Christopher I. Byrnes

Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems

Conditions under which a nonlinear system can be rendered passive via smooth state feedback are derived. It is shown that, as in the case of linear systems, this is possible if and only if the system in question has relative degree one and is weakly minimum phase. It is proven that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied. This result incorporates and extends a number of stabilization schemes recently proposed for global asymptotic stabilization of certain classes of nonlinear systems. >

Asymptotic stabilization of minimum phase nonlinear systems

How a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown. More precisely, conditions under which, for every compact set of initial states, it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described. The theory includes the development of globally defined transformations of the system equations to their global normal form. >

New results and examples in nonlinear feedback stabilization

Using the general methodology of nonlinear zero dynamics we derive globally stabilizing state feedback laws for broad classes of nonlinear systems. While it is tempting to reinterpret this design philosophy in terms of high gain feedback, we present an example of a globally stabilizable nonlinear feedback system which cannot be stabilized, even locally, by any output feedback law - in particular, by a high gain law.

On the attitude stabilization of rigid spacecraft

Abstract In this paper, we settle in the negative a longstanding problem concerning the existence of a smooth (static or dynamic) state variable feedback law locally asymptotically stabilizing a rigid spacecraft with two controls about a desired reference attitude. Modelling a spacecraft actuated by three thruster jets, one of which has failed, this well studied system is known to be locally reachable and locally asymptotically null controllable. We obtain our result as a corollary of a surprising result which asserts, for a class of nonlinear systems containing several examples of interest, that such a system is locally asymptotically stabilizable precisely when it can be linearized via state feedback transformations. We give a further result on the instability (in the sense of Lyapunov) of rigid spacecraft for certain feedback laws, but we are able to construct a feedback law locally asymptotically driving the closed-loop trajectories to a motion about the third principal axis. This law is derived using general principles comprising a nonlinear enhancement of root-locus design principles.

Local stabilization of minimum-phase nonlinear systems

Abstract In this note we briefly review the notion of a nonlinear minimum-phase system, i.e. a system which when constrained in such a way as to produce zero output, evolves with an asymptotically stable dynamics. The main purpose of the paper is to show that any minimum-phase nonlinear system can always be locally stabilized by smooth state-feedback

Structurally stable output regulation of nonlinear systems

We address a number of issues that were left open in earlier works on the subject of output regulation of nonlinear systems, and we describe an approach to structurally stable regulation that unifies and extends a number of existing results. Moreover, we also address in part the issue of robust regulation, i.e. the issue of achieving regulation in the presence of parameter uncertainties ranging within a prescribed set.

Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems

In this paper a necessary and sufficient condition for a nonlinear system of the form /spl Sigma/, given by x(k+1)=f(x(k))+g(x(k))u(k), y(k)=h(x(k))+J(x(k))u(k), to be lossless is given, and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero-state observable. Then, we investigate conditions under which /spl Sigma/ can be rendered lossless via smooth state feedback. In particular, we show that this is possible if and only if the system in question has relative degree /spl lcub/0,...,0/spl rcub/ and has lossless zero dynamics. Under suitable controllability-like rank conditions, we prove that nonlinear systems having relative degree /spl lcub/0,...,0/spl rcub/ and lossless zero dynamics can be globally stabilized by smooth state feedback. As a consequence, we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable. The global stabilization problem of the nonlinear system /spl Sigma/ without output is also investigated in this paper by means of feedback equivalence. Some of the results are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear. >

A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint

We present a generalized entropy criterion for solving the rational Nevanlinna-Pick problem for n+1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in a state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

H/sub /spl infin//-control of discrete-time nonlinear systems

This paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback, state feedback, and dynamic output feedback, respectively, for discrete-time nonlinear systems. The H/sub /spl infin//-control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality, differential game, and LaSalles invariance principle in discrete time. A substantial difficulty that V(A(x)+B(x)u+E(x)w) [respectively, V(f(x,u,w))] is no longer quadratic in [/sub w//sup u/] arising in the case of discrete-time nonlinear systems has been surmounted in the paper. In the case of a linear system, we show how the results reduce to the well-known ones recently proposed in the literature.

A new approach to spectral estimation: a tunable high-resolution spectral estimator

Traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates. Here, we present a new approach to spectral estimation, which is based on the use of filter banks as a means of obtaining spectral interpolation data. Such data replaces standard covariance estimates. A computational procedure for obtaining suitable pole-zero (ARMA) models from such data is presented. The choice of the zeros (MA-part) of the model is completely arbitrary. By suitable choices of filter bank poles and spectral zeros, the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum.