Robust stability and ${\mathcal{H}}_{\infty }$ control for networked control systems with transmission delay and its application to 2 DoF laboratory helicopter

Abstract

This article explores the stability analysis and control synthesis of networked control systems (NCSs) with uncertainty and transmission delay via a new quadratic polynomial negative-definiteness method. First, a novel negative-definiteness method of quadratic polynomials is proposed by considering two independent parameters. Several existing methods in the literature are encompassed as particular examples. Furthermore, a novel delay-product Lyapunov–Krasovskii functional(LKF) is presented to exploit more delay convexity information. Then, new hierarchical ${\mathcal{H}}_{\infty }$ stability and stabilization conditions are developed according to the proposed quadratic polynomial negative-definiteness method and the delay-product LKF. Finally, the generality and effectiveness of the established conditions are experimentally shown for a 2-degrees of freedom (DOF) laboratory helicopter system.

Introduction

NCSs represent a control system where feedback and measurement information is transmitted through communication networks. Unlike conventional point-to-point systems, the system components, such as sensing devices, control units, and actuators, can be spatially distributed [1]. NCSs offer remarkable advantages in installation cost, installation and maintenance, low power requirements, and transmission reliability. Meanwhile, the rapid expansion of network transmission technologies has further facilitated the adoption of NCSs. NCSs have attracted much consideration from academia and industry, while serving a wide range of applications in various fields, including remotely operated equipment, smart grids, and sensor networks [2], [3], [4], [5].

The characteristics of the networks, such as vulnerability, time delay, and data dropouts, inevitably introduce constraints in the analysis and synthesis of NCSs. Fig. 1 illustrates a typical architecture of NCSs where data packets from a sensor are transmitted to a controller and then to an actuator to drive the controlled plant. To ensure data transmission accuracy, most network protocols employ handshaking mechanisms to prevent unexpected packet loss and checksum algorithms from detecting data errors [6]. The retransmission function is activated immediately in the event of a data drop or transmission error. Although the retransmission function reduces the loss of feedback signals and sensing information, it can potentially increase the communication-induced time delay during transmission. Randomly occurring retransmissions are likely to cause congestion in the communication channel. In other words, retransmissions may cause a shortfall in communication bandwidth. The employment of communication networks has created information transmission problems and forced the opening of interfaces to the cyber world. Specifically, NCSs with communication networks endure more severe security problems than traditional systems. Fig. 1 shows the basic process of hacking the NCSs through a cyber network. An attacker can surreptitiously launch an attack through the communication network to compromise the NCSs [7]. Malicious packet dropout (MPD) is a straightforward method that blocks communication links to prevent the transmission of control and sensing information between system components. As a result, the control signals of NCSs may experience delays due to data retransmission or MPD attacks.

Over the past decade, the issues of how to reduce the effect of time delay on the stability of NCSs and improve the stability performance of the system have been a hot research topic in the field of control. Time-varying delays introduced by the communication network effortlessly cause performance fluctuation or instability. In [8], the NSC with constant transmission delay was studied based on a looped LKF. The authors in Zheng et al. [3] investigated the fuzzy NCS with multiple mixed time-varying delays. In [9], $H_{\infty }$ stability conditions with less conservatism were developed for NSCs with Markovian jump delay. In [2], the stability of a NCS with additive time delays was studied using a suitable LKF. The $H_{\infty }$ stability issues of a NCS with time-varying communication delay were studied in Zheng et al. [10]. The distributed delay with a probability density function is considered in Yan et al. [11] based on event-triggered control. A linear multivariable NCS with uncertain time delays was studied by utilizing an adaptive control approach in Steinberger et al. [6]. Large time-varying communication delays were considered for NCSs in Nonomura and Fujii [12], and the results were explanted to the servo system. In [13], the stability conditions for NCSs with random time delays were established by using a novel LKF. Reviewing the existing research results on the communication delay problem, the LKF approach is a powerful tool to analyze and eliminate the effect of time delay on NCSs.

