Electrical and Electronic Engineering - Research Publications
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ItemExploiting homogeneity for the optimal control of discrete-time systems: Application to value iteration(IEEE, 2021)To investigate solutions of (near-)optimal control problems, we extend and exploit a notion of homogeneity recently proposed in the literature for discrete-time systems. Assuming the plant dynamics is homogeneous, we first derive a scaling property of its solutions along rays provided the sequence of inputs is suitably modified. We then consider homogeneous cost functions and reveal how the optimal value function scales along rays. This result can be used to construct (near-)optimal inputs on the whole state space by only solving the original problem on a given compact manifold of a smaller dimension. Compared to the related works of the literature, we impose no conditions on the homogeneity degrees. We demonstrate the strength of this new result by presenting a new approximate scheme for value iteration, which is one of the pillars of dynamic programming. The new algorithm provides guaranteed lower and upper estimates of the true value function at any iteration and has several appealing features in terms of reduced computation. A numerical case study is provided to illustrate the proposed algorithm.
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ItemOptimistic planning for the near-optimal control of nonlinear switched discrete-time systems with stability guarantees(IEEE, 2020-03-12)Originating in the artificial intelligence literature, optimistic planning (OP) is an algorithm that generates near-optimal control inputs for generic nonlinear discrete-time systems whose input set is finite. This technique is therefore relevant for the near-optimal control of nonlinear switched systems, for which the switching signal is the control. However, OP exhibits several limitations, which prevent its application in a standard control context. First, it requires the stage cost to take values in [0, 1], an unnatural prerequisite as it excludes, for instance, quadratic stage costs. Second, it requires the cost function to be discounted. Third, it applies for reward maximization, and not cost minimization. In this paper, we modify OP to overcome these limitations, and we call the new algorithm OPmin. We then make stabilizability and detectability assumptions, under which we derive near-optimality guarantees for OPmin and we show that the obtained bound has major advantages compared to the bound originally given by OP. In addition, we prove that a system whose inputs are generated by OPmin in a receding-horizon fashion exhibits stability properties. As a result, OPmin provides a new tool for the near-optimal, stable control of nonlinear switched discrete-time systems for generic cost functions.
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ItemStability guarantees for nonlinear discrete-time systems controlled by approximate value iteration(IEEE, 2020-03-12)Value iteration is a method to generate optimal control inputs for generic nonlinear systems and cost functions. Its implementation typically leads to approximation errors, which may have a major impact on the closed-loop system performance. We talk in this case of approximate value iteration (AVI). In this paper, we investigate the stability of systems for which the inputs are obtained by AVI. We consider deterministic discrete-time nonlinear plants and a class of general, possibly discounted, costs. We model the closed-loop system as a family of systems parameterized by tunable parameters, which are used for the approximation of the value function at different iterations, the discount factor and the iteration step at which we stop running the algorithm. It is shown, under natural stabilizability and detectability properties as well as mild conditions on the approximation errors, that the family of closed-loop systems exhibit local practical stability properties. The analysis is based on the construction of a Lyapunov function given by the sum of the approximate value function and the Lyapunov-like function that characterizes the detectability of the system. By strengthening our conditions, asymptotic and exponential stability properties are guaranteed.