# American Institute of Mathematical Sciences

January 2019, 24(1): 83-107. doi: 10.3934/dcdsb.2018162

## Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

 1 School of Mechanical, Industrial, & Manufacturing Engineering, Oregon State University, Corvallis, OR 97331-6011, USA 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

* Corresponding author

† H. Shoori and J. Singler were supported in part by National Science Foundation grant DMS-1217122; B. Batten was supported in part by the Department of Energy under Award Number DE-FG36-08GO18179.

Received  February 2017 Revised  January 2018 Published  June 2018

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Citation: Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162
##### References:
 [1] D. Amsallem and U. Hetmaniuk, Error estimates for Galerkin reduced-order models of the semi-discrete wave equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 135-163. doi: 10.1051/m2an/2013099. [2] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, vol. 6 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005, With a foreword by Jan C. Willems. doi: 10.1137/1.9780898718713. [3] H. T. Banks, A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering, CRC Press, Boca Raton, FL, 2012. doi: 10.1201/b12209. [4] B. A. Batten and K. A. Evans, Reduced-order compensators via balancing and central control design for a structural control problem, Internat. J. Control, 83 (2010), 563-574. doi: 10.1080/00207170903301234. [5] B. A. Batten, H. Shoori, J. R. Singler and M. H. Weerasinghe, Model reduction of a nonlinear cable-mass PDE system with dynamic boundary input, in Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, (2016), 327–334. [6] P. Benner and P. Goyal, Balanced truncation model order reduction for quadratic-bilinear control systems, arXiv: 1705.00160. [7] P. Benner, E. Sachs and S. Volkwein, Model order reduction for PDE constrained optimization, in Trends in PDE constrained optimization, vol. 165 of Internat. Ser. Numer. Math., Birkhäuser/Springer, Cham, 2014,303–326. doi: 10.1007/978-3-319-05083-6_19. [8] T. Bui-Thanh, K. Willcox and O. Ghattas, Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications, AIAA Journal, 46 (2008), 2520-2529. [9] J. A. Burns and E. M. Cliff, Control of hyperbolic PDE systems with actuator dynamics, in Proceedings of IEEE Conference on Decision and Control, 2014, 2864–2869. doi: 10.1109/CDC.2014.7039829. [10] J. A. Burns and B. B. King, A reduced basis approach to the design of low-order feedback controllers for nonlinear continuous systems, J. Vib. Control, 4 (1998), 297-323. doi: 10.1177/107754639800400305. [11] J. A. Burns and L. Zietsman, On the inclusion of actuator dynamics in boundary control of distributed parameter systems, in IFAC Proceedings Volumes, 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, 45 (2012), 138–142. doi: 10.3182/20120829-3-IT-4022.00039. [12] J. A. Burns and L. Zietsman, Control of a thermal fluid heat exchanger with actuator dynamics, in Proceedings of IEEE Conference on Decision and Control, 2016, 3131–3136. doi: 10.1109/CDC.2016.7798738. [13] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737-2764. doi: 10.1137/090766498. [14] F. Conrad and A. Mifdal, Uniform stabilization of a hybrid system with a class of nonlinear feedback laws, Adv. Math. Sci. Appl., 11 (2001), 549-569. [15] R. F. Curtain and A. J. Sasane, Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems, Internat. J. Control, 74 (2001), 1260-1270. doi: 10.1080/00207170110061059. [16] D. N. Daescu and I. M. Navon, Efficiency of a POD-based reduced second-order adjoint model in 4D-var data assimilation, Internat. J. Numer. Methods Fluids, 53 (2007), 985-1004. doi: 10.1002/fld.1316. [17] F. Fang, T. Zhang, D. Pavlidis, C. Pain, A. Buchan