This paper deals with stability of discrete-time switched linear systems whose all subsystems are unstable. We present sufficient conditions on the subsystems matrices such that a switched system is globally exponentially stable under a set of purely time-dependent switching signals that are allowed to activate all subsystems. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-theoretic arguments. We present a numerical experiment to demonstrate our results.
Indian Institute of Science Bangalore（班加罗尔印度科学研究所）
We present an algorithm to compute stabilizing minimum dwell times for discrete-time switched linear systems without the explicit knowledge of state-space models of their subsystems. Given a set of finite traces of state trajectories of the subsystems that satisfies certain properties, our algorithm involves the following tasks: first, multiple Lyapunov functions are designed from the given data; second, a set of relevant scalars is computed from these functions; and third, a stabilizing minimum dwell time is determined as a function of these scalars. A numerical example is presented to demons…
This paper deals with classes of (de)stabilizing switching signals for switched systems. Most of the available conditions for stability of switched systems are sufficient in nature, and consequently, their violation does not conclude instability of a switched system. The study of instability is, however, important for obvious reasons. Our contributions are twofold: Firstly, we propose a class of switching signals under which a continuous-time switched linear system is unstable. Our characterization of instability depends solely on the asymptotic behaviour of frequency of switching, frequency o…
This paper deals with stability of discrete-time switched linear systems whose all subsystems are unstable and the set of admissible switching signals obeys pre-specified restrictions on switches between the subsystems and dwell times on the subsystems. We derive sufficient conditions on the subsystems matrices such that a switched system is globally exponentially stable under a set of purely time-dependent switching signals that obeys the given restrictions. The main apparatuses for our analysis are (matrix) commutation relations between certain products of the subsystems matrices and graph-t…
Stability Analysis of Continuous-Time Switched Systems with a Random Switching Signal
University of Science and Technology of China（中国科学技术大学）；University of Hong Kong（香港大学）；University of Southampton（南安普顿大学）；University of Strathclyde（思克莱德大学）
The dynamical behavior of switched affine systems is known to be more intricate than that of the well-studied switched linear systems, essentially due to the existence of distinct equilibrium points for each subsystem. First, under arbitrary switching rules, the stability analysis must be generally carried out with respect to a compact set with non-empty interior rather than to a singleton. We provide a novel proof technique for existence and outer approximation of attractive invariant sets of a switched affine system, under the hypothesis of global uniform stability of its linearization. On t…
We study the synthesis of mode switching protocols for a class of discrete-time switched linear systems in which the mode jumps are governed by Markov decision processes (MDPs). We call such systems MDP-JLS for brevity. Each state of the MDP corresponds to a mode in the switched system. The probabilistic state transitions in the MDP represent the mode transitions. We focus on finding a policy that selects the switching actions at each mode such that the switched system that follows these actions is guaranteed to be stable. Given a policy in the MDP, the considered MDP-JLS reduces to a Markov j…
A novel procedure for the online identification of a class of discrete-time switched linear systems, which simultaneously estimates the parameters and switching manifolds of the systems, is proposed in this paper. Firstly, to estimate the parameters of the subsystems, a discrete-time concurrent learning-based recursive parameter estimator is designed to guarantee the exponential convergence of the estimation errors to zero. Secondly, as an assistant procedure of the identification framework, an online switching detection method is proposed by making use of the history stacks produced by the co…
Stabilizing switching signals: a transition from point-wise to asymptotic conditions
Robert Bosch Centre for Cyber-Physical Systems, Indian Institute of Science（印度科学学院罗伯特博世网络物理系统中心）；Systems & Control Engineering, Indian Institute of Technology Bombay（孟买印度理工学院系统与控制工程系）
In this work, a data-driven modeling framework of switched dynamical systems under time-dependent switching is proposed. The learning technique utilized to model system dynamics is Extreme Learning Machine (ELM). First, a method is developed for the detection of the switching occurrence events in the training data extracted from system traces. The training data thus can be segmented by the detected switching instants. Then, ELM is used to learn the system dynamics of subsystems. The learning process includes segmented trace data merging and subsystem dynamics modeling. Due to the specific lear…
Distributed switched large-scale systems are composed by dynamically coupled subsystems, in which interactions among subsystems vary over time according to an exogenous input signal named switching signal. In this paper, we present a distributed robust switched model-based predictive control (DSwMPC) to control such systems that are subject to local state and input constraints. The proposed method guarantees stabilizing the origin of the whole closed-loop system and ensures the constraints satisfaction in the presence of an unknown switching signal. In the distributed model-based predictive co…
Indian Institute of Science Bangalore（印度科学理工学院班加罗尔）
This paper deals with stabilization of discrete-time switched linear systems when explicit knowledge of the state-space models of their subsystems is not available. Given the set of admissible switches between the subsystems, the admissible dwell times on the subsystems and a set of finite traces of state trajectories of the subsystems that satisfies certain properties, we devise an algorithm that designs periodic switching logics which preserve stability of the resulting switched system. We combine two ingredients: (a) data-based stability analysis of discrete-time linear systems and (b) mult…
Delft University of Technology（代尔夫特理工大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
For the identification of switched systems with a measured switching signal, this work aims to analyze the effect of switching strategies on the estimation error. The data for identification is assumed to be collected from globally asymptotically or marginally stable switched systems under switches that are arbitrary or subject to an average dwell time constraint. Then the switched system is estimated by the least-squares (LS) estimator. To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this work. The obt…
This paper deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$-switching-cycle. These two sufficient conditions can be viewed as gener…
Feedback stabilization of switched systems under arbitrary switching: A convex characterization
Aeronautics Institute of Technology (ITA)（巴西空军理工学院）；Politecnico di Torino（都灵理工大学）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes（巴黎萨克雷大学、法国国家科学研究中心、中央苏佩莱克学院、信号与系统实验室）
This paper deals with learning stability of partially observed switched linear systems under arbitrary switching. Such systems are widely used to describe cyber-physical systems which arise by combining physical systems with digital components. In many real-world applications, the internal states cannot be observed directly. It is thus more realistic to conduct system analysis using the outputs of the system. Stability is one of the most frequent requirement for safety and robustness of cyber-physical systems. Existing methods for analyzing stability of switched linear systems often require th…
