Composite Systems with Uncertain Couplings of Fixed Structure: Scaled Riccati Equations and the Problem of Quadratic Stability

We consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure. The couplings are viewed as structured perturbations of the block-diagonal system representing the collection of the separate nominal subsystems (the “nominal system”). We define spectral value sets and stability radii for these time-invariant structured perturbations and derive formulas for their computation. Scaled Riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time-varying and possibly nonlinear perturbations of the given structure. From these we derive necessary and sufficient conditions under which the stability radii with respect to time-invariant and time-varying perturbations are equal. These results are obtained by constructing joint quadratic Liapunov functions of optimal robustness. With their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized Aizerman conjecture.

[1]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[2]  J. Neuberger.  Qualitative analysis of large scale dynamical systems , 2007 .

[3]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[4]  D. Hinrichsen,et al.  Stability radius for structured perturbations and the algebraic Riccati equation , 1986 .

[5]  Dragoslav D. Šiljak,et al.  Large-Scale Dynamic Systems: Stability and Structure , 1978 .

[6]  R. Varga.  Geršgorin And His Circles , 2004 .

[7]  David A. Schoenwald,et al.  Decentralized control of cooperative robotic vehicles: theory and application , 2002, IEEE Trans. Robotics Autom..

[8]  Fabian Wirth.  Asymptotic Behavior of the Value Functions of Discrete-Time Discounted Optimal Control , 2001 .

[9]  Mathukumalli Vidyasagar,et al.  Input-Output Analysis of Large-Scale Interconnected Systems , 1981 .

[10]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[11]  M. Corless,et al.  Systems with structured uncertainty: relations between quadratic and robust stability , 1993, IEEE Trans. Autom. Control..

[12]  Y. Pyatnitskiy,et al.  Criteria of asymptotic stability of differential and difference inclusions encountered in control theory , 1989 .

[13]  D. Hinrichsen,et al.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness , 2010 .

[14]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[15]  Wolfgang Kliemann,et al.  The dynamics of control , 2000 .

[16]  M. Vidyasagar.  Input-Output Analysis of Large-Scale Interconnected Systems: Decomposition, Well-Posedness, and Stability , 1981 .

[17]  John A. Jacquez,et al.  Qualitative Theory of Compartmental Systems , 1993, SIAM Rev..

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[19]  Edward J. Davison.  Connectability and structural controllability of composite systems , 1977, Autom..

[20]  Diederich Hinrichsen,et al.  Interconnected Systems with Uncertain Couplings: Explicit Formulae for mu-Values, Spectral Value Sets, and Stability Radii , 2006, SIAM J. Control. Optim..

[21]  Arturo Roman Messina,et al.  Robust stability analysis of large power systems using the structured singular value theory , 2005 .

[22]  Eduardo D. Sontag,et al.  A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks , 2007, 2007 46th IEEE Conference on Decision and Control.