Hierarchical Performance Analysis of Uncertain Large Scale Systems using Uncertain Phase IQC Characterization

In this report, we investigate the performance analysis of uncertain large scale systems. Due to their complexity, the usual robustness analysis methods based on e.g: $\mu$ or Integral Quadratic Constraints (IQC) cannot be practically applied. In order to address this problem, in [1], we propose to represent a large scale system as an interconnection of sub-systems and to perform a hierarchical analysis by propagating the IQC characterization of each uncertain sub-system through the interconnection. For a given computational time, the conservatism of the analysis dramatically depends on the class of IQC under consideration. In this report, we propose a new class of IQC which characterizes the phase of uncertain system. An application to the robustness analysis of a PLL network reveals that the use of this class of IQC improves the trade-off between conservatism and computation time.

[1]  E. Jonckheere,et al.  Spectral factor reduction by phase matching: the continuous-time single-input single-output case† , 1985 .

[2]  Ulf T. Jönsson,et al.  A Popov criterion for networked systems , 2007, Syst. Control. Lett..

[3]  Anders Rantzer,et al.  Robust Stability Analysis of Sparsely Interconnected Uncertain Systems , 2013, IEEE Transactions on Automatic Control.

[4]  Gérard Scorletti,et al.  Convex hierarchical analysis for the performances of uncertain large-scale systems , 2014, 53rd IEEE Conference on Decision and Control.

[5]  Anders Rantzer,et al.  Distributed robust stability analysis of interconnected uncertain systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[6]  Glenn Vinnicombe,et al.  Scalable Decentralized Robust Stability Certificates for Networks of Interconnected Heterogeneous Dynamical Systems , 2006, IEEE Transactions on Automatic Control.

[7]  A. Tits,et al.  Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics , 1991 .

[8]  D. Owens.  The numerical range: a tool for robust stability studies? , 1984 .

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

[10]  Li Lee,et al.  Robustness under bounded uncertainty with phase information , 1999, IEEE Trans. Autom. Control..

[11]  G. Scorletti,et al.  Robustness analysis with time-delays , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[12]  A. Rantzer,et al.  System analysis via integral quadratic constraints , 1997, IEEE Trans. Autom. Control..

[13]  Gérard Scorletti,et al.  Phase IQC for the hierarchical performance analysis of uncertain large scale systems , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[14]  A. Korniienko,et al.  Control law synthesis for distributed multi-agent systems: Application to active clock distribution networks , 2011, Proceedings of the 2011 American Control Conference.

[15]  John C. Doyle.  Analysis of Feedback Systems with Structured Uncertainty , 1982 .

[16]  Ulf Jönsson,et al.  A scalable robust stability criterion for systems with heterogeneous LTI components , 2009, ACC.

[17]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[18]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in System and Control Theory , 1994, Studies in Applied Mathematics.

[19]  J. R. Bar-on,et al.  Gain margins for multivariable control systems , 1990, 29th IEEE Conference on Decision and Control.

[20]  Altug Iftar,et al.  Modeling of uncertain dynamics for robust controller design in state space , 1991, Autom..

[21]  Hisaya Fujioka,et al.  Characterization of Robust Stability of a Class of Interconnected Systems , 2007, ACC.

[22]  Yung-Shan Chou,et al.  Stability multipliers and μ upper bounds: connections and implications for numerical verification of frequency domain conditions , 1999, IEEE Trans. Autom. Control..

[23]  J. Edmunds,et al.  Principal gains and principal phases in the analysis of linear multivariable feedback systems , 1981 .

[24]  Michael G. Safonov.  Propagation of conic model uncertainty in hierarchical systems , 1983 .

[25]  Wassim M. Haddad,et al.  The structured phase margin for robust stability analysis of linear systems with phase and time delay uncertainties , 2008, Int. J. Control.

[26]  Gérard Scorletti,et al.  Performance control for interconnection of identical systems: Application to PLL network design , 2016 .

[27]  A. Korniienko,et al.  Embedding of Uncertainty Propagation: Application to Hierarchical Performance Analysis , 2013 .

[28]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .