Suboptimal decentralized control over noisy communication channels

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Abstract

In this paper we present a technique for the design of decentralized controllers for mean square stability of a large scale system with cascaded clusters of subsystems. Each subsystem is linear and time-invariant and both system and measurement are subject to Gaussian noise. For stability analysis of this system we consider the effects of Additive White Gaussian Noise (AWGN) channels used for exchanging information between subsystems.

Introduction

In recent years, the development of Micro-Electro Mechanical Systems (MEMS) has made it possible to deploy these small sized embedded devices in distributed parameter systems for efficient control. Some examples are: smart mechanical structures and distributed flow control [1]. Each micro-electro mechanical system consists of sensors, a data processor, a communication unit, and an actuator. MEMS collaborate with each other towards a common goal by exchanging observation and control signals. Due to limited power supply of MEMS, transmission in systems equipped with these devices is limited by communication constraints over short distances. This necessitates the creation of a co-design framework to integrate the control and communication requirements in systems controlled by networks of MEMS. The objective of this paper is to develop such a framework for a large scale system with cascaded clusters of subsystems which is controlled by a network of MEMS. In this system, subsystems are linear time-invariant and each subsystem is controlled by a micro-electro mechanical device attached to it. Subsystems are subject to Gaussian process noise and Gaussian measurement noise and interconnected via Additive White Gaussian Noise (AWGN) channels. For this large scale system we present a decentralized technique for design of controllers, encoders and decoders for mean square stability and reliable data reconstruction. These policies are executed by MEMS.

In contrast with similar works, in Refs. [2], [3], [4], [5], optimal stabilizing controllers of Linear Quadratic Gaussian (LQG) team decision problems were presented without considering the effects of communication constraints. In the presence of limited capacity finite alphabet channels, stabilizing controllers were also given in [6], [7], [8], [9], [10], [11]. These results are mostly concerned with deterministic systems (e.g., [6], [7], [8], [10]). In this paper we generalize the results of [2], [3], [4], [5] by considering the effects of communication imperfections. We also generalize the results of [6], [7], [8], [9], [10], [11] by addressing the stability problem of stochastic dynamic systems over AWGN channels.

The paper is organized as follows: In Section 2, the problem formulation is given. In Section 3, design techniques for decentralized controllers, encoders, and decoders are presented. In Section 4, we compare the rate requirement for stability using the proposed decentralized technique with the minimum rate requirement for stability using the centralized technique of [12]. The paper is concluded in Section 5 with a summary of proposed techniques as developed here.

Section snippets

Problem formulation

Throughout, certain conventions are used: Sequences of Random Vectors (R.V.’s) are denoted by y(T)(y0,y1,,yT) or Y(T)(Y0,Y1,,YT) for TN+{0,1,2,}. A logarithm of base 2 is denoted by log() and the Euclidean norm with weight R on any finite dimensional space is denoted by R. The space of all matrices Aq×o is denoted by M(q×o) and the transpose of A, where A can be either a matrix or a vector, is denoted by A. The identity matrix with dimension M(q×q) is denoted by Iq, the inverse of

Control through communication channels

In this section we present decentralized controllers, encoders and decoders for mean square stability and reliable communication when the communication links are AWGN channels.

Throughout this section it is assumed that subsystems si and sj in cluster Sr have overlapped communication range so that they can exchange information. Each subsystem in cluster Sr broadcasts encoded observation signal to other subsystems. Hence, there is a possibility of collision in the broadcast information. In large

Comparison

In this section, we compare the rate requirement for stability using the decentralized technique of Section 3.1 with the minimum rate requirement for stability using the centralized technique of [12]. For simplicity of comparison, we consider the large scale system of Section 3 with only two clusters S3 and S2 (S1 is omitted). That is, we consider the following system: {Xt+1=AXt+BUt+CWt,yt(i)=Fixt(i)+Givt(i),i=3,4,5,6, where Xt=(xt(3)xt(4)xt(5)xt(6)),X0N(X̄,V̄),Ut=(ut(3)ut(4)ut(5)ut(6)),Wt=(wt(

Conclusion

In this paper we considered large scale systems with cascaded clusters of linear subsystems, in which the observation signals are exchanged between subsystems via AWGN channels. It was shown that the linear policies stabilize these systems. That was shown by finding a suboptimal solution for quadratic payoff functional which results in mean square stability. A sufficient condition on the capacity of channels for mean square stability was presented. Using an example it was shown that, for the

Acknowledgements

A. Farhadi is a Post Doctoral Fellow at INRIA (the French national institute for research in computer science and control), Grenoble, France. This paper is a result of a research work done in Canada. N.U. Ahmed is a Professor at SITE, the University of Ottawa, Canada. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant no. A7109.

References (24)

  • A.S. Matveev, A.V. Savkin, Stabilization of multi sensors networked control systems with communication constraints, in:...
  • G.N. Nair, R.J. Evans, P.E. Caines, Stabilising decentralised linear systems under data rate constraints, in:...
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