We present an approach for compensating input delay of arbitrary length in nonlinear control systems. This approach, which due to the infinite dimensionality of the actuator dynamics and due to the nonlinear character of the plant results in a nonlinear feedback operator, is essentially a nonlinear version of the Smith predictor and its various predictor-based modifications for linear plants. Global stabilization in the presence of arbitrarily long delay is achieved for all nonlinear plants that are globally stabilizable in the absence of delay and that satisfy the property of forward completeness (which is satisfied by most mechanical systems, electromechanical systems, vehicles, and other physical systems). For strict-feedforward systems, one obtains the predictor-based feedback law explicitly. For the linearizable subclass of strict-feedforward systems, closed-loop solutions are also obtained explicitly. The feedback designs are illustrated through two detailed examples.

Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems

The backstepping approach is adapted to the problem of globally uniformly asymptotically stabilizing nonlinear systems in feedback form with a delay arbitrarily large in the input. The strategy of design relies on the construction of a Lyapunov-Krasovskii functional. Continuously differentiable control laws are constructed.

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Backstepping design for time-delay nonlinear systems

Sampling arises simultaneously with input and output delays in networked control systems. When the delay is left uncompensated, the sampling period is generally required to be sufficiently small, the delay sufficiently short, and, for nonlinear systems, only semiglobal practical stability is generally achieved. For example, global stabilization of strict-feedforward systems under sampled measurements, sampled-data stabilization of the nonholonomic unicycle with arbitrarily sparse sampling, and sampled-data stabilization of LTI systems over networks with long delays, are open problems. In this paper, we present two general results that address these example problems as special cases. First, we present global asymptotic stabilizers for forward complete systems under arbitrarily long input and output delays, with arbitrarily long sampling periods, and with continuous application of the control input. Second, we consider systems with sampled measurements and with control applied through a zero-order hold, under the assumption that the system is stabilizable under sampled-data feedback for some sampling period, and then construct sampled-data feedback laws that achieve global asymptotic stabilization under arbitrarily long input and measurement delays. All the results employ “nominal” feedback laws designed for the continuous-time systems in the absence of delays, combined with “predictor-based” compensation of delays and the effect of sampling.

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Nonlinear Stabilization Under Sampled and Delayed Measurements, and With Inputs Subject to Delay and Zero-Order Hold

This note addresses the problems of stability analysis and stabilization of systems presenting nested saturations. Depending on the open-loop stability assumption, the global stability analysis and stabilization problems are considered. In the (local) analysis problem, the objective is the determination of estimates of the basin of attraction of the system. Considering the stabilization problem, the goal is to design a set of gains in order to enlarge the basin of attraction of the closed-loop system. Based on the modelling of the system presenting nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. From these conditions, convex optimization strategies are proposed to solve both problems

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Stability analysis and stabilization of systems presenting nested saturations

We introduce and solve stabilization problems for linear and nonlinear systems with state-dependent input delay. Since the state dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed, the key design challenge is how to determine the predictor state. We resolve this challenge and establish closed-loop stability of the resulting infinite-dimensional nonlinear system for general non-negative-valued delay functions of the state. Due to an inherent limitation on the allowable delay rate in stabilization of systems with time-varying input delays, in the case of state-dependent delay, where the delay rate becomes dependent on the gradient of the delay function and on the state and control input, only regional stability results are achievable. For forward-complete systems, we establish an estimate of the region of attraction in the state space of the infinite-dimensional closed-loop nonlinear system and for linear systems we prove exponential stability. Global stability is established under a restrictive Lyapunov-like condition, which has to be a priori verified, that the delay rate be bounded by unity, irrespective of the values of the state and input. We also establish local asymptotic stability for locally stabilizable systems in the absence of the delay. Several illustrative examples are provided, including unicycle stabilization subject to input delay that grows with the distance from the reference position.

Compensation of State-Dependent Input Delay for Nonlinear Systems

The problem of globally uniformly asymptotically and locally exponentially stabilizing a family of nonlinear feedforward systems when there is a delay in the input is solved. No limitation on the size of the delay is imposed. Explicit expressions of bounded control laws are determined.

Global asymptotic stabilization of feedforward systems with delay in the input

A model of PVTOL aircraft with two delayed inputs is considered. The origin of this system is globally asymptotically and locally exponentially stabilized by bounded control laws. The explicit expressions of the control laws applied are determined through recent extensions of the forwarding approach to systems with a delay in the input. In a second step, the output feedback stabilization problem consisting in globally asymptotically stabilizing the origin of the PVTOL when the variables of velocity are unmeasured is solved through a recent technique which extensively exploits the presence of positive delays in the inputs.

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Global Asymptotic Stabilization of a PVTOL Aircraft Model with Delay in the Input

Feedforward System with Delay in the Input and Vanishing Perturbations 1

This paper presents an electronic circuit that exhibits parametric resonance, namely a parametric RLC oscillator. The circuit exploits parametric resonance in an analog application, namely, a gain-frequency dependent switch. In such circuit, when the amplitude and frequency of the parametric excitation varies then a transition curves, in the parameter space, are computed by using an algorithm that calculates the symbolic monodromy matrix. Such curves are interpreted to implement an analog application.

R. Francisco

Papers

Global asymptotic stabilization of feedforward systems with delay in the input

Global Asymptotic Stabilization of a PVTOL Aircraft Model with Delay in the Input

Global asymptotic stabilization of feedforward systems with delay in the input

Feedforward System with Delay in the Input and Vanishing Perturbations 1

On the implementation of a parametric oscillator in analog applications