2015, 12(5): 1017-1035. doi: 10.3934/mbe.2015.12.1017

Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering

1. 

Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece

2. 

Dept. of Paediatric Haematology-Oncology, Athens Children Hospital Aghia Sofia, 11527, Athens, Greece

3. 

Department of Physics, University of Ngaoundere, P.O. Box 454 Ngaoundere, Cameroon

Received  February 2015 Revised  February 2015 Published  June 2015

A method for early diagnosis of parametric changes in intracellular protein synthesis models (e.g. the p53 protein - mdm2 inhibitor model) is developed with the use of a nonlinear Kalman Filtering approach (Derivative-free nonlinear Kalman Filter) and of statistical change detection methods. The intracellular protein synthesis dynamic model is described by a set of coupled nonlinear differential equations. It is shown that such a dynamical system satisfies differential flatness properties and this allows to transform it, through a change of variables (diffeomorphism), to the so-called linear canonical form. For the linearized equivalent of the dynamical system, state estimation can be performed using the Kalman Filter recursion. Moreover, by applying an inverse transformation based on the previous diffeomorphism it becomes also possible to obtain estimates of the state variables of the initial nonlinear model. By comparing the output of the Kalman Filter (which is assumed to correspond to the undistorted dynamical model) with measurements obtained from the monitored protein synthesis system, a sequence of differences (residuals) is obtained. The statistical processing of the residuals with the use of $\chi^2$ change detection tests, can provide indication within specific confidence intervals about parametric changes in the considered biological system and consequently indications about the appearance of specific diseases (e.g. malignancies)
Citation: Gerasimos G. Rigatos, Efthymia G. Rigatou, Jean Daniel Djida. Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1017-1035. doi: 10.3934/mbe.2015.12.1017
References:
[1]

W. Abou-Jaoudé, M. Chavés and J. L. Gouzé, A Theoretical Exploration Of Birhythmicity in the p53-mdm2 Network,, INRIA Research Report No 7406, (7406).   Google Scholar

[2]

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S. Bououden, D. Boutat, G. Zheng, J. P. Barbot and F. Kratz, A triangular canonical form for a class of 0-flat nonlinear systems,, International Journal of Control, 84 (2011), 261.  doi: 10.1080/00207179.2010.549844.  Google Scholar

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F. N. Chaudhury, Ordinary and neural chi-squared tests for fault detection in multioutput stochastic systems,, IEEE Transactions on Control Systems Technology, 8 (2000), 372.   Google Scholar

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T. L. Chien, C. C. Chen and C.J. Huang, Feedback linearization control and its application to MIMO cancer immunotherapy,, IEEE Transactions on Control Systems Technology, 18 (2008), 953.  doi: 10.1109/TCST.2009.2029089.  Google Scholar

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J. Elias, L. Dimitrio, J. Clairambault and R. Natalini, The p53 Protein and its Molecular Network: Modelling a Missing Link between DNA Damage and Cell Fate,, Biochimica and Biophysica Acta - Proteins and proteomics, (2013).   Google Scholar

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M. Fliess and H. Mounier, Tracking control and $\pi$-freeness of infinite dimensional linear systems, In: G. Picci and D.S. Gilliam Eds.,, Dynamical Systems, 258 (1999), 41.  doi: 10.1007/978-3-0348-8970-4_3.  Google Scholar

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N. Geva-Zatansky, E. Gekel, E. Batchelor, G. Lahav and U. Alan, Fourier Analysis and Systems Identification of the p53 Feedback Loop,, Proceedings of the National Academy of Sciences, (2010).   Google Scholar

[10]

J. Hale and V. Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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M. Honiguchi, S. Koyanagi, A. M. Hamden, K. Kakimoto, N. Matsunaga, C. Yamashita and S. Ohda, Rhythmic control of the ARF-MDM2 pathway by ATF4 underlies circadian accumulation of p53 malignant cells,, Cancer Research, 73 (2013), 2639.   Google Scholar

