非線性未知系統之適應性模糊─類神經控制研究(I)

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非線性未知系統之適應性模糊-類神經研究(1/3)

計劃編號:NSC 89-2213-E-011-072 執行期限:88 年 8 月 1 日至 89 年 7 月 31 日 計劃主持人:李祖添台灣科技大學電機系教授

一、中文摘要

本研究報告主要針對狀態變數不可量測的非線性連續系統，發展以估測器(observer) 為基礎的強健適應模糊類神經控制器。在設計輸出控制律時，假設非線性系統只具有輸出可量測，而不允許對輸出作微分，以避免數值微分造成雜訊的放大。本研究報告主要可分成二個部份，第一部份，我們利用模糊類神經網路，針對某一組單一輸入輸出非線性系統，設計適應估測器，此適應模糊類神經估測器並不像文獻上[44-45]所提的適應估測器，受制於受估測系統中的非線性項必須是輸出的函數，換言之，此適應模糊類神經估測器可針對具有非線性項是狀態變數的函數之非線性系統進行估測。第二部份，針對某一組單一輸入輸出非線性系統，發展以估測器為基礎的強健適應模糊類神經控制器，此適應模糊類神經控制器保證具有閉迴路系統穩定與所含有訊號皆為有限的特性。

最後，藉由一些如倒單擺穩定和追蹤的問題、機械手臂軌跡追蹤等問題的模擬，來驗證本文所提的方法確實可行。

二、計劃緣由與目的

Adaptive control of systems has been an active area of research for at least a quarter of a century [1-12]. For linear systems, there has been some research on the stability analysis of adaptive control systems, design of adaptive observers and adaptive control of plants [2-3].

Also, many researchers focus on robust adaptive control that it guarantees signal boundedness in the presence of modeling errors and bounded disturbances [4-6]. For nonlinear systems, some adaptive control schemes via feedback linearization have been reported [7-11]. The fundamental ideal of feedback linearization is to transform a nonlinear system dynamics into a linear one. Then, linear control techniques are employed to acquire desired performance. Based on feedback linearization, some preliminary results on the adaptive control have been presented in [8] for a class of nonlinear continuous-time systems. Since feedback linearization technique critically depends on exact cancellation of the nonlinear terms in input-output map, some approaches [12-13] have

been developed to avoid this difficulty by finding parameter-independent diffeomorphisms to transform nonlinear systems to canonical forms for which stable adaptive control methods can be developed.

Several methods have been proposed for the adaptive control of nonlinear systems in which all the system states are not accessible and information needed for the generation of the control input has to be obtained from the observed inputs and outputs [14-19]. In [14-15], the authors have addressed the problem of designing global adaptive output-feedback tracking controls for single-input single-output (SISO) nonlinear systems. [16] has presented a semiglobal adaptive output feedback controller that ensures the output of the system tracks any given reference signal that is bounded and has bounded derivatives up to the nth order, where n is the order of the system. A new set of tools proposed in [20] interlaces the design of observer and parameter estimator with the design of feedback control in order to avoid overparametrization.

The adaptive observers which require simultaneous estimation of parameters and states have been proposed for nonlinear systems [21-22]. In [22], the author has addressed the problem of necessary and sufficient conditions given for a nonlinear system to be transformed by state-space change of coordinates into the special adaptive observer form used in [21] to design adaptive observers. In the special adaptive observer form, the unknown nonlinearities are restricted to be function of the systems. The restrictive growth has been removed by [23].

Recently, since neural networks [24] and fuzzy logic [25] are universal approximators, some adaptive control schemes of nonlinear systems via fuzzy logic [26-28] or neural networks [29-32] have been proposed. Likewise, for a class of nonlinear discrete-time systems, adaptive control using neural networks has been proposed in [33] by feedback linearization.

Since fuzzy set was introduced by Zadeh in 1965, it has received much attention. Over the past two decades, fuzzy logic has been applied to some control problems. Specifically, some theoretical issues about fuzzy logic control have

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been discussed in [34-35]. Also, some identification algorithms via fuzzy logic [36-37]

have been proposed. A mathematical tool to fuzzily describe a system has been proposed in [37], which identifies the system by using its input-output data. This approach suffers from off-line tuning. A fuzzy logic identifier proposed in [36] does not need off-line preprocessing. An indirect adaptive fuzzy controller presented in [26] guarantees stability of the overall system.

