Delaylowerboundbased adaptive fuzzy memory control for uncertain nonlinear systems with state and input delays
 Shuni Song^{1}Email author,
 Jingyi Liu^{1} and
 Dan Liu^{2}
https://doi.org/10.1186/s1366201609753
© Song et al. 2016
Received: 28 May 2016
Accepted: 19 September 2016
Published: 18 October 2016
Abstract
This paper is concerned with the problem of adaptive fuzzy control for a class of uncertain nonlinear strictfeedback systems. The considered systems contain uncertain state delay and input delay simultaneously. By utilizing the mean value theorem, the unknown timedelayed functions related to all state variables are dealt with. A novel adaptive filter is designed to eliminate the effect of the timevarying input delay. Based on the backstepping technique, a delaylowerboundbased adaptive fuzzy memory control scheme is developed. It is proved that the proposed adaptive memory control method can guarantee that all the signals in the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin. A simulation example is given to demonstrate the effectiveness of the proposed control scheme.
Keywords
1 Introduction
Over the past decades, adaptive control has become a mature and wellestablished research area within control systems society. Especially, the adaptive backstepping control technique as a powerful tool has received attractive attention for controlling strictfeedback nonlinear systems. This technique has a number of advantages over the conventional approaches, such as dealing with those nonlinear systems without satisfying the matching condition, providing a promising way to improve the transient performance of adaptive systems by tuning the design parameters. Recently, various considerable and significant results on adaptive backstepping control for nonlinear systems have been reported in [1–6], and the references therein. Among them, [1, 4–6] investigated the adaptive backstepping control technique for singleinput and singleout (SISO) nonlinear systems, [2, 3] studied for multipleinput and multipleoutput (MIMO) nonlinear systems. Recently, approximationbased adaptive fuzzy control has been developed to deal with the control problem of nonlinear systems with unknown nonlinear functions [7–9]. Reference [7] tried to consider a class of stochastic purefeedback nonlinear systems with unknown hysteresis by utilizing adaptive fuzzy control method. Reference [8] developed a fuzzy adaptive control approach for nonlinear systems with unknown control gain sign. The backstepping control technique has become one of the most popular design methods for a large class of nonlinear systems. In [10–12], the adaptive fuzzy backstepping control method is applied to deal with switched nonlinear systems. Although the backsteppingbased adaptive technique has been used widely and developed extensively, there exists the issue of ‘explosion of complexity’ in the design process. In order to avoid the problem, dynamic surface control (DSC) technique was first introduced in [13]. The DSC technique was utilized to eliminate the problem of explosion of complexity by introducing a firstorder filter at each step of the traditional backstepping approach [14, 15]. Among them, [14] investigated adaptive dynamic surface control method for nonlinear systems with an unknown dead zone in pure feedback form. Reference [15] proposed adaptive fuzzy backstepping dynamic surface control for uncertain nonlinear systems based on filters. More recently, the DSC approach has been further developed in [16, 17].
Time delays are often found in various engineering systems, such as electrical networks, microwave oscillators, nuclear reactors, etc. It is well known that time delays may destroy the stability or affect the performance of control systems. Therefore, the investigation of the stability and control design of nonlinear timedelay systems is a challenging and meaningful issue, and has received a great deal of attention in the control community in recent years. In order to deal with the stability analysis and controller design for timedelay systems, the common methods for nonlinear systems are to construct the LyapunovRazumikhin functions [18, 19] or appropriate LyapunovKrasovskii functionals [20–24]. In [25], the adaptive robust tracking control problem is considered for an EulerLagrange system possessing secondorder dynamics. In [26], an adaptive fuzzy predictive sliding mode control is presented for nonlinear systems with uncertain dynamics and unknown input delay. In [27], the robust control problem is investigated for a class of uncertain TS fuzzy systems. Then to improve the transient performance of adaptive systems by tuning the design parameters, by further combining appropriately LyapunovKrasovskii functionals with adaptive backstepping technique, adaptive control method is developed for uncertain nonlinear strictfeedback systems with time delays [28–30]. Reference [28] has developed an observerbased adaptive fuzzy control scheme for a class of nonlinear timedelay systems, and [30] tried to solve the timedelay problem in face of a class of perturbed strictfeedback nonlinear systems. Among them, the main idea is to develop the adaptive memoryless controllers, which utilizes appropriately LyapunovKrasovskii functionals to compensate for the unknown timedelay terms [29]. Although much progress has been made for the timedelay systems in the adaptive control field, some challenging difficulties still remain. The main difficulties lie in two folds: First, how to develop adaptive backstepping control scheme for nonlinear strictfeedback systems with time delays by relaxing the restrictions of the timedelayed functions with all state variables. Second, how to design adaptive memoryreliable controller with less conservative for a class of nonlinear delay systems.