and I. Navon, Reduced order modelling of an unstructured mesh air pollution model and application in 2d/3d urban street canyons, Atmospheric Environment, 96 (2014), 96-106. doi: 10.1016/j.atmosenv.2014.07.021. [18] N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667. doi: 10.3934/eect.2013.2.631. [19] K. Glover, R. F. Curtain and J. R. Partington, Realisation and approximation of linear infinite-dimensional systems with error bounds, SIAM J. Control Optim., 26 (1988), 863-898. doi: 10.1137/0326049. [20] Y. Gong, Q. Wang and Z. Wang, Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems, Comput. Methods Appl. Mech. Engrg., 315 (2017), 780-798. doi: 10.1016/j.cma.2016.11.016. [21] C. Guiver and M. R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, SIAM J. Control Optim., 52 (2014), 1366-1401. doi: 10.1137/110846981. [22] M. Gunzburger, N. Jiang and M. Schneier, An ensemble-proper orthogonal decomposition method for the nonstationary Navier-Stokes equations, SIAM J. Numer. Anal., 55 (2017), 286-304. doi: 10.1137/16M1056444. [23] M. Gunzburger and H.-C. Lee, Reduced-order modeling of Navier-Stokes equations via centroidal Voronoi tessellation, in Recent advances in adaptive computation, vol. 383 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2005, 213–224. doi: 10.1090/conm/383/07166. [24] S. Herkt, M. Hinze and R. Pinnau, Convergence analysis of Galerkin POD for linear second order evolution equations, Electron. Trans. Numer. Anal., 40 (2013), 321-337. [25] D. B. P. Huynh, D. J. Knezevic and A. T. Patera, A Laplace transform certified reduced basis method; application to the heat equation and wave equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 401-405. doi: 10.1016/j.crma.2011.02.003. [26] M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 1284-1302. doi: 10.1137/100787350. [27] B. B. King, Modeling and control of a multiple component structure, J. Math. Systems Estim. Control, 4 (1994), 36pp. [28] V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. [29] B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [30] O. Morgül, Stabilization and disturbance rejection for the wave equation, IEEE Trans. Automat. Control, 43 (1998), 89-95. doi: 10.1109/9.654893. [31] K. Morris, H∞-control of acoustic noise in a duct with a feedforward configuration, in Proceedings of 15th International Symposium on Mathematical Theory of Networks and Systems, 2002, 12–16. [32] M. R. Opmeer, Nuclearity of Hankel operators for ultradifferentiable control systems, Systems Control Lett., 57 (2008), 913-918. doi: 10.1016/j.sysconle.2008.04.007. [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [34] L. Peng and K. Mohseni, Symplectic model reduction of Hamiltonian systems, SIAM J. Sci. Comput., 38 (2016), A1-A27. doi: 10.1137/140978922. [35] V. Pereyra, Model order reduction with oblique projections for large scale wave propagation, J. Comput. Appl. Math., 295 (2016), 103-114. doi: 10.1016/j.cam.2015.01.029. [36] J. M. A. Scherpen, Balancing for nonlinear systems, Systems Control Lett., 21 (1993), 143-153. doi: 10.1016/0167-6911(93)90117-O. [37] H. E. Shoori J., An Approach to Reduced-Order Modeling and Feedback Control for Wave Energy Converters, Master's thesis, Oregon State University, 2014, URL http://ir.library.oregonstate.edu/xmlui/handle/1957/50823. [38] A. Varshney, S. Pitchaiah and A. Armaou, Feedback control of dissipative PDE systems using adaptive model reduction, AIChE journal, 55 (2009), 906-918. doi: 10.1002/aic.11770. [39] Z. Zhang, Stabilization of the wave equation with variable coefficients and a dynamical boundary control, Electron. J. Differential Equations, 2016 (2016), Paper No. 27, 10pp. [40] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice-Hall, 1996.

