Strong stability of a coupled system composed of impedance-passive linear systems which may both have imaginary eigenvalues

We consider coupled systems consisting of a well-posed and impedance passive linear system (that may be infinite dimensional), with semigroup generator <tex>$A$</tex> and transfer function G, and an internal model controller (IMC), connected in feedback. The IMC is finite dimensional, minimal and impedance passive, and it is tuned to a finite set of known disturbance frequencies ω<inf>j</inf>, where <tex>$j$</tex> E {1, … <tex>$n$</tex> }, which means that its transfer function g has poles at the points <tex>$i$</tex>ω<inf>j</inf>. We also assume that g has a feedthrough term <tex>$d$</tex> with Re d > 0. We assume that Re G(iω<inf>j</inf>) > 0 for all <tex>$j$</tex> ∊ {1, … n} and the points <tex>$i$</tex>ω<inf>j</inf> are not eigenvalues of A. We can show that the closed-loop system is well-posed and input-output stable (in particular, (I + gG)<sup>−1</sup> E <tex>$H$</tex> <sup>∞</sup> and also G(1 + gG)<sup>-l</sup> E <tex>$H$</tex> <sup>∞</sup>). It is also easily seen that the closed-loop system is impedance passive. We show that if <tex>$A$</tex> has at most a countable set of imaginary eigenvalues, that are all observable, and <tex>$A$</tex> has no other imaginary spectrum, then the closed-loop system is strongly stable. This result is illustrated with a wind turbine tower model controlled by an IMC.

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

[2]  Olof J. Staffans.  Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems , 2002, Math. Control. Signals Syst..

[3]  George Weiss,et al.  Regular linear systems with feedback , 1994, Math. Control. Signals Syst..

[4]  E. Davison,et al.  Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[5]  Hans Zwart,et al.  Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces , 2012 .

[6]  A. Isidori.  Nonlinear Control Systems , 1985 .

[7]  Daniele Astolfi,et al.  Nonlinear Output Regulation by Post-Processing Internal Model for Multi-Input Multi-Output Systems , 2013, NOLCOS.

[8]  Richard Rebarber,et al.  Internal model based tracking and disturbance rejection for stable well-posed systems , 2003, Autom..

[9]  Zhe Chen,et al.  Effect of Tower Shadow and Wind Shear in a Wind Farm on AC Tie-Line Power Oscillations of Interconnected Power Systems , 2013 .

[10]  Lorenzo Marconi,et al.  Semiglobal robust output regulation of minimum‐phase nonlinear systems , 2000 .

[11]  Timo Hämäläinen,et al.  A finite-dimensional robust controller for systems in the CD-algebra , 2000, IEEE Trans. Autom. Control..

[12]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[13]  Alessandro Astolfi,et al.  Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[14]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[15]  W. Wonham,et al.  The internal model principle for linear multivariable regulators , 1975 .

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

[17]  O. Staffans.  Well-Posed Linear Systems , 2005 .

[18]  George Weiss,et al.  Strong Stabilization of a Non-uniform SCOLE Model , 2008 .

[19]  Xiaowei Zhao,et al.  Controllability and Observability of a Well-Posed System Coupled With a Finite-Dimensional System , 2011, IEEE Transactions on Automatic Control.

[20]  Bayu Jayawardhana,et al.  State Convergence of Passive Nonlinear Systems With an $L^{2}$ Input , 2009, IEEE Transactions on Automatic Control.

[21]  Xiaowei Zhao,et al.  Suppression of the vibrations of wind turbine towers , 2011, IMA J. Math. Control. Inf..

[22]  Hassan K. Khalil,et al.  Robust servomechanism output feedback controllers for feedback linearizable systems , 1994, Autom..

[23]  A. Isidori,et al.  Output regulation of nonlinear systems , 1990 .

[24]  Xiaowei Zhao,et al.  Well-posedness and controllability of a wind turbine tower model , 2011, IMA J. Math. Control. Inf..

[25]  Xiaowei Zhao,et al.  Strong stabilization of a wind turbine tower model , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[26]  Olof J. Staffans,et al.  A Physically Motivated Class of Scattering Passive Linear Systems , 2012, SIAM J. Control. Optim..

[27]  L. Markus,et al.  Stabilization of a hybrid system of elasticity by feedback boundary damping , 1988 .

[28]  Christopher I. Byrnes,et al.  Structurally stable output regulation of nonlinear systems , 1997, Autom..

[29]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[30]  B. Guo,et al.  Boundary Controllability and Observability of a One-Dimensional Nonuniform SCOLE System , 2005 .

[31]  Lassi Paunonen,et al.  Stability and Robust Regulation of Passive Linear Systems , 2017, SIAM J. Control. Optim..

[32]  Olof J. Staffans,et al.  Passive and Conservative Infinite-Dimensional Impedance and Scattering Systems (From a Personal Point of View) , 2003, Mathematical Systems Theory in Biology, Communications, Computation, and Finance.

[33]  Zhengtao Ding,et al.  Global adaptive output regulation of a class of nonlinear systems with nonlinear exosystems , 2007, Autom..

[34]  Xiaowei Zhao,et al.  Well-Posedness and Controllability of a Class of Coupled Linear Systems , 2009, SIAM J. Control. Optim..

[35]  Xiaowei Zhao,et al.  Stability Properties of Coupled Impedance Passive LTI Systems , 2017, IEEE Transactions on Automatic Control.

[36]  Wolfgang Arendt,et al.  Tauberian theorems and stability of one-parameter semigroups , 1988 .

[37]  G. Weiss,et al.  Spectral properties of infinite-dimensional closed-loop systems , 2005, Math. Control. Signals Syst..

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

[39]  Lassi Paunonen,et al.  Robust Regulation of Infinite-Dimensional Port-Hamiltonian Systems , 2017, IEEE Transactions on Automatic Control.

[40]  Hassan K. Khalil,et al.  Nonlinear Systems Third Edition , 2008 .

[41]  B. Francis.  The linear multivariable regulator problem , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[42]  Mark R. Opmeer,et al.  Transfer functions of infinite-dimensional systems: positive realness and stabilization , 2017, Math. Control. Signals Syst..

[43]  Xiaowei Zhao,et al.  Load reduction of a monopile wind turbine tower using optimal tuned mass dampers , 2017, Int. J. Control.

[44]  Marius Tucsnak,et al.  Well-posed systems - The LTI case and beyond , 2014, Autom..