# Finite-Horizon H-Infinity Control for Time-Varying State-Saturated Systems under Stochastic Communication Protocol

Authors: Kewang Huang; Feng Pan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the H∞ control problem is investigated for a class of time-varying state-saturated systems with both stochastic parameters and nonlinearities under the stochastic communication protocol (SCP). The sensor-to-controller network is considered where only one sensor can obtain access to the communication network at each transmission instant. The SCP is adopted to mitigate the undesirable data collision phenomenon. The model transformation technique is employed to simplify the addressed problem, and then, the completing squares method is carried out to obtain a sufficient condition for the existence of the finite-horizon H∞ controller. The controller parameters are characterized by solving two coupled backward recursive Riccati-like difference equations. A simulation example is finally utilized to illustrate the effectiveness of the proposed controller design scheme.


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## Body

## 1. Introduction

Due to its advantages of easy sharing, strong flexibility, and easy installation and maintenance, networked control systems (NCSs) have attracted a large number of scholars’ attention in recent years [1–4].Most existing methods of the controller and the filter design are only for infinite time-invariant controlled object. In the actual NCSs, due to the operating point offset, equipment aging, and environmental factors, many systems are time varying. Therefore, it is of more practical significance to study the transient characteristics of time-varying systems in the finite horizon than the steady-state characteristics of time-invariant systems. The corresponding fuzzy control problem, finite-time domain filtering problem, and envelope constraint problem for continuous time-varying system, discrete time-varying system, and discrete time-varying nonlinear system are discussed in [5–7], respectively.In NCSs, the bandwidth of communication network is limited, which tends to cause data conflicts of control or measurement signals in the process of network transmission, resulting in packet loss and network-induced delay [8, 9]. If multiple nodes in the network attempt to simultaneously transfer data, these phenomena caused by nonideal networks may be even more serious. Therefore, an effective way to avoid data conflicts is to join the communication protocol to manage the rights of each node to access the network. Random communication protocols have been widely studied and applied for their advantages of random access and strong expansibility [10, 11]. In [12], the filtering problem of a kind of linear time-invariant system under a random communication protocol was studied, which described the communication process between the sensor and the filter as a Markov chain. In [13], the auxiliary vector and a new state estimation structure were proposed to the problem of finite-time state estimation of delayed artificial neural network under the random communication protocol. In [14], a random pulse delay model was proposed under the consideration of packet loss, network-induced delay, and random communication protocol. Furthermore, an output feedback controller was designed to ensure the stability of the system in the mean square sense. In [15, 16], the sliding mode control and robust finite-time stability problem with time-varying delays were investigated.It is important to note that the above literature implicitly assumes that the system state is not saturated. However, in practice, due to the physical limitations or protection measures of the equipment, state saturation often occurs. For example, mechanical systems with position and speed limits, digital filters implemented in finite word length formats, and neural networks with saturated transfer functions [17, 18]. In these cases, the state of the system is constrained within a bounded set. If state saturation is not considered in the comprehensive analysis, the desired performance of the closed-loop system cannot be guaranteed. Although the control and filtering of NCSs in infinite time domain and random communication protocol have been studied in depth, the control problem in finite-time domain of state-saturated time-varying system has not attracted attention under the influence of random communication protocol, which arouses the author’s research interest.In summary, this paper is concerned with the control problem of the state saturation system in the finite-time domain based on the random communication protocol and considering the influence of random parameters and random nonlinearity. The main contributions in this paper are as follows: (1) both stochastic parameters and nonlinearities under the stochastic communication protocol are considered in the design of the output feedback controllers guaranteeing the desiredH∞ performance, (2) a sufficient condition is provided for an auxiliary index which is closely related to the desired finite-horizon H∞ performance, and (3) a suboptimal controller design scheme is provided by solving two coupled backward recursive Riccati-like difference equations.The rest of this paper is organized as follows. Section2 introduces the problem description. Sufficient conditions for the H∞ performance analysis and controller design are given in Sections 3 and 4. In Section 5, a numerical example is presented. Finally, Section 6 concludes the paper.Notations: the notations used in this paper are as follows. Rn denotes then-dimensional Euclidean space. ℒ20,N is the space of square summable vectors. E⋅ stands for the mathematical expectation and Prob⋅ means the occurrence probability of the event. In symmetric block matrices or complex matrix expressions, we use ∗ to represent a term that is induced by symmetry and diag⋅ denotes the block diagonal matrix. If their dimensions are not explicitly stated, they are assumed to be compatible for algebraic operations. The notation P>0 (≥0) means that P is a symmetric and positive definite (semidefinite) matrix. 0 and I represent the zero matrix and identity matrix with compatible dimensions.


