On global asymptotic controllability of planar affine nonlinear systems

~~On global asymptotic controllability of planar affine nonlinear systems1. Isidon, A., Nonlinear Control Systems, 3rd ed., London: Springer-Verlag, 1995. 2. Khalil, H. K., Nonlinear Systems, 2nd ed., Upper Saddle River, NJ: Prentice-Hall, 1996. 3. Brockett, R. W., Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. Brockett, R. W., Millmann, R. S., Sussmann, H. J.), Boston: Birkhauser Inc., 1983, 181-191. 4. Nikitin, S., Top…     

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.     

On small-controllability and controllability of a class of nonlinear systems

In this paper, small-controllability and controllability of a class of nonlinear systems are studied. Both the discrete-time case and continuous-time case are considered. By proposing an implicit function approach, sufficient conditions for the systems respectively to be small-controllable and controllable are obtained. Examples are also provided to illustrate the results of the paper.     

On global controllability for a class of polynomial affine nonlinear systems

In this paper,a necessary and sufficient condition of the global controllability for a class of low dimensional polynomial affine nonlinear systems with special structure is obtained.The condition is imposed on the coefficients of the system only and the methods are based on Green’s formula and the trajectory analysis of planar linear system.Furthermore, I point out that the global controllability does not hold for the corresponding high dimensional polynomial system.     

On the controllability of nonlinear systems with symmetry

Sufficient and necessary conditions are given for the controllability of nonlinear systems possesing symmetry.     

Local and global controllability for nonlinear systems

A technique to study local and global controllability properties for a wide class of nonlinear systems is presented. In addition to an extensive study of systems on ² we propose — as an application of our method — a new criterion for global controllability of systems with polynomial drift term, defined on ⁿ. In the case of linear systems, the criterion reduces to the classical Kalman condition.A study of local controllability at the equilibrium point of the drift term of systems defined on ² enables us not only to rederive a local controllability result derived by Hermes [4,5], but also to extend this result when no assumptions are made on the boundedness of the controls.     

Local controllability of nonlinear systems

The local controllability of control systems with an arbitrary number of controls is considered, first on an open set and then at a given point; necessary conditions are derived concerning the Lie algebra T′ generated by the input vector fields, and applied to gradient systems.The main results are a geometric sufficient condition for local controllability at a point, and an equivalent condition based on the computation of Lie brackets, assuming T′ to be (n − 1)-dimensional.     

Null controllability of planar bimodal piecewise linear systems

This article investigates the null controllability of planar bimodal piecewise linear systems, which consist of two second order LTI systems separated by a line crossing through the origin. It is interesting to note that even when both subsystems are controllable in the classical sense, the whole piecewise linear system may be not null controllable. On the other hand, a piecewise linear system could be null controllable even when it has uncontrollable subsystems. First, the evolution directions from any non-origin state are studied from the geometric point of view, and it turns out that the directions usually span an open half space. Then, we derive an explicit and easily verifiable necessary and sufficient condition for a planar bimodal piecewise linear system to be null controllable. Finally, the article concludes with several numerical examples and discussions on the results and future work.     

On the constrained small-time controllability of linear systems

This note presents a necessary and sufficient condition for small-time controllability of a linear system with respect to a cone. This result extends the controllability conditions for linear systems to the case of control and phase constraints.     

On controllability of two-dimensional discrete-time bilinear systems

In this paper, controllability of two-dimensional discrete-time bilinear systems is studied. Sufficient conditions for the systems to be controllable are presented. In particular, the control inputs are computed to achieve the state transition for the controllable systems by using the root locus technique. Examples are also provided to illustrate the results of the paper.     

Regional controllability for infinite-dimensional bilinear systems: approach and simulations

In this study, we consider a regional controllability problem for a class of distributed bilinear systems evolving in a spatial domain Ω. A feedback control is used to steer the system state close to a desired profile at a final time T, only on ω a subregion of the system domain which may be interior or on the boundary of Ω. Our purpose is to prove that an optimal control exists, and characterised as a solution to an optimality system. Numerical algorithm is given and successfully illustrated by simulations.     

Planar nonlinear systems: Practical stabilization and Hermes controllability condition

We analyze further some results from our recent work (1991) concerning the practical stabilizability problem by means of smooth feedback laws. These results are used to establish that if a smooth planar nonlinear system satisfies the Hermes controllability condition at its equilibrium, then it can be practically stabilized.     

On tracking control for affine nonlinear systems by sliding mode

The stability problem of output tracking of a bounded time-varying reference by decouplable affine nonlinear systems using sliding mode control is investigated. It is shown that when the error dynamics on the sliding surface is chosen to be linear time-invariant, closed loop stability of systems under the presented sliding mode control can be guaranteed only if the systems are minimum phase.     

On the controllability of a class of nonlinear stochastic systems

We study the controllability properties of the class of stochastic differential systems characterized by a linear controlled diffusion perturbed by a smooth, bounded, uniformly Lipschitz nonlinearity. We obtain conditions that guarantee the weak and strong controllability of the system. Also, given any open set in the state space we construct a control, depending only on the Lipschitz constant and the infinity-norm of the nonlinear perturbation, such that the hitting time of the set has a finite expectation with respect to all initial conditions.     

Approximate controllability for abstract measure differential systems

In this paper, we investigate approximate controllability for abstract measure differential systems based on generalizing knowledge for ordinary differential systems. We first introduce new concepts of the reachable set and approximate controllability for abstract measure differential systems. Then based on the nonlinear alternative for α-condensing mapping in Banach space, we present sufficient conditions of approximate controllability for a class of abstract measure differential systems. Our results in dealing with approximate controllability are less conservative than those in the previous literature. Finally, an example is given to illustrate the availability of our results for approximate controllability.     

On null controllability of linear systems in Banach spaces

Our aim here is to give characterizations for the null controllability of linear systems in general Banach spaces. The starting point of this paper is the work of Pengnian Chen and Huashu Qin (Systems Control Lett. 45 (2002) 155), which solves the problem of exact controllability. In fact, the results of (Systems Control Lett. 45 (2002) 155) say that, in case of setting in general Banach spaces, there are the same characterizations of exact controllability as in the case of reflexive Banach spaces. An open problem raised in (Systems Control Lett. 45 (2002) 155) is the validity of a similar result for null controllability. We give here a positive answer to this problem. However, it seems to us that the technique of (Systems Control Lett. 45 (2002) 155) does not work for null controllability, and, consequently, our approach is completely different.     

Connectability and structural controllability of composite systems

The notion of connectability for multivariable composite systems consisting of a number of subsystems interconnected in an arbitrary way is introduced in this paper. It is shown that connectability plays a fundamental role in composite systems; in particular, it is shown that under certain mild conditions, almost all composite interconnected systems are controllable and observable from any nontrivial input and output if and only if the resultant composite system is connectable. A class of composite systems called general input-output hierarchical systems, which has the property of always being connectable is then defined. Since such systems are almost always controllable, this observation perhaps gives some insight in explaining why so many real world systems have as their basis a hierarchical structure. An application of the previous results is then made to show that a system (C, A, B) is structurally controllable and observable if and only if it is connectable and a certain non-pathological rank condition holds.     

Approximate controllability of nonlinear stochastic impulsive systems with control acting on the nonlinear terms

This paper is concerned with the approximate controllability of the stochastic impulsive system with control acting on the nonlinear terms. In the case that the nonlinear terms are dependent on the control, the control cannot be expressed explicitly and analysed. In this situation, we generate the control sequence by the approximate equations and give the properties of the control sequence and the driven solution sets. Some discussions on the assumptions are given to impose on the system. The Hausdorff measure is adopted to relax the requirement of the compactness condition. It is also shown that under some sufficient conditions the stochastic impulsive system is approximately controllable without the requirement of the controllability of the associated linear system. The results of this paper can be degraded into special cases and coincide with some existing ones.