#### Abstract

This paper investigates the stochastic finite-time boundedness problem for nonlinear discrete time networked systems with randomly occurring multi-distributed delays and missing measurements. The randomly occurring multi-distributed delays and missing measurements are described as Bernoulli distributed white noise sequence. The goal of this paper is to design a full-order output-feedback controller to guarantee that the corresponding closed-loop system is stochastic finite-time bounded and with desired performance. By constructing a new Lyapunov-Krasovskii functional, sufficient conditions for the existence of output-feedback are established. The desired full-order output-feedback controller is designed in terms of the solution to linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the validity of the designed method.

#### 1. Introduction

Networked control system (NCS) is a new research field originated from control engineering practice, which is composed of controller, actuator, and sensor. Due to low cost, low power consumption, and practical resource sharing, the dynamic behavior of NCSs has been widely concerned in recent years [1–5]. However, network is not a reliable communication medium. Due to the physical limitations of network bandwidth and service capability, time delay and missing measurement are inevitable problems in network transmission. These problems are an important reason for the deterioration of system performance and the instability of NCSs. Therefore, in the past few decades, a large number of new methods have been used to deal with NCSs with various time delays. At present, the following three models are mainly adopted: constant delay model [6], the upper and lower bounds with time-varying delay model [7], and random time delay with a certain probability distribution delay model (such as finite state markov process, Bernoulli distribution, etc. [8, 9]). In [8], the filtering problem of the Markov delay network system with a partially unknown distribution is studied. By using Lyapunov theory, the authors analyze the robust hybrid controller, controller design, and synchronous dissipation state estimation of the network control system with Bernoulli distribution in [10–13], respectively. On the other hand, missing measurement is also a major cause of NCSs performance degradation and system instability. Study of missing measurement: one of the most common ways to describe the missing measurement is probability method; this method assumes the missing measurement to satisfy a certain probability distribution, such as Bernoulli distribution. Many control problems are considered based on the Bernoulli distribution, such as [14–17]. In [14], recursive state estimation for time-varying complex networks with probability distribution missing measurements is investigated by using the mean square constraint method. The rapid state estimation of discrete nonlinear systems with random missing measurement is studied in [15]. In [16], the author extends the above research to the network control system with multiple missing measurement and studies the extended Kalman filter problem of the NCSs with random nonlinearity and multiple packet loss. In [17–19], the authors further extended the above results to the study of robust control, quantised recursive filtering, and event-based filtering of the NCSs with Bernoulli distribution and concluded that the use of Bernoulli distribution to describe missing measurement is more general.

It should be pointed out that the above-mentioned literature on system stability considers Lyapunov’s stability at an infinite interval. In practice, however, we are concerned not only with the stability of the system on an infinite interval, but with the stability on a finite interval. The finite time stability (FTS) is different from the stability in the general sense. It studies the state behavior in the finite time; that is, the system should meet certain transient performance requirements (for example, meeting the requirements of the system orbit for a certain degree of deviation from the equilibrium point). In order to study the transient performance of the system, Dorato proposed the concept of FTS in 1961 and obtained extensive research [20]. In recent years, some scholars have extended FTS to studies on finite time boundedness (FTB) and finite time control. In [21–25], the authors studied the finite time problem of discrete time system based on linear matrix inequality. In [26–29], the authors extended the FTB and finite time control problem to the NCSs. Further, in [30], finite time control problem is extened to fuzzy discrete system. In [31, 32], the authors discussed FTB and finite time state estimation of neural networks with missing measurement based on the semini-research approach and gave the design method of the state estimator. Recently, the issues of nonfragile finite time state estimators and event-driven finite time state estimators for neural networks have been investigated [33–35].

In addition, it is worth noting that, due to the influence of internal and external environmental factors, nonlinearity always exists in the actual system. At present, the control problem of the network system with nonlinearity is widely concerned [36]. Since the nonlinear perturbation phenomena may vary arbitrarily, Bernoulli distribution is more appropriate to describe the randomly occurring nonlinearity. In recent years, the problem of randomly occurring nonlinearities has been widely discussed. For example, in [37], the authors studied the control problem for randomly occurring nonlinear systems with saturation and channel fading. In [38, 39], the authors studied the reliable finite time filtering and quantized fault detection filters problem for discrete time systems, respectively.

