Self-structuring fuzzy-neural backstepping control with a B-spline-based compensator
Introduction
Over the past two decades, neural networks-based (NNs) control has attracted growing popularity as an adaptive control approach for engineering problems [1], [2], [3]. With the property of approximating nonlinearity to arbitrary accuracy by sufficient hidden neurons and learning, the NN-based controller design has been applied to cope with the increasing demands for handling complexity, uncertainty, or unknown dynamics. Based on the interactions within layers, the NN-based method has widely been recognized as a powerful tool for industrial applications [1], [4], [5], [6]. The characteristics of approximation, parallelism, and adaptation also imply their potentials for real-time machine intelligence realization [4], [5], [6], [7]. Recently, the alternative of fuzzy neural network (FNN) aroused much interest for the enhancement of network performance. With the proved ability of universal approximation [8], [9], FNN is regarded as a complementary approach and possesses the advantages from low-level learning and computation to high-level human-like reasoning. The property of linguistic expression also enables the control strategy being formed in a rule-based manner which is derived from observation and expert knowledge. With the fuzzified nodes embedded in the layers, the learning abilities of network can be upgraded [10], [11], [12].
Although it is widely accepted that the parameterized neural network is capable of approximating linear or nonlinear mapping by adequately choosing network structures and training methods, however, a challenging problem for designers is to select an appropriate structure for balancing the number of rules and the approximation accuracy. If the network size is chosen too small, it is impossible to assure the approximation to converge to an acceptable level due to the limited nodes. On the other hand, if over-determined nodes are given, the computational burden is huge and the waste of computation resource implies its impracticality for real-world applications. To tackle the problem of structure optimization, the self-organizing (or self-learning) methods have aroused much interest [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. For those approaches, the increasing rule construction concept was the first proposed [13], [14], [15], [16], where the new node is generated when the current structure fails to cover the input space. Park et al. [17] presented a self-structuring neural network, where the over-activated nodes in the parameter learning will be split into two separated nodes to cover the input space as evenly as possible. The similar concept was also adopted in [12]. However, the increasing node evolution implies the redundancy since it does not remove the ineffective nodes. In order to find the optimal or sub-optimal rule structure through rule adding and rule pruning procedures, Wu et al. proposed a dynamic mechanism for adjusting the nodes dynamically for giving the appropriate input space interpretation [18], [19] and other similar approaches were also proposed [20], [21]. However, the heavy computational load for node significance determination is unavoidable. In [13], [14], [15], [16], [21], the discussion of control stability during system evolution was not provided.
From the control point of view, if the exact model of the controlled system is well-defined there will be an ideal control scheme to achieve favorable performance by canceling the system dynamics. However, the ideal model is difficult to obtain with the involvements of internal parameters variation and external disturbances for real applications. To compensate for the approximation imperfection, the compensation mechanism such as the chattering control and robust compensation are introduced. However, the chattering phenomenon of the sliding mode control strategy is unfavorable. To reduce the adverse effect, the saturation function was proposed but it loses the guarantee of system stability. In the past decades, an approximation error bound estimation mechanism was proposed to estimate the bound of approximation error so that the chattering phenomenon of the control effort can be reduced [1]. However, the adaptive law for estimating the error bound tends to make the bound going to infinity if the main controller is ill-designed. For the NN-based control collaborates with the robust compensation control [10], the control performance is subject to the predetermined attenuation level. If an inappropriate attenuation level is given, the control effort may lead to undesirable large signal which is hard for realization.
Backstepping control is known as a systematic and recursive design methodology with the flexibility to avoid cancellations of useful nonlinearities [22], [23], [24], [25], [26]. It provides an alternative feedback design in terms of pseudo-control, where the appropriate state and virtual control are derived from the state variables in a recursive way to form smaller subsystems. Based on this manipulation, the final Lyapunov functions are represented by these subsystems associated with each individual pseudo-control stage [27]. The rapid development of backstepping design was also witnessed where the backstepping method is combined with other methods such as sliding mode control, robust control, fuzzy control, neural network, and so on [28], [29], [30], [31]. For these approaches, the backstepping-based control method provides the flexibility to accommodate unmodeled nonlinearity, where the asymptotic stability and robustness to some unmatched uncertainties can be guaranteed.
