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1488 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 Adaptive Tracking Control for Nonaffine Nonlinear Systems with Zero Dynamics Hui Hu Dept of Electrical and Information Engineering, Hunan Institute of Engineering, Hunan Xiangtan, China Email: onlymyhui@126.com Peng Guo Dept of Computer Science, Hunan Institute of Engineering, Hunan Xiangtan, China Email: da_peng219@126.com Abstract—A direct adaptive neural network tracking control scheme is presented for a class of nonaffine nonlinear systems with zero dynamics. The method does not assume boundedness on the time derivative of a control effectiveness term. Parameters in neural networks are updated using a gradient descent method which designed in order to minimize a quadratic cost function of the error between the unknown ideal implicit controller and the used neural networks controller. The final updated law is a nonlinear function of output error. No robust control term is used in controller. The convergence of parameters and the uniformly ultimately bounded of tracking error and all states of the corresponding closed-loop system are demonstrated by Lyapunov stability theorem.Simulation results illustrate the availability of this method. Index Terms—nonaffine nonlinear, neural network, tracking control, gradient descent method, zero dynamic I. INTRODUCTION Modern mechanical or electrical systems that are to be controlled become more and more complicated and, thus, their mathematical models are often hard to be established. In recent years, adaptive neural network [1, 2, 3, 4, 7, 8, 9, 10, 12, 15] that model the functional mechanism of the human brain and fuzzy logic control [5, 6, 11, 13, 14, 16, 17, 18] that can cooperate with human expert knowledge have been successfully applied to many control problems because they need no accurate mathematical models of the system under control. These methodologies become especially more helpful if control of highly uncertain, nonlinear and complex systems is the design issue. The main philosophy that is exploited heavily in system theory applications is the universal function approximation property of neural networks or fuzzy logic. Benefits of using neural networks or fuzzy logic for control applications include its ability to effectively control nonlinear plants while adapting to unmodeled dynamics. In fact, most of the works [1-4, 6, 9, 11, 12, 13, 15-18, 22-25]are devoted to the control problem of the affine-incontrol nonlinear systems, i.e., systems characterized by inputs appearing linearly in the system state equation. Few results are available for nonaffine nonlinear systems where the control input appears in a nonlinear fashion [5, 7, 8, 10, 14, 19, 21]. In general, a two-step procedure is taken in nonaffine nonlinear system. First, based on implicit function theorem an ideal controller is developed to stabilize the underlying system and makes the tracking approach a neighborhood of zero. Then, a neural network or fuzzy logic to approximate this ideal controller is designed. Based on the Lyapunov stability analysis, an adaptation law is devised to update the adjustable parameters. However a bounding controller may also be added for more performance robustness. In the above most methods the parameter adaptation laws are designed based on a Lyapunov approach, where an error signal between the desired output and the actual output is used to update the adjustable parameters and the control laws are composed of three control terms: a linear control term, an adaptive neural network control term and a robust control term used to compensate for disturbances and approximation errors. On the other hand, almost all of the above works don’t consider the zero dynamics, though it plays an important role in nonlinear system control. Considering that zero dynamics exist in many practical systems, including isothermal continuous stirred tank reactors, aircraft trajectory tracking control systems and others, it is necessary to investigate their influence on nonlinear system. In the paper, according to [5], we introduce a direct adaptive neural network control approach for a class of nonaffine nonlinear systems with zero dynamics. The basic idea is to use neural network to adaptively construct an unknown ideal controller and the parameter adaptive laws is designed, based on the gradient descent method, to directly minimizing the error between the unknown ideal controller and the neural network controller And no robust control term is used in controller. This paper proves the availability of the method in both theory and simulation experiment. The paper is organized as follows. First, the problem is formulated in Section II. Zero dynamics is given in III. Designing a control law with on-line tuning of neural network weighting factors is given in Section IV. In Section V, convergence and stability analysis of control © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.6.1488-1495
