Event-triggered robust H ∞ control for uncertain networked control systems with time delay

Abstract

H_∞ control for networked control systems with exogenous disturbances and norm-bounded parameter uncertainties has been extensively investigated. However, how to better use the limited network capacity and computation resources while reducing the conservativeness of the H_∞ control is still not fully understood. This paper presents a new dynamic discrete event-triggered scheme with improved modelling and control design to tackle this problem. The event-triggering is designed with periodic data sampling, and consequently the closed-loop system is formulated as a unified time-delayed model with the sampled data. From this model, an augmented Lyapunov–Krasovskii functional is constructed with triple-integral terms. A new free-weight matrix technique and the Wirtinger-based inequality are utilized over the Lyapunov–Krasovskii functional to derive a less conservative controller. This leads to two delay-range-dependent stability criteria in terms of linear matrix inequalities. Integrating all these components forms a co-design method for the minimum H_∞ performance index and our event-triggered scheme. Simulation experiments are conducted to demonstrate the approach presented in this paper.

Keywords

Networked control systems event-triggering Wirtinger inequality

Introduction

Networked control systems (NCSs) are a class of control systems in which sensing devices, control facilities and actuating agencies are interconnected via a digital communication network (Xu et al. (2013)). In comparison with traditional centralized control systems, NCSs show a number of advantages such as their flexibility, low cost in installation and maintenance and high reliability. Therefore, they have been widely deployed in various application areas. Typical examples include transportation systems, robot control, remote surgery, smart grids and unmanned aerial vehicles (Dos Santos et al., 2011; Vu and Turitsyn, 2015; Zhang et al., 2016b).

However, NCSs also demonstrate some challenges. In general, the communication networks of NCSs have limited bandwidth, network-induced delays, packet disorder and losses. These characteristics may cause degradation of the control performance of the systems (Du et al., 2017; Pang et al., 2016; Tang et al., 2015; Zhang et al., 2013). Therefore, increasing attention has been paid to control design of NCSs. Among various control methods, the periodic sampling procedure has been extensively developed for analysis and synthesis of NCSs. It has been realized that periodic sampling, data transmission and system control with a constant period may lead to a waste of limited network capacity and computing resources of NCSs. This demands new theories which are capable of more efficient use of NCS resources (Donkers and Heemels, 2012; Du et al., 2015; Peng et al., 2016b; Xia et al., 2015).

For this purpose, it is desired to reduce the amount of sampled data to be transmitted while still maintaining a pre-specified control performance of NCSs. This can be achieved through better design of data transmission strategies. Event-triggered control or event-based control has been developed to reduce the frequency of data transmission between NCS components. It is aperiodic with a variable transmission interval. As the networks do not transmit any data if the event-triggered condition is not met, a well-designed event-triggered NCS reduces the communication traffic significantly in comparison with conventional periodic control. The conservatism of the control system may also be improved.

Practically, a trade-off is required between the conservatism and the computational complexity of the NCS controller when time-delay bounds are considered in the NCSs. Recently, a new type of control methodology, known as a delay-range-dependent scheme, has been developed for a time-delay system. It considers a delay-interval from a nonzero lower bound to finite upper bounds (Hua et al., 2016; Hussain and Rehan, 2016; Majeed et al., 2015; Rafique et al., 2015; Rehan et al., 2016; Takaraoglu et al., 2015). The problem of disturbance noise suppression by $H_{\infty}$ control has been investigated for a class of discrete-time systems with time-varying delay (Hua et al., 2016; Takaraoglu et al., 2015). Majeed et al. (2015) and Rafique et al. (2015) have studied feedback controller design for nonlinear systems with interval time-delays in the state vectors by exploiting lower and upper bounds on time delays. Rehan et al. (2016) and Hussain and Rehan (2016) have presented a delay-range-dependent state feedback controller synthesis approach and a linear matrix inequality (LMI)-based controller design method for dynamic nonlinear time-delay systems with continuous interval time-varying delays under input saturation.

Event-triggered control has been increasingly considered in the literature, e.g. (Abdelrahim et al., 2016; Heemels and Donkers, 2013; Hu et al., 2012; Peng et al., 2017; Postoyan et al., 2015; Sahoo et al., 2014; Shi et al., 2016). Many existing event-triggered schemes are advocated, such as self-triggered sampling (Fan et al., 2015; Memon and Mahmoud, 2016; Peng et al., 2016b; Tolić et al., 2015), level-crossing sampling (Moser and Natschlager,2014) and sending on area sampling (Liu et al., 2014; Peng et al., 2016a). It has been shown that event-triggered control can greatly reduce the sampled-data packet transmission rate in comparison with periodic control in time-delay systems (Ge and Han, 2015; Li et al., 2015; Seuret et al., 2016; Wu et al., 2015; Yue et al., 2014; Zhu and Jiang, 2015). Zhu and Jiang (2015) have presented an effective event-based control strategy for distributed control of multi-agent systems with limited on-board resources. Seuret et al. (2016) demonstrated a simultaneous design of a state feedback law and an event-triggered condition for ensuring local exponential stability and linear quadratic (LQ) performance in the presence of plant input saturation. In the work by Wu et al. (2015), event-triggered conditions and controller synthesis approaches have been presented for delayed linear systems, and the resulting optimization problems are formulated in terms of linear matrix inequalities (LMIs) to minimize the upper bound of quadratic cost functions. Ge and Han (2015) investigated distributed event-triggered $H_{\infty}$ filtering over sensor networks with communication delays. An adaptive event-triggered communication scheme is proposed for NCSs with randomly occurring nonlinearities and uncertainties (Zhang et al., 2016a). In the work by Yue et al. (2014), event-triggered predictive control algorithms are developed on the controller node for compensating communication delays actively, leading to achievement of a desired control performance as well as a reduced use of network resources. Li et al. (2015) addressed event-triggered control for discrete-time linear parameter-varying systems by jointly designing a mixed event-triggered mechanism and a set of state feedback controller.

In spite of the progress reviewed above, how to better use the limited network capacity and computational resources while reducing the conservatism of NCS control is still not well understood. For example, an NCS is sampled in discrete-time instants while the system state normally changes in continuous-time. How do we describe the time-varying delays and exogenous disturbances in an integrated model with both discrete- and continuous-time dynamics? With such an integrated model, how do we improve the stability and control performance of the NCS through stability analysis and control synthesis?

This paper considers the control design of NCSs under exogenous disturbances and norm-bounded parameter uncertainties. It establishes an integrated closed-loop model for NCSs, designs a robust $H_{\infty}$ controller, and presents delay-range-dependent stability criteria in terms of linear matrix inequalities by utilizing a Lyapunov–Krasovskii functional and the Wirtinger-based inequality. The contributions of our work include:

Considering the network-induced delay, a new event-triggered strategy is proposed and an integrated closed-loop model is built for NCSs with time-varying delays.

New sufficient conditions are established for solving the robust $H_{\infty}$ control problem of NCSs. They are derived theoretically through using the augmented Lyapunov–Krasovskii functional approach and other equations, which guarantee asymptotic stability and less conservative results of the closed-loop NCSs. The triggering parameters and explicit expression of robust $H_{\infty}$ state feedback gain are obtained by integrating the event-triggered scheme and network-induced delays.

A co-design method is derived based on LMIs for the minimum $H_{\infty}$ performance index and event-triggered parameters.

The paper is organized as follows. We begin with establishment of an integrated model for NCSs to couple the considered system and the event-triggered scheme in a unified framework. Then, stabilization criteria and $H_{\infty}$ performance analysis are investigated, and event-triggered parameters and feedback controller gain are derived. After that, simulation experiments are carried out for two examples to demonstrate the effectiveness of our approach presented in this paper. Finally, conclusions are drawn from our research.

System description and problem statement

System plant

The considered NCS control framework is shown in Figure 1. For NCSs with continuous-time plant dynamics and limited network capacity, a non-uniform-sampling-based event-triggered control scheme will be proposed in this paper. It generates data transmission events through our ETDTP (event-triggered data transmission protocol). Other components of the NCS include a plant, a sensor, a controller, an actuator, a ZOH (zero-order-holder) and communication networks.

Consider the following continuous-time plant with parametric uncertainty and exogenous disturbance

{\begin{matrix} \overset{\cdot}{x} (t) = A (t) x (t) + B (t) u (t) + B_{w} w (t) \\ z (t) = C (t) x (t) + D (t) u (t) + D_{w} w (t) \end{matrix}

(1)

where $x (t) \in R^{n}$ is the plant state vector, $u (t) \in R^{m}$ denotes the control input vector, $w (t) \in L_{2} [0, \infty)$ is the exogenous disturbance input vector, $z (t) \in R^{p}$ stands for the control output vector and $x (0) \in R^{n}$ represents the initial condition. $A (t)$ , $B (t)$ , $C (t)$ and $D (t)$ are uncertain matrices that satisfy $A (t) = A + Δ A (t)$ , $B (t) = B + Δ B (t)$ , $C (t) = C + Δ C (t)$ and $D (t) = D + Δ D (t)$ , where A, B, $B_{w}$ , C, D and $D_{w}$ are known constant matrices with appropriate dimensions, $Δ A (t)$ , $Δ B (t)$ , $Δ C (t)$ and $Δ D (t)$ are time-varying parametric uncertainties, which are norm-bounded and can be described as

[\begin{matrix} Δ A (t) & Δ B (t) \\ Δ C (t) & Δ D (t) \end{matrix}] = [\begin{matrix} G_{1} \\ G_{2} \end{matrix}] F (t) [\begin{matrix} E_{1} & E_{2} \end{matrix}]

(2)

where $G_{1}$ , $G_{2}$ , $E_{1}$ and $E_{2}$ are real known constant matrices with appropriate dimensions, and $F (t)$ is an unknown, real, and time-varying matrix with Lebesgue measurable elements and satisfies $F^{T} (t) F (t) \leq I$ .

Remark 1. The uncertainties $Δ B (t)$ and $Δ D (t)$ are also perturbations to the plant input matrices $B (t)$ and $D (t)$ , respectively. From equation (2) , $Δ B (t) = Δ D (t) = 0$ when $E_{2} = 0$ .

Remark 2. Uncertain dynamics are usually unavoidable in practical systems. They may result from measurement errors and/or parameter perturbations. The robust stability of uncertain systems with time-varying delay can be derived by using the augmented Lyapunov–Krasovskii functional approach. However, a conservatism reduction is often achieved with increased computational complexity, resulting in difficulties in control synthesis. Similarly, if we aim to reduce the computation complexity, the conservatism of the system will increase in general. Intuitively, it is a difficult task to reduce both the computational complexity and the system conservatism at the same time.

In order to conserve network resources such as network bandwidth, an event-triggered scheme is proposed to replace conventional periodic data transmissions. It is assumed that both the controller and actuator are event-driven while the sensor is time-driven. The physical plant is sampled periodically by the sensor at each sampling period h, where $h > 0$ is a constant. Network transmission delay may not necessarily be integer multiples of the sampling period h. The ZOH information may be updated between sampling instants. The sensor must transmit its sampling data to the controller whenever the event-triggered condition is violated.

Assume that the system states can be measured and transmitted with a single packet. For time-driven sampling with a constant sampling period h, the sampled data at the kth sampled instant is denoted by $x (k h)$ , $k \in N$ . The sampling sequence instants are denoted by $M_{k} = {0, h, \dots, k h}$ . The transmission instants of the sampled data are described as $M_{i_{k}} = {0, i_{1} h, \dots, i_{k} h}$ , where $i_{k} \in N$ , and $M_{i_{k}} \subset M_{k}$ .

After the event-triggered condition is violated, successful data transmission instants are represented by $t_{k}$ , the transmission sequence instants are denoted by $M_{t_{k}} = {t_{k} h, t_{k} = 0, 1, \dots,}$ , satisfying $M_{t_{k}} \subset M_{i_{k}}$ .

The plant states $x (t_{k} h)$ are transmitted to the controller via the NCS communication network, $\bar{x} (t)$ is generated by the ZOH with the holding time $t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$ , $τ_{t_{k}}$ is the time delay induced by network communications for the states $x (t_{k} h)$ to arrive at the ZOH.

Event-triggered data transmission protocol

This subsection presents the problem event-based state feedback control with our event-triggered data transmission strategy. In our design, the sampled data packets are transmitted directly to the event generator. The event generator monitors the event-triggered condition continuously. In order to reduce network traffic and enhance network resource utilization, an event-triggered data transmission protocol (ETDTP) is designed to determine whether or not the current sampled data packet should be transmitted. It is composed of buffer and an event generator. The buffer stores the latest transmitted data packet $x (t_{k} h)$ , where ${t_{k} | k = 1, 2, \dots}$ represents the release instant sequence of the ETDTP. The event generator checks the current sampled-data packet $x (t_{k} h + j h)$ with regard to the following event-triggered condition

\begin{matrix} {[x ((t_{k} + j) h) - x (t_{k} h)]}^{T} Ω_{1} [x ((t_{k} + j) h) - x (t_{k} h)] \\ \leq σ x^{T} ((t_{k}) h) Ω_{2} x ((t_{k}) h) \end{matrix}

(3)

where $Ω_{j}$ are the positive matrices, $j \in {1, 2}$ , $σ \in [0, 1)$ .

