Observer-based guaranteed cost control for networked control systems with packet dropout and nonlinear disturbance

Abstract

In this article, the observer-based guaranteed cost control method is investigated for the networked control systems (NCSs) with packet dropout and nonlinear disturbance involving event-triggered mechanism (ETM). The packet dropout processes, which appear in both the sensor-to-controller and the controller-to-actuator links, are represented by two mutually independent Bernoulli distributions, and the nonlinear disturbance is supposed to satisfy the Lipschitz condition. Since the systems states are unmeasurable, a state observer is designed to estimate the values of system state. The sufficient condition for the stability of the closed-loop system is addressed by applying the Lyapunov theorem, and a guaranteed cost controller is obtained based on the stability condition with the satisfaction of some given specified value of cost function. Furthermore, the controller design issue can be formulated as a convex optimization problem that is addressed by the linear matrix inequality (LMI) technique using the cone complementarity linearization algorithm. Finally, the availability of above method is proved by a simulation example.

Keywords

Event-triggered mechanism networked control systems nonlinear disturbance output feedback control

Introduction

Networked control systems (NCSs), which have the characteristics of networking, digitization, hierarchization, and intelligence, refer to the feedback control system formed by the components of the control loop exchanging data through the communication network and have been widely applied in the fields of factory automation, household robots and remote operations, and so on (Lin et al., 2019; Shen et al., 2019; Truong and Ahn, 2015; Yang et al., 2014; Yuan and Wu, 2017). However, the communication network in control system brings some new problems, such as signal quantization, packet dropout, and network-induced delay, which cause the analysis and synthesis of NCSs extremely complicated. Especially, packet dropout, which occurs due to the unreliability of network transmission, makes the structure and parameters of the NCSs change greatly (Hu and Mu, 2020; Shah and Mehta, 2017; Su and Chesi, 2017; Tan et al., 2015; Yu et al., 2018). Thus, during the past decade, many scholars have focused on the issue of packet dropout. For example, Yang et al. (2011) proposed a compensation scheme to deal with the problem of random packet dropout in the network, effectively reducing the impact of packet dropout on system stability. In Zou et al. (2015), the missing data compensation strategy was introduced to address the problem of predictive control for NCSs affected by quantization and packet dropout. Qian et al. (2018) designed a new feedback control law, which added effective free control to reduce the impact of packet dropout on system performance. Similar works can be found in Lu et al. (2016) and Zhao et al. (2019). It should be mentioned that the above conclusions are on condition that the packet dropout occurs in sensor-to-controller link. Recently, some scholars gradually pay attention to the double-ended packet dropouts occurring in both the sensor-to-controller link and controller-to-actuator link. For example, Qiu et al. (2016) took into account of the output feedback guaranteed cost control for NCSs with double-ended packet dropouts that were described by two different independent Markov chains, different from general packet dropouts that were described by Bernoulli distribution, which had more possible states. However, Bernoulli process is clear enough to describe the double-ended packet dropouts phenomenon considered in this article. In Li et al. (2021), a method to suppress double-ended packet dropouts for the NCSs was designed, which it was new idea to regard packet dropout as an artificial disturbance, and in Yang et al. (2017), using zoom strategy and Lyapunov theorem, the stability conditions for the system with random double-ended packet dropouts were obtained. Obviously, the handling of double-ended packet dropouts also is a difficult point that should be studied by this article.

Although the phenomena of packet dropout have been well dealt with in the above articles, there are still the problems of low utilization of channels and computation resources due to the use of time-triggered mechanism (TTM) in which the measured data are transmitted into the communication network at each sampling time (Borgers and Heemels, 2014; Rahnama et al., 2018; Shi et al., 2016; Varma et al., 2020; Zhang and Han, 2014). Hence, the event-triggered mechanism (ETM) in NCSs is increasingly introduced and studied. In comparison to TTM, the ETM, which only transmits the current data when the pre-described condition is satisfied, decreases the transmission times so that it can save the communication resources of NCSs (Dolk and Heemels, 2017; Li et al., 2017; Meng and Chen, 2014; Yang et al., 2018; Zhang and Peng, 2020). Lately, a lot of works have been done on ETM for NCSs. Peng and Yang (2013) designed a new ETM for NCSs with packet dropout and network-induced delay and provided the maximum allowable continuous packet dropouts. In Zhang et al. (2017), an ETM based on sampled data was designed, which can guarantee the positive minimum inter-event time to avoid zeno phenomenon compared with the general ETM. Lu and Yang (2019) considered denial-of-service (DoS) attack, the decentralized ETM was adopted to balance the transmission efficiency and tolerable attack intensity as far as possible under the premise of guaranteeing the system stability. Although there have been a large number of studies considering the NCS with ETM and packet dropouts and nonlinear disturbance, limited works focus on the observer-based guaranteed cost control, which we think could be useful for readers. Meanwhile, Lu and Yang (2019) use an observer to deal with the problem of unmeasurable system state, which gives us a new idea.

Control method based on the observer is an effective way to solve the control problem that state variables are not easy to measure in many practical applications. In recent years, the control problem for NCSs with unmeasurable system states has attracted much attention. Li et al. (2019) addressed the controller design problem of observer-based NCSs in which the stochastic communication protocol (SCP) was considered. In Li et al. (2019), to guarantee the stability of the considered system under the ETM, an observer-based controller was designed. In Selivanov and Fridman (2016), according to whether there were communication links in sensor-to-controller and controller-to-actuator, different observers were constructed to estimate the system state so as to reduce the impact of network-induced delays on system performance. It is shown that aforementioned articles reveal the merits of observer-based control scheme, but Lyapunov theory is used to address the closed-loop stability of the NCSs which explicitly considers the nonlinear disturbance, and a sufficient condition for the stability of the NCS is derived and the guaranteed cost controller is designed, which we think it is the major difficulty overcame in this paper.

Inspired by these studies, the guaranteed cost control based on observer for uncertain discrete-time NCSs with nonlinear disturbance and double-ended packet dropouts via ETM is studied in this article. The main contents of the research are as follows.

Double-ended packet dropouts, which mean that packet dropouts occur in both sensor-to-controller and controller-to-actuator links, are described by Bernoulli distribution, and the ETM is introduced to improve the utilization of the communication channels.

A sufficient condition for the stability of the NCS is derived by Lyapunov theorem, and on the basis of this condition, the observer-based guaranteed cost controller is provided by considering some given specified value of cost function.

