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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 504356, 15 pages
http://dx.doi.org/10.1155/2012/504356
Research Article

Robust Finite-Time Control for Uncertain Systems Subject to Intermittent Measurements

School of Automation, Northwestern Polytechnical University, Shaanxi Province, Xi’an 710072, China

Received 7 June 2012; Accepted 16 August 2012

Academic Editor: Beatrice Paternoster

Copyright © 2012 Zhenghong Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the robust finite-time controller design problem of discrete-time systems with intermittent measurements. It is assumed that the system is subject to the norm-bounded uncertainties and the measurements are intermittent. The Bernoulli process is used to describe the phenomenon of intermittent measurements. By substituting the state-feedback controller into the system, a stochastic closed-loop system is obtained. Based on the analysis of the robust stochastic finite-time stability and the performance, the controller design method is proposed. The controller gain can be calculated by solving a sequence of linear matrix inequalities. Finally, a numerical example is used to show the design procedure and the effectiveness of the proposed design methodology.

1. Introduction

In the real world, system models are unavoidable to contain uncertainties which can result from the modeling error or variations of the system parameters. During the past 20 years, the norm-bounded uncertainties have been widely used in the system modeling and control for practical plants [110]. In [11], the authors studied the time-delay linear systems with the norm-bounded uncertainties. In [10], the robust memoryless controller design for linear time-delay systems with norm-bounded time-varying uncertainty was studied. In [3], the robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty was explored.

Since the late 1980s, the strategy has attracted a lot of attention due to the fact that this control strategy can be easily utilized to deal with the uncertainties and attenuation the effect from the external input to the controlled output [1216]. The strategy was original from [17]. From then on, the useful tool has been applied to different kinds of systems. In [18], the controller of linear parameter-varying systems with parameter-varying delays was exploited. In [19], the strategy was used to the filter design for uncertain Markovian jump systems with mode-dependent time delays. While in [20], the strategy was used in the robust controller design of discrete-time Markovian jump linear systems with mode-dependent time-delays. Gain-scheduled controller design problem for time-varying systems was investigated in [21].

In the literature, the strategy was always based on the Lyapunov asymptotic stability which is with an infinite interval. However, in some practical applications, the asymptotic stability is inadequate if large values of the state are not acceptable and there exists saturation [15, 2237]. Although the finite-time stability was proposed in 1960s [22], it only attracted the researchers' attention very recently. In [38], observer-based finite-time stabilization for extended Markov jump systems was studied. The observer-based finite-time control of time-delayed jump systems was studied in [39]. By considering the partially known transition jump rates, the finite-time filtering for non-linear stochastic systems was explored in [40]. For time-varying singular impulsive systems, the finite-time stability conditions were obtained in [41]. At the application side, the finite-time stability has been used in [25].

On another research frontier, the intermittent measurements have been paid a great number of efforts. In an ideal sampling, the measurements are consecutive. But, in a harsh sampling environment, the sampling may not be consecutive but intermittent [4247]. If the phenomenon of intermittent measurements is not considered during the controller and filter design period, the actual missing measurements may deteriorate the designed systems. Although, there are many results on the control, uncertain systems, and finite-time stability, there are few results on the control for uncertain systems subject to intermittent measurements. This fact motivates the research. In this paper, the contributions of this work are summarized as follows: (1) The intermittent measurements are considered the finite-time framework. Due to the induced stochastic system, the robust stochastic finite-time boundedness is studied. (2) The performance with the robust stochastic finite-time stability is investigated.

Notation.𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝑚×𝑛 represents the set of all 𝑚×𝑛 real matrices. 𝔼{} is the expectation operator with respect to some probability measure. 𝜆max{} and 𝜆min{} are the maximum eigenvalue and the minimum eigenvalue of the matrix, respectively.

2. Problem Formulation

In this paper, the following uncertain discrete-time linear system is considered: 𝑥𝑘+1=(𝐴+Δ𝐴)𝑥𝑘+𝐵1𝑢𝑘+𝐵2+Δ𝐵2𝜔𝑘,𝑧𝑘=𝐸𝑥𝑘+𝐹1𝑢𝑘+𝐹2𝜔𝑘,(2.1) where 𝑥𝑘𝑛 denotes the state vector, 𝑢𝑘𝑚 is the control input, 𝑧𝑘𝑝 is the controlled output, and 𝜔𝑘𝑟 is the time-varying disturbance which satisfies 𝑘=1𝜔T𝑘𝜔𝑘𝑑2,𝑘0,(2.2) where 𝑑>0 is a given scalar.