In the last several years, for the study of stabilization of timedelay systems, both controllers with or without memory were proposed; see [31–34]. The memoryless controllers have feedback of the current state only, and they are designed to guarantee the delayindependent stability of the closedloop systems. Although the memoryless controllers [31, 32] are easy to implement, they tend to be conservative. In contrast, the memory controllers have a feedback including not only the current states but also the past ones. Hence, it is obvious that the memory controller is less conservative and may achieve a better performance than the memoryless case [34, 35]. Nevertheless, the memory control schemes are only developed for linear timedelay systems [33, 34]. To the best of our knowledge, there are few results for the case where an adaptive fuzzy memory controller is designed for uncertain nonlinear strictfeedback systems with state and input delays.
Motivated by the aforementioned observations, in this paper, the problem of adaptive fuzzy memory control is investigated for a class of uncertain nonlinear strictfeedback systems with state and input delays. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and hyperbolic tangent function is introduced to avoid the controller singularity problem. Compared with the existing work, the main advantages of the proposed control scheme are as follows: (i) By using the mean value theorem, delaylowerbound states are separated from delayed state functions. Based on delaylowerbound states, a novel adaptive fuzzy memory controller is developed. In contrast with the previous memoryless control approach [30, 36–38], by exploiting the history information of states, therefore, the proposed controller is prone to be less conservative. (ii) Timevarying input delay is considered in this paper. By constructing a novel adaptive filter, the effects from input delay are compensated. Note that in existing results [39–41], the considered delayed input is always timeinvariant. Hence the considered systems in this paper are more general and more practical.
Utilizing the mean value theorem, the unknown timedelayed functions related to all state variables are dealt with. A novel adaptive filter is constructed to eliminate the effect of timevarying input delay. For the analysis of stability, a priori knowledge of the bound on the control is put forward. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and hyperbolic tangent function is introduced to avoid the controller singularity problem. Combining an adaptive control methodology with the backstepping technique, a delaylowerboundbased adaptive fuzzy memory controller by considering the history information of states is designed. Finally, the proposed adaptive control scheme can guarantee that all the signals in the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin.
The paper is organized as follows. In Section 2, some preliminaries are presented and the problem is formulated. Adaptive fuzzy memory control design and stability analysis are given in Section 3. A simulation study for a practical example verifies the main results in Section 4. Finally, we conclude this paper in Section 5.
2 Preliminaries and problem statement
2.1 System description
The control objective presented in this paper will design an adaptive fuzzy memory controller to guarantee that all the signals in the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin.
To facilitate the control system design, the following assumptions are proposed.
Assumption 1
Assumption 2
Assumption 3
The desired reference signal \(y_{r}\) and its derivatives \(\dot{y}_{r}\), \(\ddot{y}_{r}\) are bounded, i.e., there exists a compact set \(Y = \{ (y_{r},\dot{y}_{r}, \ddot{y}_{r}): y_{r}^{2}+\dot{y}_{r}^{2}+\ddot{y}_{r}^{2}\leq Y_{R}\}\), where \(Y_{R}\) is a positive constant.
