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

Abstract

This paper investigates the robust finite-time ℋ_∞ 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 ℋ_∞ 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 [1–10]. In [11], the authors studied the time-delay linear systems with the norm-bounded uncertainties. In [10], the robust memoryless ℋ_∞ 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 ℋ_∞ 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 [12–16]. The strategy was original from [17]. From then on, the useful tool has been applied to different kinds of systems. In [18], the ℋ_∞ 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 ℋ_∞ controller design problem for time-varying systems was investigated in [21].

In the literature, the ℋ_∞ 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, 22–37]. 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 [42–47]. 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 ℋ_∞ control, uncertain systems, and finite-time stability, there are few results on the ℋ_∞ 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 ℋ_∞ performance with the robust stochastic finite-time stability is investigated.

Notation.ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 represents the set of all 𝑚×𝑛 real matrices. 𝔼{⋅} is the expectation operator with respect to some probability measure. 𝜆_max{⋅} and 𝜆_min{⋅} 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: 𝑥_𝑘+1=(𝐴𝑧+_𝑘Δ𝐴)𝑥_𝑘+𝐵₁𝑢_𝑘+𝐵₂+Δ𝐵₂𝜔_𝑘,=𝐸𝑥_𝑘+𝐹₁𝑢_𝑘+𝐹₂𝜔_𝑘,(2.1) where 𝑥_𝑘∈ℝ^𝑛 denotes the state vector, 𝑢_𝑘∈ℝ^𝑚 is the control input, 𝑧_𝑘∈ℝ^𝑝 is the controlled output, and 𝜔_𝑘∈ℝ^𝑟 is the time-varying disturbance which satisfies ^∞_𝑘=1𝜔^T_𝑘𝜔_𝑘≤𝑑²,𝑘∈ℕ₀,(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=𝐻𝐺^𝑘𝑀₁𝑀₂,(2.3) where 𝐻, 𝑀₁, and 𝑀₂ are known real constant matrices, which characterize how the uncertain parameters in 𝐺_𝑘 enter the nominal matrices 𝐴  and  𝐵₁, and 𝐵₂ is an unknown time-varying matrix function satisfying‖‖𝐺_𝑘‖‖≤𝐼,∀𝑘∈ℕ₀.(2.4)

Consider an ideal state feedback controller as follows: 𝑢_𝑘̂=𝐾𝑥_𝑘,(2.5) where 𝑢_𝑘̂ is the ideal control signal which is obtained with the ideal state measuring, 𝐾 is the state feedback gain to be designed, and 𝑥_𝑘 is the system state. In an ideal sensing environment, it is always assumed that 𝑥_𝑘 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 𝑢_𝑘=𝐾𝑥_𝑘,themeasurementisavailable,𝑢_𝑘=0,themeasurementismissing.(2.6)

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

By substituting the actual control signal into the state-space model of (2.1), one gets 𝑥_𝑘+1=𝐴+Δ𝐴+𝛽𝐵₁𝐾+𝛼_𝑘−𝛽𝐵₁𝐾𝑥_𝑘+𝐵₂+Δ𝐵₂𝜔_𝑘𝑧_𝑘=𝐴Δ𝐴,𝛼_𝑘𝑥_𝑘+𝐵Δ𝐵₂𝜔_𝑘,=𝐸+𝛽𝐹₁𝐾+𝛼_𝑘−𝛽𝐹₁𝐾𝑥_𝑘+𝐹₂𝜔_𝑘=𝐸𝛼_𝑘𝑥_𝑘+𝐹𝜔_𝑘.(2.8) It is obvious that there is a stochastic variable 𝛼_𝑘 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 𝑥_𝑘+1=𝐴𝑥_𝑘,𝑘∈ℕ₀,(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 𝑁∈ℕ₀, if 𝑥^T₀𝑅𝑥₀≤𝑐²¹, then 𝑥^T_𝑘𝑅𝑥_𝑘≤𝑐²², for all 𝑘∈{1,2,…,𝑁}.

Definition 2.2 (Robustly Finite-Time Stable (RFTS) [23]). For a class of discrete-time linear uncertain systems 𝑥_𝑘+1=(𝐴+Δ𝐴)𝑥_𝑘,𝑘∈ℕ₀,(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 𝑁∈ℕ₀; if for all admissible uncertainties Δ𝐴, 𝑥^T₀𝑅𝑥₀≤𝑐²¹, then 𝑥^T_𝑘𝑅𝑥_𝑘≤𝑐²², for all 𝑘∈{1,2,…,𝑁}.

Definition 2.3 (Robustly Stochastically Finite-Time Stable (RSFTS)). For a class of discrete-time linear uncertain systems 𝑥_𝑘+1=𝐴Δ𝐴,𝛼_𝑘𝑥_𝑘,𝑘∈ℕ₀,(2.11) is said to be RSFTS with respect to (𝑐₁,𝑐₂,𝑅,𝑁), where the system matrix 𝐴(Δ𝐴,𝛼_𝑘) has the uncertainty and the stochastic variable, 𝑅 is a positive definite matrix, 0<𝑐₁<𝑐₂, and 𝑁∈ℕ₀; if for all admissible uncertainties Δ𝐴, stochastic variable 𝛼_𝑘, 𝑥^T₀𝑅𝑥₀≤𝑐²¹, then 𝔼{𝑥^T_𝑘𝑅𝑥_𝑘}≤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 𝑥_𝑘+1=𝐴Δ𝐴,𝛼_𝑘𝑥_𝑘+𝐵(Δ𝐵)𝜔_𝑘,𝑘∈ℕ₀(2.12) is said to be RSFTB with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑁), where the system matrix 𝐴(Δ𝐴,𝛼_𝑘) 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 𝑁∈ℕ₀; if for all admissible uncertainties Δ𝐴 and Δ𝐵, stochastic variable 𝛼_𝑘, 𝑥^T₀𝑅𝑥₀≤𝑐²¹, then 𝔼{𝑥^T_𝑘𝑅𝑥_𝑘}≤𝑐²², for all 𝑘∈{1,2,…,𝑁}.

With the above definitions, the robust finite-time ℋ_∞ 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 𝑧_𝑘 satisfies 𝔼^𝑁_𝑖=1𝑧^T_𝑘𝑧_𝑘<𝛾^2𝑁_𝑖=1𝜔^T_𝑘𝜔_𝑘,(2.13)for all 𝑙₂-bounded 𝜔_𝑘, where prescribed value 𝛾 is the ℋ_∞ 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=^Ξ11Ξ12_Ξ21Ξ22, the following three conditions are equivalent to each other: (i)Ξ<0; (ii)Ξ₁₁<0, Ξ₂₂−Ξ^T₁₂Ξ⁻¹¹¹Ξ₁₂<0; (iii)Ξ₂₂<0, Ξ₁₁−Ξ₁₂Ξ⁻¹²²Ξ^T₁₂<0.

Lemma 2.6 (see [48, 49]). Let Θ=Θ^T, 𝐻 and 𝑀 be real matrices with compatible dimensions, and 𝐺_𝑘 be time varying and satisfy (2.4). Then it concludes that the following condition: Θ+𝐻𝐺_𝑘𝑀+𝐻𝐺_𝑘𝑀^T<0(2.14) holds if and only if there exists a positive scaler 𝜀>0 such that ⎡⎢⎢⎢⎢⎣Θ𝜀𝐻𝑀^T∗∗−∗𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎦<0(2.15) is satisfied.

3. Main Results

3.1. Stability and ℋ_∞ 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 𝑥_𝑘+1=𝐴𝑥_𝑘,𝑘∈ℕ₀,(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 𝜃≥1 such that the following conditions hold: co𝐴n^Td𝑃𝐴−𝜃𝑃<0,(3.2)𝑃<𝜃^𝑁𝑐²²𝑐²¹,(3.3) where 𝑃=𝑅^−1/2𝑃𝑅^−1/2 and cond(𝑃)=𝜆_max(𝑃)/𝜆_min(𝑃).

Lemma 3.2 (sufficient conditions for robust finite-time stability). For a class of discrete-time linear systems 𝑥_𝑘+1=(𝐴+Δ𝐴)𝑥_𝑘,𝑘∈ℕ₀,(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 𝜃≥1 such that (3.3) and the following condition hold: ⎡⎢⎢⎢⎢⎢⎢⎣−∗𝑃−𝑃𝜃𝐴𝑃𝜀𝑃0𝐻𝑀0_T1∗∗∗∗−∗𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎦<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−𝜃+𝑃Δ𝐴)⎤⎥⎥⎦<0.(3.7) Since Δ𝐴=𝐻𝐺_𝑘𝑀₁, 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=𝑃^T₁, 𝑃₂=𝑃^T₂, and two scalars 𝜃≥1, and 𝜀>0 such that the following conditions hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝑃₁−0𝑃₁ℎ𝑃₁𝐵₁𝐾000∗∗𝑃₁𝐴−+𝜃𝑃𝛽₁𝐵₁𝐾𝑃₁𝐵₂𝜀𝑃𝐻000𝑀^T₁∗∗∗−𝜃𝑃₂0𝑀^T₂∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.8)𝜆_max𝑃¹𝑐²¹+𝜆_max𝑃₂𝑑²<𝑐²²𝜆_min𝑃¹𝜃^𝑁,(3.9) where 𝑃₁=𝑅^−1/2𝑃₁𝑅^−1/2 and ℎ=√𝛽(1−𝛽).

