arXiv subjects

Sachin Shivakumar

Publications and source records attributed to Sachin Shivakumar.

24 records · Page 2Linked to original sources

arXiv2020

Duality and $H_{\infty}$-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.

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arXiv2020

Computing Input-Output Properties of Coupled PDE systems

In this paper, we propose an LMI-based approach to analyze input-output properties of coupled linear PDE systems. This work expands on a newly developed state-space theory for coupled PDEs and extends the positive-real and bounded-real lemmas to infinite dimensional systems. We show that conditions for passivity and bounded L2 gain can be expressed as linear operator inequalities on RxL2. A method to convert these operator inequalities to LMIs by using parameterization of the operator variables is proposed. This method does not rely on discretization and as such, the properties obtained are prima facie provable. We use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that the bounds are computable on desktop computers for systems consisting of up to 20 coupled PDEs.

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arXiv2020

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.

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arXiv2020

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

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. +, *, ' e tc.) 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 RxL2. These operators frequently appear in analysis and control of infinite-dimensional systems such as Partial Differential equations (PDE) and Time-delay 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 (LOIs) 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.

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arXiv2020

H\infty-Optimal Observer Design for Linear Systems with Delays in States, Outputs and Disturbances

This paper considers the H\infty-optimal estimation problem for linear systems with multiple delays in states, output, and disturbances. First, we formulate the H\infty-optimal estimation problem in the Delay-Differential Equation (DDE) framework. Next, we construct an equivalent Partial Integral Equation (PIE) representation of the optimal estimator design framework. We then show that in the PIE framework, the H\infty-optimal estimator synthesis problem can be posed as a Linear PI Inequality (LPI). LPIs are a generalization of LMIs to the algebra of Partial Integral (PI) operators and can be solved using the PIETOOLS toolbox. Finally, we convert the PIE representation of the optimal estimator back into an ODE-PDE representation - a form similar to a DDE, but with corrections to estimates of the infinite-dimensional state (the time-history). Numerical examples show that the synthesis condition we propose produces an estimator with provable H\infty-gain bound which is accurate to 4 decimal places when compared with results obtained using Pade-based discretization.

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arXiv2018

Representation and Stability Analysis of PDE-ODE Coupled Systems

In this work, we present a scalable Linear Matrix Inequality (LMI) based framework to verify the stability of a set of linear Partial Differential Equations (PDEs) in one spatial dimension coupled with a set of Ordinary Differential Equations (ODEs) via input-output based interconnection. Our approach extends the newly developed state space representation and stability analysis of coupled PDEs that allows parametrizing the Lyapunov function on $L_2$ with multipliers and integral operators using polynomial kernels of semi-separable class. In particular, under arbitrary well-posed boundary conditions, we define the linear operator inequalities on $\mathbb{R}^n \times L_2$ and cast the stability condition as a feasibility problem constrained by LMIs. In this framework, no discretization or approximation is required to verify the stability conditions of PDE-ODE coupled systems. The developed algorithm has been implemented in MATLAB where the stability of example PDE-ODE coupled systems are verified.

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