Assumption 4
[39]
Assume that a priori knowledge of the bound on the control is proposed and the finite integral of past control values is bounded by a known constant, i.e., \(\int_{t\tau}^{t}u(s)\,ds\leq u_{0}\).
2.2 Fuzzy logic systems
 \(R^{i}\)::

If \(x_{1}\) is \(F^{i}_{1} \) and \(x_{2} \) is \(F^{i}_{2}\) and … and \(x_{n}\) is \(F^{i}_{q}\), then y is \(B^{i}\), \(i=1,2,\ldots,\iota\),
Let \(\varphi_{i} = \frac{\prod^{q}_{j=1}\mu_{F^{i}_{j}}(x_{j})}{\sum^{\iota}_{i=1}(\prod^{q}_{j=1}\mu_{F^{i}_{j}}(x_{j}))}\), and denote \(\theta=[ \overline{y}_{1},\overline{y}_{2},\ldots,\overline{y}_{\iota}]^{T}=[\theta_{1},\theta_{2},\ldots,\theta_{\iota}]^{T} \) and \(\varphi ^{T}(x)=[\varphi_{1}(x),\ldots,\varphi_{\iota}(x)]\), then FLS (6) can be rewritten as \(y(x)=\theta^{T}\varphi(x)\).
Lemma 1
[42]
Also the fuzzy minimum approximation errors \(\varepsilon_{i}\) and \(\delta _{i}\) are defined as \(\varepsilon_{i}=f_{i}(\underline{x}_{i})\hat{f}_{i}(\underline {x}_{i}\theta_{fi})\), \(\delta_{i}=h_{i}(\underline{ x}_{\tau_{i}}) \hat{h}_{i}(\underline{ x}_{\tau _{i}}\theta_{hi})\), \(1\leq i\leq n\), where \(\varepsilon_{i}\) satisfies \(\varepsilon_{i}\leq\varepsilon_{i}^{*}\), and \(\varepsilon_{i}^{*}\) is an unknown constant and \(\delta_{i}\) satisfies \(\delta_{i}\leq\delta_{i}^{*}\), and \(\delta_{i}^{*}\) is an unknown constant.
2.3 Adaptive filter design
Remark 1
Several results have been developed for inputdelayed nonlinear systems with exact timedelayed knowledge [43, 44], but few results examine the timevarying input delay problem for uncertain nonlinear systems. The considered system in this paper is with timevarying input delay. The adaptive filter is designed with exact timedelayed knowledge, regardless of the effect of the timevarying input delay.
3 Control design and stability analysis
3.1 Adaptive fuzzy memory control design
In this section, an adaptive fuzzy memory control scheme is designed by combining the backstepping method with the DSC technique, which can guarantee that all the signals in the closedloop system are bounded and the tracking error is as small as desired.
Remark 2
Note that \({H_{1}}/{S_{1}}\) is discontinuous at \(S_{1}=0\). Based on the property of the hyperbolic tangent function, i.e., \(\lim_{S_{1}\rightarrow0}\frac{\tanh^{2}(S_{1}/\varepsilon _{1})}{S_{1}}=0\), the function \(\tanh(S_{1}/\varepsilon _{1})\) is introduced to compensate for the singularity problem.
Remark 3
For the design of control and the analysis of the performance, it is always indispensable that a priori knowledge is introduced; see [39, 45]. Reference [39] tried to introduce a priori knowledge for the design of saturated control for an uncertain nonlinear system with input delay. A priori known tracking accuracy is proposed for the design of fuzzyapproximationbased global adaptive control in [45]. In this paper, for the a priori knowledge of the bound on the control one is required to develop an adaptive control methodology.
