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

Abstract

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.Thenwhere P eq = P J 0 , J 1 , J 2 J 3 , J 4 , J 5 . From the Theorem statement, P eq + P * eq 0. Then, all the conditions of Theorem 1 are satisfied. Hence y L2 ≤ γ w L2 .

Delay-robust stabilization of an n+m hyperbolic PDE-ODE system

“…The control of interconnected ODEs and hyperbolic PDEs is an extremely active research topic [8], [10], [16], [27]. This class of systems naturally appears when modeling delays (that can be seen as first-order hyperbolic PDEs) in the actuating and sensing paths of ODEs [8], [9], [17], [29], [30], [33]. For instance, one can consider the problem of the attenuation of mechanical vibrations in drilling applications (see [28] for a review of drilling vibrations models).…”

Representation of Networks and Systems with Delay: DDEs, DDFs, ODE-PDEs and PIEs

“…If we define the subalgebra of PI operators parameterized by polynomials, then the software package PIETOOLS [12] allows for: manipulation of PI operators as a class object; declaration of PI operator variables; enforcement of PI operators positivity constraints; and solution of convex optimization problems defined by linear operator inequality constraints. For a more extensive discussion of the optimization of PI operators and their use in analysis and optimal estimation and control of infinite dimensional systems, we refer to the PIETOOLS manual [12] or any of the recent papers on analysis and control in the PIE framework [19,8,10,20,11,9]. Without embarking on an exhaustive discussion of these results, we note that the consensus seems to be that analysis and control in the PIE framework is possible when the distributedparameter part of the state is in L N 2 where N ≤ 50.…”

“…Unlike Dirac and differential operators, PI operators are bounded and form an algebra. Furthermore PIE models do not require boundary conditions or continuity constraints [8,9,10] -simplifying analysis and control problems. Indeed, it has been shown in several recent papers [8,9,10,11] that many problems in analysis and optimal estimation and control of coupled ODE-PDE models can be formulated as optimization over the cone of positive PI operators.…”

“…Furthermore PIE models do not require boundary conditions or continuity constraints [8,9,10] -simplifying analysis and control problems. Indeed, it has been shown in several recent papers [8,9,10,11] that many problems in analysis and optimal estimation and control of coupled ODE-PDE models can be formulated as optimization over the cone of positive PI operators. The further development of efficient software tools for manipulation and optimization of PI operators (See PIETOOLS [12]) means that representation of delay systems using PIEs may allow for the development of new algorithms for control of network models such as in Eqn.…”

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

“…Recently, the development of Partial Integral Equation (PIE) representations of PDE systems has created a framework for the extension of LMI-based methods to infinite-dimensional systems. This PIE representation encompasses a broad class of distributed parameter systems and is algebraic -eliminating the use of boundary conditions and continuity constraints [1], [2], [3]. Such PIE representations have the form T ẋ(t) + B d1 ẇ(t) + B d2 u(t) = Ax(t) + B 1 w(t) + B 2 u(t) z(t) = C 1 x(t) + D 11 w(t) + D 12 u(t), y(t) = C 2 x(t) + D 21 w(t) + D 22 u(t) where the T , A, B i , C i , D ij are Partial Integral (PI) operators and have the form…”