Robust Tracking Control for Lower Limb Rehabilitation Exoskeleton Robots with Multi-Performances

1. Introduction

Lower limb rehabilitation exoskeleton robots have attracted a great deal of attention due to the fact that LLRER have good precision performance as well as effectively help physically handicapped patients to implement rehabilitation training process [1,2]. Physically handicapped patients move according to a pre-specified gait trajectory with the driving force proposed by LLRER. After possessing some moving abilities, physically handicapped patients implement the walking ability with the aid of LLRER [3,4]. Among above investigations, the most significant topic is focused on robustly tracking a pre-specified gait trajectory and then it is important to construct the desired tracking control for robots.

Nowadays, investigations on tracking topic for LLRER are mainly focused on dynamic modelling, that is similar to robotic investigations [5]. Most widely used effective control approaches address the global stability problem of LLRER, such as the proportional-integral-derivative control [6], the model predictive control [7,8], the adaptive control [9,10] and the sliding mode control [11,12]. Proportional-integral-derivative control needs to transform the dynamic equations of LLRER into “linearized” systems by compensating nonlinear parts and is realized by linear system theory. However, LLRER are highly nonlinear systems, which dynamic equations cannot be completely represented by the transformed linearization systems. Model predictive control is widely used for LLRER due to its appealing properties [13]. However, the model predictive control effectively works only for linear range near the equilibrium point. Adaptive control is often utilized to design the tracking controller for LLRER [14]. The adaptive control is largely restricted owing to the complicated updating rule. Sliding mode control proposes appealing robust performance for the LLRER [15]. However, the severe chattering property is the significant drawback that may destroy the overall stability of the LLRER.

For the trajectory tracking control of LLRER, the external disturbance rejection performance and the stability performance will be required to be simultaneously solved [5]. In order to simultaneously address both above performances for LLRER, some effective methods have been widely used, such as the linear parameter varying control [16,17], the model predictive control [7], the deep reinforcement learning [18,19,20], the backstepping control [21] and so on. Nevertheless, the aforementioned methods are effective only for a linear region near the operating quiet point of the practical nonlinear LLRER via the Jacobian linearization technique. To solve the severe restriction of the practical nonlinear LLRER, a lot of engineers use the function approximators technique to improve practical drawback, such as the neural network approach [15,22] and the fuzzy logic approach [9,23,24]. The neural network approach possesses significant performances for nonlinear LLRER [22]. Nevertheless, it has the following impractical shortcomings: (1)the complicated coupling neural network rules is impractical; (2)the related input control of the LLRER is constructed only by applying current states of neural network; (3)the neural network approach belongs to a supervised learning method and then needs the LLRER to support lots of sampling points. The main shortcoming of the fuzzy logic control is that the fuzzy classified data-mapping rules are completely built by many accumulated expert experiences in the past [9]. Motivated by aforementioned analysis and discussion, it reveals an important fact that the robustness of controlling the LLRER still is a challenging topic for the external disturbance rejection performance and the stability performance. In this article, we “firstly” apply the feedback linearization approach to solve well above traditional difficulties with multi-performances, such as the disturbance rejection, the freely adjustable exponential convergence rate and the better closed-loop stability. Recently, the feedback linearization approach has drawn a great of attention for many industrial applications including the synchronous motor [25], the psychrometric testing chamber [26], the hydraulic cylinder [27], the micro tandem helicopter [28] and the electrical vehicle DC charging pile [29].

More rigorous analysis and discussion for the impact of disturbance, more stricter definition of the tracking error, called as almost disturbance decoupling, had been proposed in [30,31]. Nevertheless, both above references exploit the fact that some control systems cannot obtain the almost disturbance decoupling performance due to the sufficient condition that the considered functions should have “complete” characteristic property such as the following system: ${\dot{x}}_{\mathrm{st}\text{\_}1}(t)={\tan }^{-1}\left(x_{\mathrm{st}\text{\_}2}\right)+{\mathrm{\Phi }}_n(t)$, ${\dot{x}}_{\mathrm{st}\text{\_}2}(t)=I_{\mathit{np}}$, y_{op_1} = x_{st_1}≡O_{op_1}, where Φ_n,O_{op_1} and I_np are the disturbance, output and input, respectively. Conversely, the presented feedback linearization approach will well address the almost disturbance decoupling performance for the trajectory tracking control of LLRER in this article and main contributions are totally listed as follows:

(i)

This study has “firstly” proposed the significant result of the exponential convergence rate for the highly nonlinear LLRER system.

