# Cross Gramian & Empirical Gramian References #


## Cross Gramian References ##

+ K.V. Fernando and H. Nicholson. [Minimality of SISO linear systems](http://dx.doi.org/10.1109/PROC.1982.12460). Proceedings of the IEEE, 70(10): 1241--1242, 1982.
+ K.V. Fernando and H. Nicholson. [On the Cauchy index of linear systems](http://dx.doi.org/10.1109/TAC.1983.1103200). IEEE Transactions on Automatic Control, 28(2): 222--224, 1983.
+ K.V. Fernando and H. Nicholson. [On the structure of balanced and other principal representations of SISO systems](http://dx.doi.org/10.1109/TAC.1983.1103195). IEEE Transactions on Automatic Control, 28(2): 228--231, 1983.
+ A.J. Laub, L.M. Silverman and M. Verma. [A note on cross-Grammians for symmetric realizations](http://dx.doi.org/10.1109/PROC.1983.12688). Proceedings of the IEEE, 71(7): 904--905, 1983.
+ K.V. Fernando and H. Nicholson. [On a fundamental property of the cross-Gramian matrix](http://dx.doi.org/10.1109/TCS.1984.1085524). IEEE Transactions on Circuits and Systems, 31(5): 504--505, 1984.
+ K.V. Fernando and H. Nicholson. [Reachability, observability, and minimality of MIMO systems](http://dx.doi.org/10.1109/PROC.1984.13094). Proceedings of the IEEE, 72(12): 1820--1821, 1984.
+ K.V. Fernando and H. Nicholson. [On the cross-Gramian for symmetric MIMO systems](http://dx.doi.org/10.1109/TCS.1985.1085737). IEEE Transactions on Circuits and Systems, 32(5): 487--489, 1985.
+ A.M. Davidson. [Balanced systems and model reduction](http://dx.doi.org/10.1049/el:19860362). Electronics Letters, 22(10): 531--532, 1986.
+ J.A. De Abreu-Garcia and F.W. Fairman. [A note on cross Grammians for orthogonally symmetric realizations](http://dx.doi.org/10.1109/TAC.1986.1104421). IEEE Transactions on Automatic Control 31(9): 866--868, 1986
+ L. Fortuna and A. Gallo and C. Guglielmino and G. Nunnari. [On the solution of a nonlinear matrix equation for MIMO symmetric realizations](http://dx.doi.org/10.1016/0167-6911%2888%2990115-6). System & Control Letters, 11(1):  79--82, 1988.
+ R.W. Aldhaheri. [Model order reduction via real Schur-form decomposition](http://dx.doi.org/10.1080/00207179108953642). International Journal of Control, 53(3): 709--716, 1991.
+ W.Q. Liu, V. Sreeram and K.L. Teo. [Model reduction for state-space symmetric systems](http://dx.doi.org/10.1016/S0167-6911%2898%2900024-3). Systems & Control Letters, 34(4): 209--215, 1998. 
+ A. Gheondea and R.J. Ober: [A Trace Formula for the Hankel Operator](http://dx.doi.org/10.1090/S0002-9939-99-04669-9). Proceedings of the American Mathematical Society, 127(7): 2007--2012, 1999.
+ D.C. Sorensen and A.C. Antoulas. [Projection methods for balanced model reduction](http://www.caam.rice.edu/caam/trs/2001/TR01-03.pdf). Technical Report: 1--18, 2001.
