臺灣博碩士論文加值系統

本論文主要在探討連續時間參數識別的兩大主題，一為遭受非隨機
干擾下的參數識別，另一為時變參數的識別。除了量側雜訊外，一般
系統的輸出通常會被干擾所汙染，而這些干擾包含感測器的誤差、模
型誤差以及系統遭受外部擾動而引發。大部分的系統識別方法僅考慮
干擾為白雜訊，當出現以上的干擾時會造成參數估側偏差。在作參數
化時，我們可以將不同來源的干擾總和於一個干擾於輸出端。在本論
文中， 我們將提出一個離線識別以及兩個線上及時識別的方法來處理
此問題。
在離線識別的方法中，未知的干擾將由一有限項的傅立葉餘弦級數
來表示。此級數係數為未知。藉由結合級數的基底函數及原本的回歸
向量(regressor) 可得到一組擴展的回歸向量，藉由最小平方法做批量計
算，即可得到包含係數及參數在內的估測值。最後並提出此擴展的回
歸向量於持續刺激性(persistent excitation)的必要條件。
在第一個線上估測方法中，其架構是建立在梯度演算法上。在干擾
的影響下，為了使參數的誤差方程式可以收斂到零，必須作額外的控
制補償。於控制設計上，將使用平均法來得到近似系統，並利用H1頻
率成型來合成控制器。此控制訊號將可追蹤干擾訊號並將之抵消，因
此可保證估測參數可收斂到正確值。
第二個線上估測的方法為使用狀態估測器。為了將干擾納入估測器
作估測，於此我們提出一種干擾產生濾波器，並將其模型加入原參數
狀態方程式中。利用卡曼濾波器(Kalman filter)作狀態估測。相對於傳
統的內部模型法(internal model approach)，此新方法將可適用於更廣泛
類型的干擾。以上提出的三個方法可同時估測出參數及干擾。
以上兩種線上估測的設計方法經過一些調整後可運用在時變參數識
別的問題上。其細節將於本文中作描述。

關鍵字: 強韌參數識別，時變參數識別，干擾識別，卡曼濾波器。

Two subjects of continuous-time parameter identification problems expressed in linear regression form are discussed in this thesis. One is the time-invariant parameter identification while subject to non-stochastic disturbances termed as the robust identification. The other is the time-variant parameter identification.

In addition to the measurement stochastic noise, the output signal of a system is usually contaminated with the non-stochastic disturbances which are usually resulted from errors of measure devices, system unmodled dynamics or the process disturbances acting on the system. Most identifications considering the disturbance as a white noise will have biased estimates while subject to these kinds of disturbances. In the parameterization, one can lump all the disturbances into one disturbance term at the output expressed in linear regression form. We proposes one off-line approach and two on-line approaches to deal with this problem.

In the off-line approach, the unknown disturbance will be approximately expanded by a finite Fourier cosine series with unknown coefficients. The unknown coefficients and the known basis functions will be augmented to the original parameter vector and the regressor respectively. With the expanded regressor, one can obtain the estimates of the expanded parameter vector by adopting the least-squares batch calculation. A necessary condition on persistent excitation of the expanded regressor is proposed too.


In the first of the two on-line approaches, the estimation scheme is built under the structure of gradient algorithm. A compensation is made to reject the effect of the disturbance in the estimation error dynamics by designing a stabilized controller. In the design procedure, the averaging method is used for system approximation and the $H_{infty}$ frequency shaping methodology is utilized to synthesize the controller. The control signal will be able to track the disturbance signal and cancel it in the estimation error dynamics and that guarantees the convergence of the parameter estimation.

In the second of the on-line approaches, an state-observer based estimator is constructed. To include the estimation of the disturbance into the estimation scheme, the system plant is augmented with the model of the proposed disturbance generating filter also termed as dynamics extension filter. The Kalman filter is adopted to perform the states estimation. Compared with the conventional internal model approach, the proposed method could be applied to a more general disturbance class. The three proposed approaches can identify both parameters and the disturbance simultaneously.

The design procedures of the above two on-line approaches can be grafted to the time-variant parameter identification problem with some modifications. Special consideration will be addressed in the context.

Keywords: Robust identification, Time-variant parameter identification, Disturbance
identification, Kalman filter.

