1 Introduction

Recently, the control problem of uncertain nonlinear system has received great attention since many practical systems possess uncertain and nonlinear characteristics. Since many types of neural networks have been proved to have excellent function approximation ability, such as, radial basis function neural network(RBFNN)[1], self-recurrent wavelet neural network [2], recurrent wavelet Elman neural network [3], a series of neural network-based control schemes have been proposed to tackle system uncertainty. In [4], an adaptive neural network tracking control scheme was presented for remotely operated vehicles with unknown dynamic model. In [5], an event-triggered controller using sampled-data neural network was presented for a class of continuous-time nonlinear systems. In [6], an adaptive projection neural network was designed to control redundant manipulators with unknown parameters. In [7], a neural network proportional-derivative control strategy was developed for a teleoperation system. In [8], an adaptive neural network backstepping control method was proposed for nonlower triangular nonlinear systems with unmodeled dynamics. Among these control schemes, adaptive neural network control via backstepping design is particularly attractive and intensive efforts have been devoted to adaptive neural network control via backstepping design. In [9], neural network control was combined with backstepping method to develop a composite controller for a class of strict-feedback uncertain nonlinear systems. In [10], an adaptive neural network control was proposed for uncertain multi-input–multi-output (MIMO) nonlinear systems with block-triangular form. In [11], an observer-based adaptive neural network control was presented for single-input single-output (SISO) stochastic nonlinear systems with unknown time delay. In [12], robust stabilization of nonaffine pure-feedback uncertain nonlinear systems was investigated via adaptive neural network control. In [13], an adaptive neural network backstepping tracking control was designed for an n-link robotic manipulator. In [14], an adaptive backstepping wavelet neural network control was developed for induction motor (IM) drive. In [15], adaptive backstepping-based neural network output feedback control was proposed for uncertain switched stochastic nonlinear systems.

Though adaptive neural network control via backstepping design has become one of the most popular methods to address the control problem of uncertain nonlinear system, an obvious deficiency of traditional backstepping design is the problem of “explosion of complexity” caused by repeated differentiation of virtual control. In particular, when RBFNN is utilized to approximate system uncertainties, we need to take derivative of these radial basis functions, which further increases computation burden. To overcome the “explosion of complexity”, adaptive dynamic surface neural network control was presented [16], where first-order filter was introduced to obtain the derivative of virtual control at each step of conventional backstepping. In [17], an adaptive dynamic surface neural network control was developed to achieve fault-tolerant tracking for a class of uncertain nonlinear systems. In [18], dynamic surface control was incorporated into adaptive neural network control to develop a control strategy for a class of uncertain nonlinear systems with pure-feedback form. In [19], an adaptive dynamic surface neural network control with output feedback form was presented for a class of stochastic nonlinear systems. Adaptive dynamic surface neural network control has been applied into many practical systems, such as, underactuated autonomous surface vehicles [20], hypersonic flight vehicles [21,22,23], flexible joint robots [2], power system [24] and permanent magnet synchronous motor [25]. However, these results fail to consider output constraints and input nonlinearities. In fact, output constraints and input nonlinearities exist in various practical applications.

Output constraint can be found in many practical systems due to physical barrier in the surroundings, safety requirement and system performance specification. If output constraint is ignored in the control design, system performance will be degraded and system damage may occur. Therefore, output constraint plays a very important role in control design. For constrained linear system, reference governor method [26] and convex optimization approach [27] were presented to address output constraint problem. However, these approaches rely on computationally intensive algorithms. To prevent output constraint violation, system transformation techniques were proposed in [28, 29] to transform the constrained system into an equivalent unconstrained system. Recently, Barrier Lyapunov function-based design has received great attention since Barrier Lyapunov function tends to infinity when the output approaches some constraints. In [30, 31], a logarithm-type Barrier Lyapunov function was presented to handle the issue of output constraint. In [32], a tan-type Barrier Lyapunov function was proposed to avoid output constraint violation. In [33], a cot-type Barrier Lyapunov function was incorporated into Lyapunov function design to deal with constrained state. In [34], an integral Barrier Lyapunov function was designed to prevent the violation of boundary output constraint. In [35, 36], an asymmetric time-varying Barrier Lyapunov function was constructed to ensure time-varying output constraint satisfaction. In [37], a time-varying Barrier Lyapunov function was proposed to tackle time-varying output constraint. However, these results fail to consider input nonlinearities. Due to physical constraints in actuator, input nonlinearities can be found in many control systems.

Dead zone and input saturation are common input nonlinearities, which may degrade system performance, reduce control accuracy and even lead to system instability. The existence of dead zone keeps the output of an actuator at zero until the control input exceeds a certain value and the existence of input saturation forces the actuator to give constant output if the control input exceeds a certain value. Therefore, the existence of dead zone and input saturation severely affects the function of control input and makes the control problems complicated and challenging. In order to deal with dead zone, several studies [38,39,40,41,42,43] modeled the dead zone as a disturbance term and compensated it using disturbance observer or adaptive approach. In [44, 45], dead-zone inverse method was adopted to tackle dead-zone problem. In [46, 47], neural network and fuzzy logic approximation were employed to compensate the dead zone. To handle the issue of input saturation, some results [48,49,50] introduced auxiliary system and used its state to develop a constrained control. In [51,52,53], input saturation was approximated by smooth functions and the approximation error was treated as a disturbance-like term in their control design. In [54], an adaptive scheme was developed to tackle input saturation and achieve output consensus of multiple nonlinear systems subjected to input saturation. In [55], an anti-windup compensator was employed to solve the problem of input saturation. It should be emphasized that the aforementioned control schemes are designed only for the systems that have strict-feedback form or can be transformed into strict-feedback structure, which prohibits broad applications of these methods.

From a mathematical viewpoint, nonstrict-feedback system with the whole system state in each subsystem function is a more general system form. Many practical systems, such as stirred tank reactor process system [56], mass and spring damper system [57], Brusselator [58], electromechanical system [59], have nonstrict-feedback structure. To control nonstrict-feedback system, virtual control should contain the whole system state to deal with the nonlinear function in each subsystem, which will result in algebraic loop problem. This makes the controller design very difficult and challenging. By utilizing monotonous increasing function, variable separation method was developed in [59,60,61,62] to design adaptive fuzzy or neural network control for nonstrict-feedback system. However, the above results require that the nonlinear function satisfy the monotonously increasing property. In [63], adaptive neural network tracking problem was considered for nonstrict-feedback switched nonlinear system. In [64], adaptive neural network stabilization problem was investigated for nonlinear system with nonstrict-feedback form. However, these results cannot deal with output constraint and input saturation simultaneously. To the best of our knowledge, there is no studies about nonstrict-feedback nonlinear systems subjected to input saturation, dead zone and output constraint.

Motivated by aforementioned discussion, an adaptive neural network dynamic surface control is proposed for a class of nonstrict-feedback systems subjected to input saturation, dead zone and output constraint. Tan-type Lyapunov function is incorporated into Lyapunov function design to ensure output constraint satisfaction and an anti-windup compensator is introduced to deal with input saturation. Based on backstepping method, a series of novel stabilizing virtual control functions are derived. First-order sliding mode differentiator is presented to obtain the first-order derivative of virtual control and overcome the explosion of complexity problem. Dead-zone inverse approach is adopted to obtain the control input. With the aid of finite time stability theory, the finite time convergence of the output tracking error into a small set around the origin is proved. The main contributions of this paper can be summarized as follows: (1) Based on adaptive neural network dynamic surface control, input saturation, dead zone and output constraint are considered into controller design to address tracking problem for uncertain nonstrict-feedback nonlinear system. To the best of our knowledge, this is the first time to report results about adaptive tracking control for nonstrict-feedback nonlinear system subjected to input saturation, dead zone and output constraint. (2) The proposed control scheme not only overcomes the difficulty of applying backstepping control into nonstrict-feedback system, but also removes the restrictive assumption that the nonlinear function should be monotonous increasing. (3) The finite time tracking problem for uncertain nonstrict-feedback nonlinear system is studied using a series of novel virtual control functions. (4) Different from conventional dynamic surface control, first-order sliding mode differentiator is combined with backstepping design to overcome the explosion of complexity problem, which has finite time convergence property and satisfies separation principle, thereby having superior performance.

2 Preliminary

2.1 Problem formulation

Consider the following uncertain nonlinear system with nonstrict-feedback form:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{x_{i}}=f_{i}(x)+g_{i}(\bar{x}_{i})x_{i+1}\\&\ldots \\&\dot{x_{n}}=f_{n}(x)+g_{n}(\bar{x}_{n})u\\&y=x_{1} \end{aligned} \right. \end{aligned} \end{aligned}$$
(1)

where \(\bar{x}_{i}=[x_{1},\ldots ,x_{i}]^{\mathrm{T}}\), \(x\in R^{n}\) denotes state variables. System output \(y\in R\) is constrained in open set \(|y|<k_{c}\) with \(k_{c}\) being a positive real number denoting output constraint, \(f_{i}(x)\) and \(g_{i}(\bar{x}_{i})\) are unknown smooth nonlinear functions. \(u\in R\) is control input subjected to dead zone and input saturation, which can be described as follows:

$$\begin{aligned} \begin{aligned} u=\left\{ \begin{aligned}&u_{\max }\quad \mathrm{if}~v>v_{\max }\\&m_{r}(v-b_{r})\quad \mathrm{if}~b_{r}<v\le v_{\max }\\&0\quad \mathrm{if}~b_{l}\le v\le b_{r}\\&m_{l}(v-b_{l})\quad \mathrm{if}~v_{\min }\le v<b_{l}\\&u_{\min }\quad \mathrm{if}~v< v_{\min } \end{aligned} \right. \end{aligned} \end{aligned}$$
(2)

where v is the desired control input, \(m_{l}\) and \(m_{r}\) are general slopes of dead-zone input, \(b_{l}\) and \(b_{r}\) are left and right breakpoints of dead-zone input, \(u_{\max }\) and \(u_{\min }\) are the upper and lower bound of u(t). Control input (2) can be rewritten as follows:

$$\begin{aligned} \begin{aligned} u=sat(u_{d})=\left\{ \begin{aligned}&u_{\max },\quad \mathrm{if}~v>v_{\max }\\&u_{d},\quad \mathrm{if}~v_{\min }\le v \le v_{\max }\\&u_{\min },\quad \mathrm{if}~v<v_{\min } \end{aligned} \right. \end{aligned} \end{aligned}$$

where

$$\begin{aligned}&u_{d}=K^{\mathrm{T}}(t)\varPhi (t)v(t)+\Delta \\&K(t)=[K_{r}(v(t)),K_{l}(v(t))]^{\mathrm{T}}\\&K_{r}(v(t))=\left\{ \begin{aligned}&m_{r},\quad \mathrm{if}~v\ge b_{l}\\&0,~~~~~~~\mathrm{else} \end{aligned} \right. \\&K_{l}(v(t))=\left\{ \begin{aligned}&m_{l},\quad \mathrm{if}~v\le b_{r}\\&0,~~~~~~\mathrm{else} \end{aligned} \right. \\&\varPhi (t)=[\varphi _{r}(t),\varphi _{l}(t)]^{\mathrm{T}}\\&\varphi _{r}(t)=\left\{ \begin{aligned}&1,\quad \mathrm{if}~v\ge b_{l}\\&0,~~~~~\mathrm{else} \end{aligned} \right. \\&\varphi _{l}(t)=\left\{ \begin{aligned}&0,\quad \mathrm{if}~v>b_{r}\\&1,~~~~~\mathrm{else} \end{aligned} \right. \\&\Delta =\left\{ \begin{aligned}&-m_{l}b_{l},\quad \mathrm{if}~v<b_{l}\\&-(m_{l}+m_{r})v,\quad \mathrm{if}~b_{l}\le v\le b_{r}\\&-m_{r}b_{r},\quad \mathrm{if}~v>b_{r} \end{aligned} \right. \end{aligned}$$

The error between u and \(u_{d}\) is defined as \(\Delta u=u-u_{d}\).