The structure of the article is arranged as follows. The NCS model is provided in section “Problem statement.” A new sufficient condition which guarantees the asymptotic stability of given system with nonlinear disturbance is obtained in section “Main results,” and the guaranteed cost controller design method based on the observer is also presented in this section. Section “Simulation example” verifies the availability of the presented design method by a simulation example. The concluding remark is provided in section “Conclusion.”

Notations: $ℜ^{n \times m}$ indicates the set of $n \times m$ real matrices. $A^{- 1}$ and $A^{T}$ imply the inverse and transpose of matrix $A$ , respectively. Matrix $P > 0$ indicates that $P$ is a positive definite matrix. $I$ means the identity matrix with proper dimensions, and $diag {\dots}$ means the block diagonal matrix. The notation * represents entries implied of the symmetric matrix. $E {x}$ and $Prob {x}$ are the mathematical expectation and probability of $x$ , respectively.

Problem statement

The structure for NCS with ETM is shown in Figure 1. The uncertain discrete-time system with nonlinear disturbance is represented as

\begin{matrix} x (r + 1) = (A + Δ A) x (r) + (B + Δ B) u (r) + ς (x (r)) \\ y (r) = Cx (r) \end{matrix}

(1)

where $x (r) \in ℜ^{n}$ is the state vector, $u (r) \in ℜ^{m}$ is the control input, and $y (r) \in ℜ^{p}$ is the system output. $A$ , $B$ , and $C$ are the constant matrices, $Δ A$ and $Δ B$ imply the norm-bounded parametric uncertainties and can be expressed as

[\begin{matrix} Δ A & Δ B \end{matrix}] = DF (r) [\begin{matrix} E_{a} & E_{b} \end{matrix}]

(2)

where $D$ , $E_{a}$ and $E_{b}$ are known constant matrices with appropriate dimensions, unknown matrix $F (r)$ satisfies $F (r)^{T} F (r) \leq I$ . $ς (x (r))$ indicates the nonlinear disturbance, which is the Euclidean norm-bounded and satisfies the Lipschitz condition with respect to state $x$ , that is

‖ ς (x (r)) - ς (\hat{x} (r)) ‖ \leq l ‖ x (r) - \hat{x} (r) ‖

(3)

where $\hat{x} (r)$ , $l > 0$ are the estimated states and Lipschitz constant, respectively and ∥.∥ denotes the Euclidean norm. The ETM is introduced to determine whether the measured output should be transferred to the unreliable network. The event generator receives the measured output $y (r)$ at each sampling time $r$ then determines whether the current measured output satisfies the following condition:

G_{r} = ϕ (r)^{T} Q ϕ (r) - γ y (i_{r})^{T} Qy (i_{r}) \geq 0

(4)

where $ϕ (r) = y (r) - y (i_{r})$ with $y (i_{r})$ being the latest triggered data, $Q$ is a positive matrix with proper dimensions, $γ \in (0, 1)$ is a known arbitrary scalar and $i_{r + 1}$ is the next triggered time instant. If condition (4) can be satisfied, the latest triggered data will be renewed, $y (i_{r + 1}) = y (r)$ and be transmitted; else the triggered data $y (i_{r})$ will be held.

Remark 1. The ETM reduces the transmission times in comparison to TTM. Clearly, ETM can increase the utilization of the channel transmission and computation resources of NCSs and decrease the transmission burden of NCSs.

Remark 2. Assume $y (i_{0}) = y (0)$ at initial time. If the triggered condition (4) is satisfied, the triggered data will be renewed for all $r \geq 0$ . $γ$ is used to tune the triggered frequency, the triggered frequency decreases with the increase in $γ$ .

Figure 1.

Structure of NCS with ETM.

Moreover, since the properties of communication network, when the measured output satisfies the triggered condition (4) and is released into the network link, the phenomenon of packet dropout may occur. We suppose that the packet dropout process follows the Bernoulli distribution ${ϖ (i_{r})} = {ϖ (i_{0}), ϖ (i_{1}), \dots, ϖ (i_{r}), \dots}$ . Then, the data sequence obtained by buffer is described as ${ϖ (i_{r}) y (i_{r})} = {ϖ (i_{0}) y (i_{0}), ϖ (i_{1}) y (i_{1}), \dots, ϖ (i_{r}) y (i_{r}), \dots}$ . In view of the latest data received by controller can be stored by the buffer, when the current triggered data transmitted to the network are lost, the buffer will transfer the latest stored data to the controller. Then the number of unreleased data between current time and last triggered time is described as $n_{i_{r}} = r - i_{r} - 1$ . We introduce a new sequence ${α (r)} = {\underset{n_{i_{0}} + 1}{\underset{︸}{ϖ (i_{0}), \dots, ϖ (i_{0})}}, \underset{n_{i_{1}} + 1}{\underset{︸}{ϖ (i_{1}), \dots, ϖ (i_{1})}}, \dots, \underset{n_{i_{r}} + 1}{\underset{︸}{ϖ (i_{r}), \dots, ϖ (i_{r})}}, \dots},$ when $r \in [i_{r}, i_{r + 1})$ , the data $\bar{y} (r)$ received by controller can be represented by

\bar{y} (r) = {\begin{matrix} y (i_{r}) α (r) = 0 \\ \bar{y} (r - 1) α (r) = 1 \end{matrix}

where $\bar{y} (r - 1)$ is the input of controller at time $r - 1$ . Recalling $ϕ (r) = y (r) - y (i_{r})$ when $r \in [i_{r}, i_{r + 1})$ , we can get

\bar{y} (r) = (1 - α (r)) (y (r) - ϕ (r)) + α (r) \bar{y} (r - 1) .

(5)

Supposing the probability of ${ϖ (i_{r})}$ is $\bar{α}$ , it is clear that

{\begin{matrix} Prob {α (r) = 1} = E {α (r)} = \bar{α} \\ Prob {α (r) = 0} = 1 - \bar{α} \end{matrix}

(6)

For the controller-to-actuator link, phenomenon of packet dropout can be described in a similar way

{\begin{matrix} Prob {β (r) = 1} = E {β (r)} = \bar{β} \\ Prob {β (r) = 0} = 1 - \bar{β} \end{matrix}

(7)

where $\bar{β} \in (0, 1)$ represents the probability of packet dropout between controller and actuator.

Due to function of buffer, the control input of system is expressed as follows

u (r) = (1 - β (r)) \bar{u} (r) + β (r) u (r - 1) .

