Arthur J. Krener

Nonlinear controllability and observability

The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems. This paper deals with analogous questions for nonlinear systems.

Linearization by output injection and nonlinear observers

Observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection. Necessary and sufficient conditions for the existence of such a transformation are given.

Nonlinear observers with linearizable error dynamics

We present a new method for designing asymptotic observers for a class of nonlinear systems. The error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast.

Nonlinear decoupling via feedback: A differential geometric approach

The paper deals with the nonlinear decoupling and noninteracting control problems. A complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems. The paper also includes algorithms concerned with the actual construction of the appropriate control laws.

On the Equivalence of Control Systems and the Linearization of Nonlinear Systems

Given two control systems where the control enters linearly, a necessary and sufficient condition is derived that these systems be locally diffeomorphic, i.e., that there exist a local diffeomorphism between the state spaces which carries a trajectory of the first system for each control into the trajectory of the second system for the same control. As a corollary we derive necessary and sufficient conditions for a system to be locally diffeomorphic to a linear system.

Approximate linearization by state feedback and coordinate change

A nonlinear system can always be approximated to first order by linear systems. It has been shown by Jakubczyk and Respondek [4] and Hunt and Su [3] that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback. In this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system. The use of this technique in the design of nonlinear compensators has been suggested recently by the author [6].

Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems

In this paper, a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given, which is the generalization of the Brunovsky form of linear systems. A set of invariants under the quadratic changes of coordinates and feedbacks is found. It is then proved that any linearly controllable nonlinear system is linearizable to second degree by a dynamic state feedback.

Nonlinear Observer Design in the Siegel Domain

We extend the method of Kazantzis and Kravaris [Systems Control Lett., 34 (1998), pp. 241--247] for the design of an observer to a larger class of nonlinear systems. The extended method is applicable to any real analytic observable nonlinear system. It is based on the solution of a first-order, singular, nonlinear PDE. This solution yields a change of state coordinates which linearizes the error dynamics. Under very general conditions, the existence and uniqueness of the solution is proved. Lyapunovs auxiliary theorem and Siegels theorem are obtained as corollaries. The technique is constructive and yields a method for constructing approximate solutions.

A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems

The main results of this paper are two-fold. The first, Theorem 1, is a generalization of the work of Chow and others concerning the set of locally accessible points of a nonlinear control system. It is shown that under quite general conditions, this set lies on a surface in state space and has a nonemptyinterior in the relative topology of that surface.The second result, Theorem 3, generalizes the bang-bang theorem to nonlinear control systems using higher order control variations as developed by Kelley and others. As a corollary we obtain Halkin’s bang-bang theorem for a linear piecewise analytic control system.

Modeling and estimation of discrete-time Gaussian reciprocal processes

Discrete-time Gaussian reciprocal processes are characterized in terms of a second-order two-point boundary-value nearest-neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics. This second-order model is the analog for reciprocal processes of the standard first-order state-space models for Markov processes. The model is used to obtain a solution to the smoothing problem for reciprocal processes. The resulting smoother obeys second-order equations whose structure is similar to that of the Kalman filter for Gauss-Markov processes. It is shown that the smoothing error is itself a reciprocal process. >