Remark 1

In system (1), if \(f_{i}(x)=f_{i}(\bar{x}_{i})\) with \(\bar{x}_{i}=[x_{1},\ldots ,x_{i}]^{\mathrm{T}}\), the system (1) becomes strict-feedback nonlinear system. If \(f_{i}(x)= f_{i}(\bar{x}_{i}, 0)\) and \(g_{i}(x)=\partial f_{i}(\bar{x}_{i},\) \(x_{i+1}^{0})/\partial x_{i+1}\) with \( x_{i+1}^{0}\) being a number between 0 and \(x_{i+1}\), the system (1) becomes pure-feedback nonlinear system. Therefore, nonstrict-feedback nonlinear system is a general system form which includes strict-feedback nonlinear system and pure-feedback nonlinear system as its special form [65]. Nonstrict-feedback system can be used to describe many practical systems, such as, stirred tank reactor process system [56], mass and spring damper system [57], Brusselator [58], electromechanical system [59]. Therefore, it is necessary to study control problem for nonstrict-feedback nonlinear system with constraints.

Remark 2

For strict-feedback nonlinear system and pure-feedback nonlinear system, the existing adaptive backstepping control schemes view \(x_{i+1}\) as a control input for the first i-th subsystem and a virtual control input \(\alpha _{i}\) is designed to stabilize the first i-th subsystem. To ensure the existence and the uniqueness of the virtual control, \(\alpha _{i}\) should be a function of partial state \(x_{j}\), \(j\le i\). Otherwise, algebraic loop problem will occur. However, the nonlinear function \(f_{i}(x)\) in nonstrict-feedback system contains state \(x_{j}\), \(j>i\). Therefore, the existing adaptive backstepping control schemes for strict-feedback system and pure-feedback system cannot be applied to nonstrict-feedback system. In addition, as shown in [62], it is difficult to tackle the function of \(x_{i}\) descended from previous step to the current design step. Therefore, controlling nonstrict-feedback nonlinear system with constraints is a very difficult and challenging problem.

The studied problem can be formulated as designing an adaptive backstepping control scheme for nonstrict-feedback system (1) such that the system output y can track the reference output \(y_{r}\) within finite time while output constraint is not violated and all the closed-loop signals remain bounded. To this end, the following assumptions are imposed on control parameters, control gain and reference output signal.

Assumption 1

The upper and lower bound of the dead-zone slopes and break points are known: \(0<\underline{m}\le m_{l}\le \bar{m}\), \(0<\underline{m}\le m_{r}\le \bar{m}\), \(0<\underline{b}\le |b_{l}|\le \bar{b}\), \(0<\underline{b}\le b_{r}\le \bar{b}\).

Assumption 2

The signs of \(g_{i}(\bar{x}_{i})\) are known and there exist positive constants \(\underline{g}_{i}\) and \(\bar{g}_{n}\ge \underline{g}_{n}\) such that \(\underline{g}_{i}\le |g_{i}(\bar{x}_{i})|\) and \(\underline{g}_{n}\le |g_{n}(\bar{x}_{i})|\le \bar{g}_{n}\).

Assumption 3

There exist positive constants \(B_{i}\) such that the reference output signal \(y_{r}\) satisfies \(|y_{r}^{(i)}|\le B_{i}\).

Remark 3

According to Assumption 1, there exist positive constants \(m_{0}\) and \(\bar{\Delta }\) such that \(K^{\mathrm{T}}(t)\varPhi (t)\ge m_{0}\), \(\Delta \le \bar{\Delta }\).

2.2 Neural network approximation

As we all know, RBFNN can approximate any continuous unknown nonlinear function due to its excellent approximation capability. Here, RBFNN is utilized to approximate a continuous unknown nonlinear function h(Z) as follows:

$$\begin{aligned} h(Z)=W^{\mathrm{T}}S(Z) \end{aligned}$$
(3)

where \(Z\in \varOmega _{z}\subset R^{q}\) is RBFNN input vector, \(W=[W_{1},\ldots ,W_{l}]^{\mathrm{T}}\in R^{l}\) is RBFNN weight vector, l denotes the number of neurons in hidden layer, \(S(Z)=[S_{1}(Z),\ldots ,S_{l}(Z)]^{\mathrm{T}}\) is radial basis function vector with \(S_{i}(Z)=\exp (-(Z-c_{i})^{\mathrm{T}}(Z-c_{i})/b_{i}^{2}) (i=1,\ldots ,l)\), where \(c_{i}=[c_{i1},\ldots ,c_{iq}]^{\mathrm{T}}\) and \(b_{i}\) represent the center and the width of radial basis function. The ideal weight vector \(W^{*}\) is selected as the value of W that minimizes the approximation error.

$$\begin{aligned} W^{*}=\arg \min _{W\in R^{l}}\left\{ \sup _{Z\in \varOmega _{Z}}|h(Z)-W^{\mathrm{T}}S(Z)|\right\} \end{aligned}$$
(4)

Using optimal weight vector \(W^{*}\), the continuous unknown nonlinear function h(Z) can be approximated as follows:

$$\begin{aligned} h(Z)=W^{*T}S(Z)+\varepsilon \end{aligned}$$
(5)

where \(\varepsilon \) is approximation error which is bounded, i.e., \(\Vert \varepsilon \Vert \le \bar{\varepsilon }\) with \(\bar{\varepsilon }\) being an unknown positive constant.

2.3 Mathematical lemmas

In this section, some useful Lemmas that play an important role in controller design are introduced.

Lemma 1

[66] For any positive constant \(\gamma \) and any variable \(z\in R\), the following inequality holds:

$$\begin{aligned} 0\le |z|-\frac{z^{2}}{\sqrt{z^{2}+\gamma ^{2}}}<\gamma \end{aligned}$$
(6)

Lemma 2

[67] For any \(a\in R^{+}\), \(b\in R^{+}\) and \(p\in R^{+}\), \(q\in R^{+}\) satisfying \(1/p+1/q=1\), the following inequality holds:

$$\begin{aligned} ab\le \frac{a^{p}}{p}+\frac{b^{q}}{q} \end{aligned}$$
(7)

Lemma 3

[68] For \(x_{i}\in R\) and \(0<b<1\), we have:

$$\begin{aligned} \left( \sum _{i=1}^{n}|x_{i}|\right) ^{b}\le \sum _{i=1}^{n}|x_{i}|^{b} \end{aligned}$$
(8)

Lemma 4

[68] For any real numbers \(x_{1},\ldots ,x_{n}\) and \(0<p<1\), one has:

$$\begin{aligned} \sum _{i=1}^{n}|x_{i}|^{1+p}\le \left( \sum _{i=1}^{n}|x_{i}|^{2}\right) ^{(1+p)/2} \end{aligned}$$
(9)

Lemma 5

[69] For any positive real numbers \(\alpha \), \(\beta \) and \(0<\gamma <1\), if a Lyapunov function V satisfies \(\dot{V}+\alpha V+\beta V^{\gamma }\le 0\), then the Lyapunov function V can converge to zero within finite time \(T_{0}\le \frac{1}{\alpha (1-\gamma )}\ln \frac{\alpha V^{1-\gamma }(x_{0})+\beta }{\beta }\).

Lemma 6

[70] For any \(\varepsilon >0\) and \(x\in R\), the inequality \(|x|-x\tanh (x/\varepsilon )\le \varrho \varepsilon \) holds, where \(\varrho =0.2785\).

3 Main results

In this section, a systematic design and stability analysis procedure for the proposed control scheme will be given.

Step 1 Define the error variables as \(z_{1}=x_{1}-y_{r}\) and the dynamics of the error variables can be expressed as follows:

$$\begin{aligned} \dot{z}_{1}=\dot{x}_{1}-\dot{y}_{r}=g_{1}(x_{1})x_{2}+f_{1}(x)-\dot{y}_{r} \end{aligned}$$
(10)

Since the nonlinear function \(f_{1}(x)\) is unknown, RBFNN is used to approximate it.

$$\begin{aligned} f_{1}(x)=W_{1}^{*T}S_{1}(x)+\varepsilon _{1} \end{aligned}$$
(11)

where \(W_{1}^{*}=blockdiag[W_{1k}^{*T}](k=1,\ldots ,m)\) is ideal weight vector, \(S_{1}(x)\) is radial basis function vector and \(\varepsilon _{1}\) is approximation error. The approximation error is bounded, i.e., \(|\varepsilon _{1}|\le \bar{\varepsilon }_{1}\).

Substituting (11) into (10), one has:

$$\begin{aligned} \dot{z}_{1}=g_{1}(x_{1})x_{2}+W_{1}^{*T}S_{1}(x)+\varepsilon _{1}-\dot{y}_{r} \end{aligned}$$
(12)

For notational simplicity, set \(M_{z_{1}}=z_{1}/\cos ^{2}(\frac{\pi z_{1}^{2}}{2k_{b}^{2}})\) and the updating law for neural network weight is designed as follows:

$$\begin{aligned} \dot{\hat{W}}_{1}=\varGamma _{1}(M_{z_{1}}S_{1}(x)-\sigma _{1}\hat{W}_{1}) \end{aligned}$$
(13)

where \(\varGamma _{1}\) and \(\sigma _{1}\) are positive real numbers.

The adaptation law for the upper bound of approximation error \(\bar{\varepsilon }_{1}\) is chosen as follows:

$$\begin{aligned} \dot{\hat{\varepsilon }}_{1}=\varLambda _{1} \left( \frac{M_{z_{1}}^{2}}{\sqrt{M_{z_{1}}^{2}+\gamma _{1}^{2}}}-\kappa _{1}\hat{\varepsilon }_{1}\right) \end{aligned}$$
(14)

where \(\varLambda _{1}\), \(\gamma _{1}\) and \(\kappa _{1}\) are positive real numbers.

To facilitate virtual control design, define an auxiliary function as follows:

$$\begin{aligned} \begin{aligned} \bar{\alpha }_{1}&=k_{1}\frac{k_{b}^{2}}{\pi }\frac{\sin \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \cos \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) }{z_{1}}\\&\qquad +\eta _{1} \left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}}\frac{\cos ^{2} \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) A}{z_{1}}\\&\qquad +\hat{\varepsilon }_{1}\frac{M_{z_{1}}}{\sqrt{M_{z_{1}}^{2}+\gamma _{1}^{2}}}-\dot{y}_{r} +\hat{W}_{1}^{\mathrm{T}}S_{1}(x) \end{aligned} \end{aligned}$$
(15)

where \(k_{1}\) and \(\eta _{1}\) are positive real numbers and A can be designed as follows:

$$\begin{aligned} \begin{aligned} A=\left\{ \begin{aligned}&\tan ^{3/4} \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) ,\quad \mathrm{if}~|z_{1}|\ge \upsilon \\&\tan ^{-\frac{1}{4}}\left( \frac{\pi \upsilon ^{2}}{2k_{b}^{2}}\right) \tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) ,~~~~~\mathrm{else} \end{aligned} \right. \end{aligned} \end{aligned}$$
(16)

where \(k_{b}=k_{c}-B_{0}\), \(\upsilon \) is a positive real number.

With the aid of auxiliary function (15), the virtual control can be designed as follows:

$$\begin{aligned} \alpha _{1}=-\frac{1}{\underline{g}_{1}}\frac{M_{z_{1}}\bar{\alpha }_{1}^{2}}{\sqrt{M_{z_{1}}^{2}\bar{\alpha }_{1}^{2}+\gamma _{1}^{2}}} \end{aligned}$$
(17)

Remark 4

The design of (16) is to ensure \(\lim \limits _{z_{1}\rightarrow 0}\frac{\cos ^{2}(\frac{\pi z_{1}^{2}}{2k_{b}^{2}})A}{z_{1}}\) \(=0\). In addition, using L’Hospital’s rule, we have \(\lim \limits _{z_{1}\rightarrow 0}\frac{1}{z_{1}}\) \(\sin (\frac{\pi z_{1}^{2}}{2k_{b}^{2}})\cos (\frac{\pi z_{1}^{2}}{2k_{b}^{2}})=0\). Therefore, (15) does not contain singularity term. Besides, it is worth noting that (16) is continuous, thereby avoiding chattering problem.

From system output constraint and reference output constraint, we have \(|z_{1}|<k_{b}\) with \(k_{b}+B_{0}=k_{c}\). Select the Barrier Lyapunov function (BLF) as follows:

$$\begin{aligned} V_{b}=\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \end{aligned}$$
(18)

Remark 5

Tan-type Barrier Lyapunov function has the following properties:

(1) When \(z_{1}\) approaches \(k_{b}\), \(V_{b}\) grows to infinity.

