Reduced order LQG control design for port Hamiltonian systems

The aim of this paper is to propose a reduced order control design method for large scale port Hamiltonian systems. To this end, a structure preserving reduction method and a modified LQG control design are combined to derive a reduced order model suitable for control purposes. We first recall the structure preserving reduction method for port Hamiltonian systems called effort constraint method and characterize the error bound associated to this reduction method. We then give sufficient conditions for non-standard LQG design which allow to design a passive controller equivalent to the control by interconnection of port Hamiltonian systems. This LQG method allows to define an LQG balanced realization by computing the LQG Gramians, the effort-constraint method is then used to derive a reduced order port Hamiltonian system and to design a reduced order passive LQG controller. Finally, the method is illustrated in simulation on a mass–spring–damper system. The performances of the reduced order controller are compared to the results obtained with a full order passive LQG controller.

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

[2]  A. Schaft,et al.  On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems , 1999 .

[3]  João Pedro Hespanha,et al.  Linear Systems Theory , 2009 .

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

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

[6]  M. Balas.  Active control of flexible systems , 1978 .

[7]  B. Maschke,et al.  Structure preserving reduction of port hamiltonian system using a modified LQG method , 2014, Proceedings of the 33rd Chinese Control Conference.

[8]  Edmond A. Jonckheere,et al.  A new set of invariants for linear systems--Application to reduced order compensator design , 1983 .

[9]  Arjan van der Schaft,et al.  Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems , 2012, Syst. Control. Lett..

[10]  Françoise Couenne,et al.  Structure-preserving infinite dimensional model reduction: Application to adsorption processes , 2009 .

[11]  Athanasios C. Antoulas,et al.  A new result on passivity preserving model reduction , 2005, Syst. Control. Lett..

[12]  Arjan van der Schaft,et al.  Hamiltonian discretization of boundary control systems , 2004, Autom..

[13]  Stefano Stramigioli,et al.  Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach , 2014 .

[14]  Laurent Lefèvre,et al.  Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws , 2012, J. Comput. Phys..

[15]  Yongxin Wu.  Passivity preserving balanced reduction for the finite and infinite dimensional port Hamiltonian systems , 2015 .

[16]  Arjan van der Schaft,et al.  Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos , 2011, IEEE Transactions on Automatic Control.

[17]  Timo Reis,et al.  Linear-quadratic Gaussian balancing for model reduction of differential-algebraic systems , 2011, Int. J. Control.

[18]  Y. Halevi.  Stable LQG controllers , 1994, IEEE Trans. Autom. Control..