Conditions on Shifted Passivity of Port-Hamiltonian Systems

In this paper, we examine the shifted passivity property of port-Hamiltonian systems. Shifted passivity accounts for the fact that in many applications the desired steady-state values of the input and output variables are nonzero, and thus one is interested in passivity with respect to the shifted signals. We consider port-Hamiltonian systems with strictly convex Hamiltonian, and derive conditions under which shifted passivity is guaranteed. In case the Hamiltonian is quadratic and state dependency appears in an affine manner in the dissipation and interconnection matrices, our conditions reduce to negative semidefiniteness of an appropriately constructed constant matrix. Moreover, we elaborate on how these conditions can be extended to the case when the shifted passivity property can be enforced via output feedback, thus paving the path for controller design. Stability of forced equilibria of the system is analyzed invoking the proposed passivity conditions. The utility and relevance of the results are illustrated with their application to a 6th order synchronous generator model as well as a controlled rigid body system.

[1]  John T. Wen,et al.  A unifying passivity framework for network flow control , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[2]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[3]  Romeo Ortega,et al.  Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[4]  Richard H. Middleton,et al.  Disturbance rejection via control by interconnection of port-Hamiltonian systems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[5]  Claudio De Persis,et al.  An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages , 2014, Autom..

[6]  Frank Allgöwer,et al.  Duality and network theory in passivity-based cooperative control , 2013, Autom..

[7]  Diana Bohm,et al.  L2 Gain And Passivity Techniques In Nonlinear Control , 2016 .

[8]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[9]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[10]  Paulo Tabuada,et al.  Compositional Transient Stability Analysis of Multimachine Power Networks , 2013, IEEE Transactions on Control of Network Systems.

[11]  Arjan van der Schaft,et al.  Perspectives in modeling for control of power networks , 2016, Annu. Rev. Control..

[12]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[13]  Arjan van der Schaft,et al.  Load balancing of dynamical distribution networks with flow constraints and unknown in/outflows , 2013, Syst. Control. Lett..

[14]  George H. Hines,et al.  Equilibrium-independent passivity: A new definition and numerical certification , 2011, Autom..

[15]  Nima Monshizadeh,et al.  Bregman Storage Functions for Microgrid Control , 2015, IEEE Transactions on Automatic Control.

[16]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[17]  Romeo Ortega,et al.  New results on disturbance rejection for energy-shaping controlled port-Hamiltonian systems , 2017, ArXiv.

[18]  Arjan van der Schaft,et al.  Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[19]  Stephen P. Boyd,et al.  A Primer on Monotone Operator Methods , 2015 .

[20]  J. Keenan.  Availability and irreversibility in thermodynamics , 1951 .

[21]  Arjan van der Schaft,et al.  On differential passivity of physical systems , 2013, 52nd IEEE Conference on Decision and Control.

[22]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[23]  John T. Wen,et al.  Cooperative Control Design - A Systematic, Passivity-Based Approach , 2011, Communications and control engineering.

[24]  Jacquelien M. A. Scherpen,et al.  A port-Hamiltonian approach to power network modeling and analysis , 2013, Eur. J. Control.

[25]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems on Graphs , 2011, SIAM J. Control. Optim..

[26]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[27]  Romeo Ortega,et al.  Permanent Magnet Synchronous Motors are Globally Asymptotically Stabilizable with PI Current Control , 2018, Autom..

[28]  A. Schaft,et al.  The Hamiltonian formulation of energy conserving physical systems with external ports , 1995 .

[29]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems Theory: An Introductory Overview , 2014, Found. Trends Syst. Control..

[30]  R. Ortega,et al.  A Passivation Approach to Power Systems Stabilization , 1998 .

[31]  John W. Simpson-Porco,et al.  Equilibrium-Independent Dissipativity With Quadratic Supply Rates , 2017, IEEE Transactions on Automatic Control.

[32]  Claudio De Persis,et al.  Agreeing in networks: Unmatched disturbances, algebraic constraints and optimality , 2017, Autom..

[33]  Antonio A. Alonso,et al.  Stabilization of distributed systems using irreversible thermodynamics , 2001, Autom..

[34]  Rodolphe Sepulchre,et al.  A Differential Lyapunov Framework for Contraction Analysis , 2012, IEEE Transactions on Automatic Control.

[35]  R. Ortega.  Passivity-based control of Euler-Lagrange systems : mechanical, electrical and electromechanical applications , 1998 .