(2) \(\lim _{k_{b}\rightarrow \infty } \frac{k_{b}^{2}}{\pi }\tan (\frac{\pi z_{1}^{2}}{2k_{b}^{2}})=\frac{1}{2}z_{1}^{2}\)

The first property means that tan-type Lyapunov function is a good Barrier Lyapunov function candidate. The second property shows that when there is no constraint, the tan-type Lyapunov function degrades to the commonly used quadratic form. Therefore, tan-type Lyapunov function can deal with both constrained systems and unconstrained systems.

Take BLF (18) into account and the Lyapunov function in the first step can be constructed as follows:

$$\begin{aligned} V_{1}=V_{b}+\frac{1}{2\varLambda _{1}} \tilde{\varepsilon }_{1}^{2}+\frac{1}{2\varGamma _{1}} \tilde{W}_{1}^{\mathrm{T}}\tilde{W}_{1} \end{aligned}$$
(19)

where \(\tilde{\varepsilon }_{1}=\bar{\varepsilon }_{1}-\hat{\varepsilon }_{1}\), \(\tilde{W}_{1}=W_{1}^{*}-\hat{W}_{1}\).

Define the error variable \(z_{2}=x_{2}-\alpha _{1}\) and the time derivative of \(V_{1}\) along (12–14) is:

(20)

It follows immediately from Lemma 1 that

$$\begin{aligned} M_{z_{1}}\varepsilon _{1}\le |M_{z_{1}}|\bar{\varepsilon }_{1}<\bar{\varepsilon }_{1}\gamma _{1}+\bar{\varepsilon }_{1}\frac{M_{z_{1}}^{2}}{\sqrt{M_{z_{1}}^{2}+\gamma _{1}^{2}}} \end{aligned}$$
(21)

Using Lemma 1 and Assumption 2, we have:

$$\begin{aligned} M_{z_{1}}g_{1}(x_{1})\alpha _{1}\le -\frac{M_{z_{1}}^{2}\bar{\alpha }_{1}^{2}}{\sqrt{M_{z_{1}}^{2}\bar{\alpha }_{1}^{2}+\gamma _{1}^{2}}}\le \gamma _{1}-M_{z_{1}}\bar{\alpha }_{1} \end{aligned}$$
(22)

Substituting (15), (17), (21) and (22) into (20) results in

$$\begin{aligned} \begin{aligned} \dot{V}_{1}\le&-k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\eta _{1}\left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}} \tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) +\gamma _{1}\\&+\bar{\varepsilon }_{1}\gamma _{1}+M_{z_{1}}g_{1} (x_{1})z_{2}+\kappa _{1}\hat{\varepsilon }_{1}\tilde{\varepsilon }_{1}+\sigma _{1}\tilde{W}_{1}^{\mathrm{T}}\hat{W}_{1} \end{aligned} \end{aligned}$$
(23)

Step 2 Taking time derivative of error variable \(z_{2}\) and using RBFNN to approximate unknown nonlinear function \(f_{2}(x)\), we have:

$$\begin{aligned} \begin{aligned} \dot{z}_{2}&=g_{2}(\bar{x}_{2})x_{3}+f_{2}(x)-\dot{\alpha }_{1}\\&=g_{2}(\bar{x}_{2})x_{3}+W_{2}^{*T}S_{2}(x)+\varepsilon _{2}-\dot{\alpha }_{1} \end{aligned} \end{aligned}$$
(24)

where \(W_{2}^{*}=blockdiag[W_{2k}^{*T}](k=1,\ldots ,m)\) is ideal weight vector, \(S_{2}(x)\) is radial basis function vector and \(\varepsilon _{2}\) is approximation error. The approximation error is bounded, i.e., \(|\varepsilon _{2}|\le \bar{\varepsilon }_{2}\).

In (24), differentiation of virtual control input \(\alpha _{1}\) leads to the explosion of complexity. In order to overcome the explosion of complexity, first-order sliding mode differentiator is employed to obtain the derivative of the virtual control input.

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{\nu }_{11}=-\lambda _{0}|\nu _{11}-\alpha _{1}|^{1/2}sign(\nu _{11}-\alpha _{1})+\nu _{12}\\&\dot{\nu }_{12}=-\lambda _{1}sign(\nu _{11}-\alpha _{1}) \end{aligned} \right. \end{aligned} \end{aligned}$$
(25)

where \(\nu _{11}\) and \(\nu _{12}\) are state variables of the differentiator, \(\lambda _{0}\) and \(\lambda _{1}\) are positive real numbers. According to [71, 72], \(\nu _{12}\) can approximate the first-order derivative of virtual control \(\alpha _{1}\) to arbitrary accuracy if the initial deviation \(|\nu _{11}(t_{0})-\alpha _{1}(t_{0})|\) and \(|\nu _{12}(t_{0})-\dot{\alpha }_{1}(t_{0})|\) are bounded. Therefore, we have \(|\nu _{12}-\dot{\alpha }_{1}|\le \delta _{1}\) with \(\delta _{1}\) being an unknown positive constant.

Remark 6

Conventional dynamic surface control employs first-order filter to obtain the first-order derivative of virtual control. Different from conventional dynamic surface control, in this paper, first-order sliding mode differentiator is utilized to overcome the explosion of complexity and the poor precision of first-order filter.

The weight updating law for RBFNN is selected as follows:

$$\begin{aligned} \dot{\hat{W}}_{2}=\varGamma _{2}\left( z_{2}S_{2}(x)-\sigma _{2}\hat{W}_{2}\right) \end{aligned}$$
(26)

where \(\varGamma _{2}\) and \(\sigma _{2}\) are positive real numbers.

The adaptation law for parameters \(\bar{\varepsilon }_{2}\) and \(\delta _{1}\) can be expressed as follows:

$$\begin{aligned}&\dot{\hat{\varepsilon }}_{2}=\varLambda _{2} \left( \frac{z_{2}^{2}}{\sqrt{z_{2}^{2}+\gamma _{2}^{2}}} -\kappa _{2}\hat{\varepsilon }_{2}\right) \end{aligned}$$
(27)
$$\begin{aligned}&\dot{\hat{\delta }}_{1}=\varPi _{1}\left( z_{2}\tanh (z_{2}/\varepsilon ) -\mu _{2}\hat{\delta }_{1}\right) \end{aligned}$$
(28)

where \(\varLambda _{2}\), \(\varPi _{1}\), \(\mu _{2}\), \(\gamma _{2}\) and \(\kappa _{2}\) are positive real numbers.

In order to obtain virtual control, an auxiliary function is designed as follows:

$$\begin{aligned} \begin{aligned} \bar{\alpha }_{2}&=k_{2}z_{2}+\eta _{2}z_{2}^{1/2}+\hat{\varepsilon }_{2}\frac{z_{2}}{\sqrt{z_{2}^{2}+\gamma _{2}^{2}}}\\&\quad -\nu _{12}+\hat{W}_{2}^{\mathrm{T}}S_{2}(x)\\&\quad +\hat{\delta }_{1}\tanh (z_{2}/\varepsilon )+\bar{g}_{1}\frac{M_{z_{1}}^{2}z_{2}}{\sqrt{M_{z_{1}}^{2}z_{2}^{2}+\gamma _{2}^{2}}} \end{aligned} \end{aligned}$$
(29)

where \(k_{2}\) and \(\eta _{2}\) are positive real numbers.

Using auxiliary function (29), the virtual control law is derived as follows:

$$\begin{aligned} \alpha _{2}=-\frac{1}{\underline{g}_{2}}\frac{z_{2}\bar{\alpha }_{2}^{2}}{\sqrt{z_{2}^{2}\bar{\alpha }_{2}^{2}+\gamma _{2}^{2}}} \end{aligned}$$
(30)

The Lyapunov function can be chosen as follows:

$$\begin{aligned} V_{2}=V_{1}+\frac{1}{2}z_{2}^{2}+\frac{1}{2\varLambda _{2}}\tilde{\varepsilon }_{2}^{2}+\frac{1}{2\varPi _{1}}\tilde{\delta }_{1}^{2}+\frac{1}{2\varGamma _{2}}\tilde{W}_{2}^{\mathrm{T}}\tilde{W}_{2} \end{aligned}$$
(31)

where \(\tilde{\varepsilon }_{2}=\bar{\varepsilon }_{2}-\hat{\varepsilon }_{2}\), \(\tilde{\delta }_{1}=\delta _{1}-\hat{\delta }_{1}\), \(\tilde{W}_{2}=W_{2}^{*}-\hat{W}_{2}\).

Defining \(z_{3}=x_{3}-\alpha _{2}\) and taking time derivative of \(V_{2}\) along (24), (26–28), we have:

$$\begin{aligned} \begin{aligned} \dot{V}_{2}&=\dot{V}_{1}+z_{2}\dot{z}_{2}-\frac{1}{\varLambda _{2}}\tilde{\varepsilon }_{2}\dot{\hat{\varepsilon }}_{2}-\frac{1}{\varPi _{1}}\tilde{\delta }_{1}\dot{\hat{\delta }}_{1}-\frac{1}{\varGamma _{2}}\tilde{W}_{2}^{\mathrm{T}}\dot{\hat{W}}_{2}\\&=\dot{V}_{1}+z_{2}\left( g_{2}(\bar{x}_{2})(z_{3}+\alpha _{2})\right. \\&\quad \left. +W_{2}^{*T}S_{2}(x)+\varepsilon _{2}-\dot{\alpha }_{1}\right) \\&~~~-\tilde{\delta }_{1}\left( z_{2}\tanh (z_{2}/\varepsilon ) -\mu _{2}\hat{\delta }_{1}\right) \\&\quad -\tilde{\varepsilon }_{2}\left( \frac{z_{2}^{2}}{\sqrt{z_{2}^{2}+\gamma _{2}^{2}}}-\kappa _{2}\hat{\varepsilon }_{2}\right) \\&~~~-\tilde{W}_{2}^{\mathrm{T}}(z_{2}S_{2}(x)-\sigma _{2}\hat{W}_{2})\\&=\dot{V}_{1}+z_{2}g_{2}(\bar{x}_{2})z_{3}+z_{2}g_{2}(\bar{x}_{2})\alpha _{2}+\hat{W}_{2}^{\mathrm{T}}z_{2}S_{2}(x)\\&~~~+z_{2}\varepsilon _{2}-\tilde{\varepsilon }_{2}\frac{z_{2}^{2}}{\sqrt{z_{2}^{2}+\gamma _{2}^{2}}}+\kappa _{2}\tilde{\varepsilon }_{2}\hat{\varepsilon }_{2}-z_{2}\dot{\alpha }_{1}\\&~~~-\tilde{\delta }_{1}z_{2}\tanh (z_{2}/\varepsilon )+\mu _{2}\tilde{\delta }_{1}\hat{\delta }_{1} +\sigma _{2}\tilde{W}_{2}^{\mathrm{T}}\hat{W}_{2} \end{aligned} \end{aligned}$$
(32)

Based on Lemma 1, the following inequality holds:

$$\begin{aligned} z_{2}\varepsilon _{2}\le |z_{2}|\bar{\varepsilon }_{2}<\bar{\varepsilon }_{2}\gamma _{2}+\bar{\varepsilon }_{2}\frac{z_{2}^{2}}{\sqrt{z_{2}^{2}+\gamma _{2}^{2}}} \end{aligned}$$
(33)

From Lemma 1 and Assumption 2, we have:

$$\begin{aligned} M_{z_{1}}g_{1}(x_{1})z_{2}\le & {} \bar{g}_{1}|M_{z_{1}}||z_{2}|<\bar{g}_{1}\gamma _{2}\nonumber \\&\quad +\bar{g}_{1}\frac{M_{z_{1}}^{2}z_{2}^{2}}{\sqrt{M_{z_{1}}^{2}z_{2}^{2}+\gamma _{2}^{2}}} \end{aligned}$$
(34)
$$\begin{aligned} z_{2}g_{2}(\bar{x}_{2})\alpha _{2}\le & {} -\frac{z_{2}^{2}\bar{\alpha }_{2}^{2}}{\sqrt{z_{2}^{2}\bar{\alpha }_{2}^{2}+\gamma _{2}^{2}}}\le \gamma _{2}-z_{2}\bar{\alpha }_{2} \end{aligned}$$
(35)

