H control with limited communication and message losses

https://doi.org/10.1016/j.sysconle.2007.09.007Get rights and content

Abstract

We propose an H approach to a remote control problem where the communication is constrained due to the use of a shared channel. The controller employs a periodic time sequencing scheme for message transmissions from multiple sensors and to multiple actuators of the system. It further takes into account the information on the random message losses that occur in the channel. An exact characterization for controller synthesis is obtained and is stated in terms of linear matrix inequalities. Furthermore, an analysis on the loss probabilities of the messages to accomplish stabilization is carried out. The results are illustrated through a numerical example.

Introduction

In this paper, we consider two constraints that arise in remote control systems and formulate a synthesis problem of H-type controllers. One is the time sequencing of messages transmitted over a network that connects the plant and the controller. We assume that only one message can be sent by any component sharing the network at a time, and we employ a periodic sequencing scheme for the communication. Such schemes have been commonly used in practice to meet both data rate and real-time requirements when a number of sensors and actuators share a channel.

The other constraint modeled in this work is the random losses of messages during their transmission over the channel. In a remotely controlled system, messages would often be lost or delayed in a random manner due to network congestion or errors. We adopt the strategy such that if the arrival is delayed, messages are regarded as lost. Furthermore, we assume that the information whether a message is lost or not is always communicated as an acknowledgment message to the controller with one-step delay. These messages are implemented in common network protocols such as the TCP. For more on network issues, we refer to [8], [10].

These two constraints have recently been studied in the context of networked control. In [3], time sequencing was first taken into account in a stabilization problem. This problem was further studied in [7], [21]. In [11], in a decentralized control setup, local controllers are allowed to periodically communicate to each other over a network channel. In [14], an H control design of a controller using periodic communication (with no loss) was proposed. On the other hand, the effect of message losses on stabilization was studied in [6] for a scalar case. The design of Kalman filters under lossy measurements is dealt with in [19], while an optimal LQG problem is considered in [9]. In [5], a framework for the use of general stochastic channels was introduced. A common point indicated in these works is the existence of a critical loss probability determined by the unstable poles of the plant. Another way to model such systems is via the Markovian jump systems [1], [4], [13]. Issues arising in networked control have provided with new motivations for the research of such systems. In particular, an H synthesis method is presented in [18]. See also [20], [22] for related results on random delays.

In the remote control system of this paper, the controller is assumed to be periodic and also to take account of the losses. Consequently, the proposed H design can be viewed as a periodic or multirate control problem with random switchings in the system matrices. The main result is a necessary and sufficient condition stated in terms of linear matrix inequalities (LMIs). The design problem can hence be efficiently solved by numerical methods. We note that the results are in particular based on the recent works [17], [18], where a bounded real lemma for Markovian jump linear systems and then a synthesis method for a special case are derived. In this paper, we obtain a generalization to the periodic case; this forms the basis for treatment of systems involving multirate operations as in networked control. Furthermore, we examine the critical values for the loss probabilities mentioned above. It is shown that the unstable plant poles as well as the communication rates contribute to such probabilities.

This paper is organized as follows. In Section 2, the H remote control problem under periodic sequencing and random message losses is formulated. Then, in Section 3, we present some preliminary results regarding stability and H norm characterizations for periodic systems with random switchings. In Section 4, the main result for the controller design is provided. This is followed by some examples in Section 5, where critical loss probabilities are obtained. We present a numerical example in Section 6 and then concluding remarks in Section 7.

Section snippets

Problem formulation

Consider the remote control system depicted in Fig. 1. The generalized plant G is a discrete-time system and has a state-space equation of the following form:xk+1=Axk+B1wk+B2uk,zk=C1xk+D11wk+D12uk,yk=C2xk+D21wk,where xkRn is the state, wkRm1 is the exogenous input, ukRm2 is the control input, zkRp1 is the controlled output, and ykRp2 is the measurement output. We make the standard assumptions that (A,B2) is controllable and (A,C2) is observable.

The plant is remotely controlled by the

Periodic systems with random switchings

In this section, we characterize the stochastic stability and derive an exact condition for the H norm condition.