Fuzzy rules of the indirect adaptive fuzzy controller are generated automatically during the adaptive procedure. Furthermore, if the fuzzy descriptions of the system are available, they can be incorporated into the designs.

Neural networks are also applied to several control problems. Based on the approximation capabilities of neural networks, identification and control schemes [29-32] have been proposed.

In [29], the general stability analysis for approximators has been developed. This method can be directly applicable to other types of approximators. The stability and convergence properties of recurrent high-order neural networks as models of nonlinear dynamical systems have been developed in [30]. Singular perturbation analysis employed to investigate the stability and robustness properties of dynamical neural network identifier has been presented in [31]. A dynamic recurrent neural-network-based adaptive observer for SISO nonlinear systems has been presented in [38]. In [39], an output feedback controller has been developed based on a high-gain observer used to estimate the time derivatives of the system output.

More recently, though the neural network and fuzzy logic are universal approximators, each has different characteristics. The former possesses characteristics of fault-tolerance, parallelism and learning, while the latter has characteristics of linguistic information and logic control. Therefore, both of them have the complementary characteristics. The algorithms of fuzzy-neural control [40-41] have been proposed. Neural-network-based fuzzy logic control has been developed in [41].

The H_∞ control approach provides both robust stability and disturbance attenuation with H∞ -norm bound for closed-loop uncertain linear systems and for the closed-loop uncertain nonlinear systems. Therefore, the motivations for introducing H_∞ optimal control into neural network control and fuzzy logic control have been presented in [42] and [43], respectively.

Both of them used H_∞ tracking performance design to achieve the desired attenuation of modeling errors. In [42], the influence of modeling error (and disturbance) can be attenuated to a desired value for adaptive fuzzy

control systems by using H_∞ control design technique, under the constraint that the system states must be available for measurement.

The goal of this report is to study observer-based robust adaptive fuzzy-neural control of continuous time nonlinear dynamical systems. In designing the output feedback control law, the system output is assumed to be measured, but no differentiation of the system output is allowed in order to avoid the noise amplification associated with numerical differentiation.

Consider the nth nonlinear dynamical system of the form

,

) ) g(

) (f(

x C

x x B Ax x y T

d u

=

+ + +

=

&

where













=

0 0

1 0

0 0

0 1

0 0

1 0

K K

K K K K K

K K

A ,













= 1 0 0 0

M B ,













= 0 0 0 1

M C ,

T

xn

x

x ]

[ ⁽⁻¹⁾

= &L

x =[x₁x₂Lx_n]^T∈ℜⁿ is a vector of states, d is the external bounded disturbance, and u∈ℜ and y∈ℜ are the control input and system output, respectively.

)

f(x and g(x) are uncertain (unknown) nonlinear functions. Only the system output y is assumed to be measurable. There are two primary objectives of report: 1) Constructing an adaptive fuzzy-neural observer for the nonlinear system. 2) Developing an observer-based robust adaptive fuzzy-neural controller for the nonlinear system.

三、研究方法與成果

The proposed methods are obtained by the strictly-positive-real Lyapunov (SPR- Lyapunov) design approach and the H_∞ control technique.

The primary contribution of this report is discussed as follows.

1) An adaptive observer for a class of SISO nonlinear systems is proposed by using fuzzy-neural networks.

Unlike previous results [44-45], the assumption that the system nonlinearities only are restricted to the system output is not required.

Specifically, the fuzzy-neural observer does not assume that nonlinearities in the system are restricted to the system output only.

2) We develop an observer-based robust adaptive fuzzy-neural controller for a class of SISO nonlinear systems under the constraint that only the system

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output is available for measurement.

Furthermore, an observer-based output-feedback control law and update law to tune on-line the weighting factors of the adaptive fuzzy-neural controller are derived.

The observer-based output-feedback control law and update law provide robust stability for the closed-loop system, and guarantee that all signals involved are bounded.

四、結論與討論

In this report, the adaptive fuzzy-neural observer and observer-based robust adaptive fuzzy-neural controller tuned on-line for a class of SISO nonlinear systems is developed by using SPR-Lyapunov design approach and the H_∞ control technique. The system nonlinearities approximated by fuzzy-neural networks are not assumed to be functions of the system states which are available for measurement. The overall adaptive scheme provides robust stability for the closed-loop system and guarantees that all signals involved are bounded.