Substitute (29), (30) and (33), (35) into (32) and the time derivative of \(V_{2}\) can be calculated as follows:

$$\begin{aligned} \begin{aligned} \dot{V}_{2}\le&\dot{V}_{1}-k_{2}z_{2}^{2}-\eta _{2}z_{2}^{3/2}+z_{2} \left( \nu _{12}-\dot{\alpha }_{1}\right) \\&-z_{2}\delta _{1}\tanh (z_{2}/\varepsilon ) +\kappa _{2}\hat{\varepsilon }_{2}\tilde{\varepsilon }_{2} +\mu _{2}\hat{\delta }_{1}\tilde{\delta }_{1}\\&+\sigma _{2}\tilde{W}_{2}^{\mathrm{T}}\hat{W}_{2}-\bar{g}_{1}\frac{M_{z_{1}}^{2}z_{2}^{2}}{\sqrt{M_{z_{1}}^{2}z_{2}^{2}+\gamma _{2}^{2}}}\\&+\gamma _{2}+z_{2}g_{2}(\bar{x}_{2})z_{3}+\bar{\varepsilon }_{2}\gamma _{2} \end{aligned} \end{aligned}$$
(36)

Considering \(|\nu _{12}-\dot{\alpha }_{1}|\le \delta _{1}\) and Lemma 6, we have:

$$\begin{aligned}&z_{2}(\nu _{12}-\dot{\alpha }_{1})-z_{2}\delta _{1}\tanh (z_{2}/\varepsilon )\le |z_{2}|\delta _{1}\nonumber \\&\quad -z_{2}\delta _{1}\tanh (z_{2}/\varepsilon )\nonumber \\&\quad \le \delta _{1}\varrho \varepsilon \end{aligned}$$
(37)

Taking (23) together with (34) and (37) into account, (36) becomes:

$$\begin{aligned} \begin{aligned} \dot{V}_{2}\le&-k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\eta _{1} \left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}}\tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -k_{2}z_{2}^{2}\\&-\eta _{2}z_{2}^{3/2}+\sum _{i=1}^{2} \left( \kappa _{i}\hat{\varepsilon }_{i}\tilde{\varepsilon }_{i} +\sigma _{i}\tilde{W}_{i}^{\mathrm{T}}\hat{W}_{i}\right) +\mu _{2}\hat{\delta }_{1}\tilde{\delta }_{1}\\&+z_{2}g_{2}(\bar{x}_{2})z_{3}+\bar{g}_{1}\gamma _{2} +\sum _{i=1}^{2} \left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i}\right) +\delta _{1}\varrho \varepsilon \end{aligned} \end{aligned}$$
(38)

Step i Similar to step 2, the unknown nonlinear function \(f_{i}(x)\) can be approximated using RBFNN and the dynamics of error variables \(z_{i}\) is

$$\begin{aligned} \begin{aligned} \dot{z}_{i}&=g_{i}(\bar{x}_{i})x_{i+1}+f_{i}(x)-\dot{\alpha }_{i-1}\\&=g_{i}(\bar{x}_{i})x_{i+1}+W_{i}^{*T}S_{i}(x)+\varepsilon _{i}-\dot{\alpha }_{i-1} \end{aligned} \end{aligned}$$
(39)

where \(W_{i}^{*}=blockdiag[W_{ik}^{*T}](k=1,\ldots ,m)\) is ideal weight vector, \(S_{i}(x)\) is radial basis function vector and \(\varepsilon _{i}\) is approximation error. The approximation error is bounded, i.e., \(|\varepsilon _{i}|\le \bar{\varepsilon }_{i}\).

Similar to step 2, the following first-order sliding mode differentiator is utilized to obtain the derivative of virtual control input:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{\nu }_{(i-1)1}=-\lambda _{0}|\nu _{(i-1)1}-\alpha _{i-1}|^{1/2}sign(\nu _{(i-1)1}-\alpha _{i-1})\\&+\nu _{(i-1)2}\\&\dot{\nu }_{(i-1)2}=-\lambda _{1}sign(\nu _{(i-1)1}-\alpha _{i-1}) \end{aligned} \right. \end{aligned} \end{aligned}$$
(40)

where \(\nu _{(i-1)1}\) and \(\nu _{(i-1)2}\) are state variables of the differentiator, \(\lambda _{0}\) and \(\lambda _{1}\) are positive real numbers. According to [71, 72], \(\nu _{(i-1)2}\) can approximate the first-order derivative of virtual control \(\alpha _{i-1}\) to arbitrary accuracy if the initial deviation \(|\nu _{(i-1)1}(t_{0})-\alpha _{i-1}(t_{0})|\) and \(|\nu _{(i-1)2}(t_{0})-\dot{\alpha }_{i-1}(t_{0})|\) are bounded. Therefore, we have \(|\nu _{(i-1)2}-\dot{\alpha }_{i-1}|\le \delta _{i-1}\) with \(\delta _{i-1}\) being an unknown positive constant.

The weight updating law for RBFNN is given as follows:

$$\begin{aligned} \dot{\hat{W}}_{i}=\varGamma _{i}(z_{i}S_{i}(x)-\sigma _{i}\hat{W}_{i}) \end{aligned}$$
(41)

where \(\varGamma _{i}\) and \(\sigma _{i}\) are positive real numbers.

The adaptive law for parameters \(\bar{\varepsilon }_{i}\) and \(\delta _{i-1}\) can be described as follows:

$$\begin{aligned} \dot{\hat{\varepsilon }}_{i}=\varLambda _{i} \left( \frac{z_{i}^{2}}{\sqrt{z_{i}^{2}+\gamma _{i}^{2}}}-\kappa _{i}\hat{\varepsilon }_{i}\right) \end{aligned}$$
(42)
$$\begin{aligned} \dot{\hat{\delta }}_{i-1}=\varPi _{i-1} \left( z_{i}\tanh (z_{i}/\varepsilon )-\mu _{i}\hat{\delta }_{i-1}\right) \end{aligned}$$
(43)

where \(\varLambda _{i}\), \(\varPi _{i-1}\), \(\gamma _{i}\), \(\mu _{i}\) and \(\kappa _{i}\) are positive real numbers.

The virtual control can be designed as follows:

$$\begin{aligned} \alpha _{i}=-\frac{1}{\underline{g}_{i}}\frac{z_{i}\bar{\alpha }_{i}^{2}}{\sqrt{z_{i}^{2}\bar{\alpha }_{i}^{2}+\gamma _{i}^{2}}} \end{aligned}$$
(44)

with

$$\begin{aligned} \bar{\alpha }_{i}= & {} k_{i}z_{i}+\eta _{i}z_{i}^{1/2}+\hat{\varepsilon }_{i}\frac{z_{i}}{\sqrt{z_{i}^{2}+\gamma _{i}^{2}}}-\nu _{(i-1)2}+\hat{W}_{i}^{\mathrm{T}}S_{i}(x)\nonumber \\&+\,\hat{\delta }_{i-1}\tanh (z_{i}/\varepsilon )+\bar{g}_{i-1}\frac{z_{i-1}^{2}z_{i}}{\sqrt{z_{i-1}^{2}z_{i}^{2}+\gamma _{i}^{2}}} \end{aligned}$$
(45)

where \(k_{i}\), \(\eta _{i}\) are positive real numbers.

Consider the following Lyapunov function:

$$\begin{aligned} V_{i}=V_{i-1}+\frac{1}{2}z_{i}^{2}+\frac{1}{2\varLambda _{i}}\tilde{\varepsilon }_{i}^{2}+\frac{1}{2\varPi _{i-1}}\tilde{\delta }_{i-1}^{2}+\frac{1}{2\varGamma _{i}}\tilde{W}_{i}^{\mathrm{T}}\tilde{W}_{i} \end{aligned}$$
(46)

where \(\tilde{\varepsilon }_{i}=\bar{\varepsilon }_{i}-\hat{\varepsilon }_{i}\), \(\tilde{\delta }_{i-1}=\delta _{i-1}-\hat{\delta }_{i-1}\), \(\tilde{W}_{i}=W_{i}^{*}-\hat{W}_{i}\).

Define \(z_{i+1}=x_{i+1}-\alpha _{i}\) and the time derivative of \(V_{i}\) along (39), (41–43) is

$$\begin{aligned} \begin{aligned} \dot{V}_{i}&=\dot{V}_{i-1}+z_{i}\dot{z}_{i}-\frac{1}{\varLambda _{i}}\tilde{\varepsilon }_{i}\dot{\hat{\varepsilon }}_{i}-\frac{1}{\varPi _{i-1}}\tilde{\delta }_{i-1}\dot{\hat{\delta }}_{i-1}\\&\quad -\frac{1}{\varGamma _{i}}\tilde{W}_{i}^{\mathrm{T}}\dot{\hat{W}}_{i}\\&=\dot{V}_{i-1}+z_{i}(g_{i}(\bar{x}_{i})(z_{i+1}+\alpha _{i})\\&\quad +W_{i}^{*T}S_{i}(x)+\varepsilon _{i}-\dot{\alpha }_{i-1})\\&\quad -\tilde{\delta }_{i-1} \left( z_{i}\tanh (z_{i}/\varepsilon )-\mu _{i}\hat{\delta }_{i-1}\right) \\&\quad -\tilde{\varepsilon }_{i} \left( \frac{z_{i}^{2}}{\sqrt{z_{i}^{2}+\gamma _{i}^{2}}} -\kappa _{i}\hat{\varepsilon }_{i}\right) -\tilde{W}_{i}^{\mathrm{T}}\left( z_{i}S_{i}(x)-\sigma _{i}\hat{W}_{i}\right) \\&=\dot{V}_{i-1}+z_{i}g_{i} (\bar{x}_{i})z_{i+1}+z_{i}g_{i} (\bar{x}_{i})\alpha _{i}+\hat{W}_{i}^{\mathrm{T}}z_{i}S_{i}(x)\\&\quad +z_{i}\varepsilon _{i}-\tilde{\varepsilon }_{i}\frac{z_{i}^{2}}{\sqrt{z_{i}^{2}+\gamma _{i}^{2}}} +\kappa _{i}\tilde{\varepsilon }_{i}\hat{\varepsilon }_{i}-z_{i} \dot{\alpha }_{i-1}\\&\quad -\tilde{\delta }_{i-1}z_{i}\tanh (z_{i}/\varepsilon ) +\mu _{i}\tilde{\delta }_{i-1}\hat{\delta }_{i-1} +\sigma _{i}\tilde{W}_{i}^{\mathrm{T}}\hat{W}_{i} \end{aligned} \end{aligned}$$
(47)

By Lemma 1, the following inequality is established:

$$\begin{aligned} z_{i}\varepsilon _{i}\le |z_{i}|\bar{\varepsilon }_{i}<\bar{\varepsilon }_{i}\gamma _{i}+\bar{\varepsilon }_{i}\frac{z_{i}^{2}}{\sqrt{z_{i}^{2}+\gamma _{i}^{2}}} \end{aligned}$$
(48)

According to Lemma 1 and Assumption 2, we have:

$$\begin{aligned}&z_{i-1}g_{i-1}(\bar{x}_{i-1})z_{i}\le \bar{g}_{i-1}|z_{i-1}||z_{i}|\nonumber \\&\quad <\bar{g}_{i-1}\gamma _{i}+\bar{g}_{i-1}\frac{z_{i-1}^{2}z_{i}^{2}}{\sqrt{z_{i-1}^{2}z_{i}^{2}+\gamma _{i}^{2}}} \end{aligned}$$
(49)
$$\begin{aligned}&z_{i}g_{i}(\bar{x}_{i})\alpha _{i}\le -\frac{z_{i}^{2}\bar{\alpha }_{i}^{2}}{\sqrt{z_{i}^{2}\bar{\alpha }_{i}^{2}+\gamma _{i}^{2}}}\le \gamma _{i}-z_{i}\bar{\alpha }_{i} \end{aligned}$$
(50)

