Observers for linearly unobservable nonlinear systems

We provide a method for constructing local observers for some nonlinear systems around a critical point where the linearization is not observable or not detectable. Two examples are provided to illustrate the results of the paper.     

Patchy solution of a Francis–Byrnes–Isidori partial differential equation

The solution to the nonlinear output regulation problem requires one to solve a first‐order partial differential equation, known as the Francis–Byrnes–Isidori equations. In this paper, we propose a method to compute approximate solutions to the Francis–Byrnes–Isidori equations when the zero dynamics of the plant are hyperbolic and the exosystem is two dimensional. With our method, we are able to produce approximations that converge uniformly to the true solution. Our method relies on the periodic nature of two‐dimensional analytic center manifolds. Copyright © 2012 John Wiley & Sons, Ltd.     

Backstepping design with local optimality matching

In nonlinear H^∞-optimal control design for strict-feedback nonlinear systems, our objective is to construct globally stabilizing control laws to match the optimal control law up to any desired order, and to be inverse optimal with respect to some computable cost functional. Our recursive construction of a cost functional and the corresponding solution to the Hamilton-Jacobi-Isaacs equation employs a new concept of nonlinear Cholesky factorization. When the value function for the system has a nonlinear Cholesky factorization, we show that the backstepping design procedure can be tuned to yield the optimal control law     

Nonlinear observer design for state and disturbance estimation

A new systematic framework for nonlinear observer design that allows the concurrent estimation of the process state variables together with key unknown process or sensor disturbances is proposed. The nonlinear observer design problem is addressed within a similar methodological framework as the one introduced in [N. Kazantzis, C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems Control Lett. 34 (1998) 241; A.J. Krener, M. Xiao, Nonlinear observer design in the Siegel domain, SIAM J. Control Optim. 41 (2002) 932.] for state estimation purposes only. From a mathematical standpoint, the problem under consideration is addressed through a system of first-order singular PDEs for which a rather general set of solvability conditions is derived. A nonlinear observer is then designed with a state-dependent gain that is computed from the solution of the system of singular PDEs. Under the aforementioned conditions, both state and disturbance estimation errors converge to zero with assignable rates. The convergence properties of the proposed nonlinear observer are tested through simulation studies in an illustrative example involving a biological reactor.     

The existence and uniqueness of Volterra series for nonlinear systems

Given an input-output map described by a nonlinear control systemdot{x}=f(x,u)and nonlinear outputy=h(x), we present a simple straightforward means for obtaining a series representation of the outputy(t)in terms of the inputu(t). When the control enters linearly,dot{x} =f(x)+ ug(x), the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.     

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 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.     

The Controlled Center Systems

In this correspondence, we propose a methodology to stabilize systems with control bifurcations by introducing ldquothe controlled center systems.rdquo A controlled center system is a reduced-order controlled dynamics consisting of the linearly uncontrollable dynamics with the first variable of the linearly controllable dynamics as input. The controller of the full order system is then constructed. We apply this methodology to systems with a transcontrollable, a Hopf, and a double-zero, control bifurcation.     

Modeling and estimation of discrete-time Gaussian reciprocalprocesses

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     

Linear time invariant minimax filtering

The problem of filtering a signal from a linear time invariant system with white Gaussian observation and unknown driving noise bounded at each instant of time is considered. We review the minimax filter of Johansen and Berkovitz–Pollard for the double integrator. While their solution is very elegant, the optimal filter is infinite dimensional. In a previous paper we showed that nearly the same performance can be achieved by a two dimensional filter and we generalized their approach to other linear time invariant systems. In this paper we show how to design nearly optimal filters for any linear time invariant system.