Robust Control Over a Packet-based Network

In this paper, we consider a robust network control problem. We consider linear unstable and uncertain discrete time plants with a network between the sensors and controller and the controller and plant. We investigate two defining characteristics of network controlled systems and the impact of uncertainty on these. Namely, the minimum data rates required for the two networks and the tolerable data drop out in the form of packet losses. We are able to derive sufficient conditions in terms of the minimum data rate and minimum packet arrival rate to ensure stability of the closed loop system. I. I NTRODUCTION Recently, networked control systems (NCS) have gained great attention from both the control community and the network and communication community. When compared with classical feedback control system, networked control systems have many advantages. For example, they can reduce the system wiring, make the system easy to operate and maintain and later diagnose in case of malfunctioning, and increase system agility [16]. In spite of the great advantages that the networked control architecture brings, inserting a network in between the plant and the controller introduces many problems as well. For instance, zero-delayed sensing and actuation, perfect information and synchronization are no longer guaranteed in the new system architecture as only finite bandwidth is available and packet drops and delays may occur due to network traffic conditions. These must be revisited and analyzed before any practical networked control systems are built. In the past decade, many researchers have spent effort on those issues and a number of significant results were obtained and many are in progress. Many of the aforementioned issues are studied separately.  and Sahai [11] have presented some interesting results in the area of control under communication constraints. Specifically, Tatikonda gave a necessary and sufficient condition on the channel data rate such that a noiseless LTI system in the closed loop is asymptotically stable. He also gave rate results for stabilizing a noisy LTI system over the digital channel. Sahai proposed the notion of anytime capacity to deal with real time estimation and control for a networked control system. In our companion paper [13], the authors have considered various rate issues under finite bandwidth, packet drops and finite controls. The effect of pacekt loss and delay on state Control and Dynamical Systems, California Institute of Technology; Pasadena, CA 91125. Email:{shiling, epstein, murray }@cds.caltech.edu; Tel: (626) 395-2313, Fax: (626) 796-8914. Work supported in part by NSF ITR grant CCR-0326554. The authors are equally contributed to this work. estimation was studied by the work of Sinopoli, et. al. in . It has further been investigated by many researchers including the present authors in  and . One of the hallmarks of a good control system design is that the closed loop remain stable in the presence of uncertainty , . While the researchers in  studied the problem of LQG control across a packet dropping networks, not many have considered the norm bounded uncertainty investigated in the present paper. We examine the impact of a norm bounded uncertainty on the network control system and provide sufficient conditions for stability in terms of the minimum data rates and packet arrival rates for the networks. The paper is organized as follows. In Section II, we present the mathematical model of the closed loop system and state our assumptions. In Section III, we state the sufficient conditions for closed loop stability for the case where a network connects the sensors to the controller. In Section IV, we state the sufficient stability conditions where in addition there is a network between the controller and the plant. For both sections we obtain results for scalar and general vector cases. Conclusions and future work are given in the last section. II. PROBLEM SET UP We consider linear discrete time systems with a norm bounded uncertainty in the A matrix. We will investigate two NCS that we will define by the type of networks embedded in the control loop. The first NCS considered has a network between the measurement sensors and the controller, with the controller then directly connected to the actuators/plant. The second NCS will also include a network between the controller and the actuators/plant. These two network types and depicted in Figures 1 and 2. The networks are defined in terms of their data rates and probability of dropping packets. We would consider any packet delays as losses, i. ., we do not use delayed packets for estimation or control. The following equations represent the closed loop system for NCS I . xk+1 = (A + ∆k)xk + Buk (1) yk = λkCxk (2) wherexk ∈ IR is the state of the system, uk ∈ IR is the control input,yk ∈ IR is the output of the system, and λk are Bernoulli i.i.d random variable with parameter λ, i.e., E[λk] = λ for all k. ∆k satisfies∆k ∆k ≤ KI for all k. We also assume the initial condition x0 ∈ IR is bounded. The matrix A is assumed to be unstable without loss of generality as for any matrix A, we can always do some state transformation to decompose the states into stable ones and