Substituting (44), (45), (48) and (50) into (47) yields:

$$\begin{aligned} \begin{aligned} \dot{V}_{i}\le&\dot{V}_{i-1}-k_{i}z_{i}^{2}-\eta _{i}z_{i}^{3/2}+\kappa _{i}\hat{\varepsilon }_{i}\tilde{\varepsilon }_{i}+\mu _{i}\hat{\delta }_{i-1}\tilde{\delta }_{i-1}\\&+z_{i}(\nu _{(i-1)2}-\dot{\alpha }_{i-1})-z_{i}\delta _{i-1} \tanh (z_{i}/\varepsilon )\\&+\sigma _{i}\tilde{W}_{i}^{\mathrm{T}}\hat{W}_{i} -\bar{g}_{i-1}\frac{z_{i-1}^{2}z_{i}^{2}}{\sqrt{z_{i-1}^{2}z_{i}^{2} +\gamma _{i}^{2}}}+\gamma _{i}\\&+z_{i}g_{i}(\bar{x}_{i})z_{i+1}+\bar{\varepsilon }_{i}\gamma _{i} \end{aligned} \end{aligned}$$
(51)

Similar to step 2, one has:

$$\begin{aligned} \begin{aligned}&z_{i}(\nu _{(i-1)2}-\dot{\alpha }_{i-1})-z_{i}\delta _{i-1}\tanh (z_{i}/\varepsilon )\\&\quad \le |z_{i}|\delta _{i-1}-z_{i}\delta _{i-1}\tanh (z_{i}/\varepsilon )\le \delta _{i-1}\varrho \varepsilon \end{aligned} \end{aligned}$$
(52)

Substituting (52) and (49) into (51) produces:

$$\begin{aligned} \begin{aligned} \dot{V}_{i}\le&-k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \\&-\eta _{1}\left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}} \tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\sum _{j=2}^{i}\left( k_{j}z_{j}^{2}\right. \\&\left. +\eta _{j}z_{j}^{3/2}\right) +\sum _{j=1}^{i} \left( \kappa _{j}\hat{\varepsilon }_{j}\tilde{\varepsilon }_{j} +\sigma _{j}\tilde{W}_{j}^{\mathrm{T}}\hat{W}_{j}\right) \\&+\sum _{j=2}^{i} \left( \mu _{j}\hat{\delta }_{j-1}\tilde{\delta }_{j-1} +\bar{g}_{j-1}\gamma _{j}+\delta _{j-1}\varrho \varepsilon \right) \\&+z_{i}g_{i}(\bar{x}_{i})z_{i+1} +\sum _{j=1}^{i}\left( \gamma _{j}+\bar{\varepsilon }_{j}\gamma _{j}\right) \end{aligned} \end{aligned}$$
(53)

Step n In the final design step, the actual control input will be derived. Following the same line used in (24), the dynamics of error variable \(z_{n}\) can be described as follows:

$$\begin{aligned} \begin{aligned} \dot{z}_{n}&=g_{n}(\bar{x}_{n})u(v)+f_{n}(x)-\dot{\alpha }_{n-1}\\&=g_{n}(\bar{x}_{n})u(v)+W_{n}^{*T}S_{n}(x)+\varepsilon _{n}-\dot{\alpha }_{n-1}\\&=g_{n}(\bar{x}_{n})(u_{d}+\Delta u)+W_{n}^{*T}S_{n}(x)+\varepsilon _{n}-\dot{\alpha }_{n-1} \end{aligned} \end{aligned}$$
(54)

where \(W_{n}^{*}=blockdiag[W_{nk}^{*T}](k=1,\ldots ,m)\) is ideal weight vector, \(S_{n}(x)\) is radial basis function vector and \(\varepsilon _{n}\) is approximation error. The approximation error is bounded, i.e., \(|\varepsilon _{n}|\le \bar{\varepsilon }_{n}\).

Similar to step 2, the following first-order sliding mode differentiator is utilized to obtain the derivative of virtual control input:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{\nu }_{(n-1)1}=-\lambda _{0}|\nu _{(n-1)1}-\alpha _{n-1}|^{1/2}sign(\nu _{(n-1)1}\\&-\alpha _{n-1})+\nu _{(n-1)2}\\&\dot{\nu }_{(n-1)2}=-\lambda _{1}sign(\nu _{(n-1)1}-\alpha _{n-1}) \end{aligned} \right. \end{aligned} \end{aligned}$$
(55)

where \(\nu _{(n-1)1}\) and \(\nu _{(n-1)2}\) are state variables of the differentiator, \(\lambda _{0}\) and \(\lambda _{1}\) are positive real numbers. According to [71, 72], \(\nu _{(n-1)2}\) can approximate the first-order derivative of virtual control \(\alpha _{n-1}\) to arbitrary accuracy if the initial deviation \(|\nu _{(n-1)1}(t_{0})-\alpha _{n-1}(t_{0})|\) and \(|\nu _{(n-1)2}(t_{0})-\dot{\alpha }_{n-1}(t_{0})|\) are bounded. Therefore, we have \(|\nu _{(n-1)2}-\dot{\alpha }_{n-1}|\le \delta _{n-1}\) with \(\delta _{n-1}\) being an unknown positive constant.

By repeating the same way used in step 1, step 2 and step i, we can give the weight updating law for RBFNN, the adaptive law for parameters \(\bar{\varepsilon }_{n}\) and \(\delta _{n-1}\) as follows:

$$\begin{aligned}&\dot{\hat{W}}_{n}=\varGamma _{n}(z_{n}S_{n}(x)-\sigma _{n}\hat{W}_{n})\end{aligned}$$
(56)
$$\begin{aligned}&\dot{\hat{\varepsilon }}_{n}=\varLambda _{n} \left( \frac{z_{n}^{2}}{\sqrt{z_{n}^{2}+\gamma _{n}^{2}}}-\kappa _{n}\hat{\varepsilon }_{n}\right) \end{aligned}$$
(57)
$$\begin{aligned}&\dot{\hat{\delta }}_{n-1}=\varPi _{n-1} \left( z_{n}\tanh (z_{n}/\varepsilon )-\mu _{n}\hat{\delta }_{n-1}\right) \end{aligned}$$
(58)

where \(\varGamma _{n}\), \(\sigma _{n}\), \(\gamma _{n}\), \(\varLambda _{n}\), \(\kappa _{n}\), \(\varPi _{n-1}\) and \(\mu _{n}\) are positive real numbers.

To deal with input saturation, the following anti-windup compensator is introduced:

$$\begin{aligned} \begin{aligned} \dot{w}=\left\{ \begin{aligned}&-kw-\xi sig^{1/2}w-\frac{\bar{g}_{n}\frac{z_{n}^{2}\Delta u^{2}}{\sqrt{z_{n}^{2}\Delta u^{2}+\gamma _{n}^{2}}}+\frac{1}{3}(\Delta u)^{3}}{w}\\&+\Delta u,\quad \mathrm{if}~|w|\ge \tau \\&0,\quad \mathrm{if}~|w|<\tau \end{aligned} \right. \end{aligned} \end{aligned}$$
(59)

where w is the state of auxiliary design system, k and \(\xi \) are positive real numbers to be designed, \(\tau \) is a small positive constant, \(sig^{\alpha }(\cdot )=|\cdot |^{\alpha }sign(\cdot )\).

Remark 7

\(|w|<\tau \) means that there is no input saturation and the anti-windup compensator does not work, while \(|w|\ge \tau \) means that there exists input saturation and the anti-windup compensator activates to compensate the control input error \(\Delta u\) caused by input saturation.

Considering input saturation and dead zone, the actual control input can be designed as follows:

$$\begin{aligned} v=-\frac{1}{\underline{g}_{n}m_{0}}\frac{z_{n}\bar{v}^{2}}{\sqrt{z_{n}^{2}\bar{v}^{2}+\gamma _{n}^{2}}} \end{aligned}$$
(60)

with

$$\begin{aligned} \begin{aligned} \bar{v}&=k_{n}z_{n}+\eta _{n}z_{n}^{1/2}+\hat{\varepsilon }_{n} \frac{z_{n}}{\sqrt{z_{n}^{2} +\gamma _{n}^{2}}}\\&\quad -\nu _{(n-1)2}+\hat{W}_{n}^{\mathrm{T}}S_{n}(x)\\&\quad +\hat{\delta }_{n-1}\tanh (z_{n}/\varepsilon )+\bar{g}_{n-1}\frac{z_{n-1}^{2}z_{n}}{\sqrt{z_{n-1}^{2}z_{n}^{2}+\gamma _{n}^{2}}} \end{aligned} \end{aligned}$$
(61)

where \(k_{n}\), \(\eta _{n}\) are positive real numbers.

Let us consider the first case where input saturation exists, i.e., \(|w|\ge \tau \). Define the following Lyapunov function:

$$\begin{aligned} V_{n}= & {} V_{n-1}+\frac{1}{2}z_{n}^{2}+\frac{1}{2\varLambda _{n}}\tilde{\varepsilon }_{n}^{2}+\frac{1}{2\varPi _{n-1}}\tilde{\delta }_{n-1}^{2}+\frac{1}{2}w^{2}\nonumber \\&+\frac{1}{2\varGamma _{n}}\tilde{W}_{n}^{\mathrm{T}}\tilde{W}_{n} \end{aligned}$$
(62)

where \(\tilde{\varepsilon }_{n}=\bar{\varepsilon }_{n}-\hat{\varepsilon }_{n}\), \(\tilde{\delta }_{n-1}=\delta _{n-1}-\hat{\delta }_{n-1}\), \(\tilde{W}_{n}=W_{n}^{*}-\hat{W}_{n}\).

Noting that \(u_{d}=K^{\mathrm{T}}(t)\varPhi (t)v(t)+\Delta \) and taking time derivative of \(V_{n}\) along (54), (56–59) obtain:

$$\begin{aligned} \dot{V}_{n}= & {} \dot{V}_{n-1}+z_{n}\dot{z}_{n} -\frac{1}{\varLambda _{n}}\tilde{\varepsilon }_{n}\dot{\hat{\varepsilon }}_{n}\nonumber \\&-\frac{1}{\varPi _{n-1}}\tilde{\delta }_{n-1}\dot{\hat{\delta }}_{n-1} +w\dot{w} -\frac{1}{\varGamma _{n}}\tilde{W}_{n}^{\mathrm{T}}\dot{\hat{W}}_{n}\nonumber \\= & {} \dot{V}_{n-1}+z_{n}(g_{n}(\bar{x}_{n})(u_{d}+\Delta u)\nonumber \\&+W_{n}^{*T}S_{n}(x)+\varepsilon _{n}-\dot{\alpha }_{n-1})\nonumber \\&-\tilde{\delta }_{n-1} \left( z_{n}\tanh (z_{n}/\varepsilon )-\mu _{n}\hat{\delta }_{n-1}\right) \nonumber \\&-\tilde{\varepsilon }_{n}\left( \frac{z_{n}^{2}}{\sqrt{z_{n}^{2} +\gamma _{n}^{2}}}-\kappa _{n}\hat{\varepsilon }_{n}\right) \nonumber \\&-\tilde{W}_{n}^{\mathrm{T}}\left( z_{n}S_{n}(x)-\sigma _{n}\hat{W}_{n}\right) \nonumber \\&+w(-kw-\xi sig^{1/2}w\nonumber \\&-\frac{\bar{g}_{n}\frac{z_{n}^{2}\Delta u^{2}}{\sqrt{z_{n}^{2}\Delta u^{2}+\gamma _{n}^{2}}}+\frac{1}{3}(\Delta u)^{3}}{w}+\Delta u)\nonumber \\= & {} \dot{V}_{n-1}+z_{n}g_{n}(\bar{x}_{n})\left( K^{\mathrm{T}}(t)\varPhi (t)v(t)+\Delta \right) \nonumber \\&+z_{n}g_{n}(\bar{x}_{n})\Delta u +\hat{W}_{n}^{\mathrm{T}}z_{n}S_{n}(x)+z_{n}\varepsilon _{n}\nonumber \\&-\tilde{\varepsilon }_{n}\frac{z_{n}^{2}}{\sqrt{z_{n}^{2}+\gamma _{n}^{2}}} -z_{n}\dot{\alpha }_{n-1}-\tilde{\delta }_{n-1}z_{n}\tanh (z_{n}/\varepsilon )\nonumber \\&+\,\kappa _{n}\tilde{\varepsilon }_{n}\hat{\varepsilon }_{n} +\mu _{n}\tilde{\delta }_{n-1}\hat{\delta }_{n-1} +\sigma _{n}\tilde{W}_{n}^{\mathrm{T}}\hat{W}_{n}-kw^{2}\nonumber \\&-\,\xi |w|^{3/2}-\bar{g}_{n}\frac{z_{n}^{2}\Delta u^{2}}{\sqrt{z_{n}^{2}\Delta u^{2} +\gamma _{n}^{2}}}\nonumber \\&-\frac{1}{3}(\Delta u)^{3}+w\Delta u \end{aligned}$$
(63)