Consider the system G0 in (4). The first result gives a necessary and sufficient condition for stochastic stability of this system.

Proposition 3.1

The system G0 in (4) is stochastically stable if and only if there exists an N-periodic matrix PkRn×n such that Pk=PkT>0 andiIMαiAk,iTPk+1Ak,i-Pk<0forkIN,where IN{0,,N-1}.

This proposition can be proved by following similar lines as in the

H control synthesis

In this section, we present the solution to the remote control synthesis problem. In particular, the H characterization result in Section 3 is used.

We first give the state-space equation for the closed-loop system Fl(G˜,K):x¯k+1=A¯k,θ1(k),θ2(k)x¯k+B¯k,θ1(k),θ2(k)wk,zk=C¯k,θ1(k),θ2(k)x¯k+D¯k,θ1(k),θ2(k)wk,where x¯kR2n is the state given by x¯k[xkTx^kT]T andA¯k,θ1(k),θ2(k)A+θ1,kθ2,kB2S2,kD^kS1,kC2θ2,kB2S2,kC^k,θ1(k)θ1,kB^k,θ2(k)S1,kC2A^k,θ1(k),θ2(k),B¯k,θ1(k),θ2(k)B1+θ1,kθ2,kB2S2,kD^kS1,kD21θ

Stabilizability using observer-based controllers

In this section, we consider the stabilization of the plant (1) (when wk0) employing a specific class of controllers with an observer-based structure. We derive some necessary upper bounds on the message loss probabilities α1 and α2 for this case. These bounds are shown to depend on the unstable dynamics of the plant as well as the communication rates in the two channels.

The analysis in this section is motivated by [9], [19]. It is an extension of them for the controller in the form (3), which

Numerical example

In this section, we present a numerical example that illustrates the results of the paper.

We consider the second-order plant as follows: xk+1=200.71.1xk+11wk+12uk,zk=0.5-1xk+uk,yk=1-2xk+wk.We fix the period of the switchings as N=2. First, we look at the case as in Example 5.1, where there is perfect communication from the controller to the actuator. So the switching pattern for that channel is s2=[1,1] and the loss probability is α2=0. As the other pattern, we let s1=[1,0]. Thus, messages are

Conclusion

We proposed an H design method for a remotely controlled system over a shared network. Two constraints due to the presence of the network have been incorporated: The periodic sequencing of message transmissions and the random message losses. We have developed a bounded real lemma type result for the class of periodic systems with random switchings. This has been used in the design and has led us to an exact condition consisting of LMIs, which can be solved efficiently. Future research will

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      They can be roughly categorized into the following types based on the resulting closed-loop systems: Switching systems (Ishii, 2008; Zhang & Yu, 2008), asynchronous dynamical systems (Zhang et al., 2001), and jump linear systems with Markov chains (Schenato, 2009; Seiler & Sengupta, 2005). Note that packet dropouts defined in the aforementioned references have two cases, dropped or sent successfully, which are modeled as a Bernoulli or a two-state Markov chain process based on zero-input or hold-input.

    • Fundamental limitations and intrinsic limits of feedback: An overview in an information age

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      Various extensions to more general classes of systems have been made in the more recent works (Silva, Derpich, & Østergaard, 2010a; Okano & Ishii, 2014, 2017). Control over lossy channels (Elia, 2005; Elia & Eisenbeis, 2011; Feng, Chen, & Gu, 2018a, 2018b; Hespanha et al., 2007; Ishii, 2008; Tsumura, Ishii, & Hoshina, 2009; Quevedo, Silva, & Goodwin, 2008; Schenato et al., 2007; You & Xie, 2010, 2011): Similar bounds can be obtained for control over unreliable channels where data packets drop out randomly. For losses that can be represented as an i.i.d. process, bounds on the loss probability are established in terms of the topological entropy, or more specifically, the reciprocal of the product of the unstable poles.

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    This work was supported in part by the Ministry of Education, Culture, Sports, Science and Technology, Japan, under Grant no. 17760344.

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