Output feedback control of nonlinear systems subject to sensor data losses
Introduction
Recently, an increasing number of control applications which have the control loops closed via a shared communication network have been discussed, see for example [44], [41], [31] and the references therein. These control systems are known as networked control systems (NCS) and differ from standard control systems (in which direct point-to-point links are used), in that the network introduces additional dynamics in the closed-loop system. There are different ways of modeling the dynamics introduced in the closed-loop system by the network, like time-varying delays, data losses or data quantization. In the present work, we focus on output feedback control of nonlinear systems subject to sensor data losses. This class of systems are of particular interest for wireless NCS [33], which play a prominent role in several areas of interest like sensor networks [1], [5], multi-agent systems [4], [40] and chemical processes [25], [27]. New wireless, relatively low cost, sensors are available from vendors. These sensors can be used to implement new control loops, or to add redundancy in already working plants. These sensors are implemented in a wireless setting and are susceptible to communication network interference which would result in time intervals in which readings may not be provided to the control system; this set-up leads to the control problem formulation that is considered in this work. Several applications of networked control systems based on wireless communication links have been presented in the literature, see for example [32], [43], [42], [15].
There are some recent works in the literature focusing on the analysis of the stability and robustness properties of nonlinear systems under state feedback control in the presence of data losses [38], [39], [30], [29], [25], [35], [10]. In these works it is proved that, if the maximum time in which the system operates in open-loop (i.e., without feedback) is small enough, practical stability is guaranteed. However, these results are based on the assumption that full-state measurements are available. In many systems, this assumption does not hold and an output feedback control scheme such as high-gain observers [20], [17], [37], [24], [6], [11], [12], [13] or robust finite-time convergence observers [14], [22] has to be used. However, output feedback control of nonlinear systems subject to sensor data losses has not been studied.
Motivated by the above considerations, we consider the problem of output feedback control of nonlinear systems subject to sensor data losses. Fig. 1 shows a schematic of the class of closed-loop systems under consideration. The process output is fed to the observer, which provides an estimate of the state to the controller. When sensor data losses occur, the observer does not receive new measurements to update the estimated state. In this paper, we study the stability and robustness properties of a combination of a Lyapunov-based controller with a high-gain observer in the presence of sensor data losses. Due to the nature of the fast dynamics of the observer, to obtain results that differ from output feedback control of nonlinear sampled-data systems [9], [19] (which are a degenerate case of nonlinear systems subject to sensor data losses), it is necessary to approximately decouple the dynamics of the observer from the sensor data losses. To this end, the minimum time that the control system operates in closed-loop between consecutive periods without measurements, must be bounded from below. The main idea is that the estimated state must converge to the actual state before new sensor data losses occur. Once the dynamics of the high-gain observer are approximately decoupled from the sensor data losses, practical stability is guaranteed if the maximum time without measurements is smaller than an upper bound that practically depends on the properties of the closed-loop system under state feedback control.
The paper is organized as follows: In Section 2, preliminary notation and results on Lyapunov-based control and high-gain observers are introduced. In Section 3, the main contribution of the paper is presented. In Section 4 we explicitly consider the issue of measurement-sampling in the output feedback controller. In Section 5, the results are demonstrated using a chemical process example. In Section 6, we present some concluding remarks.
Section snippets
Preliminaries
The main objective of this paper is to study the stability and robustness properties of an output feedback controller based on a combination of a Lyapunov-based controller with a high-gain observer with respect to sensor data losses. We assume that this controller has been designed a priori under the assumption of flawless communication. This approach has been followed in previous works to study state feedback control of nonlinear systems subject to sensor data losses, see for example [38], [39]
Stabilization subject to sensor losses
In this section, we consider system (1) subject to sensor data losses in closed-loop with the output feedback controller (4) introduced in the previous section. When sensor data is lost in the sensor link, the observer no longer has access to the output to update the estimated state. There are different potential actions that the controller can take when sensor data is lost. One strategy is to set the input to zero (or any fixed value) [33]. Other approaches [26], [27] use the model of the
Sampled-data output feedback controller
It is important to put into perspective the result of Theorem 1 with respect to the existing results of sampled-data high-gain observer-based output feedback control of nonlinear systems [9], [19]. In this subsection we explicitly consider the issue of measurement-sampling in the output feedback controller and study the robustness to data losses of a sampled-data output feedback controller. We start by scaling the observer variables to avoid inherent ill-conditioning of the partial differential
Application to a chemical reactor
Consider a well mixed, nonisothermal continuous stirred tank reactor where three parallel irreversible elementary exothermic reactions take place of the form , and . is the desired product and and are byproducts. The feed to the reactor consists of pure A at flow rate , temperature and molar concentration . Due to the nonisothermal nature of the reactor, a jacket is used to remove/provide heat to the reactor. Using first principles and standard modeling assumptions, the
Conclusions
In this work, we considered the problem of output feedback control of nonlinear systems subject to sensor data losses. We have studied the stability and robustness properties of an output feedback controller resulting from a combination of a Lyapunov-based controller with a high-gain observer for both the continuous and the sampled-data cases. We have proved that in order to approximately decouple the estimation of the actual state from the feedback stabilization the minimum time between two
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David Muñoz de la Peña is now with the Departmento de Ingeniería de Sistemas y Automática, Universidad de Sevilla, Sevilla, 41004, Spain. Financial support from NSF, CTS-0529295, and MEC, DPI2007-66718-C04-01, is gratefully acknowledged.