# zbMATH — the first resource for mathematics

Pseudo-predictor feedback stabilization of linear systems with time-varying input delays. (English) Zbl 1300.93141
Summary: This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a Pseudo-Predictor Feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results is well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.

##### MSC:
 93D15 Stabilization of systems by feedback 93C05 Linear systems in control theory 93B52 Feedback control
DDE-BIFTOOL
Full Text:
##### References:
 [1] Anderson, B. D.O.; IIchmann, A.; Wirth, F. R., Stabilizability of linear time-varying systems, Systems & Control Letters, 62, 9, 747-755, (2013) · Zbl 1280.93068 [2] Anderson, B. D.O.; Moore, J. B., New results in linear system stability, SIAM Journal on Control, 7, 398-414, (1969) · Zbl 0182.48402 [3] Artstein, Z., Linear systems with delayed controls: a reduction, IEEE Transactions on Automatic Control, 27, 4, 869-879, (1982) · Zbl 0486.93011 [4] Cacace, F.; Germani, A.; Manes, C., Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment, Systems & Control Letters, 69, 47-52, (2014) · Zbl 1288.93065 [5] Chen, J.; Fu, P.; Niculescu, S.-I.; Guan, Z., An eigenvalue perturbation approach to stability analysis, part II: when will zeros of time-delay systems cross imaginary axis?, SIAM Journal on Control and Optimization, 48, 5583-5605, (2010) · Zbl 1261.93061 [6] Chen, W. H.; Zheng, W. X., Exponential stability of nonlinear time-delay systems with delayed impulse effects, Automatica, 47, 5, 1075-1083, (2011) · Zbl 1233.93080 [7] Cong, S.; Zou, Y., A new delay-dependent exponential stability criterion for ito stochastic systems with Markovian switching and time-varying delay, International Journal of Systems Science, 41, 12, 1493-1500, (2010) · Zbl 1206.93105 [8] Engelborghs, K., Luzyanina, T., & Samaey, G. (2001). DDE-BIFTOOL v.2.00: a Matlab package for bifurcation analysis of delay differential equation. T.W. Rep. 330. Dept. Comput. Sci., KU Leuven. [9] Fridman, E., A descriptor system approach to $$H_\infty$$ control of linear time-delay systems, IEEE Transactions on Automatic Control, 47, 2, 253-270, (2002) · Zbl 1364.93209 [10] Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE conference on decision and control(pp. 2805-2810). [11] Gu, K., A review of some subtleties of practical relevance for time-delay systems of neutral type, ISRN Applied Mathematics, 2012, (2012), Article ID 725783 · Zbl 1264.34006 [12] Gu, K.; Chen, J.; Kharitonov, V. L., Stability of time-delay systems, (2003), Springer · Zbl 1039.34067 [13] Hale, J. K., Theory of functional differential equations, (1977), Springer New York [14] Huang, L., The theory foundation of stability and robustness, (2003), Science Press Beijing, (in Chinese) [15] Kalman, R. E., Contributions to the theory of optimal control, Boletin De la Sociedad Matematica Mexicana, 5, 2, 102-119, (1960) [16] Kleinman, D. L., On the linear regulator problem and the matrix Riccati equation. technical report, (1966), Massachusetts Institute of Technology [17] Kojima, A.; Uchida, K.; Shimemura, E.; Ishijima, S., Robust stabilization of a system with delays in control, IEEE Transactions on Automatic Control, 39, 8, 1694-1698, (1994) · Zbl 0800.93985 [18] Krstic, M., Delay compensation for nonlinear, adaptive, and PDE systems, (2009), Birkhäuser · Zbl 1181.93003 [19] Krstic, M., Lyapunov stability of linear predictor feedback for time-varying input delay, IEEE Transactions on Automatic Control, 55, 2, 554-559, (2010) · Zbl 1368.93547 [20] Lam, J.; Xu, S.; Ho, D.; Zou, Y., On global asymptotic stability for a class of delayed neural networks, International Journal of Circuit Theory and Applications, 40, 1165-1174, (2012) [21] Li, Z.; Zhou, B.; Lin, Z., On exponential stability of integral delay systems, Automatica, 49, 3368-3376, (2013) · Zbl 1315.93066 [22] Manitius, A. Z.; Olbrot, A. W., Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control, 24, 541-553, (1979) · Zbl 0425.93029 [23] Melchor-Aguilar, D., On stability of integral delay systems, Applied Mathematics and Computation, 217, 3578-3584, (2010) · Zbl 1221.45012 [24] Michiels, W.; Niculescu, S.