1 Introduction

Recently, unmanned aerial vehicles (UAVs) have been widely applied in civil and military areas, owing to the advantages of vertical take-off, hovering, flexibility, and rapid response [1, 2]. However, the helicopter is a highly nonlinear multiple-input multiple-output (MIMO) system with multiple coupling and many internal system parameters are difficult to measure accurately in practical applications [3]. Thus, a stabilized control strategy is urgently developed to ensure the stable robustness of nonlinear helicopter systems.

In recent decades, various control methodologies have been put forward for controller design of helicopter systems, such as model-based control, linear quadratic regulator (LQR) control, and Q-learning control [4,5,6,7,8]. In [4], a model-based tracking controller was introduced to track reference trajectories for small-scale unmanned helicopters. In [5], a linear quadratic controller was designed by an enhanced performance adaptive technique, and the system’s performance and robustness were discussed. In [6], a direct adaptive LQR control with the advantage of the wide operating horizon was introduced to increase the system’s stability and robustness. In [7], to handle the LQR problem, a new Q-learning algorithm was constructed utilizing an iteration policy. However, the above-mentioned studies linearized the nonlinear helicopter systems by ignoring the nonlinear portion and couple effects for ease of control design, and the instability and poor performance of the nonlinear helicopter systems in practical applications may arise. Thus, incorporating the nonlinearities into the controller design is necessary.

In such case, various nonlinear control methods, such as robust control, optimal control, adaptive neural network (NN) control and so on, have been developed to tackle the uncertainties in UAV systems [9,10,11,12,13]. In [9], a trigonometric-saturation-function-based position controller was designed to address internal and external disturbances. In [10], an adaptive control was proposed to handle the trajectory tracking issue of flapping-wing micro aerial vehicles. In [11], an adaptive NN control was introduced to approximate unknown helicopter systems with output constraints and input dead-zone. However, the control design in the above literatures resolved the issue of system uncertainties and nonlinearities, and the effect of input backlash nonlinearity was not considered. Moreover, the effectiveness of the above approach was only validated in simulations without an experimental verification. Recently, some nonlinear control methods have been developed in the practical application of helicopter systems. In [14], an explicit model predictive control was introduced for a helicopter system to track desired time-varying trajectories. In practice, the optimization problems is difficult to be solved in the case of short sampling time intervals. It can be noted that the optimization computation in the predictive control scheme was achieved offline and the effects of internal mechanical properties and external disturbances were neglected. In [15], a sliding mode control (SMC) synthesis was proposed to compensate the effect of the external disturbances and chattering phenomenon often occured under the SMC, which may result in system instabilities or even destruction under long-time operation. Hence, a control scheme with online adjustment and steady operation is necessary for practical application.

Input nonlinearities universally exist in practical engineering applications due to non-smooth mechanical connections [16,17,18,19,20,21,22]. Backlash is one of the input nonlinearities, the existence of which may affect the smoothness of input signal and then cause control performance degradation. Thus, it is imperative to develop effective control schemes to handle and compensate for the effects of backlash nonlinearity. In [20], an H-infinity observer-based feedback control was constructed for time-delayed MIMO nonlinear systems to compensate for unknown backlash and estimate unmeasurable states. In [21], a neural network-adaptive strategy was established to address the tracking control for uncertain MIMO nonlinear systems. In [22], a robust H-infinity control was proposed for UAV systems to handle parametric uncertainties including nonlinear friction force and backlash. Although the successful solution has been attained on the control design for tackling the backlash in MIMO systems, little research has been reported thus far on the adaptive NN control for unknown 2-DOF helicopter systems with input backlash, which inspires us for further study.

In practical control engineering, due to the requirement for the stability and security of the system, the signals of states and outputs should be maintained in a reasonably constrained range [23,24,25]. If the constraints are transgressed and violated, it may give rise to system performance degradation, result in instability, or even lead to serious accidents [26]. Thus, constraints are required to be incorporated into the control design. Barrier Lyapunov function (BLF), as an effective tool to address the constraint issue, was proposed to handle output constraints in nonlinear systems [27]. In [28], the BLF was introduced in the design of adaptive NN control to prevent from violating the constraint. In [29], a novel integral BLF was proposed to address the full state constraints. In [30], time-varying constraints were considered in an adaptive NN control design of n-link robot systems. However, the above study only resolved the output constrained control for nonlinear systems, and the simultaneous effects of the backlash nonlinearity, output constraints, and uncertainties were neglected. Considering the complexly coupled characteristics in 2-DOF helicopter systems will pose an increased challenge to the control design.