The matrices 𝐴,𝐵1,𝐵2,𝐸,𝐹1, and 𝐹2 are constant matrices with appropriate dimensions. Δ𝐴 and Δ𝐵2 are real time-varying matrix functions representing the time-varying parameter uncertainties. It is assumed that the uncertainties are normbounded and admissible, which can be modeled as Δ𝐴Δ𝐵2=𝐻𝐺𝑘𝑀1𝑀2,(2.3) where 𝐻, 𝑀1, and 𝑀2 are known real constant matrices, which characterize how the uncertain parameters in 𝐺𝑘 enter the nominal matrices 𝐴  and  𝐵1, and 𝐵2 is an unknown time-varying matrix function satisfying𝐺𝑘𝐼,𝑘0.(2.4)

Consider an ideal state feedback controller as follows: ̂𝑢𝑘=𝐾𝑥𝑘,(2.5) where ̂𝑢𝑘 is the ideal control signal which is obtained with the ideal state measuring, 𝐾 is the state feedback gain to be designed, and 𝑥𝑘 is the system state. In an ideal sensing environment, it is always assumed that 𝑥𝑘 is available for all the time instants 𝑘. However, in many practical applications, such as the networked control systems (NCSs), the measurements may not be consecutive but intermittent. In order to get the general case, it is assumed that the state measurements are intermittent and the actual control law is governed by 𝑢𝑘=𝐾𝑥𝑘,themeasurementisavailable,𝑢𝑘=0,themeasurementismissing.(2.6)

To better describe the intermittent measurements, a Bernoulli process 𝛼𝑘 is used to represent the intermittent measurements such that the actual control signal is 𝑢𝑘=𝛼𝑘𝐾𝑥𝑘,(2.7) where 𝛼𝑘 takes values in the set {0,1}, 𝛼𝑘=0 refers to that the measurement is missing, and 𝛼𝑘=1 means that the measurement is available. In addition, it is assumed that the probability of 𝛼𝑘=1 is 𝛽.

By substituting the actual control signal into the state-space model of (2.1), one gets 𝑥𝑘+1=𝐴+Δ𝐴+𝛽𝐵1𝛼𝐾+𝑘𝐵𝛽1𝐾𝑥𝑘+𝐵2+Δ𝐵2𝜔𝑘=𝐴Δ𝐴,𝛼𝑘𝑥𝑘+𝐵Δ𝐵2𝜔𝑘,𝑧𝑘=𝐸+𝛽𝐹1𝛼𝐾+𝑘𝐹𝛽1𝐾𝑥𝑘+𝐹2𝜔𝑘=𝐸𝛼𝑘𝑥𝑘+𝐹𝜔𝑘.(2.8) It is obvious that there is a stochastic variable 𝛼𝑘 in the closed-loop system in (2.8). Therefore, the objective of this paper is to find some sufficient conditions which can guarantee that the closed-loop system in (2.8) is robustly stochastically finite-time boundedness and reduces the effect of the disturbance input to the controlled output to a prescribed level.

Firstly, some useful definitions and lemmas are introduced, which will be used throughout the rest of the paper.

Definition 2.1 (Finite-Time Stable (FTS) [23]). For a class of discrete-time linear systems 𝑥𝑘+1=𝐴𝑥𝑘,𝑘0,(2.9) is said to be FTS with respect to (𝑐1,𝑐2,𝑅,𝑁), where 𝑅 is a positive definite matrix, 0<𝑐1<𝑐2 and 𝑁0, if 𝑥T0𝑅𝑥0𝑐21, then 𝑥T𝑘𝑅𝑥𝑘𝑐22, for all 𝑘{1,2,,𝑁}.

Definition 2.2 (Robustly Finite-Time Stable (RFTS) [23]). For a class of discrete-time linear uncertain systems 𝑥𝑘+1=(𝐴+Δ𝐴)𝑥𝑘,𝑘0,(2.10) is said to be RFTS with respect to (𝑐1,𝑐2,𝑅,𝑁), where 𝑅 is a positive definite matrix, 0<𝑐1<𝑐2, and 𝑁0; if for all admissible uncertainties Δ𝐴, 𝑥T0𝑅𝑥0𝑐21, then 𝑥T𝑘𝑅𝑥𝑘𝑐22, for all 𝑘{1,2,,𝑁}.