To the best of the authors’ knowledge, the SFTB control problem for a discrete-time NCS with randomly occurring multi-distributed delays and missing measurements has not been fully investigated, which motivates the main purpose of our study. The main contributions of the paper are summarized as follows: The discrete-time NCS under consideration is more general networked control systems that include randomly occurring nonlinearities, multi-distributed delays, and missing measurements, where the randomly occurring nonlinearities, multi-distributed delays, and missing measurements are governed by a set of Bernoulli distributed white noise sequences; definitions of SFTB and SFTB are extended to discrete-time NCSs; sufficient conditions are provided to ensure that the corresponding closed-loop system is stochastic finite-time bounded and performance constraint is met by using the Lyapunov-Krasovskii functional method and the LMIs technique; full-order output-feedback controller, a more general controller than the commonly used state feedback controller, is designed.

#### 2. Problem Formulation

Consider a nonlinear discrete-time NCS with randomly multi-distributed delays as shown in Figure 1.

The state-space equation is given as follows: where is the state vector, is the control input, is the measurement output, is the controlled output, is a nonlinear function, are external disturbances, denotes the vector-value initial condition, , , , , , , and are known real constant matrices, denotes the infinite distributed delays, are time-varying delays satisfying where and are nonnegative scalars. are uncorrelated random variables and satisfy the following probability distribution: where is a known constant and the variance .

The stochastic variable is a Bernoulli distributed white noise sequence with where is a known constant, and the variance . Throughout the paper, we assume that and are mutually independent.

For the aforementioned system, we make the following assumptions.

*Assumption 1. * is a nonlinear function, and we assume that satisfies and the sector-bounded condition where are known real matrices and is a known positive definite matrix.

*Assumption 2. *For any given positive number , external disturbances input is time varying and satisfies

In this paper, the actual system output is expressed as where the stochastic variable takes values on 0 and 1 with the following probabilities: where is a known constant and the variance .

Equation (7) can be written in the light of system (1) In this paper, the full-order dynamic output feedback controller for system (1) is described as where is the state estimation of system (1) and , , and are the controller parameters to be determined.

By combining the controller (10) with (1), we can obtain the following closed-loop system: where

In this paper, our goal is to design a full-order output feedback controller to ensure that the corresponding closed-loop dynamic system is SFTB and with a desired perfoamance. The definitions of SFTB and SFTB and a lemma are introduced before sequel.

*Definition 3 (SFTB). *Given positive numbers , and a symmetric positive definite matrix , the closed-loop system (11) is stochastic finite-time boundedness w.r.t. if

*Definition 4 (SFTB). *Given positive numbers , and a symmetric positive definite matrix , the closed-loop system (11) is stochastic finite-time boundedness w.r.t. if the closed-loop system (11) is SFTB w.r.t. and under the zero-initial condition the output satisfiesfor all satisfying Assumption 2.

Lemma 5 ([23] (Schur complement)). *Given constant matrices , , and , then if and only if where and .*

#### 3. Main Result

In this section, we first give sufficient conditions for the SFTB of closed-loop system (11) and then discuss the design of dynamic output feedback controller (10).

##### 3.1. Stochastic Finite-Time Boundedness Analysis

The following theorem provides sufficient conditions for the SFTB of closed-loop system (11) by employing Lyapunov-Krasovskii functional approach.

Theorem 6. *Given positive constants , a symmetric positive-definite matrix , closed-loop system (11) is SFTB w.r.t. if, for a scalar , there exist symmetric positive definite matrices , and positive scalars such that the following inequalities hold: where*

*Proof. *Choose the following Lyapunov-Krasovskii functional candidate: whereLet us assume that The following goal is to show that if conditions (16) and (17) hold, then First, calculating the difference variation of , and taking the mathematical expectation, one has whereSimilarly, it can be obtained that Considering , It follows from (25) that Similarly, we have It is obtained from (23), (27), and (28) that where , .

Notice that (5) implies the following inequality: Combining (29) with (30), we have where By applying Lemma 5 to (16), we deduce that . Thus, we obtain from (31) that That is, It follows from (34) that By iteration and taking Assumption 2 into account, it can be obtained that Noting definition of , we have By taking (36) and (37) into account, we have where .

On the other hand, also from the definition of , we have Consequently, we get from (38) and (39) that Note that By Lemma 5, (41) is equivalent to LMI (23); it can be verified that Therefore, according to Definition 3, closed-loop system (11) is SFTB. The above discussion completes the proof of Theorem 6.

*Remark 7. *If nonlinearities and perturbation are not considered in system (11), that is, and , the SFTB problem of system (11) is deduced by analyzing the SFTB of the following system: where system parameters are the same as before.

