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.     

Global stability of two linearly interconnected nonlinear systems

Simple and easy to use conditions are developed which guarantee global stability of two linearly interconnected nonlinear systems. The conditions consist of two scalar frequency response conditions of Popov type on the linear dynamical subsystems and inequalities involving the sector bounds and parameters in the problem; the inequalities have a simple graphical interpretation in terms of the product of the interconnection coefficients. Examples with particular numerical values are presented to illustrate application of the results.     

On stochastic controllability for nonlinear systems

Both the stochastic ε-controllability and the stochastic controllability with probability one are first defined. Second, by using a stochastic Lyapunov-like approach, several theorems are developed which give sufficient conditions for the stochastic controllability defined for an important class of nonlinear stochastic systems. A theorem of stochastic uncontrollability is also presented, giving sufficient conditions for stochastic uncontrollability for a class of nonlinear systems. Finally, the relation between the deterministic controllability and the stochastic one is comparatively discussed.     

A controllability theory for nonlinear systems

A Lyapunov-like approach to the controllability of nonlinear dynamic systems is presented. A theory is developed which yields sufficient conditions for complete controllability for some classes of nonlinear systems; feedback controllers which drive the systems to desired terminal conditions, at a specified final time, are also obtained. Well-known controllability conditions for linear dynamic systems are derived using this general controllability theory. Elliptical regions are found which contain (bound) the trajectories of a class of systems controlled according to these methods. These regions are used in synthesizing controllers for nonlinear systems and for a class of state-variable inequality constrained problems. An uncontrollability theorem, based also upon Lyapunov-like notions, is presented; this yields sufficiency conditions for uncontrollability for some types of nonlinear systems. Relationships of the theories to other nonlinear controllability approaches are indicated.     

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.     

On nonlinear controllability of homogeneous systems linear in control

This paper considers small-time local controllability (STLC) of single- and multiple-input systems, x/spl dot/=f/sub 0/(x)+/spl Sigma//sub i=1//sup m/f/sub i/u/sub i/ where f/sub 0/(x) contains homogeneous polynomials and f/sub 1/,...,f/sub m/ are constant vector fields. For single-input systems, it is shown that even-degree homogeneity precludes STLC if the state dimension is larger than one. This, along with the obvious result that for odd-degree homogeneous systems STLC is equivalent to accessibility, provides a complete characterization of STLC for this class of systems. In the multiple-input case, transformations on the input space are applied to homogeneous systems of degree two, an example of this type of system being motion of a rigid-body in a plane. Such input transformations are related via consideration of a tensor on the tangent space to congruence transformation of a matrix to one with zeros on the diagonal. Conditions are given for successful neutralization of bad type (1,2) brackets via congruence transformations.     

Relative controllability of nonlinear systems with delays in control

Relative controllability of nonlinear time-varying systems with time-variable delays in control is considered. Using Schauder's fixed point theorem, sufficient conditions for global relative controllability are given. The results obtained are a generalization of Davison [3], and Klamka [7].     

On the controllability of nonlinear systems with symmetry

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

Global null controllability of perturbed linear periodic systems

In this note we study the global null controllability with bounded controls of perturbed linear periodic systems in Rⁿ. More explicitly, a globally null controllable periodic system is perturbed by suitably small termsV(t)andC(t)to obtain a system of the formdot{x} = (A(t) + V(t))x + (B(t) + C(t))uand sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

A note on the controllability of nonlinear systems

In this paper a question of the controllability of nonlinear systems is considered. On the basis of theorems due to Graves, a new sufficient condition for local controllability is established.     

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 the controllability of a class of nonlinear systems

Sufficient conditions are given which assure that the systemfrac{dx}{dt}= A(t)x + B(t)u + f(x,u,t)is globally or locally controllable.     

Further results on controllability properties of discrete-time nonlinear systems

Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability property, and globally, if there exists a global attractor for the system.This research was supported in part by US Air Force Grant AFOSR-91-0346, and also by an INDAM (Istituto Nazionale di Alta Matematica Francesco Severi, Italy) fellowship.     

Global transformations of nonlinear systems

Recent results have established necessary and sufficient conditions for a nonlinear system of the formdot{x}(t) = f(x(t))-u(t)g(x(t)). withf(0) = 0, to be locally equivalent in a neighborhood of the origin in Rⁿto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that{gldot[fdotg],...,(ad^{n-1}fldotg)}span ann-dimensional space and that{gldot[fldot g],...,(ad^{n-2}fldotg)}is an involutive set.     

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.     

Global controllability of linear autonomous systems: A geometric consideration

The global controllability in finite time of a linear autonomous system with restrained controls is investigated. Necessary and sufficient conditions are obtained by an approach based on the consideration of geometric properties of the system.     

Global stabilization of nonlinear cascade systems

We present sufficient conditions for the global stabilizability of two cascade connected nonlinear systems. These are based on general results concerning global asymptotic stability of triangular systems which are proved in the last section. For polynomial systems, in particular, the stabilizing feedback is given explicitly.     

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.     

Local stabilization and controllability of a class of non-triangular nonlinear systems

This paper considers the problems of local asymptotic stabilization using continuous static state feedback (LCFS) and small time local controllability (STLC) of single-input nonlinear systems. A specific class of nontriangular systems, called the essentially triangular form, is introduced. For this class of systems sufficient conditions for LCFS and necessary and sufficient conditions for STLC are obtained. In particular, it is shown that STLC implies LCFS. A motivating physical example of the ETF system is presented.