Remark 3. The event-triggered scheme in equation (3) is related from the latest released sampled-data $x (t_{k} h)$ and current sampled-data $x (t_{k} h + n h)$ , $n \in {0, 1, \dots, t_{k + 1} - t_{k} - 1}$ , $k \in N$ , where $t_{k + 1}$ is the next latest release time instant. The next latest released sampled-data is used in our scheme given by equation (3) through a quadratic condition, while existing work by Yue et al. (2013) used current sampled-data on the right-hand side of their event-triggered scheme. Furthermore, the positive weighting matrices $Ω_{1}$ and $Ω_{2}$ in our scheme ( equation (3) ) present more flexibility in the solution space when the current sampled data is decided whether to be released, which could improve the system"s performance.

When the event-triggered condition in equation (3) is satisfied, current sampled-data packet is discarded directly. Otherwise, the ETDTP immediately releases the current sampled data packet to the controller via network communications. The sampled data packet is transmitted only when the state error between the current sampling state and the previously transmitted state exceeds a threshold. If the event-triggered scheme is used, the frequency of network transmissions of sampled data will be reduced.

ZOH

A released data packet reaches the ZOH with a time delay through the NCS communication network. Let $t_{k}$ ( $k = 1, 2, \dots$ ) denote the time instant at which data packet $x (t_{k} h)$ arrives at the ZOH. The ZOH sends the packet to the controller immediately and holds it until the next data packet comes. We have $t_{1} < t_{2} < \dots < t_{k} < \dots$ when data packets do not drop and data packet disorders do not occur. When the latest transmitted data packet $x (t_{k} h)$ is released from the ZOH, it actuates the controller immediately with $\bar{x} (t)$ , which is described as

\bar{x} (t) = x (t_{k} h), t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})

(4)

Modelling of NCSs with ETDTP

When a sampled-data is released by the event generator to the controller, it incurs a sensor-to-controller communication delay $τ_{t_{k}}^{s c}$ . Although the controller computation introduces a time delay $τ_{t_{k}}^{c}$ , it is not considered in this paper with the assumption that it is much smaller than network-induced delays. Also, data transmissions from the controller to the actuator induce a controller-to-actuator communication delay $τ_{t_{k}}^{c a}$ . Therefore, the total network-induced delay can be lumped together as a time-varying delay $τ_{t_{k}} = τ_{t_{k}}^{s c} + τ_{t_{k}}^{c a}$ at instant $t_{k}$ when the sampled data is released from the sensor to the actuator.

Let $τ_{m} = m i n_{t_{k} \in N} {τ_{t_{k}}} > 0$ and $τ_{M} = m a x_{t_{k} \in N} {τ_{t_{k}}}$ . $τ_{t_{k}}$ is bounded by the following inequality

τ_{m} \leq τ_{t_{k}} \leq τ_{M}

(5)

The next time instant for releasing sampled data in equation (3) is determined from the following communication scheme

\begin{matrix} t_{k + 1} h = t_{k} h \\ + m i n {j h | e_{k}^{T} (t) Ω_{1} e_{k} (t) > σ x^{T} (t_{k} h) Ω_{2} x (t_{k} h), j \in N^{+}} \end{matrix}

(6)

where $e_{k} (t) = x (t_{k} h + j h) - x (t_{k} h)$ is the state error between the next latest transmitting time instant and current real-time sampling time instant. A quadratic function is used in this paper for our event-triggered condition, which relies on monitoring intermittent sampled-data, rather than a continuous plant as shown by Wang and Lemmon (2010).

Remark 4. The term $e_{k} (t)$ in equation (3) characterizes the relationship between current sampled-data $x (t_{k} h + i h)$ and the next latest transmitted sampled-data $x (t_{k} h)$ . This is different from other existing methods, e.g. those by Hu et al. (2012) and Hu et al. (2015) . The release period $s_{k} h > 0$ is the lower bound of the inter-event time $s_{k} h = t_{k + 1} h - t_{k} h$ from equation (6) . This implies that there is no Zeno behaviour in our ETDTP scheme ( Selivanov and Fridman, 2015 ).

In this paper, our proposed state-feedback controller for the plant in equation (1) implements a linear feedback control $u (t) = K x (t)$ , where K is the controller gain. Event-triggered data packets are sent to the ZOH through network communications. With consideration of the effects of the total network-induced delay $τ_{t_{k}}$ and our ETDTP, the output of the ZOH is modelled as $\bar{x} (t) = x (t_{k} h)$ . The controller receives the state data $\bar{x} (t)$ at time instant $t_{k} h + τ_{t_{k}}$ . It holds the signal $\bar{x} (t)$ until the next sampled data is received in Figure 2. We have

u (t) = K x (t) = K \bar{x} (t), t_{k} h \leq t < t_{k + 1} h, k = 0, 1, 2, \dots

(7)

where $t_{k + 1}$ is the next updating time instant of the ZOH after $t_{k}$ . Considering the effect of the network transmission delay, we describe the model of the physical plant with ETDTP as

{\begin{matrix} \overset{\cdot}{x} (t) = A (t) x (t) + B (t) K \bar{x} (t) + B_{w} w (t) \\ z (t) = C (t) x (t) + D (t) K \bar{x} (t) + D_{w} w (t) \end{matrix}

(8)

Set

F_{k} = m i n {j | t_{k} h + j h \geq t_{k + 1} h, j \in N^{+}

(9)

The time interval $[t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$ can be divided into subintervals as

t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}}) = ⋃_{i = 0}^{F_{k}} ℧_{i}

(10)

{\begin{matrix} ℧_{0} = [t_{k} h + τ_{t_{k}}, t_{k} h + h + τ_{M}) \\ ℧_{i} = [t_{k} h + (i - 1) h + τ_{M}, t_{k} h + i h + τ_{M}), i = 2, \dots, F_{k} \\ ℧_{P_{k}} = [t_{k} h + (F_{k} - 1) h + τ_{M}, t_{k + 1} h + τ_{t_{k + 1}}) \end{matrix}

(11)

Now, define function $τ (t)$ as

τ (t) = {\begin{matrix} t - t_{k} h & t \in ℧_{0} \\ t - t_{k} h - h & t \in ℧_{1} \\ ⋮ \\ t - t_{k} h - (F_{k} - 1) h & t \in ℧_{F_{k}} \end{matrix}

(12)

$τ (t)$ is a piecewise function with $\overset{\cdot}{τ} (t) = 1$ . It follows that

t - τ (t) = {\begin{matrix} t_{k} h & t \in ℧_{0} \\ t_{k} h + h & t \in ℧_{1} \\ ⋮ \\ t_{k} h + (F_{k} - 1) h & t \in ℧_{F_{k}} \end{matrix}

(13)

When $t \in ℧_{i}, {i = 0, 1, \dots, F_{k} - 1}$ holds, implying that the following inequality satisfies

\begin{matrix} 0 < τ_{m} \leq τ_{t_{k}} \leq τ (t) \leq h + τ_{M} \end{matrix}

(14)

Because

\begin{matrix} τ_{t_{k}} \leq τ (t) < t_{k + 1} h + τ_{t_{k}} - t_{k} h - (F_{k} - 1) h \\ = t_{k + 1} h - (t_{k} + F_{k}) h + h + τ_{t_{k}} \end{matrix}

we have

t_{k + 1} h < (t_{k} + F_{k}) h

This means that there exists $τ_{t_{k}} \leq τ (t) < h + τ_{M}$ when $t \in ℧_{F_{k}}$ and thus $η_{m} \leq τ (t) < η_{M}$ in $t \in ℧_{i}, {i = 0, 1, \dots, F_{k}}$ , where $η_{m} = τ_{m}$ and $η_{M} = h + τ_{M}$ . $e_{k} (t)$ is the state error between the current real-time sampling time instant and the next latest transmitting time instant. It is mathematically defined as

e_{k} (t) = {\begin{matrix} x (t_{k} h) - x (t_{k} h) & t \in ℧_{0} \\ x (t_{k} h + h) - x (t_{k} h) & t \in ℧_{1} \\ ⋮ \\ x (t_{k} h + (F_{k} - 1) h) - x (t_{k} h) & t \in ℧_{F_{k}} \end{matrix}

(15)

It follows that

x (t_{k} h) = x (t - τ (t)) - e_{k} (t)

(16)

Thus, we have

u (t) = K x (t - τ (t)) - K e_{k} (t)

(17)

Remark 5. From the definition of $e_{k} (t)$ in equation (15) and the event-triggered scheme given by equation (3) , we can derive the following inequality for $t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$

e_{k}^{T} (t) Ω_{1} e_{k} (t) \leq σ [x (t - τ (t)) - e_{k} (t)]^{T} Ω_{2} [x (t - τ (t)) - e_{k} (t)]

(18)

Applying $τ (t)$ and $e_{k} (t)$ , we can re-write equation (8) as

{\begin{matrix} \overset{\cdot}{x} (t) = A (t) x (t) + B (t) K x (t - τ (t)) - B (t) K e_{k} (t) + B_{w} w (t) \\ z (t) = C (t) x (t) + D (t) K x (t - τ (t)) - D (t) K e_{k} (t) + D_{w} w (t) \end{matrix}

(19)

where $t \in [t_{k} h + τ_{t_{k}}, t_{k + 1} h + τ_{t_{k + 1}})$ .

The event-triggered robust $H_{\infty}$ control problem for networked control systems with parameter uncertainties and transmission delay is formulated as follows: for a given disturbance attenuation level scalar $γ > 0$ , we determine the event-triggered parameters $0 < σ < 1$ and $Ω_{i} > 0 {i = 1, 2}$ . A state feedback controller can be designed such that the event-triggered control system given by equation (19) subject to equation (18) is asymptotically stable with a prescribed $H_{\infty}$ performance level $γ$ . This means that:

the event-triggered control system given by equation (19) subject to equation (18) with $w (t) = 0$ for the physical plant

{\begin{matrix} \overset{\cdot}{x} (t) = A (t) x (t) + B (t) K x (t - τ (t)) - B (t) K e_{k} (t) \\ z (t) = C (t) x (t) + D (t) K x (t - τ (t)) - D (t) K e_{k} (t) \\ t_{k} h + τ_{t_{k}} \leq t < t_{k + 1} h + τ_{t_{k + 1}} \end{matrix}

(20)

is asymptotically stable, where the initial condition of the state $x (t)$ is: $x (t) = ϕ (t)$ , $t \in [t_{0} - η, t_{0}]$ , $ϕ (t_{0}) = x_{0}$ and $ϕ (t)$ is a continuous function on $[t_{0} - η, t_{0}]$ ;

under the zero initial condition, the controlled output $z (t)$ satisfies the inequality $∥ z (t) ∥_{2} < γ ∥ w (t) ∥_{2}$ for any nonzero $w (t) \in L_{2} [0, \infty)$ and a prescribed constant $γ > 0$ .

Remark 6. From the measurement data of the system, the event-triggered scheme given by equation (18) is usually used in discrete-time space while the system state changes in continuous-time space. Therefore, the event-triggered scheme given by equation (18) is adapted for continuous-time dynamics in an integrated model.

After the problem statement is given above, we introduce some lemmas below to facilitate the development of our main results in this paper.

Lemma 1. (Schur complement). Let A be a symmetric real matrix represented by

A = [\begin{matrix} a_{11} & a_{12} \\ a_{12}^{T} & a_{22} \end{matrix}]

where $a_{22}$ is square and nonsingular. Then $A > 0$ if and only if $a_{22} > 0$ and $a_{11} - a_{12} a_{22}^{- 1} a_{12}^{T} > 0$ .

Lemma 2. ( Min et al., 2004 ) For any vectors $X, Y \in R^{n}$ , and positive definite matrix $Θ \in R^{n \times n}$ , the following inequality holds

\pm 2 X^{T} Y \leq X^{T} Θ^{- 1} X + Y^{T} Θ Y

Lemma 3. ( Petersen,1987 ) Given matrices $M = M^{T}$ , G and E of appropriate dimensions, then

M + G F (t) E + E^{T} F^{T} (t) G^{T} < 0

for all $F (t)$ satisfying $F^{T} (t) F (t) \leq I$ , if and only if there exists a constant $λ > 0$ , such that

M + λ G G^{T} + λ^{- 1} E^{T} E < 0

Lemma 4. ( Seuret et al., 2013 ) For a given symmetric and positive definite matrix $W > 0$ of appropriate dimensions and for any differentiable signal x in $[a, b] \to R^{n}$ , the following inequality holds

\int_{a}^{b} {\overset{\cdot}{x}}^{T} (s) W \overset{\cdot}{x} (s) d s \geq \frac{1}{b - a} {[\begin{matrix} x^{T} (b) \\ x^{T} (a) \\ υ \end{matrix}]}^{T} Γ (W) [\begin{matrix} x^{T} (b) \\ x^{T} (a) \\ υ \end{matrix}]

(21)

where

\begin{matrix} υ = \frac{1}{b - a} \int_{a}^{b} x (s) d s, Γ (W) = [\begin{matrix} a_{1} W & a_{2} W & a_{3} W \\ * & a_{1} W & a_{3} W \\ * & * & a_{4} W \end{matrix}] \\ a_{1} = π^{2} / 4 + 1, a_{2} = π^{2} / 4 - 1, a_{3} = - π^{2} / 2, a_{4} = π^{2} \end{matrix}

Stability analysis of the event-triggered closed-loop system

Robust $H_{\infty}$ performance analysis for uncertain systems

In this section, we do not directly deal with uncertain systems. Instead, we first consider a nominal system without uncertainties and present stability when $F (t) \equiv 0$ . This means we consider a nominal system with $Δ A (t) = 0$ , $Δ B (t) = 0$ , $Δ C (t) = 0$ , $Δ D (t) = 0$ , implying that

{\begin{matrix} \overset{\cdot}{x} (t) = A x (t) + B K x (t - τ (t)) - B K e_{k} (t) + B_{w} w (t) \\ z (t) = C x (t) + D K x (t - τ (t)) - D K e_{k} (t) + D_{w} w (t) \end{matrix}

(22)

For this nominal system, we have the following theorem with regard to the asymptotic stability.

Theorem 1. For given parameters $γ > 0$ , $0 \leq σ < 1$ , $η_{m}$ , $η_{M}$ and K, the closed-loop system in equation (22) is asymptotically stable with $H_{\infty}$ control performance under the event-triggered scheme given by equation (18) , if there exist matrices $M_{i} > 0 (i = 1, \dots, 8)$ , and

P = [\begin{matrix} p_{11} & 0 & 0 \\ 0 & p_{22} & p_{23} \\ 0 & * & p_{33} \end{matrix}] > 0

$Q_{i} > 0 (i = 1, 2)$ , $Z_{i} > 0 (i = 1, 2, 3)$ , $R_{i} > 0 (i = 1, \dots, 6)$ and $Ω_{i} > 0 (i = 1, 2)$ are positive definite symmetric matrices, $a_{i} > 0 (i = 1, 2, 3, 4)$ , such that the following matrix inequities hold

[\begin{matrix} Φ & * & * & * \\ Γ_{1} & - γ^{2} & * & * \\ Γ_{2} & D_{w} & - I & * \\ Λ \bar{A} & Λ B_{w} & 0 & - Λ \end{matrix}] < 0

(23)

for

\begin{matrix} [\begin{matrix} M_{1} & M_{5} \\ * & Z_{1} \end{matrix}] \geq 0, [\begin{matrix} M_{2} & M_{6} \\ * & Z_{2} \end{matrix}] \geq 0, [\begin{matrix} M_{3} & M_{7} \\ * & Z_{2} \end{matrix}] \geq 0, [\begin{matrix} M_{4} & M_{8} \\ * & Z_{3} \end{matrix}] \geq 0 \\ [\begin{matrix} R_{1} & M_{1} \\ * & R_{2} \end{matrix}] \geq 0, [\begin{matrix} R_{3} & M_{2} \\ * & R_{4} \end{matrix}] \geq 0, [\begin{matrix} R_{3} & M_{3} \\ * & R_{4} \end{matrix}] \geq 0, [\begin{matrix} R_{5} & M_{4} \\ * & R_{6} \end{matrix}] \geq 0 \\ Φ = [\begin{matrix} Ψ_{1} & Ψ_{2} \\ * & Ψ_{3} \end{matrix}], Ψ_{1} = [\begin{matrix} φ_{11} & φ_{12} & φ_{13} & 0 \\ * & φ_{22} & φ_{23} & 0 \\ * & * & φ_{33} & φ_{34} \\ * & * & * & φ_{44} \end{matrix}], Ψ_{2} = [\begin{matrix} ϱ_{11} & ϱ_{12} & ϱ_{13} & ϱ_{14} \\ ϱ_{21} & ϱ_{22} & ϱ_{23} & 0 \\ 0 & ϱ_{32} & ϱ_{33} & 0 \\ ϱ_{41} & ϱ_{42} & ϱ_{43} & 0 \end{matrix}] \\ Ψ_{3} = [\begin{matrix} - \frac{a_{4}}{η_{m}} (Z_{1} + Z_{3}) & 0 & 0 & 0 \\ * & - \frac{a_{4}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) & 0 & 0 \\ * & * & - \frac{a_{4}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) & 0 \\ * & * & * & - Ω_{1} \end{matrix}] \end{matrix}

where

\begin{matrix} \bar{A} = [\begin{matrix} A & 0 & B K & 0 & 0 & 0 & 0 & - B K \end{matrix}], a_{1} = \frac{π^{2}}{4} + 1, a_{2} = \frac{π^{2}}{4} - 1, a_{3} = - \frac{π^{2}}{2}, a_{4} = π^{2} \\ Γ_{1} = [\begin{matrix} B_{w}^{T} p_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], Γ_{2} = [\begin{matrix} C & 0 & D K & 0 & 0 & 0 & 0 & - D K \end{matrix}] \\ Λ = 2 η_{m} Z_{1} + 2 (η_{M} - η_{m}) Z_{2} + 2 η_{M} Z_{3} + \frac{η_{m}^{2}}{2} R_{2} + \frac{η_{M}^{2} - η_{m}^{2}}{2} R_{4} + \frac{η_{M}^{2}}{2} R_{6} \\ φ_{11} = p_{11} A + A^{T} p_{11} - \frac{a_{1}}{η_{m}} (Z_{1} + Z_{3}) + Q_{1} + Q_{2} + η_{m} (M_{1} - M_{2} - M_{3}) + η_{M} (M_{2} + M_{3} + M_{4}) + M_{5} + M_{8} \\ + \frac{η_{m}^{2}}{2} R_{1} + \frac{η_{M}^{2} - η_{m}^{2}}{2} R_{3} + \frac{η_{M}^{2}}{2} R_{5}, φ_{12} = - \frac{a_{2}}{η_{m}} (Z_{1} + Z_{3}), φ_{13} = p_{11} B K \\ φ_{22} = - a_{1} (\frac{1}{η_{m}} (Z_{1} + Z_{3}) + \frac{1}{η_{M} - η_{m}} (Z_{2} + Z_{3})) + M_{6} - M_{5} - Q_{1}, φ_{23} = - \frac{a_{2}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) \\ φ_{33} = - \frac{2 a_{1}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) + M_{7} - M_{6} + σ Ω_{2}, φ_{34} = - \frac{a_{2}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) \\ φ_{44} = - \frac{a_{1}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) - Q_{2} - M_{7} - M_{8} \\ ϱ_{11} = η_{m} p_{22}^{T} - \frac{a_{3}}{η_{m}} (Z_{1} + Z_{3}), ϱ_{12} = (η_{M} - η_{m}) p_{23}, ϱ_{13} = (η_{M} - η_{m}) p_{23}, ϱ_{14} = - p_{11} B K \\ ϱ_{21} = η_{m} {(p_{23} - p_{22})}^{T} - \frac{a_{3}}{η_{m}} (Z_{1} + Z_{3}), ϱ_{22} = (η_{M} - η_{m}) (p_{33}^{T} - p_{23}) - \frac{a_{3}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) \\ ϱ_{23} = (η_{M} - η_{m}) (p_{33}^{T} - p_{23}), ϱ_{32} = - \frac{a_{3}}{η_{M} - η_{m}} (Z_{2} + Z_{3}), ϱ_{33} = - \frac{a_{3}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) \\ ϱ_{41} = - η_{m} p_{23}, ϱ_{42} = (η_{M} - η_{m}) {(- p_{33})}^{T}, ϱ_{43} = (η_{M} - η_{m}) {(- p_{33})}^{T} - \frac{a_{3}}{η_{M} - η_{m}} (Z_{2} + Z_{3}) \end{matrix}

Proof. Construct the following Lyapunov–Krasovskii functional

V (t) = V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t)

(24)

where

\begin{matrix} V_{1} (t) = ξ^{T} (t) P ξ (t), V_{2} (t) = \int_{t - η_{m}}^{t} x^{T} (s) Q_{1} x (s) d s \\ + \int_{t - η_{M}}^{t} x^{T} (s) Q_{2} x (s) d s \\ V_{3} (t) = 2 \int_{- η_{m}}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) Z_{1} \overset{\cdot}{x} (s) d s d θ \\ + 2 \int_{- η_{M}}^{- η_{m}} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) Z_{2} \overset{\cdot}{x} (s) d s d θ \\ + 2 \int_{- η_{M}}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) Z_{3} \overset{\cdot}{x} (s) d s d θ \\ V_{4} (t) = \int_{- η_{m}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} x^{T} (s) R_{1} x (s) d s d λ d θ \\ + \int_{- η_{m}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) d s d λ d θ \\ + \int_{- η_{M}}^{- η_{m}} \int_{θ}^{0} \int_{t + λ}^{t} x^{T} (s) R_{3} x (s) d s d λ d θ \\ + \int_{- η_{M}}^{- η_{m}} \int_{θ}^{0} \int_{t + λ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{4} \overset{\cdot}{x} (s) d s d λ d θ \\ + \int_{- η_{M}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} x^{T} (s) R_{5} x (s) d s d λ d θ \\ + \int_{- η_{M}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{6} \overset{\cdot}{x} (s) d s d λ d θ \end{matrix}

Taking the derivative of $V (t)$ in equation (24) with respect to t along the trajectory of the system given by equation (22) gives

\overset{\cdot}{V} (t) = {\overset{\cdot}{V}}_{1} (t) + {\overset{\cdot}{V}}_{2} (t) + {\overset{\cdot}{V}}_{3} (t) + {\overset{\cdot}{V}}_{4} (t)

Denote $ξ (t) = [\begin{matrix} x^{T} (t) & \int_{t - η_{m}}^{t} x^{T} (s) d s & \int_{t - η_{M}}^{t - η_{m}} x^{T} (s) d s \end{matrix}] .$ It follows that

{\overset{\cdot}{V}}_{1} (t) = 2 ξ^{T} (t) P \overset{\cdot}{ξ} (t)

(25)

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) = x^{T} (t) (Q_{1} + Q_{2}) x (t) - x^{T} (t - η_{m}) Q_{1} x (t - η_{m}) \\ - x^{T} (t - η_{M}) Q_{2} x (t - η_{M}) \end{matrix}

(26)

When $η_{m} \leq τ (t) \leq η_{M}$ , the time derivatives of $V_{3} (t)$ and $V_{4} (t)$ are respectively calculated as follows

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = 2 η_{m} {\overset{\cdot}{x}}^{T} (t) Z_{1} \overset{\cdot}{x} (t) + 2 (η_{M} - η_{m}) {\overset{\cdot}{x}}^{T} (t) Z_{2} \overset{\cdot}{x} (t) \\ + 2 η_{M} {\overset{\cdot}{x}}^{T} (t) Z_{3} \overset{\cdot}{x} (t) - \int_{t - η_{m}}^{t} {\overset{\cdot}{x}}^{T} (s) (Z_{1} + Z_{3}) \overset{\cdot}{x} (s) d s \\ - \int_{t - τ (t)}^{t - η_{m}} {\overset{\cdot}{x}}^{T} (s) (Z_{2} + Z_{3}) \overset{\cdot}{x} (s) d s \\ - \int_{t - η_{M}}^{t - τ (t)} {\overset{\cdot}{x}}^{T} (s) (Z_{2} + Z_{3}) \overset{\cdot}{x} (s) d s - \int_{t - η_{m}}^{t} {\overset{\cdot}{x}}^{T} (s) Z_{1} \overset{\cdot}{x} (s) d s \\ - \int_{t - τ (t)}^{t - η_{m}} {\overset{\cdot}{x}}^{T} (s) Z_{2} \overset{\cdot}{x} (s) d s - \int_{t - η_{M}}^{t - τ (t)} {\overset{\cdot}{x}}^{T} (s) Z_{2} \overset{\cdot}{x} (s) d s \\ - \int_{t - η_{M}}^{t} {\overset{\cdot}{x}}^{T} (s) Z_{3} \overset{\cdot}{x} (s) d s \end{matrix}