口試委員會審定書i
致謝iii
中文摘要v
Abstract vii
1 Introduction 1
1.1 Continuous-Time Identification . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Identification with The Disturbance . . . . . . . . . . . . . . . . . . . . 2
1.3 Time-Variant Parameter Identification . . . . . . . . . . . . . . . . . . . 6
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Preliminary 9
2.1 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Approximation : Perturbation Method . . . . . . . . . . . . . . . . . . . 14
2.3 Approximation : Averaging Method . . . . . . . . . . . . . . . . . . . . 16
3 Identification Problem 19
3.1 Parameterization in Linear Regression Form . . . . . . . . . . . . . . . . 19
3.2 Identifier Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Gradient and Least-Squares Algorithm . . . . . . . . . . . . . . . . . . . 22
3.4 Identifier Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Persistent Excitation and Exponential Parameter Convergence . . . . . . 26
4 Off-Line Parameter Identification Subject to Deterministic Disturbance 27
4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 End-Continuous Disturbance . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 End-Discontinuous Disturbance . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Persistent Excitation Condition of Expanded Regressor . . . . . . . . . . 39
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 On-Line Parameter Identification Subject to Deterministic Disturbance 45
5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Estimator Joining The H1 Design . . . . . . . . . . . . . . . . . . . . . 46
5.2.1 Estimator Structure . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Approximation via Averaging Process . . . . . . . . . . . . . . . 49
5.2.3 H1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Estimator Joining The Disturbance Generating filter . . . . . . . . . . . . 81
5.3.1 Estimator Structure . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.2 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 83
5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.4 Transfer Function Analysis . . . . . . . . . . . . . . . . . . . . . 114
5.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6 On-Line Time-Variant Parameter Identification 127
6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Estimator Joining The H1 Design . . . . . . . . . . . . . . . . . . . . . 128
6.2.1 Estimator Structure . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.2 Approximation via Averaging Process . . . . . . . . . . . . . . . 131
6.2.3 H1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3 Estimator Joining The Parameter Generating Filter . . . . . . . . . . . . 159
6.3.1 Estimator Structure . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3.2 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 161
6.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Bibliography 183

