Non-uniform sampling finite-time control for networked control systems via event-driven transmission

Abstract

This article investigates the networked finite-time control problem for a class of continuous-time linear systems. A novel event-driven data transmission scheme is invoked to reduce the network loads. Moreover, the non-uniform sampling triggered by sampling jitter or the fault of hold circuits is taken into account. By employing an input-delay approach, the underlying system is first reformulated to linear system with time-varying delays. Based on the Lyapunov–Krasovskii functional method, the stabilizing controller is then designed in the finite-time control sense, such that the closed-loop system is finite-time bounded. Potential and advantage of the developed techniques are demonstrated by a numerical example and an application to damping boring bar system.

Introduction

In the past decades, networked control systems (NCSs) have attracted increasing attention due to its advantages in shortening the installation period and facilitating the system diagnosis (see Gao et al., ¹ Fridman, ² Xu et al., ³ and Antsaklis and Baillieul ⁴ and references therein). For NCSs, data sampling and transmission are regarded as two main implementation procedures, where the data sampling is commonly assumed to be periodical.^5–7 However, owing to the existence of time-varying nature of the channel load or sporadic faults, periodic sampling is hard to perform, which motivates many authors to focus on the study of non-uniform sampling. ³ In addition, although the conventional scheme can employ the advantages of network-based implementation approach, it may also bring about the severe over-provisioning of NCSs. ⁸ Hence, how to reduce the transmission loads of network while guaranteeing the required dynamic performance has become a hot research topic recently. ⁸

To mitigate the otiose communication of the sampling data and reduce the transmission-network loads, event-driven control approach is developed.^9–11 Compared with conventional schemes of transmitting all the sampled data, the event-driven control technique can eliminate some trivial sampled data, that is, only the sampled data can be transmitted after the sampling implementation when its value satisfies some prescribed event-driven condition. Currently, some event-driven/self-triggered control strategies have been reported. In studies by Hristu-Varsakelis and Kumar ¹² and Heemels et al., ¹³ an event-driven control scheme is developed when the prescribed event is triggered; however, the hard ware–dependent event detector will be destroyed; another event-driven control scheme in Peng and Han ⁸ and Peng and Yang ¹¹ is developed, but the considered context is just periodic (uniform). Combination of both the non-uniform data sampling and event-driven data transmission for the NCSs has been seldom studied and few results have been published.

On the other hand, common investigations on the controller design of NCSs mainly focus on Lyapunov asymptotic stability (LAS), which is defined in the infinite-time interval.^14,15 For many engineering applications, however, the dynamical performance analysis in a finite-time interval may be paid more attention to. Specifically, the dynamical performance of the system should satisfy a given threshold in a fixed finite-time interval (i.e. finite-time boundedness). Potential examples include the boring bar control system ¹⁶ and vehicle suspension system. ¹⁷ So far, some attempts on the finite-time control have been conducted.^18–25 However, to the best of authors’ knowledge, no results for utilizing the event-driven communication scheme are reported in the finite-time control sense, which motivates us for this study.

In this article, we aim at addressing the finite-time control problem for NCSs with non-uniform sampling and event-driven communication scheme. The main contributions of the article are twofold. First, the event-driven controller design problem for NCSs is investigated in the finite-time boundedness sense. Second, a co-design algorithm is presented to obtain the optimal parameter of event-driven transmission scheme and the networked finite-time controller. The remainder of this article is organized as follows. Section “Preliminaries and problem formulation” introduces the studied model and outlines the problems statement. In section “Event-triggered sampled-data controller design,” a new continuous-time linear system is formulated, for which the sufficient criteria on finite-time boundedness are first derived and then the finite-time controller is designed. Moreover, a co-design algorithm for data communication and control design procedure is provided to find an optimal communication scheme for the lowest deployment costs. Section “Simulations” is devoted to testify the potential of the developed results by a numerical example and an application to boring bar system. Section “Conclusion” concludes this study.

Notation

${\mathrm{ℝ}}^n$ and ${\mathrm{ℝ}}^{m\times n}$ refer to, respectively, the n dimensional Euclidean space and the set of all $m\times n$ real matrices; $\Vert ·\Vert \mathrm{refers}$ to the Euclidean vector norm. $\mathrm{\mathbb{N}}$ stands for the set of nonnegative integers. In addition, the set of $n\times n$ symmetric positive definite matrices is denoted by ${\wp }_{\succ 0}^n$ . In addition, in symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry and $\mathrm{diag}\{\cdots \}$ stands for a block-diagonal matrix. For a square matrix P, ${\lambda }_{max}(P)$ and ${\lambda }_{min}(P)$ denote its maximum and minimum eigenvalue, respectively. sym $\{A\}$ is the shorthand notation for $A+A^T$ . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Preliminaries and problem formulation

Consider a class of continuous-time linear systems described by

$$\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{1}F(t,x\left(t\right))\hfill \\ \multicolumn{1}{c}{y\left(t\right)=\mathrm{Cx}\left(t\right)}\end{array}$$

(1)

where $x(t)\in {\mathrm{ℝ}}^{n_x}$ and $y(t)\in {\mathrm{ℝ}}^{n_z}$ denote the state vector and controlled output, respectively; $F(t)\in {\mathrm{ℝ}}^{n_w}$ is a nonlinear function which denote the exogenous disturbance while $u(t)\in {\mathrm{ℝ}}^{n_u}$ represents the control input. $A,B,B_1,C$ are the system matrices with appropriate dimensions.

Under the network-based transmission framework, we aim to design the state-feedback sampled-data control as follows

$$u\left(t\right)=\mathrm{Kx}\left(t_k\right),t\in [t_k,t_{k+1})$$

where $t_k$ represents the instant of successfully receiving the transmission data. The sampling of the state value $x(t)$ is non-uniform while the transmission is event-driven, that is, the sampling period is uncertain and the data transmission will be determined by the following event-driven scheme.

Proposition 1 (Event-driven transmission scheme)

Given a positive definite matrix $\Phi \succ 0$ , consider system (1) with non-uniform sampling and the event-driven communication scheme. ¹¹ The data transmission will be triggered if

$$t_{k+1}=\underset{\ell }{\min }\left\{t_{k,\ell }|e^T\left(t_{k,\ell }\right)\Phi e\left(t_{k,\ell }\right)\ge \delta x^T\left(t_k\right)\Phi x\left(t_k\right),t_{k,\ell }\ge t_k\right\}$$

(2)

where $\delta \ge 0$ is a prescribed index, $t_{k,\ell }$ denotes the $\ell \mathrm{th}$ sampling after the $k\mathrm{th}$ transmission, $e(t_{k,\ell })\stackrel{\Delta }{=}x(t_{k,\ell })-x(t_k)$ with $\ell \in \mathrm{\mathbb{N}}$ (see Figure 1).

Figure 1.

Networked sampled-data damping boring bar with non-uniform sampling and event-triggered communication.

To proceed further, the following assumptions and definitions are introduced for the development of the main results.

Assumption 1

$F(t,x(t))$ denotes the exogenous disturbance variable input, which belongs to Lipschitz space, that is, for any two given constants $t_1$ and $t_2$ , the following inequality holds

$$|F(t_1,x(t_1))-F(t_2,x(t_2))|\le L|t_1-t_2|$$

where L is a known positive constant.

Assumption 2

The sampling period is bounded by h $(h>0)$ , which yields that

$$0<t_{k,\ell +1}-t_{k,\ell }\le h$$

Remark 1

Assumption 1 describes the property of the exogenous disturbance variable input $F(t,x(t))$ in function (1), which belongs to Lipschitz space. This assumption widely exists in many practical engineering and is commonly used in many existing literature, such as trigonometric function; assumption gives the thresholds the non-uniform periods, which are generally known a priori.

Definition 1

Given the positive constants $c_1,c_2,T_f$ with $c_1\ge c_2$ and matrix $R>0$ , system (1) is said to be finite-time bounded (FTB) with respect to $(c_1,c_2,R,T_f)$ , if $\forall t\in [T_f,\infty )$

$$y^T\left(t_0\right)\mathrm{Ry}\left(t_0\right)\le c_1\Rightarrow y^T\left(t\right)\mathrm{Ry}\left(t\right)\le c_2$$

(3)

Remark 2

In the existing publications,^18,20 the objective of the finite-time control is to keep the state evolution of the system remain below a given bound in a fixed interval if the initial state values satisfy some prescribed conditions. Here, we extend the definition to another context, that is, applying finite-time control to keep the output evolution of the system always remain below a given bound after a fixed interval if the initial output values satisfy some prescribed condition. This definition is very practical for some engineering projects, such as boring bar system and vehicle suspension system.^16,17

Thus, our objective in this article is to design a state-feedback sampled-data controller K under the event-driven transmission framework, such that the resulting closed-loop system (1) is FTB.

Event-triggered sampled-data controller design

In this section, we present the sufficient conditions of finite-time boundedness for system (1), upon which the state-feedback control is designed. Previously, an input-delay approach is first employed to reformulate the system (1) for proceeding further.

Key idea

Denote a set of sampling instants between two successive event-triggered instants $t_k$ and $t_{k+1}$ of system as ${\Omega }_{t_k,t_{k+1})}\stackrel{\Delta }{=}\{t_{k,\ell }|t_{k,\ell }=\sum _{j=1}^{\ell }{h}_{k,j}+t_k,\ell \in \mathrm{\mathbb{N}}\}$ , where $h_{k,j}$ denotes the $j\mathrm{th}$ sampling period after $k\mathrm{th}$ data transmission. Following Assumption 2 yields that $0<h_{k,j}\le h$ . According to the prescribed event-driven transmission scheme in equation (2), $\forall t\in [t_{k,\ell },t_{k,\ell +1})$ , it yields that $x(t-d(t))=x(t_{k,\ell })=x(t_k)+e(k,\ell h)$ , where $d(t)=t-t_{k,\ell }$ and $0<d(t)\le h_{k,\ell +1}\le h$ . System (1) can be then redescribed by

$$\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{BK}(x(t-d\left(t\right))-e\left(\ell h\right))+B_{1}F(t,x\left(t\right))\hfill \\ \multicolumn{1}{c}{y\left(t\right)=\mathrm{Cx}\left(t\right)}\\ \multicolumn{1}{c}{x\left(\epsilon \right)=\varphi \left(\epsilon \right),\epsilon \in [-h,0]}\end{array}$$

(4)

Remark 3

Based on the input-delay approach, the continuous-time time-delay closed-loop system (4) is derived from equation (1). This technique was first introduced by Miheev et al. ²⁶ and has been widely used to solve the sampled-data control/filtering problems.^1,14 However, most of the publications mentioned are concerned with the case that the sampling and transmission are synchronous, that is, the sampled data will be immediately sent to the controller via network when sampling is done. However, some of the communication data are actually ineffective to guarantee the required system performance while occupying the network sources. In this note, whether the sampled data are transmitted or not will be determined by the prescribed event-driven condition, which is different with the existing sampling and communication scheme. Employing this scheme, some ineffective data communication will be eliminated and the network loads will be reduced.

Assume that the control gain K in system (4) is known a priori, we shall present the sufficient conditions under which the sampled-data closed system (4) is FTB with the event-driven transmission scheme.

Lemma 1

Consider system (4) and let $\varsigma ,{\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4$ be given positive constants with $\varsigma \in [\overline{\varsigma },\infty )$ . Suppose that there exist matrices $P\succ 0$ , $Q\succ 0$ , $Z\succ 0$ , $\Lambda \succ 0$ , and $\mathcal{W}$ with appropriate dimensions such that

$$\Im \prec 0$$

(5)

$${\rho }_{1}I⪯P,{\rho }_{2}I⪰P,{\rho }_{3}I⪰Q,{\rho }_{4}I⪰Z$$

(6)

and

$${\kappa }_1-{\kappa }_2-\varsigma T_f<0$$

(7)

where

$$\Im \stackrel{\Delta }{=}\left[\begin{array}{cc}{\Im }_1\hfill & {\Im }_2\hfill \\ \multicolumn{1}{c}{*}& {\Im }_3\hfill \end{array}\right],{\Im }_1\stackrel{\Delta }{=}\left[\begin{array}{cc}{\mathcal{X}}_1\hfill & {\mathcal{X}}_2\hfill \\ \multicolumn{1}{c}{*}& {\mathcal{X}}_3\hfill \end{array}\right]$$

$${\Im }_2\stackrel{\Delta }{=}\left[\begin{array}{cc}\mathcal{W}\hfill & \aleph \hfill \end{array}\right],{\Im }_3\stackrel{\Delta }{=}\mathrm{diag}\{-e^{\varsigma h}{h}^{-1}Z,-h^{-1}Z\}$$

$${\mathcal{X}}_1\stackrel{\Delta }{=}\left[\begin{array}{cc}\psi \hfill & \mathrm{PBK}+e^{-\varsigma h}{W}_2^T-e^{-\varsigma h}{W}_1\hfill \\ \multicolumn{1}{c}{*}& \delta \Phi -e^{-\varsigma h}\mathrm{sym}\left\{W_2\right\}\hfill \end{array}\right]$$

$${\mathcal{X}}_2\stackrel{\Delta }{=}\left[\begin{array}{ccc}{e}^{-\varsigma h}{W}_3^T\hfill & -\mathrm{PBK}+e^{-\varsigma h}{W}_4^T\hfill & P{B}_1+e^{-\varsigma h}{W}_5^T\hfill \\ \multicolumn{1}{c}{-e^{-\varsigma h}{W}_3^T}& -\delta \Phi -e^{-\varsigma h}{W}_4^T\hfill & -e^{-\varsigma h}{W}_5^T\hfill \end{array}\right]$$

$${\mathcal{X}}_3\stackrel{\Delta }{=}\mathrm{diag}\{-e^{-\varsigma h}Q,(\delta -1)\Phi ,-\Lambda \}$$

$$\mathcal{W}\stackrel{\Delta }{=}[\begin{array}{ccccc}{W}_1^T\hfill & W_2^T\hfill & W_3^T\hfill & W_4^T\hfill & W_5^T\hfill \end{array}{]}^T$$

$$\aleph \stackrel{\Delta }{=}[\begin{array}{ccccc}\mathrm{ZA}\hfill & \mathrm{ZBK}\hfill & 0\hfill & -\mathrm{ZBK}\hfill & Z{B}_1\hfill \end{array}{]}^T$$

$$\begin{array}{l}\psi \triangleq \mathrm{sym}\left\{A^{T}P+e^{-\varsigma h}{W}_1\right\}+Q+\varsigma P+L^T\Lambda L\mathrm{,}\\ \overline{\varsigma }\triangleq \frac{\ln {\kappa }_1-\ln {\kappa }_2}{T_f}\end{array}$$

$$\begin{array}{l}{\kappa }_1\triangleq c_1{\lambda }_2\left({\rho }_2+{\rho }_1{\rho }_3+{\rho }_2{\rho }_4\right)\mathrm{,}\\ {\kappa }_2\triangleq c_2{\lambda }_1{\rho }_1\mathrm{,}{\rho }_1\triangleq h{e}^{\alpha h}\mathrm{,}{\rho }_2\triangleq 0.5h^2{e}^{\alpha h}\end{array}$$

Then, system (1) is FTB with respect to $(c_1,c_2,R,T_f)$ .