(27)

\begin{array}{l} {\overset{\cdot}{V}}_{4} (t) = \frac{η_{m}^{2}}{2} x^{T} (t) R_{1} x (t) + \frac{η_{m}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{2} \overset{\cdot}{x} (t) \\ + \frac{η_{M}^{2} - η_{m}^{2}}{2} x^{T} (t) R_{3} x (t) + \frac{η_{M}^{2} - η_{m}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{4} \overset{\cdot}{x} (t) \\ + \frac{η_{M}^{2}}{2} x^{T} (t) R_{5} x (t) \\ + \frac{η_{m}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{6} \overset{\cdot}{x} (t) - \int_{- η_{m}}^{0} \int_{t + θ}^{t} x^{T} (s) R_{1} x (s) d s d θ \\ - \int_{- η_{m}}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) d s d θ \\ - \int_{- τ (t)}^{- η_{m}} \int_{t + θ}^{t} x^{T} (s) R_{3} x (s) d s d θ \\ - \int_{- τ (t)}^{- η_{m}} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{4} \overset{\cdot}{x} (s) d s d θ \\ - \int_{- η_{m}}^{- τ (t)} - τ (t) \int_{t + θ}^{t} x^{T} (s) R_{3} x (s) d s d θ \\ - \int_{- η_{m}}^{- τ (t)} - τ (t) \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{4} \overset{\cdot}{x} (s) d s d θ \\ - \int_{- η_{m}}^{0} \int_{t + θ}^{t} x^{T} (s) R_{5} x (s) d s d θ \\ - \int_{- η_{m}}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{6} \overset{\cdot}{x} (s) d s d θ \end{array}

(28)

For any matrices $M_{i}, i = 1, \dots, 8$ , with appropriate dimensions, the following equalities hold

\begin{matrix} η_{m} x^{T} (t) M_{1} x (t) - \int_{t - η_{m}}^{t} x^{T} (s) M_{1} x (s) d s \\ - 2 \int_{- η_{m}}^{0} \int_{t + θ}^{t} x^{T} (s) M_{1} \overset{\cdot}{x} (s) d s d θ = 0 \end{matrix}

(29)

\begin{matrix} (τ (t) - η_{m}) x^{T} (t) M_{2} x (t) - \int_{t - τ (t)}^{t - η_{m}} x^{T} (s) M_{2} x (s) d s \\ - 2 \int_{- τ (t)}^{- η_{m}} \int_{t + θ}^{t} x^{T} (s) M_{2} \overset{\cdot}{x} (s) d s d θ = 0 \end{matrix}

(30)

\begin{matrix} (η_{M} - τ (t)) x^{T} (t) M_{3} x (t) - \int_{t - η_{M}}^{t - τ (t)} x^{T} (s) M_{3} x (s) d s \\ - 2 \int_{- η_{M}}^{- τ (t)} \int_{t + θ}^{t} x^{T} (s) M_{3} \overset{\cdot}{x} (s) d s d θ = 0 \end{matrix}

(31)

\begin{matrix} η_{M} x^{T} (t) M_{4} x (t) - \int_{t - η_{M}}^{t} x^{T} (s) M_{4} x (s) d s \\ - 2 \int_{- η_{M}}^{0} \int_{t + θ}^{t} x^{T} (s) M_{4} \overset{\cdot}{x} (s) d s d θ = 0 \end{matrix}

(32)

\begin{matrix} x^{T} (t) M_{5} x (t) - x^{T} (t - η_{m}) M_{5} x (t - η_{m}) \\ - 2 \int_{t - η_{m}}^{t} x^{T} (s) M_{5} \overset{\cdot}{x} (s) d s = 0 \end{matrix}

(33)

\begin{matrix} x^{T} (t - η_{m}) M_{6} x (t - η_{m}) - x^{T} (t - τ (t)) M_{6} x (t - τ (t)) \\ - 2 \int_{t - τ (t)}^{t - η_{m}} x^{T} (s) M_{6} \overset{\cdot}{x} (s) d s = 0 \end{matrix}

(34)

\begin{matrix} x^{T} (t - τ (t)) M_{7} x (t - τ (t)) - x^{T} (t - η_{M}) M_{7} x (t - η_{M}) \\ - 2 \int_{t - η_{M}}^{t - τ (t)} x^{T} (s) M_{7} \overset{\cdot}{x} (s) d s = 0 \end{matrix}

(35)

\begin{matrix} x^{T} (t) M_{8} x (t) - x^{T} (t - η_{M}) M_{8} x (t - η_{M}) \\ - 2 \int_{t - η_{M}}^{t} x^{T} (s) M_{8} \overset{\cdot}{x} (s) d s = 0 \end{matrix}

(36)

From Lemma 4, the following inequalities are obtained

\begin{matrix} - \int_{t - η_{m}}^{t} {\overset{\cdot}{x}}^{T} (s) (Z_{1} + Z_{3}) \overset{\cdot}{x} (s) d s \\ \leq - \frac{1}{η_{m}} {[\begin{matrix} x (t) \\ x (t - η_{m}) \\ υ_{1} \end{matrix}]}^{T} \\ [\begin{matrix} a_{1} (Z_{1} + Z_{3}) & a_{2} (Z_{1} + Z_{3}) & a_{3} (Z_{1} + Z_{3}) \\ * & a_{1} (Z_{1} + Z_{3}) & a_{3} (Z_{1} + Z_{3}) \\ * & * & a_{4} (Z_{1} + Z_{3}) \end{matrix}] [\begin{matrix} x (t) \\ x (t - η_{m}) \\ υ_{1} \end{matrix}] \end{matrix}

(37)

\begin{matrix} - \int_{t - τ (t)}^{t - η_{m}} {\overset{\cdot}{x}}^{T} (s) (Z_{2} + Z_{3}) \overset{\cdot}{x} (s) d s \\ \leq - \frac{1}{τ (t) - η_{m}} {[\begin{matrix} x (t - η_{m}) \\ x (t - τ (t)) \\ υ_{2} \end{matrix}]}^{T} \\ [\begin{matrix} a_{1} (Z_{2} + Z_{3}) & a_{2} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & a_{1} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & * & a_{4} (Z_{2} + Z_{3}) \end{matrix}] [\begin{matrix} x (t - η_{m}) \\ x (t - τ (t)) \\ υ_{2} \end{matrix}] \end{matrix}

(38)

\begin{array}{l} - \int_{t - η_{M}}^{t - τ (t)} {\overset{\cdot}{x}}^{T} (s) (Z_{2} + Z_{3}) \overset{\cdot}{x} (s) d s \\ \leq - \frac{1}{η_{M} - τ (t)} {[\begin{array}{l} x (t - τ (t)) \\ x (t - η_{M}) \\ υ_{3} \end{array}]}^{T} \\ [\begin{matrix} a_{1} (Z_{2} + Z_{3}) & a_{2} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & a_{1} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & * & a_{4} (Z_{2} + Z_{3} \end{matrix}] [\begin{array}{l} x (t - τ (t)) \\ x (t - η_{M}) \\ υ_{3} \end{array}] \end{array}

(39)

From equations (25) to (28), (29) to (36) and (37) to (39), $\overset{\cdot}{V} (t)$ in equation (24) can be expressed as

\begin{matrix} \overset{\cdot}{V} (t) \leq 2 ξ^{T} (t) P \overset{\cdot}{ξ} (t) + x^{T} (t) (Q_{1} + Q_{2}) x (t) - x^{T} (t - η_{m}) Q_{1} x (t - η_{m}) - x^{T} (t - η_{M}) Q_{2} x (t - η_{M}) \\ + 2 η_{m} {\overset{\cdot}{x}}^{T} (t) Z_{1} \overset{\cdot}{x} (t) + 2 (η_{M} - η_{m}) {\overset{\cdot}{x}}^{T} (t) Z_{2} \overset{\cdot}{x} (t) + 2 η_{M} {\overset{\cdot}{x}}^{T} (t) Z_{3} \overset{\cdot}{x} (t) + \frac{η_{m}^{2}}{2} x^{T} (t) R_{1} x (t) + \frac{η_{m}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{2} \overset{\cdot}{x} (t) \\ + \frac{η_{M}^{2} - η_{m}^{2}}{2} x^{T} (t) R_{3} x (t) + \frac{η_{M}^{2} - η_{m}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{4} \overset{\cdot}{x} (t) + \frac{η_{M}^{2}}{2} x^{T} (t) R_{5} x (t) + \frac{η_{M}^{2}}{2} {\overset{\cdot}{x}}^{T} (t) R_{6} \overset{\cdot}{x} (t) \\ + η_{m} x^{T} (t) M_{1} x (t) + (τ (t) - η_{m}) x^{T} (t) M_{2} x (t) + (η_{M} - τ (t)) x^{T} (t) M_{3} x (t) + η_{M} x^{T} (t) M_{4} x (t) \\ + x^{T} (t) M_{5} x (t) - x^{T} (t - η_{m}) M_{5} x (t - η_{m}) + x^{T} (t - η_{m}) M_{6} x (t - η_{m}) - x^{T} (t - τ (t)) M_{6} x (t - τ (t)) \\ + x^{T} (t - τ (t)) M_{7} x (t - τ (t)) - x^{T} (t - η_{M}) M_{7} x (t - η_{M}) + x^{T} (t) M_{8} x (t) - x^{T} (t - η_{M}) M_{8} x (t - η_{M}) \\ - \int_{t - η_{m}}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{1} & M_{5} \\ * & Z_{1} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s - \int_{t - τ (t)}^{t - η_{m}} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{2} & M_{6} \\ * & Z_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{t - η_{M}}^{t - τ (t)} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{3} & M_{7} \\ * & Z_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s - \int_{t - η_{M}}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{4} & M_{8} \\ * & Z_{3} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{- η_{m}}^{0} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{1} & M_{1} \\ * & R_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ - \int_{- τ (t)}^{- η_{m}} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{3} & M_{2} \\ * & R_{4} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ - \int_{- η_{M}}^{- τ (t)} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{3} & M_{3} \\ * & R_{4} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ - \int_{- η_{M}}^{0} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{5} & M_{4} \\ * & R_{6} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ - \frac{1}{η_{m}} {[\begin{matrix} x (t) \\ x (t - η_{m}) \\ υ_{1} \end{matrix}]}^{T} [\begin{matrix} a_{1} (Z_{1} + Z_{3}) & a_{2} (Z_{1} + Z_{3}) & a_{3} (Z_{1} + Z_{3}) \\ * & a_{1} (Z_{1} + Z_{3}) & a_{3} (Z_{1} + Z_{3}) \\ * & * & a_{4} (Z_{1} + Z_{3}) \end{matrix}] [\begin{matrix} x (t) \\ x (t - η_{m}) \\ υ_{1} \end{matrix}] \end{matrix}

(40)

\begin{matrix} - \frac{1}{τ (t) - η_{m}} {[\begin{matrix} x (t - η_{m}) \\ x (t - τ (t)) \\ υ_{2} \end{matrix}]}^{T} [\begin{matrix} a_{1} (Z_{2} + Z_{3}) & a_{2} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & a_{1} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & * & a_{4} (Z_{2} + Z_{3}) \end{matrix}] [\begin{matrix} x (t - η_{m} \\ x (t - τ (t)) \\ υ_{2} \end{matrix}] \\ - \frac{1}{η_{M} - τ (t)} {[\begin{matrix} x (t - τ (t)) \\ x (t - η_{M}) \\ υ_{3} \end{matrix}]}^{T} [\begin{matrix} a_{1} (Z_{2} + Z_{3}) & a_{2} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & a_{1} (Z_{2} + Z_{3}) & a_{3} (Z_{2} + Z_{3}) \\ * & * & a_{4} (Z_{2} + Z_{3} \end{matrix}] [\begin{matrix} x (t - τ (t)) \\ x (t - η_{M}) \\ υ_{3} \end{matrix}] \end{matrix}

where

\begin{matrix} υ_{1} = \frac{1}{η_{m}} \int_{t - η_{m}}^{t} x (s) d s, υ_{2} = \frac{1}{τ (t) - η_{m}} \int_{t - τ (t)}^{t - η_{m}} x (s) d s, \\ υ_{3} = \frac{1}{η_{M} - τ (t)} \int_{t - η_{M}}^{t - τ (t)} x (s) d s \end{matrix}

Then, considering the event-triggered scheme given by equation (18), we have

\begin{matrix} \overset{\cdot}{V} (t) \leq {\overset{\cdot}{V}}_{1} (t) + {\overset{\cdot}{V}}_{2} (t) + {\overset{\cdot}{V}}_{3} (t) + {\overset{\cdot}{V}}_{4} (t) \\ - e^{T} (t) Ω_{1} e (t) + σ x^{T} (t - τ (t)) Ω_{2} x (t - τ (t)) \\ = ζ_{1}^{T} (t) [Ω + {\bar{A}}^{T} Λ \bar{A}] ζ_{1} - \int_{t - η_{m}}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{1} & M_{5} \\ * & Z_{1} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{t - τ (t)}^{t - η_{m}} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{2} & M_{6} \\ * & Z_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{t - η_{M}}^{t - τ (t)} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{3} & M_{7} \\ * & Z_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{t - η_{M}}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} M_{4} & M_{8} \\ * & Z_{3} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s \\ - \int_{- η_{m}}^{0} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{1} & M_{1} \\ * & R_{2} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ - \int_{- τ (t)}^{- η_{m}} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{3} & M_{2} \\ * & R_{4} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ - \int_{- η_{M}}^{- τ (t)} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{3} & M_{3} \\ * & R_{4} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ - \int_{- η_{M}}^{0} \int_{t + θ}^{t} {[\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}]}^{T} [\begin{matrix} R_{5} & M_{4} \\ * & R_{6} \end{matrix}] [\begin{matrix} x (s) \\ \overset{\cdot}{x} (s) \end{matrix}] d s d θ \\ \leq ζ_{1}^{T} (t) Θ ζ_{1} \end{matrix}

where

ζ_{1} = {[\begin{matrix} x^{T} (t) & x^{T} (t - η_{m}) & x^{T} (t - τ (t)) & x^{T} (t - η_{M}) & υ_{1}^{T} & υ_{2}^{T} & υ_{3}^{T} & e^{T} (t) \end{matrix}]}^{T}, Θ = [\begin{matrix} Φ & {\bar{A}}^{T} Λ \\ * & - Λ \end{matrix}]

According to the Schur complement, equation (23) ensures $Θ < 0$ , implying that $\overset{\cdot}{V} (t) < 0$ . Therefore, the closed-loop nominal system given by equation (22) is asymptotically stable.

Then, for the $H_{\infty}$ performance of the network-based system given by equation (22) with $w (t) \neq 0$ , the following performance index is introduced

\begin{matrix} \overset{\cdot}{V} (t) \leq {[\begin{matrix} ζ_{1} (t) \\ w (t) \end{matrix}]}^{T} [\begin{matrix} Θ & Γ_{1}^{T} + {\bar{A}}^{T} Λ B_{w} \\ * & B_{w}^{T} Λ B_{w} \end{matrix}] [\begin{matrix} ζ_{1} (t) \\ w (t) \end{matrix}] \\ J_{z w} = \int_{0}^{\infty} [Z^{T} (t) Z (t) - γ^{2} w^{T} (t) w (t)] d t \end{matrix}

Under a zero initial condition, the performance index $T_{z w}$ is transformed into the following inequality

\begin{matrix} J_{z w} = \int_{0}^{\infty} [Z^{T} (t) Z (t) - γ^{2} w^{T} (t) w (t) + \overset{\cdot}{V} (t)] d t - V (\infty) \\ \leq \int_{0}^{\infty} [Z^{T} (t) Z (t) - γ^{2} w^{T} (t) w (t) + \overset{\cdot}{V} (t)] d t \\ \leq \int_{0}^{\infty} ζ^{T} (t) [\begin{matrix} Θ + Γ_{2}^{T} Γ_{2} & Γ_{1}^{T} + {\bar{A}}^{T} Λ B_{w} + Γ_{2}^{T} D_{w} \\ * & - γ^{2} I + B_{w}^{T} Λ B_{w} + D_{w}^{T} D_{w} \end{matrix}] ζ (t) \end{matrix}

with $ζ (t) = [ζ_{1} (t), w (t)]$ . According to the Schur complement and from equation (23), we have

Ξ = [\begin{matrix} Θ + Γ_{2}^{T} Γ_{2} & Γ_{1}^{T} + {\bar{A}}^{T} Λ B_{w} + Γ_{2}^{T} D_{w} \\ * & - γ^{2} I + B_{w}^{T} Λ B_{w} + D_{w}^{T} D_{w} \end{matrix}] < 0

This means that $J_{z w} < 0$ . Therefore, under zero initial conditions and for any nonzero $w (t) \in L_{2} [0, \infty]$ , the control output $z (t)$ satisfies $‖ z (t) ‖_{2} \leq γ ‖ w (t) ‖_{2}$ . Hence, the closed-loop system given by equation (22) is asymptotically stable with $H_{\infty}$ norm bound $γ$ if the LMI given by equation (23) is feasible. This completes the proof. □

Remark 7. Some parameters and matrices need to be determined in Theorem 1. Our objective is to derive delay-range-dependent stability and stabilization conditions within the LMI framework. The LMIs of Theorem 1 can be solved efficiently through numerical computation. From the LMI approach, for given scalars $γ > 0$ , $0 \leq σ < 1$ , $η_{m}$ , $η_{M}$ and K, it is possible to determine the asymptotic stability of the considered time-delay system. Theorem 1 provides new delay-range-dependent conditions for stability and stabilization of event-triggered robust $H_{\infty}$ control of the nominal system given by equation (22) . The system is stabilizable under our control scheme only when the LMIs have a solution.

Remark 8. An appropriate Lyapunov–Krasovskii functional will help reduce the system conservatism, which may result from some techniques to deal with cross items for the stability of the time-delay systems. In this paper, a sufficient condition is derived for stability criteria together with new zero equalities in equations (29) to (36) , which increase the flexibility in dealing with the cross items in the Lyapunov–Krasovskii functional. The two terms $- \int_{- η_{m}}^{0} \int_{t + θ}^{t} x^{T} (s) V x (s) d s d θ$ and $- \int_{- η_{m}}^{0} \int_{t + θ}^{t} {\overset{\cdot}{x}}^{T} (s) W \overset{\cdot}{x} (s) d s d θ$ can be obtained for the time-derivative of the triple-integral terms $\int_{- η_{m}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} x^{T} (s) R_{1} x (s) d s d λ d θ$ and $\int_{- η_{m}}^{0} \int_{θ}^{0} \int_{t + λ}^{t} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) d s d λ d θ$ about $V_{4} (t)$ . Introducing equations (29) to (32) helps handle the above double-integral terms, which are generated from the triple-integral terms. But one-integral term, such as $- \int_{t - η_{m}}^{t} x^{T} (s) M x (s) d s$ , will be generated during the processing. Therefore, the derivation result of $V_{3} (t)$ is utilized to dispose the above one-integral term from equations (33) to (36) .

Remark 9. The results in Theorem 1 refines the inequality from Lemma 4. Wirtinger’s inequality may introduce an extra term $\int_{a}^{b} x (s) d s$ , in addition to the terms $x (a)$ and $x (b)$ , for estimation of the integral of the derivative function. Therefore, an improvement is obtained by combining this inequality with a Lyapunov–Krasovskii functional where the term $\int_{a}^{b} x (s) d s$ appears explicitly. The inequality given by equation (21) is related to Jensen’s inequality in essence. Jensen’s inequality can be considered a special case of Wirtinger’s inequality. However, Wirtinger’s inequality is more effective in conservatism reduction.

When considering uncertain systems, we have the following corollary for a condition to satisfy robust stability and a $H_{\infty}$ norm bound constraint.

Corollary 1. For given parameters $γ > 0$ , $0 \leq σ < 1$ , $η_{m}$ , $η_{M}$ and K, the closed-loop system in equation (19) with time varying structured uncertainties is robustly stable with $H_{\infty}$ control performance under the event-triggered scheme given by equation (18) , if there exist matrices $M_{i} > 0 (i = 1, \dots, 8)$ and symmetric matrices P, $Q_{i} > 0 (i = 1, 2)$ , $Z_{i} > 0 (i = 1, 2, 3)$ , $R_{i} > 0 (i = 1, \dots, 6)$ , $Ω_{i} > 0 (i = 1, 2)$ and $a_{i} > 0 (i = 1, 2, 3, 4)$ are positive definite and scalar $ϵ > 0$ , such that the following matrix inequity holds

[\begin{matrix} Φ & * & * & * & * \\ Γ_{1} & - γ^{2} & * & * & * \\ Γ_{2} & D_{w} & - I & * & * \\ Λ \bar{A} & Λ B_{w} & 0 & - Λ & * \\ Δ_{1} & 0 & Δ_{2} & Δ_{3} & Δ_{4} \end{matrix}] < 0

(41)

where

Δ_{1} = [\begin{array}{l} G_{1}^{T} p_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ ϵ E_{1} & 0 & ϵ E_{2} K & 0 & 0 & 0 & 0 & - ϵ E_{2} K \end{array}], Δ_{2} = [\begin{array}{l} G_{2}^{T} \\ 0 \end{array}], Δ_{3} = [\begin{array}{l} G_{1}^{T} Λ \\ 0 \end{array}] a n d Δ_{4} = [\begin{matrix} - ε & 0 \\ 0 & - ε \end{matrix}]

Proof. If A, B, C and D in equation (23) are replaced with $A + G_{1} F (t) E_{1}$ , $B + G_{1} F (t) E_{2}$ , $C + G_{2} F (t) E_{1}$ and $D + G_{2} F (t) E_{2}$ , respectively, and define the matrix in the left side of equation (23) to be $Ξ$ , then equation (23) for the uncertain system is changed to the following condition

Ξ + \sum_{g}^{T} F (T) \sum_{e} + \sum_{e}^{T} F^{T} (t) \sum_{g} < 0

(42)

where

\begin{matrix} \sum_{g} = [\begin{matrix} G_{1}^{T} p_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{2}^{T} & G_{1}^{T} Λ \end{matrix}] \\ \sum_{e} = [\begin{matrix} E_{1} & 0 & E_{2} K & 0 & 0 & 0 & 0 & - E_{2} K & 0 & 0 & 0 \end{matrix}] \end{matrix}

By Lemma 3, equation (42) holds if there exists an $ϵ > 0$ satisfying

Ξ + ϵ^{- 1} \sum_{g}^{T} \sum_{g} + ϵ \sum_{e}^{T} \sum_{e} < 0

(43)

Applying Schur complements, we see that that equation (43) is equivalent to equation (41). This completes the proof. □

Remark 10. Corollary 1 investigates the stability and stabilization of the event-triggered robust $H_{\infty}$ control of the uncertain system given by equation (19) with interval time-varying delays. The obtained condition is formulated in terms of LMIs. The stability test is presented as LMIs, which can be solved numerically. The LMI toolbox makes it easy to examine whether or not the uncertain system with delay-range-dependent conditions is quadratically stabilizable under our control scheme.

Event-triggered $H_{\infty}$ controller design for uncertain systems

From Theorem 1, an $H_{\infty}$ controller is designed for the nominal system given by equation (22) under the event-triggered scheme given by equation (18).

Theorem 2. For given parameters $γ > 0$ , $0 \leq σ < 1$ , $η_{m}$ and $η_{M}$ , $μ_{i}$ (i=1, …, 6), the closed-loop system given by equation (22) is stable with $H_{\infty}$ control performance under the event-triggered scheme given by equation (18) , if there exists a nonsingular matrix X, matrices ${\tilde{M}}_{i} > 0 (i = 1, \dots, 8)$ and symmetric matrices $\tilde{P}$ , ${\tilde{Q}}_{i} > 0 (i = 1, 2)$ , ${\tilde{Z}}_{i} > 0 (i = 1, 2, 3)$ , ${\tilde{R}}_{i} > 0 (i = 1, \dots, 6)$ , $Ω_{i} > 0 (i = 1, 2)$ , $a_{i} > 0 (i = 1, 2, 3, 4)$ are positive definite, such that the following matrix inequalities hold

[\begin{matrix} \tilde{Φ} & * & * & * & * & * & * & * & * \\ {\tilde{Γ}}_{1} & - γ^{2} & * & * & * & * & * & * & * \\ {\tilde{Γ}}_{2} & D_{w} & - I & * & * & * & * & * & * \\ \tilde{A} & B_{w} & 0 & Π_{1} & * & * & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & Π_{2} & * & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & Π_{3} & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & Π_{4} & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & 0 & Π_{5} & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & 0 & 0 & Π_{6} \end{matrix}] < 0