Proof. Consider the following Lyapunov function: 𝑉(𝑘)=𝑥^T_𝑘𝑃₁𝑥_𝑘,(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 𝔼𝑉(𝑘+1)∣𝑥_𝑘=𝑥^T_𝑘𝐴+Δ𝐴+(𝛽+ℎ)𝐵₁𝐾^T𝑃₁𝐴+Δ𝐴+(𝛽+ℎ)𝐵₁𝐾𝑥_𝑘+2𝑥^T_𝑘𝐴+Δ𝐴+𝛽𝐵₁𝐾^T𝑃₁𝐵₂+Δ𝐵₂𝜔_𝑘+𝜔^T_𝑘𝐵₂+Δ𝐵₂^T𝑃₁𝐵₂+Δ𝐵₂𝜔_𝑘=⎡⎢⎢⎣𝑥^𝑘𝜔_𝑘⎤⎥⎥⎦^T⎡⎢⎢⎣Ω₁₁Ω₁₂∗Ω₂₂⎤⎥⎥⎦⎡⎢⎢⎣𝑥_𝑘𝜔_𝑘=⎡⎢⎢⎣𝑥_𝑘⎤⎥⎥⎦𝜔_𝑘⎤⎥⎥⎦^TΩ⎡⎢⎢⎣𝑥^𝑘𝜔_𝑘⎤⎥⎥⎦,(3.11) where ΩΩ₁₁₁₂=𝐴+Δ𝐴+(𝛽+ℎ)𝐵₁𝐾^T𝑃₁𝐴+Δ𝐴+(𝛽+ℎ)𝐵₁𝐾,Ω₂₂=𝐴+Δ𝐴+𝛽𝐵₁𝐾^T𝑃₁𝐵₂+Δ𝐵₂,=𝐵₂+Δ𝐵₂^T𝑃₁𝐵₂+Δ𝐵₂.(3.12)
On the other hand, by using Schur complement, the condition in (3.8) implies that Ω<⎡⎢⎢⎣𝜃𝑃¹0𝜃0𝑃₂⎤⎥⎥⎦,(3.13) since the condition in (3.8) is equivalent with Θ+𝐻𝐺_𝑘𝑀+𝐻𝐺_𝑘𝑀^T<0,(3.14) where Θ=⎡⎢⎢⎢⎢⎢⎢⎣−𝑃¹∗−0𝑃₁ℎ𝑃𝐵₁𝐾0𝑃𝐴+𝛽𝐵₁𝐾𝑃𝐵₂∗∗∗∗−𝜃∗𝑃₁−𝜃0𝑃₂⎤⎥⎥⎥⎥⎥⎥⎦,𝐻=⎡⎢⎢⎢⎢⎢⎢⎣𝑃000𝐻⎤⎥⎥⎥⎥⎥⎥⎦,𝑀=00𝑀₁𝑀₂.(3.15) With the condition (3.13), one gets 𝔼𝑉(𝑘+1)∣𝑥_𝑘<𝜃𝑉(𝑘)+𝜃𝜔^T_𝑘𝑃₂𝜔_𝑘.(3.16) Taking the iterative operation with respect to the time instant 𝑘, one obtains 𝔼𝑉(𝑘)∣𝑥₀<𝜃^𝑘𝑉(0)+^𝑘_𝑖=1𝜃^{𝑘−𝑖+1}𝜔^T_𝑗−1𝑃₂𝜔_𝑗−1<𝜃^𝑁𝜆^max𝑃¹𝑐²¹+𝜆_max𝑃₂𝑑².(3.17) Recalling the Lyapunov function, there is 𝔼𝑉(𝑘)∣𝑥₀>𝜆_min𝑃¹𝑥^T𝑘𝑅𝑥_𝑘.(3.18) Combing (3.17) and (3.19), one gets 𝔼𝑥^T_𝑘𝑅𝑥_𝑘<𝜃^𝑁𝜆_min𝑃¹𝜆^max𝑃¹𝑐²¹+𝜆_max𝑃₂𝑑².(3.19) With the condition (3.9), it concludes that 𝔼𝑥^T_𝑘𝑅𝑥_𝑘<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 ℋ_∞ 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=𝑃^T, and three scalars 𝜃≥1, 𝜀>0, and 𝛾>0 such that the following conditions hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝑃−0𝑃𝑃𝐴ℎ𝑃+𝐵𝛽₁𝐵𝐾₁000∗∗−𝜃𝑃₁𝐾𝑃𝐵₂𝜀𝑃𝐻000𝑀^T₁∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.21)𝜆_max𝑃𝑐²¹+𝛾²𝑑²<𝑐²²𝜆_min𝑃𝜃^𝑁,(3.22) where 𝑃=𝑅^−1/2𝑃𝑅^−1/2 and ℎ=√𝛽(1−𝛽).

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 ℋ_∞ 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 ℋ_∞ attenuation level 𝛾, if there exist positive-definite matrix 𝑃=𝑃^T, and three scalars 𝜃≥1, 𝜀>0, and 𝛾>0 such that the following conditions hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝐼−0𝑃0000ℎℎ𝑃𝐹𝐵₁₁𝐾000∗∗−𝑃0𝑃𝐴+𝛽𝐵𝐾₁000∗∗∗−𝐼𝐸+𝛽𝐹₁𝐾𝑃𝐵₂𝜀𝑃𝐻0∗∗∗∗−𝜃𝑃𝐾𝐹0₂00𝑀0_T1∗∗∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.23)𝛾²𝑑²<𝑐²²𝜆_min𝑃𝜃^𝑁,(3.24) where 𝑃=𝑅^−1/2𝑃𝑅^−1/2 and ℎ=√𝛽(1−𝛽).