Following the same line as step 1, step 2 and step i, we obtain the following inequalities:

$$\begin{aligned}&z_{n}\varepsilon _{n}\le z_{n}\bar{\varepsilon }_{n}<\bar{\varepsilon }_{n}\gamma _{n}+\bar{\varepsilon }_{n}\frac{z_{n}^{2}}{\sqrt{z_{n}^{2}+\gamma _{n}^{2}}}\end{aligned}$$
(64)
$$\begin{aligned}&z_{n}g_{n}(\bar{x}_{n})\Delta u\le \bar{g}_{n}|z_{n}||\Delta u|<\bar{g}_{n}\gamma _{n}+\bar{g}_{n}\frac{z_{n}^{2}\Delta u^{2}}{\sqrt{z_{n}^{2}\Delta u^{2}+\gamma _{n}^{2}}} \nonumber \\\end{aligned}$$
(65)
$$\begin{aligned}&z_{n-1}g_{n-1}(\bar{x}_{n-1})z_{n}\le \bar{g}_{n-1}|z_{n-1}||z_{n}|\nonumber \\&\quad <\bar{g}_{n-1}\gamma _{n}+\bar{g}_{n-1}\frac{z_{n-1}^{2}z_{n}^{2}}{\sqrt{z_{n-1}^{2}z_{n}^{2}+\gamma _{n}^{2}}} \end{aligned}$$
(66)
$$\begin{aligned}&z_{n}g_{n}(\bar{x}_{n})K^{\mathrm{T}}\varPhi v\le -\frac{z_{n}^{2}\bar{v}^{2}}{\sqrt{z_{n}^{2}\bar{v}^{2}+\gamma _{n}^{2}}}\le \gamma _{n}-z_{n}\bar{v} \end{aligned}$$
(67)

Invoking Lemma 2, Assumption 2 and Remark 3, we have:

$$\begin{aligned}&w\Delta u \le \frac{2}{3}|w|^{3/2}+\frac{1}{3}|\Delta u|^{3} \end{aligned}$$
(68)
$$\begin{aligned}&z_{n}g_{n}\Delta \le \bar{g}_{n}|z_{n}|\bar{\Delta }\le \frac{\bar{g}_{n}}{2} \left( z_{n}^{2}+\bar{\Delta }^{2}\right) \end{aligned}$$
(69)

Substituting (60), (61), (64), (65), (67), (68) and (69) into (63) leads to

$$\begin{aligned} \dot{V}_{n}\le & {} \dot{V}_{n-1}-\left( k_{n}-\frac{\bar{g}_{n}}{2}\right) z_{n}^{2}-\eta _{n}z_{n}^{3/2}\nonumber \\&+z_{n}\left( \nu _{(n-1)2}-\dot{\alpha }_{n-1}\right) -z_{n}\delta _{n-1}\tanh (z_{n}/\varepsilon )\nonumber \\&+\gamma _{n}+\bar{\varepsilon }_{n}\gamma _{n}+\bar{g}_{n}\gamma _{n} +\kappa _{n}\tilde{\varepsilon }_{n}\hat{\varepsilon }_{n}+\mu _{n}\tilde{\delta }_{n-1}\hat{\delta }_{n-1}-kw^{2} \nonumber \\&- \left( \xi -\frac{2}{3}\right) |w|^{3/2} +\,\sigma _{n}\tilde{W}_{n}^{\mathrm{T}}\hat{W}_{n} +\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2}\nonumber \\&-\bar{g}_{n-1}\frac{z_{n-1}^{2}z_{n}^{2}}{\sqrt{z_{n-1}^{2}z_{n}^{2} +\gamma _{n}^{2}}} \end{aligned}$$
(70)

Similar to step 2 and step i, the following inequality holds:

$$\begin{aligned} \begin{aligned}&z_{n}(\nu _{(n-1)2}-\dot{\alpha }_{n-1})-z_{n}\delta _{n-1}\tanh (z_{n}/\varepsilon )\\&\le |z_{n}|\delta _{n-1}-z_{n}\delta _{n-1}\tanh (z_{n}/\varepsilon )\le \delta _{n-1}\varrho \varepsilon \end{aligned} \end{aligned}$$
(71)

Substituting (71) into (70), we have:

$$\begin{aligned} \dot{V}_{n}\le & {} -k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\eta _{1} \left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}}\tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \nonumber \\&-\sum _{i=2}^{n-1}\left( k_{i}z_{i}^{2}+\eta _{i}z_{i}^{3/2}\right) \nonumber \\&-\left( k_{n}-\frac{\bar{g}_{n}}{2}\right) z_{n}^{2}-\eta _{n}z_{n}^{3/2} +\sum _{i=1}^{n}\left( \kappa _{i}\hat{\varepsilon }_{i}\tilde{\varepsilon }_{i}\right. \nonumber \\&\left. +\,\sigma _{i}\tilde{W}_{i}^{\mathrm{T}}\hat{W}_{i}\right) +\sum _{i=2}^{n}\mu _{i}\hat{\delta }_{i-1}\tilde{\delta }_{i-1}-kw^{2}\nonumber \\&-\left( \xi -\frac{2}{3}\right) |w|^{3/2}\nonumber \\&+\sum _{i=2}^{n}\left( \bar{g}_{i-1}\gamma _{i} +\delta _{i-1}\varrho \varepsilon \right) +\bar{g}_{n}\gamma _{n}\nonumber \\&+\sum _{i=1}^{n} \left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i}\right) +\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2} \end{aligned}$$
(72)

Using Lemma 2, we have:

$$\begin{aligned} \begin{aligned} \tilde{\varepsilon }_{i}\hat{\varepsilon }_{i}&\le -\frac{\tilde{\varepsilon }_{i}^{2}}{2}+\frac{\varepsilon _{i}^{2}}{2}\\&=-\frac{\tilde{\varepsilon }_{i}^{2}}{4} -\frac{1}{4}\left( |\tilde{\varepsilon }_{i}| -\sqrt{|\tilde{\varepsilon }_{i}|}\right) ^{2}\\&\quad +\frac{1}{4}|\tilde{\varepsilon }_{i}|-\frac{1}{2}|\tilde{\varepsilon }_{i}|^{3/2}+\frac{\varepsilon _{i}^{2}}{2}\\&\le -\frac{\tilde{\varepsilon }_{i}^{2}}{4}+\frac{1}{8}|\tilde{\varepsilon }_{i}|^{2}+\frac{1}{8}-\frac{1}{2}|\tilde{\varepsilon }_{i}|^{3/2}+\frac{\varepsilon _{i}^{2}}{2}\\&\le -\frac{\tilde{\varepsilon }_{i}^{2}}{8}+\frac{1}{8}-\frac{1}{2}|\tilde{\varepsilon }_{i}|^{3/2}+\frac{\varepsilon _{i}^{2}}{2} \end{aligned} \end{aligned}$$
(73)

By repeating the same way used in (73), the following inequalities can be obtained:

$$\begin{aligned}&\tilde{\delta }_{i-1}\hat{\delta }_{i-1}\le -\frac{\tilde{\delta }_{i-1}^{2}}{8}+\frac{1}{8}-\frac{1}{2}|\tilde{\delta }_{i-1}|^{3/2}+\frac{\delta _{i-1}^{2}}{2} \end{aligned}$$
(74)
$$\begin{aligned}&\tilde{W}_{i}^{\mathrm{T}}\hat{W}_{i}\le -\frac{\Vert \tilde{W}_{i}\Vert ^{2}}{8}+\frac{1}{8}-\frac{1}{2}\Vert \tilde{W}_{i}\Vert ^{3/2}+\frac{\Vert W_{i}\Vert ^{2}}{2}\nonumber \\ \end{aligned}$$
(75)

Substituting (73–75) into (72), one has:

$$\begin{aligned} \dot{V}_{n}\le & {} -k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \nonumber \\&-\eta _{1}\left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}} \tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\sum _{i=2}^{n-1}\left( k_{i}z_{i}^{2}\right. \nonumber \\&\left. +\eta _{i}z_{i}^{3/2}\right) - \left( k_{n}-\frac{\bar{g}_{n}}{2}\right) z_{n}^{2}\nonumber \\&-\eta _{n}z_{n}^{3/2} +\sum _{i=1}^{n}\left( -\frac{\kappa _{i}\tilde{\varepsilon }_{i}^{2}}{8}\right. -\frac{\kappa _{i}}{2}|\tilde{\varepsilon }|^{3/2} -\frac{\sigma _{i}\Vert \tilde{W}_{i}\Vert ^{2}}{8}\nonumber \\&\left. -\frac{\sigma _{i}}{2}\Vert \tilde{W}_{i}\Vert ^{3/2}\right) -\sum _{i=2}^{n}\left( \frac{\mu _{i}\tilde{\delta }_{i-1}^{2}}{8}+\frac{\mu _{i}}{2}|\tilde{\delta }_{i-1}|^{3/2}\right) \nonumber \\&-kw^{2}-\left( \xi -\frac{2}{3}\right) |w|^{3/2}\nonumber \\&+\sum _{i=2}^{n}\left( \bar{g}_{i-1}\gamma _{i}+\frac{\mu _{i}}{8}+\frac{\mu _{i}\delta _{i-1}^{2}}{2} +\delta _{i-1}\varrho \varepsilon \right) +\bar{g}_{n}\gamma _{n}\nonumber \\&+\sum _{i=1}^{n}\left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i}+\frac{\kappa _{i}}{8}+\kappa _{i}\frac{\varepsilon ^{2}}{2} +\frac{\sigma _{i}}{8}+\frac{\sigma _{i}\Vert W_{i}\Vert ^{2}}{2}\right) \nonumber \\&+\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2} \end{aligned}$$
(76)

