Sojoudi, Somayeh, Lavaei, Javad and Aghdam, Amir G. (2009) Robust controllability and observability degrees of polynomially uncertain systems. Automatica, 45 (11). pp. 26402645. ISSN 00051098

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Official URL: http://dx.doi.org/10.1016/j.automatica.2009.07.017
Abstract
This paper deals with the class of polynomially uncertain continuoustime linear timeinvariant (LTI) systems whose uncertainties belong to a semialgebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sumofsquares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any largescale interconnected system in order to identify those inputs and outputs that are more effective in controlling the system, in a robust manner. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results.
Divisions:  Concordia University > Gina Cody School of Engineering and Computer Science > Electrical and Computer Engineering 

Item Type:  Article 
Refereed:  Yes 
Authors:  Sojoudi, Somayeh and Lavaei, Javad and Aghdam, Amir G. 
Journal or Publication:  Automatica 
Date:  2009 
Digital Object Identifier (DOI):  10.1016/j.automatica.2009.07.017 
Keywords:  Analysis of systems with uncertainties; Optimization under uncertainties; Sumofsquares; Large scale systems 
ID Code:  975171 
Deposited By:  DANIELLE DENNIE 
Deposited On:  22 Jan 2013 14:16 
Last Modified:  18 Jan 2018 17:39 
References:
Chesi et al., 2005 G. Chesi, A. Garulli, A. Tesi, A. Vicino Polynomially parameterdependent Lyapunov functions for robust stability of polytopic systems: An LMI approach IEEE Transactions on Automatic Control, 50 (3) (2005), pp. 365–370Davison and Chang, 1990 E.J. Davison, T.N. Chang Decentralized stabilization and pole assignment for general proper systems IEEE Transactions on Automatic Control, 35 (6) (1990), pp. 652–664
Dullerud and Paganini, 2005 G.E. Dullerud, F. Paganini A course in robust control theory: A convex approach Texts in applied mathematics, Springer (2005)
Hardy et al., 1952 G.H. Hardy, J.E. Littlewood, G. Polya Inequalities (second edition)Cambridge University Press, Cambridge, UK (1952)
Lavaei and Aghdam, 2007a J. Lavaei, A.G. Aghdam A graph theoretic method to find decentralized fixed modes of LTI systems Automatica, 43 (12) (2007), pp. 2129–2133
Lavaei and Aghdam, 2007b J. Lavaei, A.G. Aghdam Optimal periodic feedback design for continuoustime LTI systems with constrained control structure International Journal of Control, 80 (2) (2007), pp. 220–230
Lavaei and Aghdam, 2008a J. Lavaei, A.G. Aghdam Robust stability of LTI systems over semialgebraic sets using sumofsquares matrix polynomials IEEE Transactions on Automatic Control, 53 (1) (2008), pp. 417–423
Lavaei and Aghdam, 2008b J. Lavaei, A.G. Aghdam Control of continuoustime LTI systems by means of structurally constrained controllers Automatica, 44 (1) (2008), pp. 141–148
Lofberg, 2004 Lofberg, J. (2004). A toolbox for modeling and optimization in MATLAB. In Proc. of the CACSD conf. Available online at: http://control.ee.ethz.ch/~joloef/yalmip.php
Oliveira and Geromel, 2005 M.C.de Oliveira, J.C. Geromel A class of robust stability conditions where linear parameter dependence of the Lyapunov function is a necessary condition for arbitrary parameter dependence Systems & Control Letters, 54 (11) (2005), pp. 1131–1134
Oliveira and Peres, 2006 R.C.L.F. Oliveira, P.L.D. Peres LMI conditions for robust stability analysis based on polynomially parameterdependent Lyapunov functions Systems & Control Letters, 55 (1) (2006), pp. 52–61
Prajna et al., 2004 Prajna, S., Papachristodoulou, A., Seiler, P., & Parrilo, P. A. (2004). SOSTOOLS sum of squares optimization toolbox for MATLAB. Users guide. Available online at: http://www.cds.caltech.edu/sostools
Savkin and Petersen, 1999 A.V. Savkin, I.R. Petersen Weak robust controllability and observability of uncertain linear systems IEEE Transactions on Automatic Control, 44 (5) (1999), pp. 1037–1041
Scherer and Hol, 2006 C.W. Scherer, C.W.J. Hol Matrix sumofsquares relaxations for robust semidefinite programs Mathematical Programming, 107 (1–2) (2006), pp. 189–211
Siljak, 1991 D.D. Siljak Decentralized control of complex systems Academic Press, Cambridge (1991)
Sojoudi and Aghdam, 2007 Sojoudi, S., & Aghdam, A. G. (2007). Characterizing all classes of LTI stabilizing structurally constrained controllers by means of combinatorics. In Proc. 46th IEEE conf. decision and contr. (pp. 4415–4420)
Sojoudi et al., in press Sojoudi, S., Lavaei, J., & Aghdam, A. G. (2009). Controllability and observability of uncertain systems: A robust measure. In Proceedings of 48th IEEE Conference on Decision and Control (in press)
Ugrinovskii, 2005 V.A. Ugrinovskii Robust controllability of linear stochastic uncertain systems Automatica, 41 (5) (2005), pp. 807–813
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