Minimum sensitivity realizations of networks of linear systems

Previous Article
Robust and flexible landmarks detection for uncontrolled frontal faces in the wild
NACO Home
This Issue
Next Article
Partial stabilizability and hidden convexity of indefinite LQ problem

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

Minimum sensitivity realizations of networks of linear systems

Uwe Helmke ^1, and Michael Schönlein ^2,

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

Abstract
Related Papers

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.

Keywords: linear systems, sensitivity, balancing, optimization., Networks.

Mathematics Subject Classification: Primary: 93B35; Secondary: 93C0.

Citation: Uwe Helmke, Michael Schönlein. Minimum sensitivity realizations of networks of linear systems. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 241-262. doi: 10.3934/naco.2016010

References:

[1]	J. B. Cruz and W. R. Perkins, A new approach to the sensitivity problem in multivariable feedback system design, IEEE T. Automat. Contr., 9 (1964), 216-223.
[2]	D. F. Delchamps, New geometric approaches to parameter sensitivity in feedback systems, in Modelling, Identification and Robust Control(eds. C. I. Byrnes and A. Lindquist), Elsevier Science Publishers B. V. (North-Holland), (1986), 445-456.
[3]	R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634.
[4]	P. A. Fuhrmann and U. Helmke, Reachability, observability and strict equivalence of networks of linear systems, Math. Control Signal., 25 (2013), 437-471. doi: 10.1007/s00498-012-0104-0.
[5]	P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Cham: Springer, 2015. doi: 10.1007/978-3-319-16646-9.
[6]	M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects, Springer, Berlin, 1993. doi: 10.1007/978-1-4471-2039-1.
[7]	S. Hara, A unified approach to decentralized cooperative control for large-scale networked dynamical systems, in Perspectives in Mathematical System Theory, Control, and Signal Processing (eds. J. C. Willems et al.), Springer, Berlin, (2010), 61-72. doi: 10.1007/978-3-540-93918-4_6.
[8]	S. Hara, T. Hayakawa and H. Sugata, LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems, SICE Journal of Control, Measurement, and Systems Integration, 2 (2009), 299-306.
[9]	U. Helmke, I. Kurniawan, P. Lang and M. Schönlein, Sensitivity optimal design of networks of identical linear systems, in Proc. Mathematical Theory of Networks and Systems (MTNS2012), Melbourne, Australia, 5-9 July 2012, paper 0283.
[10]	U. Helmke and J. B. Moore, L² sensitivity minimization of linear system representations via gradient flows, J. Math. Syst. Estim. Control, 5 (1995), 79-98.
[11]	U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1.
[12]	R. A. Horn and R. Mathias, Block-matrix generalizations of Schur's basic theorems on Hadamard products, Linear Algebra Appl., 172 (1992), 337-346. doi: 10.1016/0024-3795(92)90033-7.
[13]	S. Koshita, M. Abe, and M. Kawamata, Analysis of second-order modes of linear discrete-time systems under bounded-real transformations, IEICE T. Fund. Electr., 90 (2007), 2510-2515.
[14]	S. Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289 (1998), 267-277. doi: 10.1016/S0024-3795(98)10209-4.
[15]	J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8.
[16]	M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355.
[17]	B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Contr., 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.
[18]	C. T. Mullis and R. A. Roberts, Roundoff noise in digital filters: frequency transformations and invariants, IEEE T. Acoust. Speech, 24 (1976), 538-550.
[19]	C. T. Mullis and R. A. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE T. Circuits Syst., 23 (1976), 551-562.
[20]	R. Olfati-Saber, J. Fax, and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233.
[21]	L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE T. Automat. Contr., 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945.
[22]	V. Tavsanoglu and L. Thiele, Optimal design of state-space digital filters by simultaneous minimization of sensitivity and roundoff noise, IEEE T. Automat. Contr., 31 (1984), 884-888. doi: 10.1109/TCS.1984.1085426.
[23]	L. Thiele, On the sensitivity of linear state-space systems, IEEE T. Circuits Syst., 33 (1986), 502-510. doi: 10.1109/TCS.1986.1085951.
[24]	W.-Y. Yan, J. B. Moore and U. Helmke, Recursive algorithms for solving a class of nonlinear matrix equations with applications to certain sensitivity optimization problems, SIAM J. Control Optim., 32 (1994), 1559-1576. doi: 10.1137/S0363012992226855.
[25]	G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404.

show all references

References:

[1]	J. B. Cruz and W. R. Perkins, A new approach to the sensitivity problem in multivariable feedback system design, IEEE T. Automat. Contr., 9 (1964), 216-223.
[2]	D. F. Delchamps, New geometric approaches to parameter sensitivity in feedback systems, in Modelling, Identification and Robust Control(eds. C. I. Byrnes and A. Lindquist), Elsevier Science Publishers B. V. (North-Holland), (1986), 445-456.
[3]	R. Fornaro, Numerical evaluation of integrals around simple closed curves, SIAM J. Numer. Anal., 10 (1973), 623-634.
[4]	P. A. Fuhrmann and U. Helmke, Reachability, observability and strict equivalence of networks of linear systems, Math. Control Signal., 25 (2013), 437-471. doi: 10.1007/s00498-012-0104-0.
[5]	P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Cham: Springer, 2015. doi: 10.1007/978-3-319-16646-9.
[6]	M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects, Springer, Berlin, 1993. doi: 10.1007/978-1-4471-2039-1.
[7]	S. Hara, A unified approach to decentralized cooperative control for large-scale networked dynamical systems, in Perspectives in Mathematical System Theory, Control, and Signal Processing (eds. J. C. Willems et al.), Springer, Berlin, (2010), 61-72. doi: 10.1007/978-3-540-93918-4_6.
[8]	S. Hara, T. Hayakawa and H. Sugata, LTI systems with generalized frequency variables: A unified framework for homogeneous multi-agent dynamical systems, SICE Journal of Control, Measurement, and Systems Integration, 2 (2009), 299-306.
[9]	U. Helmke, I. Kurniawan, P. Lang and M. Schönlein, Sensitivity optimal design of networks of identical linear systems, in Proc. Mathematical Theory of Networks and Systems (MTNS2012), Melbourne, Australia, 5-9 July 2012, paper 0283.
[10]	U. Helmke and J. B. Moore, L² sensitivity minimization of linear system representations via gradient flows, J. Math. Syst. Estim. Control, 5 (1995), 79-98.
[11]	U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer, London, 1994. doi: 10.1007/978-1-4471-3467-1.
[12]	R. A. Horn and R. Mathias, Block-matrix generalizations of Schur's basic theorems on Hadamard products, Linear Algebra Appl., 172 (1992), 337-346. doi: 10.1016/0024-3795(92)90033-7.
[13]	S. Koshita, M. Abe, and M. Kawamata, Analysis of second-order modes of linear discrete-time systems under bounded-real transformations, IEICE T. Fund. Electr., 90 (2007), 2510-2515.
[14]	S. Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra Appl., 289 (1998), 267-277. doi: 10.1016/S0024-3795(98)10209-4.
[15]	J. Lunze, Control Theory of Digitally Networked Dynamic Systems, Cham: Springer, 2014. doi: 10.1007/978-3-319-01131-8.
[16]	M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355.
[17]	B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE T. Automat. Contr., 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568.
[18]	C. T. Mullis and R. A. Roberts, Roundoff noise in digital filters: frequency transformations and invariants, IEEE T. Acoust. Speech, 24 (1976), 538-550.
[19]	C. T. Mullis and R. A. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE T. Circuits Syst., 23 (1976), 551-562.
[20]	R. Olfati-Saber, J. Fax, and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233.
[21]	L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE T. Automat. Contr., 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945.
[22]	V. Tavsanoglu and L. Thiele, Optimal design of state-space digital filters by simultaneous minimization of sensitivity and roundoff noise, IEEE T. Automat. Contr., 31 (1984), 884-888. doi: 10.1109/TCS.1984.1085426.
[23]	L. Thiele, On the sensitivity of linear state-space systems, IEEE T. Circuits Syst., 33 (1986), 502-510. doi: 10.1109/TCS.1986.1085951.
[24]	W.-Y. Yan, J. B. Moore and U. Helmke, Recursive algorithms for solving a class of nonlinear matrix equations with applications to certain sensitivity optimization problems, SIAM J. Control Optim., 32 (1994), 1559-1576. doi: 10.1137/S0363012992226855.
[25]	G. Zames, Functional analysis applied to nonlinear feedback systems, IEEE T. Circuits Syst., 10 (1963), 392-404.

[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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 1998, 4 (1) : 131-140. doi: 10.3934/dcds.1998.4.131
[15]	Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078
[16]	Ö. Uğur, G. W. Weber. Optimization and dynamics of gene-environment networks with intervals. Journal of Industrial and Management Optimization, 2007, 3 (2) : 357-379. doi: 10.3934/jimo.2007.3.357
[17]	Michael Herty. Modeling, simulation and optimization of gas networks with compressors. Networks and Heterogeneous Media, 2007, 2 (1) : 81-97. doi: 10.3934/nhm.2007.2.81
[18]	Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343
[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 and Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Uwe Helmke Michael Schönlein
on Google Scholar
Uwe Helmke Michael Schönlein

[Back to Top]