Reachability analysis of constrained switched linear systems

In this note, we investigate the reachability of switched linear systems with switching/input constraints. We prove that, under a mild assumption of the feasible switching signals, the reachability set is the reachable subspace of the unconstrained system. We also address the local reachability for switched linear systems with input constraints and present a complete criterion for a general class of switched linear systems.     

基于多面体包含的非线性混成系统可达性分析

针对一类非线性混成系统的可达性问题,提出了一种基于多面体包含的分析方法。首先介绍了混成系统及其可达性,讨论了如何应用多面体包含对多项式混成系统进行线性近似,并采用量词消去和非线性优化方法来构造相应的线性混成系统,然后运用验证工具SpaceEx求得原非线性混成系统的过近似可达集,并应用于验证系统的安全性。     

Reachability of linear switched systems: Differential geometric approach

The paper investigates the structure of the reachable set of linear switched systems. The structure of the reachable set is determined using techniques from classical nonlinear systems theory, namely, the theory of orbits developed by H. Sussman and the realization theory for nonlinear systems developed by B. Jakubczyk.     

Reachability and stabilization of discrete-time affine systems with disturbances

This paper studies the fundamental problems: whether an affine system affected by additive disturbances is robustly transferable from a source set (simplex) to a target set (polytope) and whether it is robustly stabilizable with its state constrained in a simplex. First, a necessary and sufficient condition is derived for the existence of affine feedback control that solves the robust reachability problem. Further investigation is provided for two situations relying on whether the union of the source set and the target set is convex or non-convex. For the former one, a necessary and sufficient condition is obtained in the form of linear inequalities, while for the latter, several computationally feasible sufficient conditions are found. Second, we show that robust stabilization subject to a state constraint is equivalent to find a feasible solution to a linear equation. Once it is known that either of the problems has a solution by checking the derived conditions, design of control laws is then straightforward.     

Perturbation analysis of fuzzy linear systems

This paper deals with the perturbation analysis of fuzzy linear systems. Three cases of perturbation are considered: (a) the right hand side is perturbed while the coefficient matrix remains unchanged; (b) the coefficient matrix is perturbed while the right hand side remains unchanged, and (c) both the coefficient matrix and the right hand side are perturbed. For all of the three cases, the respective relative error bounds for solutions of fuzzy linear system are derived. The results are illustrated by numerical examples.     

Reachability and observability of linear impulsive systems

Linear impulsive systems constitute a class of hybrid systems in which the state propagates according to linear continuous-time dynamics except for a countable set of times at which the state can change instantaneously. While in general these impulsive effects can be time-driven and/or event-driven, here we focus our attention on the time-driven case. For this class of systems, we address the fundamental concepts of reachability and observability. In particular, we present a geometric characterization of the reachable and unobservable sets in terms of invariant subspaces and provide algorithms for their construction.     

Reachability analysis of continuous-time piecewise affine systems

This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system.     

极小-极大-加系统(F,G,H)的能达能观性

在数字电路中,两个时间信号通过逻辑电路的与门相当于极大运算,或门相当于极小运算.因此,极小-极大-加系统可用于数字电路的时间分析.对于非线性极强的极小-极大-加系统(F,G,H),引入了分别能达和上限能观的概念.利用图论的方法给出了极小-极大-加系统(F,G,H)的状态变量xt为分别能达分量的充要条件,同时,还得到了xt为上限能观分量的充要条件.     

The direct solution of banded linear systems by a new matrix factorization procedure

In the finite difference/element discretisation of boundary value problems involving partial differential equations there occurs the problem of solving large order systems of banded linear systems.

Previously, direct methods of solution considered were the usual Gaussian elimination and LU decomposition strategies. In this paper an alternative factorization strategy is presented.     

Reachability analysis of complex planar hybrid systems

Hybrid systems are systems that exhibit both discrete and continuous behavior. Reachability, the question of whether a system in one state can reach some other state, is undecidable for hybrid systems in general. In this paper we are concerned with GSPDIs, 2-dimensional systems generalizing SPDIs (planar hybrid systems based on “simple polygonal differential inclusions”), for which reachability have been shown to be decidable. GSPDIs are useful to approximate 2-dimensional control systems, allowing the verification of safety properties of such systems.     

Reachability analysis of switched linear discrete singular systems

This paper studies the reachability problem of the switched linear discrete singular （SLDS） systems. Under the condition that all subsystems are regular, the reachability of the SLDS systems is characterized based on a peculiar repeatedly introduced switching sequence. The necessary and sufficient conditions are obtained for the reachability of the SLDS systems.     

A theorem on global stabilization of nonlinear systems by linear feedback

In this paper we investigate the global stabilizability problem for a wide class of single-input nonlinear systems whose the linearization at the equilirrium is controllable. We show that under general assumptions there exists a linear feedback law which globally exponentially stabilizes the system at its equilibrium. The proof of our main theorem is based on some ideas from a previous paper. We use the theorem to recover a recent result of Gauthier et al. concerning the observer design problem.     

When does the local controllability of the general model of 2-D linear systems imply its local reachability?

A relationship between the local controllability of the general model of two-dimensional linear systems and its local reachability is considered. Sufficient conditions are established under which the local controllability of the general model implies its local reachability.     

Componentwise stability of linear systems: a non-symmetrical case

In this paper, we present an extension of the concept of componentwise asymptotic (exponential) stability of linear systems in the non-symmetrical case. The main motivation for these results is the need for a more detailed evaluation of the dynamical behaviour of linear systems in electrical engineering and biology. Necessary and sufficient conditions for componentwise asymptotic (exponential) stability in the non-symmetrical case are given. The symmetrical case is obtained as a particular case. © 1997 by John Wiley & Sons, Ltd.     

An Automata-Theoretic Approach to the Reachability Analysis of RPPS Systems

We show how the tree-automata techniques proposed by Lugiez and Schnoebelen apply to the reachability analysis of RPPS systems. Using these techniques requires that we express the states of RPPS systems in a tailor-made process rewrite system where reachability is a relation recognizable by finite tree-automata.     

Partial unknown input reconstruction for linear systems

The problem of partial unknown input (UI) reconstruction is addressed. It is considered that a linear functional of the UI vector has to be reconstructed using output information only. Necessary and sufficient conditions are given allowing for the reconstruction in finite time of the required UI’s; analogous conditions are obtained for the asymptotic reconstruction of the required UI’s. The solution of the problem under consideration provides a means to solve the problem of fault detection and isolation for disturbed linear systems.     

Towards an Efficient Path-Oriented Tool for Bounded Reachability Analysis of Linear Hybrid Systems using Linear Programming

The existing techniques for reachability analysis of linear hybrid automata do not scale well to problem sizes of practical interest. Instead of developing a tool to perform reachability check on all the paths of a linear hybrid automaton, a complementary approach is to develop an efficient path-oriented tool to check one path at a time where the length of the path being checked can be made very large and the size of the automaton can be made large enough to handle problems of practical interest. This approach of symbolic execution of paths can be used by design engineers to check important paths and thereby, increase the faith in the correctness of the system. Unlike simple testing, each path in our framework represents a dense set of possible trajectories of the system being analyzed. In this paper, we develop the linear programming based techniques towards an efficient path-oriented tool for the bounded reachability analysis of linear hybrid systems.