Robust Adaptive Fuzzy Controller Design for Wing Rock Syetems
Chih-Min Lin* Chun-Fei Hsu
* Department of Electrical Engineering, Yuan-Ze University,
Chung-Li, Tao-Yuan, 320, Taiwan, Republic of China
e-mail:
Abstract: In this study, a robust adaptive fuzzy control (RAFC) system is proposed. The RAFC system is comprised of a computation controller and a robust controller. The computation controller containing a neural-network-estimator is the principal controller and the robust controller is designed to achieve tracking performance. An on-line tuning method is derived to tune the parameters of the neural network for estimating the controlled system dynamic function. To investigate the effectiveness of the RAFC, the design methodology is applied to control a wing rock system. The phenomenon of the wing rock system is manifested by a limit cycle oscillation predominantly in rolling motion for an aircraft operating at subsonic speeds and high angles of attack. Simulation results demonstrate that the proposed RAFC system can achieve favorable tracking performances for the wing rock system.
Key-Words: Fuzzy control, Robust control, Adaptive law, Neural network, Limit cycle, Wing rock system.
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1 Introduction
Many high-performance aircraft often require for operating at subsonic speeds and high angles of attack. They may become unstable or enter into a self-induced limit cycle oscillation, mainly rolling motion known as wing rock [2,3]. This wing rock is a concern because it may have adverse effects on maneuverability, tracking accuracy, and operational safety. A series of papers have considered the control of the wing rock system based on adaptive control technique [6]. In the adaptive techniques, the knowledge of the structure of the aerodynamic functions is required; however, the structure is difficult to obtain.
The application of fuzzy set theory to control problems has been the focus of numerous studies [4]. The fuzzy control is a model free design method; however, it has not been view as a rigorous approach due to the lack of formal synthesis techniques that can guarantee the system stability. To tackle this problem, some researches have been directed by the use of the Lyapunov synthesis approach to construct a stable adaptive fuzzy control [1, 8, 10]. The adaptive fuzzy control proposed in [1] has combined the attenuation technique in the fuzzy controller design. In recently developed adaptive fuzzy control schemes, only the consequence parts of fuzzy rules are tuned. To increase the learning ability, some researches used a radial basis function (RBF) neural network to tune the premise and consequence parts of the fuzzy rules simultaneously [5, 9].
This paper develops a robust adaptive fuzzy control (RAFC) system to achieve tracking performance for the wing rock system. Both the control design and the adaptive fuzzy control approach have been employed together to design the new control system. By the adaptive fuzzy control approach, this study uses a RBF neural network to approximate the system dynamic function. All the neural network parameters are tuned based on the Lyapunov function, thus the stability of the system can be guaranteed. By the control design technique, the approximation error can attenuate to a desired small level. Finally, simulation results demonstrate that the developed RAFC system can achieve robust tracking performance for the wing rock system with unknown system dynamic.
2 Description of Radial Basis Function Neural Network
A RBF neural network maps according to
(1)
where , and contain the input variables and the output variable of the RBF neural network, respectively; represents the connective weight between the hidden layer and the output layer; represents the firing weight of the k-th neuron in the hidden layer; and and are the center and width of the activation function, respectively. The firing weight can be represented as
(2)
where
(3)
or
(4)
in which is the inverse radius of the radial basis function. For simplicity of discussion, define the vectors and collecting all parameters of the hidden layer as
(5)
. (6)
Then, the output of the RBF neural network can be represented in a vector form as
(7)
where , and . It has been proven that there exists an RBF approximator of (7) such that it can uniformly approximate a nonlinear even time-varying function . By the universal approximation theorem, there exists an optimal RBF approximator such that [8]
(8)
where denotes an approximation error, and are the optimal parameter vectors of and , respectively, and and are the optimal parameter vectors of and , respectively. The optimal weighting vectors , and that are needed to best approximate a given nonlinear function are difficult to determine. An estimated RBF approximator is defined as
(9)
where and are the estimated vectors of and , respectively, and and are the estimated vectors of and , respectively. Define the estimated error as
(10)
where and . In the following, some tuning laws will be developed to on-line tune the parameters of the RBF approximator to achieve favorable estimation. The Taylor expansion linearization technique is employed to transform the nonlinear function into a partially linear form, i.e. [9]
(11)
or
(12)
where , , is a vector of higher-order terms, , , and and are defined as
(13)
. (14)
Substituting (12) into (10), gives
(15)
where and are used since they are scales; and the sum of matching error .
3 Problem Statement of Wing Rock Systems
The nonlinear motion equation for an 80-deg slender delta wing has been developed by Elzebda et al. as [2]
(16)
where is the roll angle, an over-dot denotes a derivative with respect to time, is the control effort which represents the aileron deflection, is the density of air, is the freestream velocity, is the wing reference area, is the chord, is the mass moment of inertia, and is the roll moment coefficient. The roll-moment coefficient is written as [2]
.
(17)
The aerodynamic parameters have been provided that they are nonlinear functions for different angles of attack [2]. Substituting (17) into (16), system (16) can be rewritten as
(18)
where and the parameters , are given by . To observe the dynamic behavior, the open-loop system time response with was simulated for two initial conditions: a small initial condition (, ) and a large initial condition (, ). The phase-plane plot is shown in Fig. 1. For the small initial condition a limit cycle oscillation exists, and for the large initial condition the roll angle is divergent.
Fig. 1. Phase-plane portraits of the uncontrolled wing rock system.
4 Robust Adaptive Fuzzy Control of Wing Rock Systems
The control problem is to find a control law so that the output can track the command trajectory , where the tracking error vector is defined as
. (19)
Assume that the parameters of the wing rock system in (18) are well known, there exits an ideal controller as [7]
. (20)
Substituting (20) into (18), gives the following equation
. (21)
If and are chosen to correspond to the coefficients of a Hurwitz polynomial, it implies . However, the system dynamic function in (20) is a nonlinear time-varying function and it can’t be exactly obtained, so can not be implemented. An RBF approximator will be used to observe the unknown system dynamic function . By the universal approximation theorem, there exists an optimal RBF neural network such that [8]
(22)
where is the optimal weight vector of the RBF neural network and denotes the approximation error. However, the optimal weight vector is difficult to determine. Define the weight estimated error vector as
(23)
where is the estimated vector of the optimal vector . While appears, the following tracking performance is requested [1]
(24)
where and . The and are given positive weighting matrices, is a design gain, and is a prescribed attenuation level. If the system starts with initial conditions and , then the performance in (24) can be rewritten as
(25)
where the -gain from to the tracking error must be equal to or less than .
For achieving a favorable tracking performance and an arbitrarily small attenuation level simultaneously, a RAFC system is shown in Fig. 2 with the controller
(26)
in which the computation controller is chosen as
(27)
with the dynamic approximator chosen as
. (28)
The estimated vectors , and are the optimal vectors of, and , respectively. By substituting (26) into (18) and using (20), the tracking error dynamic equation can be obtained as follows:
(29)
where , and . Using (15), equation (29) can be rewritten as
.
(30)
Therefore, the following theorem can be stated.
Theorem 1: Consider a wing rock system represented by (18), if the robust adaptive fuzzy control system is designed as (26), in which the adaptation laws of the system dynamic function approximator are designed as in (31) ~ (33), and the robust controller is designed as in (34), then the stability of the system can be guaranteed.
(31)
(32)
(33)
(34)
where , and are the learning rates with positive constants, is a positive weighting factor, and positive matrix is the solution of the following Riccati-like equation
(35)
if and only if or .
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Proof: Refer to [1], [5] and [9].
Therefore, for a prescribed in tracking control, in order to guarantee the solvability of tracking performance, the weight on control law of (34) must satisfy the above inequality. Then, the tracking performance in (24) can be achieved for a prescribed attenuation level .
5 Simulation Results
The parameters for the aerodynamic coefficients in (18) are given in [6]. Solve the Riccati-like equation (35) with , it is obtained that
and. (36)
It should be emphasized that the derivation of RAFC system does not need to use the aerodynamic parameters and the structure of the aerodynamic functions. The system parameters are used only for simulations. To investigate the effectiveness of the developed control system, two initial conditions are simulated. The RBF neural network with 7 hidden layer neurons in rule layer is utilized to approach the wing rock system dynamics. The learning rates are selected as , , . To attenuate to a small level via tracking design technique, the simulation results of the RAFC system with for small and large initial conditions are shown in Fig. 3. The state responses are shown in Figs. 3(a) and 3(b); and the associated control efforts are shown in Figs. 3(b) and 3(d) for small and large initial conditions, respectively. From these simulation results, it can be seen that robust tracking performance can be achieved without any knowledge of system dynamic functions.
Fig. 2. Block diagram of robust Adaptive fuzzy control wing rock system.
Fig. 3. The time responses of robust adaptive fuzzy control wing rock system with .
6 Conclusions
This paper developed a robust fuzzy adaptive control (RAFC) system to attenuate the effects of the dynamic function approximation error on the tracking performance using tracking technique. The RAFC system is comprised of a computation controller and a robust controller. The computation controller including a neural-network-estimator is used to mimic the system dynamic function and the robust controller is used to attenuate the effects of the approximation error. Finally, the developed RAFC system is applied to control a wing rock system to demonstrate its effectiveness.
Acknowledgment
This work was supported by the National Science Council of the Republic of China under Grant NSC 90-2213-E-155-016.
References:
[1] B.S. Chen, C.H. Lee, tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach, IEEE Trans. Fuzzy Systems, Vol. 4, 1996, pp. 32-43.
[2] J.M. Elzebda, A.H. Nayfeh, D.T. Mook, Development of an analytical model of wing rock for slender delta wings,” Journal of Aircraft, Vol. 26, 1989, pp. 737-743.
[3] C.E. Lan, Y. Chen, K.J. Lin, Experimental and analytical investigations of transonic limit-cycle oscillations of a flaperon, Journal of Aircraft, Vol. 32, 1995, pp. 905-910.
[4] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller-part I/II, IEEE Trans. Systems, Man, and Cybernetic, Vol. 20, 1990, pp. 404-435.
[5] Y. Li, N. Sundararajan, P. Saratchandran, Stable neuro-flight-controller using fully tuned radial basis function neural networks, Journal of Guidance, Control, and Dynamics, Vol. 24, 2001, pp. 665-674.
[6] S.N. Singh, W. Yim, W.R. Wells, Direct adaptive and neural control of wing-rock motion of slender delta wings, Journal of Guidance, Control, and Dynamics, Vol. 18, 1995, pp. 25-30.
[7] J.E. Slotine, W. Li, Applied Nonlinear Control, Englewood Cliffs: NJ: Prentice-Hall, 1991.
[8] L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1994.
[9] S.D. Wang, C.K. Lin, Adaptive tuning of fuzzy controller for robots, Fuzzy Set and Systems, Vol. 110, 2000, pp. 351-363.
[10] C.C. Wong, B.C. Huang, J.Y. Chen, Rule regulation indirect adaptive fuzzy controller design, IEE Proc. Control Theory Applications, Vol. 145, 1998, pp. 513-518.
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