Direct adaptive fuzzy control for nonlinear systems with time-varying delays☆
Introduction
In recent years, considerable attention has been paid to fuzzy logic control. Some fuzzy logic control techniques have been developed, for instance, see [3], [15], [16], [23], [35], [36] and the references therein. Approximation-based adaptive fuzzy control or neural control has emerged as a popular and convenient tool in analysis and synthesis of complex and ill-defined systems, to which the application of conventional control techniques is not straightforward or feasible. The main idea of such a control methodology is to utilize fuzzy logic systems or neural networks to approximate the unknown nonlinearities in dynamic systems and the adaptive backstepping technique to construct the controllers. Following such an idea, some adaptive neural control or fuzzy control schemes were proposed for nonlinear delay-free systems with strict-feedback structure. In [1], [4], [6], [37], the adaptive neural tracking control was addressed for nonlinear delay-free systems in strict-feedback form. The corresponding adaptive fuzzy control schemes were proposed in [11], [20], [31], [32], [33], [38], [25].
All the aforementioned studies do not deal with state time-delays, which often occur in many dynamic systems, such as rolling mill systems, biological systems, metallurgical processing systems, network systems, and so on. It is well known that the existence of delays usually becomes the source of instability and performance deterioration of systems. Stability analysis and control synthesis of Takagi–Sugeno (T–S) fuzzy delayed systems were proposed in [2], [12], [17], [26], [24], respectively. Especially, approximation-based adaptive control has also been addressed for nonlinear systems with time-delay. In [5], the problem of adaptive fuzzy tracking control was addressed for nonlinear strict-feedback systems with time-delays. The proposed tracking controller guarantees that all the closed-loop signals remain bounded, while the output converges to a neighborhood of the reference signal. By using a wavelet fuzzy network online approximation model, the stabilization problem was discussed for nonlinear strict-feedback systems with time-delay in [7]. In [21], the adaptive control was investigated via backstepping and fuzzy networks technique. In [8], the observer-based adaptive neural controller was designed for a special class of nonlinear time-delay systems. By using fuzzy logic systems to approximate the unknown nonlinear functions, the adaptive fuzzy control scheme was proposed in [28] for nonlinear delayed systems with lower-triangular form. The main advantage of the results proposed in [28] is that the controllers contain fewer adaptation laws.
Usually, there are two approaches for stability analysis of time-delay systems. The first one is based on the Lyapunov–Krasovskii functionals [5], [7], [28], and the second one on the Lyapunov-Razumikhin functionals [9], [13], [18], [34], [19], [27]. The Lyapunov–Krasovskii functional methods usually make stability analysis and controller design more complex than the Razumikhin functional approach. All the existing results on the approximation-based adaptive control are obtained by using the Lyapunov–Krasovskii functional approach. To the authors’ knowledge, there have not been any reported results on the use of the Razumikhin functional approach for this purpose. In addition, all the existing adaptive fuzzy or neural controllers were proposed only for the nonlinear systems with constant delays. Therefore, in theory, the existing adaptive fuzzy controllers are not capable of dealing with systems with time-varying delays. Thus, it is of importance to develop a new adaptive fuzzy control approach for nonlinear systems with time-varying delays.
The above observations motivate the present research, in which we investigate adaptive fuzzy control for nonlinear systems with time-varying delays. By utilizing the Razumikhin functional approach and fuzzy backstepping technique, a novel direct adaptive fuzzy controller is proposed. The proposed adaptive fuzzy controller guarantees that all the signals of the closed-loop system are bounded, while the system output converges to a small neighborhood of the reference signal. Three main contributions are made in this paper. (1) A Razumikhin lemma-based adaptive fuzzy controller design procedure is first proposed. Compared with the existing results, the proposed fuzzy controllers can handle systems with both constant and time-varying delays. (2) A novel direct adaptive fuzzy control approach is presented. Unlike the existing approximation-based adaptive control methods, the direct adaptive fuzzy control technique employs fuzzy logic systems to approximate the desired but unknown control input signals rather than the nonlinear functions in dynamic systems. As a result, the direct adaptive fuzzy controllers have a simpler form and fewer tuning parameters. Furthermore, it is more convenient to implement the controllers in practice. (3) Different from the existing adaptive fuzzy control methods, for an nth order system the proposed control scheme contains only n adaption laws, no matter how many fuzzy rule bases are used. Therefore, the computation burden is reduced, and the algorithm is easily realized in real-time. In addition, the prior knowledge of the upper bound of the virtual control coefficients is not required.
The rest of the paper is organized as follows. Section 2 provides preliminaries and the formulation of the problem. Section 3 develops an adaptive fuzzy controller design procedure based on the Razumikhin functional approach and backstepping technique. Section 4 presents four examples to illustrate the effectiveness of the proposed controllers. These are followed by conclusions in Section 5.
Section snippets
Problem formulation and preliminaries
Consider the nonlinear time-delay system described bywhere and are the state variable, the control input, and the output of the system (1), respectively, denotes the time-varying delay in the ith equation, and and are unknown smooth nonlinear functions satisfying . For ,
Control design and stability analysis
In this section, a backstepping-based adaptive fuzzy controller design procedure will be developed. For the purpose of simplicity, the time variable t is omitted from the corresponding functions and is denoted as .
Step 1. Let . It follows from the first differential equation of the system (1) that:
Consider a Lyapunov function candidate aswhere . Differentiating yields:
Simulation
In this section, four examples are used to illustrate the effectiveness of the method developed in this paper. In all the four examples, the reference signal is chosen as . The proposed adaptive fuzzy controller will be used to control the systems to track the given reference signal. Example 1 Consider the following 2nd-order nonlinear system proposed in [7].
In [7], stabilization was addressed for the case of and being
Conclusion
In this paper, the problem of adaptive fuzzy tracking control has been addressed for a class of nonlinear strict-feedback systems with time-varying delays. Based on the Razumikhin lemma, an adaptive fuzzy controller has been constructed. The proposed controller ensures that all the signals of the resulting closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin. A main feature of the proposed fuzzy control design scheme is that the proposed
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This work was supported in part by the national science foundation of China (Nos. 60674055 and 60774047).