Sufficient Conditions For Global Controllability Research Articles

Necessary and sufficient condition for global controllability of nonlinear systems

AbstractThis paper concerns the global controllability of general nonlinear control systems, which is intimately related to their defining vector fields. We give a necessary and sufficient condition by studying how to enlarge the defining set of vector fields without changing its controllability. Four such approaches are obtained. First, for a given set of vector fields, if one of them is reversible, then the images of these given vector fields under the flow of it can be included without changing the controllability. Second, if two of them are reversible, then their Lie bracket can be included to the given set without changing its controllability. Third, the controllability of a given set of vector fields is preserved by positive span. Finally, the controllability of a given set of vector fields is preserved under closure. For a given nonlinear control system, its controllability is determined as follows. Begin with the defining set of vector fields, step by step, we enlarge it by the aforementioned approaches until its controllability can be readily verified. The power of the proposed approaches is demonstrated by examples.

Controllability and stabilization of Boolean control networks by the auxiliary function of flipping

SummaryIn this article, a hybrid method combining flip matrix approach and an open (or closed) loop control is proposed to study global controllability and stabilization of Boolean control networks (BCNs). First, the necessary and sufficient condition for global controllability of BCNs, by flipping some members of a perturbation set and under free control sequences, is proposed. After that, using a search algorithm, the minimal perturbation sets for global controllability (MS‐GCs) of the BCN are obtained. Next, we propose a necessary and sufficient criterion for global stabilization of BCNs by flipping some members of a perturbation set and under a state feedback control. Similarly, an algorithm is given to search for the minimal perturbation sets for global stabilization (MS‐GSs) of the BCN. Moreover, the time‐optimal MS‐GSs of the BCN are also obtained by an algorithm. Some examples are given to illustrate the effectiveness of the results.

On the Controllability Properties of Hinged Closed Mechanisms

The mechanisms obtained from pendulum systems with a single pivot point by adding a holonomic restraint connecting the kinematic chain to the second pivot point have been considered. The features of the configuration space that affect the dynamics of such objects have been discussed. Sufficient conditions for global controllability in the entire phase space have been formulated and an example of a mechanism (with configurations on a sphere with two handles) that is globally controlled under the action of a single limited control force has been given.

On impulsive isoenergetic control in systems with gyroscopic forces

We consider the problem of transfer of a Lagrangian system with gyroscopic forces to a given state in the configuration space by means of an impulsive isoenergetic control. This type of control can change instantly the direction of the generalized velocity, but preserve the total energy of the system. We present sufficient conditions for global controllability by a finite number of impulsive switchings, and some results related to the estimation of this number.

On global controllability of affine nonlinear systems with a triangular-like structure

In this paper, we investigate a class of affine nonlinear systems with a triangular-like structure and present its necessary and sufficient condition for global controllability, by using the techniques developed by Sun Yimin and Guo Lei recently. Furthermore, we will give two examples to illustrate its application.

Necessary and Sufficient Condition for Global Controllability of a Class of Affine Nonlinear Systems

In this paper, we investigate the global controllability of a class of n-dimensional affine nonlinear systems with n – 1 controls and constant control matrix. A necessary and sufficient condition for its global controllability has been obtained by using the methods recently developed. Furthermore, we generalize the above result to a class of affine nonlinear systems with a block-triangular-like structure. Finally, we will give three examples to show the applications of our results.

Necessary and Sufficient Condition for Global Controllability of Planar Affine Nonlinear Systems

In this correspondence, we will investigate the global controllability of general planar affine nonlinear systems, and present a simple necessary and sufficient condition for global controllability. The analysis is carried out in detail, which is based on some basic facts in planar topology such as the Jordan curve Theorem, and in the geometrical theory of ordinary differential equations including the Poincare-Bendixson Theorem. Furthermore, we will also discuss the global controllability for higher dimensional affine nonlinear system with a triangular-like structure.

Controllability for a class of area-preserving twist maps

In this paper, we study controllability of two-dimensional integrable twist maps with bounded area-preserving time-dependent (control) perturbations. In contrast to the time-independent perturbation case of the Kolmogorov–Arnold–Moser theorem, there are no invariant sets other than the whole phase space if the perturbation is made a function of time. We give necessary and sufficient conditions for global controllability of these maps.

On the Kreĭn-Rutman theorem and its applications to controllability

This paper extends Krein-Rutman’s theorem on linear operators leaving an invariant cone in infinite-dimensional Banach spaces to multivalued convex functions. The result is applied to obtain necessary and sufficient conditions for global controllability and reachability of nonlinear discrete-time systems described by convex processes.

Some remarks on controllability and reachability of nonstationary discrete-time processes*

In this paper. some criteria for controllability and reachability of nonstationary discrete-time systems described by difference inclusions in Banach spaces are given. The results are applied to obtaining necessary and sufficient conditions for global controllability and reachability of linear discrete-time nonautonomous systems with restrictions not only on controls, but also on states

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ ( p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.

Controlling a constrained linear system to an affine target

We consider the problem of steering the state of a linear system to an affine target when the admissible controls are required to satisfy magnitude constraints. A necessary and sufficient condition for the existence of an admissible control which steers the system to the target from a specified initial condition is presented, as well as a necessary condition and a sufficient condition for global controllability to the target. The conditions are similar to those available in the literature in that they involve a finite dimensional search. Here, however, the affine nature of the target is exploited to reduce the dimensionality of the resulting optimization problem. Hence, the results are more easily applied than those developed in [13].

On the global controllability of perturbed nonlinear systems

Sufficient conditions for global controllability of certain types of perturbed nonlinear time-varying systems are given. The results are obtained by using Schauder's fixed-point theorem.