Boundary control design for conservation laws in the presence of measurement disturbances

Boundary feedback control design for systems of linear hyperbolic conservation laws in the presence of boundary measurements affected by disturbances is studied. The design of the controller is performed to achieve input-to-state stability (ISS) with respect to measurement disturbances with a minimal gain. The closed-loop system is analyzed as an abstract dynamical system with inputs. Sufficient conditions in the form of dissipation functional inequalities are given to establish an ISS bound for the closed-loop system. The control design problem is turned into an optimization problem over matrix inequality constraints. Semidefinite programming techniques are adopted to devise systematic control design algorithms reducing the effect of measurement disturbances. The effectiveness of the approach is extensively shown in several numerical examples.

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

[2]  Hans Zwart,et al.  C0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain , 2014, 1409.6524.

[3]  M. Krstić,et al.  Boundary Control of PDEs , 2008 .

[4]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[5]  Hiroshi Ito,et al.  Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions , 2014, 1406.2458.

[6]  Nicolas Marchand,et al.  Event-based control of linear hyperbolic systems of conservation laws , 2016, Autom..

[7]  Pierre Apkarian,et al.  Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequalities (LMI) characterizations , 2001, IEEE Trans. Autom. Control..

[8]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[9]  P. Pepe,et al.  A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems , 2006, Syst. Control. Lett..

[10]  L. Vandenberghe,et al.  Extended LMI characterizations for stability and performance of linear systems , 2009, Syst. Control. Lett..

[11]  Felix L. Schwenninger,et al.  On continuity of solutions for parabolic control systems and input-to-state stability , 2017, Journal of Differential Equations.

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

[13]  Christophe Prieur,et al.  Input-to-State Stability of Infinite-Dimensional Systems: Recent Results and Open Questions , 2020, SIAM Rev..

[14]  Joseph J. Winkin,et al.  Boundary feedback control of linear hyperbolic systems: Application to the Saint-Venant-Exner equations , 2018, Autom..

[15]  Alexandre Seuret,et al.  Tractable sufficient stability conditions for a system coupling linear transport and differential equations , 2017, Syst. Control. Lett..

[16]  Mauro Garavello,et al.  Traffic Flow on a Road Network , 2005, SIAM J. Math. Anal..

[17]  Robert Shorten,et al.  ISS Property with Respect to Boundary Disturbances for a Class of Riesz-Spectral Boundary Control Systems , 2019, Autom..

[18]  Matthieu Barreau,et al.  Lyapunov Stability Analysis of a String Equation Coupled With an Ordinary Differential System , 2017, IEEE Trans. Autom. Control..

[19]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[20]  Christophe Prieur,et al.  Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control , 2013, Autom..

[21]  Sergey Dashkovskiy,et al.  Input-to-state stability of interconnected hybrid systems , 2010, Autom..

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

[23]  Jonathan R. Partington,et al.  Infinite-Dimensional Input-to-State Stability and Orlicz Spaces , 2016, SIAM J. Control. Optim..

[24]  G. Bastin,et al.  Stability and Boundary Stabilization of 1-D Hyperbolic Systems , 2016 .

[25]  Fabian R. Wirth,et al.  An ISS small gain theorem for general networks , 2007, Math. Control. Signals Syst..

[26]  Sergey Dashkovskiy,et al.  Input-to-state stability of infinite-dimensional control systems , 2012, Mathematics of Control, Signals, and Systems.

[27]  L. Shampine.  Solving Hyperbolic PDEs in MATLAB , 2005 .

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

[29]  Fabian R. Wirth,et al.  Lyapunov characterization of input-to-state stability for semilinear control systems over Banach spaces , 2017, Syst. Control. Lett..

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

[31]  Christophe Prieur,et al.  Necessary and Sufficient Conditions on the Exponential Stability of Positive Hyperbolic Systems , 2016, IEEE Transactions on Automatic Control.

[32]  Pierdomenico Pepe,et al.  Input-to-State Stabilization of Stabilizable, Time-Delay, Control-Affine, Nonlinear Systems , 2009, IEEE Transactions on Automatic Control.

[33]  Vincent Andrieu,et al.  Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations , 2017, IEEE Transactions on Automatic Control.