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PIETOOLS 2024: User Manual

Sachin Shivakumar, Declan Jagt, Danilo Braghini, Amritam Das, Yulia Peet, Matthew Peet

TL;DR

PIETOOLS 2024 provides a comprehensive MATLAB-based framework for modeling, analyzing, and controlling linear distributed-parameter systems via Partial Integral Equations (PIEs). It unifies PDE/DDE/ODe representations into PIE form, enabling convex optimization through Linear PI Inequalities (LPIs) and providing integrated simulation with PIESIM. The toolbox supports 1D and 2D PI operators (opvar/opvar2d) and offers GUI, multiple input formats, and interconnection capabilities for building closed-loop PIE systems, alongside robust LPI-based design for controllers and estimators. Together, these capabilities streamline stability analysis, H2/H-infinity performance bounds, and closed-loop synthesis for complex PDE and DDE models, with visualization and benchmarking through demonstrations. The release emphasizes 2D PDE support, enhanced H2 analysis, and streamlined workflows that extend SDP techniques to infinite-dimensional systems with practical tools for simulation and control design.

Abstract

The PIETOOLS 2024 User Manual describes all the features of version 2024 of the MATLAB toolbox PIETOOLS for the analysis and control of Partial Integral Equations (PIEs). The manual is aimed to guide, with examples, first-time users to four fundamental features of PIETOOLS: converting coupled ODE-PDEs, DDEs, DDFs, etc., to PIE representation; analysis of stability and input-output properties of PIEs; design of optimal observers and controllers for PIEs; simulation of open- and closed-loop PIE systems. The use of PIETOOLS is not limited to the features described above. However, the manual focuses on these features to provide a holistic understanding of the workflow of PIETOOLS, which will serve as a foundation to develop more complicated programs, for example, the design of boundary feedback controllers, robust observers, robust controllers, etc..

PIETOOLS 2024: User Manual

TL;DR

PIETOOLS 2024 provides a comprehensive MATLAB-based framework for modeling, analyzing, and controlling linear distributed-parameter systems via Partial Integral Equations (PIEs). It unifies PDE/DDE/ODe representations into PIE form, enabling convex optimization through Linear PI Inequalities (LPIs) and providing integrated simulation with PIESIM. The toolbox supports 1D and 2D PI operators (opvar/opvar2d) and offers GUI, multiple input formats, and interconnection capabilities for building closed-loop PIE systems, alongside robust LPI-based design for controllers and estimators. Together, these capabilities streamline stability analysis, H2/H-infinity performance bounds, and closed-loop synthesis for complex PDE and DDE models, with visualization and benchmarking through demonstrations. The release emphasizes 2D PDE support, enhanced H2 analysis, and streamlined workflows that extend SDP techniques to infinite-dimensional systems with practical tools for simulation and control design.

Abstract

The PIETOOLS 2024 User Manual describes all the features of version 2024 of the MATLAB toolbox PIETOOLS for the analysis and control of Partial Integral Equations (PIEs). The manual is aimed to guide, with examples, first-time users to four fundamental features of PIETOOLS: converting coupled ODE-PDEs, DDEs, DDFs, etc., to PIE representation; analysis of stability and input-output properties of PIEs; design of optimal observers and controllers for PIEs; simulation of open- and closed-loop PIE systems. The use of PIETOOLS is not limited to the features described above. However, the manual focuses on these features to provide a holistic understanding of the workflow of PIETOOLS, which will serve as a foundation to develop more complicated programs, for example, the design of boundary feedback controllers, robust observers, robust controllers, etc..

Paper Structure

This paper contains 185 sections, 8 theorems, 209 equations, 25 figures, 9 tables.

Table of Contents

  1. About PIETOOLS
  2. PIETOOLS 2024 Release Notes
  3. Major Updates:
  4. Minor Updates
  5. Installing PIETOOLS
  6. Installation
  7. Scope of PIETOOLS
  8. Motivation
  9. PIETOOLS for Analysis, Control, and Simulation of ODE-PDE Systems
  10. Defining Models in PIETOOLS
  11. Declaring Boundary Conditions
  12. Simulating ODE-PDE Models
  13. Analysis and Control of the ODE-PDE Model Using PIEs
  14. Summary
  15. PI Operators in PIETOOLS
  16. ...and 170 more sections

Key Result

Lemma 4

For any PI parameters $Q,R\in\mathcal{ N}^{\text{m}\times\text{n}}\bigl[[a,b],[c,d]\bigr]$, there exist unique parameters $P\in\mathcal{ N}^{\text{m}\times\text{n}}\bigl[[a,b],[c,d]\bigr]$ such that That is, for any $\mathbf{ x}\in Z^{\textnormal{n}}\bigl[[a,b],[c,d]\bigr]$,

Figures (25)

  • Figure 1: Transient response of the state variable $\dot{\mathbf{ x}}(t,s)$ and regulated output $r(t)$ by simulating the ODE-PDE model \ref{['part1:ode']} and \ref{['part1:pde']} with $u(t) = 0$ for external disturbance $w(t) = sin(5t) e^{-t}$.
  • Figure 2: Transient response of the state variable $\dot{\mathbf{ x}}(t,s)$ and regulated output $r(t)$ on the closed-loop system for external disturbance $w(t) = sin(5t) e^{-t}$.
  • Figure 3: Open- $(r(t))$ and closed-loop ($r_{cl}(t)$) transient response of the regulated output $r(t)$ for external disturbance $w(t) = sin(5t) e^{-t}$.
  • Figure 4: Example of empty GUI window.
  • Figure 5: GUI window after loading the file PIETOOLS_PDE_Ex_Heat_Eq_with_Distributed_Disturbance_GUI from the library of PDE examples.
  • ...and 20 more figures

Theorems & Definitions (18)

  • Definition 1: 3-PI Operator
  • Definition 2: 9-PI Operator
  • Definition 3: PI Operator
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 8 more