## 2. Problem Description

Consider the state-saturated random time-varying system in the finite horizonk∈0,N:(1)xk+1=σAkμkxk+fxk+Bkuk+Dkwk,yk=Ckxk+Ekvk,zk=Lkxk,where xk∈ℝnx, uk∈ℝnu, zk∈ℝnz, and yk∈ℝny, respectively, represent system state, control input, controlled output, and measured output vector before transmission from the sensor-to-controller network. wk∈ℒ20,N and vk∈ℒ20,N are process noise and measurement interference of the system, respectively. The matrices Bk, Dk, Ck, Ek, and Lk are time-varying parameter matrices with known appropriate dimensions. σ⋅ is a saturation function, Akμk is a random parameter matrix, μk is a zero-mean Gaussian white noise sequence with variance 1, and fxk is a random nonlinear function of the system. The random parameter matrix Akμk and the random nonlinear function fxk are independent of each other.As [19] shows, the random parameter matrix Akμk has the following statistical characteristics:(2)EAkμk=A¯k,Covamuk,astk=Tamuk,astk,where Tamuk,astk is a known parameter and astk is s,t th element of matrix Akμk, i.e., astk=Akμks,t.For allxk, the first moment and covariance of the random nonlinear function fxk satisfy(3)Efxk|xk=0,(4)EfxkfTxj|xk=0,k≠j,(5)EfxkfTxk|xk=∑n=1qℓn,kℓn,kTxkTΓn,kxk,where q is a known nonnegative integer and ℓn,k and Γn,kn=1,2,…,q are known appropriate dimension matrices.Define saturation functionσ⋅:ℝnx⟶ℝnx:(6)σxk=σ1xk1σ2xk2⋯σnxxknxT,(7)σmxkm=signxkmminxkm,xk,maxm,m=1,2,…,nx,where xkm is the component of vector xk and xk,maxm>0 is the saturation level.In order to solve the problem of network congestion and data conflict caused by multiple transmission nodes using the communication network at the same time, random communication protocol is adopted in the network. Under the SCP scheduling, only one node is allowed to access the network and transmit data at each transport moment.ξk is defined as the node that obtains access to the network at the instant k, and its value is determined by the Markov chain. The transition probability matrix is Pk=pkijny×ny, specifically defined as(8)pkij=Probξk+1=j|ξk=i,where pkij∈0,1i,j∈1,2,…,ny represents the transition probability of k moment from state i to state j and satisfies ∑j=1nypkij=1.Set the measurement output after transmission from the sensor-to-controller network asy¯k=y¯1,ky¯2,k…y¯ny,kT, and y¯i,k represents the measurement output of the i sensor node received by the controller at the transmission time k. Zero input processing strategy is adopted, and the updating rules of y¯i,k are(9)y¯i,k=yi,k,ifi=ξk,0,Others.On the basis of formula (9), there are(10)y¯k=Φξkyk,where Φξk=diagδξk−1,δξk−2,…,δξk−ny and δξk−i is the Kronecker Delta function.For system (1), the following output feedback controller is designed:(11)uk=Kξk,ky¯k=Kξk,kΦξkCkxk+Kξk,kΦξkEkvk,where Kξk,k is the controller gain to be designed.According to formula (2), Akμk can be expressed as Akμk=A¯k+A˜kμk, and w¯k=wkvkT is defined for ease of analysis. The following closed-loop system is obtained by substituting equations (10) and (11) into system (1):(12)xk+1=σA¯k+BkKξk,kΦξkCkxk+A˜kμkxk+fxk+DkBkKξk,kΦξkEkw¯k,y¯k=Φξkyk,zk=Lkxk.The purpose of this study is to design a finite-time domain output feedback controller (11) so that, for a given disturbance suppression level γ, positive definite matrix S, and system initial state x0, the controlled output zk of the system meets the following H∞ performance requirements:(13)J1≜E∑k=0Nzk2−γ2w¯k2−γ2x0TSx0<0,∀w¯k≠0.


## 3.H∞ Performance Analysis

System (12) contains BkKξk,kΦξkEkvk, so to facilitate the solution of controller gain Kξk,k, we define D¯k=Dkεk−1Bk and Wk=wkεkKξk,kΦξkEkvkT. From the definition of Wk, perturbation Wk∈ℒ20,N rewrites formulae (12) to (14). The positive real number εk is introduced to describe the scale of measurement interference and to provide more degrees of freedom [20] for the design of the controller:(14)xk+1=σA¯k+BkKξk,kΦξkCkxk+A˜kμkxk+fxk+D¯kWk,y¯k=Φξkyk,zk=Lkxk.Next, similar to equation (13), for system (14), the following H∞ performance indicators are defined:(15)J2≜E∑k=0Nzk2−γ2Wk2−γ2x0TSx0<0,∀Wk≠0.Lemma 1.
For performance indicators (13) and (14), if εk2EkTΦξkTKξk,kTKξk,kΦξkEk≤I and J2<0 are satisfied, then J1≤J2<0.Proof.
Consideringεk2EkTΦξkTKξk,kTKξk,kΦξkEk≤I and J2<0, then(16)J1−J2=E∑k=0Nγ2Wk2−w¯k2=E∑k=0Nγ2vkTεk2EkTΦξkTKξk,kTKξk,kΦξkEk−Ivk≤0.
So,J1≤J2<0. Proving is completed.Lemma 2.
For∀xk∈ℝnx, if the matrix F is a positive definite symmetric matrix, σTxkFσxk≤λmaxFxkTxk is invariant, where λmaxF represents the maximum eigenvalue of the matrix F.Proof.
According to formula (7), σmxkm≤xkm can be obtained, and then, σTxkσxk−xkTxk=∑m=1nxσmxkm−xkm≤0, or σTxkσxk≤xkTxk can be obtained. Because F is a positive definite symmetric matrix, then σTxkFσxk≤λmaxFσTxkσxk≤λmaxFxkTxk. The proof is completed.Lemma 3 (see [21]).
For giving nonzero matricesU, V, and G, the optimal solution to minimize problem minXUXV−GF is U†VG†, where U† and G† represent the Moore–Penrose pseudoinverse of U and G, respectively.Theorem 1.
Consider the state-saturated system (1) over finite-time domain k∈0,N and its SCP model parameter ξk. Given the controller gain Ki,k, positive scalar εk, H∞ performance index γ>0, and positive definite matrix S, the sufficient condition for the system to meet the H∞ performance requirements is that the existence of positive definite symmetric matrix Pki makes the following recursive Riccati-like difference equations