(8)

Then, the model of NCS is deduced as

\begin{matrix} x (r + 1) = (A + Δ A) x (r) + (1 - β (r)) (B + Δ B) \bar{u} (r) + β (r) \\ (B + Δ B) u (r - 1) + ς (x (r)), \\ \bar{y} (r) = (1 - α (r)) (Cx (r) - ϕ (r)) + α (r) \bar{y} (r - 1) . \end{matrix}

For the purpose of facilitating the description of the above system, by defining

$z (r) = {[\begin{matrix} x {(r)}^{T} & u {(r - 1)}^{T} \end{matrix}]}^{T}$ and combining equations (1), (5), and (8), we can obtain

\begin{matrix} z (r + 1) = A_{1} (r) z (r) + B_{1} (r) \bar{u} (r) + A_{2} (r) z (r) + B_{2} (r) \bar{u} (r) \\ + ς (x (r)) \bar{y} (r) = C (r) z (r) - (1 - α (r)) ϕ (r) + α (r) \bar{y} (r - 1) \end{matrix}

(9)

where

\begin{matrix} A_{1} (r) = [\begin{matrix} A & β (r) B \\ 0 & β (r) I \end{matrix}], A_{2} (r) = [\begin{matrix} Δ A & β (r) Δ B \\ 0 & 0 \end{matrix}], \\ B_{1} (r) = (1 - β (r)) [\begin{matrix} B \\ I \end{matrix}], \end{matrix}

B_{2} (r) = (1 - β (r)) [\begin{matrix} Δ B \\ 0 \end{matrix}], C (r) = (1 - α (r)) [\begin{matrix} C & 0 \end{matrix}] .

In this article, the observer-based control scheme is introduced and can be represented by

Observer : {\begin{matrix} \hat{z} (r + 1) = A_{1} (r) \hat{z} (r) + B_{1} (r) \bar{u} (r) + ς (\hat{x} (r)) + L_{P} (\bar{y} (r) - \hat{y} (r)) \\ \hat{y} (r) = C (r) \hat{z} (r) - (1 - α (r)) ϕ (r) + α (r) \bar{y} (r - 1) \end{matrix}

(10)

Controller : \bar{u} (r) = K_{S} \hat{z} (r) = K_{S 1} \hat{x} (r) + K_{S 2} u (r - 1)

(11)

where $\hat{z} (r) = {[\begin{matrix} \hat{x} {(r)}^{T} & u {(r - 1)}^{T} \end{matrix}]}^{T} \in ℜ^{n + m}$ means the estimate of state, $\hat{y} (r) \in ℜ^{p}$ is the output of observer, $L_{P} \in ℜ^{(n + m) \times p}$ and $K_{S} \in ℜ^{m \times (n + m)}$ indicate the observer gain and feedback gain, respectively. Then, by defining the evaluated error

e (r) : = z (r) - \hat{z} (r),

(12)

the system can be obtained

\begin{matrix} z (r + 1) = {\bar{A}}_{1} z (r) + {\bar{A}}_{2} e (r) + (β (r) - \bar{β}) [{\bar{A}}_{3} z (r) + {\bar{A}}_{4} e (r)] \\ + ς (x (r)) e (r + 1) = {\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r) + (α (r) - \bar{α}) {\bar{B}}_{3} e (r) \\ + (β (r) - \bar{β}) [{\bar{B}}_{4} e (r) + {\bar{B}}_{5} z (r)] + ς (x (r)) - ς (\hat{x} (r)) \end{matrix}

(13)

where

\begin{matrix} {\bar{A}}_{1} = [\begin{matrix} A & \bar{β} B \\ 0 & \bar{β} I \end{matrix}] + (1 - \bar{β}) [\begin{matrix} B \\ I \end{matrix}] K_{S} + [\begin{matrix} Δ A & \bar{β} Δ B \\ 0 & 0 \end{matrix}] + (1 - \bar{β}) [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, \\ {\bar{A}}_{2} = - (1 - \bar{β}) [\begin{matrix} B \\ I \end{matrix}] K_{S} - (1 - \bar{β}) [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, \\ {\bar{A}}_{3} = [\begin{matrix} 0 & B \\ 0 & I \end{matrix}] - [\begin{matrix} B \\ I \end{matrix}] K_{S} + [\begin{matrix} 0 & Δ B \\ 0 & 0 \end{matrix}] - [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, \\ {\bar{A}}_{4} = [\begin{matrix} B \\ I \end{matrix}] K_{S} + [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, {\bar{B}}_{1} = [\begin{matrix} A & \bar{β} B \\ 0 & \bar{β} I \end{matrix}] \\ - (1 - \bar{α}) L_{P} [\begin{matrix} C & 0 \end{matrix}] - (1 - \bar{β}) [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, \end{matrix}

\begin{matrix} {\bar{B}}_{2} = [\begin{matrix} Δ A & \bar{β} Δ B \\ 0 & 0 \end{matrix}] + (1 - \bar{β}) [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, {\bar{B}}_{3} = L_{P} [\begin{matrix} C & 0 \end{matrix}], \\ {\bar{B}}_{4} = [\begin{matrix} 0 & B \\ 0 & I \end{matrix}] + [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S}, {\bar{B}}_{5} = [\begin{matrix} 0 & Δ B \\ 0 & 0 \end{matrix}] - [\begin{matrix} Δ B \\ 0 \end{matrix}] K_{S} . \end{matrix}

Relative to system (13), the cost function is defined as

J = \sum_{r = 0}^{\infty} (x (r)^{T} R_{1} x (r) + u (r)^{T} R_{2} u (r))

(14)

where $R_{1}$ and $R_{2}$ are known positive definite weighting matrices.

The following definition is applied to analyze the stability of system (13) in this article.

Definition 1. The closed-loop system (13) is stable, if there exists a Lyapunov function $V (r)$ such that

Δ V (r) : = E {V (r + 1) | r} - V (r) < 0 .

(15)

Main results

Stability analysis

For the purpose of guaranteeing the asymptotic stability of the considered system (13), we provide a sufficient condition in this section by applying Lyapunov theorem. Before moving on, the following lemmas are needed to be introduced.

Lemma 1. Qiu et al. (2016) Let $Y$ be a symmetric matrix and $H$ , $E$ be given matrices with proper dimensions, then

Y + HF (k) E + E^{T} F (k)^{T} H^{T} < 0

holds for all $F (k)$ that satisfy $F (k)^{T} F (k) \leq I$ if and only if there exists a scalar $ε > 0$ such that

Y + ε H H^{T} + ε^{- 1} E^{T} E < 0 .

Lemma 2. Zemzemi et al. (2015) Suppose X and Y are vectors or matrices with suitable dimensions, then for any positive $μ > 0$ , we have

X^{T} Y + Y^{T} X \leq μ X^{T} X + μ^{- 1} Y^{T} Y .

Then, the following theorem can be obtained.

Theorem 1. The system (13) is asymptotically stable, if there exist scalars $μ_{1} > 0, μ_{2} > 0,$ $ε_{1} > 0, ε_{2} > 0,$ matrices $K_{S}, L_{P}$ and positive matrices $P$ and $S$ , such that

Φ = [\begin{matrix} Φ_{11} & * & * \\ Φ_{21} & Φ_{22} & * \\ Φ_{31} & Φ_{32} & Φ_{33} \end{matrix}] < 0

(16)

where

Φ_{11} = [\begin{matrix} - P + χ_{11} & * & * \\ 0 & - S & * \\ χ_{21} & 0 & χ_{22} \end{matrix}],

Φ_{21} = [\begin{matrix} \bar{A} + (1 - \bar{β}) \bar{B} K_{S} & - (1 - \bar{β}) \bar{B} K_{S} & 0 \\ b_{1} & 0 & 0 \\ P b_{1} & 0 & 0 \\ \bar{A} + (1 - \bar{β}) \bar{B} K_{S} & - (1 - \bar{β}) \bar{B} K_{S} & 0 \\ 0 & \bar{A} - (1 - \bar{α}) L_{P} \bar{C} & 0 \\ 0 & b_{1} & 0 \\ 0 & S b_{1} & 0 \\ 0 & \bar{A} - (1 - \bar{α}) L_{P} \bar{C} & 0 \\ \hat{A} - \bar{B} K_{S} & \bar{B} K_{S} & 0 \\ 0 & L_{P} \bar{C} & 0 \\ 0 & \hat{A} & 0 \end{matrix}],

\begin{matrix} Φ_{22} = diag {- P^{- 1}, - P^{- 1}, - {μ_{1}}^{- 1} I, - μ_{1} I, - S^{- 1}, - S^{- 1}, \\ - {μ_{2}}^{- 1} I, - μ_{2} I, (P^{*})^{- 1}, (S_{1}^{*})^{- 1}, (S_{2}^{*})^{- 1}}, \end{matrix}

Φ_{31} = [\begin{matrix} 0 & 0 & 0 \\ {\bar{E}}_{1} + (1 - \bar{β}) E_{b} K_{S} & - (1 - \bar{β}) E_{b} K_{S} & 0 \\ 0 & 0 & 0 \\ {\bar{E}}_{2} - E_{b} K_{S} & E_{b} K_{S} & 0 \end{matrix}], Φ_{32} = [\begin{matrix} D_{11} & D_{12} \end{matrix}],

Φ_{33} = [\begin{matrix} - ε_{1}^{- 1} I & * & * & * \\ 0 & - ε_{1} I & * & * \\ 0 & 0 & - ε_{2}^{- 1} I & * \\ 0 & 0 & 0 & - ε_{2} I \end{matrix}], D_{11} = [\begin{matrix} {\bar{D}}^{T} & 0 & 0 & {\bar{D}}^{T} & {\bar{D}}^{T} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}],

\begin{matrix} D_{12} = [\begin{matrix} 0 & 0 & {\bar{D}}^{T} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\bar{D}}^{T} & 0 & {\bar{D}}^{T} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ χ_{11} = [\begin{matrix} γ C^{T} QC & 0 \\ 0 & 0 \end{matrix}], χ_{21} = [\begin{matrix} - γ QC & 0 \end{matrix}], \end{matrix}

\begin{matrix} χ_{22} = (γ - 1) Q, \bar{A} = [\begin{matrix} A & \bar{β} B \\ 0 & \bar{β} I \end{matrix}], \hat{A} = [\begin{matrix} 0 & B \\ 0 & I \end{matrix}], \\ {\bar{A}}^{*} = [\begin{matrix} Δ A & \bar{β} Δ B \\ 0 & 0 \end{matrix}], {\hat{A}}^{*} = [\begin{matrix} 0 & Δ B \\ 0 & 0 \end{matrix}], \end{matrix}

\begin{matrix} \bar{B} = [\begin{matrix} B \\ I \end{matrix}], {\bar{B}}^{*} = [\begin{matrix} Δ B \\ 0 \end{matrix}], \bar{C} = [\begin{matrix} C & 0 \end{matrix}], b_{1} = [\begin{matrix} l & 0 \\ 0 & 0 \end{matrix}], \\ σ_{1} = \bar{α} (1 - \bar{α}), σ_{2} = \bar{β} (1 - \bar{β}), \end{matrix}

\begin{matrix} P^{*} = - σ_{2} P, S_{1}^{*} = - σ_{1} S, S_{2}^{*} = - σ_{2} S, \bar{D} = {[\begin{matrix} D^{T} & 0 \end{matrix}]}^{T}, \\ {\bar{E}}_{1} = [\begin{matrix} E_{a} & \bar{β} E_{b} \end{matrix}], {\bar{E}}_{2} = [\begin{matrix} 0 & E_{b} \end{matrix}] . \end{matrix}

Proof. We define a Lyapunov function as

V (r) = z (r)^{T} Pz (r) + e (r)^{T} Se (r) .

(17)

Considering equation (13) and $E {α (r) - \bar{α}} = 0, E {β (r) - \bar{β}} = 0, E {(α (r) - \bar{α}) (β (r) - \bar{β})} = 0$ , the following is obtained

\begin{matrix} Δ V (r) = [{\bar{A}}_{1} z (r) + {\bar{A}}_{2} e (r)]^{T} P [{\bar{A}}_{1} z (r) + {\bar{A}}_{2} e (r)] + [{\bar{A}}_{1} z (r) + {\bar{A}}_{2} e (r)]^{T} \\ P ς (x (r)) + E {(β (r) - β)^{2}} \times [{\bar{A}}_{3} z (r) + {\bar{A}}_{4} e (r)]^{T} P [{\bar{A}}_{3} z (r) + {\bar{A}}_{4} e (r)] \\ + ς (x (r))^{T} P [{\bar{A}}_{1} z (r) + {\bar{A}}_{2} e (r)] + ς (x (r))^{T} P \times ς (x (r)) + [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)]^{T} \\ S [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)] + [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)]^{T} S [ς (x (r)) - ς (\hat{x} (r))] \\ + [ς (x (r)) - ς (\hat{x} (r))]^{T} S [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)] + E {(α (r) - α)^{2}} [{\bar{B}}_{3} e (r)]^{T} \\ S [{\bar{B}}_{3} e (r)] + E {(β (r) - β)^{2}} [{\bar{B}}_{4} e (r) + {\bar{B}}_{5} z (r)]^{T} S [{\bar{B}}_{4} e (r) + {\bar{B}}_{5} z (r)] \\ + [ς (x (r)) - ς (\hat{x} (r))]^{T} S [ς (x (r)) - ς (\hat{x} (r))] - z (r)^{T} Pz (r) - e (r)^{T} Se (r) . \end{matrix}

(18)

According to condition (3), we know that

‖ ς (x (r)) ‖^{2} \leq l^{2} ‖ x (r) ‖^{2},

(19)

‖ ς (x (r)) - ς (\hat{x} (r)) ‖^{2} \leq l^{2} ‖ x (r) - \hat{x} (r) ‖^{2},

(20)

then considering Lemma 2 and (20), we have

\begin{matrix} [ς (x (r)) - ς (\hat{x} (r))]^{T} S [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)] + [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)]^{T} \\ S [ς (x (r)) - ς (\hat{x} (r))] \\ \leq μ_{2} [S (ς (x (r)) - ς (\hat{x} (r)))]^{T} [S (ς (x (r)) - ς (\hat{x} (r)))] + {μ_{2}}^{- 1} [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)]^{T} \\ [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)] \\ \leq μ_{2} [b_{1} e (r)]^{T} S^{T} S [b_{1} e (r)] + [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)]^{T} {μ_{2}}^{- 1} [{\bar{B}}_{1} e (r) + {\bar{B}}_{2} z (r)] . \end{matrix}

(21)

Other similar formulas have the same solution. Hence, according to $E {(α (r) - \bar{α})^{2}} = \bar{α} (1 - \bar{α}) = σ_{1}$ and $E {(β (r) - \bar{β})^{2}} = \bar{β} (1 - \bar{β}) = σ_{2}$ , equation (18) is further derived to obtain

Δ V (r) \leq δ (r) Ξ δ (r),

(22)

where

Ξ = [\begin{matrix} Ξ_{11} & * \\ Ξ_{21} & Ξ_{22} \end{matrix}], δ (r) = {[\begin{matrix} z {(r)}^{T} & e {(r)}^{T} \end{matrix}]}^{T},

\begin{matrix} Ξ_{11} = {\bar{A}}_{1}^{T} P {\bar{A}}_{1} + μ_{1} b_{1}^{T} P^{T} P b_{1} + {\bar{A}}_{1}^{T} {μ_{1}}^{- 1} {\bar{A}}_{1} + σ_{2} {\bar{A}}_{3}^{T} P {\bar{A}}_{3} + b_{1}^{T} P b_{1} \\ + σ_{2} {\bar{B}}_{5}^{T} S {\bar{B}}_{5} + {\bar{B}}_{2}^{T} {μ_{2}}^{- 1} {\bar{B}}_{2} + {\bar{B}}_{2}^{T} S {\bar{B}}_{2} - P, \end{matrix}

\begin{matrix} Ξ_{21} = {\bar{A}}_{2}^{T} P {\bar{A}}_{1} + {\bar{A}}_{2}^{T} {μ_{1}}^{- 1} {\bar{A}}_{1} + σ_{2} {\bar{A}}_{4}^{T} P {\bar{A}}_{3} + {\bar{B}}_{1}^{T} S {\bar{B}}_{2} + {\bar{B}}_{1}^{T} {μ_{2}}^{- 1} {\bar{B}}_{2} \\ + σ_{2} {\bar{B}}_{4}^{T} S {\bar{B}}_{5}, \end{matrix}

\begin{matrix} Ξ_{22} = {\bar{A}}_{2}^{T} P {\bar{A}}_{2} + {\bar{A}}_{2}^{T} {μ_{1}}^{- 1} {\bar{A}}_{2} + σ_{2} {\bar{A}}_{4}^{T} P {\bar{A}}_{4} + {\bar{B}}_{1}^{T} S {\bar{B}}_{1} + μ_{2} b_{1}^{T} S^{T} S b_{1} \\ + {\bar{B}}_{1}^{T} {μ_{2}}^{- 1} {\bar{B}}_{1} + σ_{1} {\bar{B}}_{3}^{T} S {\bar{B}}_{3} + σ_{2} {\bar{B}}_{4}^{T} S {\bar{B}}_{4} + b_{1}^{T} S b_{1} - S . \end{matrix}

Considering the event-triggered condition (4), we can get for all $r \in [i_{r}, i_{r + 1})$ , $γ (y (r) - ϕ (r))^{T} Q (y (r) - ϕ (r)) - ϕ (r)^{T} Q ϕ (r) \geq 0$ is obtained. Therefore, we have

\begin{matrix} Δ V (r) \leq δ (r) Ξ δ (r) + γ (y (r) - ϕ (r))^{T} Q (y (r) - ϕ (r)) - ϕ (r)^{T} Q ϕ (r) \\ = δ (r) Ξ δ (r) + γ x (r)^{T} C^{T} QCx (r) - γ x (r)^{T} C^{T} Q ϕ (r) - γ ϕ (r)^{T} QCx (r) \\ + (γ - 1) ϕ (r)^{T} Q ϕ (r) = η (r)^{T} Λ η (r) \end{matrix}

(23)

where

Λ = [\begin{matrix} Ξ_{11} + χ_{11} & * & * \\ Ξ_{21} & Ξ_{22} & * \\ χ_{21} & 0 & χ_{22} \end{matrix}], η (r) = {[\begin{matrix} z {(r)}^{T} & e {(r)}^{T} & ϕ {(r)}^{T} \end{matrix}]}^{T} .

By the Schur complement, it is shown that $Δ V (r) < 0$ can be guaranteed by

Λ = [\begin{matrix} Φ_{11} & * \\ 21 & Φ_{22} \end{matrix}] < 0

where

F_{21} = {[\begin{matrix} Δ_{11} & Δ_{12} \end{matrix}]}^{T}, Δ_{11} = [\begin{matrix} {\bar{A}}_{1}^{T} & b_{1}^{T} & {(P b_{1})}^{T} & {\bar{A}}_{1}^{T} & {\bar{ℬ}}_{2}^{T} \\ {\bar{A}}_{2}^{T} & 0 & 0 & {\bar{A}}_{2}^{T} & {\bar{ℬ}}_{1}^{T} \end{matrix}],

Δ_{12} = [\begin{matrix} 0 & 0 & {\bar{B}}_{2}^{T} & {\bar{A}}_{3}^{T} & 0 & {\bar{B}}_{5}^{T} \\ b_{1}^{T} & {(S b_{1})}^{T} & {\bar{B}}_{1}^{T} & {\bar{A}}_{4}^{T} & {\bar{B}}_{3}^{T} & {\bar{B}}_{4}^{T} \end{matrix}] .

