Conditions For Reachability Research Articles

Reachability of time‐varying fractional dynamical systems with prescribed control

AbstractThe focus of this article revolves around investigating reachability results in the context of time‐varying fractional dynamical systems with prescribed or predetermined control by employing Riemann–Liouville's fractional derivative. The investigation delves into the necessary and sufficient conditions for reachability results concerning time‐varying linear fractional dynamical systems, employing both the Gramian technique and fractional calculus. Additionally, Schauder's fixed‐point technique is employed to establish the sufficient conditions for the reachability results of time‐varying nonlinear fractional dynamical systems. The validity of the theoretical findings is confirmed through the successive approximation technique, accompanied by a series of numerical examples.

Evolution Function Based Reach-Avoid Verification for Time-varying Systems with Disturbances

In this work, we investigate the reach-avoid problem of a class of time-varying analytic systems with disturbances described by uncertain parameters. Firstly, by proposing the concepts of maximal and minimal reachable sets, we connect the avoidability and reachability with maximal and minimal reachable sets respectively. Then, for a given disturbance parameter, we introduce the evolution function for exactly describing the reachable set, and find a series representation of this evolution function with its Lie derivatives, which can also be regarded as a series function with respect to the uncertain parameter. Afterward, based on the partial sums of this series, over- and under-approximations of the evolution function are constructed, which can be realized by interval arithmetics with designated precision. Further, we propose sufficient conditions for avoidability and reachability and design a numerical quantifier elimination based algorithm to verify these conditions; moreover, we improve the algorithm with a time-splitting technique. We implement the algorithms and use some benchmarks with comparisons to show that our methodology is both efficient and promising. Finally, we additionally extend our methodology to deal with systems with complex initial sets and time-dependent switchings. The performance of our extended method for these systems is also shown by four examples with comparisons and discussions.

Polynomial methods to construct inputs for uniformly ensemble reachable linear systems

This paper is concerned with linear parameter-dependent systems and considers the notion of uniform ensemble reachability. The focus of this work is on constructive methods to compute suitable parameter-independent open-loop inputs for such systems. In contrast to necessary and sufficient conditions for ensemble reachability, computational methods have to distinguish between continuous-time and discrete-time systems. Based on recently derived sufficient conditions and techniques from polynomial approximation, we present two methods for discrete-time single-input linear systems. Moreover, we illustrate that one method can also be applied to certain continuous-time single-input systems.

Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control.

This paper studies the sliding mode control method for coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks on a directed non-strongly connected topology. A novel fractional integral sliding mode surface and the corresponding control law are designed to realize global Mittag-Leffler synchronization. The sufficient conditions for synchronization and reachability of the sliding mode surface are derived via the hierarchical method and the Lyapunov method. Finally, simulations are provided to verify our theoretical findings.

Reach control problem for a class of convex differential inclusions on simplices

Abstract This paper focuses on the reach control problem for systems expressed as a class of convex differential inclusions. The purpose is to find affine feedback for the trajectories of the systems to reach and leave an unrestricted facet of a given simplex in a finite time. It is proved that the condition for strong reachability is sufficient and necessary. The sufficient condition for weak reachability is obtained based on the solvability of the reach control problem for linear affine systems. At last, algorithms are designed and numerical examples are given to verify the validity of the results.

Feedback equivalence and uniform ensemble reachability

This paper considers feedback methods for ensemble reachability of parameter-dependent linear systems (A(θ),B(θ)), where the parameter θ is varying over a compact Jordan arc in the complex plane. Recently, pointwise testable sufficient conditions for uniform ensemble reachability have been developed. Beside the necessity of pointwise reachability these conditions put restrictions on the spectra of the matrices A(θ) and the Hermite indices of the pair (A(θ),B(θ)). In this paper we show that these conditions can be ensured by applying a suitable feedback transformation if the pair (A(θ),B(θ)) is pointwise reachable and its Kronecker indices are independent of the parameter.

Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks

This study focuses on the finite-time set reachability of probabilistic Boolean multiplex control networks (PBMCNs). Firstly, based on the state transfer graph (STG) reconstruction technique, the PBMCNs are extended to random logic dynamical systems. Then, a necessary and sufficient condition for the finite-time set reachability of PBMCNs is obtained. Finally, the obtained results are effectively illustrated by an example.

Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

<p style='text-indent:20px;'>The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.</p>

Ensemble reachability of homogenous parameter‐depedent systems

AbstractIn this paper we consider the class (θA, B) of parameter‐dependent linear systems given by matrices A ∈ ℂn×n and B ∈ ℂn×m. This class is of interest for several applications and the frequently met task for such systems is to steer the origin toward a given target family f(θ) by using an input that is independent from the parameter. This paper provides a collection of necessary and sufficient conditions for ensemble reachability for these systems.

Set reachability of Markovian jump Boolean networks and its applications

This study investigates the set reachability of Markovian jump Boolean networks (MJBNs). Firstly, through comprehensive analysis, two kinds of set reachability, strong set reachability and weak set reachability, are defined. By using semi-tensor product of matrices, the considered MJBN is converted into a Markov chain which can not only describe the state transition of the MJBN synchronously, but also maintain its transition probability. Based on this, the set reachability problem of the MJBN is transformed into the reachability and convergence problems of the corresponding Markov chain. Then some necessary and sufficient conditions for set reachability of MJBNs are obtained. Finally, the observability and output stability problems of MJBNs are discussed as two applications of set reachability.

Output Monomial Reachability and Zero Output Controllability of Positive Switched Systems

In this paper, we present a sufficient condition for the output reachability of discrete-time positive switched systems. Besides, necessary and sufficient conditions for output monomial reachability and zero output controllability are provided. Further, some examples are shown.