Definition 2.3 (Robustly Stochastically Finite-Time Stable (RSFTS)). For a class of discrete-time linear uncertain systems 𝑥𝑘+1=𝐴Δ𝐴,𝛼𝑘𝑥𝑘,𝑘0,(2.11) is said to be RSFTS with respect to (𝑐1,𝑐2,𝑅,𝑁), where the system matrix 𝐴(Δ𝐴,𝛼𝑘) has the uncertainty and the stochastic variable, 𝑅 is a positive definite matrix, 0<𝑐1<𝑐2, and 𝑁0; if for all admissible uncertainties Δ𝐴, stochastic variable 𝛼𝑘, 𝑥T0𝑅𝑥0𝑐21, then 𝔼{𝑥T𝑘𝑅𝑥𝑘}𝑐22, for all 𝑘{1,2,,𝑁}.

Definition 2.4 (Robustly Stochastically Finite-Time Bounded (RSFTB)). For a class of discrete-time linear uncertain systems 𝑥𝑘+1=𝐴Δ𝐴,𝛼𝑘𝑥𝑘+𝐵(Δ𝐵)𝜔𝑘,𝑘0(2.12) is said to be RSFTB with respect to (𝑐1,𝑐2,𝑑,𝑅,𝑁), where the system matrix 𝐴(Δ𝐴,𝛼𝑘) has the uncertainty and the stochastic variable, the input matrix contains the norm-bounded uncertainty, 𝑅 is a positive definite matrix, 0<𝑐1<𝑐2, and 𝑁0; if for all admissible uncertainties Δ𝐴 and Δ𝐵, stochastic variable 𝛼𝑘, 𝑥T0𝑅𝑥0𝑐21, then 𝔼{𝑥T𝑘𝑅𝑥𝑘}𝑐22, for all 𝑘{1,2,,𝑁}.

With the above definitions, the robust finite-time control problem in this paper can be summarized as follows: for the uncertain system in (2.1), the objective is to design a state feedback controller in (2.7) such that for all the admissible uncertainties and the intermittent measurements: (i)the closed-loop system (2.8) is RSFTS; (ii)under the zero-initial condition, the controlled output 𝑧𝑘 satisfies 𝔼𝑁𝑖=1𝑧T𝑘𝑧𝑘<𝛾2𝑁𝑖=1𝜔T𝑘𝜔𝑘,(2.13)for all 𝑙2-bounded 𝜔𝑘, where prescribed value 𝛾 is the attenuation level.

If the above conditions are both satisfied, the designed controller is called a RSFTB state-feedback controller. To achieve the design objectives, the following lemmas are introduced.

Lemma 2.5 (Schur complement). Given a symmetric matrix Ξ=Ξ11Ξ12Ξ21Ξ22, the following three conditions are equivalent to each other: (i)Ξ<0; (ii)Ξ11<0, Ξ22ΞT12Ξ111Ξ12<0; (iii)Ξ22<0, Ξ11Ξ12Ξ122ΞT12<0.

Lemma 2.6 (see [48, 49]). Let Θ=ΘT, 𝐻 and 𝑀 be real matrices with compatible dimensions, and 𝐺𝑘 be time varying and satisfy (2.4). Then it concludes that the following condition: Θ+𝐻𝐺𝑘𝑀+𝐻𝐺𝑘𝑀T<0(2.14) holds if and only if there exists a positive scaler 𝜀>0 such that Θ𝜀𝐻𝑀T𝜀𝐼0𝜀𝐼<0(2.15) is satisfied.

3. Main Results

3.1. Stability and Performance Analysis

In this section, the finite-time stability, robust finite-time stability, and robust stochastic finite-time stability will be analyzed by assuming the controller gain is given.

Lemma 3.1 (sufficient conditions for finite-time stability [23]). For a class of discrete-time linear systems 𝑥𝑘+1=𝐴𝑥𝑘,𝑘0,(3.1) they are finite-time stable with respect to (𝑐1,𝑐2,𝑅,𝑁) if there exist a positive-definite matrix 𝑃 and a scalar 𝜃1 such that the following conditions hold: 𝐴T𝑃𝐴𝜃𝑃<0,(3.2)cond𝑃<𝑐22𝜃𝑁𝑐21,(3.3) where 𝑃=𝑅1/2𝑃𝑅1/2 and cond(𝑃)=𝜆max(𝑃)/𝜆min(𝑃).

