PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators

• Published in: 2020 American Control Conference (ACC) pp. 2667 - 2672
• Main Authors: Shivakumar, Sachin, Das, Amritam, Peet, Matthew M.
• Format: Conference Proceeding
• Language: English
• Published: AACC 01-07-2020
• Subjects: Aerospace electronics; Control systems; Matlab; Optimization; Stability analysis; Syntactics
• ISSN: 2378-5861
• DOI: 10.23919/ACC45564.2020.9147712

In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g. +, *, ' etc.) is defined. PI operators are a generalization of bounded linear operators on infinite-dimensional spaces that form a *-subalgebra with two binary operations (addition and composition) on the space ℝ × L 2 . These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential Equations (PDE) and Timedelay systems (TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add operator positivity constraints, declare an objective function, and solve the resulting optimization problem using a syntax similar to the sdpvar class in YALMIP. Use of the resulting Linear Operator Inequalities (LOI) are demonstrated on several examples, including stability analysis of a PDE, bounding operator norms, and verifying integral inequalities. The result is that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable, user-friendly and computationally efficient toolbox for parsing, performing algebraic operations, setting up and solving convex optimization problems on PI operators.