+ D.C. Sorensen and A.C. Antoulas. [The Sylvester equation and approximate balanced reduction](http://dx.doi.org/10.1016/S0024-3795%2802%2900283-5). Linear Algebra and its Applications, 351-352: 671--700, 2002.
+ B. Moaveni and A. Khaki-Sedigh. [Input-Output Pairing based on Cross-Gramian Matrix](http://dx.doi.org/10.1109/SICE.2006.314989). International Joint Conference SICE-ICASE: 2378--2380, 2006.
+ T.C. Ionescu and J.M.A. Scherpen. [Cross gramians for nonlinear systems](http://ifip2007.agh.edu.pl/abstracts/48.pdf). 23rd TC7 Conference on System Modeling and Optimization: 322--323, 2007.
+ T.C. Ionescu and J.M.A. Scherpen. [Nonlinear cross Gramians and gradient systems](http://dx.doi.org/10.1109/CDC.2007.4434707). 2007 46th IEEE Conference on Decision and Control: 3745--3750, 2007. 
+ U. Baur and P. Benner. [Cross-gramian based model reduction for data-sparse systems](http://eudml.org/doc/130548). Electronic Transactions on Numerical Analysis, 31: 256--270, 2008.
+ B. Moaveni and A. Khaki-Sedigh. [Input-output pairing analysis for uncertain multivariable processes](http://dx.doi.org/10.1016/j.jprocont.2007.10.015). Journal of Process Control, 18(6): 527--532, 2008.
+ B. Moaveni and A. Khaki-Sedigh. [A new approach to compute the cross-Gramian matrix and its application in input-output pairing of linear multivariable plants](http://dx.doi.org/10.3923/jas.2008.608.614). Journal of Applied Sciences, 8(4): 608--614, 2008.
+ T.C. Ionescu and J.M.A. Scherpen. [Nonlinear Cross Gramians](http://dx.doi.org/10.1007/978-3-642-04802-9_16). System Modeling and Optimization: 293--306, 2009.
+ T.C. Ionescu, K. Fujimoto and J.M.A. Scherpen. [The Cross Operator and the Singular Value Analysis for Nonlinear Symmetric Systems](http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7074629). Proceedings of the European Control Conference: 1565--1570, 2009.
+ T.C. Ionescu, K. Fujimoto and J.M.A. Scherpen. [Singular value analysis of nonlinear symmetric systems](http://dx.doi.org/10.1109/TAC.2011.2126630). IEEE Transactions on Automatic Control, 56(9): 2073--2086, 2011.
+ H.R. Shaker. [Generalized cross-gramian for linear systems](http://dx.doi.org/10.1109/ICIEA.2012.6360824). 7th IEEE Conference on Industrial Electronics and Applications: 749--751, 2012.
+ K. Fujimoto and J.M.A. Scherpen. [On balanced truncation for symmetric nonlinear systems](http://fwn06.housing.rug.nl/mtns2014-papers/fullPapers/0313.pdf). 21st International Symposium on Mathematical Theory of Networks and Systems: 1498--1502, 2014.
+ L.A. Mironovskii and T.N. Solveva. [Analysis of Multiplicity of Hankel Singular Values of Control Systems](http://dx.doi.org/10.1134/S0005117915020022). Automation and Remote Control, 76(2): 205--218, 2015.



