On reachable set bounding for discrete-time switched nonlinear positive systems with mixed time-varying delays and disturbance

This paper addresses the reachable set bounding for discrete-time switched nonlinear positive systems with mixed time-varying delays and disturbance, which contains switched linear positive systems as a special case. By resorting to a new method that does not involve the common Lyapunov–Krasovskii functional one, explicit criteria to ensure any state trajectory of the system converges exponentially into a prescribed sphere are obtained under average dwell time switching. The results can then be extended to more general time-varying systems. Finally, two numerical examples are used to demonstrate the effectiveness of the obtained results.     

A geometric approach for reachability and observability of linear switched impulsive systems

This paper is concerned with the reachability and observability of linear switched impulsive systems with singular impulse matrices. First some new concepts with respect to the reachability and unobservability are introduced. Especially, span reachability is proposed because the reachable sets of switched impulsive systems do not always constitute subspaces. Then the geometric characterization of the span reachable and unobservable sets is presented. Moreover, the relations between the span reachable set, unobservable set and the invariant subspaces of such systems are discussed. Finally, corresponding criteria applied to linear impulsive systems and linear switched systems are also discussed.     

A Survey of Reachability and Controllability for Positive Linear Systems

This paper is a survey of reachability and controllability results for discrete-time positive linear systems. It presents a variety of criteria in both algebraic and digraph forms for recognising these fundamental system properties with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control, manpower planning, scheduling and other areas of operations research) but also in studying properties of reachable sets, in feedback control problems, and others. The paper highlights the intrinsic combinatorial structure of reachable/controllable positive linear systems and reveals the monomial components of such systems. The system matrix decomposition into monomial components is demonstrated by solving some illustrative examples.     

Approximating Reachable Sets for n-dimensional Linear Discrete Systems

We present a technique for approximating the reachable set fromthe origin for n-dimensional discrete-time linear systems subjectto bounded control, where the state matrix is stable and diagonalizable.The procedure is based on decomposing the system into one- andtwo-dimensional subsystems for which the reachable set of eachof the subsystems can be over-estimated by a polygon. Thesepolygons are then used to create an n-dimensional polyhedronwhich contains the reachable set of the original system     

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.     

Output controllability and optimal output control of positive fractional order linear discrete system with multiple delays in state, input and output

The article concerns output controllability and optimal output control of positive fractional order discrete linear systems with multiple delays in state, input and output. Necessary and sufficient conditions for output reachability (output controllability from zero initial conditions) and null output controllability (output controllability to zero final output) are given and proven. We also prove that the positive system is output controllable if it is output reachable and null output controllable with the output reachability index is equal or less than the null output controllability index. Sufficient conditions for the solvability of the optimal output control problem are given. Numerical examples are presented to illustrate the theoretical results.     

Robust tube-based MPC of constrained piecewise affine systems with bounded additive disturbances

In this article, we propose a robust tube-based MPC formulation for a class of hybrid systems, namely autonomously switched PWA systems, with bounded additive disturbances. The term tube-based refers to those control techniques whose objective is to maintain all possible trajectories of the uncertain system inside a tube which is a set around the nominal (or reference) system trajectory, that is free from disturbances. Common methods in tube-based control systems consider an error dynamical system as the difference between the state of the nominal system and the state of the perturbed system. However, this definition of the error dynamical system leads to a complicated switched affine system for PWA systems. Therefore, we use a new notion of the reference system similar to the nominal system except that the switching between the various modes of the PWA system is driven by the state of the real system. Using this reference system instead of the nominal system leads us to an error dynamical system that can be modeled as a switched linear system. We employ a switched linear controller to stabilize this error system under arbitrary switching. This auxiliary controller forces the states of the uncertain system to remain in a tube confined to the invariant set around the state of the reference system. We add new constraints and tighten some other constraints of the nominal hybrid MPC for the reference system, in order to ensure convergence of the uncertain system and to guarantee robust exponential stability of the closed-loop system.     

Asymptotic stability and stabilizability of special classes of discrete-time positive switched systems

In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by monomial matrices or by circulant matrices. Necessary and sufficient conditions are provided guaranteeing either (global uniform) asymptotic stability or stabilizability (i.e. the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence). Such conditions lead to simple algorithms that allow to easily detect, under suitable conditions, whether a given positive switched system is not stabilizable.     

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.     