valid:(17)Pki=λP2A¯k+BkKkΦξkCkTD¯kΔki−1D¯kTA¯k+BkKi,kΦξkCk+λPA¯k+BkKi,kΦξkCkT⋅A¯k+BkKi,kΦξkCk+λP∑m=1nxTam,k+λP∑n=1qΓn,ktrℓn,kℓn,kT+LkTLk,and subjects to constraints(18)Δki>0,P0i≤γ2S,εk2EkTΦξkTKi,kTKi,kΦξkEk≤I,where λP=λmaxP¯k+1i, P¯k+1i=∑j=1nypkijPk+1j, Δki=γ2I−λPD¯kTD¯k, and Tamsk,amtk is the s,t element of the symmetric matrix Tam,k.Proof.
Define the following function:(19)Jki=Exk+1TPk+1ξk+1xk+1−xkTPkixk|i=ξk.
According to equations (14) and Lemma 2, it can be obtained that(20)Jki=EσTA¯k+BkKi,kΦξkCkxk+A˜kμkxk+fxk+D¯kWkP¯k+1iσA¯k+BkKi,kΦξkCkxk+A˜kμkxk+fxk+D¯kWk−xkTPkixk|i=ξk≤EλPxkTA¯k+BkKi,kΦξkCkTA¯k+BkKi,kΦξkCkxk+2xkTA¯k+BkKi,kΦξkCkTD¯kWk+xkTA˜kTμkA˜kμkxk+fTxkfxk+WkTD¯kTD¯kWk−xkTPkixk|i=ξk.
According to properties of matrix covariance and trace, equations (2)∼(5) can be obtained:(21)ExkTA˜kTμkA˜kμkxk=xkT∑m=1nxTam,kxk,EfTxkfxk=xkT∑n=1qΓn,ktrℓn,kℓn,kTxk.
Therefore, equation (20) is equivalent to(22)Jki≤ExkTλPA¯k+BkKi,kΦξkCkTA¯k+BkKi,kΦξkCk+λP∑m=1nxTam,k+λP∑n=1qΓn,ktrℓn,kℓn,kT−Pkixk+2λPxkTA¯k+BkKi,kΦξkCkTD¯kWk+λPWkTD¯kTD¯kWk|i=ξk.
Add zero itemzk2−γ2Wk2−zk2+γ2Wk2 to the right side of equation (22), and use perfect plane method to change equation (22) into(23)Jki≤ExkTλPA¯k+BkKi,kΦξkCkTA¯k+BkKi,kΦξkCk+λP∑m=1nxTam,k+λP∑n=1qΓn,ktrℓn,kℓn,kT+LkTLk−Pkixk+λPWkTD¯kTD¯kWk+2λPxkTA¯k+BkKi,kΦξkCkTD¯kWk−γ2WkTWk−zk2+γ2Wk2|i=ξk=ExkTλPA¯k+BkKi,kΦξkCkTA¯k+BkKi,kΦξkCk+λP∑m=1nxTam,k+λP∑n=1qΓn,ktrℓn,kℓn,kT+LkTLk−Pkixk+W¯kTΔkiW¯k−Wk−W¯kTΔkiWk−W¯k−zk2+γ2Wk2|i=ξk,where W¯k=λPΔki−1D¯kTA¯k+BkKi,kΦξkCkxk.
According to the properties of conditional expectation, inequality (23) is accumulated from 0 to N, and the difference equation (17) such as Riccati is obtained(24)ExN+1TPN+1ξN+1xN+1−x0TP0ix0≤E−∑k=0NWk−W¯kTΔkiWk−W¯k−∑k=0Nzk2−γ2Wk2.
ConsideringΔki>0, P0i≤γ2S, εk2EkTΦξkTKi,kTKi,kΦξkEk≤I, and the matrix Pk+1i is positively definite symmetry, it is not difficult to obtain(25)J2≤ExkTP0i−γ2Sxk−∑k=0NWk−W¯kTΔkiWk−W¯k<0.
Therefore,J1≤J2<0; the system meets the H∞ performance requirements. Proof is completed.


