Verification of the control system performance using viability theory

Abstract

Abstract-The development of efficient methods for the control system performance verification has drawn a lot of attention recently. In this paper, the use of viability theory for this purpose is investigated in case of non-linear systems. In particular, verification algorithms based on the use of the computation of invariance and viability kernels and capture basin are proposed. A Lagrangian method has been used in order to approximate these sets for nonlinear systems. Because of simplicity and efficient computations, zonotopes are adopted for set representation. An application example based on a well known control benchmark is provided in order to show the effectiveness of the proposed method.

Process Performance Verification Using Viability Theory

“…To illustrate the effectiveness of the process verification approach proposed in this paper, two application examples are provided. A preliminary version of the results of this paper is presented in [23]. Here, an algorithm based This paper is organized as follows.…”

Computation of discriminating kernel and robust capture basin with regulation map by interval methods

“…Researchers started to apply viability theory on control technology since 1990 . Related works compute the viability kernels and the capture basins for systems without uncertainties, whereas the works of Zarch et al handled fault tolerance evaluation. Most of them compute the viability kernels and the capture basins by reachability analysis, and the analytic methods are mainly categorized to Hamilton‐Jacobi equations, zonotopes, and interval methods.…”

“…Most of them compute the viability kernels and the capture basins by reachability analysis, and the analytic methods are mainly categorized to Hamilton‐Jacobi equations, zonotopes, and interval methods. The algorithms of zonotope have been adopted in the works of Zarch et al, whereas tools of interval methods have been taken in related works, and comparatively, Hamilton‐Jacobi equations have been used in other works . As both control and uncertain parametric variables, in viability theory, are regarded as parameters in a parameterized dynamic system, prior to 1994, discriminating kernel proposed in the work of Cardaliaguet et al has distinguished them.…”

“…If the uncertainty is considered, propagating the initial sets would be hard due to overestimation of interval computation, assumptions on propagation time and initial sets, fixed control inputs during propagation, and the random selection among control inputs. Apart from these, algorithms relying zonotopes adopted in the work of Zarch et al are limited to symmetric convex set approximations, and their computational complexity increases exponentially with the iteration of the propagation and the dimension of systems. In comparison to the methods mentioned earlier, the accuracy of computation using Hamilton‐Jacobi equations for reachability analysis is lower than the aforementioned methods, and it would be hard to obtain guaranteed solutions .…”