Proof. The ℋ_∞ 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=𝔼𝑉(𝑘+1)∣𝑥_𝑘+𝔼𝑧^T_𝑘𝑧_𝑘−𝛾²𝐼.(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 ℋ_∞ 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 ℋ_∞ attenuation level 𝛾, if there exist positive-definite matrix 𝑄=𝑄^T, two scalars 𝜃≥1, and 𝜀>0 and 𝐿 such that the following conditions hold: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝐼−0𝑄0000ℎℎ𝐵𝐹₁₁𝐿000∗∗−𝑄0𝐴𝑄+𝛽𝐿𝐵₁000∗∗∗−𝐼𝐸𝑄+𝛽𝐹₁𝐿𝐵₂𝜀𝐻0∗∗∗∗−𝜃𝑄𝐿𝐹0₂00𝑄𝑀0_T1∗∗∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.26)𝛾²𝑑²<𝑐²²𝜃^𝑁𝜆_max𝑄,(3.27) where 𝑄=𝑅^1/2𝑄𝑅^1/2 and ℎ=√𝛽(1−𝛽). Moreover, the controller gain can be calculated as 𝐾=𝐿𝑄⁻¹.

Proof. In Theorem 3.5, pre- and postmultiplying (3.25) by diag{𝐼,𝑃⁻¹,𝑃⁻¹,𝐼,𝐼,𝐼,𝐼,𝐼}, the following equivalent condition is obtained: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝐼−𝑃0₋₁00ℎ𝐹₁𝐾000∗∗−𝑃0₋₁0ℎ𝐵₁𝐾000∗∗∗−0𝐼𝐴𝐸++𝛽𝛽𝐵𝐹₁₁𝐾𝐵₂𝜀𝐻0∗∗∗∗−𝜃𝑃𝐾𝐹0₂00𝑀0_T1∗∗∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.28) Letting 𝑄 denote 𝑃⁻¹, the condition (3.28) is equivalent with ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝐼−0𝑄0000ℎℎ𝐵𝐹₁₁𝐾000∗∗−𝑄0𝐴+𝛽𝐵𝐾₁000∗∗∗−𝐼𝐸+𝛽𝐹₁𝐾𝐵₂𝜀𝐻0∗∗∗∗−𝜃𝑄₋₁𝐾𝐹₂0000𝑀^T₁∗∗∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<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: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−∗𝐼−0𝑄0000ℎℎ𝐵𝐹₁₁𝐾𝑄000∗∗−𝑄0𝐴𝑄+𝛽𝐾𝐵𝑄₁000∗∗∗−𝐼𝐸𝑄+𝛽𝐹₁𝐾𝑄𝐵₂𝜀𝐻0∗∗∗∗−𝜃𝑄𝐾𝑄𝐹0₂00𝑄𝑀0_T1∗∗∗∗∗−𝛾²𝐼0𝑀^T₂∗∗∗∗∗∗∗∗∗∗∗∗−0𝜀𝐼−0𝜀𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0.(3.30) Defining 𝐿=𝐾𝑄, the condition (3.30) is equivalent with (3.26). With the fact that 𝜆_min𝑃=1𝜆_max𝑄,(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<𝑐²²𝜃^𝑁.(3.32) In addition, the ℋ_∞ 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 ℋ_∞ 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=⎡⎢⎢⎣−𝐸11=−311⎤⎥⎥⎦,0,𝐵₁𝐹=₁⎡⎢⎢⎣−11⎤⎥⎥⎦,𝐵₂=⎡⎢⎢⎣−00.1.2⎤⎥⎥⎦𝐻=⎡⎢⎢⎣00..0021⎤⎥⎥⎦,𝑀₁=1,𝐹₂=0,=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.

Figure 1 

Trajectories of the system state in the simulation without the designed controller.

Figure 2 

Trajectories of the system state in the simulation with the designed controller.

Figure 3 

Stochastic values of the intermittent measurements in the simulation.

5. Conclusion

In this paper, the robust finite-time ℋ_∞ 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 ℋ_∞ 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.