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1489 system is given. In Section VI, simulation results are presented to confirm the effectiveness and applicability of the proposed method. Finally, conclusions are included. A. Notations and Preliminaries The following notations and definitions will extensively be used throughout the paper. Let be the real number, n n× m and represent the real n-vectors and the real n× m matrices, respectively. i denotes the usual Euclidean norm of a vector. In the case where y is a scalar, y denotes its absolute value and if Y is a matrix, Y means Frobenious norm defined as Y T { } = tr Y Y .where tr{ i} stands for trace operator. Implicit Function Theorem: Assume that n m n h : × → is continuously differentiable at each n m ab of an open set S ⊂ × a , b be a point ( , ) . Let ( 0 0) point in S for which ( , ) h a b and for which the 0 0 Jacobian matrix ⎡∂h ⎤( a , b ) ⎣ ∂a⎦ 0 0 is nonsingular. Then n m there exist neighborhoods U ⊂ of a0 and V ⊂ of b0 such that for each b∈V the equation h( a, b ) = 0has a unique solution a∈ U . Moreover, the solution can be given as a = g( b) where g is continuously differentiable at b= b0 . II. PROBLEM FORMALATION Consider the following SISO nonaffine nonlinear system [8]: where [ ] ⎧ dξi ⎪ = ξi+ 1 i =1, , r−1 dt ⎪ dξr ⎪ = h( ξη , , u) ⎨ dt ⎪ dη ⎪ = q(, ξη, u) ⎪ dt ⎪ ⎩ y = ξ1 , , T r ξ = ξ1 ξ r ∈ Rξ ⊂ , (1) n−r η∈R η ⊂ are system states and u ∈Ωu ⊂ , y ∈ are system input and output respectively. h(, ξη, u) is a smooth partially known function, and q(, ξη , u) is a smooth partially known vector field. The control objective is to design an adaptive neural network controller for a class of SISO nonaffine nonlinear systems (1) such that the system output follows a desired trajectory while all signals in the closed-loop system remain bounded. ∂h(, ξη, u) Assumption 1: The function hu (, ξη , u) = ∂u is nonzero and bounded for all (, ξη, u) ∈ × × R R . This implies that h (, ξη , u) is strictly either positive or u ξ η negative for all ( ξη , , u) ∈ × × R R .Without loss of generality, it is assumed that it exists a constant c such that hu (, ξη , u) ≥ c > 0. Define the reference vector ( r−1) T r y = ( y y y ) ∈R ξ d d d d The reference signal y d and its time derivative are assumed to be smooth and bounded. We also define the tracking error as e= yd − y and corresponding error vector as ( r−1) T r e = (,, e e e ) ∈ R . Assumption 2: When the desired output y d and its r- order derivative are of known bound, there exists a positive constant bd to satisfy (1) ( r−1) T ( yd yd yd ) ≤ bd Then the error equation is as follows: η (2) ( r) e = A0 e + b⎡ ⎣yd −h(, ξη, u) ⎤ ⎦ (3) ⎡ 0 1 0 0⎤ ⎡0⎤ ⎢ 0 0 1 0 ⎥ ⎢ ⎢ ⎥ 0 ⎥ ⎢ ⎥ where A0 = ⎢ ⎥ , b = ⎢0⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 1⎥ ⎢ ⎥ ⎢ ⎣ 0 0 0 0 0⎥ ⎦ ⎢ r × r ⎣1⎥ ⎦ r × 1 A b is controllable, then there will exist a Obviously, ( , 0 ) constant matrix [ , , ] T 0 1 r 1 T c K = k k k − which makes eigenvalues of matrix A = A0 − bK all have negative real part. Thus, for any given positive definite symmetric matrix Q , there exists a unique positive definite symmetric solution P to the following Lyapunov algebraic equation: A P+ PA = − Q (4) T c Define a signal T ( r) T ⎛b Pe ⎞ ν= yd + K e + λtanh⎜ ⎟ ⎝ Ξ ⎠ where tanh( •) ∈− ( 1,1) is the hyperbolic tangent function, Ξ,λ are the positive design parameters, when error T ⎛bPe⎞ e →+ ∞ , the value of tanh ⎜ ⎟ → + ∞ , and when ⎝ Ξ ⎠ T ⎛b Pe ⎞ error e →-∞ the value of tanh ⎜ ⎟ → - ∞ . ⎝ Ξ ⎠ T ⎛b Pe ⎞ When e → 0 , tanh ⎜ ⎟ → 0 .The term ⎝ Ξ ⎠ T ⎛b Pe ⎞ λ tanh ⎜ ⎟ is a smooth approximation of the ⎝ Ξ ⎠ T discontinuous term sign( b Pe ) c λ usually used in robust © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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