五、參考文獻

[1] R.V. Monopoli, "Model reference adaptive control with an augmented error signal," IEEE Trans. Auto. Control., vol.

AC-19, pp.474-484, 1974.

[2] K.S. Narendra and A.M. Annaswamy, Stable adaptive systems, Englewood Cliffs, NJ: Prentice-Hall, 1989.

[3] S.S. Sastry and M. Bodson, Adaptive control: stability, convergence, and robustness. Englewood Cliffs, NJ:

Prentice-Hall, 1989.

[4] P.A. Ioannou and K.S. Tsakalis, "A robust direct adaptive controller," IEEE Trans.

Automat. Contr., vol. AC-31, pp.

1033-1043, Nov. 1986.

[5] P.A. Ioannou and A. Datta, "Robust adaptive control: a unified approach,"

Proceedings of the IEEE, vol.79. no.12, pp.1735-1768, December, 1991.

[6] P.A. Ioannou, and J. Sun, Robust adaptive control. Englewood Cliffs, NJ:

[7] A. Isidori, Nonlinear control systems.

New York: Springer-Verlag, 1989.

[8] J.J.E. Slontine and W. Li, Applied nonlinear control. Englewood Cliffs, NJ:

[9] S.S. Sastry and A. Isidori, "Adaptive control of linearizable systems," IEEE Trans. Automat. Contr., vol. 34, pp.

1123-1131, 1989.

[10] S. Monaco and D. Normand-Cyrot,

"Minimum-phase nonlinear discrete-time

systems and feedback stabilization," in Proc. IEEE Conf. Decision and Control, pp. 979-986, 1987.

[11] B. Jakubczyk, "Feedback linearization of discrete time systems," Syst. Contr. Lett., vol. 9, pp. 411-416, 1987.

[12] D.G. Taylor, P.V. Kokotovic, R. Marino, and I. Kanellakopoulos, "Adaptive regulation of nonlinear systems with unmodeled dynamics," IEEE Trans.

Automat. Contr., vol. 34, pp. 405-412, 1989.

[13] I. Kanellakopoulos, P.V. Kokotovic, and A.S. Morse, "Systematic design of adaptive controllers for feedback linearizatble systems," IEEE Trans.

Automat. Contr., vol. 36, no. 11, pp.

1241-1253, 1991.

[14] R. Marino and P. Tomei, "Globally adaptive output-feedback control of nonlinear systems, part I: linear parameterization," IEEE Trans. Automat.

Contr., vol. 38, no. 1, pp. 17-32, 1993.

[15] R. Marino and P. Tomei, "Globally adaptive output-feedback control of nonlinear systems, part II: nonlinear parameterization," IEEE Trans. Automat.

Contr., vol. 38, no. 1, pp. 33-48, 1993.

[16] H.K. Khalil, "Adaptive output feedback control of nonlinear systems represented by input-output models," IEEE Trans.

177-188, 1996.

[17] I. Kanellakopoulos, P.V. Kokotovic, and A.S. Morse, "Adaptive output-feedback control of a class of nonlinear systems," in Proc. IEEE Conf. Decision and Control, pp. 1082-1087, 1981.

[18] I. Kanellakopoulos, P.V. Kokotovic, and A.S. Morse, "A toolkit for nonlinear feedback design," Syst. Contr. Lett., vol.19, pp.177-185,1992.

[19] B. Aloliwi and H.K. Khalil, "Adaptive output-feedback regualtion of a class of nonlinear systems: convergence and robustness," IEEE Trans. Automat. Contr., vol. 42, no. 12, pp. 1714-1716, 1997.

[20] I. Kanellakopoulos, M. Krstic, and P.V.

Kokotovic, "Interlaced controller-observer design for adaptive nonlinear control,"

Proceedings of the 1992 American Control Conference, pp. 1337-1347.

[21] R. Marino, "Adaptive observers for single output nonlinear systems," IEEE Trans.

1054-1058, 1990.

[22] G. Bastin and M. Gevers, "Stable Adaptive observers for nonlinear time varying systems," IEEE Trans. Automat.

Contr., vol. 33, pp. 650-658, 1988.

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[23] K. Krstic and P.V. Kokotovic, "Adaptive nonlinear output-feedback scheme with Marino-Tomei controller," IEEE Trans.

