Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions

Abstract This paper addresses the boundary control problem of fluid transport in a Poiseuille flow taking the actuator dynamics into account. More precisely, sufficient stability conditions are derived to guarantee the exponential stability of a linear hyperbolic differential equation system subject to nonlinear quadratic dynamic boundary conditions by means of Lyapunov based techniques. Then, convex optimization problems in terms of linear matrix inequality constraints are derived to either estimate the closed-loop stability region or synthesize a robust control law ensuring the local closed-loop stability while estimating an admissible set of initial states. The proposed results are then applied to application-oriented examples to illustrate local stability and stabilization tools.

[1]  Mamadou Diagne,et al.  A multi-model approach to Saint-Venant equations: A stability study by LMIs , 2012, Int. J. Appl. Math. Comput. Sci..

[2]  Miguel Velez-Reyes,et al.  Nonlinear control of a heating, ventilating, and air conditioning system with thermal load estimation , 1999, IEEE Trans. Control. Syst. Technol..

[3]  Antoine Girard,et al.  Stability of Switched Linear Hyperbolic Systems by Lyapunov Techniques , 2014, IEEE Transactions on Automatic Control.

[4]  Luc Dugard,et al.  Boundary observers for linear and quasi-linear and quasi-linear hyperbolic systems with application to flow control , 2013 .

[5]  Georges Bastin,et al.  On boundary feedback stabilization of non-uniform linear 2×2 hyperbolic systems over a bounded interval , 2011, Syst. Control. Lett..

[6]  Emmanuel Witrant,et al.  Dynamic Boundary Stabilization of Linear Parameter Varying Hyperbolic Systems: Application to a Poiseuille Flow , 2013, TDS.

[7]  Vincent Talon,et al.  Fresh Air Fraction Control in Engines Using Dynamic Boundary Stabilization of LPV Hyperbolic Systems , 2015, IEEE Transactions on Control Systems Technology.

[8]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[9]  A. Papachristodoulou,et al.  A tutorial on sum of squares techniques for systems analysis , 2005, Proceedings of the 2005, American Control Conference, 2005..

[10]  Christophe Prieur,et al.  Boundary Control of Open Channels With Numerical and Experimental Validations , 2008, IEEE Transactions on Control Systems Technology.

[11]  J. Geromel,et al.  Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems , 2002 .

[12]  Daniel Ferreira Coutinho,et al.  Modeling and control of flow with dynamical boundary actions , 2015, 2015 IEEE Conference on Control Applications (CCA).

[13]  Georges Bastin,et al.  Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems , 2008, SIAM J. Control. Optim..

[14]  Antoine Girard,et al.  An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems , 2016 .

[15]  Miroslav Krstic,et al.  Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays , 2007, CDC.

[16]  Georges Bastin,et al.  Lyapunov exponential stability of linear hyperbolic systems of balance laws , 2011 .

[17]  Ioan Doré Landau,et al.  Reduced order bilinear models for distillation columns , 1978, Autom..

[18]  Antoine Girard,et al.  Tikhonov theorem for linear hyperbolic systems , 2015, Autom..

[19]  Hongxia Gao,et al.  Boundary control of a flexible marine riser , 2012, Proceedings of the 31st Chinese Control Conference.

[20]  Francesco Amato,et al.  Stabilization of Bilinear Systems Via Linear State-Feedback Control , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[21]  Minyue Fu,et al.  ℒ︁2‐Gain analysis and control of uncertain nonlinear systems with bounded disturbance inputs , 2008 .

[22]  Carlos E. de Souza,et al.  Nonlinear State Feedback Design With a Guaranteed Stability Domain for Locally Stabilizable Unstable Quadratic Systems , 2012, IEEE Transactions on Circuits and Systems I: Regular Papers.

[23]  Carlo Cosentino,et al.  On the region of attraction of nonlinear quadratic systems , 2007, Autom..

[24]  Sophie Tarbouriech,et al.  State feedback design for input-saturating quadratic systems , 2010, Autom..

[25]  Georges Bastin,et al.  Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws , 2012, Autom..

[26]  Georges Bastin,et al.  A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws , 2007, IEEE Transactions on Automatic Control.

[27]  Christophe Prieur,et al.  Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[28]  Daniel Coutinho,et al.  A robust non-linear feedback control strategy for a class of bioprocesses , 2013 .

[29]  Phatiphat Thounthong,et al.  A New Control Law Based on the Differential Flatness Principle for Multiphase Interleaved DC–DC Converter , 2010, IEEE Transactions on Circuits and Systems II: Express Briefs.

[30]  Graziano Chesi,et al.  LMI Techniques for Optimization Over Polynomials in Control: A Survey , 2010, IEEE Transactions on Automatic Control.