# American Institute of Mathematical Sciences

2016, 6(3): 241-262. doi: 10.3934/naco.2016010

## Minimum sensitivity realizations of networks of linear systems

 1 Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg, Germany 2 Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg, Germany

Received  April 2015 Revised  July 2016 Published  September 2016

We investigate networks of linear control systems that are interconnected by a fixed network topology. A new class of sensitivity Gramians is introduced whose singular values measure the sensitivity of the network. We characterize the state space realizations of the interconnected node transfer functions such that the overall network has minimum sensitivity. We also develop an optimization approach to the sum of traces of the sensitivity Gramians that determine minimum sensitivity state space realizations of the network. Our work extends previous work by [6,10,11] on $L^2$-minimum sensitivity design.
Citation: Uwe Helmke, Michael Schönlein. Minimum sensitivity realizations of networks of linear systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 241-262. doi: 10.3934/naco.2016010
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##### References:
 [1] Arya Mazumdar, Ron M. Roth, Pascal O. Vontobel. On linear balancing sets. Advances in Mathematics of Communications, 2010, 4 (3) : 345-361. doi: 10.3934/amc.2010.4.345 [2] Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1 [3] Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611 [4] David Russell. Structural parameter optimization of linear elastic systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1517-1536. doi: 10.3934/cpaa.2011.10.1517 [5] Caglar S. Aksezer. On the sensitivity of desirability functions for multiresponse optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 685-696. doi: 10.3934/jimo.2008.4.685 [6] Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035 [7] Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2029-2044. doi: 10.3934/jimo.2019041 [8] Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 [9] Francisco Montes de Oca, Liliana Pérez. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2663-2690. doi: 10.3934/dcdsb.2015.20.2663 [10] Giuseppe Buttazzo, Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Networks & Heterogeneous Media, 2007, 2 (4) : 761-777. doi: 10.3934/nhm.2007.2.761 [11] Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks & Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733 [12] Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381 [13] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 [14] S.Durga Bhavani, K. Viswanath. A general approach to stability and sensitivity in dynamical systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 131-140. doi: 10.3934/dcds.1998.4.131 [15] Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343 [16] Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078 [17] Ö. Uğur, G. W. Weber. Optimization and dynamics of gene-environment networks with intervals. Journal of Industrial & Management Optimization, 2007, 3 (2) : 357-379. doi: 10.3934/jimo.2007.3.357 [18] Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks & Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81 [19] H.T. Banks, S. Dediu, H.K. Nguyen. Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. Mathematical Biosciences & Engineering, 2007, 4 (3) : 403-430. doi: 10.3934/mbe.2007.4.403 [20] Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521

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