show all references

##### References:
 [1] D. Amsallem and U. Hetmaniuk, Error estimates for Galerkin reduced-order models of the semi-discrete wave equation, ESAIM Math. Model. Numer. Anal., 48 (2014), 135-163. doi: 10.1051/m2an/2013099. [2] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, vol. 6 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005, With a foreword by Jan C. Willems. doi: 10.1137/1.9780898718713. [3] H. T. Banks, A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering, CRC Press, Boca Raton, FL, 2012. doi: 10.1201/b12209. [4] B. A. Batten and K. A. Evans, Reduced-order compensators via balancing and central control design for a structural control problem, Internat. J. Control, 83 (2010), 563-574. doi: 10.1080/00207170903301234. [5] B. A. Batten, H. Shoori, J. R. Singler and M. H. Weerasinghe, Model reduction of a nonlinear cable-mass PDE system with dynamic boundary input, in Proceedings of the International Symposium on Mathematical Theory of Networks and Systems, (2016), 327–334. [6] P. Benner and P. Goyal, Balanced truncation model order reduction for quadratic-bilinear control systems, arXiv: 1705.00160. [7] P. Benner, E. Sachs and S. Volkwein, Model order reduction for PDE constrained optimization, in Trends in PDE constrained optimization, vol. 165 of Internat. Ser. Numer. Math., Birkhäuser/Springer, Cham, 2014,303–326. doi: 10.1007/978-3-319-05083-6_19. [8] T. Bui-Thanh, K. Willcox and O. Ghattas, Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications, AIAA Journal, 46 (2008), 2520-2529. [9] J. A. Burns and E. M. Cliff, Control of hyperbolic PDE systems with actuator dynamics, in Proceedings of IEEE Conference on Decision and Control, 2014, 2864–2869. doi: 10.1109/CDC.2014.7039829. [10] J. A. Burns and B. B. King, A reduced basis approach to the design of low-order feedback controllers for nonlinear continuous systems, J. Vib. Control, 4 (1998), 297-323. doi: 10.1177/107754639800400305. [11] J. A. Burns and L. Zietsman, On the inclusion of actuator dynamics in boundary control of distributed parameter systems, in IFAC Proceedings Volumes, 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, 45 (2012), 138–142. doi: 10.3182/20120829-3-IT-4022.00039. [12] J. A. Burns and L. Zietsman, Control of a thermal fluid heat exchanger with actuator dynamics, in Proceedings of IEEE Conference on Decision and Control, 2016, 3131–3136. doi: 10.1109/CDC.2016.7798738. [13] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737-2764. doi: 10.1137/090766498. [14] F. Conrad and A. Mifdal, Uniform stabilization of a hybrid system with a class of nonlinear feedback laws, Adv. Math. Sci. Appl., 11 (2001), 549-569. [15] R. F. Curtain and A. J. Sasane, Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems, Internat. J. Control, 74 (2001), 1260-1270. doi: 10.1080/00207170110061059. [16] D. N. Daescu and I. M. Navon, Efficiency of a POD-based reduced second-order adjoint model in 4D-var data assimilation, Internat. J. Numer. Methods Fluids, 53 (2007), 985-1004. doi: 10.1002/fld.1316. [17] F. Fang, T. Zhang, D. Pavlidis, C. Pain, A. Buchan and I. Navon, Reduced order modelling of an unstructured mesh air pollution model and application in 2d/3d urban street canyons, Atmospheric Environment, 96 (2014), 96-106. doi: 10.1016/j.atmosenv.2014.07.021. [18] N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667. doi: 10.3934/eect.2013.2.631. [19] K. Glover, R. F. Curtain and J. R. Partington, Realisation and approximation of linear infinite-dimensional systems with error bounds, SIAM J. Control Optim., 26 (1988), 863-898. doi: 10.1137/0326049. [20] Y. Gong, Q. Wang and Z. Wang, Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems, Comput. Methods Appl. Mech. Engrg., 315 (2017), 780-798. doi: 10.1016/j.cma.2016.11.016. [21] C. Guiver and M. R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, SIAM J. Control Optim., 52 (2014), 1366-1401. doi: 10.1137/110846981. [22] M. Gunzburger, N. Jiang and M. Schneier, An ensemble-proper orthogonal decomposition method for the nonstationary Navier-Stokes equations, SIAM J. Numer. Anal., 55 (2017), 286-304. doi: 10.1137/16M1056444. [23] M. Gunzburger and H.