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G. Lillacci, M. Boccadoro and P. Valigi, The p53 Network and its Control Via MDM2 Inhibitors: Insights from a Dynamical Model,, Proc. 45th IEEE Conference on Decision and Control, (2006).  doi: 10.1109/CDC.2006.376908.  Google Scholar

[15]

M. Jahoor Alam, N. Fatima, G. R. Devi and R. K. Brojen, The enhancement of stability of P53 in MTBP induced p53-MDM2 regulatory network,, Biosystems, 110 (2012), 74.  doi: 10.1016/j.biosystems.2012.09.005.  Google Scholar

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G. Lillacci and M. Khammash, Parameter Identification of Biological Networks Using Extended Kalman Filtering and $\chi^2$ 2 Criteria,, 49th IEEE Conference on Decision and Control Atlanta, (2010).   Google Scholar

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[22]

P. Martin and P. Rouchon, Systèmes Plats: Planification et Suivi Des Trajectoires,, Journées X-UPS, (1999).   Google Scholar

[23]

N. Meskin, H. Nounou, M. Nounou and A. Datta, Parameter estimation of biological phenomena: An unscented kalman filter approach,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 10 (2013), 537.  doi: 10.1109/TCBB.2013.19.  Google Scholar

[24]

S. K. Peirce and H. W. Findley, Targetting the MDM2-p53 interaction as a therapeutic strategy for the treatment of cancer,, Cell Health and cytoskeleton, 2 (2010), 49.   Google Scholar

[25]

J. Qi, S. Shao, Y. Shen and X. Gu, Cellular Responding DNA Damage: A Predictive Model of P53 Gene REgulatory Networks under Continuous Ion Radiation,, Proc. 27th Chinese Control Conference, (2008).  doi: 10.1109/CHICC.2008.4605442.  Google Scholar

[26]

J. P. Qi, S. H. Shao, J. Xie and Y. Zhu, A mathematical model of P53 gene regulatory networks under radiotherapy,, Biosystems, 90 (2007), 698.  doi: 10.1016/j.biosystems.2007.02.007.  Google Scholar

[27]

M. Quach, N. Brunel and F. d'Alche-Buc, Estimating parameters and hidden variables in non-linear state-space models based on ODEs for biological networks inference,, Bioinformatics, 23 (2007), 3209.  doi: 10.1093/bioinformatics/btm510.  Google Scholar

[28]

G. G. Rigatos and S. G. Tzafestas, Extended Kalman Filtering for Fuzzy Modelling and Multi-Sensor Fusion,, Mathematical and Computer Modelling of Dynamical Systems, 13 (2007), 251.  doi: 10.1080/01443610500212468.  Google Scholar

[29]

G. Rigatos, Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotics and Industrial Engineering,, Springer, (2011).  doi: 10.1007/978-3-642-17875-7.  Google Scholar

[30]

G. G. Rigatos, A derivative-free Kalman Filtering approach to state estimation-based control of nonlinear dynamical systems,, IEEE Transactions on Industrial Electronics, 59 (2012), 3987.   Google Scholar

[31]

G. Rigatos and P. Siano, Validation of Fuzzy Kalman Filters Using the Local Statistical Approach to Fault Diagnosis, IMACS Mascot 2012,, Annual Conference of the Italian Institute for Calculus Applications, (2012).   Google Scholar

[32]

G. Rigatos and Q. Zhang, Fuzzy model validation using the local statistical approach,, Fuzzy Sets and Systems, 60 (2009), 882.  doi: 10.1016/j.fss.2008.07.008.  Google Scholar

[33]

G. Rigatos, Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons,, Springer, (2013).  doi: 10.1007/978-3-662-43764-3.  Google Scholar

[34]

G. Rigatos and E. Rigatou, A Kalman Filtering Approach to Robust Synchronization of Coupled Neural Oscillators, ICNAAM 2013,, 11th International Conference of Numerical Analysis and Applied Mathematics, (2013).   Google Scholar