Then, in consideration of the parameter uncertainties and according to equation (2), we can obtain

Λ = Y + H_{1} F (r) E_{1} + E_{1}^{T} F (r)^{T} H_{1}^{T} + H_{2} F (r) E_{2} + E_{2}^{T} F (r)^{T} H_{2}^{T} < 0

(24)

where

Y = [\begin{matrix} Φ_{11} & * \\ Φ_{21} & Φ_{22} \end{matrix}], H_{1} = {[\begin{matrix} M_{11} & M_{12} \end{matrix}]}^{T}, H_{2} = {[\begin{matrix} N_{11} & N_{12} \end{matrix}]}^{T},

\begin{matrix} E_{1} = [\begin{matrix} {\bar{E}}_{1} + (1 - \bar{β}) E_{b} K_{S} & - (1 - \bar{β}) E_{b} K_{S} & E^{*} \end{matrix}], \\ E_{2} = [\begin{matrix} {\bar{E}}_{2} - E_{b} K_{S} & E_{b} K_{S} & E^{*} \end{matrix}], \end{matrix}

\begin{matrix} E^{*} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ M_{11} = [\begin{matrix} 0 & 0 & 0 & {\bar{D}}^{T} & 0 & 0 & {\bar{D}}^{T} \end{matrix}], \end{matrix}

\begin{matrix} M_{12} = [\begin{matrix} {\bar{D}}^{T} & 0 & 0 & {\bar{D}}^{T} & 0 & 0 & 0 \end{matrix}], \\ N_{11} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ N_{12} = [\begin{matrix} 0 & 0 & 0 & 0 & {\bar{D}}^{T} & 0 & {\bar{D}}^{T} \end{matrix}] . \end{matrix}

Resorting to Lemma 1, equation (24) is satisfied for all $F (r)^{T} F (r) \leq I$ , if there exist scalars $ε_{1}$ , $ε_{2}$ such that

Y + ε_{1} H_{1} H_{1}^{T} + {ε_{1}}^{- 1} E_{1}^{T} E_{1} + ε_{2} H_{2} H_{2}^{T} + {ε_{2}}^{- 1} E_{2}^{T} E_{2} < 0 .

Applying the Schur complement, equation (16) is obtained. Then, we deduce that the system (13) should be asymptotically stable by Definition 1.

Observer-based guaranteed cost controller design

According to the conclusion of the above section, the design method of observer-based guaranteed cost controller is obtained. However, there are also some nonlinear formulations that cannot be handled by LMI technique, so the purpose of this section is to transform them into a convex optimization problem that can be dealt with LMI technique.

Theorem 2. If there exist scalars $ε_{1} > 0, {\bar{ε}}_{1} > 0,$ $ε_{2} > 0, {\bar{ε}}_{2} > 0,$ $μ_{1} > 0, {\bar{μ}}_{1} > 0,$ $μ_{2} > 0, {\bar{μ}}_{2} > 0,$ matrices $K_{S}$ , $L_{P}$ and positive matrices $P$ , $S$ , $M$ , $W$ such that the following conditions hold

Π = [\begin{matrix} Π_{11} & * & * \\ Φ_{21} & Π_{22} & * \\ Φ_{31} & Φ_{32} & Π_{33} \end{matrix}] < 0,

(25)

PM = I, SW = I, ε_{1} {\bar{ε}}_{1} = 1, ε_{2} {\bar{ε}}_{2} = 1, μ_{1} {\bar{μ}}_{1} = 1, μ_{2} {\bar{μ}}_{2} = 1,

(26)

where

Π_{11} = [\begin{matrix} - P + χ_{11} + R & * & * \\ 0 & - S & * \\ χ_{21} & 0 & χ_{22} \end{matrix}],

\begin{matrix} Π_{22} = diag {- M, - M, - {\bar{μ}}_{1} I, - μ_{1} I, - W, - W, - {\bar{μ}}_{2} I, \\ - μ_{2} I, - σ_{2}^{- 1} M, - σ_{1}^{- 1} W, - σ_{2}^{- 1} W}, \end{matrix}

Π_{33} = [\begin{matrix} - {\bar{ε}}_{1} I & * & * & * \\ 0 & - ε_{1} I & * & * \\ 0 & 0 & - {\bar{ε}}_{2} I & * \\ 0 & 0 & 0 & - ε_{2} I \end{matrix}], R = [\begin{matrix} R_{1} & * \\ 0 & R_{2} \end{matrix}] .

Then, the system (13) is asymptotically stable, and the controller based on the observer which composed of equations (10) and (11) can make the cost function (14) satisfy

J < J^{*} = z (0)^{T} Pz (0) + e (0)^{T} Se (0)

(27)

where $z (0) = {[\begin{matrix} x {(0)}^{T} & 0 \end{matrix}]}^{T}, e (0) = {[\begin{matrix} {(x (0) - \hat{x} (0))}^{T} & 0 \end{matrix}]}^{T} .$

Proof. According to Theorem 1, it is obvious that the system (13) is asymptotically stable.

Setting $u (- 1) = 0$ , the cost function (14) is also represented as

J = \sum_{r = 0}^{\infty} (x (r)^{T} R_{1} x (r) + u (r - 1)^{T} R_{2} u (r - 1)) .

From equation (23), we arrive at

\begin{matrix} E {V (r + 1) | r} - V (r) + γ (y (r) - ϕ (r))^{T} Q (y (r) - ϕ (r)) - ϕ (r)^{T} Q ϕ (r) \\ + x (r)^{T} R_{1} x (r) + u (r - 1)^{T} R_{2} u (r - 1) \\ = η (r)^{T} Λ η (r) + z (r)^{T} Rz (r) = η (r)^{T} ϒ η (r), \end{matrix}

(28)

where

ϒ = [\begin{matrix} Ξ_{11} + χ_{11} + R & * & * \\ Ξ_{21} & Ξ_{22} & * \\ χ_{21} & 0 & χ_{22} \end{matrix}] .

