp.262-283.
D. J. N. Limebeer, Anderson B. D. O., and Hendel B. (1994). A Nash game approach to
mixed H 2 / H∞ control. IEEE Trans. Aut. Control, 39(1), pp. 69-82.
K. Ogata (1987). Discrete-time control systems . Prentice Hall, 1987.
T. Pappas, Laub A. J., Sandell N. R., Jr. (1980). On the numerical solution of the discrete –
time algebraic Riccati equation. IEEE Trans. Aut. Control, 25(4), pp. 631-641.
P. L. D. Peres and Geromel J. C. (1993). H 2 control for discrete-time systems optimality and
robustness. Automatica, Vol. 29, No. 1, pp. 225-228.
J. E. Potter (1966). Matrix quadratic solution. J. SIAM App. Math., 14, pp. 496-501.
P. M. Makila (1998). Comments ″Robust, Fragile, or Optimal ?″. IEEE Trans. Aut. Control. ,
43(9), pp. 1265-1267.
M. A. Rotea, and Khargonekar P. P. (1991). H 2 -optimal control with an H∞ -constraint: the
state-feedback case. Automatica, 27(2), pp. 307-316.
H. Rotstein, and Sznaier M. (1998). An exact solution to general four-block discrete-time
mixed H 2 / H∞ problems via convex optimization, IEEE Trans. Aut. Control, 43(10),
pp. 1475-1480.
C. E. de Souza and Xie L. (1992). On the discrete-time bounded real lemma with application
in the characterization of static state feedback H∞ controllers, Systems & Control
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M. Sznaier (1994). An exact solution to general SISO mixed H 2 / H∞ problems via convex
optimization, IEEE Trans. Aut. Control, 39(12), pp. 2511-2517.
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X. Xu (1996). A study on robust control for discrete-time systems with uncertainty, A Master
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1996
11
Robust Control Design of Uncertain
Discrete-Time Systems with Delays
Jun Yoneyama, Yuzu Uchida and Shusaku Nishikawa
Aoyama Gakuin University
Japan
1. Introduction
When we consider control problems of physical systems, we often see time-delay in the
process of control algorithms and the transmission of information. Time-delay often appear in
many practical systems and mathematical formulations such as electrical system, mechanical
system, biological system, and transportation system. Hence, a system with time-delay is a
natural representation for them, and its analysis and synthesis are of theoretical and practical
importance. In the past decades, research on continuous-time delay systems has been active.
Difficulty that arises in continuous time-delay system is that it is infinite dimensional and a
corresponding controller can be a memory feedback. This class of controllers may minimize
a certain performance index, but it is difficult to implement it to practical systems due to
a memory feedback. To overcome such a difficulty, a memoryless controller is used for
time-delay systems. In the last decade, sufficient stability conditions have been given via
linear matrix inequalities (LMIs), and stabilization methods by memoryless controllers have
been investigated by many researchers. Since Li and de Souza considered robust stability
and stabilization problems in (8), less conservative robust stability conditions for continuous
time-delay systems have been obtained ((7), (11)). Recently, H∞ disturbance attenuation
conditions have also been given ((10), (15), (16)).
On the other hand, research on discrete-time delay systems has not attracted as much attention
as that of continuous-time delay systems. In addition, most results have focused on state
feedback stabilization of discrete-time systems with time-varying delays. Only a few results
on observer design of discrete-time systems with time-varying delays have appeared in the
literature(for example, (9)). The results in (3), (12), (14), (18) considered discrete-time systems
with time-invariant delays. Gao and Chen (4), Hara and Yoneyama (5), (6) gave robust
stability conditions. Fridman and Shaked (1) solved a guaranteed cost control problem.
Fridman and Shaked (2), Yoneyama (17), Zhang and Han (19) considered the H∞ disturbance
attenuation.
They have given sufficient conditions via LMIs for corresponding control
problems. Nonetheless, their conditions still show the conservatism. Hara and Yoneyama
(5) and Yoneyama (17) gave least conservative conditions but their conditions require many
LMI slack variables, which in turn require a large amount of computations. Furthermore,
to authors’ best knowledge, few results on robust observer design problem for uncertain
discrete-time systems with time-varying delays have given in the literature.
In this paper, we consider the stabilization for a nominal discrete-time system with
time-varying delay and robust stabilization for uncertain system counterpart. The system
under consideration has time-varying delays in state, control input and output measurement.