## Empirical Gramian References ##

+ B.C. Moore. [Principal component analysis in linear systems: Controllability, observability, and model reduction](http://dx.doi.org/10.1109/TAC.1981.1102568). IEEE Transactions on Automatic Control, 26(1): 17--32, 1981.
+ X. Ma, J.A. De Abreu Garcia. [On the computation of reduced order models of nonlinear systems using balancing technique](http://dx.doi.org/10.1109/CDC.1988.194502). Proceedings of the 27th IEEE Conference on Decision and Control 2: 1165--1166, 1988.
+ S. Lall, J.E. Marsden and S. Glavaski. [Empirical model reduction of controlled nonlinear systems](http://authors.library.caltech.edu/20343/2/10.1.1.123.4669.pdf). Proceedings of the IFAC World Congress, F: 473--478, 1999.
+ J. Hahn and T.F. Edgar. [Reduction of nonlinear models using balancing of empirical gramians and Galerkin projections](http://dx.doi.org/10.1109/ACC.2000.878734). Proceedings of the American Control Conference, 4: 2864--2868, 2000.
+ J. Hahn and T.F. Edgar. [A gramian based approach to nonlinearity quantification and model classification](http://dx.doi.org/10.1021/ie010155v). Industrial and engineering chemistry research, 40(24): 5724--5731, 2001.
+ S. Lall, J.E. Marsden and S. Glavaski. [A subspace approach to balanced truncation for model reduction of nonlinear control systems](http://dx.doi.org/10.1002/rnc.657). International Journal of Robust and Nonlinear Control, 12(6): 519--535, 2002.
+ J. Hahn and T.F. Edgar. [Balancing approach to minimal realization and model reduction of stable nonlinear systems](http://dx.doi.org/10.1021/ie0106175). Industrial and engineering chemistry research, 41(9): 2204--2212, 2002.
+ J. Hahn and T.F. Edgar. [An improved method for nonlinear model reduction using balancing of empirical gramians](http://dx.doi.org/10.1016/S0098-1354%2802%2900120-5). Computers and chemical engineering, 26(10): 1379--1397, 2002.
+ J. Hahn, T.F. Edgar and M. Marquardt. [Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems](http://dx.doi.org/10.1016/S0959-1524%2802%2900024-0). Journal of Process Control, 13(2): 115--127, 2003.
+ A. Keil and J.L. Gouze. [Model reduction of modular systems using balancing methods](http://campar.in.tum.de/twiki/pub/Main/AndreasKeil/keil2003modelreduction.pdf). Technical Report, 2003.
+ M. Condon and R. Ivanov. [Empirical balanced truncation of nonlinear systems](http://dx.doi.org/10.1007/s00332-004-0617-5). Journal of Nonlinear Science, 14(5):405--414, 2004.
+ M. Condon and R. Ivanov. [Model reduction of nonlinear systems](http://dx.doi.org/10.1108/03321640410510730). COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 23(2):  547--557, 2004.
+ A.K. Singh and J. Hahn. [Optimal sensor location for nonlinear dynamic systems via empirical gramians](http://www.nt.ntnu.no/users/skoge/prost/proceedings/dycops04/pdffiles/papers/17.pdf). In Proceedings of Dynamics and Control of Process Systems 7: 965--970, 2004.
+ A.K. Singh and J. Hahn. [On the use of Empirical Gramians for Controllability and Observability Analysis](http://dx.doi.org/10.1109/ACC.2005.1469922). Proceedings of the American Control Conference, 2005.
+ A.K. Singh and J. Hahn. [Determining optimal sensor locations for state and parameter estimation for stable nonlinear systems](http://dx.doi.org/10.1021/ie040212v). Industrial and Engineering Chemistry Research, 44(15): 5645--5659, 2005.
+ A.K. Singh and J. Hahn. [Sensor location for stable nonlinear dynamic systems: Multiple sensor case](http://dx.doi.org/10.1021/ie0511175). Industrial & engineering chemistry research, 45(10): 3615--3623, 2006.
+ A.K. Singh and J. Hahn. [Computing Sensor Locations for Nonlinear Systems under the Influence of Disturbances](http://www.nt.ntnu.no/users/skoge/prost/proceedings/aiche-2006/data/papers/P50021.pdf). AIChE Annual Meeting, 2006.