-I., Stability and stabilization of time-delay systems: an eigenvalue-based approach, (2007), SIAM Philadelphia [25] Mondie, S.; Melchor-Aguilar, D., Exponential stability of integral delay systems with a class of analytic kernels, IEEE Transactions on Automatic Control, 57, 484-489, (2012) · Zbl 1369.93537 [26] Mondie, S.; Michiels, W., Finite spectrum assignment of unstable time-delay systems with a safe implementation, IEEE Transactions on Automatic Control, 48, 2207-2212, (2003) · Zbl 1364.93312 [27] Richard, J. P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302 [28] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series A, 10, 863-874, (2003) · Zbl 1068.34072 [29] Rugh, W. J., Linear system theory, (1996), Prentice-Hall NJ · Zbl 0892.93002 [30] Van Assche, V., Dambrine, M., Lafay, J. F., & Richard, J. P. (1999). Some problems arising in the implementation of distributed-delay control laws. In 38th IEEE conf. decision control. Phoenix, AZ. [31] Wu, Z.; Shi, P.; Su, H.; Chu, J., Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time-delays, IEEE Transactions on Neural Networks, 22, 10, 1566-1575, (2011) [32] Wu, Z.; Shi, P.; Su, H.; Chu, J., Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data, IEEE Transactions on Cybernetics, 43, 6, 1796-1806, (2013) [33] Xu, S.; Lam, J.; Yang, C., Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay, Systems & Control Letters, 43, 77-84, (2001) · Zbl 0974.93052 [34] Zhao, W.; Chen, H., Adaptive tracking and recursive identification for Hammerstein systems, Automatica, 45, 2773-2783, (2009) · Zbl 1192.93124 [35] Zhao, W.; Zhou, T., Weighted least squares based recursive parametric identification of the submodels of a PWARX system, Automatica, 48, 1190-1196, (2012) · Zbl 1244.93168 [36] Zhong, Q. C., On distributed delay in linear control laws—part I: discrete-delay implementations, IEEE Transactions on Automatic Control, 49, 11, 2074-2080, (2004) · Zbl 1365.93155 [37] Zhou, B., Truncated predictor feedback for time-delay systems. vol. XIX, 480, (2014), Springer Heidelberg, Germany · Zbl 1306.93003 [38] Zhou, B., & Cong, S. Stabilization and consensus of linear systems with multiple input delays by truncated pseudo-predictor feedback, http://arxiv.org/abs/1409.2723. · Zbl 1333.93203 [39] Zhou, B.; Duan, G.-R., Periodic Lyapunov equation based approaches to the stabilization of continuous-time periodic linear systems, IEEE Transactions on Automatic Control, 57, 8, 2139-2146, (2012) · Zbl 1369.93515 [40] Zhou, B.; Hou, M.; Duan, G., $$L_\infty$$ and $$L_2$$ semi-global stabilisation of continuous-time periodic linear systems with bounded controls, International Journal of Control, 86, 4, 709-720, (2013) · Zbl 1278.93187 [41] Zhou, B., Li, Z., & Lin, Z. (2015). On higher-order truncated predictor feedback for linear systems with input delay, International Journal of Robust and Nonlinear Control, http://dx.doi.org/10.1002/rnc.3012. [42] Zhou, B.; Lin, Z.; Duan, G.-R., A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems, Automatica, 45, 1, 238-244, (2009) · Zbl 1154.93362 [43] Zhou, B.; Lin, Z.; Duan, G., $$L_\infty$$ and $$L_2$$ low gain feedback: their properties, characterizations and applications in constrained control, IEEE Transactions on Automatic Control, 56, 5, 1030-1045, (2011) · Zbl 1368.93235 [44] Zhou, B.; Lin, Z.; Duan, G., Truncated predictor feedback for linear systems with long time-varying input delays, Automatica, 48, 10, 2387-2399, (2012) · Zbl 1271.93123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.