H_∞ control for networked systems with multiple packet dropouts

Highlights

• The networked control system is modeled as a stochastic parameter system.

• Both sensor-to-controller and controller-to-actuator packet dropouts are considered.

• A condition for H_∞ control is provided in terms of LMI with equation constraints.

• An iterative LMI approach is developed to solve the LMI with equation constraints.

Abstract

In this paper, we present a novel H_∞ control scheme for a networked control system (NCS) with multiple data packet dropouts. Multiple data packet dropouts occur randomly in both control channel and measurement channel. The NCS with both measurement and control packet dropouts is modeled as a stochastic parameter system which contains two independent Bernoulli distributed white sequences. An H_∞ dynamic output controller is designed to exponentially stabilize the networked system in the sense of mean square, and also to achieve the prescribed H_∞ disturbance attenuation level. An iterative algorithm is developed to compute the optimal H_∞ disturbance attenuation and the controller parameters by solving the semi-definite programming problem via an interior-point approach. Two illustrative examples are provided to show the applicability of the proposed method.

Introduction

A networked control system (NCS) is a control system whose feedback control loop is based on a communication network [6], [39]. Such a system through wired communication channels has been used in a variety of industrial applications, whereas a control system through wireless communication channels has important potential application advantages. However, the insertion of a wireless network into control systems will make system analysis and synthesis more challenging [15], [22], as a network itself is a dynamic system and induces transmission delays and data packet dropouts when measurement and control signals are transmitted between plant and controller via a communication network due to limited bandwidth [1]. A realistic networked control system design should take the data packet dropouts and transmission delays into account, since they are widely known to degrade the performance of control systems or even cause instability. Therefore, in the past decade, the control problem of networked systems with data packet dropouts and transmission delays has received increasing attention, see e.g. [5], [8], [9], [10], [13], [17], [23], [29], [30], [34], [40] and references therein.

A great number of research results have been reported about modeling and synthesis for NCSs with a single data packet dropout and transmission delay [16], [22], [24], [31], [34], [35], [36], [38], [39], [41]. There have been two lines of research on this topic. The first line is to model packet dropout and transmission delay as Markov chains [22], [35], [41]. The resulting NCS is transformed to a standard jump linear system with time delays. The jump linear system approach can therefore be employed to analyze and design the NCS with data packet dropout and transmission delay. The second line is to describe packet dropout and delay as a binary switching sequence which is viewed as a Bernoulli distributed white sequence taking on values of 0 and 1 with certain probability [34], [39], [40]. The NCS with data packet dropout and transmission delay is then modeled as a stochastic parameter system. The stochastic control approach is applied to analyze and design the NCS with data packet dropout and transmission delay. However, the data packet dropouts are random by nature due to network congestion. There may exist more than one consecutive packet dropout. Therefore, it is important to investigate the filtering and control problems for systems with multiple packet dropouts.

Recently, there have been significant research efforts on the filtering problem for NCSs with multiple packet dropouts. For example, the problem of optimal H₂ filtering for NCSs with multiple packet dropout has been studied in [19], [20]. By generalization of the H₂-norm definition, a steady-state filter is designed via an LMI approach. Based on a similar model for multiple packet dropouts in [19], the finite-horizon optimal linear filtering, prediction and smoothing problems have been investigated in [27]. The solution to the problem is obtained via an innovation analysis approach. The optimal linear filter, predictor and smoother are derived in terms of the solution of a Riccati difference equation. H_∞ filtering is investigated for NCSs with mixed random delays and packet dropouts in [37]. Distributed filtering is presented for sensor networks with one packet dropout [25]. As to the control problem for NCSs with multiple packet dropouts, few results have been reported [33], [35]. It is worth to mention that the stability analysis and controller design of NCSs with single- and multiple-packet transmissions have been considered in [35]. The jump linear system approach is adopted to obtain the control laws which guarantee stochastic stability of the resulting closed-loop systems. In [33], the problems of output tracking performance analysis and controller design for networked control systems (NCSs) with both constant sampling period and time-varying sampling period have been investigated. An LMI approach has been applied to design the controllers for NCSs with multiple packet dropouts which can guarantee asymptotic tracking of prescribed reference outputs while rejecting disturbance [7].

In this paper, a novel iterative LMI approach is proposed to deal with the controller design for NCSs with multiple packet dropouts. Both sensor-to-controller packet dropouts and controller-to-actuator packet dropouts are considered. The NCS with multiple packet dropouts is modeled as a stochastic parameter system which contains two independent Bernoulli distributed white sequences. A dynamic output controller is designed such that the closed-loop networked control system is stochastically exponentially stable, and the prescribed H_∞ disturbance attenuation performance is achieved. A new iterative algorithm is developed to compute the optimal H_∞ disturbance attenuation and the controller parameters by solving the semi-definite programming problem (SDP) via interior-point approach [14], [32]. To compare with [34], [39], [40], [41], [40], [41] only consider one-step time delay for data packet transmission and an augmented approach is adopted to handle the time-delay for stability analysis and controller design. [34], [39] consider one-step packet dropout and an LMI approach is used to solve the controller design problems. In this paper, a novel iterative LMI approach is developed to deal with multiple packet dropouts. The contributions and novelty of this paper are: (i) A novel control scheme based on the iterative LMI approach is proposed to deal with multiple packet dropouts in both measurement and control channels. When there exist multiple packet dropouts, the controller design cannot transfer to the solution of an LMI problem. It will be an LMI problem with two equations constraints, see (32), (33). A novel iterative LMI approach is proposed to solve the LMI problem with equation constraints. (ii) A new Lyapunov approach based on stochastic functional is developed to analyze the stability and derive the H_∞ performance. Since the resulting closed-loop system is reformed as a stochastic parameter system, the traditional Lyapunov functional cannot be used to analyze the stability and derive the H_∞ performance.

The notation X ⩾ Y (respectively, X > Y) where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite). $\mathbb{E}\{x\}$ stands for the expectation of the stochastic variable x. Prob{·} means the occurrence probability of the event “·”. If A is a matrix, λ_max(A) (respectively, λ_min(A)) means the largest (respectively, smallest) eigenvalue of A. l₂[0,∞] is the space of square integrable vectors, and ${\mathbb{I}}^{+}$ is the set of positive integers. In symmetric block matrices, “∗” is used as an ellipsis for terms induced by symmetry.