Driven by the existing results, this study concentrates on the design of an adaptive NN control of uncertain 2-DOF helicopter systems involving the input backlash and output constraint. The main contributions are summarized as follows:

  1. 1.

    The input backlash, output constraints, and uncertainties are considered in 2-DOF helicopter systems, and the RBFNN is utilized to develop an adaptive NN constrained control to approximate the unknown function and uncertainties, and ensure no violation of constraints.

  2. 2.

    The proposed adaptive NN control can stabilize the closed-loop system at a specified region. The simulation and experiment results on the Quanser’s 2-DOF helicopter platform prove the control performance.

2 Problem formulation

A backlash nonlinearity shown in Fig. 1 is depicted as [31]

$$\begin{aligned} \tau (t)=B(u_d)=\left\{ \begin{aligned}&\kappa (u_d(t)-B_1),\quad \mathrm{if} \quad {\dot{u}_d>0} \quad \mathrm{and} \\&\quad \quad \quad \quad \quad \quad \tau (t)=\kappa (u_d(t)-B_1), \\&\kappa (u_d(t)-B_2),\quad \mathrm{if} \quad {\dot{u}_d<0} \quad \mathrm{and} \\&\quad \quad \quad \quad \quad \quad \tau (t)=\kappa (u_d(t)-B_2), \\&\tau (t\_),\quad \quad \quad \quad \mathrm{otherwise}, \end{aligned} \right. \end{aligned}$$
(1)

where \(B(u_d)\) denotes a backlash nonlinearity, \(u_d(t)\) is a desired control input, and \(\tau (t\_)\) means that the output signal \(\tau (t)\) remains unchanged. \(\kappa >0\), \(B_1 > 0\), and \(B_2 < 0\) are constant parameters. In addition, \(\kappa\) is the line’s slope.

Fig. 1
figure 1

Backlash diagram

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Assumption 1

The parameters \(\kappa\), \(B_1\), and \(B_2\) are known and bounded constants in (1) and satisfy the following constraints

$$\begin{aligned}&0< \kappa _{\min }\le \kappa \le \kappa _{\max }, \\&0< (\kappa B_1)_{\min }\le \kappa B_1 \le (\kappa B_1)_{\max }, \\&(\kappa B_2)_{\min }\le \kappa B_2 \le (\kappa B_2)_{\max }<0. \end{aligned}$$

Then, we can rewrite (1) as

$$\begin{aligned} \tau (t)=B(u_d)=\kappa u_d(t) + D(u_d), \end{aligned}$$
(2)

where \(D(u_d)\) denotes a nonlinearity error for the backlash described as

$$\begin{aligned} D(u_d)=\left\{ \begin{aligned}&-\kappa B_1,\qquad \quad \mathrm{if} \quad {\dot{u}_d>0} \quad \mathrm{and} \\&\quad \quad \quad \quad \quad \quad \tau (t)=\kappa (u_d(t)-B_1), \\&-\kappa B_2,\qquad \quad \mathrm{if} \quad {\dot{u}_d<0} \quad \mathrm{and} \\&\quad \quad \quad \quad \quad \quad \tau (t)=\kappa (u_d(t)-B_2), \\&\tau (t\_)-\kappa u_d(t),\quad \quad \mathrm{otherelse}. \end{aligned} \right. \end{aligned}$$
(3)

Remark 1

From the Assumption 1, we derive that \(\Vert D(u_d)\Vert <D^{*}\) is bounded and the unknown parameter \(D^{*}\) satisfies \(D^{*}=\max \left\{ (\kappa B_1)_{\max },-(\kappa B_2)_{\min }\right\}\).

Fig. 2
figure 2

The schematic diagram of a 2-DOF helicopter

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The schematic diagram of a 2-DOF helicopter is displayed in Fig. 2. From which, it can be observed that the pitch and yaw motions are controlled by the thrust forces \(F_p\) and \(F_y\), respectively.

By Lagrange dynamic equations, the nonlinear equations of a 2-DOF helicopter system can be modeled as [32]:

$$\begin{aligned}&(J_p+mL^2_{cm})\ddot{\theta }=K_{pp}V_p+K_{py}V_y-mgL_{cm}\cos \theta \nonumber \\&\quad -D_p{\dot{\theta }}-mL_{cm}^2{\dot{\psi }}^2\sin \theta \cos \theta , \end{aligned}$$
(4)
$$\begin{aligned}& (J_y+mL^2_{cm}\cos ^2\theta )\ddot{\psi }=K_{yp}V_p+K_{yy}V_y-D_y{\dot{\psi }}\nonumber \\&\quad +2mL^2_{cm}{\dot{\psi }}{\dot{\theta }}\sin \theta \cos \theta , \end{aligned}$$
(5)

where the specific parameters above-mentioned can be referred in Table 1.

Define the state vectors as \(x_1=[\theta ,\psi ]^T\) and \(x_2=[{\dot{\theta }},{\dot{\psi }}]^T\), and an input vector as \(\tau =[V_p,V_y]^T\). To facilitate the control design, the dynamic Eqs. (4) and (5) can be rewritten as the state-space equations formulated as

$$\begin{aligned}&{\dot{x}}_1=x_2,\nonumber \\&{\dot{x}}_2=F(x_1,x_2)+M(x_1)\tau ,\nonumber \\&y=x_1,\nonumber \\&\tau =B(u_d), \end{aligned}$$
(6)

where \(\tau =B(u_d)\) denotes the a backlash nonlinearity of the actuators and

$$\begin{aligned} F(x_1,x_2)=\left[ \begin{array}{cc} \frac{a_2}{a_1}\\ \frac{b_2}{b_1} \end{array} \right] ,~~ M(x_1)=\left[ \begin{array}{cc} \frac{Kpp}{a_1}&{}\frac{Kpy}{a_1}\\ \frac{Kyp}{b_1}&{}\frac{Kyy}{b_1} \end{array} \right] , \end{aligned}$$
(7)

with the variables \(a_1\), \(a_2\), \(b_1\) and \(b_2\) designed as

$$\begin{aligned}&a_1=J_p+mL^2_{cm},\end{aligned}$$
(8)
$$\begin{aligned}&a_2=-mgL_{cm}\cos \theta -D_p{\dot{\theta }}-mL_{cm}^{2}{\dot{\psi }}^2\sin \theta \cos \theta ,\end{aligned}$$
(9)
$$\begin{aligned}&b_1=J_y+mL^2_{cm}\cos ^2\theta ,\end{aligned}$$
(10)
$$\begin{aligned}&b_2=-D_y{\dot{\psi }}+2mL^2_{cm}{\dot{\psi }}{\dot{\theta }}\sin \theta \cos \theta . \end{aligned}$$
(11)
Table 1 Parameters of 2-DOF helicopter system
Full size table

In practical application, the system nonlinearity and parameters in 2-DOF helicopter systems are difficult to measure accurately, and the functions \(F(x_1,x_2)\) and \(M(x_1)\) are uncertain, which raises the difficulties in control design. To deal with this, (6) can be rewritten as

$$\begin{aligned}&{\dot{x}}_1=x_2,\nonumber \\&{\dot{x}}_2=F(x_1,x_2)+{\Delta }F(x_1,x_2)+\left( M(x_1)+{\Delta }M(x_1)\right) \tau ,\nonumber \\&y=x_1,\nonumber \\&\tau =B(u_d). \end{aligned}$$
(12)

For the convenience of reading, we replace \(F(x_1,x_2)\), \(M(x_1)\), \({\Delta }F(x_1,x_2)\), and \({\Delta }M(x_1)\) with F, M, \({\Delta }F\), and \({\Delta }M\), respectively.

Lemma 1

The RBFNNs can approximate any continuous unknown function \(f(\zeta )\): \({\mathbb {R}}^k \rightarrow {\mathbb {R}}\) [33] and can be described as

$$\begin{aligned}&f(\zeta )={W}^{*T}S(\zeta )+\epsilon , \end{aligned}$$
(13)

where \(\zeta \subset {\mathbb {R}}^k\) represents a RBFNN input vector, \({\hat{W}}\in {\mathbb {R}}^{n}\) is an estimated weight vector, and \(n>1\) is the number of hidden layer nodes. \(S(\zeta )\) is the basic function expressed as

$$\begin{aligned} S(\zeta )=\exp \!{\left[ \frac{{-(\zeta -C)}^{T}(\zeta -C)}{{b}^2}\right] }, \end{aligned}$$
(14)

where C denotes a RBFNN center vector of the receptive field, and b denotes the width of the Gaussian function.

Lemma 2

For any positive constant \({k_a\in {\mathbb {R}}}\) satisfying \({x\in {\mathbb {R}}}\) and \(\Vert x\Vert <\Vert k_a\Vert\), the following inequality holds [34]

$$\begin{aligned} \log \frac{k_a^2}{k_a^2-x^2}\le \frac{x^2}{k_a^2-x^2}. \end{aligned}$$
(15)

3 Control design and stability analysis

Based on backstepping approach, we design the output tracking errors \(z_1\) and \(z_2\), and an auxiliary variable \(\alpha\) as follows

$$\begin{aligned}&z_1=x_1-y_d, \nonumber \\&z_2=x_2-\alpha , \nonumber \\&\alpha =-c_1z_1+{\dot{y}}_d, \end{aligned}$$
(16)

where \(c_1=\rm{diag}[c11,c12]\) is a positive definite diagonal matrix.

Combining (12), we provide the time derivative of \(z_1\) and \(z_2\) as

$$\begin{aligned}&{\dot{z}}_1=x_2-{\dot{y}}_d=\alpha -{\dot{y}}_d+z_2=-c_1z_1+z_2, \end{aligned}$$
(17)
$$\begin{aligned}&{\dot{z}}_2={\dot{x}}_2-{\dot{\alpha }}=F+{\Delta }F+\left( M+{\Delta }M\right) \tau -{\dot{\alpha }}. \end{aligned}$$
(18)

Considering that the control gain matrix M may be irreversible [35], we define \(u_d(t)=M^{T}\upsilon\), with \(\upsilon\) being a designed control input signal, and an appropriate positive parameter \(\lambda\) is selected to make the matrix \(\left( MM^{T}+\lambda I_{2\times 2}\right)\) reversible, with \(I_{2\times 2}\) being second order identity matrix. Then, with \(\tau (t)=\kappa u_d(t) + D(u_d)\) in (2), (18) can be given by

$$\begin{aligned} {\dot{z}}_2&=F+{\Delta }F+\left( M+{\Delta }M\right) \left( \kappa u_d(t)+D(u_d)\right) -{\dot{\alpha }} \nonumber \\&=F+\kappa Mu_d(t)+\Delta F+\kappa \Delta Mu_d(t)\nonumber \\&\quad +\left( M+\Delta M\right) D(u_d)-{\dot{\alpha }} \nonumber \\&=F+\kappa \left( MM^{T}+\lambda I_{2\times 2}\right) \upsilon -\kappa \lambda I_{2\times 2}\upsilon +\Delta F \nonumber \\&\quad +\kappa \Delta Mu_d(t)+\left( M+\Delta M\right) D(u_d)-{\dot{\alpha }}. \end{aligned}$$
(19)

We consider the Lyapunov function as

$$\begin{aligned} V_1=\frac{1}{2}\sum_{i=1}^2\log \frac{k_{ai}^2}{k_{ai}^2-z_{1i}^2}. \end{aligned}$$
(20)

Invoking (17), the time derivative of \(V_1\) is described as

$$\begin{aligned} \dot{V}_1=\sum_{i=1}^2(\frac{z_{1i}z_{2i}}{k_{ai}^2-z_{1i}^2}-\frac{c_{1i}z_{1i}^2}{k_{ai}^2-z_{1i}^2}). \end{aligned}$$
(21)

To eliminate \(\sum_{i=1}^2\frac{z_{1i}z_{2i}}{k_{ai}^2-z_{1i}^2}\) in (21), we construct a new Lyapunov function as

$$\begin{aligned} V_2=\frac{1}{2}\sum_{i=1}^2\log \frac{k_{ai}^2}{k_{ai}^2-z_{1i}^2}+\frac{1}{2}z_2^Tz_2. \end{aligned}$$
(22)

Differentiating (22) yields

$$\begin{aligned} {\dot{V}}_2&=\sum_{i=1}^2(\frac{z_{1i}z_{2i}}{k_{ai}^2-z_{1i}^2}-\frac{c_{1i}z_{1i}^2}{k_{ai}^2-z_{1i}^2})+z_2^T\left[ F+\kappa \left( MM^{T}+\lambda I_{2\times 2}\right) \upsilon \right. \nonumber \\& \quad -\kappa \lambda I_{2\times 2}\upsilon +\Delta F +\kappa \Delta Mu_d(t)+\left( M+\Delta M\right) D(u_d)-{\dot{\alpha }}]. \end{aligned}$$
(23)

Invoking (23), the following RBFNN is adopted to approximate the unknown function and uncertainties of the system

$$\begin{aligned} f(\zeta )&=W^{*T}S(\zeta )+\epsilon ^{*} \nonumber \\&=-\kappa \lambda I_{2\times 2}\upsilon +\Delta F\nonumber \\&\quad +\kappa \Delta Mu_d(t)+\left( M+\Delta M\right) D(u_d)-{\dot{\alpha }}, \end{aligned}$$
(24)

where \(W^{*}\) is an optimal weight vector, \(\epsilon ^{*}\) is the optimal approximate error, and \(S(\zeta )\) is the basis function vector with the input vector \(\zeta =[x_1^T,x_2^T,{\dot{\alpha }}^{T},u_d^{T}]\).

Then, the designed control signal \(\upsilon\) can be formulated as

$$\begin{aligned} \upsilon =-\frac{1}{\kappa }(MM^{T}+\lambda I_{2\times 2})^{-1} \left( c_2z_2+F+\left[\begin{array}{c}\frac{z_{11}}{k_{a1}^2-z_{11}^2}\\\frac{z_{12}}{k_{a2}^2-z_{12}^2}\end{array}\right]+W^{*T}S(\zeta )+\epsilon ^{*}\right) . \end{aligned}$$
(25)

Considering that the optiaml weight \(W^{*}\) is unknown, the estimated weight \({\hat{W}}\) is used to approximate it. Hence, the designed control signal \(\upsilon\) can be described as follows

$$\begin{aligned} \upsilon =-\frac{1}{\kappa }(MM^{T}+\lambda I_{2\times 2})^{-1}\left( c_2z_2+F+\left[\begin{array}{c}\frac{z_{11}}{k_{a1}^2-z_{11}^2}\\\frac{z_{12}}{k_{a2}^2-z_{12}^2}\end{array}\right]+{\hat{W}}^{T}S(\zeta )\right) . \end{aligned}$$
(26)

where \(c_2\) is a positive definite diagonal matrix, and \({\hat{W}}\) is an estimated weight satisfying \({{\hat{W}}}={\tilde{W}}+{W}^{*}\) with \({\tilde{W}}\) being a weight error.

Furthermore, we propose an adaptive law as

$$\begin{aligned} \dot{\hat{W}}=-\Gamma (z_2^{T}S(\zeta )+\sigma {\hat{W}}). \end{aligned}$$
(27)

where \(\sigma\) is a small postive constant.

Remark 2

Different from the literatures [36, 37], the adaptive neural control method was applied for a class of switched nonlinear systems, to construct a finite-time fault-tolerant controller for handling the tracking problem or construct a novel mode-dependent adaptive laws and sampled-data control laws for stabilizing the controlled system. However, in this paper, we exploit the adaptive neural control method to develop a continous control law for a nonlinear 2-DOF helicopter system for achieving an excellent control performance and guarantee the output contraints satifaction. In addition, when the RBFNN was employed to approximate the unknown dynamic uncertainties and the backlash error in (24), the possiblity of singularity problem caused by the affine item was taken into considertion in (19).

Proof

The Lyapunov function for stability analysis is constructed as

$$\begin{aligned} V=\frac{1}{2}\sum_{i=1}^2\log \frac{k_{ai}^2}{k_{ai}^2-z_{1i}^2}+\frac{1}{2}z_2^{T}z_2+\frac{1}{2}tr({\tilde{W}}^{T}\Gamma ^{-1} {\tilde{W}}). \end{aligned}$$
(28)

