Aneel Tanwani

Observability for Switched Linear Systems: Characterization and Observer Design

This paper presents a characterization of observability and an observer design method for switched linear systems with state jumps. A necessary and sufficient condition is presented for observability, globally in time, when the system evolves under predetermined mode transitions. Because this characterization depends upon the switching signal under consideration, the existence of singular switching signals is studied alongside developing a sufficient condition that guarantees uniform observability with respect to switching times. Furthermore, while taking state jumps into account, a relatively weaker characterization is given for determinability, the property that concerns with recovery of the original state at some time rather than at all times. Assuming determinability of the system, a hybrid observer is designed for the most general case to estimate the state of the system and it is shown that the estimation error decays exponentially. Since the individual modes of the switched system may not be observable, the proposed strategy for designing the observer is based upon a novel idea of accumulating the information from individual subsystems. Contrary to the usual approach, dwell-time between switchings is not necessary, but the proposed design does require persistent switching. For practical purposes, the calculations also take into account the time consumed in performing computations.

Invertibility of switched nonlinear systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.

Observability implies observer design for switched linear systems

This paper presents a characterization of observability and an observer design method for a class of hybrid systems. A necessary and sufficient condition is presented for observability, globally in time, when the system evolves under predetermined mode transitions. A relatively weaker characterization is given for determinability, the property that concerns with recovery of the original state at some time rather than at all times. These conditions are then utilized in the construction of a hybrid observer that is feasible for implementation in practice. The observer, without using the derivatives of the output, generates the state estimate that converges to the actual state under persistent switching.

On observability of switched differential-algebraic equations

We investigate observability of switched differential algebraic equations. The article primarily focuses on a class of switched systems comprising of two modes and a switching signal with a single switching instant. We provide a necessary and sufficient condition under which it is possible to recover the value of state trajectory (globally in time) with the help of switching phenomenon, even though the constituent subsystems may not be observable. In case the switched system is not globally observable, we discuss the concept of forward observability which deals with the recovery of state trajectory after the switching. A necessary and sufficient condition that characterizes forward observability is presented.

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

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.

Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization ☆

We consider the problem of output feedback stabilization in linear systems when the measured outputs and control inputs are subject to event-triggered sampling and dynamic quantization. A new sampling algorithm is proposed for outputs which does not lead to accumulation of sampling times and results in asymptotic stabilization of the system. The approach for output sampling is based on defining an event function that compares the difference between the current output and the most recently transmitted output sample not only with the current value of the output, but also takes into account a certain number of previously transmitted output samples. This allows us to reconstruct the state using an observer with sample-and-hold measurements. The estimated states are used to generate a control input, which is subjected to a different event-triggered sampling routine; hence the sampling times of inputs and outputs are asynchronous. Using Lyapunov-based approach, we prove the asymptotic stabilization of the closed-loop system and show that there exists a minimum inter-sampling time for control inputs and for outputs. To show that these sampling routines are robust with respect to transmission errors, only the quantized (in space) values of outputs and inputs are transmitted to the controller and the plant, respectively. A dynamic quantizer is adopted for this purpose, and an algorithm is proposed to update the range and the centre of the quantizer that results in an asymptotically stable closed-loop system.

Invertibility of nonlinear switched systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with finding the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if 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. When all the subsystems are invertible, we demonstrate output tracking by finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.

Robust invertibility of switched linear systems

In this paper, we address the effects of uncertainties in output measurements and initial conditions on invertibility of the switched systems - the problem concerned with the recovery of the input and the switching signal using the output and the initial state. By computing the reachable sets and maximal error in the propagation of state trajectories through the inverse system, we derive conditions under which it is possible to recover the exact switching signal over a certain time interval, provided the uncertainties are bounded in some sense. In addition, we discuss separately the case where each subsystem is minimum phase and it is possible to recover the exact switching signal globally in time. The input, though, is recoverable only up to a neighborhood of the original input.

On using norm estimators for event-triggered control with dynamic output feedback

For feedback stabilization of a control system using dynamic output feedback, we consider the problem of finding two different sequences of time instants at which the sampled outputs (respectively, control inputs) must be sent to the controller (resp. the plant). Instead of static inequalities, the states of so-called norm estimators are used to determine sampling instants. Using the tools from Lyapunov theory for hybrid systems and stability of cascaded nonlinear systems, it is shown that the closed loop system is globally asymptotically stable. Additional assumptions are required on the controller and system dynamics to guarantee that the proposed sampling routines do not lead to an accumulation of sampling times over a finite interval.

Observability of Switched Linear Systems

Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. The main difference in these approaches is how the switching signal is viewed: Is it a fixed and known function of time, is it an unknown external signal, is it the result of a discrete dynamical system (an automaton) or is it controlled and is therefore an input? We will focus on the recently introduced geometric characterization of observability, which assumes knowledge of the switching signal. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times. To relieve the computational burden, some relaxed conditions that do not rely on the switching times are given; this also allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential algebraic systems is possible and presented as well.