Proof

Consider the Lyapunov–Krasovskii functional candidate expressed by

$$V(t,x\left(t\right))=V_1(t,x\left(t\right))+V_2(t,x\left(t\right))+V_3(t,x\left(t\right))$$

(8)

where

$$V_1(t,x\left(t\right))\stackrel{\Delta }{=}{x}^T\left(t\right)\mathrm{Px}\left(t\right)$$

$${\stackrel{·}{V}}_2(t,x\left(t\right))\stackrel{\Delta }{=}\underset{t-h}{\overset{t}{\int }}{e}^{-\varsigma (t-s)}{x}^T\left(s\right)\mathrm{Qx}\left(s\right)\mathrm{ds}$$

$$V_3(t,x\left(t\right))\stackrel{\Delta }{=}\underset{-h}{\overset{0}{\int }}\underset{t+\theta }{\overset{t}{\int }}{e}^{-\varsigma (t-s)}{\stackrel{·}{x}}^T\left(s\right)Z\stackrel{·}{x}\left(s\right)\mathrm{ds}$$

Taking the full derivation of $V(t)$ with the respect to t leads to

$${\stackrel{·}{V}}_1(t,x\left(t\right))=2x^T\left(t\right)P\stackrel{·}{x}\left(t\right)$$

$${\stackrel{.}{V}}_2\left(t,x\left(t\right)\right)=-\varsigma {\stackrel{.}{V}}_2\left(t,x\left(t\right)\right)+x^T\left(t\right)Qx\left(t\right)-e^{-\varsigma h}{x}^T\left(t-h\right)Qx\left(t-h\right)$$

$${\stackrel{.}{V}}_3\left(t,x\left(t\right)\right)=-\varsigma {\stackrel{.}{V}}_3\left(t,x\left(t\right)\right)+h{\stackrel{.}{x}}^T\left(t\right)Z\stackrel{.}{x}\left(t\right)-{\displaystyle \underset{t-h}{\overset{t}{\int }}{e}^{-\varsigma \left(t-s\right)}{\stackrel{.}{x}}^T}\left(s\right)Z\stackrel{.}{x}\left(s\right)ds$$

$$\le -\varsigma {\stackrel{·}{V}}_3(t,x\left(t\right))+h{\stackrel{·}{x}}^T\left(t\right)Z\stackrel{·}{x}\left(t\right)-e^{-\varsigma h}\underset{t-h}{\overset{t}{\int }}{\stackrel{·}{x}}^T\left(s\right)Z\stackrel{·}{x}\left(s\right)\mathrm{ds}$$

Denote ${\xi }^T(t)\stackrel{\Delta }{=}\begin{array}{ccccc}{x}^T(t)\hfill & x^T(t-d(t))\hfill & x^T(t-h)\hfill & e^T(\ell h)\hfill & F^T(t)\hfill \end{array}]$ and based on the Newton–Leibniz formula, it obtains that

$$X_w\stackrel{\Delta }{=}{e}^{-\varsigma h}{\xi }^T\left(t\right)W(x\left(t\right)-x(t-d\left(t\right))-\underset{t-d\left(t\right)}{\overset{t}{\int }}{\stackrel{·}{x}}^T\left(s\right)\mathrm{ds})=0$$

Moreover, by the definition of event-driven transmission scheme in Proposition 1, we have $e^T(t_{k,\ell })\Phi e(t_{k,\ell })<\delta x^T(t_k)\Phi x(t_k)$ , $t_{k,\ell }\in {\Omega }_{t_k,t_{k+1})}$ . Then, one has that

$$\begin{array}{c}{e}^T\left(t_{k,\ell }\right)\Phi e\left(t_{k,\ell }\right)\le \delta x^T\left(t_k\right)\Phi x\left(t_k\right)\hfill \\ =\delta {(x(t-d\left(t\right))-e\left(t_{k,\ell }\right))}^T\Phi (x(t-d\left(t\right))-e\left(t_{k,\ell }\right))\hfill \\ =\delta x^T(t-d\left(t\right))\Phi x(t-d\left(t\right))-\delta x^T(t-d\left(t\right))\Phi e\left(t_{k,\ell }\right)\hfill \\ -\delta e^T\left(t_{k,\ell }\right)\Phi x(t-d\left(t\right))+\delta e^T\left(t_{k,\ell }\right)\Phi e\left(t_{k,\ell }\right)\hfill \end{array}$$

which yields that

$$\begin{array}{c}\stackrel{·}{V}(t,x\left(t\right))+\varsigma V(t,x\left(t\right))\hfill \\ \le 2x^T\left(t\right)P\stackrel{·}{x}\left(t\right)+x^T\left(t\right)\mathrm{Qx}\left(t\right)-x^T(t-h)e^{-\varsigma h}\mathrm{Qx}(t-h)\hfill \\ +h{\stackrel{·}{x}}^T\left(t\right)Z\stackrel{·}{x}\left(t\right)\hfill \\ -e^{-\varsigma h}\underset{t-d\left(t\right)}{\overset{t}{\int }}{\stackrel{·}{x}}^T\left(s\right)Z\stackrel{·}{x}\left(s\right)\mathrm{ds}-e^T\left(\ell h\right)\Phi e\left(\ell h\right)+2X_w\hfill \\ \le {\xi }^T\left(t\right)\Xi \xi \left(t\right)-e^{-\varsigma h}\hfill \\ \underset{t-d\left(t\right)}{\overset{t}{\int }}({\xi }^T\left(t\right)W+{\stackrel{·}{x}}^T\left(s\right)Z)Z^{-1}(W^T\xi \left(t\right)+Z\stackrel{·}{x}\left(s\right))\mathrm{ds}\hfill \end{array}$$

where

$$\Xi \stackrel{\Delta }{=}{\Xi }_1+e^{-\varsigma h}\mathrm{sym}\left\{{\Xi }_2\right\}+h{e}^{-\varsigma h}W{Z}^{-1}{W}^T$$

with

$$x^T\left(t\right)L^T\Lambda \mathrm{Lx}\left(t\right)-F\left(t\right)\Lambda F\left(t\right)>0$$

$${\Xi }_1\triangleq \left[\begin{array}{ccccc}{\phi }_1& {\phi }_2& 0& -PB-h{A}^{T}ZB& P{B}_1+h{A}^{T}Z{B}_1\\ \ast & {\phi }_3& 0& -h{B}^{T}ZB-\delta \Phi & h{B}^{T}Z{B}_1\\ \ast & \ast & -e^{-\varsigma h}Q& 0& 0\\ \ast & \ast & \ast & \left(\delta -1\right)\Phi +h{B}^{T}ZB& -h{B}^{T}Z{B}_1\\ \ast & \ast & \ast & \ast & h{B}_1^{T}Z{B}_1-\Lambda \end{array}\right]$$

$${\Xi }_2\stackrel{\Delta }{=}[W_1{W}_2{W}_3{W}_4{W}_5]$$

$${\phi }_1\stackrel{\Delta }{=}{A}^{T}P+\mathrm{PA}+Q+\varsigma P+h{A}^T\mathrm{ZA}+L^T\Lambda L$$

$${\phi }_2\stackrel{\Delta }{=}\mathrm{PB}+h{A}^T\mathrm{ZB},{\phi }_3\stackrel{\Delta }{=}h{B}^T\mathrm{ZB}+\delta \Phi$$

Thus, we have

$$V(t,x\left(t\right))\le e^{-\varsigma (t-t_0)}V(t,x\left(t_0\right))$$

When $t>T$ , it holds that

$$V(t,x\left(t\right))\le e^{-\varsigma T}V(t,x\left(t_0\right))$$

which implies that the system is Lyapunov asymptotic stable.

Additionally, following from (8) implies that

$$\begin{array}{c}\multicolumn{1}{c}{V(t_0,x\left(t_0\right))\le \frac{{\lambda }_{max}\left(P\right)}{{\lambda }_{min}\left(R\right)}{x}^T\left(t_0\right)\mathrm{Rx}\left(t_0\right)}\\ +\underset{t-h}{\overset{t}{\int }}{e}^{-\varsigma (t-s)}\mathrm{ds}\frac{{\lambda }_{max}\left(Q\right)}{{\lambda }_{min}\left(R\right)}{x}^T\left(t_0\right)\mathrm{Rx}\left(t_0\right)\hfill \\ +\underset{-h}{\overset{0}{\int }}\underset{t+\theta }{\overset{t}{\int }}{e}^{-\varsigma (t-s)}\mathrm{ds}\frac{{\lambda }_{max}\left(T\right)}{{\lambda }_{min}\left(R\right)}{\stackrel{·}{x}}^T\left(t_0\right)R\stackrel{·}{x}\left(t_0\right)\hfill \\ \le ({\rho }_2+{\rho }_1{\rho }_3+{\rho }_2{\rho }_4)\frac{c_1}{{\lambda }_1}\hfill \end{array}$$

(9)

Besides

$$V(t,x\left(t\right))\ge \frac{{\lambda }_{min}\left(P\right)}{{\lambda }_{max}\left(R\right)}{x}^T\left(t\right)\mathrm{Rx}\left(t\right)\ge \frac{{\rho }_1}{{\lambda }_2}{x}^T\left(t\right)\mathrm{Rx}\left(t\right)$$

(10)

Then, $\forall t>T_f$ we have

$$\begin{array}{c}{x}^T\left(t\right)\mathrm{Rx}\left(t\right)\le \frac{{\lambda }_{2}V(t,x\left(t\right))}{{\rho }_1}\le \frac{{\lambda }_2}{{\rho }_1}{e}^{-\varsigma T}V(t,x\left(t_0\right))\hfill \\ \le e^{-\varsigma T}\frac{{\lambda }_2}{{\rho }_1}({\rho }_2+{\rho }_1{\rho }_3+{\rho }_2{\rho }_4)\frac{c_1}{{\lambda }_1}\le c_2\hfill \end{array}$$

That is, the underlying system is FTB. This completes the proof.

Remark 4

Sufficient conditions of finite-time boundedness for system (1) are obtained by the known control gain K. Moreover, when the controller gain is unknown and needs to be designed, a feasible control design procedure is investigated in the following section.

Finite-time controller design

The following theorem presents the linear matrix inequality (LMI)-based existence conditions for the finite-time controller design of system (4).

Theorem 1

Consider system (4) and let $\varsigma ,{\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4$ be given positive constants with $\varsigma \in [\overline{\varsigma },\infty )$ . Suppose that there exist matrices $S\succ 0$ , $T\succ 0$ , $V\succ 0$ , $\overline{\Lambda }\succ 0$ , and $\overline{\mathcal{W}}$ with appropriate dimensions such that

$$\wp <0$$

(11)

$${\rho }_{1}I⪰S,{\rho }_{2}I⪯S,{\rho }_{3}I⪯T,{\rho }_{4}I⪯V$$

(12)

and

$${\kappa }_1-{\kappa }_2-\varsigma T_f<0$$

(13)

where

$$\wp \stackrel{\Delta }{=}\left[\begin{array}{cc}{\wp }_1\hfill & {\wp }_2\hfill \\ \multicolumn{1}{c}{*}& {\wp }_3\hfill \end{array}\right],{\wp }_1\stackrel{\Delta }{=}\left[\begin{array}{cc}{\mathcal{M}}_1\hfill & {\mathcal{M}}_2\hfill \\ \multicolumn{1}{c}{*}& {\mathcal{M}}_3\hfill \end{array}\right]$$

$${\wp }_2\stackrel{\Delta }{=}{\left[\begin{array}{ccccc}\overline{\mathcal{W}}\hfill & {\mathcal{F}}_1\hfill & {\mathcal{F}}_2\hfill & {\mathcal{F}}_3\hfill & {\mathcal{F}}_4\hfill \end{array}\right]}^T$$

$${\wp }_3\stackrel{\Delta }{=}\mathrm{diag}\{e^{\varsigma h}{h}^{-1}(V-2S),-h^{-1}V,-\delta \Phi ,-T,-\overline{\Lambda }\}$$

$${\mathcal{M}}_1\stackrel{\Delta }{=}\left[\begin{array}{cc}{\overline{\phi }}_1\hfill & {\overline{\phi }}_2\hfill \\ \multicolumn{1}{c}{*}& -e^{-\varsigma h}\mathrm{sym}\left\{{\overline{W}}_2\right\}\hfill \end{array}\right]$$

$${\mathcal{M}}_2\stackrel{\Delta }{=}\left[\begin{array}{ccc}{e}^{-\varsigma h}{\overline{W}}_3^T\hfill & e^{-\varsigma h}{\overline{W}}_4^T-\mathrm{BY}\hfill & e^{-\varsigma h}{\overline{W}}_5^T+B_1\hfill \\ \multicolumn{1}{c}{-e^{-\varsigma h}{\overline{W}}_3^T}& -e^{-\varsigma h}{\overline{W}}_4^T\hfill & -e^{-\varsigma h}{\overline{W}}_5^T\hfill \end{array}\right]$$

$${\mathcal{M}}_3\stackrel{\Delta }{=}\mathrm{diag}\{e^{-\varsigma h}(T-2S),{\Phi }^{-1}-2S,\overline{\Lambda }-2I\}$$

$$\overline{\mathcal{W}}\stackrel{\Delta }{=}{\left[\begin{array}{ccccc}{\overline{W}}_1^T\hfill & {\overline{W}}_2^T\hfill & {\overline{W}}_3^T\hfill & {\overline{W}}_4^T\hfill & {\overline{W}}_5^T\hfill \end{array}\right]}^T$$

$${\mathcal{F}}_1\stackrel{\Delta }{=}{\left[\begin{array}{ccccc}\mathrm{AS}\hfill & \mathrm{BY}\hfill & 0\hfill & -\mathrm{BY}\hfill & B_1\hfill \end{array}\right]}^T$$

$${\mathcal{F}}_2\stackrel{\Delta }{=}{\left[\begin{array}{ccccc}0\hfill & \delta \Phi S\hfill & 0\hfill & -\delta \Phi S\hfill & 0\hfill \end{array}\right]}^T$$

$${\mathcal{F}}_3\stackrel{\Delta }{=}{\left[\begin{array}{cccccc}S\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$$

$${\mathcal{F}}_4\stackrel{\Delta }{=}{\left[\begin{array}{cccccc}\mathrm{LS}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$$

$${\overline{\phi }}_1\stackrel{\Delta }{=}\mathrm{sym}\left\{S{A}^T\right\}+\varsigma S+e^{-\varsigma h}\mathrm{sym}\left\{{\overline{W}}_1\right\}$$

$${\overline{\phi }}_2\stackrel{\Delta }{=}\mathrm{BY}-e^{-\varsigma h}{\overline{W}}_1+e^{-\varsigma h}{\overline{W}}_2^T,\overline{\varsigma }\stackrel{\Delta }{=}\frac{\ln {\kappa }_1-\ln {\kappa }_2}{T_f}$$

$$\begin{array}{l}{\kappa }_1\triangleq c_1{\lambda }_2\left(1/{\rho }_2+{\rho }_1/{\rho }_3+{\rho }_2/{\rho }_4\right)\mathrm{,}\\ {\kappa }_2\triangleq c_2{\lambda }_1/{\rho }_1\mathrm{,}{\rho }_1\triangleq h{e}^{\alpha h}\mathrm{,}{\rho }_2\triangleq 0.5h^2{e}^{\alpha h}\end{array}$$

Then, system (1) is FTB with respect to $(c_1,c_2,R,T_f)$ . Moreover, if equations (11)–(13) have a solution, the state-feedback controller gain can be given by

$$K=Y{S}^{-1}$$

(14)

Proof

Denoting $J\stackrel{\Delta }{=}\mathrm{diag}\{P^{-1},P^{-1},P^{-1},P^{-1},I,P^{-1},Z^{-1}\}$ and performing a congruence transformation to equation (5) by J, together with the definitions of $S\stackrel{\Delta }{=}{P}^{-1}$ , $V\stackrel{\Delta }{=}{Z}^{-1}$ , $T\stackrel{\Delta }{=}{Q}^{-1}$ , $\overline{\Lambda }\stackrel{\Delta }{=}{\Lambda }^{-1}$ , $Y\stackrel{\Delta }{=}\mathrm{KS}$ , $\overline{W}\stackrel{\Delta }{=}{\mathrm{SW}}_1^T{\mathrm{SSW}}_2^{T}S$ $S{W}_3{\mathrm{SSW}}_4^T{\mathrm{SSW}}_5^T{]}^T$ , and $-S{T}^{-1}S⪯T-2S,-S\Phi S⪯{\Phi }^{-1}-2S,-S{V}^{-1}S⪯V-2S$ , then the obtained equality can be well guaranteed by equation (11). Thus, based on Lemma 1, we can say that the finite-time boundedness of system (4) is ensured by equations (11)–(14). This completes the proof.

