Linear Programming based Lower Bounds on Average Dwell-Time via Multiple Lyapunov Functions

With the objective of developing computational methods for stability analysis of switched systems, we consider the problem of finding the minimal lower bounds on average dwell-time that guarantee global asymptotic stability of the origin. Analytical results in the literature quantifying such lower bounds assume existence of multiple Lyapunov functions that satisfy some inequalities. For our purposes, we formulate an optimization problem that searches for the optimal value of the parameters in those inequalities and includes the computation of the associated Lyapunov functions. In its generality, the problem is nonconvex and difficult to solve numerically, so we fix some parameters which results in a linear program (LP). For linear vector fields described by Hurwitz matrices, we prove that such programs are feasible and the resulting solution provides a lower bound on the average dwell-time for exponential stability. Through some experiments, we compare our results with the bounds obtained from other methods in the literature and we report some improvements in the results obtained using our method.Comment: Accepted for publication in Proceedings of European Control Conference 202

On Stability of Measure Driven Differential Equations

International audienceWe consider the problem of stability in a class of differential equations which are driven by a differential measure associated with the inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions present a trade-off between the Lebesgue-integrable and the measure-driven components of the system. In case the system is not asymptotically stable, we derive weaker conditions such that the norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems

Input-to-State Stabilization in H1H^1-Norm for Boundary Controlled Linear Hyperbolic PDEs with Application to Quantized Control

International audienceWe consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. For this system class, the problem of designing dynamic controllers for input-to-state stabilization in $H^1$-norm with respect to measurement errors is considered. The analysis is based on constructing a Lyapunov function for the closed-loop system which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived

Passivity-Based Observer Design for a Class of Lagrangian Systems with Perfect Unilateral Constraints

International audienceThis paper addresses the problem of estimating the velocity variables, using the position measurement as output, in nonlinear Lagrangian dynamical systems with perfect unilateral constraints. The dynamics of such systems are formulated as a measure differential inclusion (MDI) at velocity level which naturally encodes the relations for prescribing the post-impact velocity. Under the assumption that the velocity of the system is uniformly bounded, an observer is designed which is also a measure differential inclusion. It is proved that there exists a unique solution to the proposed observer and that solution converges asymptotically to the actual velocity

On Output Regulation in Systems with Differential Variational Inequalities (Long Version)

International audienceWe consider the problem of designing state feedback control laws for output regulation in a class of dynamical systems which are described by variational inequalities and ordinary differential equations. In our setup, these variational inequalities are used to model state trajectories constrained to evolve within time-varying, closed, and convex sets, and systems with complementarity relations. We first derive conditions to study the existence and uniqueness of solutions in such systems. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: first, a static feedback control law is derived when full state feedback is available; In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing switching signals for regulation in power converters

Supplementary material of "Back-and-forth Operation of State Observers and Norm Estimation of Estimation Error"

authors' final versionThis is a supplementary material for the paper ``Back-and-forth Operation of State Observers and Norm Estimation of Estimation Error'' that will be presented at the 51st IEEE Conference on Decision and Control, December, 2012

An Inversion-Based Approach to Fault Detection and Isolation in Switching Electrical Networks

Abstract-This paper proposes a framework for fault detection and isolation (FDI) in electrical energy systems based on techniques developed in the context of invertibility of switched systems. In the absence of faults-the nominal mode of operation-the system behavior is described by one set of linear differential equations or more in the case of systems with natural switching behavior, e.g., power electronics systems. Faults are categorized as hard and soft. A hard fault causes abrupt changes in the system structure, which results in an uncontrolled transition from the nominal mode of operation to a faulty mode governed by a different set of differential equations. A soft fault causes a continuous change over time of certain system structure parameters, which results in unknown additive disturbances to the set(s) of differential equations governing the system dynamics. In this setup, the dynamic behavior of an electrical energy system (with possible natural switching) can be described by a switched state-space model where each mode is driven by possibly known and unknown inputs. The problem of detection and isolation of hard faults is equivalent to uniquely recovering the switching signal associated with uncontrolled transitions caused by hard faults. The problem of detection and isolation of soft faults is equivalent to recovering the unknown additive disturbance caused by the fault. Uniquely recovering both switching signal and unknown inputs is the concern of the (left) invertibility problem in switched systems, and we are able to adopt theoretical results on that problem, developed earlier, to the present FDI setting. The application of the proposed framework to fault detection and isolation in switching electrical networks is illustrated with several examples. Index Terms-Electrical energy systems, fault detection and isolation (FDI), invertibility, switched linear systems, switch-singular pairs

Invertibility and observability of switched systems with inputs and outputs

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 small-scale 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 switch-singular 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 switch-singular 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 switch-singular 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 dwell-time 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 time-varying systems, and in contrast to time-invariant 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 observer-based 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

Suboptimal Filtering over Sensor Networks with Random Communication

International audienceThe problem of filter design is considered for linear stochastic systems using distributed sensors. Each sensor unit, represented by a node in an an undirected and connected graph, collects some information about the state and communicates its own estimate with the neighbors. It is stipulated that this communication between sensor nodes is time-sampled randomly and the sampling process is assumed to be a Poisson counter. Our proposed filtering algorithm for each sensor node is a stochastic hybrid system: It comprises a continuous-time differential equation, and at random time instants when communication takes place, each sensor node updates its state estimate based on the information received by its neighbors. In this setting, we compute the expectation of the error covariance matrix for each unit which is governed by a matrix differential equation. To study the asymptotic behavior of these covariance matrices, we show that if the gain matrices are appropriately chosen and the mean sampling rate is large enough, then the error covariances practically converge to a constant matrix