It is apparent that equations (25) and (26) are equivalent to $ϒ < 0$ using the Schur complement. Hence, we arrive at

\begin{matrix} E {V (r + 1) | r} - V (r) + γ (y (r) - ϕ (r))^{T} Q (y (r) - ϕ (r)) - ϕ (r)^{T} \\ Q ϕ (r) + x (r)^{T} R_{1} x (r) + u (r - 1)^{T} R_{2} u (r - 1)) < 0 \\ \Rightarrow x (r)^{T} R_{1} x (r) + u (r - 1)^{T} R_{2} u (r - 1) < V (r) - E {V (r + 1) | r} . \end{matrix}

(29)

By summing equation (29) from $r = 0$ to $r = \infty$ , we get

\sum_{r = 0}^{\infty} (x (r)^{T} R_{1} x (r) + u (r - 1)^{T} R_{2} u (r - 1)) < V (0) - E {V (\infty)} .

Owing to the asymptotic stability of above system, we can get

J < V (0) - E {V (\infty)} = V (0) = z (0)^{T} Pz (0) + e (0)^{T} Se (0) = J^{*} .

Since the equality constraints in equation (26) cannot be solved by LMI technique, the following inequality constraints are introduced to convert the controller design issue to a convex optimization issue.

\begin{matrix} [\begin{matrix} P & I \\ I & M \end{matrix}] \geq 0, [\begin{matrix} S & I \\ I & W \end{matrix}] \geq 0, [\begin{matrix} ε_{1} & 1 \\ 1 & {\bar{ε}}_{1} \end{matrix}] \geq 0, [\begin{matrix} ε_{2} & 1 \\ 1 & {\bar{ε}}_{2} \end{matrix}] \geq 0, \\ [\begin{matrix} μ_{1} & 1 \\ 1 & {\bar{μ}}_{1} \end{matrix}] \geq 0, [\begin{matrix} μ_{2} & 1 \\ 1 & {\bar{μ}}_{2} \end{matrix}] \geq 0 . \end{matrix}

(30)

Then, the problem of the guaranteed cost controller based on the observer is presented as

min trace (PM + SW + ε_{1} {\bar{ε}}_{1} + ε_{2} {\bar{ε}}_{2} + μ_{1} {\bar{μ}}_{1} + μ_{2} {\bar{μ}}_{2})

s . t . (25), (30) .

(31)

The above transformation process is applied by the cone complementarity linearization approach. Its main idea is that for the $n \times n$ matrix variables $P > 0$ and $M > 0$ , if the following LMI

[\begin{matrix} P & I \\ I & M \end{matrix}] \geq 0

is satisfied, then we can get $trace (PM) \geq n$ ; if and only if $PM = I$ , $trace (PM) = n$ .

The specific process of realizing the presented design approach is shown in Figure 2 and Table 1.

Figure 2.

The specific process of algorithm.

Table 1.

The complete procedure of algorithm.

Algorithm
Step 1, set the values of $γ$ , $z (0)$ , $\hat{z} (0)$ , $y (0)$ . Suppose that $y (0)$ is the initial triggered output.
Step 2, at $r \geq 0$ , address the convex optimization problem (31) to obtain the values of $P$ , $S$ , $K_{S}$ , $L_{P}$ .
Step 3, calculate $y (r) = Cx (r)$ and transmit it to the event generator. If the event-triggered scheme (4) can be satisfied, $y (r)$ will become the triggered data $y (i_{r})$ , then is transmitted into network link. If $y (i_{r})$ is received by buffer successfully, $\bar{y} (r) = y (i_{r})$ ; else, $\bar{y} (r) = \bar{y} (r - 1)$ . If event-triggered condition cannot be satisfied, the latest triggered data will be held and $\bar{y} (r) = \bar{y} (r - 1)$ .
Step 4, calculate $\bar{u} (r) = K \hat{z} (r)$ and transmit it into the network link. If $\bar{u} (r)$ is delivered to actuator successfully, $u (k) = \bar{u} (r)$ ; else $u (k) = u (r - 1)$ .
Step 5, according to equation (13), obtain $z (r + 1)$ , $e (r + 1)$ and calculate $\hat{z} (r + 1) = z (r + 1) - e (r + 1)$ .
Step 6, update time $r$ to $r + 1$ and go to Step 2.

Simulation example

The effectiveness of above controller design method will be verified by the longitudinal dynamics of the HIRM aircraft similar to Li and Sun (2014). Giving the sampling interval of $0.025 s$ , we get

\begin{matrix} A = [\begin{matrix} 0.9862 & 0.0243 & 0 \\ - 0.0264 & 0.9894 & 0 \\ - 0.0003 & 0.0249 & 1 \end{matrix}], \\ B = {[\begin{matrix} - 0.0038 & - 0.0180 & - 0.001 \end{matrix}]}^{T}, C = [\begin{matrix} 1 & 0 & 1 \end{matrix}], \\ D = {[\begin{matrix} 0.1 & 0 & 0.2 \end{matrix}]}^{T}, E_{a} = [\begin{matrix} 0.2 & 0 & 0.3 \end{matrix}], E_{b} = 0.4 . \end{matrix}

Choose the unknown matrix $F (r) = \sin (r)$ , the nonlinear disturbance $ς (x (r)) = 0.05 \sin (x (r))$ , and the weighting matrices of cost function $R_{1} = diag {1; 1; 1}$ , $R_{2} = 0.1$ , and $l = 0.05$ . The probabilities of packet dropouts in the sensor-to-controller and the controller-to-actuator links are given as $\bar{α} = 0.6$ and $\bar{β} = 0.5$ , respectively. Set initial states $z (0) = [- 0.17 0.2 - 0.22 0]^{T}$ and $\hat{z} (0) = [0 0 0 0]^{T}$ . To reveal the merit of the presented technique, two groups of comparative experiments have been carried out. The one is the guaranteed cost control method based on the observer for NCS via TTM, the other is the observer-based guaranteed cost control method for NCS via ETM. For the former case, the following simulation results can be obtained by addressing the convex optimization problem (31).