[35]

G. Rigatos and E. Rigatou, Control of the p53 Protein - Mdm2 Inhibitor System Using Nonlinear Kalman Filtering,, Bioinformatics 2014, (2014).   Google Scholar

[36]

G. Rigatos and E. Rigatou, Synchronization of Circadian Oscillators and Protein Synthesis Control Using the Derivative-free Nonlinear Kalman Filter,, Journal of Biology Systems, (2014).  doi: 10.1142/S0218339014500259.  Google Scholar

[37]

P. Rouchon, Flatness-based control of oscillators,, ZAMM - Journal of Applied Mathematics and Mechanics, 85 (2005), 411.  doi: 10.1002/zamm.200410194.  Google Scholar

[38]

J. Rudolph, Flatness Based Control of Distributed Parameter Systems, Examples and Computer Exercises from Various Technological Domains,, Shaker Verlag, (2003).   Google Scholar

[39]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response,, Mathematical Biosciences and Engineering, 12 (2015), 185.   Google Scholar

[40]

H. Sira-Ramirez and S. Agrawal, Differentially Flat Systems,, Marcel Dekker, (2004).   Google Scholar

[41]

J. Wagner, L. Ma, J. J. Rice, W. Hu, A. J. Levine and G. A. Stolovitzky, p53-mdm2 loop controlled by a balance of its feedback strength and effective dampening using ATM and delayed feedback,, IEE Proceedings on Systems Biology, 152 (2005), 109.  doi: 10.1049/ip-syb:20050025.  Google Scholar

[42]

Q. Wang, P. Molenaar, S. Harsh, K. Freeman, J. Xie, C. Gold, M. Rovine and J. Ulbrecht, Presonalized state-space modeling of glucose dynamics for Type 1 Diabetes using continuously monitored glucose, insulin dose and meal intake: An Extended Kalman Filter approach,, Journal of Diabetes Science and Technology, 8 (2014), 331.   Google Scholar

[43]

Z. Wang, X. Liu, Y. Liu, J. Liang and V. Vinciotti, An Extended Kalman Filtering Approach to Modeling Nonlinear Dynamic Gene Regulatory Networks via Short Gene Expression Time Series,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6 (2009), 410.   Google Scholar

[44]

J. F. Xin and Y. Jia, A Mathematical Model of a P53 Oscillation Network Triggered by DNA Damage,, Chinese Physics, (2010).   Google Scholar

[45]

J. Xiong and T. Zhou, Parameter Identification for Nonlinear State-Space Models of a Biological Network via Linearization and Robust State Estimation,, Proceedings of the 32nd Chinese Control Conference, (2010), 8235.   Google Scholar

[46]

Y. Yang and H. Lin, P53-mdm2 Core Regulation Revealed by a Mathematical Model,, 2008 IEEE Intl. Conference on Systems, (2008).  doi: 10.1109/ICSMC.2008.4811537.  Google Scholar

[47]

Y. Zhang and K. T. Chong, Discretization of Nonlinear systems with Delayed Multi-Input via Taylor Series and Scaling and Squaring Technique,, SICE-ICASE International Joint Conference 2006, (2006).  doi: 10.1109/SICE.2006.315409.  Google Scholar

[48]

Z. Zheng, S. J. Baek, D. H. Yu and K. T. Chong, Comparison study of the Taylor Series Based Discretization Method for Nonlinear Input-delay Systems,, 2013 Australian Control Conference, (2013).  doi: 10.1109/AUCC.2013.6697306.  Google Scholar

[49]

T. Zhou, Sensitivity Penalization Based Robust State Estimation for Uncertain Linear Systems,, IEEE Transactions on Automatic Control, 55 (2010), 1018.  doi: 10.1109/TAC.2010.2041681.  Google Scholar

show all references

References:
[1]