(44)

for

\begin{matrix} [\begin{matrix} {\tilde{M}}_{1} & {\tilde{M}}_{5} \\ * & {\tilde{Z}}_{1} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{M}}_{2} & {\tilde{M}}_{6} \\ * & {\tilde{Z}}_{2} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{M}}_{3} & {\tilde{M}}_{7} \\ * & {\tilde{Z}}_{2} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{M}}_{4} & {\tilde{M}}_{8} \\ * & {\tilde{Z}}_{3} \end{matrix}] \geq 0 \\ [\begin{matrix} {\tilde{R}}_{1} & {\tilde{M}}_{1} \\ * & {\tilde{R}}_{2} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{R}}_{3} & {\tilde{M}}_{2} \\ * & {\tilde{R}}_{4} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{R}}_{3} & {\tilde{M}}_{3} \\ * & {\tilde{R}}_{4} \end{matrix}] \geq 0, [\begin{matrix} {\tilde{R}}_{5} & {\tilde{M}}_{4} \\ * & {\tilde{R}}_{6} \end{matrix}] \geq 0 \\ \tilde{Φ} = [\begin{matrix} {\tilde{Ψ}}_{1} & {\tilde{Ψ}}_{2} \\ * & {\tilde{Ψ}}_{3} \end{matrix}], {\tilde{Ψ}}_{1} = [\begin{matrix} {\tilde{φ}}_{11} & {\tilde{φ}}_{12} & {\tilde{φ}}_{13} & 0 \\ * & {\tilde{φ}}_{22} & {\tilde{φ}}_{23} & 0 \\ * & * & {\tilde{φ}}_{33} & {\tilde{φ}}_{34} \\ * & * & * & {\tilde{φ}}_{44} \end{matrix}], {\tilde{Ψ}}_{2} = [\begin{matrix} {\tilde{ϱ}}_{11} & {\tilde{ϱ}}_{12} & {\tilde{ϱ}}_{13} & {\tilde{ϱ}}_{14} \\ {\tilde{ϱ}}_{21} & {\tilde{ϱ}}_{22} & {\tilde{ϱ}}_{23} & 0 \\ 0 & {\tilde{ϱ}}_{32} & {\tilde{ϱ}}_{33} & 0 \\ {\tilde{ϱ}}_{41} & {\tilde{ϱ}}_{42} & {\tilde{ϱ}}_{43} & 0 \end{matrix}] \\ {\tilde{Ψ}}_{3} = [\begin{matrix} - \frac{a_{4}}{η_{m}} ({\tilde{Z}}_{1} + {\tilde{Z}}_{3}) & 0 & 0 & 0 \\ * & - \frac{a_{4}}{η_{M} - η_{m}} ({\tilde{Z}}_{2} + {\tilde{Z}}_{3}) & 0 & 0 \\ * & * & - \frac{a_{4}}{η_{M} - η_{m}} ({\tilde{Z}}_{2} + {\tilde{Z}}_{3}) & 0 \\ * & * & * & - {\tilde{Ω}}_{1} \end{matrix}] \end{matrix}

where

\begin{matrix} {\tilde{Γ}}_{1} = [\begin{matrix} B_{w}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], {\tilde{Γ}}_{2} = [\begin{matrix} C X & 0 & D Y & 0 & 0 & 0 & 0 & - D Y \end{matrix}] \\ \tilde{A} = [\begin{matrix} A X & 0 & B Y & 0 & 0 & 0 & 0 & - B Y \end{matrix}], a_{1} = \frac{π^{2}}{4} + 1, a_{2} = \frac{π^{2}}{4} - 1, a_{3} = - \frac{π^{2}}{2}, a_{4} = π^{2} \\ {\tilde{φ}}_{i j} = X^{T} φ_{i j} X, {\tilde{ϱ}}_{i j} = X^{T} ϱ_{i j} X, (i = 1, \dots, 4; j = 1, \dots, 4) \\ {\tilde{p}}_{i j} = X^{T} p_{i j} X, (i = 2, 3; j = 2, 3, 4) \\ Π_{1} = - \frac{4 η_{m}}{μ_{1}} X + \frac{2 η_{m}}{μ_{1}^{2}} {\tilde{Z}}_{1}, Π_{2} = - \frac{4 (η_{M} - η_{m})}{μ_{2}} X + \frac{2 (η_{M} - η_{m})}{μ_{2}^{2}} {\tilde{Z}}_{2}, Π_{3} = - \frac{4 η_{M}}{μ_{3}} X + \frac{2 η_{M}}{μ_{3}^{2}} {\tilde{Z}}_{3} \\ Π_{4} = - \frac{4}{μ_{4}} X + \frac{2}{μ_{4}^{2}} {\tilde{R}}_{2}, Π_{5} = - \frac{4}{μ_{5}} X + \frac{2}{μ_{5}^{2}} {\tilde{R}}_{4}, Π_{6} = - \frac{4}{μ_{6}} X + \frac{2}{μ_{6}^{2}} {\tilde{R}}_{6} \end{matrix}

Furthermore, if the above LMI condition is feasible, a suitable state feedback controller gain in the form of equation (22) is given by $K = Y X^{- 1}$ .

Proof. According to Theorem 1 and the Schur complement, the system in equation (22) is asymptotically stable with $H_{\infty}$ control performance under the event-triggered scheme given by equation (18) if the following inequality holds

[\begin{matrix} Φ & Γ_{1}^{T} & Γ_{2}^{T} & {\bar{A}}^{T} Z_{1} & {\bar{A}}^{T} Z_{2} & {\bar{A}}^{T} Z_{3} & {\bar{A}}^{T} R_{2} & {\bar{A}}^{T} R_{4} & {\bar{A}}^{T} R_{6} \\ * & - γ^{2} & D_{w} & B_{w}^{T} Z_{1} & B_{w}^{T} Z_{2} & B_{w}^{T} Z_{3} & B_{w}^{T} R_{2} & B_{w}^{T} R_{4} & B_{w}^{T} R_{6} \\ * & * & - I & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - \frac{1}{2 η_{m}} Z_{1} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & - \frac{1}{2 (η_{M} - η_{m})} Z_{2} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - \frac{1}{2 η_{M}} Z_{3} & 0 & 0 & 0 \\ * & * & * & * & * & * & - \frac{2}{η_{m}^{2}} R_{2} & 0 & 0 \\ * & * & * & * & * & * & * & - \frac{2}{(η_{M}^{2} - η_{m}^{2})} R_{4} & 0 \\ * & * & * & * & * & * & * & * & - \frac{2}{η_{m}^{2}} R_{6} \end{matrix}] < 0

(45)

Without loss of generality, assume that $p_{11}$ is invertible, then pre- and post-multiply equation (45) with

d i a g {\underset{8}{\underset{︸}{p_{11}^{- T}, \dots, p_{11}^{- T}}}, I, I, Z_{1}^{- T}, Z_{2}^{- T}, Z_{3}^{- T}, R_{2}^{- T}, R_{4}^{- T}, R_{6}^{- T}}

and its transpose respectively, and set

\begin{matrix} X = p_{11}^{- 1}, {\tilde{Z}}_{i} = X^{T} Z_{i} X (i = 1, 2, 3), \\ {\tilde{R}}_{i} = X^{T} R_{i} X (i = 1, 2, 3, 4), {\tilde{Q}}_{i} = X^{T} Q_{i} X (i = 0, 1, 2) \\ {\tilde{M}}_{i} = X^{T} M_{i} X (i = 1, 2, 3, 4, 5, 6), {\tilde{Ω}}_{1} = X^{T} Ω_{1} X, \\ {\tilde{Ω}}_{2} = X^{T} Ω_{2} X \\ ϑ = d i a g {\underset{8}{\underset{︸}{X, \dots, X}}}, {\tilde{Γ}}_{1} = Γ_{1} ϑ, {\tilde{Γ}}_{2} = Γ_{2} ϑ, \\ \tilde{A} = \bar{A} ϑ, \tilde{Ω} = ϑ^{T} Ω ϑ, Y = K X \end{matrix}

From Lemma 2, for given positive scalars $μ_{i}$ $(i = 1, 2, 3)$ , the following inequalities can be obtained

- Z_{i}^{- 1} \leq - \frac{2}{μ_{i}} X + \frac{1}{μ_{i}^{2}} {\tilde{Z}}_{i}, - Z_{i}^{- T} \leq - \frac{2}{μ_{i}} X + \frac{1}{μ_{i}^{2}} {\tilde{Z}}_{i}

Similarly, we have

\begin{matrix} - R_{2}^{- T} \leq - \frac{2}{μ_{4}} X + \frac{1}{μ_{4}^{2}} {\tilde{R}}_{2}, - R_{4}^{- T} \leq - \frac{2}{μ_{5}} X + \frac{1}{μ_{5}^{2}} {\tilde{R}}_{4}, \\ - R_{6}^{- T} \leq - \frac{2}{μ_{6}} X + \frac{1}{μ_{6}^{2}} {\tilde{R}}_{4} \end{matrix}

From the above inequalities, the result in equation (44) can be derived. This completes the proof.

Remark 11. The event-triggered matrices $Ω_{1}$ and $Ω_{2}$ and the controller gain K can be obtained if the LMI in equation (44) is feasible.

Remark 12. The condition given by Theorem 2 is a sufficient condition. This means that when the condition is met, the stability of the closed system given by equation (22) under our event-triggered control scheme given by equation (18) is guaranteed.

Remark 13. Theorem 2 relates event-triggered parameters $σ$ , $Ω_{1}$ and $Ω_{2}$ with $H_{\infty}$ controller gain K. These parameters are associated with the utilization rate of network resources. Therefore, an algorithm can be designed to derive these parameters that decreases the use of network resources while still maintaining the desired $H_{\infty}$ performance. We have designed the following Algorithm 1 for co-design of our event-triggered scheme and the feedback controller.

Algorithm 1 (Solving Theorem 2)
Step 1: $ς$ is the step increment of $σ$ , set $σ = σ + ς$ , where $σ$ is initially set to zero, $σ \in [0, 1)$ ;
Step 2: if there exists a feasible solution satisfying the LMI given by equation (44) for a given $σ$ in step 1, go to step 3; otherwise, go to step 1;
Step 3: the maximum $η_{M}$ and corresponding $Ω_{1}$ , $Ω_{2}$ and controller gain K are obtained from Theorem 2 using the Matlab LMI Toolbox.

Remark 14. According to Lemma 2, $- 2 μ X \leq μ^{2} Z^{- 1} + X^{T} Z X$ . It follows that $- Z^{- 1} \leq \frac{1}{μ^{2}} (2 μ X + X^{T} Z X)$ . Using this form helps reduce the complexity of numerical computation for Theorem 2. The terms $- Z_{i}^{- T}$ $(i = 1, 2, 3)$ , $- R_{2}^{- T}$ , $- R_{4}^{- T}$ and $- R_{6}^{- T}$ can be obtained from the inequalities in equation (45). However, the nonlinear terms such as $X^{T} Z X$ in the inequality given by equation (44) cannot be derived directly. Therefore, Theorem 2 cannot be solved directly using the Matlab LMI Toolbox. To reduce the conservatism that may result from the derivation of the LMI based on the inequality given by equation (44) , one may apply the cone complementarity linearization (CCL) algorithm.

Remark 15. The expression in equation (44) is an LMI over not only matrix variables but also the scalar $γ$ . It is desired that weighting matrices can be determined such that the $H_{\infty}$ performance index $γ$ could be minimized at the cost of consuming a certain amount of communication resources for a given threshold parameter $σ$ . This implies that the scalar $γ$ can be included as an optimal variable to obtain reduction of the $H_{\infty}$ disturbance attention level bound. With Theorem 2, the problem to be solved in this section, for given $η_{m}$ , $η_{M}$ and $σ$ , can be transformed into the following minimization problem

\begin{matrix} \begin{matrix} \min_{σ, η_{m}, η_{M}} & γ \end{matrix} \\ s . t . {\tilde{M}}_{i} > 0, \tilde{p} > 0, {\tilde{Q}}_{i} > 0, {\tilde{Z}}_{i} > 0, {\tilde{R}}_{i} > 0, Ω_{i} > 0 a n d e q u a t i o n (44) . \end{matrix}

In practical situations, the $H_{\infty}$ performance index $γ$ should be made as small as possible so as to minimize the effects of exogenous disturbance and noise on the system output. Also, $σ$ should be made as large as possible so as to reduce the use of communication resources. It is a difficult task to directly obtain both a minimal $γ$ and a maximal $σ$ by solving Theorem 2. In this sense, applying Theorem 2, one may perform a trade-off analysis between the use of communication resources and the level of $H_{\infty}$ performance. The following co-design algorithm is proposed to deal with this trade-off problem.

Remark 16. The computational complexity of the results can be estimated from the total number N of decision variables and the total row size L of the LMIs. For Theorem 2, $N = 18 n^{2} + 7 n$ and $L = 11 n$ . It is worth mentioning that both N and L are dependent on the system dimension n. A less conservative result may be derived at the cost of higher computational complexity by using an advanced inequality. The computational complexity can be handled by the Matlab LMI toolbox and all designs can be implemented offline, making the LMI method practicable and effective.

Considering uncertain systems with time-varying delay, we extend Theorem 2 to design a robust $H_{\infty}$ controller for an uncertain system given by equation (19). Similar to the derivation of Theorem 2, the following corollary can be derived based on the results in Corollary 1.

