L2 control design of event-triggered networked control systems with quantizations

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Abstract

This paper deals with L2 controller design problem for event-triggered networked control systems (NCSs) with quantizations. The system state is periodically sampled and quantized. A new model of NCSs that involves the network conditions, state and event-triggered communication strategy is proposed. Based on this model, some novel criteria for the asymptotic stability analysis and L2 state feedback controller design method of the event-triggered NCSs with quantizations and time-varying delay are established to guarantee a prescribed L2 disturbance rejection attenuation level γ by using Lyapunov–Krasovskii function approach. Finally, a simulation example is provided to illustrate the effectiveness of the proposed method. It is shown that the average transmission interval could be increased substantially while the control performance is maintained.

Introduction

Networked control systems (NCSs) are spatially distributed systems for which communication between sensors, actuators and controllers are connected by a shared communication network. In recent years, NCSs have brought many innovative impacts to control systems. They are becoming increasingly important in industrial processes for many advantages, such as low installation and maintenance costs, high reliability, increased system flexibility, and decreased wiring [1]. As such, network-based analysis and design have many industrial applications in, for example, aircrafts, manufacturing plants, robots, automobiles, and remote surgery [2], [3], [4], [7]. However, grate challenges are also met due to the network induced imperfection, namely time delays, packet dropouts, disorder, time-varying transmission intervals and competition of multiple nodes accessing networks as well as data quantization, which can deteriorate the performance of NCSs and even destabilize the systems [2]. So far, much effort has been devoted to modeling, analysis and design of NCSs in the presence of network-induced delays, packet dropouts and disorder, see, for example [5], [6], [7], [8] and the references therein.

Notice that the network in NCSs is the shared band-limited digital communication network [2]. One common problem to be addressed when considering NCSs is whether there is sufficient communication bandwidth to feed back information to the controller and then send the control commands to the actuators and the plant. Traditionally, the control task is executed periodically, which allows the closed-loop system to be analyzed and the controller to be designed using the well-developed theory on sampled-data systems [20]. However, the control strategy obtained based on this approach is conservative in the sense that resource usage is more frequent than necessary to ensure a specified performance level, since stability is guaranteed in the worst case scenarios under sufficiently fast periodic execution of the control action. To overcome this drawback, several researchers suggested the idea of event-triggered control. Event-triggered communication scheme has been proved to be an efficient way to reduce the transmitted data in the networks, which can relieve the burden of network bandwidth occupation in comparison with a traditional periodic sampling method [9], [10], [11], [12], [13], [14], [15]. In [9], started from the paradigm that a real-time scheduler could be regarded as a feedback controller that decides which task is executed at any given instant, a simple event-triggered scheduler based on this feedback paradigm was investigated to guarantee performance thus relaxing the more traditional periodic execution requirements. In [10], a decentralized event-triggered implementation, over sensor/actuator networks, of centralized nonlinear controllers was presented. In [11], a new self-triggering scheme that ensures finite-gain L2 stability of the resulting self-triggered feedback systems was proposed. This scheme relaxes the assumptions that the magnitude of the process noise is bounded by a linear function of the norm of the system state. In [12], a novel event-triggering scheme is presented to ensure exponential stability of the resulting sampled-data system. The scheme postpones the triggering of events over previously proposed methods and therefore enlarges the inter-sampling period. Hu and Yue [15] are concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. An innovative delay system model is proposed, based on this model, the criteria for the asymptotic stability analysis and control synthesis of event-triggered networked control systems are established. Unfortunately, to the best of our knowledge, up to now, the stabilization and L2 control problems for general NCSs with simultaneous consideration of the quantizations and event-triggered communication scheme have not been adequately addressed yet, which still remains an interesting research topic. This motivates the current research.

In this paper, some criteria for the L2 state feedback controller design method for the event-triggered NCSs with quantizations are presented. Different from some existing ones, the feedback networked control system in this paper is modeled as a delay system considering the network-induced delays and event-triggering scheme. By using Lyapunov–Krasovskii function approach, new sufficient conditions that guarantee the asymptotic stability of the closed-loop NCSs are established in terms of linear matrix inequities (LMIs). Moreover, the explicit expression of feedback gain is also derived with integration of signal quantizations, event-triggering, and network-induced delays. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Notations: Throughout this paper, Rn and Z+ denote the n-dimensional Euclidean space, positive integer set, respectively. Rm×n is the set of m×n real matrices. The superscripts and −1 stand for matrix transposition and matrix inverse, respectively. sym{X} denotes the expression X+X. In denotes the n×n identity matrix. The notation X>0 (respectively, X0) denotes a real symmetric positive definite (positive semi-definite). In symmetric block matrices, “⁎” is used as ellipsis for terms induced by symmetry, diag{} denotes the block-diagonal matrix. Matrixes, if not explicitly stated, are assumed to have appropriate dimensions. The space of square-integrable vector functions over [0,) is denoted by L2[0,), and for ω=ω(t)L2[0,), its norm is denoted by ω2.

Section snippets

Problem statement and preliminaries

Consider the NCSs with event-triggering shown in Fig. 1. The physical plant is given byẋ(t)=Ax(t)+Bu(t)+Bωω(t),tt0where x(t)Rn is the state vector, u(t)Rm is the control input vector, ω(t)L2[0,) is the disturbance input. A, B, and Bω are the parameter matrices with appropriate dimensions. The initial condition of the system (1) is given by x(t0)=x0. Throughout this paper, we assume that system (1) is controlled throughout a network with a networked state feedback controller, which is

Main results

In this section, a sufficient condition for the closed-loop system (18) to be asymptotically stable and the disturbance ω(t) satisfy the L2 disturbance attenuation will be derived.

Illustrative example

Example

In this section, an example is provided to validate the effectiveness of the theoretical results. The inverted pendulum introduced by [11] is considered. The plant׳s state-space representation is given byẋ(t)=[010000mg/M0000100g/l0]x(t)+[01/M01/Ml]u(t)We choose the other parameters asBω=[1111].ω(t)={sgn(sint)ift[0,100]0otherwisewhere M=10 is the cart mass and m=1 is the mass of the pendulum bob, l=3 is the length of the pendulum arm and g=10 is the gravitational acceleration. The initial

Conclusions

In this paper, to reduce the communication load in the network, a novel event-triggered scheme has been proposed to determine when the sampling signal data will be transmitted. An event-triggered L2 control design method has been proposed for NCSs with quantizations. A delay system model has been used to describe the prosperities of the event trigger and effects of both transmission delay and signal quantizing on the system. Based on this model, new criteria for stability with an L2 norm bound

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    This work is supported by the National Natural Science Foundation of China (61272064, 61273026), the Innovation Program of Shanghai Municipal Education Commission (12zz052), Shanghai Pujiang Program (14PJ1409000) and the Fundamental Research Funds for the Central Universities.

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