Robust Control Design of Uncertain Discrete-Time Systems with Delays
191
dk
ρ’s
K
1
[0.1 0.1 − 0.1 0.5 0.1] [−1.1316 − 0.1360]
2
[0.1 0.1 − 0.1 0.5 0.1] [−0.9690 − 0.0976]
3
[0.1 0.1 − 0.1 0.5 0.1] [−0.7908 − 0.0545]
4 [0.09 0.05 − 0.1 0.55 0.1] [−0.5815 − 0.0306]
Table 1. The stabilization for time-invariant delay dk
dk
ρ’s
K
0 ≤ dk ≤ 1 [0.1 0.1 − 0.1 0.5 0.1] [−1.1209 − 0.1174]
0 ≤ dk ≤ 2 [0.1 0.1 − 0.1 0.5 0.1] [−0.9429 − 0.0839]
0 ≤ dk ≤ 3 [0.1 0.1 − 0.1 0.5 0.1] [−0.7950 − 0.0469]
0 ≤ dk ≤ 4 [0.09 0.05 − 0.1 0.55 0.1] [−0.5586 − 0.0253]
Table 2. The stabilization for time-varying delay dk
dk ¯ α
ρ’s
K
3 0.05
[0.1 0.1 − 0.1 0.5 0.1] [−0.8622 − 0.0059]
3 0.10
[0.1 0.1 − 0.1 0.5 0.1] [−0.6243 − 0.0000]
2 0.15 [0.12 0.12 − 0.1 0.5 0.05] [−1.2515 − 0.0115]
Table 3. The robust stabilization for time-invariant delay dk
dk
¯ α
ρ’s
K
0 ≤ dk ≤ 3 0.05 [0.1 0.1 − 0.1 0.5 0.1] [−0.8394 − 0.0047]
0 ≤ dk ≤ 3 0.10 [0.12 0.1 − 0.1 0.5 0.1] [−1.2539 − 0.0108]
0 ≤ dk ≤ 2 0.15 [0.12 0.12 − 0.1 0.5 0.05] [−1.1740 − 0.0015]
Table 4. The robust stabilization for time-varying delay dk
For time-invariant delay dk, Theorem 4.3 gives control gains for different ¯ α in Table 3. Table 4 provides
the result for time-varying delay dk.
Example 7.2. Consider the following discrete-time delay system:
x( k + 1) = 0.85 + 0.1 α 0
x( k) + −0.1
0
x( k − d
0
0.97
−0.1 −0.1
k),
y( k) = 0.5 0.2 x( k) + 0.1 0.1 x( k − dk)
where α satisfies | α| ≤ ¯ α for ¯ α is an upper bound of α( k) . We first consider the observer design for a nominal time-delay system with α( k) = 0 by Theorem 6.1. Table 5 shows observer gains for different
time-invariant delay dk, while Table 6 gives observer gains for different time-varying delay dk. In the
following observer design, all ρ’s are set to be zero for simplicity.
Next, we consider the robust observer design for the uncertain time-delay system with α( k) = 0 . In
this case, system matrices can be represented in the form of (1) with matrices given by
A = 0.85 0
, A
, E = ¯ α 0 , E
0 0.97
d =
−0.1 0
−0.1 −0.1
d = E 1 =
0 0 ,
C = 0.5 0.2 , Cd = 0.1 0.1 , H = 0.1 , F( k) = α( k)
0
¯ α .