(ii)

The highly nonlinear LLRER system is firstly solved well by applying the feedback linearization approach to take the place of the singular perturbation approach without the severe restriction that some functions need to have “complete” characteristic property [30,31].

(iii)

A robust tracking control of LLRER system is derived in this article to obtain the exponential stability performance without addressing the difficult Hamilton-Jacobi equation that is necessary work for the H-infinity method [32].

(iv)

The presented control algorithm will improve the severe shortcomings of traditional function approximators, including the fuzzy control and the neural network control without relying on the accumulated expert experience in the past and complicated coupling neural network rules.

(v)

The current research in this study is “globally” valid in the overall dynamic plane and takes the place of Jacobian linearization method that is only “locally” valid near the operating quiet point.

(vi)

This study designs a powerful artificial intelligence human–machine interaction package of robust controller design for nonlinear LLRER system via AI Python software based on the proposed algorithm.

2. Complete Mathematical Model of Nonlinear LLRER Robot

Based on the lower limb rehabilitation exoskeleton robots considered in this article, the LLRER kinematic model is derived to show the interconnection for the LLRER and all joint torques via the Lagrange equation and the Newton Euler equation. The Lagrange equation and the Newton Euler equation focus on the energy investigation and the force investigation, respectively. The schematic diagrams for three-link and simplified two-link of the LLRER are depicted as Figure 1, Figure 2 and Figure 3. Thus the Lagrange equation is shown by

L_a = E_kinetic − E_potential

(1)

and

$${\tau }_i={\displaystyle \frac{d}{\mathit{dt}}}\left({\displaystyle \frac{\partial L_a}{\partial {\dot{\theta }}_i}}\right)-\left({\displaystyle \frac{\partial L_a}{\partial {\theta }_i}}\right),i=1,2,3$$

(2)

where E_kinetic is the overall kinetic energy of the LLRER; E_potential is the overall potential energy of the LLRER; L_a is the Lagrange function of the LLRER; τ_i,i = 1,2,3 is the i-joint torque; θ_i,i = 1,2,3 is the angle of link1~link3; ${\dot{\theta }}_i,i=1,2,3$ is the velocity of link1~link3. Based on the geometric coordinate of the LLRER, the coordinate (x_i,y_i) and the velocity $\left({\dot{x}}_i,{\dot{y}}_i\right),i=1,2,3$ of each link will be derived to be

$$x_1=d_1\sin {\theta }_1$$

(3)

$$y_1=d_1\cos {\theta }_1$$

(4)

$$x_2=l_1\sin {\theta }_1+d_2\sin {\theta }_2$$

(5)

$$y_2=l_1\cos {\theta }_1+d_2\cos {\theta }_2$$

(6)

$$x_3=l_1\sin {\theta }_1+l_2\sin {\theta }_2+d_3\sin {\theta }_3$$

(7)

$$y_3=l_1\cos {\theta }_1+l_2\cos {\theta }_2+d_3\cos {\theta }_3$$

(8)

$${\dot{x}}_1=d_1{\dot{\theta }}_1\left(\cos {\theta }_1\right)$$

(9)

$${\dot{y}}_1=-d_1{\dot{\theta }}_1\left(\sin {\theta }_1\right)$$

(10)

$${\dot{x}}_2=l_1{\dot{\theta }}_1\left(\cos {\theta }_1\right)+d_2{\dot{\theta }}_2\left(\cos {\theta }_2\right)$$

(11)

$${\dot{y}}_2=-l_1{\dot{\theta }}_1\left(\sin {\theta }_1\right)-d_2{\dot{\theta }}_2\left(\sin {\theta }_2\right)$$

(12)

$${\dot{x}}_3=l_1{\dot{\theta }}_1\left(\cos {\theta }_1\right)+l_2{\dot{\theta }}_2\left(\cos {\theta }_2\right)+d_3{\dot{\theta }}_3\left(\cos {\theta }_3\right)$$

(13)

$${\dot{y}}_3=-l_1{\dot{\theta }}_1\left(\sin {\theta }_1\right)-l_2{\dot{\theta }}_2\left(\sin {\theta }_2\right)-d_3{\dot{\theta }}_3\left(\sin {\theta }_3\right)$$

(14)

According to (2)~(14), the overall kinetic energy and the overall potential energy of the LLRER can be obtained as

$$\begin{array}{l}{E}_{\mathit{potential}}=m_{1s}g{y}_1+m_{2s}g{y}_2+m_{3s}g{y}_3\\ =m_{1s}{d}_{1}g\left(\cos {\theta }_1\right)+m_{2s}l{}_{1}g\left(\cos {\theta }_1\right)+m_{2s}{d}_{2}g\left(\cos {\theta }_2\right)\\ +m_{3s}{l}_{1}g\left(\cos {\theta }_1\right)+m_{3s}{l}_{2}g\left(\cos {\theta }_2\right)+m_{3s}{d}_{3}g\left(\cos {\theta }_3\right)\end{array}$$

(15)

$$\begin{array}{l}{E}_{\mathit{kinetic}}=\left\{{\displaystyle \frac{1}{2}}{m}_{1s}{\left({\dot{x}}_1\right)}^2+{\displaystyle \frac{1}{2}}{m}_{1s}{\left({\dot{y}}_1\right)}^2+{\displaystyle \frac{1}{2}}{I}_1{\left({\dot{\theta }}_1\right)}^2\right\}\\ +\left\{{\displaystyle \frac{1}{2}}{m}_{2s}{\left({\dot{x}}_2\right)}^2+{\displaystyle \frac{1}{2}}{m}_{2s}{\left({\dot{y}}_2\right)}^2+{\displaystyle \frac{1}{2}}{I}_2{\left({\dot{\theta }}_2\right)}^2\right\}\\ +\left\{{\displaystyle \frac{1}{2}}{m}_{3s}{\left({\dot{x}}_3\right)}^2+{\displaystyle \frac{1}{2}}{m}_{3s}{\left({\dot{y}}_3\right)}^2+{\displaystyle \frac{1}{2}}{I}_3{\left({\dot{\theta }}_3\right)}^2\right\}\end{array}$$

(16)

i.e.,

$$\begin{array}{l}{E}_{\mathit{kinetic}}=\left\{{\displaystyle \frac{1}{2}}{m}_{1s}{d}_1^2{\left({\dot{\theta }}_1\right)}^2+{\displaystyle \frac{1}{2}}{I}_1{\left({\dot{\theta }}_1\right)}^2\right\}\\ +\left\{{\displaystyle \frac{1}{2}}{m}_{2s}{l}_1^2{\left({\dot{\theta }}_1\right)}^2+\right.{\displaystyle \frac{1}{2}}{m}_{2s}{d}_2^2{\left({\dot{\theta }}_2\right)}^2\\ +m_{2s}{l}_1{d}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\cos ({\theta }_1-{\theta }_2)\right)\left.+{\displaystyle \frac{1}{2}}{I}_2{\left({\dot{\theta }}_2\right)}^2\right\}\\ +\left\{{\displaystyle \frac{1}{2}}\right.m_{3s}{l}_1^2{\left({\dot{\theta }}_1\right)}^2+{\displaystyle \frac{1}{2}}{m}_{3s}{l}_2^2{\left({\dot{\theta }}_2\right)}^2+{\displaystyle \frac{1}{2}}{m}_{3s}{d}_3^2{\left({\dot{\theta }}_3\right)}^2\\ +m_{3s}{l}_1{l}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\cos ({\theta }_1-{\theta }_2)\right)+m_{3s}{l}_1{d}_3{\dot{\theta }}_1{\dot{\theta }}_3\left(\cos ({\theta }_1-{\theta }_3)\right)\\ \left.+m_{3s}{l}_2{d}_3{\dot{\theta }}_2{\dot{\theta }}_3\left(\cos ({\theta }_2-{\theta }_3)\right)+{\displaystyle \frac{1}{2}}{I}_3{\left({\dot{\theta }}_3\right)}^2\right\}\end{array}$$

(17)

where m_1s(i.e. m₁+M₁),m_2s(i.e. m₂+M₂),m_3s(i.e. m₃+M₃) are the summing masses of link1~link3, m₁ denotes the mass of exoskeleton thigh, M₁ denotes the mass of patient thigh, m₂ denotes the mass of exoskeleton shank, M₂ denotes the mass of patient shank, M₃ denotes the summing mass of patient foot, l₁,l₂,l₃ are the lengths of link1 ~ link3, d₁,d₂,d₃ are the centroid position distances of link1 ~ link3, I₁,I₂,I₃ are the rotational inertias of link1 ~ link3 and g is the acceleration of gravity.