Remark 5

When $\delta =0$ in Proposition 1, the event-driven transmission scheme will recover a conditional communication one, that is, all the sampled data will be mandatorily transmitted to the controller via network.^7,8 Hence, we can say the control scheme with event-driven transmission is a kind of more general network-based sampled-data control strategies.

Remark 6

When $\Phi$ is fixed in equation (2), the data transmission frequency via event-driven scheme is actually determined by the index $\delta$ . When the larger value of $\delta$ is chosen, the data transmission event will be less triggered; however, the conservatism of the sufficient criteria is also increased; conversely, when the value of $\delta$ is smaller, more data will be transmitted, but the less conservatism of the sufficient criteria will be obtained. When $\delta$ reduces to zero, the event-driven transmission will change to synchronous transmission one. Moreover, the parameters $h,c_1,c_2,T_f$ also determine the conservatism of the sufficient criteria. Consequently, how to find the optimal $\delta$ with a feasible controller is the main objective of the developed methods.

Given a matrix $\Phi \succ 0$ and other prescribed parameters $h,c_1,c_2,T_f$ , a co-design algorithm of event-driven transmission and state-feedback controller is given in Algorithm 1.

Algorithm 1. Co-design of the event-driven transmission scheme and the state-feedback controller gain K
Step 1. Assign the initial parameters of $\delta >0$ , ${\rho }_2>0$ , ${\rho }_3>0$ , and ${\rho }_4>0$ with small values and then arbitrarily choose $\varsigma \in [\overline{\varsigma },\infty )$ . Moreover, assume ${\epsilon }_1$ and ${\epsilon }_2$ are the step increments of the given $\delta$ and $\varsigma$ , respectively. Then, assign the initial ${\rho }_1={\rho }_2+{\epsilon }_1$ .
Step 2. For the prescribed $c_1,c_2,\varsigma ,T_f,{\rho }_1,{\rho }_2,{\rho }_3$ and ${\rho }_4$ , if the sufficient conditions (11)–(13) are solvable, keep ${\rho }_1$ and go to the Step 4; otherwise, reset ${\rho }_1\stackrel{\Delta }{=}{\rho }_1+{\epsilon }_1$ and repeat Step 3.
Step 3. Calculate the state-feedback controller gain K and then redefine $\delta \stackrel{\Delta }{=}\delta +{\epsilon }_0$ . If $\delta <1$ , go to Step 4; otherwise, keep $\delta =\delta -{\epsilon }_0$ and controller gain K.
Step 4. For given $c_1,c_2,\varsigma ,T_f,{\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4$ and $\delta$ , if there exist feasible solution satisfying equations (11)–(13), go back to Step 3; otherwise, keep $\delta =\delta -{\epsilon }_0$ and controller gain K.

Simulations

In this section, we use two illustrative examples to testify the advantages and effectiveness of the co-design of the event-driven transmission scheme and the controller, where Example 1 is first analyzed to show the properties of the event-driven transmission scheme. Then an application to a damping boring bar system in Example 2 is investigated to testify the potential of finite-time boundedness for the developed co-design of the event-driven transmission and controller.

Example 1

Consider a linear system is described by

$$\begin{array}{l}A=\left[\begin{array}{cc}0.6& 0.1\\ 0.2& 0.5\end{array}\right]\mathrm{,}B=\left[\begin{array}{c}0.2\\ 0.1\end{array}\right]\mathrm{,}\\ B_1=\left[\begin{array}{c}0.3\\ 0.2\end{array}\right]\mathrm{,}C=\left[\begin{array}{cc}-0.2& 0.1\end{array}\right]\end{array}$$

Assume that $c_1=1.4,c_2=1.2,T_f=50,h=15\mathrm{ms},\varsigma =0.01,R=I_{1\times 1}$ and $\Phi =I_{2\times 2}$ . Here, our objectives are to find the maximum $\delta$ for event-driven transmission scheme with the given $\Phi$ and obtain a set of state-feedback controllers such that the underlying system is FTB with respect to $(c_1,c_2,R,T_f)$ . Performing Algorithm 1, the maximum value of $\delta$ can be obtained as $\delta =0.1$ with ${\rho }_1=1.0$ , ${\rho }_2=0.9$ , ${\rho }_3=1.0$ , and ${\rho }_4=0.95$ . Then, for an assigned $F(t)=0.7+0.6\sin (t)$ , the output responses and the event-driven communication instants with $\delta =0.1,0.05,\mathrm{and}0.01$ are, respectively, shown in Figures 2–4, where the communication numbers are listed in Table 1, including the general non-event-driven approach with $\delta =0$ . ⁷ It is apparent that, for the larger $\delta$ , less sampling data will be transmitted, which effectively spares the network sources while ensuring the prescribed finite-time boundedness.