\begin{matrix} P = [\begin{matrix} 1.7801 & 0.0195 & 0.0268 & 0.0043 \\ 0.0195 & 1.6899 & 0.0322 & - 0.0022 \\ 0.0268 & 0.0322 & 1.8263 & - 0.0098 \\ 0.0043 & - 0.0022 & - 0.00988 & 1.6873 \end{matrix}], \\ S = [\begin{matrix} 1.8646 & - 0.0097 & - 0.0194 & 0.0056 \\ - 0.0097 & 1.8039 & - 0.0049 & - 0.0019 \\ - 0.0194 & - 0.0049 & 1.708 & - 0.0052 \\ 0.0056 & - 0.0019 & - 0.0052 & 1.8046 \end{matrix}], \\ M = [\begin{matrix} 0.6205 & - 0.0073 & 0.0144 & - 0.0015 \\ - 0.0073 & 0.5943 & 0.0016 & 0.0007 \\ 0.0144 & 0.0016 & 0.6223 & 0.0030 \\ - 0.0015 & 0.0007 & 0.0030 & 0.5927 \end{matrix}], \\ W = [\begin{matrix} 0.5510 & - 0.0024 & - 0.0095 & - 0.0016 \\ - 0.0024 & 0.5563 & 0.0071 & 0.0005 \\ - 0.0095 & 0.0071 & 0.5778 & 0.0015 \\ - 0.0016 & 0.0005 & 0.0015 & 0.5542 \end{matrix}] . \end{matrix}

The feedback gain $K_{S} = [0.0010 0.0183 0.0099 0.3931]$ and observer gain $L_{P} = [0.5431 0.0044 0.5675 - 0.000006]^{T}$ . The upper bound of the cost function value $J^{*} = 4.3197$ . Then, for the later case of ETM presented in this paper, we choose the parameters of ETM $γ = 0.25$ , $Q = 0.018$ and the initial triggered data $y (i_{0}) = y (0) = 2.05$ . Similarly, we can get

\begin{matrix} P = [\begin{matrix} 1.8416 & 0.0209 & 0.0876 & 0.0055 \\ 0.0209 & 1.6886 & 0.0335 & - 0.0022 \\ 0.0876 & 0.0335 & 1.8866 & - 0.0091 \\ 0.0055 & - 0.0022 & - 0.0091 & 1.6873 \end{matrix}], \\ S = [\begin{matrix} 1.8647 & - 0.0098 & - 0.0194 & 0.0056 \\ - 0.0098 & 1.8039 & - 0.0049 & - 0.0019 \\ - 0.0194 & - 0.0049 & 1.7807 & - 0.0052 \\ 0.0056 & - 0.0019 & - 0.0052 & 1.8046 \end{matrix}], \\ M = [\begin{matrix} 0.6200 & - 0.0076 & 0.0141 & - 0.0018 \\ - 0.0076 & 0.5946 & 0.0013 & 0.0007 \\ 0.0141 & 0.0013 & 0.6221 & 0.0028 \\ - 0.0018 & 0.0007 & 0.0028 & 0.5927 \end{matrix}], \\ W = [\begin{matrix} 0.5510 & - 0.0024 & - 0.0095 & - 0.0016 \\ - 0.0024 & 0.5563 & 0.0071 & 0.0005 \\ - 0.0095 & 0.0071 & 0.5778 & 0.0015 \\ - 0.0016 & 0.0005 & 0.0015 & 0.5542 \end{matrix}] . \end{matrix}

The corresponding $K_{S} = [0.0009 0.0205 0.0104 0.365$ ], $L_{P} = [0.5431 0.0044 0.5677 - 0.00002]^{T}$ and the upper bound of cost function value $J^{*} = 4.3074$ . Data transmissions in the sensor-to-controller and the controller-to-actuator links are shown in Figures 3 and 4, respectively. In Figure 3, the variable values $1$ , $0$ , and $- 1$ denote that the packets satisfy triggered condition but lost, the packets satisfy triggered condition and transmitted to buffer successfully, and the packets cannot satisfy triggered condition, respectively. Obviously, only a fraction of the packets can be released into the network due to the ETM. It can be clearly known that the total number of trigger data is 56 as shown in Table 2, of which 9 data are lost, which means the ETM saves about 92% of network communication resources when the parameters of ETM $γ = 0.25$ . Moreover, combining Figure 5, it is shown that in all outputs of system, only some of data can be transmitted because of ETM. This is shown in Figures 6 and 7, both ETM and TTM can ensure the stability of system. However, from the perspective of energy, ETM is more acceptable since the needed transmitted data packages are less and the value of $J^{*}$ is very close to the value of TTM.

Figure 3.

Data transmissions in the sensor-to-controller link.

Figure 4.

Data transmissions in the controller-to-actuator link.

Table 2.

Event-triggered times under different triggered parameters.

$γ$	0.05	0.15	0.25	0.45
Event-triggered times/freq	101	62	56	38

Figure 5.

Output of system and triggered data.

Figure 6.

State responses with TTM and ETM.

Figure 7.

Control input with TTM and ETM.

In addition, four groups of comparison experiments of different $γ$ ( $0.05$ , $0.15$ , $0.25$ , $0.45$ ) are conducted to explain the effect of parameter $γ$ on the event-triggered frequency which has been discussed in Remark 2. According to the release instant and release interval in Figure 8, which the solid circles represent the successful transmission of triggered data, and the trigger times in Table 2, we can see that the triggered times decrease with the increase in $γ$ .

Figure 8.

The release instant and release interval of ETM.

To further prove the effectiveness of the proposed method, we introduce the article Li and Sun (2014), which proposed the full-order observer-based $H_{\infty}$ control problem for NCS with bounded random delays and continuous packet dropouts. The biggest difference between Li and Sun (2014) and this article is that we adopt the ETM to save network resources, which is not involved in Li and Sun (2014). By comparison, it can be seen that the control method in this paper has similar stability and performance as Li and Sun (2014). However, the ETM is adopted in this paper, which greatly reduces the release of redundant data and effectively saves network resources, as shown in Figure 3. From the above results, it is obvious that the observer-based guaranteed cost controller can effectively achieve the goal.

Conclusion

This article mainly presents the approach of event-triggered guaranteed cost controller based on the observer for NCSs with nonlinear disturbance and packet dropouts. The introduction of ETM aims to reduce transmission times so as to save communication resources. The packet dropouts occur in both the sensor-to-controller and the controller-to-actuator links are expressed by two mutually independent Bernoulli distributions. A new sufficient condition is obtained to guarantee the asymptotic stability of the system with nonlinear disturbance, and the cost function value is smaller than a specified upper bound. A simulation example verifies the superiority and practicability of the proposed method. Finally, we guess the trigger parameter of the ETM may not be able to adapt to the rapidly changing system in the future. Meanwhile, this article does not take into account the application of the double-end ETM, which are the directions of future research and the difficulties that need to be overcome.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge funding received from the Key Research Program of the Science Foundation of Shandong Province (ZR2020KE001). This is also a publication of the Enroll Plan of Young Innovative Talents of Shandong Province (Big Data and Ecological Security Research and Innovation Team Project).

ORCID iD

Hongchun Qu

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