W. Abou-Jaoudé, M. Chavés and J. L. Gouzé, A Theoretical Exploration Of Birhythmicity in the p53-mdm2 Network,, INRIA Research Report No 7406, (7406).   Google Scholar

[2]

M. Basseville and I. Nikiforov, Detection of Abrupt Changes: Theory and Applications,, Prentice-Hall, (1993).   Google Scholar

[3]

S. Bououden, D. Boutat, G. Zheng, J. P. Barbot and F. Kratz, A triangular canonical form for a class of 0-flat nonlinear systems,, International Journal of Control, 84 (2011), 261.  doi: 10.1080/00207179.2010.549844.  Google Scholar

[4]

F. N. Chaudhury, Ordinary and neural chi-squared tests for fault detection in multioutput stochastic systems,, IEEE Transactions on Control Systems Technology, 8 (2000), 372.   Google Scholar

[5]

T. L. Chien, C. C. Chen and C.J. Huang, Feedback linearization control and its application to MIMO cancer immunotherapy,, IEEE Transactions on Control Systems Technology, 18 (2008), 953.  doi: 10.1109/TCST.2009.2029089.  Google Scholar

[6]

C. H. Chuang and C. L. Lin, Estimation of Noisy Gene Regulatory Networks,, SICE Annual Conference, (2010).   Google Scholar

[7]

J. Elias, L. Dimitrio, J. Clairambault and R. Natalini, The p53 Protein and its Molecular Network: Modelling a Missing Link between DNA Damage and Cell Fate,, Biochimica and Biophysica Acta - Proteins and proteomics, (2013).   Google Scholar

[8]

M. Fliess and H. Mounier, Tracking control and $\pi$-freeness of infinite dimensional linear systems, In: G. Picci and D.S. Gilliam Eds.,, Dynamical Systems, 258 (1999), 41.  doi: 10.1007/978-3-0348-8970-4_3.  Google Scholar

[9]

N. Geva-Zatansky, E. Gekel, E. Batchelor, G. Lahav and U. Alan, Fourier Analysis and Systems Identification of the p53 Feedback Loop,, Proceedings of the National Academy of Sciences, (2010).   Google Scholar

[10]

J. Hale and V. Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[11]

M. Honiguchi, S. Koyanagi, A. M. Hamden, K. Kakimoto, N. Matsunaga, C. Yamashita and S. Ohda, Rhythmic control of the ARF-MDM2 pathway by ATF4 underlies circadian accumulation of p53 malignant cells,, Cancer Research, 73 (2013), 2639.   Google Scholar

[12]

A. Isidori, The zero dynamics of a nonlinear system: From the origin to the latest progresses of a long successful story,, European Journal of Control, 19 (2013), 369.  doi: 10.1016/j.ejcon.2013.05.014.  Google Scholar

[13]

G. B. Leenders and J. A. Tuszynski, Stochastic and deterministic models cellular p53 regulation,, Frontiers of Oncology, (2013).   Google Scholar

[14]

G. Lillacci, M. Boccadoro and P. Valigi, The p53 Network and its Control Via MDM2 Inhibitors: Insights from a Dynamical Model,, Proc. 45th IEEE Conference on Decision and Control, (2006).  doi: 10.1109/CDC.2006.376908.  Google Scholar

[15]

M. Jahoor Alam, N. Fatima, G. R. Devi and R. K. Brojen, The enhancement of stability of P53 in MTBP induced p53-MDM2 regulatory network,, Biosystems, 110 (2012), 74.  doi: 10.1016/j.biosystems.2012.09.005.  Google Scholar

[16]

B. Laroche, P. Martin and N. Petit, Commande Par Platitude: Equations Différentielles Ordinaires et Aux Derivées Partielles,, Ecole Nationale Supérieure des Techniques Avancées, (2007).   Google Scholar

[17]

J. Lévine, On necessary and sufficient conditions for differential flatness,, Applicable Algebra in Engineering, 22 (2011), 47.  doi: 10.1007/s00200-010-0137-x.  Google Scholar

[18]

