Louis R. Hunt

Stable inversion for nonlinear systems

Abstract There have been two recent major developments in output tracking for nonlinear systems, and our first main contribution is to relate these. Under appropriate assumptions, we show that the bounded.solution of the partial differential equation of Isidori and Byrnes for each trajectory of an exosystem must be given by an integral representation formula of Devasia, Chen and Paden. Under restrictive hypotheses, Devasia, Chen and Paden develop a Picard process that converges to the solution of the integral equation. This solution to the integral equation is also a bounded solution to a dynamical equation driven by the desired outputs. In aircraft applications our nonlinear systems are perturbations of ‘pure-feedback systems’ with outputs, and we find a solution to the stable inversion problem i.e. finding bounded controls and bounded state trajectories in response to bounded output signals, in two steps. The first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory. The second step computes the remaining control and states by finding a noncausal and stable solution to an ‘error-driven dynamical equation’. In other words, the method of Devasia, Chen and Paden is applied to an ‘error system’ and not to the full system. This two-step procedure is the second main contribution of this paper. Throughout this paper it is assumed that our nonlinear systems have vector relative degree.

Right coprime factorizations and stabilization for nonlinear systems

The problem of constructing right coprime factorizations (which are based on the graph of an input-output map instead of the Bezout identity) of nonlinear input-output maps is considered. The map is assumed to arise from a state variable realization with a fixed initial state. The main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map. The technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable. A notion of input-output stability that requires a bound on the magnitude of the input signals is introduced. Methods of constructing such bounds are developed. For a locally feedback linearizable system, the problem of input bounds is transferred to the equivalent linear system. This leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system. >

Brief Parameter variations, relative degree, and stable inversion

Motivated by our prior research on automatic aircraft guidance, we address stable inversion for output tracking and prove that this process is continuous with respect to parameter variations, even when these variations cause a change in relative degree. Our earlier simulations indicated that the stable inversion process is extremely accurate and that only a linear regulator about the desired trajectory is required in the face of reasonable modeling error. The principal novelty in our technique is that a differential equations point of view is taken as opposed to a state-space approach on the (driven) zero dynamics of the system. This is the situation that arises in many applications and it also enables handling the question of changes in relative degree, without having to be encumbered by the change in state space dimension as the parameters change. Linear systems are first studied, since the corresponding result is unknown. Next, a corresponding theorem for nonlinear systems proved by using the Picard process in conjunction with the result for the linear case. The principal contribution of this paper is a result concerning the continuous dependence of a generalized steady state solution of nonlinear driven differential equations, with respect to parameter variations which cause the order of the differential equation to change. Since the notion of steady state is of paramount importance to innumerable engineering situations, the contents of this paper have wider scope.

Stable inversion for nonlinear discrete-time systems

We solve the exact output tracking problem for square nonlinear discrete-time systems through stable inversion. This stable inversion problem involves finding a bounded solution of a dynamical system driven by a bounded signal. Such a solution is given by a difference representation formula, whose existence, under the proper assumptions, can be proven using a Picard process.

Finding maximum linear subsystems of nonlinear systems with outputs

The focus of this paper is to find the maximum input-output linearized subsystem. Necessary and sufficient conditions are presented under which there exist a static state feedback and a coordinate change to input-output linearize the system so that the linear subsystem has the largest dimension. A systematic method is also developed to compute the desired feedback and coordinate change.

Stable inversion and parameter variations

Abstract As part of the process of automatically guiding an aircraft, we have been successful in using stable inversion to compute a desired bounded state trajectory and corresponding bounded control. In addition to this feedforward control, we must also construct a regulator to address modeling errors and disturbances. With respect to modeling errors we find that the stable inversion procedures used are so accurate that the regulator can assume a simple form, say a linear regulator about the desired trajectory. We show that under the appropriate assumptions, the bounded state trajectory and bounded control computed through stable inversion depend continuously on the parameters of the system. This is a consequence of a mathematical result that we prove about the continuous dependence of the “particular solution” of a time varying nonlinear system driven by a bounded input. This is distinct from the usual continuous dependence of the initial value problem for systems.

Active optical lattice filters

Highly integrated, novel architectures for optical filtering can leverage structure, gain and variable delays to provide a multi-use photonic platform. Hardware and software results are presented.

Linear dynamics hidden by input-output linearization

Abstract The input-output linearization method has been quite useful in determining the linear input-output responses of non-linear systems. However, research has shown that this technique can often hide part of the non-linear system which is linearizable. We prove sufficient conditions that a non-linear system contain a k-dimensional linear system after the input-output linearization control is applied. Here p ≤ k ≤ n, where p is the relative degree of the system and n is the dimension of the non-linear system. The results for k = n are known from the literature

Lattice filter with adjustable gains and its application in optical signal processing

In this paper, we consider a class of N-th order lattice filter with gains as an extension to the traditional lattice filter structures and develop analysis and synthesis of such filters. In the analysis-synthesis parts, we present recursive methods to derive the input-output transfer function of the filter in terms of parameters of the lattice structure such as the time delay, gains, transmission and reflection coefficients and vice versa. Stability of the filter is analyzed and an algorithm to test the stability is proposed. This class of MIMO lattice filters with adjustable gains models integrated photonic devices under development in our labs for optical communication and high speed signal processing applications. Our signal processing approach to characterizing this type of lattice structures can also be used in filter realization by VLSI, FPGA, or programmable processors for acoustic or speech applications

Output tracking and steady state for nonlinear systems

The problem driving this work is exact output tracking using stable inversion for nonlinear systems (models of aircraft). The problem of stable inversion evolves to a time varying nonlinear system with inputs and the search for a unique bounded continuous solution on (-/spl infin/,/spl infin/) in response to a bounded input on (-/spl infin/,/spl infin/). We want to show that all bounded continuous solutions on 0/spl les/t</spl infin/ converge to this unique one as t/spl rarr//spl infin/ under the appropriate assumptions. The bounded and continuous solution on (-/spl infin/,/spl infin/) can be thought of as a nonlinear steady state. This exactly parallels the idea of steady state for a time invariant linear system whose homogenous part has no eigenvalues on the imaginary axis. In response to a bounded input on (-/spl infin/,/spl infin/) is the unique bounded and continuous solution on (-/spl infin/,/spl infin/). All bounded and continuous solutions on 0/spl les/t</spl infin/ must converge to this unique one, which is called the steady state. Hence, under appropriate assumptions, steady state solutions exist for nonlinear systems even though the principle of superposition does not hold.