Combining (23)–(27), we obtain

$$\begin{aligned}\dot{V}&=\sum_{i=1}^2\frac{z_{1i}\dot{z}_{1i}}{k_{ai}^2-z_{1i}^2}+z_2^{T}\dot{z}_2+tr(\tilde{W}^{T}\Gamma^{-1}\dot{\tilde{W}})\nonumber\\&=-\sum_{i=1}^2\frac{c_{1i}z_{1i}^2}{k_{ai}^2-z_{1i}^2}+tr(\tilde{W}^{T}\Gamma^{-1}\dot{\hat{W}})+\sum_{i=1}^2\frac{z_{1i}z_{2i}}{k_{ai}^2-z_{1i}^2}+\nonumber\\&{\quad}z_2^T[F+\kappa(MM^{T}+{\lambda}I_{2\times 2})\upsilon-\kappa{\lambda}I_{2\times 2}\upsilon+\nonumber\\&\quad{\Delta}F+\kappa{\Delta}Mu_d(t)+(M+{\Delta}M)D(u_d)-{\dot{\alpha}}]\nonumber\\&\le-\sum_{i=1}^2{\log}c_{1i}\frac{k_{ai}^2}{k_{ai}^2-z_{1i}^2}-z_2^{T}(c_2-\frac{1}{2}I_{2\times2})z_2+\frac{1}{2}\Vert\epsilon{*}\Vert^{2}\nonumber\\&{\quad}-\frac{\sigma}{2\lambda_{\max}(\Gamma^{-1})}tr(\tilde{W}^{T}\Gamma^{-1}\tilde{W})+\frac{\sigma}{2}\Vert{W}^{*}\Vert^{2}\nonumber\\&\le-{\rho}V+\mu\end{aligned}$$
(29)

where

$$\begin{aligned}&\rho =\min \left( \lambda _{\min }(c_1),\lambda _{\min }(c_2-\frac{1}{2}I_{2\times 2}),\frac{\sigma }{2\lambda _{\max}(\Gamma^{-1} )}\right) , \nonumber \\&\mu =\frac{1}{2}\Vert \epsilon ^{*}\Vert ^{2}+\frac{\sigma }{2}\Vert {W}^{*}\Vert ^{2}. \end{aligned}$$
(30)

If the corresponding design matrices \(c_1\), \(c_2-\frac{1}{2}I_{2\times 2}\), and \(\frac{\sigma }{2\lambda _{\max }(\Gamma )}\) are chosen to make \(\rho >0\) and the bounded initial states exist, the variables \(z_1\), \(z_2\), and \({\tilde{W}}\) are semi-globally uniformly ultimately bounded (SGUUB) [38]. We then obtain

$$\begin{aligned} 0\le V \le \frac{\mu }{\rho }+\left[ V(0)-\frac{\mu }{\rho }\right] e^{-\rho t}. \end{aligned}$$
(31)

Then, we further derive

$$\begin{aligned}&\Vert z_{1i}\Vert ^2 \le k_{ai}^2(1-e^{-2(V(0)+\frac{\mu }{\rho })}) , \nonumber \\&\Vert z_2\Vert ^2\le 2\left(V(0)+\frac{\mu }{\rho }\right) , \nonumber \\&\Vert {\tilde{W}}\Vert ^2\le \frac{2\left( V(0)+\frac{\mu }{\rho }\right)}{\lambda_{min}(\Gamma^{-1})} . \end{aligned}$$
(32)

At this moment, we can conclude that the controlled system is SUGGB.

4 Simulation and experiment results

To verify the feasibility and effectiveness of the designed control scheme, we carry out a simulation on the Matlab platform. To further prove the validity and efficacy of the derived control, comparative experiments of two control schemes are conducted on a Quanser’s 2-DOF helicopter platform shown in Fig. 8. The two schemes for comparison include adaptive NN with input backlash and output constraint control (ANNIBOCC) and SMC.

Fig. 3
figure 3

Simulated tracking performance under the ANNIBOCC: a \(\theta\), b \(\psi\)

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Fig. 4
figure 4

Simulated input signal under the ANNIBOCC a \(u_d\), b \(\tau\)

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Fig. 5
figure 5

Simulated tracking performance under the SMC: a \(\theta\), b \(\psi\)