Lemma 3.2 (sufficient conditions for robust finite-time stability). For a class of discrete-time linear systems 𝑥𝑘+1=(𝐴+Δ𝐴)𝑥𝑘,𝑘0,(3.4) they are robustly finite-time stable with respect to (𝑐1,𝑐2,𝑅,𝑁) if there exist a positive-definite matrix 𝑃, a scalar 𝜀>0, and a scalar 𝜃1 such that (3.3) and the following condition hold: 𝑃𝑃𝐴𝜀𝑃𝐻0𝜃𝑃0𝑀T1𝜀𝐼0𝜀𝐼<0.(3.5)

Proof. According to Lemma 3.1, the uncertain system is RFTS if the following condition holds: (𝐴+Δ𝐴)T𝑃(𝐴+Δ𝐴)𝜃𝑃<0.(3.6) Using the Schur complement, the above condition is equivalent with 𝑃𝑃(𝐴+Δ𝐴)𝜃𝑃<0.(3.7) Since Δ𝐴=𝐻𝐺𝑘𝑀1, by using Lemma 2.6, (3.7) is equivalent with (3.5).

Now, let us study the RSFTB of the closed-loop system in (2.8) and deal with the uncertainty and the stochastic variable by using the skills mentioned above.

Theorem 3.3. The closed-loop system in (2.8) is RSFTB with respect to (𝑐1,𝑐2,𝑑,𝑅,𝑁), if there exist positive-definite matrices 𝑃1=𝑃T1, 𝑃2=𝑃T2, and two scalars 𝜃1, and 𝜀>0 such that the following conditions hold: 𝑃10𝑃1𝐵1𝐾000𝑃1𝑃1𝐴+𝛽𝐵1𝐾𝑃1𝐵2𝜀𝑃𝐻0𝜃𝑃100𝑀T1𝜃𝑃20𝑀T2𝜀𝐼00𝜀𝐼<0,(3.8)𝜆max𝑃1𝑐21+𝜆max𝑃2𝑑2<𝑐22𝜆min𝑃1𝜃𝑁,(3.9) where 𝑃1=𝑅1/2𝑃1𝑅1/2 and =𝛽(1𝛽).

Proof. Consider the following Lyapunov function: 𝑉(𝑘)=𝑥T𝑘𝑃1𝑥𝑘,(3.10) where 𝑃1 is a symmetric positive-definite matrix. For the closed-loop system in (2.8), the expectation of one step advance of the Lyapunov function can be obtained as 𝔼𝑉(𝑘+1)𝑥𝑘=𝑥T𝑘𝐴+Δ𝐴+(𝛽+)𝐵1𝐾T𝑃1𝐴+Δ𝐴+(𝛽+)𝐵1𝐾𝑥𝑘+2𝑥T𝑘𝐴+Δ𝐴+𝛽𝐵1𝐾T𝑃1𝐵2+Δ𝐵2𝜔𝑘+𝜔T𝑘𝐵2+Δ𝐵2T𝑃1𝐵2+Δ𝐵2𝜔𝑘=𝑥𝑘𝜔𝑘TΩ11Ω12Ω22𝑥𝑘𝜔𝑘=𝑥𝑘𝜔𝑘TΩ𝑥𝑘𝜔𝑘,(3.11) where Ω11=𝐴+Δ𝐴+(𝛽+)𝐵1𝐾T𝑃1𝐴+Δ𝐴+(𝛽+)𝐵1𝐾,Ω12=𝐴+Δ𝐴+𝛽𝐵1𝐾T𝑃1𝐵2+Δ𝐵2,Ω22=𝐵2+Δ𝐵2T𝑃1𝐵2+Δ𝐵2.(3.12)
On the other hand, by using Schur complement, the condition in (3.8) implies that Ω<𝜃𝑃100𝜃𝑃2,(3.13) since the condition in (3.8) is equivalent with Θ+𝐻𝐺𝑘𝑀+𝐻𝐺𝑘𝑀T<0,(3.14) where Θ=𝑃10𝑃𝐵1𝐾0𝑃1𝑃𝐴+𝛽𝐵1𝐾𝑃𝐵2𝜃𝑃10𝜃𝑃2,000,𝐻=𝑃𝐻𝑀=00𝑀1𝑀2.(3.15) With the condition (3.13), one gets 𝔼𝑉(𝑘+1)𝑥𝑘<𝜃𝑉(𝑘)+𝜃𝜔T𝑘𝑃2𝜔𝑘.(3.16) Taking the iterative operation with respect to the time instant 𝑘, one obtains 𝔼𝑉(𝑘)𝑥0<𝜃𝑘𝑉(0)+𝑘𝑖=1𝜃𝑘𝑖+1𝜔T𝑗1𝑃2𝜔𝑗1<𝜃𝑁𝜆max𝑃1𝑐21+𝜆max𝑃2𝑑2.(3.17) Recalling the Lyapunov function, there is 𝔼𝑉(𝑘)𝑥0>𝜆min𝑃1𝑥T𝑘𝑅𝑥𝑘.(3.18) Combing (3.17) and (3.19), one gets 𝔼𝑥T𝑘𝑅𝑥𝑘<𝜃𝑁𝜆min𝑃1𝜆max𝑃1𝑐21+𝜆max𝑃2𝑑2.(3.19) With the condition (3.9), it concludes that 𝔼𝑥T𝑘𝑅𝑥𝑘<𝑐22.(3.20) Therefore, if the conditions in (3.8) and (3.9) are satisfied, the closed-loop system (2.8) is RSFTB. The proof is completed.