192
Discrete Time Systems
dk
ˆ ρ’s
¯
K
1 [−22.6 2.5 − 2.1 − 1.9 − 1.9]
0.4835
0.3622
2 [−21.9 6.7 − 0.3 − 2.6 − 1.9]
0.5372
0.3348
3
[−23.4 8.6 0.2 − 1.9 − 1.9]
0.5174
0.3910
4
[−9.0 26.4 0.2 − 2.5 − 1.9]
−3.2281
0.0459
5
[−9.0 25.9 0.2 − 2.5 − 1.9]
−2.3508
−0.7232
Table 5. The observer design for time-invariant delay dk
dk
ˆ ρ’s
¯
K
0 ≤ dk ≤ 1 [−20.2 6.5 − 2.1 − 2.5 − 1.9] 0.6295
0.3777
0 ≤ dk ≤ 2 [−21.9 6.7 − 0.3 − 2.6 − 1.9] 0.5817
0.3475
0 ≤ dk ≤ 3 [−22.0 8.6 0.2 − 1.9 − 1.9]
0.5490
0.3037
0 ≤ dk ≤ 4 [−22.0 8.6 0.2 − 2.5 − 1.9]
0.5157
0.2921
0 ≤ dk ≤ 5 [−22.5 8.6 0.2 − 1.9 − 3.1]
0.5170
0.2956
Table 6. The observer design for time-varying delay dk
dk ¯ α
ˆ ρ’s
¯
K
1 0.5 [−22.6 2.5 − 2.1 − 1.9 − 1.9]
0.5047
0.3813
2 0.4 [−21.9 6.7 − 0.3 − 2.6 − 1.9]
0.5625
0.3633
3 0.3 [−23.4 8.6 0.2 − 1.9 − 1.9]
0.5264
0.3641
4 0.3 [−9.0 26.4 0.2 − 2.5 − 1.9]
−2.6343
−1.6768
5 0.2 [−9.0 25.9 0.2 − 2.5 − 1.9]
−2.5959
−1.5602
Table 7. The observer design for time-invariant delay dk
For time-invariant delay dk, Theorem 6.2 gives observer gains for different ¯ α in Table 7. Table 8 provides
observer gains for time-varying delay dk by the same theorem.
8. Conclusions
In this paper, we proposed stabilization and robust stabilization method for discrete-time
systems with time-varying delay.
Our conditions were obtained by introducing new
Lyapunov function and using Leibniz-Newton formula and free weighting matrix method.
Robust Control Design of Uncertain Discrete-Time Systems with Delays
193
dk
¯ α
ˆ ρ’s
¯
K
0 ≤ dk ≤ 1 0.5 [−20.2 6.5 − 2.1 − 2.5 − 1.9] 0.6345
0.3520
0 ≤ dk ≤ 2 0.4 [−21.9 6.7 − 0.3 − 2.6 − 1.9] 0.5870
0.3153
0 ≤ dk ≤ 3 0.3 [−22.0 8.6 0.2 − 1.9 − 1.9]
0.5675
0.3003
0 ≤ dk ≤ 4 0.3 [−22.0 8.6 0.2 − 2.5 − 1.9]
0.5375
0.3444
0 ≤ dk ≤ 5 0.2 [−22.5 8.6 0.2 − 1.9 − 3.1]
0.5077
0.3425
Table 8. The observer design for time-varying delay dk
Similarly, we also gave observer design and robust observer design methods. Numerical
examples were given to illustrate our proposed design method.
9. References
[1] Fridman, E. & Shaked, U. (2005). Stability and guaranteed cost control of uncertain
discrete delay systems, International Journal of Control, Vol.78, 235-246.
[2] Fridman, E. & Shaked, U. (2005). Delay-dependent H∞ control of uncertain discrete delay
systems, European Journal of Control, Vol.11, 29-37.
[3] Gao, H.; Lam, J.:Wang, C. & Wang, Y. (2004). delay-dependent output feedback
stabilization of discrete-time systems with time-varying state delay, IEE Proc. Control
Theory Appl. , Vol.151, 691-698.
[4] Gao, H. & Chen, T. (2007). New results on stability of discrete-time systems with
time-varying state delay, IEEE Transactions on Automatic Control, Vol.52, 328-334.
[5] Hara, M. & Yoneyama, J. (2008), New robust stability condition for uncertain
discrete-time systems with time-varying delay, in SICE Annual Conference 2008, 743-747,
Tokyo, August 2008.
[6] Hara, M. & Yoneyama, J. (2009). An improved robust stability condition for uncertain
discrete time-varying delay systems, Journal of Cybernetics and Systems, Vol.2, 23-27.
[7] He, Y.; Wang, Q.; Xie, L. & Lin, C. (2007). Further improvement of free-weighting matrices
technique for systems with time-varying delay, IEEE Transactions on Automatic Control,
Vol.52, 293-299.
[8] Li, X. & de Souza, C. E. (1997). Delay dependent robust stability and stabilization of
uncertain linear delay systems: a linear matrix inequality approach, IEEE Transactions on
Automatic Control, Vol.42, 1144-1148.
[9] Liu, Y.; Wang, Z. & Liu, X. (2008). Robust H∞ filtering for discrete nonlinear stochastic
systems with time-varying delay, Journal of Mathematical Analysis and Applications,
Vol.341, 318-336.