Asymptotical feedback controllability of continuous-time probabilistic logic control networks

In this article, we study the controllability of continuous-time probabilistic logic control networks (CT-PLCNs) under sampled-data feedback controls (SDFCs). First, we demonstrate that the concept of finite-time controllability with probability one for discrete-time probabilistic logic control networks cannot be generalized to CT-PLCNs. Then, we propose the concepts of asymptotical feedback reachability and asymptotical feedback controllability for CT-PLCNs. Based on the invariant subsets, we prove that a target state is asymptotically feedback reachable if and only if the target state is a control equilibrium point and any initial state has an admissible path to the target state. Moreover, we introduce the concept of reachability matrix and propose an easily verifiable criterion for asymptotical feedback reachability expressed in terms of the reachability matrix. Based on these, we prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is a control equilibrium point and there is an admissible path between any pair of initial and target states. The relation between controllability and stabilizability is also discussed. We prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is asymptotically feedback stabilizable. For a controllable CT-PLCN, we propose an algorithm of designing a stabilizing SDFC for any given target state. Additionally, we discuss the asymptotical feedback controllability of CT-PLCNs under time-varying nonuniform SDFCs. Finally, an illustrative example is presented to explain the proposed methods and verify the controllability criteria.     

Computing reachable sets for uncertain nonlinear monotone systems

We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work Ramdani (2008) [22], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of off-diagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields component-wise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, component-wise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.     

Commuting and stable feedback design for switched linear systems

In the paper, commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching.     

Interval techniques for enclosures of regions of reachability and controllability and for guaranteed state and parameter estimation of dynamical systems

Interval techniques are a powerful means for calculation of enclosures of the regions of reachability and controllability of dynamical systems with uncertainties during analysis and design of controllers. In this contribution, both discrete-time and continuous-time dynamical systems are considered. Using suitable algorithms, guaranteed state enclosures can be determined for systems with uncertain parameters, uncertain initial conditions, nonlinearities, and time-varying characteristics. Although both uncertain system parameters and bounded control variables are assumed to be represented by interval boxes in the following, they have to be distinguished in reachability and controllability analysis. Typically, robustness specifications for controllers of dynamical systems are given in terms of bounds on the system's time response which must not be violated for any possible operating condition. Hence, reachability as well as controllability of states have to be proven for all possible parameter values but for at least one admissible control sequence. Robust control strategies for nonlinear systems usually rely on knowledge of all current states. However, the complete state vector is not always directly accessible for measurement. In this case, observers are applicable to reconstruct non-measurable state variables. Furthermore, they can reduce the uncertainties of the measured quantities by model-based recursive computation of estimates and fusion of information gathered by different measurement devices. If guaranteed bounds of all uncertain parameters of a dynamical system (including the sensor characteristics) and conservative bounds of all disturbances can be specified, the presented interval observer provides guaranteed enclosures of all reachable states. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear stationary control systems over a Boolean semiring: Geometric properties and the isomorphism problem

In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability, identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in this class of systems.     

Disturbance decoupling and model matching problems for discrete-time systems with time-varying delays

In this paper, the disturbance decoupling problem and the model matching problem for discrete-time linear systems with time-varying delays are considered. Solvability of the above problems is characterized by means of structural necessary and sufficient conditions that can be checked by algorithmic procedures. The basic method used to analyze the considered problems consists in representing the discrete-time linear systems with time-varying delays as switching linear systems, whose properties can be studied by a powerful structural approach. In this way, the considered control problems can be reduced to the corresponding problems for switched linear systems, whose solvability has been recently characterized.     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Controllability properties of constrained linear systems

In this paper, we present results on constrained controllability for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1.We prove that controllability of an arbitrary pointp inR ⁿ is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties ofG, the set of all nonnegative times at whichp is controllable, and ofC, the set of all controllable points. We characterize the setC for the given system and provide additional spectrum-dependent structure.We show that, for the given linear system, several notions of constrained controllability of the pointp are the same, and thus the setC is open. We also provide a necessary condition for small-time (differential or local) constrained controllability ofp.This work was supported in part by NSF Grant ECS-86-09586.     

Module structure of constant linear systems and its applications to controllability

We shall introduce a new module structure to a large class of continuous-time constant linear systems. This is done as a natural extension of the classical k[z]-module structure of finite-dimensional constant linear systems. This module action is used to investigate the relationship between reachability and controllability of linear systems. After introducing the notion of K-controllability due to Kamen [12], we give the following result in Section 5: If a constant linear system is described by a functional differential equation $\text{x}\text{?}\text{= Fx + Gu,}\text{where}\text{x}\text{and}\text{G}$ belong to a Banach space X, and if G is K-controllable to zero, then every reachable state is reachable and controllable in bounded time. (The result given in Section 5 is a little more general than this.) We also give a simple example in Section 6 to illustrate this result.     

Switching systems and entropy

Switching systems are non-autonomous dynamical systems obtained by switching between two or more autonomous dynamical systems as time goes on. They can be mainly found in control theory, physics, economy, biomathematics, chaotic cryptography and of course in the theory of dynamical systems, in both discrete and continuous time. Much of the recent interest in these systems is related to the emergence of new properties by the mechanism of switching, a phenomenon known in the literature as Parrondo's paradox. In this paper we consider a discrete-time switching system composed of two affine transformations and show that the switched dynamics has the same topological entropy as the switching sequence. The complexity of the switching sequence, as measured by the topological entropy, is fully transferred, for example, to the switched dynamics in this particular case.     

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.