Computing Input-Ouput Properties of Coupled Linear 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 R× L2. 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.

[1]  Matthew M. Peet.  LMI parametrization of Lyapunov functions for infinite-dimensional systems: A framework , 2014, 2014 American Control Conference.

[2]  Mohamadreza Ahmadi,et al.  Dissipation inequalities for the analysis of a class of PDEs , 2016, Autom..

[3]  P. Christofides,et al.  Finite-dimensional approximation and control of non-linear parabolic PDE systems , 2000 .

[4]  Mohamadreza Ahmadi,et al.  Input-output analysis of distributed parameter systems using convex optimization , 2014, 53rd IEEE Conference on Decision and Control.

[5]  D. Henningson,et al.  Input–output analysis, model reduction and control of the flat-plate boundary layer , 2009, Journal of Fluid Mechanics.

[6]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[7]  Mohamadreza Ahmadi,et al.  Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming , 2016, IEEE Transactions on Automatic Control.

[8]  Yury Orlov,et al.  H∞H∞ stabilization of the current profile in tokamak plasmas via an LMI approach , 2013, Autom..

[9]  P. Daoutidis,et al.  Feedback Control of Hyperbolic PDE Systems , 1996 .

[10]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[11]  Chanin Panjapornpon,et al.  Control of coupled PDEs–ODEs using input–output linearization: Application to a cracking furnace , 2012 .

[12]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[13]  Antonios Armaou,et al.  Analysis and control of parabolic PDE systems with input constraints , 2003, Autom..

[14]  Emilia Fridman,et al.  An LMI approach toH∞ boundary control of semilinear parabolic and hyperbolic systems , 2009 .

[15]  Matthew M. Peet,et al.  A Convex Reformulation of the Controller Synthesis Problem for Infinite-Dimensional Systems using Linear Operator Inequalities (LOIs) with Application to MIMO Multi-Delay Systems , 2018, 2018 Annual American Control Conference (ACC).

[16]  Miroslav Krstic,et al.  PDE Boundary Control for Flexible Articulated Wings on a Robotic Aircraft , 2013, IEEE Transactions on Robotics.

[17]  Emilia Fridman,et al.  An LMI approach to Hinfinity boundary control of semilinear parabolic and hyperbolic systems , 2009, Autom..

[18]  Emilia Fridman,et al.  Network-based H∞ filtering of parabolic systems , 2014, Autom..