No Arabic abstract
This article treats three problems of sparse and optimal multiplexing a finite ensemble of linear control systems. Given an ensemble of linear control systems, multiplexing of the controllers consists of an algorithm that selects, at each time (t), only one from the ensemble of linear systems is actively controlled whereas the other systems evolve in open-loop. The first problem treated here is a ballistic reachability problem where the control signals are required to be maximally sparse and multiplexed, the second concerns sparse and optimally multiplexed linear quadratic control, and the third is a sparse and optimally multiplexed Mayer problem. Numerical experiments are provided to demonstrate the efficacy of the techniques developed here.
In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^0$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.
This paper proposes a data-driven framework to solve time-varying optimization problems associated with unknown linear dynamical systems. Making online control decisions to regulate a dynamical system to the solution of an optimization problem is a central goal in many modern engineering applications. Yet, the available methods critically rely on a precise knowledge of the system dynamics, thus mandating a preliminary system identification phase before a controller can be designed. In this work, we leverage results from behavioral theory to show that the steady-state transfer function of a linear system can be computed from data samples without any knowledge or estimation of the system model. We then use this data-driven representation to design a controller, inspired by a gradient-descent optimization method, that regulates the system to the solution of a convex optimization problem, without requiring any knowledge of the time-varying disturbances affecting the model equation. Results are tailored to cost functions satisfy the Polyak-L ojasiewicz inequality.
This paper deals with the fault detection and isolation (FDI) problem for linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. In this paper, we follow a geometric approach to verify solvability of the FDI problem for such systems. To do so, we first develop a necessary and sufficient condition under which the FDI problem for a given particular linear time-invariant system is solvable. Next, we establish a necessary condition for solvability of the FDI problem for linear structured systems. In addition, we develop a sufficient algebraic condition for solvability of the FDI problem in terms of a rank test on an associated pattern matrix. To illustrate that this condition is not necessary, we provide a counterexample in which the FDI problem is solvable while the condition is not satisfied. Finally, we develop a graph-theoretic condition for the full rank property of a given pattern matrix, which leads to a graph-theoretic condition for solvability of the FDI problem.
This paper presents a sparse solver based on the alternating direction method of multipliers algorithm for a linear model predictive control (MPC) formulation in which the terminal state is constrained to a given ellipsoid. The motivation behind this solver is to substitute the typical polyhedral invariant set used as a terminal constraint in many nominal and robust linear MPC formulations with an invariant set in the form of an ellipsoid, which is (typically) much easier to compute and results in an optimization problem with significantly fewer constraints, even for average-sized systems. However, this optimization problem is no longer the quadratic programming problem found in most linear MPC approaches, thus meriting the development of a tailored solver. The proposed solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint and requires no external libraries. We show the results of its implementation in an embedded system to control a simulated multivariable plant, comparing it against other alternatives.
Linear-Rate Multi-Mode Systems is a model that can be seen both as a subclass of switched linear systems with imposed global safety constraints and as hybrid automata with no guards on transitions. We study the existence and design of a controller for this model that keeps the state of the system within a given safe set for the whole time. A sufficient and necessary condition is given for such a controller to exist as well as an algorithm that finds one in polynomial time. We further generalise the model by adding costs on modes and present an algorithm that constructs a safe controller which minimises the peak cost, the average-cost or any cost expressed as a weighted sum of these two. Finally, we present numerical simulation results based on our implementation of these algorithms.