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Model reduction of time-delay systems using position balancing and delay Lyapunov equations. (English) Zbl 1272.93036
Summary: Balanced truncation is a standard and very natural approach to approximate dynamical systems. We present a version of balanced truncation for model order reduction of linear time-delay systems. The procedure is based on a coordinate transformation of the position and preserves the delay structure of the system. We therefore call it (structure-preserving) position balancing. To every position, we associate quantities representing energies for the controllability and observability of the position. We show that these energies can be expressed explicitly in terms of the solutions to corresponding delay Lyapunov equations. Apart from characterizing the energies, we show that one block of the (operator) controllability and observability Gramians in the operator formulation of the time-delay system can also be characterized with the delay Lyapunov equation. The delay Lyapunov equation undergoes a contragredient transformation when we apply the position coordinate transformation and we propose to truncate it in a classical fashion, such that positions which are only weakly connected to the input and the output in the sense of the energy concepts are removed.

##### MSC:
 93B11 System structure simplification 93B17 Transformations 93C15 Control/observation systems governed by ordinary differential equations
Algorithm 432
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##### References:
 [1] Antoulas A (2005) Approximation dynamical systems. In: Society for industrial and applied mathematics (SIAM). Philadelphia, PA · Zbl 1112.93002 [2] Antoulas, A; Sorensen, D; Gugercin, S, A survey of model reduction methods for large-scale systems, Contemp Math, 280, 193-219, (2001) · Zbl 1048.93014 [3] Bartels, R; Stewart, GW, Solution of the matrix equation $$AX+XB=C$$, Comm ACM, 15, 820-826, (1972) · Zbl 1372.65121 [4] Beattie, C; Gugercin, S, Interpolatory projection methods for structure-preserving model reduction, Syst Control Lett, 58, 225-232, (2009) · Zbl 1159.93317 [5] Benner P, Mehrmann V, Sorensen D (eds) (2005) Dimension reduction of large-scale systems. Springer, Berlin · Zbl 1066.65004 [6] Benner, P; Saak, J, Efficient balancing based MOR for large scale second order systems, Math Comput Model Dyn Syst, 17, 123-143, (2011) · Zbl 1221.93039 [7] Chahlaoui, Y; Lemonnier, D; Vandendorpe, A; Dooren, PV, Second-order balanced truncation, Linear Algebra Appl, 415, 373-384, (2006) · Zbl 1102.93008 [8] Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, New York · Zbl 0839.93001 [9] Hale J, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, Berlin · Zbl 0787.34002 [10] Harkort, C; Deutscher, J, Krylov subspace methods for linear infinite-dimensional systems, IEEE Trans Autom Control, 56, 441-447, (2011) · Zbl 1368.93071 [11] Huesca E, Mondié S, Santos J (2009) Polynomial approximations of the Lyapunov matrix of a class of time delay systems. In: Proceedings of the 8th IFAC workshop on time-delay systems, Sinaia, Romania · Zbl 1247.65086 [12] Jarlebring E, Vanbiervliet J, Michiels W (2011) Characterizing and computing the $${\cal H}_{2}$$ norm of time-delay systems by solving the delay Lyapunov equation. IEEE Trans Autom Control 56(4):814-825 · Zbl 1368.93094 [13] Kharitonov, V, Lyapunov-krasovskii functionals for scalar time delay equations, Syst Control Lett, 51, 133-149, (2004) · Zbl 1157.34354 [14] Kharitonov, V, Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: a single delay case, Int J Control, 78, 783-800, (2005) · Zbl 1097.93027 [15] Kharitonov, V, Lyapunov matrices for a class of time delay systems, Syst Control Lett, 55, 610-617, (2006) · Zbl 1129.34331 [16] Kharitonov, V; Hinrichsen, D, Exponential estimates for time delay systems, Syst Control Lett, 53, 395-405, (2004) · Zbl 1157.34355 [17] Kharitonov, V; Plischke, E, Lyapunov matrices for time-delay systems, Syst Control Lett, 55, 697-706, (2006) · Zbl 1100.93045 [18] Kharitonov, V; Zhabko, AP, Lyapunov-krasovskii approach to the robust stability analysis of time-delay systems, Automatica, 39, 15-20, (2003) · Zbl 1014.93031 [19] Mäkilä, P; Partington, J, Laguerre and kautz shift approximations of delay systems, Int J Control, 72, 932-946, (1999) · Zbl 0963.93042 [20] Mäkilä, P; Partington, J, Shift operator induced approximations of delay systems, SIAM J Control Optim, 37, 1897-1912, (1999) · Zbl 0935.93047 [21] Mastinšek, M, Adjoints of solution semigroups and identifiability of delay differential equations in Hilbert spaces, Acta Math Univ Comen LXIII, 2, 193-206, (1994) · Zbl 0822.34053 [22] Meyer, DG; Srinivasan, S, Balancing and model reduction for second-order form linear systems, IEEE Trans Autom Control, 41, 1632-1644, (1996) · Zbl 0859.93015 [23] Michiels W (2002) Stability and stabilization of time-delay systems. Ph.D. thesis, Katholieke universiteit Leuven [24] Michiels, W; Jarlebring, E; Meerbergen, K, Krylov-based model order reduction of time-delay systems, SIAM J Matrix Anal Appl, 32, 1399-1421, (2011) · Zbl 1247.65086 [25] Niculescu SI (2001) Delay effects on stability. A robust control approach. Springer, London · Zbl 0997.93001 [26] Ochoa G, Kharitonov V (2005) Lyapunov matrices for neutral type time delay systems. In: Proceedings of the 2nd International Conference on Electrical and Electronics Engineering, Mexico City, Mexico · Zbl 1097.93027 [27] Ochoa G, Velázquez-Velázquez J, Kharitonov V, Mondié S (2007) Lyapunov matrices for neutral type time delay systems. In: Proceedings of the 7th IFAC workshop on time delay systems, Nantes, France · Zbl 1048.93014 [28] Ouellette, DV, Schur complements and statistics, Linear Algebra Appl, 36, 187-295, (1981) · Zbl 0455.15012 [29] Partington, J; Blondel, V (ed.); Megretski, A (ed.), Model reduction of delay systems, 29-32, (2004), Princeton [30] Plischke E (2005) Transient effects of linear dynamical systems. Ph.D. thesis, Universität Bremen · Zbl 1325.37001 [31] Reis, T; Stykel, T, Balanced truncation model reduction of second-order systems, Math Comput Model Dyn Syst, 14, 391-406, (2008) · Zbl 1151.93010 [32] Saadvandi, M; Meerbergen, K; Jarlebring, E, On dominant poles and model reduction of second order time-delay systems, Appl Numer Math, 62, 21-34, (2012) · Zbl 1236.93080 [33] Vanbiervliet J, Michiels W, Jarlebring E (2011) Using spectral discretisation for the optimal $${\cal H}_{2}$$ design of time-delay systems. Int J Control 84(2):228-241 · Zbl 1222.93177 [34] Velázquez-Velázquez J, Kharitonov V (2009) Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Syst Control Lett 58(1):17-25 · Zbl 1154.93026 [35] Yan, B; Tan, S; McGaughy, B, Second-order balanced truncation for passive order reduction of RLCK circuits, IEEE Trans Circuits Syst II Analog Digit Signal Process, 55, 942-946, (2008) [36] Zigic, D; Watson, L; Beattie, C, Contragredient transformations applied to optimal projection equations, Linear Algebra Appl, 188-189, 665-676, (1993) · Zbl 0781.65059
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