Dilated LMI characterization for the robust finite time control of discrete-time uncertain linear systems

This paper provides new dilated linear matrix inequalities (LMIs) characterizations for the finite time boundedness (FTB) and the finite time stability (FTS) analysis of discrete-time uncertain linear systems. The dilated LMIs are later used to design a robust controller for the finite time control of discrete-time uncertain linear systems. The relevant feature of the proposed approach is the decoupling between the Lyapunov and the system matrices, that allows considering a parameter-dependent Lyapunov function. In this way, the conservativeness with respect to previous results is decreased. Numerical examples are used to illustrate the results.Peer ReviewedPostprint (author's final draft

Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem

An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed

Discretization of control law for a class of variable structure control systems

A new method for the discretization of a class of continuous-time variable structure control systems, based on the linear complementarity theory, is proposed. The proposed method consists two steps. In the first step, the motion projected on the sliding manifold (the fast dynamics) is discretized by means of backward Euler time-step method. In the second step, the sampled and hold control law is determined such that the trajectories of the discrete-time closed loop system projected on the sliding manifold coincide with the trajectories of discretized fast dynamics. The discrete-time closed-loop system exhibits discrete-time sliding motion. It means that the trajectories of the discrete-time closed loop system reach the sliding manifold in a finite number of steps and stay on it after that. Also, it is proved that control law is a continuous function. Therefore, the closed loop system is chattering free. The theoretically obtained results are verified on the example of the non-holonomic integrator. \u

Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discrete-time controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming problem on measures, which is then relaxed as finite-dimensional semidefinite programming problems on moments of measures and their duals on sums-of-squares polynomials. Nonlinear controllers can be extracted from the solutions to the relaxed problems. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and the computational complexity is polynomial in the state and input dimensions, and hence the approach is more scalable. In addition, we show that the approach can be applied to over-approximating the backward reachable set of discrete-time autonomous polynomial systems and the controllable set of discrete-time polynomial systems under known state feedback control laws. We illustrate our approach on several dynamical systems