June  2017, 10(3): 413-444. doi: 10.3934/dcdss.2017020

State transformations of time-varying delay systems and their applications to state observer design

1. 

Department of Mathematics, Quynhon University, Quynhon, Binhdinh, Vietnam

2. 

Department of Mathematics and Informatics, Thainguyen University of Science, Thainguyen, Vietnam

* Corresponding author

Received  June 2016 Revised  January 2017 Published  February 2017

In this paper, we derive new state transformations of linear systems with a time-varying delay in the state vector. We first provide a new algebraic and systematic method for computing forward state transformations to transform time-delay systems into a novel form where time-varying delay appears in the input and output vectors, but not in the state vector. In the new coordinate system, a Luenberger-type state observer with a guaranteed $ β $-exponential stability margin can be designed. Then, a backward state transformation problem which allows us to reconstruct the original state vector of the system is investigated. By using both the forward and the backward state transformations, state observers for time-varying delay systems can be systematically designed. Conditions for ensuring the existence of the forward and backward state transformations and an effective algorithm for computing them are given in this paper. We illustrate our results by three examples and simulation results.

Citation: Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020
References:
[1]

M. Boutayeb, Observer design for linear time-delay systems, Syst. & Contr. Letters, 44 (2001), 103-109. doi: 10.1016/S0167-6911(01)00129-3.

[2]

D. BoutatA. BenaliH. Hammouri and K. Busawon, New algorithm for observer error linearization with a diffeomorphism on the outputs, Automatica, 45 (2009), 2187-2193. doi: 10.1016/j.automatica.2009.05.030.

[3]

D. BoutatL. Boutat-Baddas and M. Darouach, A new reduced-order observer normal form for nonlinear discrete time systems, Systems & Control Letters, 61 (2012), 1003-1008. doi: 10.1016/j.sysconle.2012.07.007.

[4]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777.

[5]

F. CacaceA. Germani and C. Manes, An observer for a class of nonlinear systems with time varying observation delay, Systems & Control Letters, 59 (2010), 305-312. doi: 10.1016/j.sysconle.2010.03.005.

[6]

M. Darouach, Linear functional observers for systems with delays in state variables, IEEE Trans. Automat. Control, 46 (2001), 491-496. doi: 10.1109/9.911430.

[7]

F. W. Fairman and A. Kumar, Delayless observers for systems with delay, IEEE Trans. Automat. Control, 31 (1986), 258-259. doi: 10.1109/TAC.1986.1104228.

[8]

H. Gao and X. Li, $ H_{∞} $ filtering for discrete-time state-delayed systems with finite frequency specifications, IEEE Trans. Automat. Control, 56 (2001), 2935-2941. doi: 10.1109/TAC.2011.2159909.

[9] K. GuV. L. Kharitonov and J. Chen, Stability of Time-delay Systems, Springer, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0039-0.
[10] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[11]

M. Hou and A. C. Pugh, Observer with linear error dynamics for nonlinear multi-output systems, Systems & Control Letters, 37 (1999), 1-9. doi: 10.1016/S0167-6911(98)00105-4.

[12]

M. HouP. Zitek and R. J. Patton, An observer design for linear time-delay systems, IEEE Transactions on Automatic Control, 47 (2002), 121-125. doi: 10.1109/9.981730.

[13]

D. C. HuongH. TrinhH. M. Tran and T. Fernando, Approach to fault detection of time-delay systems using functional observers, Electronic Letters, 50 (2014), 1132-1134. doi: 10.1049/el.2014.1480.

[14]

D. C. Huong and H. Trinh, Method for computing state transformations of time-delay systems, IET Control Theory & Applications, 9 (2015), 2405-2413. doi: 10.1049/iet-cta.2015.0108.

[15]

A. J. Krener, Linearization by output injection and nonlinear observers, Syst. & Contr. Letters, 3 (1983), 47-52. doi: 10.1016/0167-6911(83)90037-3.

[16]

A. J. Krener and W. Respondek, Nonlinear observers with Linearization error dynamics, Siam J. Control Optimization, 23 (1985), 197-216. doi: 10.1137/0323016.

[17]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.

