# System Identification: Formulation

• Qi He
• Le Yi Wang
• G. George Yin
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

Consider a single-input–single-output (SISO) linear time-invariant (LTI) stable discrete-time system
$$y(t) =\displaystyle\sum\limits_{ i=0}^{\infty }a_{ i}u(t - i) + d(t),\quad t = t_{0} + 1,\ldots,$$
(2.1)
where {y(t)} is the noise corrupted observation, {d(t)} is the disturbance, {u(t)} is the input with u(t) = 0 for t < 0, and $$a =\{ a_{i},i = 0,1,\ldots \},$$ satisfying
$$\|a\|_{1} =\displaystyle\sum\limits_{ i=0}^{\infty }\vert a_{ i}\vert < \infty.$$
To proceed, we define
$$\begin{array}{ll} & \theta = (a_{0},a_{1},\ldots,a_{m_{0}-1})^{\prime} \in {\mathbb{R}}^{m_{0} }, \\ & \widetilde{\theta } = (a_{m_{0}},a_{m_{0}+1},\ldots )^{\prime}, \end{array}$$
(2.2)
where z′ denotes the transpose of z.

## References

1. 1.
K. Aström and B. Wittenmark, Adaptive Control, Addison-Wesley, 1989.Google Scholar
2. 2.
R.R. Bahadur, Large deviation of the maximum likelihood estimate in the Markov chain case, in Recent Advances in Statistics, M.H. Rizvi, J.S. Rostag, and D. Siegmund eds., Academic Press, NY, 1983, 273–286.Google Scholar
3. 3.
R.R. Bahadur, S.L. Zabell, and J.C. Gupta, Large deviations, tests and estimates, in Asymptotic Theory of Statistical Tests, I.M. Chakravarti, ed., Academic Press, NY, 1980, 33–64.Google Scholar
4. 4.
C. Barbier, H. Meyer, B. Nogarede, S. Bensaoud, A battery state of charge indicator for electric vehicle, in Proc. International Conference of the Institution of Mechanical Engineers, Automotive Electronics, London, UK, May 17–19, 29–34, 1994.Google Scholar
5. 5.
E. Barsoukov, J. Kim, C. Yoon, H. Lee, Universal battery parameterization to yield a non-linear equivalent circuit valid for battery simulation at arbitrary load, J. Power Sour., 83 (1999), 61–70.
6. 6.
B.S. Bhangu, P. Bentley, D.A. Stone, and C.M. Bingham, Nonlinear observers for predicting state-of-charge and state-of-health of lead-acid batteries for hybrid-electric vehicles, IEEE Trans. Veh. Tech., 54 (2005), 783–794.
7. 7.
W. Bryc and A. Dembo. Large deviations and strong mixing, Ann. Inst. H. Poincare Probab. Stat., 32 (1996), 549–569.
8. 8.
P.E. Caines, Linear Stochastic Systems, Wiley, New York, 1988.
9. 9.
H. Chan, D. Sutanto, A new battery model for use with battery energy storage systems and electric vehicle power systems, in Proceedings of the 2000 IEEE Power Engineering Society Winter Meeting, Singapore, 470–475, January 23–27, 2000.Google Scholar
10. 10.
H.-F. Chen and L. Guo, Identification and Stochastic Adaptive Control, Birkhäuser, Boston, 1991.
11. 11.
K.L. Chung, On a stochastic approximation method, Ann. Math. Statist., 25 (1954), 463–483.
12. 12.
O. Craiu, A. Machedon, et al., 3D finite element thermal analysis of a small power PM DC motor, 12th International Conference on Optimization of Electrical and Electronic Equipment (OPTIM), 2010.Google Scholar
13. 13.
H.F. Dai, X.Z. Wei, Z.C. Sun, Online SOC estimation of high-power lithium-ion batteries used on HEVs, Proceedings of IEEE ICVES 2006, 342–347.Google Scholar
14. 14.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer-Verlag, New York, 1998.
15. 15.
P. Dupuis and H.J. Kushner, Stochastic approximation via large deviations: asymptotic properties, SIAM J. Control Optim., 23 (1985) 675–696.
16. 16.
P. Dupuis and H.J. Kushner, Stochastic approximation and large deviations: upper bounds and w.p.1 convergence, SIAM J. Control Optim., 27 (1989) 1108–1135.Google Scholar
17. 17.
W. Feller, An Introduction to Probability Theory and Its Applications, vol. I, 3rd ed. Wiley, New York, 1968.Google Scholar
18. 18.
A.E. Fitzgerald, C. Kingsley Jr., and S.D. Umans, Electric Machinery, McGraw-Hill Science/Engineering/Math, 6th ed., 2002.Google Scholar
19. 19.
J. Gärtner, On large deviations from the invariant measure, Theory Probab. Appl., 22 (1977), 24–39.
20. 20.
R. Giglioli, P. Pelacchi, M. Raugi, and G. Zini, A state of charge observer for lead-acid batteries, Energia Elettrica 65 (1988), 27–33.Google Scholar
21. 21.
W.B. Gu and C.Y. Wang, Thermal electrochemical modeling of battery systems, J. Electrochem. Soc., 147 (2000), 2910–2922.
22. 22.
B.S. Guru and H.R. Hiziroglu, Electric Machinery and Transformers, Oxford University Press, 2001.Google Scholar
23. 23.
S. Haykin, Unsupervised Adaptive Filtering: Volume I Blind Source Separation, John Wiley Sons, Inc., USA, 2000.Google Scholar
24. 24.
V. Johnson, A. Pesaran, and T. Sack, Temperature-dependent battery models for high-power lithium-ion batteries, Proceedings of the 17th Electric Vehicle Symposium, Montreal, Canada, October 2000.Google Scholar
25. 25.
V. Johnson, M. Zolot, and A. Pesaran, Development and validation of a temperature-dependent resistance/capacitance battery model for ADVISOR, Proceedings of the 18th Electric Vehicle Symposium, Berlin, Germany, October 2001.Google Scholar
26. 26.
I.-S. Kim, A technique for estimating the state of health of lithium batteries through a dual-sliding-mode observer, IEEE Trans. Power Elec., 25 (2010), 1013–1022.
27. 27.
A.D.M. Kester and W.C.M. Kallenberg, Large deviations of estimators, Ann. Statist., 14 (1986) 648–664.
28. 28.
P.R. Kumar and P. Varaiya, Stochastic Systems: Estimation, Identification and Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ, 1986.
29. 29.
H.J. Kushner, Asymptotic behavior of stochastic approximation and large deviation, IEEE Trans. Automat. Control, 29 (1984), 984–990.
30. 30.
H.J. Kushner and H. Huang, Rates of convergence for stochastic approximation type algorithms, SIAM J. Control Optim., 17 (1979) 607–617.
31. 31.
H.J. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications, 2nd ed., Springer-Verlag, New York, 2003.
32. 32.
Z.Y. Liu and C.R. Lu, Limit Theory for Mixing Dependent Random Variables, Science Press, Kluwer Academic, New York, Dordrecht, 1996.Google Scholar
33. 33.
L. Liu, L.Y. Wang, Z. Chen, C. Wang, F. Lin, and H. Wang, Integrated system identification and state-of-charge estimation of battery systems, IEEE Transactions on Energy Conversion, (2013), 1–12.Google Scholar
34. 34.
D. Linden and T. Reddy, Handbook of Batteries, 3rd ed., McGraw Hill, 2001.Google Scholar
35. 35.
L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987.
36. 36.
L. Ljung, H. Hjalmarsson, and H. Ohlasson, Four encounters with systems identification, Euro. J. Control, 17 (2011) 449–471.
37. 37.
L. Ljung and T. Söderström, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983.
38. 38.
M. Milanese and A. Vicino, Optimal estimation theory for dynamic systems with set membership uncertainty: an overview, Automatica, 27 (1991), 997–1009.
39. 39.
40. 40.
K. Ogata, Discrete-Time Control Systems, Prentice-Hall, Inc. Upper Saddle River, NJ, 1987.Google Scholar
41. 41.
B.T. Polyak, New method of stochastic approximation type, Automation Remote Control 7 (1991), 937–946.Google Scholar
42. 42.
G. Plett, Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs. Part 1. Background, J. Power Sour., 134 (2004), 252–261.Google Scholar
43. 43.
M.A. Roscher, J. Assfalg, and O.S. Bohlen, Detection of utilizable capacity deterioration in battery systems, IEEE Trans. Vehicular Tech., 60 (2011), 98–103.
44. 44.
S. Rodrigues, N. Munichandraiah, and A. Shukla, A review of state-of-charge indication of batteries by means of AC impedance measurements, J. Power Sour., 87 (2000), 12–20.
45. 45.
D. Ruppert, Stochastic approximation, in Handbook in Sequential Analysis, B.K. Ghosh and P.K. Sen, eds., Marcel Dekker, New York, 1991, 503–529.Google Scholar
46. 46.
M. Sitterly, L.Y. Wang, G. Yin, and C. Wang, Enhanced identification of battery models for real-time battery management, IEEE Transactions on Sustainable Energy, 2 (2011), 300–308.
47. 47.
A.M. Sharaf, E. Elbakush, and I.H. Atlas, A predictive dynamic controller for PMDC motor drives, Fifth International Conference on Industrial Automation, Montréal, Quebec, Canada, June 11–13, 2007.Google Scholar
48. 48.
T. Soderstom and P. Stoica, System Identification, Prentice-Hall, 1989.Google Scholar
49. 49.
V. Solo, Robust identification and large deviations, in Proc. 35th IEEE CDC, Kobe Japan, Dec. 1996, 4202–4203.Google Scholar
50. 50.
V. Solo, More on robust identification and large deviations, in Proc. IFAC99, Beijing, P.R. China, June, 1999, 451–455.Google Scholar
51. 51.
V. Solo and X. Kong, Adaptive Signal Processing Algorithms, Prentice-Hall, Englewood Cliffs, NJ, 1995.Google Scholar
52. 52.
K. Takano, K. Nozaki, Y. Saito, A. Negishi, K. Kato, and Y. Yamaguchi, Simulation study of electrical dynamic characteristics of lithium-ion battery, J. Power Sour., 90 (2000), 214–223.
53. 53.
O. Tremblay and L.A. Dessaint, Experimental validation of a battery dynamic model for EV applications, World Electric Vehicle Journal, vol. 3, at 2009 AVERE, EVS24 Stavanger, Norway, May 13–16, 2009.Google Scholar
54. 54.
V.I. Utkin and H.-C. Chang, Sliding mode control on electromechanical systems, Mathmatical Problems in Engineering, 8 (2002), 451–473.
55. 55.
S.R. Venkatesh and M.A. Dahleh, Identification in the presence of classes of unmodelled dynamics and noise, IEEE Trans. Automatic Control, 42 (1997), 1620–1635.
56. 56.
A. Widodo, M.-C. Shim, W. Caesarendra, and B.-S. Yang, Intelligent prognostics for battery health monitoring based on sample entropy, Expert Systems Appl., 38 (2011) 11763–11769.
57. 57.
L.Y. Wang and G. Yin, Persistent identification of systems with unmodeled dynamics and exogenous disturbances, IEEE Trans. Automat. Control, 45 (2000), 1246–1256.
58. 58.
L.Y. Wang and G. Yin, Asymptotically efficient parameter estimation using quantized output observations, Automatica, 43 (2007), 1178–1191.
59. 59.
L.Y. Wang and G. Yin, Quantized identification with dependent noise and Fisher information ratio of communication channels, IEEE Trans. Automat. Control, 53 (2010), 674–690.
60. 60.
L.Y. Wang, G. Yin, and J.F. Zhang, Joint identification of plant rational models and noise distribution functions using binary-valued observations, Automatica, 42 (2006), 535–547.
61. 61.
L.Y. Wang, G. Yin, J.F. Zhang, and Y.L. Zhao, Space and time complexities and sensor threshold selection in quantized identification, Automatica, 44 (2008), 3014–3024.
62. 62.
L.Y. Wang, G. Yin, J.-F. Zhang, and Y.L. Zhao, System Identification with Quantized Observations: Theory and Applications, Birkhäuser, Boston, 2010.
63. 63.
L.Y. Wang, J.-F. Zhang, and G. Yin, System identification using binary sensors, IEEE Trans. Automat. Control, 48 (2003), 1892–1907.
64. 64.
C.Z. Wei, Multivariate adaptive stochastic approximation, Ann. Statist. 15 (1987), 1115–1130.
65. 65.
B. Widrow, J. Glover, J. McCool, J. Kaunitz, C. Williams, R. Hearn, J. Zeidler, E. Dong and R. Goolin, Adaptive noise cancelling: principles and applications, IEEE Proc., 63 (1975), 1692–1716.
66. 66.
G. Yin, S. Kan, L.Y. Wang, and C.Z. Xu, Identification of systems with regime switching and unmodelled dynamics, IEEE Trans. Automat. Control, 54 (2009), 34–47.
67. 67.
G. Yin, V. Krishnamurthy, and C. Ion, Regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization, SIAM J. Optim., 14 (2004), 1187–1215.
68. 68.
G. Yin, L.Y. Wang, and S. Kan, Tracking and identification of regime-switching systems using binary sensors, Automatica, 45 (2009), 944–955.
69. 69.
G. Yin and Q. Zhang, Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications, Springer, New York, 2005.
70. 70.
G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010.
71. 71.
G. Zames, On the metric complexity of causal linear systems: ε-entropy and ε-dimension for continuous time, IEEE Trans. Automatic Control, 24 (1979), 222–230.
72. 72.
H. Zheng, H. Wang, L.Y. Wang, and G. Yin, Time-shared channel identification for adaptive noise cancellation in breath sound extraction, J. Control Theory Appl., 2 (2004), 209–221.
73. 73.
H. Zheng, H. Wang, L.Y. Wang, and G. Yin, Cyclic system reconfiguration and time-split signal separation with applications to lung sound pattern analysis, IEEE Trans. Signal Processing, 55 (2007), 2897–2913.
74. 74.
X.Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: a continuous-time model, SIAM J. Control Optim., 42 (2003), 1466–1482.
75. 75.
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155–1179.

© Qi He, Le Yi Wang, and G. George Yin 2013

## Authors and Affiliations

• Qi He
• 1
• Le Yi Wang
• 2
• G. George Yin
• 3
1. 1.Department of MathematicsUniversity of CaliforniaIrvineUSA
2. 2.Department of Electrical and Computer EngWayne State UniversityDetroitUSA
3. 3.Department of MathematicsWayne State UniversityDetroitUSA