+ C. Sun and J. Hahn. [Parameter reduction for stable dynamical systems based on Hankel singular values and sensitivity analysis](http://dx.doi.org/10.1016/j.ces.2006.04.027). Chemical engineering science, 61(16): 5393--5403, 2006.
+ C. Sun and J. Hahn. [Model reduction in the presence of uncertainty in model parameters](http://dx.doi.org/10.1016/j.jprocont.2005.10.001). Journal of Process Control, 16(6):645--649, 2006.
+ S. Streif, R. Findeisen and E. Bullinger. [Relating cross Gramians and sensitivity analysis in systems biology](http://eprints.nuim.ie/1768/1/HamiltonGramian.pdf). Theory of Networks and Systems, 10.4: 437--442, 2006.
+ M. Condon. [Model reduction of nonlinear systems](http://dx.doi.org/10.1002/pamm.200701084). Proceedings in Applied Mathematics and Mechanics, 7(1): 2130011--2130012, 2007.
+ D. Geffen, R. Findeisen, M. Schliemann, F. Allgöwer and M. Guay. [Observability based parameter identifiability for biochemical reaction networks](http://dx.doi.org/10.1109/ACC.2008.4586807). Proceedings of the American Control Conference, pages 2130--2135, 2008.
+ A. Barbagallo, V.F. De Felice and K.K. Nagarajan. [Reduced order modelling of a couette flow using Balanced Proper Orthogonal Decomposition](http://dokumente.unibw.de/pub/bscw.cgi/S50cf6918/d2700941/BarbaDefelNagar.pdf). 2nd Young ERCOFTAC Workshop, 2008.
+ S. Streif, S. Waldherr, F. Allgöwer and R. Findeisen. [Steady state sensitivity analysis of biochemical reaction networks: a brief review and new methods](http://books.google.de/books?id=Haod3KR-tR8C&lpg=PA129&ots=mr5bWYnbVy&dq=%22Steady%20state%20sensitivity%20analysis%20of%20biochemical%20reaction%20networks%3A%20a%20brief%20review%20and%20new%20methods%22&pg=PA129#v=onepage&q&f=false). Methods in Bioengineering Systems Analysis of Biological Networks: 129--148, 2009.
+ H.R. Shaker and M. Komareji. [Control Configuration Selection for Multivariable Nonlinear Systems](http://dx.doi.org/10.1021/ie301137k). Industrial and Engineering Chemistry Research, 51(25): 8583--8587, 2012.
+ C. Eberle and C. Ament. [Identifiability and online estimation of diagnostic parameters with in the glucose insulin homeostasis](http://dx.doi.org/10.1016/j.biosystems.2011.11.003). Biosystems, 107(3): 135--141, 2012.
+ S. Melchior, V. Legat, and P. Van Dooren. [Gramian based model reduction of nonlinear MIMO systems](http://www.mtns2012.conference.net.au/Extended%20Abstract/MTNS2012_0170_paper.pdf). Mathematical Theory of Networks and Systems, 2012.
+ C. Himpe and M. Ohlberger. [A Unified Software Framework for Empirical Gramians](http://dx.doi.org/10.1155/2013/365909). Hindawi Journal of Mathematics, 2013: 1--6, 2013.
+ C. Himpe and M. Ohlberger. [Cross-Gramian Based Combined State and Parameter Reduction for Large-Scale Control Systems](http://dx.doi.org/10.1155/2014/843869). Mathematical Problems in Engineering: 1--13, 2014.
+ C. Himpe and M. Ohlberger. [Model Reduction for Complex Hyperbolic Networks](http://dx.doi.org/10.1109/ECC.2014.6862188). Proceedings of the European Control Conference: 2739--2743, 2014.
+ C. Himpe and M. Ohlberger. [The Empirical Cross Gramian for Parametrized Nonlinear Systems](http://www.ifac-papersonline.net/Detailed/69193.html). In Mathematical Modelling, 8: 727--728, 2015.


## Model Reduction ##

+ A.C. Antoulas and D.C. Sorensen. [Approximation of large-scale dynamical systems: an overview](http://eudml.org/doc/207547). International Journal of Applied Mathematics and Computer Science, 11(5):1093--1121, 2001.
+ A.C. Antoulas. [Approximation of large-scale dynamical systems](http://dx.doi.org/10.1137/1.9780898718713). Society for Industrial Mathematics Volume 6, 2005.
+ L. Fortuna and M. Frasca. [Optimal and Robust Control: Advanced Topics With MATLAB](http://books.google.de/books?id=WM3OzyHKlD4C&lpg=PP1&dq=isbn%3A146650191X&pg=PP1#v=onepage&q&f=false). Taylor & Francis, 2012.


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