First, we obtain a stability condition for a nominal time-delay system. To this end, we define
180
Discrete Time Systems
a Lyapunov function and use Leibniz-Newton formula and free weighting matrix method.
These methods are known to reduce the conservatism in our stability condition, which are
expressed as linear matrix inequality. Based on such a stability condition, a state feedback
design method is proposed. Then, we extend our stabilization result to robust stabilization for
uncertain discrete-time systems with time-varying delay. Next, we consider observer design
and robust observer design. Similar to a stability condition, we obtain a condition such that
the error system, which comes from the original system and its observer, is asymptotically
stable. Using a stability condition of the error system, we proposed an observer design
method. Furthermore, we give a robust observer design method for an uncertain time-delay
system. Finally, we give some numerical examples to illustrate our results and to compare
with existing results.
2. Time-delay systems
Consider the following discrete-time system with a time-varying delay and uncertainties in
the state and control input.
x( k + 1) = ( A + Δ A) x( k) + ( Ad + Δ Ad) x( k − dk) + ( B + Δ B) u( k)
+( Bd + Δ Bd) u( k − dk)
(1)
where x( k) ∈ n is the state and u( k) ∈ m is the control. A, Ad, B and Bd are system matrices with appropriate dimensions. dk is a time-varying delay and satisfies 0 ≤ dm ≤ dk ≤ dM and
dk+1 ≤ dk where dm and dM are known constants. Uncertain matrices are of the form
Δ A Δ Ad Δ B Δ Bd = HF( k) E Ed E 1 Eb
(2)
where F( k) ∈ l× j is an unknown time-varying matrix satisfying FT( k) F( k) ≤ I and H, E, Ed, E 1 and Eb are constant matrices of appropriate dimensions.
Definition 2.1. The system (1) is said to be robustly stable if it is asymptotically stable for all
admissible uncertainties (2).
When we discuss a nominal system, we consider the following system.
x( k + 1) = Ax( k) + Adx( k − dk) + Bu( k) + Bdu( k − dk).
(3)
Our problem is to find a control law which makes the system (1) or (3) robustly stable. Let us
now consider the following memoryless feedback:
u( k) = Kx( k)
(4)
where K is a control gain to be determined. Applying the control (4) to the system (1), we have
the closed-loop system
x( k + 1) = (( A + Δ A) + ( B + Δ B) K) x( k) + (( Ad + Δ Ad) + ( Bd + Δ Bd) K) x( k − dk).
(5)
For the nominal case, we have
x( k + 1) = ( A + BK) x( k) + ( Ad + BdK) x( k − dk).
(6)
In the following section, we consider the robust stability of the closed-loop system (5) and the
stability of the closed-loop system (6).
The following lemma is useful to prove our results.
Robust Control Design of Uncertain Discrete-Time Systems with Delays
181
Lemma 2.2. ((13)) Given matrices Q = QT, H, E and R = RT > 0 with appropriate dimensions.
Q + HF( k) E + ET FT( k) HT < 0
for all F( k) satisfying FT( k) F( k) ≤ R if and only if there exists a scalar ε > 0 such that Q + 1 ε HHT + εETRE < 0.
3. Stability analysis
This section analyzes the stability and robust stability of discrete-time delay systems.
Section 3.1 gives a stability condition for nominal systems and Section 3.2 extends the stability
result to a case of robust stability.
3.1 Stability for nominal systems
Stability conditions for discrete-time delay system (6) are given in the following theorem.
Theorem 3.1. Given integers dm and dM, and control gain K. Then, the time-delay system (6) is
asymptotically stable if there exist matrices P 1 > 0 , P 2 > 0 , Q 1 > 0 , Q 2 > 0 , S > 0 , M > 0 ,
⎡ ⎤
⎡ ⎤
⎡ ⎤
L 1
N 1
T 1
⎢
⎢ L 2⎥
⎥
⎢
⎢ N 2⎥
⎥
⎢
⎢ T 2⎥
⎥
L = ⎢
⎣ L 3⎥
⎦ , N = ⎢
⎣ N 3⎥
⎦ , T = ⎢
⎣ T 3⎥
⎦
L 4
N 4
T 4
L 5
N 5
T 5
satisfying
√
Φ = Φ1 + Ξ L + Ξ T + Ξ
+ Ξ
L
N + Ξ