[24] K. Hornik, M. Stinchcombe, and H. White,

"Multilayer feedforward networks are universal approximators," Neural Networks, no. 2, pp. 359-366, 1989.

[25] L.X. Wang and J.M. Mendel, "Fuzzy basis functions, universal approximation, and orthogonal least squares learning," IEEE Trans. on Neural Networks, vol. 3, no. 5, pp. 807-814, 1992.

[26] L.X. Wang, Adaptive fuzzy systems and control: design and stability analysis. Englewood Cliffs, NJ: Prentice-Hall, 1994.

[27] L.X. Wang, "Stable adaptive fuzzy control of nonlinear systems," IEEE Trans. Fuzzy.

Syst. vol. 1, no. 2, pp. 146-160, May.

1993.

[28] M. Jamshidi, N. Vadiee, and T. J. Ress, Fuzzy logic and control. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[29] M.M. Polycarpou and P.A. Ioannou,

"Modelling, identification and stable adaptive control of continuous-time nonlinear dynamical systems using neural networks," Proc. American Control Conf, pp. 36-40,1992.

[30] E.B. Kosmatopoulos, P.A. Ioannou, and M.A. Christodoulou, "Identification of nonlinear systems using new dynamic neural network structures," in Proc. IEEE Conf. Decision and Control, pp. 20-25, 1992.

[31] G.A. Rovithakis, and M.A. Christodoulou,

"Adaptive control of unknown plants using dynamical neural networks," IEEE Trans. Syst. Man, Cyber., vol. 24, no. 3, pp. 400-411, 1995.

[32] F.C. Chen and H.K. Khalil, "Adaptive control of nonlinear systems using neural networks," Int.J.Contr., vol. 55, no.3, pp.

1299-1317, 1992.

[33] R.M. Sanner and J.J.E. Slontine,

"Guassian networks for direct adaptive control," IEEE Trans. on Neural Networks, vol. 3, no. 6, pp. 837-863, 1992.

[34] C.C. Lee, "Fuzzy logic in contreol systems: fuzzy logic controller-part I and part II," IEEE Trans. Syst. Man, Cyber., vol. 20, no. 2, pp. 404-435, 1990.

[35] K. Liu and F.L. Lewis, "Some issues about fuzzy logic control," in Proc. IEEE Conf. Decision and Control, pp.

1743-1748, 1993.

[36] K. Liu and F.L. Lewis, "Adaptive tuning of fuzzy logic identifier for unknown

nonlinear systems," Int. J. Adaptive control and Signal Processing, vol. 8, pp.

573-586, 1994.

[37] T. Takagi, and M.Sugeono, "Fuzzy identification of systems and its applications to modeling and control,"

IEEE Trans. Syst. Man, Cyber.,15, pp.116-132, 1985.

[38] Y.H. Kim, F.L. Lewis, and C.T. Abdallah,

"A dynamic recurrent

neural-network-based adaptive observer for a class of nonlinear systems,"

Automatica, vol. 33, no, 8, pp. 1539-1543, 1997.

[39] T. Zhang, S.S. Ge, and C.C. Hang,

"Adaptive output feedback control for general nonlinear systems using multilayer neural networks," Proceedings of the 1998 American Control Conference, pp. 520-524.

[40] S. Horikawa, T. Furuhashi, and Y.

Uchikawa, "On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm," IEEE Trans.

on Neural Networks, vol. 3, no. 5, September, 1992.

[41] C.T. Lin and C. S. Georege Lee,

"Neural-network-based fuzzy logic control and decision system," IEEE Trans.

on Computer, vol. 40, no. 12, pp.

1320-1336, December, 1991.

[42] B.S. Chen, C.H. Lee, and F.Y. Shiau, "An H∞ tracking performance design via adaptive neural network control in uncertain nonlinear systems," IEEE Interna. Symposium on Artificial neural networks, Tainan, Taiwan, 1994.

[43] B.S. Chen, C.H. Lee, and Y.C. Chang,

" H_∞ tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach," IEEE Trans. Fuzzy System, vol.

4, no. 1, pp. 32-43, February, 1996.

[44] R. Marino and P. Tomei, "Global adaptive observers for nonlinear systems via filtered transformations," IEEE Trans.

[45] R. Marino and P. Tomei, "Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems,"

IEEE Trans. Automat. Contr., vol. 40, no.

7, pp. 1301-1304, 1995.