The following criterion for system (43) is given based on Theorem 6.

Corollary 8. *Given positive constants , a symmetric positive-definite matrix , closed-loop system (43) is SFTB w.r.t. if, for a scalar , there exist symmetric positive definite matrices , and positive scalars such that the following inequalities hold: *

##### 3.2. Stochastic Finite-Time Boundedness Analysis

In this subsection, we will provide sufficient condition to ensure SFTB of closed-loop system (11) based on Theorem 6.

Theorem 9. *Given positive constants , , , , , a symmetric positive-definite matrix , closed-loop system (11) is SFTB w.r.t. if, for a scalar , there exist symmetric positive definite matrices and , and positive scalars such that the following inequalities hold: whereOther parameters are shown in Theorem 6.*

*Proof. *Obviously LMI (45) implies LMI (16); according to Theorem 6, we can easily conclude that the closed-loop system (11) is SFTB w.r.t. .

In the following, we shall prove that inequality (14) holds. By introducing the same Lyapunov-Krasovskii functional, we can get from (33) And by iteration, we obtain that Under the zero initial condition , and noting that , we get Letting , we then have Notice that ; (51) immediately yieldswhich implies that system (11) is SFTB. This completes proof of Theorem 9.

##### 3.3. Stochastic Finite-Time Output Feedback Control

In this subsection, we will deal with the problem of the design of output-feedback controller for system (1).

Theorem 10. *Consider the discrete-time NCS (1) with randomly occuring multi-distrbuted delays and missing measurement. For given positive constants , , and , and a symmetric positive-definite matrix , closed-loop system (11) is SFTB w.r.t. if, for a scalar , there exist symmetric positive definite matrices , , , matrices , and positive scalars , such that the following inequalities hold: where** and are symmetric positive matrices.**In this case, the parameters of the output feedback controller can be designed as where and are any nonsingular matrices satisfying .*

*Proof. *Evidently, from Theorem 9, system (11) is SFTB if, for a scalar , LMIs (45) and (46) hold. To do this, let and be as follows: where the partitioning of and is compatible with that of , , , and and are symmetric positive matrices.

Define and this implies that Thus It is clear that, by Lemma 5, if LMI (53) is feasible, then we have ; thus and therefore is nonsingular, which implies that .

First, it can be verified that the LMI (53) can be rewritten as where , , are defined in (57),That is, whereNow letting and applying the congruence transformations to (64), we can obtain LMI (45).

In addition, since and are fringed matrices of and , by the property of fringed matrices, we have Thus, (46) follows from inequalities (54) and (55). This completes the proof

*Remark 11. *Being different from literature [25], on the one hand, we consider NCSs with the stochastic and nonlinearity. On the other hand, state feedback controller is designed in [25]. However when not all the state variables are available, the results in [25] cannot be applicable. In this paper, output feedback controllers are designed in the stochastic context.

*Remark 12. *In order to design a controller independent of the matrix , we can make ; then , , and become

*Remark 13. *The feasibility of Theorem 10 can be represented by the following basic LMIs with a parameter :

*Remark 14. *So far, we have discussed problem of SFTB analysis for a class of nonlinear discrete networked systems with randomly occurring multi-distributed delays and missing measurements. Also, some sufficient conditions are given to reflect the impacts from the involved phenomena. When the error analysis of the proposed theoretical result becomes a concern, additional efforts will be made with some criterion which will appear in the near future.

#### 4. Numerical Example

In this section, a numerical example is used to verify the validity of the controller design approach proposed. Consider a discrete-time NCS described by system (1) and its parameters are selected as

We let and the stochastic parameters are chosen as Moreover, the time-varying delays satisfy , the exogenous disturbance is chosen as , and the nonlinear function is selected as

For given , , , and , it can be easily checked that is satisfied under the initial condition and .

By solving the minimization problem involving LMIs conditions (53) and (54) and using LMI toolbox in Matlab, we can obtain the optimal value and the optimal performance . By Theorem 10, the resulting controller gain matrices are expressed as

Figures 2 and 3 show the system state trajectories and the curve of . From Figures 2 and 3, we can get that all the state variables converge to zero quickly in the finite-time, which means that system (1) is stochastic finite-time stable. Moreover, for all that satisfy the initial condition , we can get that the system state trajectory cannot exceed the given optimal value , which means the obtained output-feedback controller can guarantee that system (11) is SFTB with prescribed properties. This proves the validity of the theoretical results.

In the following, for further assessment of the control method, we consider