K. C. Liang and X. Wang, Gene regulatory network reconstrunction using conditional mutual information, EURASIP, Journal on Bioinformatics and Systems Biology, (2538).   Google Scholar

[19]

G. Lillacci and M. Khammash, Parameter Identification of Biological Networks Using Extended Kalman Filtering and $\chi^2$ 2 Criteria,, 49th IEEE Conference on Decision and Control Atlanta, (2010).   Google Scholar

[20]

G. Lillacci and M. Khammash, Parameter Estimation and Model Selection in Computational Biology,, PLoS Computational Biology, (2010).  doi: 10.1371/journal.pcbi.1000696.  Google Scholar

[21]

B. Liu, S. Yan, Q. Wang and S. Liu, Oscillatory expression and variability in p53 regulatory network,, Physica D, 240 (2011), 259.  doi: 10.1016/j.physd.2010.09.004.  Google Scholar

[22]

P. Martin and P. Rouchon, Systèmes Plats: Planification et Suivi Des Trajectoires,, Journées X-UPS, (1999).   Google Scholar

[23]

N. Meskin, H. Nounou, M. Nounou and A. Datta, Parameter estimation of biological phenomena: An unscented kalman filter approach,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 10 (2013), 537.  doi: 10.1109/TCBB.2013.19.  Google Scholar

[24]

S. K. Peirce and H. W. Findley, Targetting the MDM2-p53 interaction as a therapeutic strategy for the treatment of cancer,, Cell Health and cytoskeleton, 2 (2010), 49.   Google Scholar

[25]

J. Qi, S. Shao, Y. Shen and X. Gu, Cellular Responding DNA Damage: A Predictive Model of P53 Gene REgulatory Networks under Continuous Ion Radiation,, Proc. 27th Chinese Control Conference, (2008).  doi: 10.1109/CHICC.2008.4605442.  Google Scholar

[26]

J. P. Qi, S. H. Shao, J. Xie and Y. Zhu, A mathematical model of P53 gene regulatory networks under radiotherapy,, Biosystems, 90 (2007), 698.  doi: 10.1016/j.biosystems.2007.02.007.  Google Scholar

[27]

M. Quach, N. Brunel and F. d'Alche-Buc, Estimating parameters and hidden variables in non-linear state-space models based on ODEs for biological networks inference,, Bioinformatics, 23 (2007), 3209.  doi: 10.1093/bioinformatics/btm510.  Google Scholar

[28]

G. G. Rigatos and S. G. Tzafestas, Extended Kalman Filtering for Fuzzy Modelling and Multi-Sensor Fusion,, Mathematical and Computer Modelling of Dynamical Systems, 13 (2007), 251.  doi: 10.1080/01443610500212468.  Google Scholar

[29]

G. Rigatos, Modelling and Control for Intelligent Industrial Systems: Adaptive Algorithms in Robotics and Industrial Engineering,, Springer, (2011).  doi: 10.1007/978-3-642-17875-7.  Google Scholar

[30]

G. G. Rigatos, A derivative-free Kalman Filtering approach to state estimation-based control of nonlinear dynamical systems,, IEEE Transactions on Industrial Electronics, 59 (2012), 3987.   Google Scholar

[31]

G. Rigatos and P. Siano, Validation of Fuzzy Kalman Filters Using the Local Statistical Approach to Fault Diagnosis, IMACS Mascot 2012,, Annual Conference of the Italian Institute for Calculus Applications, (2012).   Google Scholar

[32]

G. Rigatos and Q. Zhang, Fuzzy model validation using the local statistical approach,, Fuzzy Sets and Systems, 60 (2009), 882.  doi: 10.1016/j.fss.2008.07.008.  Google Scholar

[33]

G. Rigatos, Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons,, Springer, (2013).  doi: 10.1007/978-3-662-43764-3.  Google Scholar

[34]