By Lemma 3 and Lemma 4, (76) can be written as follows:

$$\begin{aligned} \dot{V}_{n}\le & {} -k_{1}\frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) -\sum _{i=2}^{n-1}k_{i}z_{i}^{2}- \left( k_{n}-\frac{\bar{g}_{n}}{2}\right) z_{n}^{2}\nonumber \\&-\sum _{i=1}^{n}\frac{\kappa _{i}\tilde{\varepsilon }_{i}^{2}}{8} -\sum _{i=1}^{n}\frac{\sigma _{i}\Vert \tilde{W}_{i}\Vert ^{2}}{8} -\sum _{i=2}^{n}\frac{\mu _{i}\tilde{\delta }_{i-1}^{2}}{8}-kw^{2}\nonumber \\&-\eta _{1}\left( \frac{k_{b}^{2}}{\pi }\right) ^{\frac{3}{4}} \tan ^{\frac{3}{4}}\left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) \nonumber \\&-\sum _{i=2}^{n}\eta _{i}z_{i}^{3/2} -\sum _{i=1}^{n}\frac{\kappa _{i}|\tilde{\varepsilon }_{i}|^{3/2}}{2}\nonumber \\&-\sum _{i=1}^{n}\frac{\sigma _{i}\Vert \tilde{W}_{i}\Vert ^{3/2}}{2} -\sum _{i=2}^{n}\frac{\mu _{i}|\tilde{\delta }_{i-1}|^{3/2}}{2}\nonumber \\&-\left( \xi -\frac{2}{3}\right) |w|^{3/2}\nonumber \\&+\sum _{i=2}^{n} \left( \bar{g}_{i-1}\gamma _{i}+\frac{\mu _{i}}{8} +\frac{\mu _{i}\delta _{i-1}^{2}}{2}+\delta _{i-1}\varrho \varepsilon \right) \nonumber \\&+\,\bar{g}_{n}\gamma _{n}\nonumber \\&+\sum _{i=1}^{n} \left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i} +\frac{\kappa _{i}}{8}+\kappa _{i}\frac{\varepsilon ^{2}}{2} +\frac{\sigma _{i}}{8}+\frac{\sigma _{i}\Vert W_{i}\Vert ^{2}}{2}\right) \nonumber \\&+\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2}\nonumber \\\le & {} -\beta _{1}\left( \frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) +\frac{1}{2}\sum _{i=2}^{n}z_{i}^{2} +\sum _{i=1}^{n}\frac{1}{2\varLambda _{i}}\tilde{\varepsilon }_{i}^{2}\right. \nonumber \\&\left. +\sum _{i=1}^{n}\frac{1}{2\varGamma _{i}} \tilde{W}_{i}^{\mathrm{T}}\tilde{W}_{i} +\sum _{i=2}^{n}\frac{1}{2\varPi _{i-1}}\tilde{\delta }_{i}^{2} +\frac{1}{2}w^{2}\right) \nonumber \\&-\beta _{2}\left( \frac{k_{b}^{2}}{\pi }\tan \left( \frac{\pi z_{1}^{2}}{2k_{b}^{2}}\right) +\frac{1}{2}\sum _{i=2}^{n}z_{i}^{2} +\sum _{i=1}^{n}\frac{1}{2\varLambda _{i}}\tilde{\varepsilon }_{i}^{2}\right. \nonumber \\&\left. +\sum _{i=1}^{n} \frac{1}{2\varGamma _{i}}\tilde{W}_{i}^{\mathrm{T}}\tilde{W}_{i}\right. \nonumber \\&\left. +\sum _{i=2}^{n}\frac{1}{2\varPi _{i-1}}\tilde{\delta }_{i}^{2} +\frac{1}{2}w^{2}\right) ^{3/4}+C\nonumber \\= & {} -\beta _{1}V_{n}-\beta _{2}V_{n}^{3/4}+C \end{aligned}$$
(77)

where

$$\begin{aligned}&\beta _{1}=\min \left\{ k_{1},2k_{i},\kappa _{i}\varLambda _{i}/4,\sigma _{i}\varGamma _{i}/4,\mu _{i}\varPi _{i-1}/4,2\left( k_{n} -\frac{\bar{g}_{n}}{2}\right) ,\right. \\&\qquad \qquad \left. 2k\right\} \\&\beta _{2}=\min \left\{ \eta _{1},2^{\frac{3}{4}}\eta _{i},2^ {-\frac{1}{4}}\kappa _{i}\varLambda _{i}^{\frac{3}{4}},2^{ -\frac{1}{4}}\sigma _{i}\varGamma _{i}^{\frac{3}{4}},2^{ -\frac{1}{4}}\mu _{i}\varPi _{i-1}^{\frac{3}{4}},\right. \\&\quad \left. 2^{\frac{3}{4}}\left( \xi -\frac{2}{3}\right) ,2^{\frac{3}{4}} \eta _{n}\right\} \\&C=\sum _{i=2}^{n}\left( \bar{g}_{i-1}\gamma _{i} +\frac{\mu _{i}}{8}+\frac{\mu _{i}\delta _{i-1}^{2}}{2} +\delta _{i-1}\varrho \varepsilon \right) +\bar{g}_{n}\gamma _{n}\\&\qquad +\sum _{i=1}^{n}\left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i} +\frac{\kappa _{i}}{8}+\kappa _{i}\frac{\varepsilon ^{2}}{2} +\frac{\sigma _{i}}{8}+\frac{\sigma _{i}\Vert W_{i}\Vert ^{2}}{2}\right) \\&\qquad +\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2} \end{aligned}$$

If input saturation does not exist, i.e. \(|w|<\tau \), the anti-windup compensator does not work and the state of anti-windup compensator keeps zero. In this case, the considered Lyapunov function can be written as follows:

$$\begin{aligned} V_{n}=V_{n-1}+\frac{1}{2}z_{n}^{2}+\frac{1}{2\varLambda _{n}}\tilde{\varepsilon }_{n}^{2}+\frac{1}{2\varPi _{n-1}}\tilde{\delta }_{n-1}^{2}+\frac{1}{2\varGamma _{n}}\tilde{W}_{n}^{\mathrm{T}}\tilde{W}_{n} \end{aligned}$$
(78)

By following the same line as the case where input saturation exists, the time derivative of Lyapunov function (78) can be obtained as follows:

$$\begin{aligned} \dot{V}_{n}\le -\beta _{1}V_{n}-\beta _{2}V_{n}^{3/4}+C \end{aligned}$$
(79)

where

$$\begin{aligned}&\beta _{1}=\min \left\{ k_{1},2k_{i},\kappa _{i}\varLambda _{i}/4,\sigma _{i}\varGamma _{i}/4,\mu _{i}\varPi _{i-1}/4,2\left( k_{n}-\frac{\bar{g}_{n}}{2}\right) \right\} \\&\beta _{2}=\min \left\{ \eta _{1},2^{\frac{3}{4}}\eta _{i},2^{-\frac{1}{4}}\kappa _{i}\varLambda _{i}^{\frac{3}{4}},2^{-\frac{1}{4}}\sigma _{i}\varGamma _{i}^{\frac{3}{4}},2^{-\frac{1}{4}}\mu _{i}\varPi _{i-1}^{\frac{3}{4}}, 2^{\frac{3}{4}}\eta _{n}\right\} \\&C=\sum _{i=2}^{n}\left( \bar{g}_{i-1}\gamma _{i}+\frac{\mu _{i}}{8} +\frac{\mu _{i}\delta _{i-1}^{2}}{2}+\delta _{i-1}\varrho \varepsilon \right) +\sum _{i=1}^{n}\left( \gamma _{i}+\bar{\varepsilon }_{i}\gamma _{i}\right. \\&\qquad \,\, \left. +\frac{\kappa _{i}}{8}+\kappa _{i} \frac{\varepsilon ^{2}}{2}+\frac{\sigma _{i}}{8} +\frac{\sigma _{i}W_{i}^{2}}{2}\right) +\frac{\bar{g}_{n}}{2}\bar{\Delta }^{2} \end{aligned}$$

The above controller design and stability analysis can be summarized as follows.

Theorem 1

Consider uncertain nonstrict-feedback nonlinear system (1) with input saturation, dead zone and output constraint. Suppose that Assumptions 1-3 hold. The neural network weight updating law is selected as (13), (26), (41) and (56). The adaptive law for unknown parameters is chosen as (14), (27), (28), (42), (43), (57) and (58). The virtual control laws are designed as (17), (30), (44) and (60). Then, the following properties can be guaranteed: (1) All the closed-loop signals are bounded. (2) The system output constraint will not be violated; (3) System output tracking error will converge to a compact set \(\{z_{1}:|z_{1}|<\max \{\upsilon ,\sqrt{\tan ^{-1}(\frac{C\pi }{\beta _{1}\varpi k_{b}^{2}})\frac{2k_{b}^{2}}{\pi }}\}\}\) within finite time, which can be designed arbitrarily small.

Proof

From (77) and (79), we have:

$$\begin{aligned} \dot{V}_{n}<-\beta _{1}V_{n}+C \end{aligned}$$
(80)

Integrating both side of (80) over [0, t], we obtain:

$$\begin{aligned} V_{n}<\left( V_{n}(0)-\frac{C}{\beta _{1}}\right) e^{-\beta _{1}t}+\frac{C}{\beta _{1}} \end{aligned}$$
(81)

The boundedness of Lyapunov function \(V_{n}\) means that Barrier Lyapunov function \(V_{b}\), the error variables \(z_{i}\), \(\tilde{\varepsilon }_{i}\), \(\tilde{W}_{i}\), \(\tilde{\delta }_{i}\) and anti-windup state w are bounded. Since \(\bar{\varepsilon }_{i}\) and \(\delta _{i}\) are constants, we have that \(\hat{\varepsilon }_{i}\) and \(\hat{\delta }_{i}\) are bounded. Given the fact that \(z_{1}\) and \(y_{r}\) are bounded, we have that \(x_{1}\) is bounded. Due to the boundedness of radial basis function \(S_{i}(x)\) and the error variables \(z_{i}\), from (13), (26), (41) and (56), we have that the neural network weight estimation \(\hat{W}_{i}\) is bounded. From Assumption 3, we have that \(\dot{y}_{r}\) is bounded. Since \(z_{1}\), \(\hat{\varepsilon }_{1}\), \(\dot{y}_{r}\), \(S_{1}(x)\) and \(\hat{W}_{1}\) are bounded and \(\underline{g}_{1}\) is a constant, from (15) and (17), we can see that \(\bar{\alpha }_{1}\) and \(\alpha _{1}\) are bounded. Due to the boundedness of \(z_{2}\) and \(\alpha _{1}\), we have that \(x_{2}\) is bounded. Since \(\frac{\partial \alpha _{1}}{\partial \hat{\varepsilon }_{1}}\dot{\hat{\varepsilon }}_{1}\), \(\frac{\partial \alpha _{1}}{\partial \dot{y}_{r}}\ddot{y}_{r}\), \(\frac{\partial \alpha _{1}}{\partial \hat{W}_{1}}\dot{\hat{W}}_{1}\) and \(\frac{\partial \alpha _{1}}{\partial z_{1}}\dot{z}_{1}\) are all continuous functions with bounded arguments and \(\delta _{1}\) is a constant, it can be concluded that \(\dot{\alpha }_{1}\) and \(\nu _{12}\) are bounded. Further, from (25), we have that \(\nu _{11}\) is bounded. Given the fact that \(z_{1}\), \(z_{2}\), \(\hat{\varepsilon }_{2}\), \(\hat{\delta }_{1}\), \(\nu _{12}\) and \(\hat{W}_{2}\) are bounded and \(\bar{g}_{1}\), \(\underline{g}_{2}\) are constants, we have that \(\bar{\alpha }_{2}\) and \(\alpha _{2}\) are bounded. Since \(\alpha _{2}\) and \(z_{3}\) are bounded, we have that \(x_{3}\) is bounded. Following the similar analysis procedure, we have that \(\alpha _{i}\), \(x_{i}\), \(\nu _{i1}\), \(\nu _{i2}\) and v are bounded. This follows that the output constraint will not be violated and all the closed-loop signals are bounded.

In order to prove finite time convergence, note that when \(C\le \beta _{1}\varpi V_{n}\), we have:

$$\begin{aligned} \dot{V}_{n}\le -\beta _{1}(1-\varpi )V_{n}-\beta _{2}V_{n}^{3/4} \end{aligned}$$
(82)

Using Lemma 5, we can deduce that Lyapunov function \(V_{n}\) will converge into a compact set \(\{V_{n}:V_{n}< \frac{C}{\beta _{1}\varpi }\}\) within finite time and the convergence time can be estimated as

$$\begin{aligned} T\le \frac{4}{\beta _{1}(1-\varpi )}\ln \frac{\beta _{1}(1-\varpi )V_{n}(0)^{1/4}+\beta _{2}}{\beta _{2}} \end{aligned}$$
(83)

If \(V_{n}\) converges to the set \(\{V_{n}:V_{n}< \frac{C}{\beta _{1}\varpi }\}\), we have \(\frac{k_{b}^{2}}{\pi }\tan (\frac{\pi z_{1}^{2}}{2k_{b}^{2}})\le V_{n}< \frac{C}{\beta _{1}\varpi }\), which means that the output tracking error will converge to a compact set \(\{z_{1}:|z_{1}|<\max \{\upsilon ,\sqrt{\tan ^{-1}(\frac{C\pi }{\beta _{1}\varpi k_{b}^{2}})\frac{2k_{b}^{2}}{\pi }}\}\}\) within finite time T. \(\square \)

Remark 8

In stability analysis, we only consider the case where \(|z_{1}|\ge \upsilon \) holds in (16) and Theorem 1 shows that the ultimate bound of tracking error \(z_{1}\) is determined by \(\max \{\upsilon , \sqrt{\tan ^{-1}(\frac{C\pi }{\beta _{1}\varpi k_{b}^{2}})\frac{2k_{b}^{2}}{\pi }}\}\). If \(\upsilon \) is selected sufficiently small such that \(\upsilon \le \sqrt{\tan ^{-1}(\frac{C\pi }{\beta _{1}\varpi k_{b}^{2}})\frac{2k_{b}^{2}}{\pi }}\), the ultimate bound of \(z_{1}\) is independent of \(\upsilon \).

