Abstract
The development of efficient methods for process performance verification has drawn a lot of attention in the research community. Viability theory is a mathematical tool to identify the trajectories of a dynamical system which remains in a constraint set. In this paper, viability theory is investigated for this purpose in the case of nonlinear processes that can be represented in Linear Parameter Varying (LPV) form. In particular, verification algorithms based on the use of invariance and viability kernels and capture basin are proposed. The difficulty with the application of this theory is the computation of these sets. A Lagrangian method has been used to approximate these sets. Because of simplicity and efficient computations, zonotopes are adopted for set representation. Two new sets called Safe Work Area (SWA) and Required Performance (RP) are defined and an algorithm is proposed to use these concepts for the verification purpose. Finally, two application examples based on well-known case studies, a two-tank system and PH neutralization plant, are provided to show the effectiveness of the proposed method.
Highlights
The objective of process design is that the process achieves the desired dynamical behavior specified in terms of a set of specifications
Based on [38], a method of expressing finite horizon viability kernels in terms of reachable sets is presented. This provides a modified version of Saint-Pierre’s viability kernel algorithm that can be implemented using efficient and scalable techniques developed within the context of reachability analysis. We can reformulate this recursive definition of the finite horizon viability kernels Kn that is defined as ViabS in (2) but with T = [0, n], in terms of the backward reach set over one discrete time step ReachtB(X)
Considering the process dynamics (1) and the desired requirements, at each time iteration in the time horizon selected for the analysis: Determine viability kernel using Algorithm 2 Determine Safe Work Area (SWA) from the viability kernel using Equation (12) Determine invariance kernel using Algorithm 1 Determine capture basin from the capture basin based on invariance kernel and desired time steps to reach the target using Algorithm 3 Determine Required Performance (RP) using Equation (13) if SWA ∩ RP = ∅
Summary
The objective of process design is that the process achieves the desired dynamical behavior specified in terms of a set of specifications. The use of sets (and in particular, zonotopes) has been proposed for systems verification, but in general restricted to the reachability analysis, as e.g., the works of Althoff and Girard, see [16] for a recent review This paper extends this idea to a more general framework, the viability theory. The main contribution of this paper is an approach for process performance verification using these concepts from viability theory. The representation of the nonlinear process model in this way facilitates the set computations based on zonotopes that are required to apply the viability theory to the process performance verification problem. Few of the evaluation and verification algorithms existing presently take into account bounds and constraints of the system explicitly Another contribution of the proposed method is that bounds and constraints of the system has been considered in computing the viability sets. Some properties that have been used are introduced in the Appendix A
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