سنتز کنترل برای تغییر ناپذیری قوی از سیستم های دینامیکی چند جمله ای با استفاده از برنامه ریزی خطی

In this paper, we consider a control synthesis problem for a class of polynomial dynamical systems subject to bounded disturbances and with input constraints. More precisely, we aim at synthesizing at the same time a controller and an invariant set for the controlled system under all admissible disturbances. We propose a computational method to solve this problem. Given a candidate polyhedral invariant, we show that controller synthesis can be formulated as an optimization problem involving polynomial cost functions over bounded polytopes for which effective linear programming relaxations can be obtained. Then, we propose an iterative approach to compute the controller and the polyhedral invariant jointly. Each iteration of the approach mainly consists in solving two linear programs (one for the controller and one for the invariant) and is thus computationally tractable. Finally, we show with several examples the usefulness of our method in applications.

The design of nonlinear systems remains a challenging problem in control science. In the past decade, building on spectacular breakthroughs in optimization over polynomial functions [1] and [2], several computational methods have been developed for synthesizing controllers for polynomial dynamical systems [3] and [4]. These approaches have shown themselves to be successful for several synthesis problems such as stabilization or optimal control in which Lyapunov functions and cost functions can be represented or approximated by polynomials. However, these approaches are not suitable for some other problems, such as those involving polynomial dynamical systems with constraints on states and inputs, and subject to bounded disturbances. In this paper, we consider a control synthesis problem for this class of systems. More precisely, given a polynomial dynamical system with input constraints and bounded disturbances, given a set of initial states View the MathML sourceP¯ and a set of safe states View the MathML sourceP¯, we aim at synthesizing a controller satisfying the input constraints and such that trajectories starting in View the MathML sourceP¯ remain in View the MathML sourceP¯ for all possible disturbances. This problem can be solved by computing jointly the controller and an invariant set for the controlled system which contains View the MathML sourceP¯ and is included in View the MathML sourceP¯ (see e.g. [5]). Here, we should mention that, even in the linear case, the problem of designing jointly a controller and an invariant is not trivial [6] and [7], and it is known that it can lead to a nonlinear problem. In the following, we propose a computational method based on the use of parameterized template expressions for the controller and the invariant. Given a candidate polyhedral invariant, we show that controller synthesis can be formulated as an optimization problem involving polynomial objective functions over bounded polytopes. Recently, using various tools such as the blossoming principle [8] for polynomials, multi-affine functions [9], and Lagrangian duality, it has been shown how effective linear programming relaxations can be obtained for such optimization problems [10]; these relaxations were then used for the computation of invariants for autonomous polynomial dynamical systems. The improvement over the work of [10] is that, in this paper, constrained inputs and bounded disturbances are considered; also, an iterative approach is given to compute jointly a controller and a polyhedral invariant. Each iteration of the approach mainly consists in solving two linear programs and is thus computationally tractable. Finally, we show applications of our approach to several examples.

In this paper, we have considered the problem of synthesizing controllers ensuring robust invariance of polynomial dynamical systems. Using the recent results of [10] on polynomial optimization over bounded polytopes, we have developed an iterative approach to solve this problem. It is mainly based on linear programming, and therefore it is effective. We have presented applications to several examples, showing the usefulness of the approach. Future work will focus on a deeper theoretical analysis of the properties of the linear programming relaxations of polynomial optimization problems as well as their application to other classes of problems in control.