Remark 9

Theorem 1 provides a guideline for the designer to select appropriate design parameters such that the ultimate bound of tracking error can be reduced. The ultimate bound of tracking error can be made as small as possible if we choose small C and large \(\beta _{1}\). To ensure C as small as possible and \(\beta _{1}\) as large as possible, we needs to select small \(\gamma _{i}\) and large \(k_{i}\), \(\varLambda _{i}\), \(\varGamma _{i}\), \(\varPi _{i-1}\), k.

Remark 10

The approach presented in this paper is partly motivated by the work in [32]. However, the results in [32] can only be used for strict-feedback system with output constraint. In this paper, we address tracking problem for nonstrict-feedback system with input saturation, dead zone and output constraint. To the best of our knowledge, this is the first time to report results about adaptive tracking control for nonstrict-feedback nonlinear system subjected to input saturation, dead zone and output constraint. In addition, the results in [32] are based on backstepping technique which has the problem of explosion of complexity. In this paper, first-order sliding mode differentiator is employed to obtain the first derivative of virtual control, which overcomes explosion of complexity.

4 Comparison with existing results

To highlight the novelty of the obtained results, we make a comparison with the existing results in this section.

  1. (1)

    The previous studies [9, 11, 12, 16,17,18,19, 29, 31,32,33, 35,36,38, 44,45,46, 48,49,50, 52,53,54] present adaptive backstepping neural network control schemes for strict-feedback system or pure-feedback system. However, these results cannot be extended to nonstrict-feedback system. As shown in Remark 2, it is very difficult and challenging to apply backstepping design into nonstrict-feedback system. This paper overcomes the difficulty of applying backstepping control into nonstrict-feedback system and proposes a novel adaptive dynamic surface neural network control scheme for nonstrict-feedback system. Since nonstrict-feedback system includes strict-feedback system and pure-feedback system as its special form, the results obtained in this paper are more general and can be applied to address adaptive neural network control problem for strict-feedback system or pure-feedback system.

  2. (2)

    The previous studies [59,60,61,62] employ variable separation technique to overcome the design difficulty arising from nonstrict-feedback structure. However, these results require that the nonlinear function \(f_{i}(x)\) in system (1) satisfies the monotonously increasing property. The results obtained in this paper remove this restrictive assumption and can be applied into more general nonstrict-feedback system.

  3. (3)

    The previous studies [56,57,58,59,60,61,62,63] can only achieve asymptotical tracking for uncertain nonstrict-feedback nonlinear system. The results obtained in this paper achieve finite time tracking for uncertain nonstrict-feedback nonlinear system.

  4. (4)

    The previous studies [57,58,59,60,61,62,63] present adaptive backstepping neural network control for uncertain nonstrict-feedback nonlinear system. However, these results suffer from the problem of “explosion of complexity”. In order to overcome the “explosion of complexity’, adaptive dynamic surface neural network control was developed in [56]. However, it employs command filter to obtain the derivative of virtual control, which has asymptotical convergence property. Different from previous work, in this paper, first-order sliding mode differentiator is combined with backstepping design to overcome the explosion of complexity problem, which has finite time convergence property and satisfies separation principle, thereby having superior performance.

  5. (5)

    To the best of our knowledge, there are no results about adaptive tracking control for nonstrict-feedback nonlinear system subjected to input saturation, dead zone and output constraint. This paper proposes an adaptive neural network dynamic surface control to address this problem.

5 Simulation results

In this section, two examples are carried out to demonstrate the effectiveness of the proposed control scheme.

Example 1

Consider the following second-order nonstrict-feedback nonlinear system:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{x_{1}}=f_{1}(x)+g_{1}(x_{1})x_{2}\\&\dot{x_{2}}=f_{2}(x)+g_{2}(\bar{x}_{2})u\\&y=x_{1} \end{aligned} \right. \end{aligned} \end{aligned}$$
(84)
Fig. 1
figure 1

Time response of system output y and its reference \(y_{r}\)

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Fig. 2
figure 2

Time response of tracking error

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where \(f_{1}(x)=x_{1}x_{2}+x_{1}^{2}\sin (x_{2})\), \(g_{1}(x_{1})=1.5+0.5\sin (x_{1})\), \(f_{2}(x)=x_{1}x_{2}e^{x_{2}}\), \(g_{2}(\bar{x}_{2})=1.5+\sin (x_{1}x_{2})\). Here, it is assumed that \(f_{1}(x)\), \(g_{1}(x_{1})\), \(f_{2}(x)\) and \(g_{2}(\bar{x}_{2})\) are unknown. Reference output is given as \(y_{r}=\sin (t)\) and the output constraint is selected as \(|y|<\pi /2\). It can be easily verified that there exist \(\underline{g}_{1}=1\), \(\bar{g}_{1}=2\), \(\underline{g}_{2}=0.5\), \(\bar{g}_{2}=2.5\) such that Assumption 2 holds. Dead zone and input saturation parameters are taken to be \(m_{r}=1\), \(b_{r}=0.1\), \(m_{l}=1.05\), \(b_{l}=-\,0.15\), \(u_{\max }=5\), \(u_{\min }=-\,4\). The design parameters in virtual control, parameter adaptive law and first-order sliding mode differentiator are chosen as \(k_{1}=k_{2}=3\), \(\eta _{1}=\eta _{2}=2.5\), \(\varLambda _{1}=\varLambda _{2}=2\), \(\kappa _{1}=\kappa _{2}=5\), \(\varGamma _{1}=\varGamma _{2}=0.4\), \(\sigma _{1}=\sigma _{2}=10\), \(\varPi _{1}=10\), \(\mu _{2}=4.12\), \(k=10\), \(\xi =5\), \(\tau =0.1\), \(\gamma _{1}=\gamma _{2}=0.1\), \(\lambda _{0}=1.5\), \(\lambda _{1}=1.1\). The results are shown in Figs. 1, 2, 3, 4, 5 and 6. As shown in Fig. 1 that the system output follows the trajectory of reference signal closely without violation of output constraint. As shown in Fig. 2, the tracking error reaches a small neighbor of the origin within finite time and remains in it thereafter. Figure 3 shows the curve of control input u. The estimations of parameters \(\bar{\varepsilon }_{i}\) and \(\delta \) and the time response of anti-windup compensator state w are depicted in Figs. 4, 5 and 6. It is clear that all the parameter estimations and the anti-windup compensator state are bounded. All the results show that the proposed control scheme can guarantee all the closed-loop signals are bounded and tracking error converges to a small neighbor of the origin within finite time, which verifies the effectiveness of the proposed control scheme.

Fig. 3
figure 3

Curve of control input u

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Fig. 4
figure 4

Time response of RBFNN approximation error estimation \(\bar{\varepsilon }_{i}\)

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Fig. 5
figure 5

Time response of estimation error for first-order differentiation error \(\delta \)

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Fig. 6
figure 6

Time response of anti-windup compensator state w

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Example 2

Consider the following system which describes the dynamics of a one-link manipulator driven by a brush dc motor [73, 74]:

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\bar{M}\ddot{q}+\bar{N}\sin (q)+\bar{B}\dot{q}=I+\Delta I\\&L\dot{I}=-RI-K_{B}\dot{q}+V\\&y=q \end{aligned} \right. \end{aligned} \end{aligned}$$
(85)

where q, \(\dot{q}\) and \(\ddot{q}\) are angular position, angular velocity and angular acceleration, I is motor armature current, \(\Delta I\) is current disturbance, V is input control voltage and the physical meaning of other parameters can be found in [73]. Let \(x_{1}=q\), \(x_{2}=\dot{q}\), \(x_{3}=I\), \(u=V\) and the system (85) can be written as

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&\dot{x_{1}}=x_{2}\\&\dot{x_{2}}=-\frac{\bar{N}}{\bar{M}}sin(x_{1})-\frac{\bar{B}}{\bar{M}}x_{2}+\frac{1}{\bar{M}}x_{3}+\frac{\Delta I}{\bar{M}}\\&\dot{x_{3}}=-\frac{R}{L}x_{3}-\frac{K_{B}}{L}x_{2}+\frac{u}{L}\\ y&=x_{1} \end{aligned} \right. \end{aligned} \end{aligned}$$
(86)
Fig. 7
figure 7

Time response of system output q and its reference \(q_{r}\)

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Fig. 8
figure 8

Time response of the motor armature current I

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The parameters in system (86) can be selected as \(\bar{N}=10\), \(\bar{B}=1\), \(\bar{M}=1\), \(L=0.05\), \(K_{B}=0.5\), \(R=0.5\) and the current disturbance is assumed to be \(\Delta I=0.1x_{1}\sin (x_{2}x_{3})\). Since the disturbance contains all the state variables, system (86) is a nonstrict-feedback system. The reference output is \(y_{r}=\pi /2\sin (t)(1-e^{-0.1t^{2}})\) and the output constraint is selected as \(|y|<\pi /2\). In this example, \(\underline{g}_{1}=\bar{g}_{1}=\underline{g}_{2}=\bar{g}_{2}=1\), \(\underline{g}_{3}=\bar{g}_{3}=20\). Dead zone and input saturation parameters are selected to be the same as Example 1. The design parameters for virtual control, parameter adaptive law and first-order sliding mode differentiator are chosen as \(k_{1}=k_{2}=k_{3}=5\), \(\eta _{1}=\eta _{2}=\eta _{3}=3\), \(\varLambda _{1}=\varLambda _{2}=\varLambda _{3}=2.5\), \(\kappa _{1}=\kappa _{2}=5\), \(\varGamma _{1}=\varGamma _{2}=\varGamma _{3}=0.8\), \(\sigma _{1}=\sigma _{2}=\sigma _{3}=10\), \(\gamma _{1}=\gamma _{2}=\gamma _{3}=0.1\), \(\varPi _{1}=\varPi _{2}=10\), \(\mu _{1}=\mu _{2}=4.5\), \(k=10\), \(\xi =5\), \(\tau =0.1\), \(\lambda _{0}=1.5\), \(\lambda _{1}=1.1\). The results are shown in Figs. 7, 8 and 9. Figures 7 and 8 show the time response of states q and I, and Fig. 9 depicts the curve of control voltage u. It can be seen from these figures that the system output track the trajectory of reference output within finite time while the violation of output constraint is avoided and desired control performance is obtained.

Fig. 9
figure 9

Curve of control voltage u

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6 Conclusion

In this paper, an adaptive neural network dynamic surface control is presented for a class of nonstrict-feedback uncertain nonlinear systems subjected to input saturation, dead zone and output constraint. By designing anti-windup compensator, the input saturation problem is solved. Tan-type Barrier Lyapunov function is introduced to prevent output constraint violation. Furthermore, using adaptive backstepping technique, a series of novel stabilizing virtual control functions are derived. In order to overcome the explosion of complexity, first-order sliding mode differentiator is employed to obtain the derivative of virtual control. Using dead zone inverse method, the real control input is obtained. With the aid of finite time stability theory, it is proved that the proposed control scheme can drive the output tracking error into a small neighbor of the origin within finite time and keep all the closed-loop signals bounded. Simulation results demonstrate the effectiveness of the proposed control scheme. In the future, there are many researches to be done, for example, how to develop a control scheme to take hysteresis effects into account, how to extend the proposed control scheme to time delay system, switched system, multi-input and multi-output system and stochastic system, how to extend the proposed control scheme to address fault-tolerant control problem, how to utilize fuzzy system to approximate the studied system and design controller and observer with the aid of the existing results [75,76,77,78], how to extend the proposed control scheme to study consensus problem of nonlinear multi-agent system using the idea of the existing result [79]. In addition, the control of bifurcation has received great attention and many control schemes have been proposed, such as PD control [80], state feedback control [81] and sliding mode control [82]. Recently, adaptive neural network backstepping control scheme [83] was presented to control bifurcation. Therefore, the extension of the proposed control scheme to bifurcation control is another future research direction.