[18]

M. Malek-Zavarei and M. Jamshidi, Time-delay Systems: Analysis Optimization and Applications North-Holland Systems and Control Series, 9. North-Holland Publishing Co. , Amsterdam, 1987.

[19]

S. Mondie and V. L. Kharitonov, Exponential estimates for retarded time-delay systems: An LMI approach, IEEE Trans. Automat. Control, 50 (2005), 268-273. doi: 10.1109/TAC.2004.841916.

[20]

P. T. NamP. N. Pathirana and H. Trinh, ϵ-bounded state estimation for time-delay systems with bounded disturbances, Int. J. Control, 87 (2014), 1747-1756. doi: 10.1080/00207179.2014.884727.

[21]

P. T. NamP. N. Pathirana and H. Trinh, Linear functional state bounding for perturbed time-delay systems and its application, IMA J. Math. Control Inf., 32 (2015), 245-255. doi: 10.1093/imamci/dnt039.

[22]

P. Niamsup and V. N. Phat, A Novel Exponential Stability Condition for a Class of Hybrid Neural Networks with Time-varying Delay, Vietnam Journal of Mathematics, 38 (2010), 341-351.

[23]

P. Niamsup and V. N. Phat, State Feedback Guaranteed Cost Controller for Nonlinear Time-Varying Delay Systems, Vietnam Journal of Mathematics, 43 (2015), 215-228. doi: 10.1007/s10013-014-0108-9.

[24]

R. M. PalharesC. E. de Souza and P. L. D. Peres, Robust $ {{H}_{\infty }} $ filtering for uncertain discretetime state-delayed systems, IEEE Trans. Signal Processing, 49 (2001), 1696-1703. doi: 10.1109/78.934139.

[25]

P. PalumboS. Panunzi and A. De Gaetano, Qualitative behavior of a family of delay-differential models of the glucose-insulin system, Discrete Continuous Dynam. Systems -B, 7 (2007), 399-424.

[26]

P. PalumboP. PepeP. Panunzi and A. De Gaetano, Time-delay model-based control of the glucose -insulin system, by means of a state observer, Eur J Control, 18 (2012), 591-606. doi: 10.3166/EJC.18.591-606.

[27]

J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[28] S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3108-8.
[29]

R. TamiD. Boutat and G. Zheng, Extended output depending normal form, Automatica, 49 (2013), 2192-2198. doi: 10.1016/j.automatica.2013.03.025.

[30]

M. V. Thuan and V. N. Phat, New criteria for stability and stabilization of neural networks with mixed interval time-varying delay, Vietnam Journal of Mathematics, 40 (2012), 79-93.

[31]

M. V. ThuanV. N. PhatT. Fernando and H. Trinh, Exponential stabilization of time-varying delay systems with non-linear perturbation, IMA J. Math. Control Inf., 31 (2014), 441-464. doi: 10.1093/imamci/dnt022.

[32]

H. Trinh, Linear functional state observer for time-delay systems, Int. J. Control, 72 (1999), 1642-1658. doi: 10.1080/002071799219986.

[33] H. Trinh and T. Fernando, Functional Observers for Dynamical Systems, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-24064-5.
[34]

Z. WangJ. Lam and X. Liu, Filtering for a class of nonlinear discrete-time stochastic systems with state delays, Journal of Computational and Applied Mathematics, 201 (2007), 153-163. doi: 10.1016/j.cam.2006.02.009.

[35]

Z. XiangS. Liu and M. S. Mahmoud, Robust $ H_{∞} $ reliable control for uncertain switched neutral systems with distributed delay, IMA J. Math. Control Inf., 32 (2015), 1-19. doi: 10.1093/imamci/dnt031.

[36]

H. Zhang and J. Wang, State estimation of discrete-time Takagi-Sugeno fuzzy systems in a network environment, IEEE Trans. Cybern., 45 (2015), 1525-1536. doi: 10.1109/TCYB.2014.2354431.

[37]

G. Zhao and J. Wang, Reset observers for linear time-varying delay systems: Delay-dependent approach, J. Frankl. Inst., 351 (2014), 5133-5147. doi: 10.1016/j.jfranklin.2014.08.011.

[38]

Y. Zhao and Z. Feng, Desynchronization in synchronous multi-coupled chaotic neurons by mix-adaptive feedback control, J. Biol. Dyn., 7 (2013), 1-10. doi: 10.1080/17513758.2012.733426.

show all references

References:
[1]

M. Boutayeb, Observer design for linear time-delay systems, Syst. & Contr. Letters, 44 (2001), 103-109. doi: 10.1016/S0167-6911(01)00129-3.

[2]

D. BoutatA. BenaliH. Hammouri and K. Busawon, New algorithm for observer error linearization with a diffeomorphism on the outputs, Automatica, 45 (2009), 2187-2193. doi: 10.1016/j.automatica.2009.05.030.

[3]

D. BoutatL. Boutat-Baddas and M. Darouach, A new reduced-order observer normal form for nonlinear discrete time systems, Systems & Control Letters, 61 (2012), 1003-1008. doi: 10.1016/j.sysconle.2012.07.007.

[4]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777.

[5]

F. CacaceA. Germani and C. Manes, An observer for a class of nonlinear systems with time varying observation delay, Systems & Control Letters, 59 (2010), 305-312. doi: 10.1016/j.sysconle.2010.03.005.

[6]

M. Darouach, Linear functional observers for systems with delays in state variables, IEEE Trans. Automat. Control, 46 (2001), 491-496. doi: 10.1109/9.911430.

[7]

F. W. Fairman and A. Kumar, Delayless observers for systems with delay, IEEE Trans. Automat. Control, 31 (1986), 258-259. doi: 10.1109/TAC.1986.1104228.

[8]

H. Gao and X. Li, $ H_{∞} $ filtering for discrete-time state-delayed systems with finite frequency specifications, IEEE Trans. Automat. Control, 56 (2001), 2935-2941. doi: 10.1109/TAC.2011.2159909.

[9] K. GuV. L. Kharitonov and J. Chen, Stability of Time-delay Systems, Springer, Birkhäuser Boston, 2003. doi: 10.1007/978-1-4612-0039-0.
[10] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[11]

M. Hou and A. C. Pugh, Observer with linear error dynamics for nonlinear multi-output systems, Systems & Control Letters, 37 (1999), 1-9. doi: 10.1016/S0167-6911(98)00105-4.

[12]

M. HouP. Zitek and R. J. Patton, An observer design for linear time-delay systems, IEEE Transactions on Automatic Control, 47 (2002), 121-125. doi: 10.1109/9.981730.

[13]

D. C. HuongH. TrinhH. M. Tran and T. Fernando, Approach to fault detection of time-delay systems using functional observers, Electronic Letters, 50 (2014), 1132-1134. doi: 10.1049/el.2014.1480.

[14]

D. C. Huong and H. Trinh, Method for computing state transformations of time-delay systems, IET Control Theory & Applications, 9 (2015), 2405-2413. doi: 10.1049/iet-cta.2015.0108.

[15]

A. J. Krener, Linearization by output injection and nonlinear observers, Syst. & Contr. Letters, 3 (1983), 47-52. doi: 10.1016/0167-6911(83)90037-3.

[16]

A. J. Krener and W. Respondek, Nonlinear observers with Linearization error dynamics, Siam J. Control Optimization, 23 (1985), 197-216. doi: 10.1137/0323016.

[17]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.

[18]

M. Malek-Zavarei and M. Jamshidi, Time-delay Systems: Analysis Optimization and Applications North-Holland Systems and Control Series, 9. North-Holland Publishing Co. , Amsterdam, 1987.

[19]

S. Mondie and V. L. Kharitonov, Exponential estimates for retarded time-delay systems: An LMI approach, IEEE Trans. Automat. Control, 50 (2005), 268-273. doi: 10.1109/TAC.2004.841916.

[20]

P. T. NamP. N. Pathirana and H. Trinh, ϵ-bounded state estimation for time-delay systems with bounded disturbances, Int. J. Control, 87 (2014), 1747-1756. doi: 10.1080/00207179.2014.884727.

[21]

P. T. NamP. N. Pathirana and H. Trinh, Linear functional state bounding for perturbed time-delay systems and its application, IMA J. Math. Control Inf., 32 (2015), 245-255. doi: 10.1093/imamci/dnt039.

[22]

P. Niamsup and V. N. Phat, A Novel Exponential Stability Condition for a Class of Hybrid Neural Networks with Time-varying Delay, Vietnam Journal of Mathematics, 38 (2010), 341-351.

[23]

P. Niamsup and V. N. Phat, State Feedback Guaranteed Cost Controller for Nonlinear Time-Varying Delay Systems, Vietnam Journal of Mathematics, 43 (2015), 215-228. doi: 10.1007/s10013-014-0108-9.

[24]

R. M. PalharesC. E. de Souza and P. L. D. Peres, Robust $ {{H}_{\infty }} $ filtering for uncertain discretetime state-delayed systems, IEEE Trans. Signal Processing, 49 (2001), 1696-1703. doi: 10.1109/78.934139.

[25]

P. PalumboS. Panunzi and A. De Gaetano, Qualitative behavior of a family of delay-differential models of the glucose-insulin system, Discrete Continuous Dynam. Systems -B, 7 (2007), 399-424.

[26]

P. PalumboP. PepeP. Panunzi and A. De Gaetano, Time-delay model-based control of the glucose -insulin system, by means of a state observer, Eur J Control, 18 (2012), 591-606. doi: 10.3166/EJC.18.591-606.

[27]

J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[28] S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3108-8.
[29]

R. TamiD. Boutat and G. Zheng, Extended output depending normal form, Automatica, 49 (2013), 2192-2198. doi: 10.1016/j.automatica.2013.03.025.

[30]

M. V. Thuan and V. N. Phat, New criteria for stability and stabilization of neural networks with mixed interval time-varying delay, Vietnam Journal of Mathematics, 40 (2012), 79-93.

[31]

M. V. ThuanV. N. PhatT. Fernando and H. Trinh, Exponential stabilization of time-varying delay systems with non-linear perturbation, IMA J. Math. Control Inf., 31 (2014), 441-464. doi: 10.1093/imamci/dnt022.

[32]

H. Trinh, Linear functional state observer for time-delay systems, Int. J. Control, 72 (1999), 1642-1658. doi: 10.1080/002071799219986.

[33] H. Trinh and T. Fernando, Functional Observers for Dynamical Systems, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-24064-5.
[34]

Z. WangJ. Lam and X. Liu, Filtering for a class of nonlinear discrete-time stochastic systems with state delays, Journal of Computational and Applied Mathematics, 201 (2007), 153-163. doi: 10.1016/j.cam.2006.02.009.

[35]

Z. XiangS. Liu and M. S. Mahmoud, Robust $ H_{∞} $ reliable control for uncertain switched neutral systems with distributed delay, IMA J. Math. Control Inf., 32 (2015), 1-19. doi: 10.1093/imamci/dnt031.

[36]

H. Zhang and J. Wang, State estimation of discrete-time Takagi-Sugeno fuzzy systems in a network environment, IEEE Trans. Cybern., 45 (2015), 1525-1536. doi: 10.1109/TCYB.2014.2354431.

[37]

G. Zhao and J. Wang, Reset observers for linear time-varying delay systems: Delay-dependent approach, J. Frankl. Inst., 351 (2014), 5133-5147. doi: 10.1016/j.jfranklin.2014.08.011.

[38]

Y. Zhao and Z. Feng, Desynchronization in synchronous multi-coupled chaotic neurons by mix-adaptive feedback control, J. Biol. Dyn., 7 (2013), 1-10. doi: 10.1080/17513758.2012.733426.

Figure 1.  Responses of $\hat{x}_2(t-\tau(t))$ and $x_2(t-\tau(t))$
Figure 2.  Responses of $\hat{x}_3(t)$ and $x_3(t)$
Figure 3.  Responses of $\hat{x}_3(t)$ and $x_3(t)$
Figure 4.  Responses of $\hat{x}_4(t)$ and $x_4(t)$
Figure 5.  Responses of $\hat{x}_3(t)$ and $x_3(t)$
Figure 6.  Responses of $\hat{x}_4(t)$ and $x_4(t)$
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