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Fig. 6
figure 6

Simulated tracking errors under the SMC and the ANNIBOCC

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Fig. 7
figure 7

Simulated input signal \(\tau\) under the SMC and the ANNIBOCC

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4.1 Simulation

In the simulation, the initial states are chosen as \(x_1=[0,0]^{T}\) and \(x_2=[0,0]^{T}\). \(y_d=[10\pi \sin (t)/180,20\pi \sin (t)/180]^{T}\) is a desired tracking trajectory. The relevant parameters for the input backlash are set as \(\kappa =0.8\), \(B_1=0.2\), and \(B_2=-0.25\). The adaptive law parameters are \(\Gamma =3\) and \(\sigma =0.01\). The control gains are selected as \(c_1=\mathrm {diag}[100,100]\) and \(c_2=\mathrm {diag}[150,100]\). The control parameter is given as \(\lambda =0.1\). According the approximated function and the input vector of RBFNN, the nodes are chosen as 150 and the center parameters are chosen as either 1 or -1. The variances are chosen as \(b=2\) and the intial weights \({\hat{W}}\) are set as zero. The 2-DOF helicopter system parameters are chosen as \(J_p=0.0219 \text { kg$\cdot$m}^2\), \(J_y=0.022 \text { kg$\cdot$m}^2\), \(D_p=0.0071 \text { N/V}\), \(D_y=0.022\text { N/V}\), \(K_{pp}=0.0011 \text { N$\cdot$m/V}\), \(K_{py}=0.0022 \text { N$\cdot$m/V}\), \(K_{yp}=-0.0027 \text { N$\cdot$m/V}\), \(K_{yy}=0.0022 \text { N$\cdot$m/V}\), \(L_{cm}=0.0071 \text { m}\) and \(m=1.075 \text { kg}\). In Fig. 3, we observe that the output signals of \(\theta\) and \(\psi\) can realize the trajectory tracking with small tracking errors shown in Fig. 6. In other words, the proposed control scheme achieves an excellent control performance in the 2-DOF helicopter system. In addition, the trajectories of \(\theta\) and \(\psi\) are kept within the specified constraint range, which denotes that the output constraints are not violated. It can be depicted from Fig. 4 that the input signals become smoother, which illustrates the presented control scheme is capable of eliminating the input backlash nonlinearity. Combining the tracking trajectories under the SMC in Fig. 5 and tracking errors in Fig. 6, we can conclude that the ANNIBOCC scheme achieved a superior control perforamance. As shown in Fig. 7, the appearance of chattering is inevitable under the SMC, which results in more complex input nonlinearity compared with the proposed scheme.

4.2 Experiment

In this part, Figs. 9, 10, 11, 12 and 13 show the experiment results of two nonlinear control schemes, including the ANNIBOCC and SMC. In addition, Fig. 9 depicts the tracking trajectories of \(\theta\) and \(\psi\) under the ANNIBOCC. From which, we can conclude that the proposed scheme ensures no violation of output constraints. In Fig. 10, by comparing the input \(u_d\) and \(\tau\), we can observe that the processed input \(\tau\) with backlash nonlinearity is smoother. Figure 11 displays the tracking trajectories under the SMC. It can be seen from Figs. 12 and 13 that the tracking errors and input signals demonstrate that the yaw action and pitch action have a drastic chattering, which is caused by the nonlinear mechanism and the characteristics of the SMC algorithm. Under the hybrid effects of chattering and input backlash nonlinearity, rapidly changing inputs are required under the SMC to minimize the tracking errors. This may cause damage to objects with limited input. To sum up, the proposed strategy can effectively eliminate the backlash of input signals and the constrained output ensures that the output signals are retained within a desired range. From the perspective of practical application, the ANNIBOCC with less chattering and smoother input, is more suitable than the SMC.

Fig. 8
figure 8

The Quanser’s 2-DOF helicopter platform

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Fig. 9
figure 9

Tracking performance under the ANNIBOCC: a \(\theta\), b \(\psi\)

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Fig. 10
figure 10

The contrast picture of \(u_d\) and \(\tau\) under the ANNIBOCC

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Fig. 11
figure 11

Tracking performance under the SMC: a \(\theta\), b \(\psi\)

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Fig. 12
figure 12

Tracking errors under the SMC and the ANNIBOCC

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Fig. 13
figure 13

Input signal \(\tau\) under the SMC and the ANNIBOCC

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5 Conclusion

The framework of the adaptive NN control of 2-DOF helicopter systems with input backlash and output constraints was presented in this study. To compensate for the hybrid effects of input backlash nonlinearity and system uncertainties, the unknown function and nonlinearity error were approximated using the RBFNN. The proposed control scheme realized that the tracking errors converged to a small neighborhood around zero and the backlash was eliminated. Furthermore, the BLF function was adopted to tackle the output constraints, and guarantee the system safety and effectiveness in the operation. Exploiting the rigorous Lyapunov analysis, the designed control realized the controlled system’s uniform stability. The simulation and experimental results validated that the designed control revealed a superior control performance. Considering that the output constraints may be asymmetric under certain circumstances and the compensation of the backlash nonlinearity can be futhur improved, the future research will therefore come up with a new approach to address this challenge.