Figure 2.

(a) Event-driven communication instants with $\delta =0.1$ and (b) the history trajectory of $y^T(t)\mathrm{Ry}(t)$ .

Figure 3.

(a) Event-driven communication instants with $\delta =0.05$ and (b) the history trajectory of $y^T(t)\mathrm{Ry}(t)$ .

Figure 4.

(a) Event-driven communication instants with $\delta =0.01$ and (b) the history trajectory of $y^T(t)\mathrm{Ry}(t)$ .

Table 1.

Event-driven communication numbers with different index $\delta$ .

$\delta$	0.1	0.05	0.01	07
$\mathrm{Number}$	70	105	229	5334

Example 2

Consider a class of damping boring bar system in Figure 5. The ideal dynamic equations of the sprung and unsprung masses are given by

$$M{\stackrel{··}{z}}_1+k{z}_2+c{\stackrel{·}{z}}_2+K{z}_1+F\left(t\right)=-u\left(t\right)$$

$$m{\stackrel{··}{z}}_2+k{z}_2+c{\stackrel{·}{z}}_2=u\left(t\right)$$

$$y\left(t\right)=z_1\left(t\right)$$

(15)

where m and M, respectively, denote absorber equivalent mass and boring bar equivalent mass; K and k stand for the stiffness and c is the damping; and $z_1$ is the displacement of boring bar and the control output.

Figure 5.

(a) Kinetic model of damping boring bar with sampled-data control scheme and (b) the equivalent model.

Denoting $x(t)\stackrel{\Delta }{=}\begin{array}{cccc}{x}_1^T(t)\hfill & x_2^T(t)\hfill & x_3^T(t)\hfill & x_4^T(t)\hfill \end{array}{]}^T$ with $x_1(t)\stackrel{\Delta }{=}{z}_1-z_2$ , $x_2(t)\stackrel{\Delta }{=}{\stackrel{·}{z}}_1$ , $x_3={\stackrel{·}{z}}_2$ , and $x_4(t)\stackrel{\Delta }{=}{z}_1$ , then we can construct the state-space model with the matrices as follows

$$\begin{array}{l}A\triangleq \left[\begin{array}{cccc}0& 1& -1& 0\\ \frac{k}{M}& 0& -\frac{c}{M}& -\frac{k+K}{M}\\ \frac{k}{m}& 0& -\frac{c}{m}& -\frac{k}{m}\\ 0& 1& 0& 0\end{array}\right]\mathrm{,}\\ B\triangleq \left[\begin{array}{c}0\\ -\frac{1}{M}\\ \frac{1}{m}\\ 0\end{array}\right]\mathrm{,}{B}_1\triangleq \left[\begin{array}{c}0\\ -\frac{1}{M}\\ 0\\ 0\end{array}\right]\end{array}$$

$$C\stackrel{\Delta }{=}\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

The damping boring bar system is analyzed in this subsection with the following parameters

$$\begin{array}{c}m=0.88\mathrm{kg},M=4.4\mathrm{kg},\hfill \\ K=k=9.25\times {10}^2,c=143.8705\hfill \\ c_1=0.0049,c_2=0.0012,\hfill \\ T_f=15,h=10\mathrm{ms},\varsigma =0.95\hfill \end{array}$$

Then, we have

$$\begin{array}{c}A\stackrel{\Delta }{=}\left[\begin{array}{cccc}0\hfill & 1\hfill & -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0.5}& 0\hfill & -0,3\hfill & -0.2\hfill \\ \multicolumn{1}{c}{0.4}& 0\hfill & -0.3\hfill & -0.1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \end{array}\right],\hfill \\ B\stackrel{\Delta }{=}\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-0.4}\\ \multicolumn{1}{c}{0.3}\\ \multicolumn{1}{c}{0}\end{array}\right],B_1\stackrel{\Delta }{=}\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-0.4}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \\ C\stackrel{\Delta }{=}\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right],F\left(t\right)=0.5+0.6\sin \left(t\right)\hfill \end{array}$$

By employing Algorithm 1, we can obtain controller gain as $K=[0.5030.329]$ and $\delta =0.2$ . The event-driven control input $u(t)$ and the history trajectory of $y^T(t)\mathrm{Ry}(t)$ are shown in Figure 6. It can be well observed from Figure 6(b) that even the initial value of $y^T(0)\mathrm{Ry}(0)=0.0038$ is larger than $c_2=0.0012$ , it can be well guaranteed that $y^T(t)\mathrm{Ry}(t)\le c_2$ for any $t\ge T_f$ . That is to say, as long as $y^T(0)\mathrm{Ry}(0)\le c_1=0.0049$ . Therefore, the event-driven transmission scheme and the finite-time control design technique are effective in this article.

Figure 6.

(a) Event-driven control input $u(t)$ with $\delta =0.1$ and (b) the history trajectory of $y^T(t)\mathrm{Ry}(t)$ .

Conclusion

The problem of finite-time control for NCSs with non-uniform sampling and event-driven transmission scheme has been investigated. Employing an input-delay approach, the network-based closed-loop control system with non-uniform sampling and event-driven transmission scheme is formulated to new time-delay system. The developed event-driven transmission scheme can effectively reduce the network load of communication. The provided two examples have well demonstrated the effectiveness and the advantages of the developed techniques in this article. The developed results are expected to be applicable to filtering for NCSs.