[1] K. J. Astr‥om and P. Eykhoff, “System identification - a survey,” Automatica, vol. 7,
no. 2 pp.123-162, 1971.
[2] P. C. Young, “Parameter identification of continuous-time models,” Automatica, vol.
17, no. 1 pp.23-39, 1981.
[3] H. Unbehauen and G. P. Rao “Continuous-time approaches to system identification
- a survey,” Automatica, vol. 26, no. 1 pp.23-35, 1990.
[4] H. Unbehauen and G. P. Rao “Identification of continuous-time systems - a tutorial,”
Proc. 11th IFAC Symposium on Identification (SYSID’97), Fukuoka, Japan, pp.1023-
1049, July 1997.
[5] H. Unbehauen and G. P. Rao “A review of identification in continuous systems,”
IFAC Annu. Rev. Control, vol. 22 pp.145-171, 1998.
[6] H. Unbehauen and G. P. Rao “Identification of continuous systems - a survey,” Syst.
Anal. Model. Simul., vol. 33 pp.99-155, 1998.
[7] D. C. Saha and G. P. Rao, Identification of Continuous Dynamical Systems - The
Poisson Moment Functional (PMF) approach, LNCIS, vol. 56, Springer Verlag,
Berlin, 1983.
[8] H. Unbehauen and G. P. Rao Identification of Continuous systems, North Holland,
Amsterdam 1987.
[9] N. K. Sinha and G. P. Rao Identification of Continuous systems-methodology and
computer implementation, Kluwer, Doredrecht 1991.
[10] S. Sagara , Z. J. Yang, and K. Wada, “Identification of Continuous systems from
noisy sampled input-output data,” Proc. 9th IFAC/IFORS Symp. Identification Syt.
Parameter Estimation, Budapest, Hungary, pp.603-608, 1991.
[11] Z. Y. Zhao, S. Sagara ,and K. Kumamaru “On-line identification of time delay and
system parameters of continuous system based on discrete-time measurements,”
Proc. 9th IFAC/IFORS Symp. Identification Syt. Parameter Estimation, Budapest,
Hungary, pp.721-726, 1991.
[12] S. Mukhopadhyay, A. Patra, and G. P. Rao “A new class of discrete-time models for
continuous-time systems,” Int. J. Control, vol. 55, no. 5, pp. 1161-1187, 1992.
[13] T. S‥odestrom, H. Fan, B. Carlson, and S. Bigi “Least squares parameter estimation
of continuous-time ARX models from discrete-time data,” IEEE Trans. Autom.
Control, vol. 42, no. 5 pp. 659-673, 1997.
[14] K. R. Godfrey “Correlation method,” Automatica, vol. 16, pp. 527-534, 1980.
[15] P. E. Wellstead “Nonparametric methods of system identification,” Automatica, vol.
17, 1981.
[16] I. D. Landau Adaptive Control (The Model Reference Approach) Marcel Dekker,
New York.
[17] L. Ljung, System Identification, Theory for The User, Prentice-Hall, 1987.
[18] T. S‥oderstr‥om and P. Stoica, System Identification, Prentice Hall, 1989.
[19] L. Y. Wang and G. G. Yin, “Closed-loop persistent identification of linear systems
with unmodeled dynamics and stochastic disturbances,” Automatica, vol. 38,
pp.1463-1474, 2002.
[20] G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons, Chapter 3.7
and 3.8, 2003.
[21] F. M. Lee, I. K. Fong, and L. C. Fu, “ Stable on-line parameter identification algorithms
for systems with non-parametric uncertainties and disturbances,” International
Journal of Control, vol. 65, no. 2, pp. 329-345, 1996.
[22] E. Walter and H. Piet-Lahanier, “ Estimation of parameter bounds from boundederror
data: a survey,” Mathematics and Computers in Simulation, vol. 32, pp. 449-
468, 1990.
[23] M. Milanese and A. Vicino, “Optimal estimation theory for dynamic systems with
set membership uncertainty: An overview,” Automatica, vol. 27, pp. 997-1009,
1991.
[24] E. W. Bai, Y. Ye, and R. Tempo, “Bounded error parameter estimation: a sequential
analytic center approach,” IEEE Transactions on Automatic Control, vol. 44, no. 6,
pp. 1107-1117, June 1999.
[25] S. S. Wilson and C. L. Carnal, “System identification with disturbances,” Proceedings
of the 26th Southeastern Symposium on System Theory, pp. 502-506, 1994.
[26] K. J. Astrom and B. Wittenmarkm, Adaptive Control, Addison Wesley, Chapter 3,
1989.
[27] P. O’Neil, Advanced Engineering Mathematics, Thomson, Brooks/Cole, Chapter 13,
2003.
[28] P. Eykhoff, System Identification, Wieley, New York, 1974.
[29] Z. Y. Zhao, S. Sagara, and M. Tomizuka, “A new bias compensating LS method for
continuous system identification in the presence of coloured noise,” Int. J. Control,
vol. 56, no. 6, pp. 1441-1452, 1992.
[30] Z. J. Yang, S. Sagara, and K. Wada, “Identification of continuous time systems
from sampled input-output data using biased eliminating techniques,” Control Theory
Adv. Tech, vol. 9, no. 1, pp. 53-75, 1993.
[31] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness,
Prentice-Hall, 1989-1994.
[32] Enrique A. Gonz’alez-Velasco Fourier Analysis and Boundary Value Problems, Elsevier,
1995.
[33] R. Lozano, D. Dimoqianopoulos, and R. Mahony, “Identification of linear timevarying
systems using a modified least-squares algorithm,” Automatica, vol. 36, pp.
1009-1015, 2000.
[34] M. Salgado, G. C. Goodwin, and R. Middleton, “Modified least squares algorithm
incorporating exponential resetting and forgetting,” International Journal of Control,
vol. 47 no. 2, pp. 477-491, 1988.
[35] S. Haykin, Adaptive Filter Theory (4th edition) N.J. , Prentice Hall, 2002.
[36] Maciej Nied’zwiecki, Identification of Time-varying Processes, New York, Wiley,
2000.
[37] R. E. Lawrence and H. Kaufman, “The Kalman Filter for the Equalization of a Digital
Communications Channel,” IEEE Transactions on Communication Technology,
vol. COM-19, pp. 1137-1141, 1971.
[38] S. Ungarala and T. B. Co, “Time-varying system identification using modulating
functions and spline models with application to bio-processes,” Computers and
Chemical Engineering, vol. 24 pp. 2739-2753, 2000.
[39] M. Nied’zwiecki, “Recursive functional series modeling estimators for identification
of time-varying plants : more bad news than good ?” IEEE Trans. Automat. Control,
vol. 35, pp. 610-616, 1990.
[40] M. Nied’zwiecki and T. Klaput, “Fast Recursive Basis Function Estimators for Identification
of Time-Varying Processes,” IEEE Transactions on signal processing, vol.
50, pp. 1925-1934, 2002.
[41] G. Davidov, A. Shavit, and Y. Korean, “Estimation of dynamical-varying parameters
by the internal model principle,” IEEE Trans. Automat. Control, vol. 37, pp. 498-
503, 1992
[42] B. A Francis and W.M. Wonham, “The internal model principle of control theory,”
Automatica, vol. 12, pp. 457-465, 1976.
[43] M. K. Tsatsanis and G. B. Giannakis, “Modeling and equalization of rapidly fading
channels,” Int. J. Adaptive Contr. Signal Process., vol. 10, pp. 159–176, 1996.
[44] K. H. Chon, H. Zhao, R. Zou, and K. Ju, “Multiple time-varying dynamic analysis
using multiple sets of basis functions,” IEEE Trans. Biomed. Eng., vol. 52, no. 5,
pp. 956–960, May 2005.
[45] M. Abu-Naser and G. A. Williamson, “Convergence properties of adaptive estimators
of time-varying linear systems using basis functions,” Proc. 12th IEEE Digital
Signal Processing Workshop, pp. 336-341, 2006.
[46] M. Abu-Naser and G. A. Williamson, “Convergence of adaptive estimators of timevarying
linear systems using basis functions: continuous-time results,” Acoustics,
Speech and Signal Processing, 2007. ICASSP 2007. IEEE International Conference
on, vol. 3 pp. III-1361-III-1364 April 2007.
[47] B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods,
Prentice Hall, 1990.
[48] H. K. Khalil, Nonlinear Systems, Prentice Hall, 2002.
[49] K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice Hall, 1998.
[50] S. Skogestad and I. Postlethwaite, Multivariable feedback control: analysis and design,
John Wiley, 2005.