In this paper, a self-structuring fuzzy-neural backstepping control system (SSFNBS) is proposed. The SSFNBS comprises two elements: the self-structuring fuzzy neural network-based (SSFNN) backstepping controller and the B-spline-based compensation controller. Based on the backstepping control theory, the SSFNN-based observer is exploited as the observer to approximate the controlled system dynamics and the B-spline-based compensation controller is used to compensate for the approximation error caused by SSFNN. To facilitate the network design, a structure learning mechanism is introduced, where the node-adding and node-pruning are involved to network structure determination. In this paper, a new B-spline-based compensator is also introduced. With the recurrence operation of the given knot vector and control points, the generated B-spline functions are exploited to generate the compensation signal. With the adoption of B-spline concept and control point adjusting mechanism, the drawbacks of the conventional compensation controllers can be freed. Based on the evolution mechanism, the structure learning and the parameter estimation can be conducted simultaneously. In the SSFNBS, the parameters are evolved by the means of the Lyapunov function to ensure the system stability. Thus, an intelligent control approach is presented. To validate the capabilities of the proposed approach, the SSFNBS is applied to several control problems. Through the simulation results the advantages of the proposed SSFNBS can be observed.
Section snippets
Problem formulation
For simplicity, the general form of a second order nonlinear system is considered in this paper:where is the state vector of the system, which is assumed to be available for measurement, f(x) is the unknown continuous functions, and u is the input. The control objective is to find a control law so that the state trajectory x can track a trajectory command xc closely. To achieve successful backstepping control, some appropriate functions of stale variables are selected as
Fuzzy neural network observer
The fuzzy neural network-based observer is shown in Fig. 1. Layer 1 accepts the input variables. Layer 2 is used to calculate the membership values, where the Gaussian function is embedded in this layer as the membership function. The nodes of layer 3 are regarded as the fuzzy rules. The links before layer 3 represent the premise. Layer 4 is the output layer, where the links of layer 4 represent the consequence and the node in this layer is the output of the fuzzy neural network. Based on the
Simulation results
In this section, the proposed SSFNBS is applied to nonlinear system, chaotic system and wing rock motion control problems to verify its effectiveness. It should be emphasized that the development of the SSFNBS does not need to know the system dynamics of the controlled system. For practical implementation, the structure and parameters of SSFNN can be online tuned by the proposed methods to estimate the controlled system dynamics. In the simulation, the parameters for SSFNBS are selected with n
Conclusions
The determination of the network size is difficult because the designed structure varies from the experience of designers. The consequential approximation imperfection of an ill-designed controller also increases the designing difficulty of the compensation controller. In this paper, the developed SSFNN observer is capable of dealing with the tradeoff between the approximation accuracy and computation complexity. With the proposed B-spline-based compensator, the disadvantages of the traditional
Acknowledgment
The author appreciates the financial support from the National Science Council of Republic of China under grant NSC 100–2917-I-564-017. The authors would like to express their gratitude to the Reviewers for their valuable comments and suggestions.
Kuo-Hsiang Cheng was born in Taipei, Taiwan, ROC, in 1978. He received B.S. degree in automatic control engineering from the Feng Chia University and the Ph.D. degree in electrical engineering from the Chang Gung University, Taiwan, in 2000 and 2006, respectively. During 2006–2011, he was a researcher of the Mechanical and Systems Research Laboratories (MSL), Industrial Technology Research Institute (ITRI), Hsinchu, Taiwan, ROC. In 2011, he received the grant from National Science Council (NSC)
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Kuo-Hsiang Cheng was born in Taipei, Taiwan, ROC, in 1978. He received B.S. degree in automatic control engineering from the Feng Chia University and the Ph.D. degree in electrical engineering from the Chang Gung University, Taiwan, in 2000 and 2006, respectively. During 2006–2011, he was a researcher of the Mechanical and Systems Research Laboratories (MSL), Industrial Technology Research Institute (ITRI), Hsinchu, Taiwan, ROC. In 2011, he received the grant from National Science Council (NSC) and currently he is conducting his postdoctoral research in the Department of Mechanical Engineering, University of California, Berkeley. His research interests include fuzzy logic, neural networks, intelligent systems and control, and intelligent vehicles.