Stability Analysis of Hybrid Jump Linear Systems With Markov Inputs

In the past two decades, the number of applications that make use of supervisory algorithms to control complex continuous-time or discrete-time systems has increased steadily. Typical examples include air traffic management, digital control systems over networks, and flexible manufacturing systems. A common feature of these applications is the intermixing of the continuous dynamics of the controlled plant with the logical and discrete dynamics of the supervising algorithms. These so-called hybrid systems are the focus of much ongoing research. To improve the performance of these systems, it is important to analyze the interactions between the supervising algorithms and the plant. Few papers have studied this interaction when the plant is represented by a discrete-time system. Thus, this dissertation fixes this deficiency by addressing the following three main objectives: to introduce a new modeling framework for discrete-time stochastic hybrid systems suitable for stability analysis; to derive testable stability conditions for these models; and to demonstrate that these models are suitable to study real-world applications. To achieve the first objective, the Hybrid Jump Linear System model is introduced. Although it has many of the same modeling capabilities as other formalisms in the literature (e.g., Discrete Stochastic Hybrid Automata), it possesses the unique advantage of representing the dynamics of both the controlled plant and the supervising algorithm in the same analytical framework: stochastic difference equations. This enables the study of their joint properties such as, for example, mean square stability. The second objective is addressed by developing a collection of testable sufficient mean square stability conditions. These tests are developed by applying, successively, switched systems\u27 techniques, singular value analysis, a second moment lifting technique, and Mark off kernel methods. The final objective is achieved by developing a hybrid jump linear system model of an AFTI-F16 flight controller deployed on a fault tolerant computer with rollback and cold-restart capabilities, and analyzing its stability properties

Square-root filtering via covariance SVD factors in the accurate continuous-discrete extended-cubature Kalman filter

This paper continues our research devoted to an accurate nonlinear Bayesian filters' design. Our solution implies numerical methods for solving ordinary differential equations (ODE) when propagating the mean and error covariance of the dynamic state. The key idea is that an accurate implementation strategy implies the methods with a discretization error control involved. This means that the filters' moment differential equations are to be solved accurately, i.e. with negligible error. In this paper, we explore the continuous-discrete extended-cubature Kalman filter that is a hybrid method between Extended and Cubature Kalman filters (CKF). Motivated by recent results obtained for the continuous-discrete CKF in Bayesian filtering realm, we propose the numerically stable (to roundoff) square-root approach within a singular value decomposition (SVD) for the hybrid filter. The new method is extensively tested on a few application examples including stiff systems

Qualitative Properties of Hybrid Singular Systems

A singular system model is mathematically formulated as a set of coupled differential and algebraic equations. Singular systems, also referred to as descriptor or differential algebraic systems, have extensive applications in power, economic, and biological systems. The main purpose of this thesis is to address the problems of stability and stabilization for singular hybrid systems with or without time delay. First, some su cient conditions on the exponential stability property of both continuous and discrete impulsive switched singular systems with time delay (ISSSD) are proposed. We address this problem for the continuous system in three cases: all subsystems are stable, the system consists of both stable and unstable subsystems, and all subsystems are unstable. For the discrete system, we focus on when all subsystems are stable, and the system consists of both stable and unstable subsystems. The stability results for both the continuous and the discrete system are investigated by first using the multiple Lyapunov functions along with the average-dwell time (ADT) switching signal to organize the jumps among the system modes and then resorting the Halanay Lemma. Second, an optimal feedback control only for continuous ISSSD is designed to guarantee the exponential stability of the closed-loop system. Moreover, a Luenberger-type observer is designed to estimate the system states such that the corresponding closed-loop error system is exponentially stable. Similarly, we have used the multiple Lyapunov functions approach with the ADT switching signal and the Halanay Lemma. Third, the problem of designing a sliding mode control (SMC) for singular systems subject to impulsive effects is addressed in continuous and discrete contexts. The main objective is to design an SMC law such that the closed-loop system achieves stability. Designing a sliding surface, analyzing a reaching condition and designing an SMC law are investigated throughly. In addition, the discrete SMC law is slightly modi ed to eliminate chattering. Last, mean square admissibility for singular switched systems with stochastic noise in continuous and discrete cases is investigated. Sufficient conditions that guarantee mean square admissibility are developed by using linear matrix inequalities (LMIs)

Hybrid dynamical control based on consensus algorithms for current sharing in DC-bus microgrids

Abstract|The main objective of this work is to propose a novel paradigm for the design of two layers of control laws for DC-bus microgrids in islanded mode. An intensive attention will be paid to the inner control level for the regulation of DC-DC electronic power converters, where the use of Hybrid Dynamical System theory will be crucial to formulate and ex- ploit switching control signals in view of reducing the dissipated energy and improving system performance. Indeed, this recent theory is well suited for analysis of power electronic converters, since they combine continuous (voltage and currents) and discrete (on-o state of switches) signals avoiding, in this way, the use of averaged models. Likewise, an outer control level for controlling DC-bus microgrids will be developed to provide a distributed strategy that makes the microgrid scalable and robust with respect to black- outs of sources and/or loads, following the principle of t Multi-Agent System theory. In this distributed strategy, they are several crucial and innovative as- pects to be regarded such as the di erent converter architectures, the hybrid and nonlinear nature of these converters. Stability properties are guaranteed by using singular perturbation analysis.Grant “Hybrid self-adaptive multi-agent systems for microgrids (HISPALIS

Quasi-optimal robust stabilization of control systems

In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise

Sampling from a system-theoretic viewpoint: Part II - Noncausal solutions

This paper puts to use concepts and tools introduced in Part I to address a wide spectrum of noncausal sampling and reconstruction problems. Particularly, we follow the system-theoretic paradigm by using systems as signal generators to account for available information and system norms (L2 and L∞) as performance measures. The proposed optimization-based approach recovers many known solutions, derived hitherto by different methods, as special cases under different assumptions about acquisition or reconstructing devices (e.g., polynomial and exponential cardinal splines for fixed samplers and the Sampling Theorem and its modifications in the case when both sampler and interpolator are design parameters). We also derive new results, such as versions of the Sampling Theorem for downsampling and reconstruction from noisy measurements, the continuous-time invariance of a wide class of optimal sampling-and-reconstruction circuits, etcetera

Stability of singular jump-linear systems with a large state space : a two-time-scale approach

This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the modulating force has a large state space. We focus on stability of such hybrid singular systems. To carry out the analysis, we use a two-time-scale formulation, which is based on the rationale that, in a large-scale system, not all components or subsystems change at the same speed. To highlight the different rates of variation, we introduce a small parameter ε>0. Under suitable conditions, the system has a limit. We then use a perturbed Lyapunov function argument to show that if the limit system is stable then so is the original system in a suitable sense for ε small enough. This result presents a perspective on reduction of complexity from a stability point of view

Weak Singular Hybrid Automata

The framework of Hybrid automata, introduced by Alur, Courcourbetis, Henzinger, and Ho, provides a formal modeling and analysis environment to analyze the interaction between the discrete and the continuous parts of cyber-physical systems. Hybrid automata can be considered as generalizations of finite state automata augmented with a finite set of real-valued variables whose dynamics in each state is governed by a system of ordinary differential equations. Moreover, the discrete transitions of hybrid automata are guarded by constraints over the values of these real-valued variables, and enable discontinuous jumps in the evolution of these variables. Singular hybrid automata are a subclass of hybrid automata where dynamics is specified by state-dependent constant vectors. Henzinger, Kopke, Puri, and Varaiya showed that for even very restricted subclasses of singular hybrid automata, the fundamental verification questions, like reachability and schedulability, are undecidable. In this paper we present \emph{weak singular hybrid automata} (WSHA), a previously unexplored subclass of singular hybrid automata, and show the decidability (and the exact complexity) of various verification questions for this class including reachability (NP-Complete) and LTL model-checking (PSPACE-Complete). We further show that extending WSHA with a single unrestricted clock or extending WSHA with unrestricted variable updates lead to undecidability of reachability problem