J Control Theory Appl 2010 8 (2) 233–238
DOI 10.1007/s11768-010-8082-6
Nonfragile guaranteed cost control for Delta
operator-formulated uncertain time-delay systems
Ruiquan LIN, Silian CHEN, Xuwei DING
(
College of Electrical Engineering and Automation, Fuzhou University, Fuzhou Fujian 350108, China)
Abstract: With consideration that the controller parameters may vary from the designed value when the controller
is realized, based on Lyapunov stability theory, a design method of nonfragile guaranteed cost control for a class of Delta
operator-formulated uncertain time-delay systems is studied. A sufficient condition for the existence of the nonfragile
guaranteed cost controller is given. A numeric example is then given to illustrate the effectiveness and the feasibility of the
designed method. The results show that even if the parameters of the designed controller are of variations, the closed-loop
system is still asymptotically stable and the super value of the cost function can also be obtained, while the closed-loop
system will be unstable if the variations of the controller parameters are not considered when the controller is designed. The
nonfragile guaranteed cost controller derived from the traditional shift operator method may cause the closed-loop system
to be unstable, while the nonfragile guaranteed cost controller based on Delta operator method can avoid the unstable
problem of the closed-loop system.
Keywords: Delta operator; Time-delay system; Nonfragile control; Guaranteed cost control; LMI
1 Introduction
In contrast with the traditional shift operator, Delta op-
erator is a new discretization method. If the sampling pe-
riod approaches zero, the Delta operator-formulated sys-
tem is regarded as a continuous one, else the system is re-
garded as a discrete one. Especially, in the fast sampling
field, the controller derived from the traditional shift oper-
ator method may cause the controlled system to be unsta-
ble, while the controller based on Delta operator method
can avoid the unstable problem of the controlled system [1].
Recently, with consideration that the controller parameters
can be exactly realized, many researchers studied the guar-
anteed cost control for the Delta operator-formulated sys-
tems [1∼3]. The design of state feedback guaranteed cost
controller for a class of Delta operator-formulated certain
systems was investigated in reference [1]. Based on the LMI
technology, the optimal guaranteed cost control for a class
of Delta operator-formulated nonlinear T-S fuzzy systems
with norm-bounded uncertainty was illustrated in reference
[2]. Also, based on the LMI technology, Liu [3] studied
the design of optimal guaranteed cost controller with pole
constraints for Delta operator-formulated uncertain linear
systems. At present, all the authors who studied the guar-
anteed cost control for Delta operator-formulated systems
supposed that the controller parameters could be exactly re-
alized, and they did not consider the case that the controller
parameters may vary from the designed value when the con-
troller was realized. In fact, when the controller is realized,
the parameters of the controller do have perturbations due
to the finite word length and the round off error of the digi-
tal processor, also the imprecision inherent in A/D and D/A
conversion and so on [4]. The variations of the controller
parameters may result in the fragile problem of the con-
trolled system [5∼9]. Therefore, when the controller is de-
signed, the variations of the controller parameters must be
considered. On the other hand, many researchers have stud-
ied the nonfragile guaranteed cost control for the time-delay
systems. For example, literature [10] studied the nonfragile
guaranteed cost control problem for a class of uncertain dis-
crete time-delay linear systems. The nonfragile guaranteed
cost control problem for a class of uncertain neutral system
with time-varying delays in both state and control input was
dicussed in reference [11]. The problem of optimal nonfrag-
ile guaranteed cost control for a class of linear time-delay
systems with structured uncertainties was considered in ref-
erence [12], and Yin [13] discussed the nonfragile guaran-
teed cost control problem for a class of nonlinear time-delay
systems described by T-S fuzzy model, yet few results have
been reported on how to design a nonfragile guaranteed cost
controller for the Delta operator-formulated time-delay sys-
tems.
In this paper, with LMI technology, the state feedback
nonfragile guaranteed cost control for a class of Delta
operator-formulated uncertain time-delay systems is dis-
cussed, where the parameters of the nonfragile guaranteed
cost controller and the controlled object are assumed to have
additive norm-bounded variations.
2 Problem statement
Consider the following Delta operator-formulated uncer-
tain time-delay system[14]:
⎧
⎨
⎩
δx(k)=(A
δ
+ΔA
δ
)x(k)+(A
δd
+ΔA
δd
)x(k − d)
+(B
δ
+ΔB
δ
)u(k),
x(0) = x
0
,
(1)
where d is the time-delay in the state, x(k) ∈ R
n
is the state
vector, and u(k) ∈ R
m
is the controlled input vector, ΔA
δ
,
ΔA
δd
, and ΔB
δ
are uncertain real matrices, which satisfy
[ΔA
δ
ΔA
δd
ΔB
δ
]=EF
1
[H
1
H
2
H
3
], (2)
Received 6 May 2008; revised 11 November 2008.
This work was supported by the Natural Science Foundation of Fujian Province (No.2008J04016).
c
South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010