The approach of LKF combined with linear matrix inequality(LMI) is an impressive framework to explore the stability of NCSs with time delay (as seen in Lin et al. [14], Zeng et al. [15], Liu and Yang [16]). The key to reducing the conservatism of system criteria is to design suitable LKFs that contain enriched information. For example, simple LKF [17], complete quadratic LKF [18] and augmented LKF [19]. Consequently, a suitable LKF can abound the coupling relationship among system states and time-varying delays. In [8], an improved LKF with loop-functional was proposed to study the NSC with constant transmission delay. The event-triggered problem of NCSs was investigated according to a simple LKF in Wang et al. [17]. In [20], a discontinuous LKF was introduced to investigate the NCSs with time delay and sampled-data control. Augmented LKFs and high-order integral inequalities were employed in Fang et al. [21] to derive relaxed stability conditions for NCSs. The delay-product LKFs were developed to derive LMI-based conditions with less conservatism in Cai et al. [22]. The authors in Zeng et al. [23] proposed a novel two-sided looped LKF to study the stability problem of the NCS with sample control. Therefore, an improved delay-product type augmented LKF is considered to derive less conservative results.

The stability and stabilization criteria based on the LKF approach are commonly presented as LMI. Augmented LKFs and high-order integral inequalities have been commonly developed to obtain stability criteria for NCSs. Then, some $d^2\left(t\right)$-related terms are produced in the stability conditions. The conservatism of LKF methods also comes from determining the negative definite matrix inequalities. For stability criteria in the form of quadratic polynomials with time-varying delay, it is an arduous problem to solve by LMI directly. Numerous approaches have been developed to guarantee the negative-definiteness of quadratic polynomials (as seen [24], [25], [26], [27], [28], [29], [30], [31], [32]). A new hierarchical-type approach was presented in Zeng et al. [28] via using tangent information to guarantee the negative-definiteness of quadratic polynomials. In [27], a negative condition for quadratic polynomials was derived based on the characteristics of interval endpoint tangents. A relaxed negative-definiteness approach was presented in Long et al. [32] using Taylor’s formula, which contains the results in Long et al. [27], Zhang et al. [31] as special cases. The authors in Lee [30] proposed geometry-based constraints for quadratic polynomials with time delay. The conservatism of stability conditions can be improved by adjusting parameters. The authors in Wang et al. [33] investigated the delayed networked systems by utilizing a negative-determination lemma in Zhang et al. [31]. A stable result of the delayed neural network system was obtained in Chen et al. [29] based on a quadratic polynomial negative-definiteness method. By reviewing the results of existing studies on this issue, most of them ignore the problem of interval optimization and the number of iterations. For these reasons, there is still room for improvement of the quadratic polynomial negative-definiteness method for NCSs.

The paper investigates the stability and stabilization issues of NCSs with time-varying delay based on the quadratic polynomial negative-definiteness method and the delay-product LKF. The main contributions of the paper are summarized as follows.

• A new quadratic polynomial negative-definiteness method is proposed, which encompasses some existing methods in Zeng et al. [28], Lee [30], Zeng et al. [34] as exceptional cases.

• A delay-product augmented LKF is established that employs more delay convexity information. Some quadratic delay-product and single integral terms are included in the constructed LKF.

• The hierarchical ${\mathcal{H}}_{\infty }$ stability and stabilization conditions for NCSs are achieved via employing the proposed negative-definiteness lemma and delay-product augmented LKF. By adjusting the introduced parameters, the conservatism of the achieved stability criteria can be reduced. Moreover, the effectiveness of the established criteria is experimentally shown by a 2-DOF laboratory helicopter system.

Notation: ${\mathbb{R}}^n$ is $n$-dimensional real Euclidean space. ${\mathbb{R}}^{n\times m}$ represent the set of all $n\times m$ real matrices. $\mathcal{Q}>0$ denotes that matrix $\mathcal{Q}$ is a positive definite. $He\left\{\mathcal{Q}\right\}={\mathcal{Q}}^T+\mathcal{Q}$. The superscript $T$ is the transposition of a matrix. $*$ stands for the symmetric term.