The indirect adaptive regulation of unknown nonlinear dynamical systems is considered in this paper. The method is based on a new neuro-fuzzy dynamical system (neuro-FDS) definition, which uses the concept of adaptive fuzzy systems (AFSs) operating in conjunction with high-order neural network functions (FHONNFs). Since the plant is considered unknown, we first propose its approximation by a special form of an FDS and then the fuzzy rules are approximated by appropriate HONNFs. Thus, the identification scheme leads up to a recurrent high-order neural network (RHONN), which however takes into account the fuzzy output partitions of the initial FDS. The proposed scheme does not require a priori experts' information on the number and type of input variable membership functions making it less vulnerable to initial design assumptions. Once the system is identified around an operation point, it is regulated to zero adaptively. Weight updating laws for the involved HONNFs are provided, which guarantee that both the identification error and the system states reach zero exponentially fast, while keeping all signals in the closed loop bounded. The existence of the control signal is always assured by introducing a novel method of parameter hopping, which is incorporated in the weight updating law. Simulations illustrate the potency of the method and comparisons with conventional approaches on benchmarking systems are given. Also, the applicability of the method is tested on a direct current (dc) motor system where it is shown that by following the proposed procedure one can obtain asymptotic regulation.
In this paper we analyze the identification problem which consists of choosing an appropriate identification model and adjusting its parameters according to some adaptive law, such that the response of the model to an input signal (or a class of input signals), approximates the response of the real system for the same input. For identification models we use fuzzy-recurrent high order neural networks. High order networks are expansions of the first-order Hopfield and Cohen-Grossberg models that allow higher order interactions between neurons. The underlying fuzzy model is of Mamdani type assuming a standard defuzzification procedure such as the weighted average. Learning laws are proposed which ensure that the identification error converges to zero exponentially fast or to a residual set when a modeling error is applied. There are two core ideas in the proposed method: (1) Several high order neural networks are specialized to work around fuzzy centers, separating in this way the system into neuro-fuzzy subsystems, and (2) the use of a novel method called switching parameter hopping against the commonly used projection in order to restrict the weights and avoid drifting to infinity.
This paper proposes a direct adaptive neuro-fuzzy controller to address the adaptive tracking problem for a class of affine nonlinear multi-variable multi-input (MVMI) unknown systems that are linearizable by nonlinear state feedback. The proposed control scheme uses a recurrent neuro-fuzzy model to approximate the system, which combines an underlying fuzzy model with the approximation abilities of high-order neural networks to produce the fuzzy recurrent high-order neural network approximator. A hybrid control scheme that combines an adaptive feedback linearization term and a sliding mode term improves system performance by suppressing the influence of external disturbances and approximation errors. The existence and boundedness of the control signal is always assured by employing the novel method of parameter modified hopping and incorporating it in weight updating laws making the closed-loop system Lyapunov stable. The case of SISO systems is considered separately by following similar analysis with the more general MVMI case. Simulations performed on well-known benchmarks demonstrate the effectiveness of the proposed control scheme.Ã .The aforementioned expression will be negative by appropriately selecting K r so that K r > N , assuming k k 2 =.k k 2 C ı ' B) We also consider the following alternative tracking signal x d D .sin.2t /, cos.t /, 2/ T with initial conditionwhereas for (69),The disturbances added in (70) were selected as random values in the interval OE 1, 1.The main parameters for the control law (17) and the learning laws (33) and (35) are selected as A D d iag. 4.17, 0.96, 6.59/ K D d iag.143.77, 146.75, 106.64/ DIRECT ADAPTIVE NEURO-FUZZY TRAJECTORY TRACKING 683We also consider the following initial conditions for (69) and (72):x 0 D OE 1.5, 1, 4 T and´d 0 D OE 1, 11, 3 T . The disturbances added in 70 were selected as random values in the interval OE 1, 1. The main parameters for the control law (17) and the learning laws (33) and (35) are selected as A D d iag. 3.31, 0.22, 0.04/ K D d iag.66.97, 130.31, 145.65/The parameters of the sigmoidal that have been used are˛1 D 0.68,ˇ1 D 0.35, and 1 D 0.91. Also, the learning rate is selected appropriately as d f l D 0.058 and the outer boundary for the adaptation of weights as " l i D 250 with Ä l,outer f i
In this paper, we investigate the indirect adaptive regulation problem of unknown affine in the control nonlinear systems. The proposed approach consists of choosing an appropriate system approximation model and a proper control law, which will regulate the system under the certainty equivalence principle. The main difference from other relevant works of the literature lies in the proposal of a potent approximation model that is bilinear with respect to the tunable parameters. To deploy the bilinear model, the components of the nonlinear plant are initially approximated by Fuzzy subsystems. Then, using appropriately defined fuzzy rule indicator functions, the initial dynamical fuzzy system is translated to a dynamical neuro-fuzzy model, where the indicator functions are replaced by High Order Neural Networks (HONNS), trained by sampled system data. The fuzzy output partitions of the initial fuzzy components are also estimated based on sampled data. This way, the parameters to be estimated are the weights of the HONNs and the centers of the output partitions, both arranged in matrices of appropriate dimensions and leading to a matrix to matrix bilinear parametric model. Based on the bilinear parametric model and the design of appropriate control law we use a Lyapunov stability analysis to obtain parameter adaptation laws and to regulate the states of the system. The weight updating laws guarantee that both the identification error and the system states reach zero exponentially fast, while keeping all signals in the closed loop bounded. Moreover, introducing a method of "concurrent" parameter hopping, the updating laws are modified so that the existence of the control signal is always assured. The main characteristic of the proposed approach is that the a priori experts information required by the identification scheme is extremely low, limited to the knowledge of the signs of the centers of the fuzzy output partitions. Therefore, the proposed scheme is not vulnerable to initial design assumptions. Simulations on selected examples of well-known benchmarks illustrate the potency of the method.
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