## 4.H∞ Controller Design

Theorem1 gives a sufficient condition for the closed-loop system (1) to satisfy the H∞ performance index. Next, the solution of controller gain matrix Ki,k under the worst disturbance case of the system is Wk=W¯k=λPΔki−1D¯kTA¯k+BkKi,kΦξkCkxk. To do this, define Fk=λPΔki−1D¯kTA¯k+BkKi,kΦξkCk, u¯k=Ki,kΦξkCkxk, and the cost function JW¯k≜E∑k=0Nzk2+u¯k2. Rewrite the closed-loop system (14) as(26)xk+1=σA¯k+BkKi,kΦξkCkxk+A˜kμkxk+fxk+D¯kFkxk,y¯k=Φξkyk,zk=Lkxk.Theorem 2.
Considering the state saturation system (1) and its SCP model parameter ξk on finite-time domain k∈0,N and given the positive scalar εk, H∞ performance index γ>0, and positive definite matrix S, the sufficient condition for the system to meet the H∞ performance requirements is that there are Ki,k and positive definite symmetric matrices Pki and Qki, so the Riccati-like difference equation recursively can be established as follows:(27)Pki=λP2A¯k+BkKi,kΦξkCkTD¯kΔki−1D¯kTA¯k+BkKi,kΦξkCk+λPA¯k+BkKi,kΦξkCkT⋅A¯k+BkKi,kΦξkCk+λP∑m=1nxTam,k+λP∑n=1qΓn,ktrℓn,kℓn,kT+LkTLk,Qki=−λQ2A¯k+D¯kFkTBkΔ¯ki−1BkTA¯k+D¯kFk+λQA¯k+D¯kFkT⋅A¯k+D¯kFk+λQ∑m=1nxTam,k+λQ∑n=1qΓn,ktrℓn,kℓn,kT+LkTLk.And, it subjects to constraints(28)Δki>0,Δ¯ki>0,P0i≤γ2S,εk2EkTΦξkTKi,kTKi,kΦξkEk≤I.At this point, the gain matrix of the controller can be calculated by the following formula:(29)Ki,k=ℳk†NkΦξkCk†,where λQ=λmaxQ¯k+1i, Q¯k+1i=∑j=1nypkijQk+1j, Δ¯ki=λQBkTBk+Iℳk=λPλQΔ¯ki−1BkTD¯kΔki−1D¯kTBkT+I, and Nk=−λQΔ¯ki−1BkTλPD¯kΔki−1D¯kT+IA¯k.Proof.
Theorem2 ensures that Theorem 1 is true; it can be known that the closed-loop system (1) meets the H∞ performance requirements. Furthermore, under the worst interference W¯k=Fkxk, the following functions are defined:(30)J¯ki=Exk+1TQk+1ξk+1xk+1−xkTQkixk|i=ξk.
According to equation (26) and Lemma 2, it can be obtained as(31)J¯ki=EσTA¯k+D¯kFkxk+A˜kμkxk+BkKi,kΦξkCkxk+fxkQ¯k+1iσA¯k+D¯kFkxk+A˜kμkxk+BkKi,kΦξkCkxk+fxk−xkTQkixk|i=ξk≤ExkTλQA¯k+D¯kFkTA¯k+D¯kFk+λQ∑m=1nxTam,k+λQ∑n=1qΓn,ktrℓn,kℓn,kT−Qkixk+λQxkTCkTΦξkTKi,kTBkTBkKi,kΦξkCkxk+2λQxkTA¯k+D¯kFkTBkKi,kΦξkCkxk|i=ξk.
Add the zero termE∑k=0NJ¯ki−∑k=0NJ¯ki to the right of the cost function JW¯k, then define u˜k=−λQΔ¯ki−1BkTA¯k+D¯kFkxk, use the complete square for u¯k, and combine equation (27) and constraint (28), and we obtain(32)JW¯k=E∑k=0Nzk2+u¯k2+J¯ki+xkTQ0ixk−xN+1TQN+1ξN+1xN|i=ξk≤E∑k=0Nu¯k−u˜kTΔ¯kiu¯k−u˜k+ExkTQ0ixk≤E∑k=0NKi,kΦξkCk+λQΔki−1BkTA¯k+D¯kFkF2Δ¯kiF2xk2+ExkTQ0ixk.
It is noted thatFk=λPΔki−1D¯kTA¯k+BkKi,kΦξkCk, in order to reduce the cost function, the gain matrix of the controller is selected as follows:(33)Ki,k=argminKi,kKi,kΦξkCk+λQΔki−1BkTA¯k+D¯kFkF=argminKi,kℳKi,kΦξkCk−Nk.
Therefore, according to Lemma3, equation (29) is the optimal solution to the minimization problem (33). Proving is complete.
Based on Theorem2, the following H∞ controller design algorithm on finite-time domain 0,N is obtained (Algorithm 1).Algorithm 1:H∞ controller design algorithm on finite time domain [0, N].
Step 1. Set k=N, for all i∈1,2,…,ny; given disturbance suppression level γ>0 positive definite matrices S, PN+1i and QN+1i are appropriate positive definite symmetric matrices.Step 2. Calculate Δki and Δ¯ki. If Δki>0 and Δ¯ki>0, go to the next step, otherwise go to Step 6.Step 3.ℳk and Nk were calculated, and controller gain matrix Ki,k was solved according to equation (29). If εk2EkTΦξkTKi,kTKi,kΦξkEk≤I, go to the next step, otherwise go to Step6.Step 4.Pki and Qki were obtained by solving Riccati-like difference equation (27), and then, λP and λQ were obtained. If Pki>0 and Qki>0, go to the next step, otherwise go to Step 6.Step 5. If k≠0, make k=k−1 and go to Step2; otherwise, judge whether P0i≤γ2S is established; then, the calculation terminates; if not, then proceed to the next step.Step 6. The algorithm is unsolvable and the calculation is terminated.


