鲁棒非脆弱保证成本控制：Delta算子不确定时滞系统

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"Nonfragile guaranteed cost control for Delta operator-formulated uncertain time-delay systems (2010年)，作者：Ruiquan LIN, Silian CHEN, Xuwei DING，发表于《Journal of Control Theory and Applications》2010年第8卷第2期，DOI：10.1007/s11768-010-8082-6" 本文探讨了非脆弱保证成本控制在处理Δ算子形式的不确定时滞系统中的应用。Δ算子常用于描述具有快速动态特性的系统，如微电子机械系统（MEMS）和高速机械系统。设计控制器时，考虑到实际实现过程中控制器参数可能会偏离预设值，这使得系统的稳定性和性能难以保证。基于李亚普诺夫稳定性理论，研究者提出了一种非脆弱保证成本控制的设计方法。李亚普诺夫稳定性理论是分析和设计线性或非线性系统稳定性的重要工具，它通过构造一个李亚普诺夫函数来确保系统的稳定性。非脆弱控制旨在设计控制器，即使在参数变化的情况下也能保持系统的稳定性和性能。文章中给出了存在非脆弱保证成本控制器的充分条件。这个条件意味着在满足特定约束条件下，可以找到一个控制器，使得系统在参数变化时仍能保持渐近稳定，并且系统的运行成本函数的上界也可以得到保证。这一点对于实际工程应用至关重要，因为实际系统中的参数往往难以精确预知，且可能受到环境因素的影响而发生变化。通过一个数值实例，作者展示了所提方法的有效性和可行性。该示例证明，即使设计控制器的参数有所变化，闭环系统仍然能够保持渐近稳定，而且可以计算出成本函数的最大值。相反，如果在设计控制器时不考虑参数变化，那么当这些变化发生时，系统可能会变得不稳定。这篇文章提供了处理不确定时滞系统的一种稳健控制策略，强调了在设计阶段考虑参数变化的重要性，这对于提高系统鲁棒性和应对实际工程问题具有深远意义。这项工作对控制系统设计、优化以及系统稳定性分析等领域具有重要的理论和实践价值。

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

+ΔA

)x(k)+(A

δd

+ΔA

δd

)x(k − d)

+(B

+ΔB

)u(k),

x(0) = x

(1)

where d is the time-delay in the state, x(k) ∈ R

is the state

vector, and u(k) ∈ R

is the controlled input vector, ΔA

ΔA

δd

, and ΔB

are uncertain real matrices, which satisfy

[ΔA

ΔA

δd

ΔB

]=EF

], (2)

Received 6 May 2008; revised 11 November 2008.

This work was supported by the Natural Science Foundation of Fujian Province (No.2008J04016).

 South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010

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