Corollary 2. For given parameters $γ > 0$ , $0 \leq σ < 1$ , $η_{m}$ and $η_{M}$ , $μ_{i}$ $(i = 1, \dots, 6)$ , the closed-loop uncertain system in equation (19) with time-varying delays is robustly stable with $H_{\infty}$ control performance under the event-triggered scheme given by equation (18) , if there exists a nonsingular matrix X, matrices ${\tilde{M}}_{i} > 0 (i = 1, \dots, 8)$ and symmetric matrices $\tilde{P}$ , ${\tilde{Q}}_{i} > 0 (i = 1, 2)$ , ${\tilde{Z}}_{i} > 0 (i = 1, 2, 3)$ , ${\tilde{R}}_{i} > 0 (i = 1, \dots, 6)$ , $Ω_{i} > 0 (i = 1, 2)$ and $a_{i} > 0 (i = 1, \dots, 4)$ are positive definite and scalar $λ > 0$ , satisfying the following matrix inequality

[\begin{matrix} \tilde{Φ} & * & * & * & * & * & * & * & * & * \\ {\tilde{Γ}}_{1} & - γ^{2} & * & * & * & * & * & * & * & * \\ {\tilde{Γ}}_{2} & D_{w} & - I & * & * & * & * & * & * & * \\ \tilde{A} & B_{w} & 0 & Π_{1} & * & * & * & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & Π_{2} & * & * & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & Π_{3} & * & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & Π_{4} & * & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & 0 & Π_{5} & * & * \\ \tilde{A} & B_{w} & 0 & 0 & 0 & 0 & 0 & 0 & Π_{6} & * \\ Δ_{1} & 0 & Δ_{2} & Δ_{3} & Δ_{3} & Δ_{3} & Δ_{3} & Δ_{3} & Δ_{3} & Δ_{4} \end{matrix}] < 0

(46)

where

\begin{array}{l} Δ_{1} = [\begin{matrix} λ G_{1}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{1} X & 0 & E_{2} Y & 0 & 0 & 0 & 0 & E_{2} Y \end{matrix}], \\ Δ_{2} = [\begin{matrix} λ G_{2}^{T} \\ 0 \end{matrix}], Δ_{3} = [\begin{matrix} - λ G_{1}^{T} \\ 0 \end{matrix}], Δ_{4} = [\begin{matrix} - λ & 0 \\ 0 & - λ \end{matrix}] \end{array}

Other parameters are set similarly to those in Theorem 2. Furthermore, if the above LMI condition is feasible, the state feedback controller gain in equation (19) is given by $K = Y X^{- 1}$ .

Proof. A similar approach to those in the proofs of Theorem 2 and Corollary 1 could be used to prove this corollary.

As in the proof process of Theorem 2, replace A, B, C and D in equation (44) with $A + G_{1} F (t) E_{1}$ , $B + G_{1} F (t) E_{2}$ , $C + G_{2} F (t) E_{1}$ and $D + G_{2} F (t) E_{2}$ , respectively. Also, define the matrix on the left-hand side of equation (44) to be $\bar{Ξ}$ . Then, equation (44) for the uncertain system is changed to the following condition

\bar{Ξ} + {\sum^{¯}}_{g}^{T} F (T) {\sum^{¯}}_{e} + {\sum^{¯}}_{e}^{T} F^{T} (t) {\sum^{¯}}_{g} < 0

(47)

where

\begin{matrix} {\sum^{¯}}_{g} = [\begin{matrix} λ G_{1}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & λ G_{2}^{T} & - λ G_{1}^{T} & - λ G_{1}^{T} & - λ G_{1}^{T} & - λ G_{1}^{T} & - λ G_{1}^{T} & - λ G_{1}^{T} \end{matrix}] \\ {\sum^{¯}}_{e} = [\begin{matrix} E_{1} X & 0 & E_{2} Y & 0 & 0 & 0 & 0 & E_{2} Y & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

By Lemma 3, equation (47) holds if there exists a $λ > 0$ satisfying

\bar{Ξ} + ϵ^{- 1} {\sum^{¯}}_{g}^{T} {\sum^{¯}}_{g} + ε {\sum^{¯}}_{e}^{T} {\sum^{¯}}_{e} < 0

(48)

Applying the Schur complement, we conclude that equation (48) is equivalent to equation (46). Thus, the feedback controller gain K in equation (19) is obtained under our event-triggered scheme given by equation (18). This completes the proof. □

Remark 17. Corollary 2 extends Theorem 2 for uncertain systems. The controller gain and trigger matrices can be obtained, which guarantee the stability of the system in equation (19) . Corollary 2 is used to co-design controller gain and triggering parameters $σ$ , $Ω_{1}$ and $Ω_{2}$ in terms of the LMI equation (46) . For given information of the trigger parameter $σ$ and transmission delay, by solving the LMI in Corollary 2, the controller gain K and trigger matrices $Ω_{1}$ and $Ω_{2}$ can be obtained, which guarantee that the system in equation (22) is stable.

Simulation experiments

This section demonstrates our method presented in this paper through two numerical examples.

Example 1. Consider the following uncertain system controlled over networks with the following parameters

\begin{matrix} \overset{\cdot}{x} (t) = [A + Δ A (t)] x (t) + [B + Δ B (t)] u (t) + B_{w} w (t) \\ z (t) = [C + Δ C (t)] x (t) + [D + Δ D (t)] u (t) + D_{w} w (t) \end{matrix}

(49)

where

\begin{array}{l} A = [\begin{matrix} - 1 & 0 & - 0.5 \\ 1 & - 0.5 & 0 \\ 0 & 0 & 0.5 \end{matrix}], B = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}], B_{w} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}], \\ C = [\begin{matrix} 1 & 0 & 1 \end{matrix}], D = 0.1, D_{w} = 0 \end{array}

System uncertainties are described by equation (2) with

\begin{matrix} G_{1} = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}], E_{1} = [\begin{matrix} 0.1 & 0.1 & 0 \\ 0 & 0 & 0.1 \\ 0 & - 0.1 & 0 \end{matrix}], \\ E_{2} = [\begin{matrix} 0.05 \\ 0 \\ 0.05 \end{matrix}], G_{2} = 0 \end{matrix}

Because $∥ F (t) ∥ \leq 1$ , the uncertain matrices satisfy $‖ Δ A (t) ‖ \leq 0.1$ and $‖ Δ B (t) ‖ \leq 0.1$ .

The exogenous disturbance is assumed to be

w (t) = {\begin{matrix} s g n (\sin (t)), & t \in [2 s, 4 s] \\ 0, & o t h e r w i s e \end{matrix}

As $η_{M} = h + τ_{M}$ , in order to reduce the conservatism, the lower bound of the delay $τ_{m}$ cannot be zero. Meanwhile, the upper bound of the time delay must be greater than or equal to the lower bound of the delay. The lower bound of the network-induced delay is assumed to be $τ_{m} = 0.1$ . Choose the sampling period $h = 0.1$ . The following feedback gain $K = [\begin{matrix} - 0.5425 & - 0.0014 & - 1.3858 \end{matrix}]$ is the same as the one from the literature (Jiang et al., 2008; Yue et al., 2005). Applying Theorem 1 with $σ = 0.1$ and $γ = 200$ , we obtain the upper bound of the time delay $τ_{M} = 0.5892$ , which is larger than the results from the work by Jiang et al. (2008) and Yue et al. (2005). Table 1 lists the $τ_{M}$ values obtained from different methods under various $τ_{m}$ values. For all scenarios listed in the table, the $τ_{M}$ values obtained from our method are larger than those from the other two existing methods (Jiang et al., 2008; Yue et al., 2005).

Table 1.

Upper bounds of the time delay $τ_{M}$ from different methods for $h = 0.1$ , $σ = 0.1$ , $γ = 200$ and various $τ_{m}$ values.

$τ_{m}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7
Our work in this paper	0.5892	0.6126	0.6433	0.6791	0.6965	0.7239	0.7463
Jiang et al. (2008)	0.5000	0.5536	0.5862	0.6134	0.6512	0.6821	0.7028
Yue et al. (2005)	0.3000	0.3567	0.3853	0.4349	—	—	—

Using Corollary 2, for $τ_{m} = 0.1$ , $τ_{M} = 0.58$ and $σ = 0.1$ , we obtain the feedback controller gain $K = [\begin{matrix} - 1.7821 & - 0.0015 & - 2.7722 \end{matrix}]$ and the event-triggered matrices

Ω_{1} = [\begin{matrix} 2.7899 & 0.0063 & 4.1943 \\ - 0.0063 & 0.0032 & 0.0134 \\ 4.1943 & 0.0134 & 6.4374 \end{matrix}]

and

Ω_{2} = [\begin{matrix} 1.2908 & - 0.0155 & 3.6842 \\ - 0.0155 & 0.0195 & 0.0093 \\ 3.6842 & 0.0093 & 2.7780 \end{matrix}]

From Algorithm 2 (co-design), the minimum $H_{\infty}$ performance $γ_{\min}$ is given as 1.4762. As shown in Table 2, $γ_{\min} = 1.4762$ , which is smaller than the 1.9 obtained by Yue et al. (2005) and 1.6242 obtained by Jiang et al. (2008). Therefore, for this example, the result from our method is less conservative than those from the other two existing methods by Jiang et al. (2008) and Yue et al. (2005).

Algorithm 2 (Co-design)
For a given h, $η_{m}$ , $η_{M}$ and pre-set $γ^{*}$ , determine $σ$ , $\tilde{p}$ , ${\tilde{M}}_{i}$ , ${\tilde{Q}}_{i}$ , ${\tilde{Z}}_{i}$ , ${\tilde{R}}_{i}$ and $Ω_{i}$ . It is desired that $γ$ is minimized.
Step 1: choose an expected value $\bar{γ} = γ^{*}$ .
Step 2: choose a sufficiently small initial $σ^{(0)} > 0$ and $σ = σ^{(0)}$ . Probe a suitable $ε$ as the step increment of $σ$ .
Step 3: if $σ \geq 1$ , go to step 8.
Step 4. for given positive scalars h, $η_{m}$ and $η_{M}$ , solve the LMI given by equation (44) from Theorem 2 with $γ = \bar{γ}$ . If there exists a feasible solution, determine $\tilde{p}$ , ${\tilde{M}}_{i}$ , ${\tilde{Q}}_{i}$ , ${\tilde{Z}}_{i}$ , ${\tilde{R}}_{i}$ and $Ω_{i}$ and go to step 5; otherwise reset $\bar{γ}$ , h, $η_{m}$ and $η_{M}$ , and repeat step 3.
Step 5: apply equation (18) to determine $σ^{}$ with the obtained $Ω_{i}$ , and set $σ = σ^{}$ .
Step 6: by using the solver feasp in the Matlab LMI Toolbox with the obtained $σ$ , ${\tilde{M}}_{i}$ , $\tilde{p}$ , ${\tilde{Q}}_{i}$ , ${\tilde{Z}}_{i}$ , ${\tilde{R}}_{i}$ and $Ω_{i}$ , return the global minimum solution $γ$ . Set $\bar{γ} = γ$ and $σ^{(0)} = σ$ , go to step 7.
Step 7: Update $σ = σ + ε$ , reset h, $η_{m}$ and $η_{M}$ , repeat step 3.
Step 8: Output $σ^{(0)}$ and $\bar{γ}$ .

Table 2.

The minimum $H_{\infty}$ performance from different methods with uncertain parameters for $τ_{m} = 0.1$ , $τ_{M} = 0.58$ and $σ = 0.1$ .

Method	Our work in this paper	Jiang et al. (2008)	Yue et al. (2005)
$γ_{\min}$	1.4762	1.6242	1.9

Selecting the sampling period $h = 0.1$ and considering the effect of the upper bound 0.5892 of the time delay, we obtain an average release period of 0.3769. The state response, release time instants and release intervals of the system are shown together with K, $Ω_{1}$ and $Ω_{2}$ in Figure 3(a) and (b).

Example 2. The inverted pendulum is considered in this example. It is modelled as a thin rod, and the surface is assumed to be friction free. Let y denote the position of the center of mass of the cart. Use M and m to represent the mass of the cart and the mass of the pendulum, respectively. l represents the length of the pendulum and $θ$ is the angle of the pendulum from vertical. Then, the equations of motion for this system ( Cannon,1967 ) are derived from Newton’s second law as

\begin{matrix} (M + m) \overset{\cdot\cdot}{y} + m l \overset{\cdot\cdot}{θ} \cos θ - m l {\overset{\cdot}{θ}}^{2} \sin θ = u \\ m l \overset{\cdot\cdot}{y} \cos θ + l \overset{\cdot\cdot}{θ} - g \sin θ = 0 \end{matrix}

The control of the inverted pendulum is to allow the motion of the cart to be dominated while maintaining the balance of the pendulum. In a practical control system, environmental noises are ubiquitous and the uncertainties in the parameters m, M and l inevitably exist in various disturbances ( Stoorvoge, 1992 ). The dynamics of the inverted pendulum of a cart have been modelled under noises and parametric uncertainties in the literature ( Peng and Yang, 2013 ; Tang et al., 2016 ; Yue et al., 2011 , 2013 )

\begin{matrix} \overset{\cdot}{x} (t) = ([\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & - \frac{m g}{M} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & \frac{g}{l} & 0 \end{matrix}] + Δ A (t)) x (t) \\ + ([\begin{matrix} 0 \\ \frac{1}{M} \\ 0 \\ - \frac{1}{M l} \end{matrix}] + Δ B (t)) u (t) + B_{ω} ω (t) \\ z (t) = (C + Δ C (t)) x (t) + (D + Δ D (t)) u (t) + D_{ω} ω (t) \end{matrix}

(50)

where parameter settings are $M = 10$ kg, $m = 1$ kg, $l = 3$ m and $g = 10 m / s^{2}$ , which is the gravitational acceleration. $x (t) = [x_{1}, x_{2}, x_{3}, x_{4}] = [y, \overset{\cdot}{y}, θ, \overset{\cdot}{θ}]$ is the state vector, the system’s initial state vector is $x_{0} = [\begin{matrix} 0.98 & 0 & 0.2 & 0 \end{matrix}]$ .

We investigate the system in equation (50) under parameter matrices in two cases with and without parameter uncertainties and disturbances, respectively.

Case 1. Consider the system given by equation (50) without uncertain parameters and disturbance, i.e. $Δ A (t) = Δ B (t) = Δ C (t) = Δ D (t) = 0$ and $ω (t) = 0$ . We have

A = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & \frac{10}{3} & 0 \end{matrix}], B = [\begin{matrix} 0 \\ 0.1 \\ 0 \\ - \frac{1}{30} \end{matrix}]

and set $C = [\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]$ and $D = 0$ . In order to compare our method with existing methods (Peng and Yang, 2013; Yue et al., 2011, 2013), the same feedback controller gain $K = [\begin{matrix} 2 & 12 & 378 & 210 \end{matrix}]$ from the work by Peng and Yang (2013) is used. Applying Theorem 1 with $h = 0.01$ and $σ = 0$ , we obtain a solution of the upper bound of the time delay $τ_{M} = 0.2112$ . As $η_{M} = h + τ_{M}$ , in order to reduce the system conservatism, the upper bound of time delay should be greater than or equal to the lower delay bound that must be nonzero. The maximum allowed sampling period is 0.2112, when $η_{m} = 0.01$ and $η_{M} = 0.2212$ . By Theorem 1 with $h = 0.01$ , $σ = 0.2$ and $γ = 200$ , we get $η_{M} = 0.1626$ and the event-triggered matrices

Ω_{1} = [\begin{matrix} 0.04964 & 1.0976 & 68.63 & 37.907 \\ 1.0976 & 24.284 & 1515.9 & 837.29 \\ 68.63 & 1515.9 & 96214 & 53135 \\ 37.907 & 837.29 & 53135 & 29344 \end{matrix}]

and

Ω_{2} = [\begin{matrix} 0.0078397 & 0.038208 & 0.5339 & 0.30332 \\ 0.038208 & 0.18661 & 2.6104 & 1.4829 \\ 0.5339 & 2.6104 & 37.041 & 21.03 \\ 0.30332 & 1.4829 & 21.03 & 11.941 \end{matrix}]

The upper bound of the time delay $τ_{M} = 0.1526$ . The state response of the system is shown in Figure 4(a). The release instants and release intervals are shown in Figure 4(b).

If $h = 0.01$ is kept unchanged, using Theorem 2 with $σ = 0.3$ gives the upper bound $η_{M} = 0.1451$ and the corresponding feedback gain $K = [\begin{matrix} 2.9816 & 14.647 & 382.13 & 213.42 \end{matrix}]$ , and weighting matrices

Ω_{1} = 10^{- 3} \times [\begin{matrix} 0.15595 & 0.64305 & 17.332 & 9.6766 \\ 0.64305 & 3.2826 & 82.784 & 46.387 \\ 17.332 & 82.784 & 2176.4 & 1215.6 \\ 9.6766 & 46.387 & 1215.6 & 679.86 \end{matrix}]

and

Ω_{2} = 10^{- 3} \times [\begin{matrix} 0.12413 & 0.46398 & 10.574 & 5.9215 \\ 0.46398 & 2.1272 & 45.592 & 25.614 \\ 10.574 & 45.592 & 1027.3 & 574.86 \\ 5.9215 & 25.614 & 574.86 & 322.19 \end{matrix}]

The closed-loop system is able to tolerate the transmission upper bound of the time delay $τ_{M} = 0.1351$ . The simulation result for $t \in [0, 30]$ shows that 19.6% of the sampled signals need to be sent to the controller. The average release period is 0.5456, which is larger than the results obtained from existing methods (Peng and Yang, 2013; Wang and Lemmon, 2008, 2009; Yue et al., 2013) as shown in Table 3. The state response of the system is shown in Figure 5(a). The release instants and release intervals are shown in Figure 5(b).

Table 3.

Average release period of $σ = 0.3$ by different methods.

Method	Average release period
Theorem 2	0.5456
Yue et al. (2013)	0.5131
Wang and Lemmon (2008)	0.4816
Peng and Yang (2013)	0.43
Wang and Lemmon (2009)	0.3917

For $h = 0.01$ and the simulation time $T = 30$ s, we choose $σ$ values in different event-triggered schemes. Table 4 compares the upper bounds of the time delay $τ_{M}$ and the packet transmission rates (PTRs) obtained from our method and existing methods (Peng and Yang, 2013; Yue et al., 2011, 2013). It is seen from Table 4 that the packet transmission rate from our method is smaller than those from existing methods. It is also seen that our method gives less conservative results than these existing methods.

Table 4.

Upper bounds of the time delay $τ_{M}$ and packet transmission rates from different methods under various $σ$ values.

Method	$σ$	0	0.1	0.2	0.3	0.5
In this paper	$τ_{M}$	0.2112	0.1707	0.1526	0.1351	0.0832
Yue et al. (2013)	$τ_{M}$	0.1989	—	0.11	—	—
Peng and Yang (2013)	$τ_{M}$	0.17	0.12	—	0.08	0.04
Yue et al. (2011)	$τ_{M}$	0.1989	—	—	0.13	—
In this paper	PTR	100%	28.2%	23.1%	19.6%	12.07%
Peng and Yang (2013)	PTR	100%	31%	—	22%	14%

Case 2. Consider the system given by equation (50) without uncertain parameters and disturbance. The uncertainties are described by equation (2) and the uncertain parameters are set to be

\begin{array}{l} G_{1} = d i a g {1, 1, 1, 1}, E_{1} = [\begin{matrix} 0 & 0.1 & 0 & 0 \\ 0 & 0 & - 0.1 & 0 \\ 0 & 0 & 0 & 0.1 \\ 0 & 0 & 0.1 & 0.1 \end{matrix}], \\ E_{2} = [\begin{matrix} 0 \\ 0.2 \\ 0 \\ 0 \end{matrix}], and G_{2} = [\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}] \end{array}

From $∥ F (t) ∥ \leq 1$ , we have uncertain matrices $‖ Δ A (t) ‖ \leq 0.2$ , $‖ Δ B (t) ‖ \leq 0.2$ . The disturbance is modelled as

w (t) = {\begin{matrix} s g n (\sin (t)), & t \in [0, 10] \\ 0, & otherwise \end{matrix}

A and B are the same as in Case 1. Set $C = [\begin{matrix} 1 & 1 & 1 & 1 \end{matrix}]$ and $D = 0.1$ . In addition, using Corollary 2 with $h = 0.01$ , $γ = 200$ and $σ = 0.1$ , we get the maximum allowed delay $η_{M} = 0.1896$ , the feedback gain $K = [\begin{matrix} 12.3027 & 26.3804 & 444.7712 & 249.8956 \end{matrix}]$ , and weighting matrices

Ω_{1} = [\begin{matrix} 3.8449 & 10.517 & 207.71 & 116.33 \\ 10.517 & 28.977 & 573.55 & 321.19 \\ 207.71 & 573.55 & 11411 & 6389.5 \\ 116.33 & 321.19 & 6389.5 & 3577.8 \end{matrix}]

and

Ω_{2} = [\begin{matrix} 5.4388 & 14.827 & 296.09 & 165.79 \\ 14.827 & 40.828 & 817.22 & 457.52 \\ 296.09 & 817.22 & 16459 & 9213.2 \\ 165.79 & 457.52 & 9213.2 & 5157.2 \end{matrix}]

The state response of the system is shown in Figure 6(a), and the release instants and release intervals are shown in Figure 6(b). The average release period is obtained to be 0.4763 when $h = 0.1796$ . Table 5 compares the upper bounds of the time delay $τ_{M}$ obtained from our method and other existing methods. It is seen from this table that the $τ_{M}$ values obtained from our method are larger than those from previous works (Peng and Yang, 2013; Yue et al., 2011, 2013). This means that Corollary 2 leads to less conservative results.

Table 5.

Case 2 of Example 2: upper bounds of the time delay $τ_{M}$ from different methods for for $h = 0.01$ , $σ = 0.1$ and $γ = 200$ and various $τ_{m}$ values.

Method	$τ_{m}$	0.01	0.05	0.09	0.13	0.17	0.19	0.21
In this paper	$τ_{M}$	0.1796	0.1832	0.1879	0.1928	0.2015	0.2053	0.2111
Yue et al. (2013)	$τ_{M}$	0.16	0.1721	0.1805	0.1834	0.1851	—	—
Yue et al. (2011)	$τ_{M}$	0.144	0.1563	0.1631	0.1706	0.1722	—	—
Peng and Yang (2013)	$τ_{M}$	0.13	0.1451	0.1526	0.1595	—	—	—

Using Corollary 2 with $h = 0.01$ , $γ = 200$ and $τ_{m} = 0.01$ , according to values of various $σ$ , the diverse upper bound of time delay $τ_{M}$ are listed in Table 6. Under Algorithm 2, the different $γ_{\min}$ can be obtained about values of various $σ$ . but the upper bounds of $τ_{M}$ are gradually reduced when values of $γ_{\min}$ gradually increase.

Table 6.

Case 2 of Example 2: upper bounds of the time delay $τ_{M}$ and minimum $H_{\infty}$ performance.

$σ$	0	0.1	0.2	0.3	0.4	0.5
$τ_{M}$	0.1903	0.1796	0.1576	0.1482	0.1391	0.1256
$γ_{\min}$	21.35	28.15	35.62	46.37	68.73	196.98

Now, we select the desired performance $γ = 50$ , $h = 0.01$ and $σ = 0.2$ , and consider the effect of the transmission delays $τ_{m} = 0.01$ and $τ_{M} = 0.1$ . the average release period is obtained to be 0.3926, which is larger than the 0.3098 obtained from the work by Yue et al. (2013). The state response, release instants and release interval of the system are shown in Figure 7(a) and (b).

The distribution of the average transmission interval at 4 s, 8 s, 12 s … 30 s is shown in Figure 8. The average transmission intervals fluctuate in a small range when the closed-loop event-triggered feedback system converges asymptotically to its equilibrium.

Conclusion

A new event-triggered discrete-time control scheme has been presented in this paper for NCSs with a linear continuous-time plant. By employing a Wirtinger-based inequality and designing an appropriate augmented Lyapunov–Krasovskii functional, the closed-loop system has been made asymptotically stable. Sufficient conditions expressed in the LMI constraints have been derived for the NCSs. From the conditions, a co-design algorithm has been designed for our event-triggered communication scheme. The minimum $H_{\infty}$ performance level has been achieved by solving the convex optimization problem, and the $H_{\infty}$ controller has also been obtained. Results from numerical examples have shown that our stability criteria presented in this paper are less conservative than those from existing methods, and the resulting communication scheme reduces the utilization of the NCS network resources greatly.

Footnotes

Appendix

Declaration of Conflicting Interest

The authors declare that there is no conflict of interest.

Funding

This work was supported in by the Project of Science and Technology Commission of Shanghai Municipality (grant numbers 5220710400,14JC1402200,15JC1401900), the National Science Foundation of China (grant numbers 61633016,61473182,61533010), and the Australian Council (ARC) under the Discovery Projects Scheme to Y-C Tian (grant number DP170103305).

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Event-triggered robust H ∞ control for uncertain networked control systems with time delay

Abstract

Keywords

Introduction

System description and problem statement

System plant

Event-triggered data transmission protocol

ZOH

Modelling of NCSs with ETDTP

Stability analysis of the event-triggered closed-loop system

Robust H ∞ performance analysis for uncertain systems

Event-triggered H ∞ controller design for uncertain systems

Simulation experiments

Conclusion

Footnotes

Appendix

Declaration of Conflicting Interest

Funding

References

Robust $H_{\infty}$ performance analysis for uncertain systems

Event-triggered $H_{\infty}$ controller design for uncertain systems