## 5. Numerical Example

A state-saturated time-varying system (1) with random parameters and random nonlinearity is considered in finite-time domain k∈0,30, and its system matrix parameters are shown as below [22]:(34)Akμk=A¯k+A˜kμk=0.42+sin2k−1−0.4−0.4+e−5k0.85+μk0.1000.2,Bk=0.85−0.65,Ck=0.65−0.7−0.60.65,Dk=−0.020.015T,Ek=0.010.01T,Lk=0.20.2.The random nonlinear function is(35)fxk=0.1signxk1xk1μ¯1k+0.2signxk2xk2sinμ¯2k0.060.09,where μk, μ¯1k, and μ¯2k are three Gaussian white noise sequences whose mean value is 0 and variance is 1, and it is not difficult to verify that the random nonlinear function satisfies(36)Efxk|xk=0,EfxkfTxk|xk=0.060.090.060.09TxkT0.01000.0173xk.In addition, it is known that the initial statex0=0.50.1T, xk,max1, xk,max2, and εk of the system is 1, the state transition matrix of the random communication protocol Pk=0.4+0.1−1k0.6−0.1−1k0.750.25, and the process noise and measurement interference of the system are wk=1.5sink and vk=0.7cos0.7k, respectively.On the premise that the algorithm in this paper has a solution and theH∞ performance index is as small as possible, the given values of γ=0.92, positive definite matrix S=0.2I, and PN+1i and QN+1i are given as 0.01I. By using the algorithm proposed in this paper, the gain matrix of the controller is obtained, as shown in Table 1. The corresponding simulation results of the system are shown in Figures 1–3. Figure 1 depicts the changes of sensor nodes that obtain network permissions under the influence of random communication protocol with the state transition matrix. Figures 2 and 3, respectively, show the state response curves of the open-loop system and the closed-loop system. The simulation results demonstrate the effectiveness of the proposed algorithm for the finite-time domain controller design.Table 1
Gain matrix ofH∞ controller in finite-time domain.
k01234Ki,k−0.0779000.7157−0.51680−0.2590−0.65350k56⋯2930Ki,k−0.5704000.2589⋯00.1248−0.01510Figure 1
Changes of the selected sensor nodes.Figure 2
The state trajectories of the open-loop system with state saturation.Figure 3
The state trajectories of the closed-loop system with state saturation.In order to highlight the advantages of the controller design algorithm proposed in this paper in the aspect of state saturation system, the algorithm in [22] is compared with the algorithm proposed in this paper. Regardless of random parameters and random nonlinearity, only for the state saturation system under the influence of random communication protocol, the parameters set are exactly the same as those in the above simulation. It is worth noting that the algorithm in [22] is not applicable to networked control systems under the influence of random communication protocols, so Pk and Qk in [22] need to be adjusted. The controller gain matrix Ki,k obtained according to the algorithm proposed in this paper and the controller gain matrix Ki,k∗ obtained according to the algorithm in [22] are shown in Table 2. The state response curves of the corresponding closed-loop system are shown in Figures 4 and 5, respectively. It can be seen from Figure 4 that the controller gain matrix Ki,k obtained based on the algorithm proposed in this paper makes the state-saturated system enter a stable state at the 6th transmission moment. It can be seen from Figure 5 that the controller gain matrix Ki,k∗ obtained from [22] makes the system enter a stable state at the 20th transmission moment. For this system, other parameters remain unchanged. On the premise that the controller has a solution, γ in this algorithm can be taken as the minimum γ=0.88, and in [22], the minimum γ can be taken as γ=0.80. To sum up, for the state saturation system under the influence of random communication protocol, the algorithm proposed in this paper takes into account many system factors, and the value of γ is relatively large, but the dynamic performance of the system is better.Table 2
Gain matrix of finite-time domainH∞ controller obtained by two algorithms.
k01234Ki,k−0.0793000.7189−0.52010−0.26310−0.66050Ki,k∗−0.0478000.3120−0.16800−0.17790−0.30100k56⋯2930Ki,k−0.5716000.2619⋯00.1270−0.01510Ki,k∗−0.1809000.1970⋯0−0.000100Figure 4
The state trajectories of the state-saturated system with gain matrixKi,k.Figure 5
The state trajectories of the state-saturated system with gain matrixKi,k∗.


## 6. Conclusion

In this paper, the finite-time domainH∞ control problem of state-saturated time-varying systems with random parameters and random nonlinearity under the influence of random communication protocol is studied. Firstly, a state saturation system model with random parameters and random nonlinearity is constructed. Then, in the sensor-controller transmission network, random communication protocol is added to avoid data conflict. Then, the sufficient conditions for the system to meet the H∞ performance requirements are obtained by using the model transformation technology and matching method. On this basis, the controller design algorithm based on the reverse-recursive Riccati difference equation technology is presented. Finally, numerical simulation shows the effectiveness of the proposed algorithm.



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