G. Rigatos and E. Rigatou, A Kalman Filtering Approach to Robust Synchronization of Coupled Neural Oscillators, ICNAAM 2013,, 11th International Conference of Numerical Analysis and Applied Mathematics, (2013).   Google Scholar

[35]

G. Rigatos and E. Rigatou, Control of the p53 Protein - Mdm2 Inhibitor System Using Nonlinear Kalman Filtering,, Bioinformatics 2014, (2014).   Google Scholar

[36]

G. Rigatos and E. Rigatou, Synchronization of Circadian Oscillators and Protein Synthesis Control Using the Derivative-free Nonlinear Kalman Filter,, Journal of Biology Systems, (2014).  doi: 10.1142/S0218339014500259.  Google Scholar

[37]

P. Rouchon, Flatness-based control of oscillators,, ZAMM - Journal of Applied Mathematics and Mechanics, 85 (2005), 411.  doi: 10.1002/zamm.200410194.  Google Scholar

[38]

J. Rudolph, Flatness Based Control of Distributed Parameter Systems, Examples and Computer Exercises from Various Technological Domains,, Shaker Verlag, (2003).   Google Scholar

[39]

H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response,, Mathematical Biosciences and Engineering, 12 (2015), 185.   Google Scholar

[40]

H. Sira-Ramirez and S. Agrawal, Differentially Flat Systems,, Marcel Dekker, (2004).   Google Scholar

[41]

J. Wagner, L. Ma, J. J. Rice, W. Hu, A. J. Levine and G. A. Stolovitzky, p53-mdm2 loop controlled by a balance of its feedback strength and effective dampening using ATM and delayed feedback,, IEE Proceedings on Systems Biology, 152 (2005), 109.  doi: 10.1049/ip-syb:20050025.  Google Scholar

[42]

Q. Wang, P. Molenaar, S. Harsh, K. Freeman, J. Xie, C. Gold, M. Rovine and J. Ulbrecht, Presonalized state-space modeling of glucose dynamics for Type 1 Diabetes using continuously monitored glucose, insulin dose and meal intake: An Extended Kalman Filter approach,, Journal of Diabetes Science and Technology, 8 (2014), 331.   Google Scholar

[43]

Z. Wang, X. Liu, Y. Liu, J. Liang and V. Vinciotti, An Extended Kalman Filtering Approach to Modeling Nonlinear Dynamic Gene Regulatory Networks via Short Gene Expression Time Series,, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6 (2009), 410.   Google Scholar

[44]

J. F. Xin and Y. Jia, A Mathematical Model of a P53 Oscillation Network Triggered by DNA Damage,, Chinese Physics, (2010).   Google Scholar

[45]

J. Xiong and T. Zhou, Parameter Identification for Nonlinear State-Space Models of a Biological Network via Linearization and Robust State Estimation,, Proceedings of the 32nd Chinese Control Conference, (2010), 8235.   Google Scholar

[46]

Y. Yang and H. Lin, P53-mdm2 Core Regulation Revealed by a Mathematical Model,, 2008 IEEE Intl. Conference on Systems, (2008).  doi: 10.1109/ICSMC.2008.4811537.  Google Scholar

[47]

Y. Zhang and K. T. Chong, Discretization of Nonlinear systems with Delayed Multi-Input via Taylor Series and Scaling and Squaring Technique,, SICE-ICASE International Joint Conference 2006, (2006).  doi: 10.1109/SICE.2006.315409.  Google Scholar

[48]

Z. Zheng, S. J. Baek, D. H. Yu and K. T. Chong, Comparison study of the Taylor Series Based Discretization Method for Nonlinear Input-delay Systems,, 2013 Australian Control Conference, (2013).  doi: 10.1109/AUCC.2013.6697306.  Google Scholar

[49]

T. Zhou, Sensitivity Penalization Based Robust State Estimation for Uncertain Linear Systems,, IEEE Transactions on Automatic Control, 55 (2010), 1018.  doi: 10.1109/TAC.2010.2041681.  Google Scholar

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