[10] Ma, S.; Zhang, C. & Cheng, Z. (2008). Delay-dependent Robust H∞ Control for Uncertain
Discrete-Time Singular Systems with Time-Delays, Journal of Computational and Applied
Mathematics, Vol.217, 194-211.
[11] Mahmoud, M.S. (2000). Robust Control and Filtering for Time-Delay Systems, New York:
Marcel Dekker, Inc.
194
Discrete Time Systems
[12] Palhares, R.M.; Campos, C.D.; Ekel, P. Ya.; Leles, M.C.R. & D’Angelo, M.F.S.V. (2005).
Delay-dependent robust H∞ control of uncertain linear systems with lumped delays,
IEE Proc. Control Theory Appl. , Vol.152, 27-33
[13] Xie, L. (1996). Output Feedback H∞ Control of systems with parameter uncertainty,
International Journal of Control, Vol.63, 741-750.
[14] Xu, S.; Lam, J. & Zou, Y. (2005). Improved conditions for delay-dependent robust
stability and stabilization of uncertain discrete-time systems, Asian Journal of Control,
Vol.7, 344-348.
[15] Xu, S.; Lam, J. & Zou, Y. (2006). New results on delay-dependent robust H∞ control for
systems with time-varying delays, Automatica, Vol.42, 343-348.
[16] Ye, D. & Yang, G. H. (2008). Adaptive robust H∞ state feedback control for linear
uncertain systems with time-varying delay, International Journal of Adaptive Control and
Signal Processing, Vol.22, 845-858.
[17] Yoneyama, J. (2008). H∞ disturbance attenuation for discrete-time systems with time
varying delays, SICE Transactions, Vol.44, 285-287(in Japanese).
[18] Yoneyama, J. & Tsuchiya, T. (2008). New delay-dependent conditions on robust stability
and stabilisation for discrete-time systems with time-delay, International Journal of Systems
Science, Vol.39, 1033-1040.
[19] Zhang, X.-M. & Han, Q.-L. (2008). A new finite sum inequality approach to
delay-dependent H∞ control of discrete-time systems with time-varying delay,
International Journal of Robust and Nonlinear Control, Vol.18, 630-647.
12
Quadratic D Stabilizable
Satisfactory Fault-tolerant Control with
Constraints of Consistent Indices for
Satellite Attitude Control Systems
Han Xiaodong1 and Zhang Dengfeng2
1Institute of Telecommunication Satellite, CAST, Beijing 100094
2Nanjing University of Science and Technology, Nanjing 210094,
P.R. China
1. Introduction
In the last twenty some years, much attention has been paid to the problem of fault-tolerant
control in satellite attitude control systems and many methods have been developed and
proven to be capable of tolerating certain types of system faults (see e.g., [1~5] and the
references therein). However, these solutions focused mainly on keeping stability of the faulty
systems and in less consideration of other performance indices. Actually, the performance
requirements of practical satellite control systems are usually multi-objective even in faulty
cases and it is desirable for fault tolerant systems to keep the required performance indices in a
satisfactory and admissible region rather than the optimization of single index [6~7].
As is well known, in many practical applications, it is desirable to construct systems to
achieve better transient property, strong anti-disturbance ability and adequate level of cost
function performance. To this end, optimal controllers have been designed by assigning pole
in a desired region (see e.g., [8] and [9]), using H∞ norm-bound constraint on disturbance
attenuation [10, 11] and the guaranteed cost control (see [12] and [13]), respectively.
Unfortunately, few results have been considered such performance indices simultaneously.
Meanwhile, once some components of satellite attitude control systems go wrong, it is
difficult to confirm desired multiple performances by the existing fault-tolerant control.
Thus, it is necessary to investigate the problem of fault-tolerant control with multiple
performance constraints.
Therefore, it is our motivation to investigate the quadratic D stabilizable satisfactory fault-
tolerant control problem with consistent indices constraints for a class of satellite attitude
control uncertain discrete-time systems subject to actuator failures. In view of possible
actuator failure as well as uncertainties which do not satisfy matching conditions existing in
both the state and control input matrices, we first derive the existence conditions of
satisfactory fault-tolerant state-feedback control law. Then by LMI technique, a convex
optimization problem is formulated to find the corresponding controller. The state-feedback
controller is designed to guarantee the closed-loop system satisfying the pre-specified
quadratic D stabilizability index, H∞ norm-bound