-C. Lee, Reduced-order modeling of Navier-Stokes equations via centroidal Voronoi tessellation, in Recent advances in adaptive computation, vol. 383 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2005, 213–224. doi: 10.1090/conm/383/07166. [24] S. Herkt, M. Hinze and R. Pinnau, Convergence analysis of Galerkin POD for linear second order evolution equations, Electron. Trans. Numer. Anal., 40 (2013), 321-337. [25] D. B. P. Huynh, D. J. Knezevic and A. T. Patera, A Laplace transform certified reduced basis method; application to the heat equation and wave equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 401-405. doi: 10.1016/j.crma.2011.02.003. [26] M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 1284-1302. doi: 10.1137/100787350. [27] B. B. King, Modeling and control of a multiple component structure, J. Math. Systems Estim. Control, 4 (1994), 36pp. [28] V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method. [29] B. C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32. doi: 10.1109/TAC.1981.1102568. [30] O. Morgül, Stabilization and disturbance rejection for the wave equation, IEEE Trans. Automat. Control, 43 (1998), 89-95. doi: 10.1109/9.654893. [31] K. Morris, H∞-control of acoustic noise in a duct with a feedforward configuration, in Proceedings of 15th International Symposium on Mathematical Theory of Networks and Systems, 2002, 12–16. [32] M. R. Opmeer, Nuclearity of Hankel operators for ultradifferentiable control systems, Systems Control Lett., 57 (2008), 913-918. doi: 10.1016/j.sysconle.2008.04.007. [33] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [34] L. Peng and K. Mohseni, Symplectic model reduction of Hamiltonian systems, SIAM J. Sci. Comput., 38 (2016), A1-A27. doi: 10.1137/140978922. [35] V. Pereyra, Model order reduction with oblique projections for large scale wave propagation, J. Comput. Appl. Math., 295 (2016), 103-114. doi: 10.1016/j.cam.2015.01.029. [36] J. M. A. Scherpen, Balancing for nonlinear systems, Systems Control Lett., 21 (1993), 143-153. doi: 10.1016/0167-6911(93)90117-O. [37] H. E. Shoori J., An Approach to Reduced-Order Modeling and Feedback Control for Wave Energy Converters, Master's thesis, Oregon State University, 2014, URL http://ir.library.oregonstate.edu/xmlui/handle/1957/50823. [38] A. Varshney, S. Pitchaiah and A. Armaou, Feedback control of dissipative PDE systems using adaptive model reduction, AIChE journal, 55 (2009), 906-918. doi: 10.1002/aic.11770. [39] Z. Zhang, Stabilization of the wave equation with variable coefficients and a dynamical boundary control, Electron. J. Differential Equations, 2016 (2016), Paper No. 27, 10pp. [40] K. Zhou, J. C. Doyle and K. Glover, Robust and Optimal Control, Prentice-Hall, 1996.
The cable mass system
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = \alpha_{l} = 0.1$, $k_{0} = k_{l} = 1$, and $\alpha_{0} = \alpha = 0$
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = 0$ and $\alpha = \alpha_{0} = \alpha_{l} = k_{0} = k_{l} = 0.01$
Example 1, Input 2: Output of the ROM and FOM for $\alpha_{0} = \alpha = 0$, $\alpha_{l} = k_{0} = k_{l} = 0.1$, and $\gamma = 0.001$
Example 5, Input 4: Output of the nonlinear ROM and FOM for $\alpha = \alpha_{0} = \alpha_{l} = 0$, $\gamma = 0.001$, and $k_{0} = k_{l} = 0.1$
Example 2, Input 1: Output of the ROM and FOM for $\gamma = 0$, $\alpha = \alpha_{0} = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 1, Input 4: Output of the ROM and FOM for $\alpha = \alpha_0 = 0$, $\gamma = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 5, Input 4: Output of the ROM and FOM for $\gamma = 0.1$, $\alpha = \alpha_{0} = \alpha_{l} = 0$, and $k_{0} = k_{l} = 0.001$
Example 3: Output of the nonlinear ROM and FOM for $\alpha_{0} = \alpha_{l} = 0$ and $\alpha = \gamma = k_{0} = k_{l} = 0.001$
Fixed simulation parameters
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
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