3.2 Stability analysis
From the definition and the boundedness of \(\alpha_{i}\), one can obtain \(\dot{\alpha}_{i}\leq\Phi_{i}(S_{1},\ldots,S_{i},\tilde{\theta} _{f1}, \ldots,\tilde{\theta}_{fi}, y_{r},\dot{y}_{r},\ddot{y}_{r},\tilde{\theta}_{h1},\ldots,\tilde{\theta} _{hi},\chi_{2},\ldots,\chi_{i})\), where \(\Phi_{i}\) (\(i=2,\ldots,n1\)) are nonnegative continuous functions.
For the proof of Theorem 1, the following lemma is introduced.
Lemma 2
For \(1\leq j\leq n1\), consider the set \(G_{\varepsilon }\) by \(G_{\varepsilon }=\{S_{j}S_{j}< 0.2554\varepsilon _{j}\}\). Then, for \(S_{j}\notin G_{\varepsilon _{j}}\), the inequality \(116\tanh^{2}(S_{j}/\varepsilon _{j})< 0\) is satisfied.
Proof
By following the same line as in the stability analysis of [37], it can be shown that all the signals in the closedloop systems are bounded, and the steadystate error can be made arbitrarily small. □
From the above design procedures and analysis, the following theorem is deduced.
Theorem 1
Consider the system (1) consisting of the adaptive laws (19), (27), (35), (40), the virtual control (18), (26), (34), and the actual controller (39). Under Assumptions 14, by using the above design procedures, for the bounded initial conditions, the boundedness of all the signals in the closedloop system can be guaranteed by the proposed control scheme and the tracking error can converge to a small neighborhood of the origin.
4 Simulation example
Example 1
(Numerical examples)
Case 1: \(f_{1}(x_{1})=0.1x_{1}e^{0.5x_{1}^{2}}\cos(x_{1})0.1x_{1}e^{0.5x_{1}^{2}}\cos (x_{1})\), \(f_{2}(x_{2})=x_{2}\sin\frac{2}{1+x_{1}^{2}}\), \(h_{1}(x_{1}(t\tau_{1}))=\frac{x_{1}^{3}(t\tau_{1})}{1+x_{1}^{2}(t\tau_{1})}\), \(h_{2}(x_{2}(t\tau_{2}))=\frac{x_{2}^{4}(t\tau_{2})\sin(x_{2})}{1+x_{2}^{2}(t\tau_{2})}\).
Case 2: \(f_{1}(x_{1})=0.2x_{1}\), \(f_{2}(x_{2})=0.2\sin(x_{1})+0.2\cos(t)\), \(h_{1}(x_{1}(t\tau_{1}))=0.5x_{1}(t\tau_{1})\), \(h_{2}(x_{2}(t\tau _{2}))=0.8x_{1}^{2}(t\tau_{2})\).
The initial conditions of states are chosen as \(x_{1}(0)=0.2\) and \(x_{2}(0)=0.2\), and the others initial values are chosen as zeros. Choose \(\tau_{m}=3~\mbox{s}\), \(\tau_{M}=5~\mbox{s}\).
Figure 1 shows the response trajectories of the control output y (state variable \(x_{1}\)) and the desired reference tracking signal \(y_{r}\). As can be seen, the control output y can track the reference signal \(y_{r}\) satisfactorily in the presence of different uncertainties.
Example 2
(Practical example)
5 Conclusion
In this paper, a delaylowerboundbased adaptive fuzzy memory control scheme has been proposed for a class of uncertain nonlinear systems with unknown time delays in state and input. By utilizing the mean value theorem, the unknown timedelay functions with all state variable have been dealt with. A novel adaptive filter has been designed to eliminate the effect of timevarying input delay. Based on a backstepping technique, a delaylowerbounddependent adaptive fuzzy memory controller has been designed. It is analyzed that the proposed control scheme guarantees that all the signals in the closedloop system are bounded and the tracking error converges to a small neighborhood of the origin. A practical simulation example also demonstrates the theoretical results. Further investigation should include more practical and general systems such as largescale nonlinear systems and purefeedback systems, and discretetime systems.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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