In order to incorporate the performance 𝛾, the following theorem provides other sufficient conditions for the RSFTB of the closed-loop system (2.8).

Theorem 3.4. The closed-loop system in (2.8) is RSFTB with respect to (𝑐1,𝑐2,𝑑,𝑅,𝑁), if there exist positive-definite matrix 𝑃=𝑃T, and three scalars 𝜃1, 𝜀>0, and 𝛾>0 such that the following conditions hold: 𝑃0𝑃𝐵1𝐾000𝑃𝑃𝐴+𝛽𝐵1𝐾𝑃𝐵2𝜀𝑃𝐻0𝜃𝑃100𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0,(3.21)𝜆max𝑃𝑐21+𝛾2𝑑2<𝑐22𝜆min𝑃𝜃𝑁,(3.22) where 𝑃=𝑅1/2𝑃𝑅1/2 and =𝛽(1𝛽).

Proof. Suppose that 𝑃1 and 𝑃2 in Theorem 3.3 are substituted by 𝑃 and 𝛾2𝐼/𝜃, respectively. Then, the condition (3.8) turns to (3.21). The maximum eigenvalue of 𝛾2𝐼/𝜃 is no more than 𝛾2. Therefore, (3.22) can guarantee the holdness of (3.9). The proof is completed.

Now, consider the controlled output and the attenuation level 𝛾.

Theorem 3.5. The closed-loop system in (2.8) is RSFTB with respect to (0,𝑐2,𝑑,𝑅,𝑁) and with an attenuation level 𝛾, if there exist positive-definite matrix 𝑃=𝑃T, and three scalars 𝜃1, 𝜀>0, and 𝛾>0 such that the following conditions hold: 𝐼000𝐹1𝐾000𝑃00𝑃𝐵1𝐾000𝑃0𝑃𝐴+𝛽𝐵1𝐾𝑃𝐵2𝜀𝑃𝐻0𝐼𝐸+𝛽𝐹1𝐾𝐹200𝜃𝑃00𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0,(3.23)𝛾2𝑑2<𝑐22𝜆min𝑃𝜃𝑁,(3.24) where 𝑃=𝑅1/2𝑃𝑅1/2 and =𝛽(1𝛽).

Proof. The attenuation level 𝛾 refers to the zero-initial value of the state. Therefore, 𝑐1 is set to be zero. Consider the following cost function: 𝐽=𝔼𝑉(𝑘+1)𝑥𝑘𝑧+𝔼T𝑘𝑧𝑘𝛾2𝐼.(3.25) The cost function can be revaluated with similar lines in Theorem 3.3. The proof is omitted.

3.2. Controller Design

The robust stochastic finite-time stability and the performance have been investigated in the above subsection. In this subsection, the controller design will be proposed.

Theorem 3.6. Given a positive constant 𝛾, the closed-loop system in (2.8) is RSFTB with respect to (0,𝑐2,𝑑,𝑅,𝑁) and with a prescribed attenuation level 𝛾, if there exist positive-definite matrix 𝑄=𝑄T, two scalars 𝜃1, and 𝜀>0 and 𝐿 such that the following conditions hold: 𝐼000𝐹1𝐿000𝑄00𝐵1𝐿000𝑄0𝐴𝑄+𝛽𝐵1𝐿𝐵2𝜀𝐻0𝐼𝐸𝑄+𝛽𝐹1𝐿𝐹200𝜃𝑄00𝑄𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0,(3.26)𝛾2𝑑2<𝑐22𝜃𝑁𝜆max𝑄,(3.27) where 𝑄=𝑅1/2𝑄𝑅1/2 and =𝛽(1𝛽). Moreover, the controller gain can be calculated as 𝐾=𝐿𝑄1.

