Héctor J. Sussmann

Controllability of nonlinear systems.

Discussion of the controllability of nonlinear systems described by the equation dx/dt - F(x,u). Concepts formulated by Chow (1939) and Lobry (1970) are applied to establish criteria for F and its derivatives to obtain qualitative information on sets which can be obtained from x which denotes a variable of state in an arbitrary, real, analytical manifold. It is shown that controllability implies strong accessibility for a large class of manifolds including Euclidean spaces.-

A general theorem on local controllability

We prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point. Earlier results of Brunovsky, Hermes, Jurdjevic, Crouch and Byrnes, Sussmann and Grossmann, are shown to be particular cases of this result. Also, a number of new sufficient conditions are obtained. All these results follow from one simple general principle, namely, that local controllability follows whenever brackets with certain symmetries can be “neutralized,” in a suitable way, by writing them as linear combinations of brackets of a lower degree. Both the class of symmetries and the definition of “degree” can be chosen to suit the problem.

The peaking phenomenon and the global stabilization of nonlinear systems

The problem of global stabilization is considered for a class of cascade systems. The first part of the cascade is a linear controllable system and the second part is a nonlinear system receiving the inputs from the states of the first part. With zero input, the equilibrium of the nonlinear part is globally asymptotically stable. In linear systems, a peaking phenomenon occurs when high-gain feedback is used to produce eigenvalues with very negative real parts. It is established that the destabilizing effects of peaking can be reduced if the nonlinearities have sufficiently slow growth. A detailed analysis of the peaking phenomenon is provided. The tradeoffs between linear peaking and nonlinear growth conditions are examined. >

Control systems on Lie groups

The controllability properties of systems which are described by an evolution equation in a Lie group are studied. The revelant Lie algebras induced by a right invariant system are singled out, and the basic properties of attainable sets are derived. The homogeneous case and the general case are studied, and results are interpreted in terms of controllability. Five examples are given.

A maximum principle for hybrid optimal control problems

Presents a version of the maximum principle for hybrid optimal control problems under weak regularity conditions. In particular, we only consider autonomous systems, in which the dynamical behavior and the cost are invariant under time translations. The maximum principle is stated as a general assertion involving terms that are not yet precisely defined, and without a detailed specification of technical assumptions. One version of the principle, where the terms are precisely defined and the appropriate technical requirements are completely specified, is stated for problems where all the basic objects-the dynamics, the Lagrangian and the cost functions for the switchings and the end-point constraints-are differentiable along the reference arc. Another version, involving nonsmooth maps, is also stated, and some brief remarks on even more general versions are given. To illustrate the use of the maximum principle, two very simple examples are shown, involving problems that can easily be solved directly. Our results are stronger than the usual versions of the finite-dimensional maximum principle. For example, even the theorem for classical differentials applies to situations where the maps are not of class C/sup 1/, and can fail to be Lipschitz-continuous. The nonsmooth result applies to maps that are neither Lipschitz-continuous nor differentiable in the classical sense. In each case, it would be trivial to construct hybrid examples of a similar nature. On the other hand, the results presented in this paper are much weaker than what can actually be proved by our methods.

Nonlinear output feedback design for linear systems with saturating controls

The existence of nonlinear smooth dynamic feedback stabilizers for linear time-invariant systems under input constraints is shown, assuming only that open-loop asymptotic controllability and detectability hold. A proof for state stabilization is sketched out.<<ETX>>

A positive real condition for global stabilization of nonlinear systems

Abstract We study the possibility of globally stabilizing, by means of a smooth state feedback, systems obtained by cascading a linear controllable system and a general nonlinear system. Our main result is that global stabilization can be achieved if the output of the linear system can be chosen to be ‘feedback positive real’ (FPR). Some recent stabilization conditions appear as special cases of the new FPR condition. Examples of systems with the FPR property are given.

Motion planning for controllable systems without drift

A general strategy for solving the motion planning problem for real analytic, controllable systems without drift is proposed. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added. Using formal calculations with a product expansion relative to P. Hall basis, another control is produced that achieves the desired result on the formal level. This provides an exact solution of the original problem if the given system is nilpotent. For a general system, an iterative algorithm is derived that converges very quickly to a solution. For nonnilpotent systems which are feedback nilpotentizable, the algorithm, in cascade with a precompensator, produces an exact solution. Results of simulations which illustrate the effectiveness of the procedure are presented.<<ETX>>

On the stabilizability of multiple integrators by means of bounded feedback controls

It is known that a linear system x=Ax+Bu can be stabilized by means of a smooth bounded control if and only if it has no eigenvalues with positive real part, and all the uncontrollable modes have a negative real part. The authors investigate, for single-input systems, the question of whether such systems can be stabilized by means of a feedback u= sigma (h(x)), where h is linear and sigma (s) is a saturation function such as sign(s) min( mod s mod ,1). A stabilizing feedback of this particular form exists if A has no multiple eigenvalues, and also in some other special cases such as the double integrator. It is shown that for the multiple integrator of order n, with n>or=3, no saturation of a linear feedback can be globally stabilizing.<<ETX>>

Existence and uniqueness of minimal realizations of nonlinear systems

In the theory of finite dimensional linear systems, it is well known that every input-output map that can be realized by one such system can also be realized by a system which is “minimal”, i.e. both controllable and observable. Moreover, the minimal realization of a given map is unique up to isomorphism. It is shown here that similar results hold for the class of all systems whose state space is a real analytic manifold, whose dynamics is given by a family of complete real analytic vector fields, and whose output is an arbitrary real analytic function on the state space.