Robustness of Nonlinear Predictor Feedback Laws to Time- and State-Dependent Delay Perturbations

Much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state---provided the dependence is known. In this paper we consider the delay variations as unknown and study robustness of nominal constant-delay predictor feedbacks under delay variations that depend on time and the state. We show that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the special case of linear systems, and under only time-varying delay perturbations, we prove robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics. We present two examples, one that is concerned with the control of a DC motor through a network and one of a bilateral teleoperation between two robotic systems.Comment: Submitte

Stabilization of Nonlinear Delay Systems Using Approximate Predictors and High-Gain Observers

We provide a solution to the heretofore open problem of stabilization of systems with arbitrarily long delays at the input and output of a nonlinear system using output feedback only. The solution is global, employs the predictor approach over the period that combines the input and output delays, addresses nonlinear systems with sampled measurements and with control applied using a zero-order hold, and requires that the sampling/holding periods be sufficiently short, though not necessarily constant. Our approach considers a class of globally Lipschitz strict-feedback systems with disturbances and employs an appropriately constructed successive approximation of the predictor map, a high-gain sampled-data observer, and a linear stabilizing feedback for the delay-free system. The obtained results guarantee robustness to perturbations of the sampling schedule and different sampling and holding periods are considered. The approach is specialized to linear systems, where the predictor is available explicitly.Comment: 14 pages, 3 figures, contains the technical proofs of a paper which is going to appear in Automatica. arXiv admin note: substantial text overlap with arXiv:1108.449

Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold

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.Comment: 32 pages. 3 figures, submitted for possible publication to IEEE Transactions on Automatic Contro

Predictor-Based Output Feedback for Nonlinear Delay Systems

We provide two solutions to the heretofore open problem of stabilization of systems with arbitrarily long delays at the input and output of a nonlinear system using output feedback only. Both of our solutions are global, employ the predictor approach over the period that combines the input and output delays, address nonlinear systems with sampled measurements and with control applied using a zero-order hold, and require that the sampling/holding periods be sufficiently short, though not necessarily constant. Our first approach considers general nonlinear systems for which the solution map is available explicitly and whose one-sample-period predictor-based discrete-time model allows state reconstruction, in a finite number of steps, from the past values of inputs and output measurements. Our second approach considers a class of globally Lipschitz strict-feedback systems with disturbances and employs an appropriately constructed successive approximation of the predictor map, a high-gain sampled-data observer, and a linear stabilizing feedback for the delay-free system. We specialize the second approach to linear systems, where the predictor is available explicitly. We provide two illustrative examples-one analytical for the first approach and one numerical for the second approach.Comment: 31 pages, 2 figures. To be submitted to Automatic