Resilient Reachability for Linear Systems

A fault-tolerant system is able to reach its goal even when some of its components are malfunctioning. This paper examines tolerance to a specific type of malfunction: the loss of control authority over actuators. Namely, we investigate whether the desired target set for a linear system remains reachable under any undesirable input. Contrary to robust control, we assume that the undesirable inputs can be observed in real time, and subsequently allow the control inputs to depend on these undesirable inputs. Building on previous work on reachability with undesirable inputs, this paper develops a reachability condition for linear systems, and obtains a formula that describes reachability of the goal set for driftless linear systems by computing the minimum of a concave-convex objective function. Prom this formulation we establish two novel sufficient conditions for resilient reachability.

On Reachability and Null-Controllability of Nonstrict Convex Processes

This letter studies reachability and null-controllability for difference inclusions involving convex processes. Such difference inclusions arise, for instance, in the study of linear discrete-time systems whose inputs and/or states are constrained to lie within a convex cone. After developing a geometric framework for convex processes relying on invariance properties, we provide necessary and sufficient conditions for both reachability and null-controllability in terms of the spectrum of dual processes.

Set reachability and observability of probabilistic Boolean networks

In this paper, the set reachability and observability of probabilistic Boolean networks (PBNs) are investigated. Using a parallel extension technique, we proved that the observability problem of a PBN can be recast as a set reachability problem of an interconnected PBN. For set reachability analysis, we designed a random logic dynamical system (RLDS) from the PBN under consideration by reconstructing the state transfer graph (STG). We proved that, for a PBN, a target subset is reachable from an initial subset if and only if all solutions to the corresponding RLDS starting from the initial subset converge to the zero state. Based on the STG reconstruction technique and using the largest invariant subset algorithm, the necessary and sufficient conditions for finite-time set reachability with probability one and asymptotical set reachability in distribution were obtained. All the results are expressed in terms of the transition probability matrix between non-zero states of the RLDS. Further, the results related to set reachability were applied to the observability problem of PBNs. The necessary and sufficient conditions for finite-time observability with probability one and asymptotical observability in distribution were obtained. Finally, examples were presented to illustrate the effectiveness of the proposed method.

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.

Modeling, Reachability and Controllability of Bounded Petri Nets Based on Semi‐Tensor Product of Matrices

AbstractIn this paper, we present a comprehensive modeling technique for bounded Petri net systems (BPNSs) in the framework of the semi‐tensor product (STP) of matrices. The two dynamic properties of BPNSs, namely, reachability and controllability, are investigated systematacially. First, the dynamics of a bounded Petri net system (BPNS), by resorting to the STP of matrices, are expressed in the form of a discrete‐time bilinear equation, which is called the marking evolution equation (MEE) of BPNSs. Second, controllability and transition‐marking adjacency matrix (TMAM) of BPNSs are defined, respectively. Further, several necessary and sufficient conditions for reachability and controllability of BPNSs are given in terms of the MEE and TMAM. Third, an efficient algorithm to verify reachability property of BPNSs, in this paper, is provided, as well as its computational complexity. Finally, an example is presented to illustrate the theoretical results in this paper. The main contribution of this paper is the presentation of a precise mathematical model for BPNSs. The main advantage of the proposed approach is that not only it can be applied to verify whether or not any given marking is reachable from the other in state space, but also it is very convenient to find all firing sequences between any two reachable markings.

Algorithms of reduced complexity to design control sequences for untimed Petri nets in varying and uncertain environments

Petri nets have been widely used for the modelling, analysis, control and optimization of discrete event systems with shared resources in the domains of engineering. This article concerns the design of control sequences for such systems modelled with untimed Petri nets. The aim of the controller is to incrementally compute sequences of transition firings with minimal size. Such sequences aim to move the marking from an initial value to a reference value. The resulting trajectory must avoid some forbidden markings and limit as possible the exploration of non-promising branches. For this purpose, the approach explores a small part of the reachability graph in the neighbourhood of the current marking. Then from the explored markings, it estimates a distance to the reference. The main contributions are (a) to reduce the explored part of the reachability graph according to a double limitation in breadth and in depth in order to provide solutions with a low computational effort; (b) to provide conditions to ensure the converge and optimality of the proposed algorithms and derive necessary and sufficient conditions for reachability; and (c) to include the firing sequence design in a global control schema suitable for reactive scheduling problems in uncertain and perturbed environments. The main application concerns deadlock-free scheduling problems in the domain of flexible manufacturing systems, but the approach is also applicable for systems in computer science and transportation.

A Lyapunov theorem certifying global weak reachability for stochastic difference inclusions with random inputs

In this paper, a Lyapunov condition certifying global weak reachability of a compact set is given for stochastic, set–valued, discrete-time systems. With respect to such systems, under mild regularity assumptions, it is shown that the existence of a lower semicontinuous function that decreases in expected value for at least one measurable selection from the set-valued mapping outside a compact set is a sufficient condition for global weak reachability of such a set.

Fractional variable order discrete-time systems, their solutions and properties

ABSTRACTIn this paper, fractional variable order discrete state-space systems based on different definitions of fractional variable order difference are investigated.The general solution of these systems is derived. Moreover, necessary and sufficient conditions for reachability and observability are given and proven. The sufficient conditions for controllability are proposed too.

On the Output Controllability of Positive Discrete Linear Delay Systems

Necessary and sufficient conditions for output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output are established. It is also shown that output reachability and null output controllability together imply output controllability.