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Title:  Invertibility and observability of switched systems with inputs and outputs 
Author(s):  Tanwani, Aneel 
Director of Research:  Liberzon, Daniel M. 
Doctoral Committee Member(s):  Kumar, P.R.; DomínguezGarcía, Alejandro D.; Mitra, Sayan 
Department / Program:  Electrical & Computer Eng 
Discipline:  Electrical & Computer Engr 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  switched systems
invertibility observability observer design switching signal reconstruction. 
Abstract:  Hybrid dynamical systems or switched systems can operate in several different modes, with some discrete dynamics governing the mode changes. Each mode of operation is described by a dynamical subsystem having an internal state, an external input (which can be thought of as a disturbance or a control signal), and a measured output. Hybrid/switched systems may arise in practice because of the interaction of digital devices with physical components in order to implement control schemes, or due to integration of smallscale systems to form a large network, or due to transitions occurring in the model of some physical phenomenon. Because of the richness of their application, switched systems have attracted the attention of many researchers over the past decade for the study of analysis and control design problems. In this thesis, we analyze the properties of invertibility and observability for switched systems and study their related applications in system design. The common facet to both these problems involves the extraction of unknown variables from the knowledge of the output. It is well known that, under certain assumptions, the state trajectory and the output response of any dynamical system are uniquely defined once the initial condition and the input are fixed. Broadly speaking, if the output is assumed to be known, the problems considered in our work deal with: (a)~the reconstruction of the input when the initial state is known, or (b)~the recovery of the initial state when the inputs are known; the former is called the invertibility problem and the latter is called observability. Invertibility is an important property in system design and system security analysis, and has only recently been studied for switched systems. Since we treat the switching signal as an exogenous signal, invertibility of switched systems relates to the ability to reconstruct the unknown input and the unknown switching signal from the knowledge of the measured output and the initial state. The thesis addresses the invertibility problem of switched systems where the subsystem dynamics are nonlinear but affine in controls. The novel concept of switchsingular pairs, which arises in the reconstruction of the switching signal, is extended to nonlinear systems and a formula is developed for checking if the given state and output form a switchsingular pair. We give a necessary and sufficient condition for a switched system to be invertible, which says that the subsystems should be invertible and there should be no switchsingular pairs. In case a switched system is invertible, one can build a switched inverse system to reconstruct the switching signal and the input. The setup naturally leads to an algorithm for output generation where a prescribed reference signal is generated using the system dynamics. In practice, the exact knowledge of the initial condition and the output may be an overly stringent requirement for invertibility of the system. We relax this requirement by allowing disturbances in the output and uncertainties in the knowledge of the initial condition. Using the theory of reachable sets, an alternative formulation for reconstruction of the switching signal is presented. To relieve the computational burden, we utilize the notion of a gap between subspaces for mode detection that involves merely coarse spherical approximation of the reachable set. This approach of using the reachable sets, though applicable to a general class of linear systems, may not reconstruct switching over large time intervals as the uncertainties in the state may grow to an extent that the outputs of the subsystems become indistinguishable. However, if the individual subsystems are assumed to be minimum phase, which is the same as assuming the stability of the minimal order inverse system in the linear case, then the switching signal can be reconstructed for all times under the dwelltime assumption. Another important property for diagnostic applications and system design is the observability of switched systems. It is seen that the switched systems essentially act as timevarying systems, and in contrast to timeinvariant systems, the ability to recover the state either instantaneously or after some time has different meanings as the information available after switching, from another subsystem, may reveal more knowledge about the state. This idea of gathering information from all the active subsystems is formalized to yield a characterization of observability for switched linear systems. A related, but relatively weaker, notion of determinability deals with recovering the value of the state at some time in the future rather than the initial time. This turns out to be particularly useful in the construction of observers, as the estimates generated by the observers are shown to converge asymptotically to the true state when the switched system is determinable. Similar concepts are studied for another class of switched systems where the underlying subsystems are modeled with differential algebraic equations instead of ordinary differential equations, but the observer design remains a topic of further study in such systems. The problem of observability is also studied in the context of switched nonlinear systems. Because of the rich nature of the dynamics of such systems and the fact that analytical solutions of the nonlinear ordinary differential equations are not always available, the framework of linear systems is not easily extendable. We therefore propose an alternate approach to derive a sufficient condition for observability in nonlinear switched systems. This condition naturally leads to an observer design, and with the help of analysis, it is shown that the corresponding state estimate indeed converges to the actual state of the system. An effort is made to obtain a characterization in the form of a necessary and sufficient condition for observability. Examples are included throughout the text to help understand the underlying concepts. Having discussed the properties of invertibility and observability from an analytical perspective, we then discuss an application of these theoretical concepts to study the problem of fault detection in electrical energy systems. The tools developed for solving the invertibility and observability problem have been tailored to address the models of voltage converters and their networks. Categorizing soft faults as unknown disturbances and hard faults as unknown mode transitions, we show that such faults can be recovered if the switched system under consideration is invertible. An algorithm for fault detection and results of simulation are included to demonstrate the utility of the proposed framework. Since the invertibility approach requires the knowledge of the initial condition and the derivatives of the output to reconstruct the soft faults, an alternative observerbased approach is presented for detection of soft faults. Because the initial condition is no longer assumed to be known, the observer dynamics first estimate the state of the system, and then we define auxiliary observer outputs that are only sensitive to faults so that the effect of a nonzero fault is reflected in those new outputs. A significant aspect of structural properties is their utility in solving some of the prominent design problems, and the concepts related to invertibility of switched systems are utilized in designing switching signals and control inputs for generating desired output trajectories. We conclude the document by proposing some synthesis problems using the system inversion tools. A desired property for the control input in output generation and tracking is its boundedness relative to the size of the output. Classically, this is achieved by requiring the inverse system to be stabilizable. We extend this idea to switched systems to propose a preliminary result for computing bounded inputs that generate a desired bounded output trajectory. If the initial condition is not known, then exact output generation may not be possible and in that case, tracking the output asymptotically is the problem of interest. We present our initial approach on how to achieve output tracking in switched systems and outline the methods for our future work related to this problem. 
Issue Date:  20120206 
Genre:  thesis 
URI:  http://hdl.handle.net/2142/29755 
Rights Information:  Copyright 2011 Aneel Tanwani 
Date Available in IDEALS:  20120206 
Date Deposited:  201112 
This item appears in the following Collection(s)

Dissertations and Theses  Electrical and Computer Engineering
Dissertations and Theses in Electrical and Computer Engineering 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois