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

Shivakumar, Sachin; Das, Amritam; Peet, Matthew M.

doi:10.23919/acc45564.2020.9147712

Cited by 15 publications

(23 citation statements)

References 7 publications

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“…where the functions H ij are as defined in (9). Lemma 15: Suppose T is as defined in Theorem 13, and let…”

Section: B Pde To Pie Conversionmentioning

confidence: 99%

“…Lyapunov stability analysis [8], robust stability analysis [12], and H ∞ -optimal control [13] of systems involving 1D PDEs. For a detailed manual of the PIETOOLS toolbox we refer to [9].…”

Section: Pietools Implementationmentioning

confidence: 99%

“…Thus, the use of PIEs and PI operators also resolves the difficulty with parameterizing positive LFs, which we now take to be of the form V (u) = u, Pu . In 1D, the Matlab toolbox PIETOOLS [9] can be used for parsing and solving LPI optimization problems.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, through use of the LPI and PIE framework, stability of almost any 1D PDE can be efficiently tested using convex optimization (see PIETOOLS 2021 [9]). However, as of yet, none of this architecture is available for 2D systems.…”

Section: Introductionmentioning

confidence: 99%

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A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Declan¹,

Peet²

2021

Preprint

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We introduce a Partial Integration Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L 2 [x, y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R n × L 2 [x] × L 2 [y] × L 2 [x, y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism.

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“…where the functions H ij are as defined in (9). Lemma 15: Suppose T is as defined in Theorem 13, and let…”

Section: B Pde To Pie Conversionmentioning

confidence: 99%

Section: Pietools Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Declan¹,

Peet²

2021

Preprint

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“…Use of the PIE system representation, defined by the algebra of Partial Integral (PI) operators, allows us to generalize LMIs developed for ODEs to infinitedimensional systems. These generalizations are referred to as Linear PI Inequalities (LPIs) and can be solved efficiently using the Matlab toolbox PIETOOLS [10]. In previous work, LPIs have been proposed for stability [7], H ∞ -gain [9] and H ∞ -optimal estimation [2] of PIE systems.…”

Section: Introductionmentioning

confidence: 99%

Duality and $H_{\infty}$-Optimal Control Of Coupled ODE-PDE Systems

Shivakumar,

Das,

Weiland

et al. 2020

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In this paper, we present a convex formulation of H∞-optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and H∞ performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and H∞ performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and H∞-performance problems using the Linear PI Inequality (LPI) framework. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Next, we use our duality results to formulate the stabilization and H∞-optimal statefeedback control problems as LPIs. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.

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H_∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

Peet

Sun

et al. 2023

Intl J Robust & Nonlinear

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This article investigates the H ∞ -optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free-obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The H ∞ -optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)-a form of convex optimization using operator variables and inequlities. Such LPI-based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non-conservtism to 4 decimal places when compared with Pad é-based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach.

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

Cited by 15 publications

References 7 publications

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Duality and $H_{\infty}$-Optimal Control Of Coupled ODE-PDE Systems

H_∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

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

Cited by 15 publications

References 7 publications

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

Duality and $H_{\infty}$-Optimal Control Of Coupled ODE-PDE Systems

H∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

Contact Info

Product

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H_∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach