Control and Learning in Uncertain Nonlinear Systems with Theoretical Guarantees
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2024-05
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Melbourne Authors
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Electrical and Electronic Engineering
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PhD thesis
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© 2024 Seth Siriya
Abstract
Control theory is a powerful tool that ensures the desired behaviour of systems through the use of feedback. While many popular controllers offer nominal robustness against minor disturbances, their performance can deteriorate when confronted with more significant and complex uncertainties, compromising theoretical guarantees like safety and stability. This thesis explores novel strategies for the design and analysis of control schemes for nonlinear systems, that uphold theoretical guarantees in the face of such uncertainties.
In the first part of the thesis, we consider the challenge of developing algorithms to safely navigate time-varying systems with bounded uncertainties and online constraints in real time. This is motivated by robot navigation problems in a priori unknown environments, where both real-time performance and theoretical safety guarantees for collision avoidance are essential. We propose a method which uses an offline-computed Hamilton-Jacobi reachability-based safety-guaranteed controller to track a virtual system replanned online, accounting for both time-varying dynamics and constraints updated in real time. Our method is illustrated on a case study of an autonomous underwater vehicle subject to wave disturbances.
In the second part of the thesis, we investigate learning-based approaches for stabilising adaptive control of discrete-time nonlinear stochastic systems. We start by turning our attention to one of the simplest nonlinear stochastic problems to formulate, but yet to be addressed in the adaptive setting --- input-constrained linear systems subject to unbounded stochastic disturbances. We first propose a simple certainty equivalence adaptive control scheme that achieves mean square boundedness of the closed-loop states on unknown scalar systems. We then generalise our method to multi-dimensional systems, and derive high probability stability bounds on the closed-loop states by combining stochastic control theory and non-asymptotic system identification.
With a view towards generalising our approach to broader classes of nonlinear systems, we analyse the problem of least squares estimation from single-trajectory data for learning discrete-time, unstable nonlinear stochastic systems with linearly parameterised uncertainty. Assuming the system is sub-exponentially unstable, and there is a region of the state space which produces informative data, we derive non-asymptotic guarantees on the least squares estimation error of the unknown parameter that hold over times where the system remains in this region. By instead assuming that the entire state space produces informative data, we obtain high probability guarantees on the estimation error that hold over all time.
Finally, we generalise the methods and analysis from previous chapters to the adaptive control problem for discrete-time, nonlinear stochastic systems with linearly parameterised uncertainty. Assuming access to a parameterised family of controllers, that successfully stabilises the system in a bounded, informative region of the state space when the parameter is well-chosen, we propose a certainty equivalence learning-based adaptive control strategy, and subsequently derive stability bounds on the closed-loop system that hold for some probabilities. We then show that if the entire state space is informative, and the family of controllers is globally stabilising with appropriately chosen parameters, high probability stability guarantees can be derived.
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control theory, nonlinear control, Hamilton-Jacobi reachability, stochastic control, adaptive control, learning