A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs

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

In this paper, we present a convex formulation of $H_{\infty}$-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 $H_{\infty}$ 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 $H_{\infty}$ 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 $H_{\infty}$-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 $H_{\infty}$-optimal state-feedback control problems as LPIs. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples

A New Treatment of Boundary Conditions in PDE Solution with Galerkin Methods via Partial Integral Equation Framework

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing equation. The framework is based on a Partial Integral Equation (PIE) representation of PDEs, where a PDE equation is transformed into an equivalent PIE formulation that does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized PIE-Galerkin approximation methodology for a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries. The significance of this result is that solution to almost any linear PDE can now be constructed in a form of an analytical approximation based on a series expansion using a suitable set of basis functions, such as, e.g., Chebyshev polynomials of the first kind, irrespective of the boundary conditions. In many cases involving homogeneous or simple time-dependent boundary inputs, an analytical integration in time is also possible. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.Comment: 42 pages, 9 figure

Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems

Partial Integral Equations (PIEs) are an alternative way to model systems governed by Partial Differential Equations (PDEs). PIEs have certain advantages over PDEs in that they are defined by integral (not differential) operators and do not include boundary conditions or continuity constraints on the solution -- a convenience when computing system properties, designing controllers, or performing simulation. In prior work, PIE representations were proposed for a limited class of $2^{nd}$-order PDEs in a single spatial variable. In this paper, we extend the PIE representation to a more general class of PDE systems including, e.g., higher-order spatial derivatives ($N^{th}$-order), PDEs with inputs and outputs, PDEs coupled with ODEs, PDEs with distributed input and boundary effects, and boundary conditions which combine boundary values with inputs and integrals of the state. These extensions are presented in a unified way by first proposing a unified parameterization of PDE systems, which we refer to as a Generalized PDE (GPDE). Given a PDE system in GPDE form, we next propose formulae that takes the GPDE parameters and constructs the Partial Integral (PI) operators that are used to define a PIE system. This formula includes a unitary (and hence invertible) map that converts solutions of the PIE to solutions of the GPDE. This unitary map is then used to show that the original GPDE and PIE have equivalent system properties, including well-posedness and stability. These representations, conversions, and mappings are illustrated through several diverse examples, including beams, mixing problems, entropy modeling, wave equations, etc. Finally, we illustrate the significance of the PIE representation by solving analysis, simulation, and control problems for several representative PDE systems