Proof. In Theorem 3.5, pre- and postmultiplying (3.25) by diag{𝐼,𝑃1,𝑃1,𝐼,𝐼,𝐼,𝐼,𝐼}, the following equivalent condition is obtained: 𝐼000𝐹1𝐾000𝑃100𝐵1𝐾000𝑃10𝐴+𝛽𝐵1𝐾𝐵2𝜀𝐻0𝐼𝐸+𝛽𝐹1𝐾𝐹200𝜃𝑃00𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0.(3.28) Letting 𝑄 denote 𝑃1, the condition (3.28) is equivalent with 𝐼000𝐹1𝐾000𝑄00𝐵1𝐾000𝑄0𝐴+𝛽𝐵1𝐾𝐵2𝜀𝐻0𝐼𝐸+𝛽𝐹1𝐾𝐹200𝜃𝑄100𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0.(3.29)
Pre- and post-multiplying (3.29) by the symmetric matrix diag{𝐼,𝐼,𝐼,𝐼,𝑄,𝐼,𝐼,𝐼}, the following equivalent condition is obtained: 𝐼000𝐹1𝐾𝑄000𝑄00𝐵1𝐾𝑄000𝑄0𝐴𝑄+𝛽𝐵1𝐾𝑄𝐵2𝜀𝐻0𝐼𝐸𝑄+𝛽𝐹1𝐾𝑄𝐹200𝜃𝑄00𝑄𝑀T1𝛾2𝐼0𝑀T2𝜀𝐼00𝜀𝐼<0.(3.30) Defining 𝐿=𝐾𝑄, the condition (3.30) is equivalent with (3.26). With the fact that 𝜆min1𝑃=𝜆max𝑄,(3.31) the condition (3.27) is equivalent with (3.24).

It is noted that the condition (3.27) is not an linear matrix inequality. However, it is easy to check that the condition (3.27) is guaranteed by imposing the following conditions [50]: 𝛾0<𝑄<𝐼,2𝑑2<𝑐22𝜃𝑁.(3.32) In addition, the performance 𝛾 refers to the attenuation level from the external noise to the controlled output. Therefore, it is desired that the performance 𝛾 should be as smaller as possible. The controller with the optimal 𝛾 is called the optimal controller. For fixed 𝜃 and 𝑐2, the optimal 𝛾 can be obtained by min𝛾2,s.t.(3.26)and(3.32).(3.33)

Remark 3.7. The results in this paper are obtained by using the Lyapunov method and only sufficient. In the future work, more techniques will be used to reduce the possible conservativeness of the results.

4. Numerical Example

Consider the system in (2.1) with the following matrix: 𝐴=1311,𝐵1=11,𝐵2=0.20.1𝐸=10,𝐹1=1,𝐹2=0,𝐻=0.020.01,𝑀1=11,𝑀2=0.5.(4.1)

For the finite-time stability test, it is assumed that 𝑅=𝐼,𝑁=5,𝑐2=5,𝑑=0.1,𝜃=1.2.(4.2) The probability of the available measurements is 0.95. With the proposed optimization problem in (3.33), the obtained minimum performance index is 𝛾=0.5034 and the corresponding controller gain is .𝐾=1.03411.5465(4.3)

In the simulation, the initial system state is chosen as [0.5;0.5]. Without the designed controller, Figure 1 depicts the trajectories of the system state. It is obvious that the open-loop system is unstable. However, with the designed controller, Figure 2 shows the trajectories of the system state with the intermittent measurements in Figure 3. The trajectories converge to zeros even though the system is subject to uncertainties, external disturbance and intermittent measurements.

504356.fig.001
Figure 1: Trajectories of the system state in the simulation without the designed controller.
504356.fig.002
Figure 2: Trajectories of the system state in the simulation with the designed controller.
504356.fig.003
Figure 3: Stochastic values of the intermittent measurements in the simulation.

5. Conclusion

In this paper, the robust finite-time controller design problem of discrete-time systems with intermittent measurements has been investigated. The uncertainties are assumed to be norm bounded. The measurements of the system state are intermittent and Bernoulli process is used to describe the phenomenon of intermittent measurements. Based on the results of the robust stochastic finite-time stability and the performance, the controller design approach was proposed. Finally, an illustrative example was used to show the design procedure and the effectiveness of the proposed design methodology.

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