Substituting (15) ~ (17) into (1) ~ (2) yields

$$\begin{array}{l}{\tau }_1=m_{1s}{d}_1^2{\ddot{\theta }}_1+I_1{\ddot{\theta }}_1+m_{2s}{l}_1^2{\ddot{\theta }}_1-m_{2s}{l}_1{d}_2{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)\left({\dot{\theta }}_1-{\dot{\theta }}_2\right)\\ +m_{2s}{l}_1{d}_2\left(\cos ({\theta }_1-{\theta }_2)\right){\ddot{\theta }}_2+m_{3s}{l}_1^2{\ddot{\theta }}_1-m_{3s}{l}_1{l}_2{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)\\ \left({\dot{\theta }}_1-{\dot{\theta }}_2\right)+m_{3s}{l}_1{l}_2\left(\cos ({\theta }_1-{\theta }_2)\right){\ddot{\theta }}_2-m_{3s}{l}_1{d}_3{\dot{\theta }}_3\left(\sin ({\theta }_1-{\theta }_3)\right)\\ \left({\dot{\theta }}_1-{\dot{\theta }}_3\right)+m_{3s}{l}_1{d}_3{\ddot{\theta }}_3\left(\cos ({\theta }_1-{\theta }_2)\right)+m_{3s}{l}_1{l}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)\\ +m_{3s}{l}_1{d}_3{\dot{\theta }}_1{\dot{\theta }}_3\left(\sin ({\theta }_1-{\theta }_3)\right)-m_{1s}{d}_{1}g\left(\sin {\theta }_1\right)-m_{2s}l{}_{1}g\left(\sin {\theta }_1\right)\\ -m_{3s}l{}_{1}g\left(\sin {\theta }_1\right)+m_{2s}{l}_1{d}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)\end{array}$$

(18)

$$\begin{array}{l}{\tau }_2=m_{2s}{d}_2^2{\ddot{\theta }}_2-m_{2s}{l}_1{d}_2{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_2)\right)\left({\dot{\theta }}_1-{\dot{\theta }}_2\right)\\ +m_{2s}{l}_1{d}_2\left(\cos ({\theta }_1-{\theta }_2)\right){\ddot{\theta }}_1+I_2{\ddot{\theta }}_2+m_{3s}{l}_2^2{\ddot{\theta }}_2\\ -m_{3s}{l}_1{l}_2{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_2)\right)\left({\dot{\theta }}_1-{\dot{\theta }}_2\right)+m_{3s}{l}_1{l}_2\left(\cos ({\theta }_1-{\theta }_2)\right){\ddot{\theta }}_1\\ -m_{3s}{l}_2{d}_3{\dot{\theta }}_3\left(\sin ({\theta }_2-{\theta }_3)\right)\left({\dot{\theta }}_2-{\dot{\theta }}_3\right)+m_{3s}{l}_2{d}_3\left(\cos ({\theta }_2-{\theta }_3)\right){\ddot{\theta }}_3\\ -m_{2s}{l}_1{d}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)-m_{3s}{l}_1{l}_2{\dot{\theta }}_1{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)\\ +m_{3s}{l}_2{d}_3{\dot{\theta }}_2{\dot{\theta }}_3\left(\sin ({\theta }_2-{\theta }_3)\right)-m_{2s}{d}_{2}g\left(\sin {\theta }_2\right)-m_{3s}l{}_{2}g\left(\sin {\theta }_2\right)\end{array}$$

(19)

$$\begin{array}{l}{\tau }_3=m_{3s}{d}_3^2{\ddot{\theta }}_3-m_{3s}{l}_1{d}_3{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_3)\right)\left({\dot{\theta }}_1-{\dot{\theta }}_3\right)\\ +m_{3s}{l}_1{d}_3\left(\cos ({\theta }_1-{\theta }_3)\right){\ddot{\theta }}_1-m_{3s}{l}_2{d}_3{\dot{\theta }}_2\left(\sin ({\theta }_2-{\theta }_3)\right)\left({\dot{\theta }}_2-{\dot{\theta }}_3\right)\\ +m_{3s}{l}_2{d}_3\left(\cos ({\theta }_2-{\theta }_3)\right){\ddot{\theta }}_2+I_3{\ddot{\theta }}_3-m_{3s}{l}_1{d}_3{\dot{\theta }}_1{\dot{\theta }}_3\left(\sin ({\theta }_1-{\theta }_3)\right)\\ -m_{3s}{l}_2{d}_3{\dot{\theta }}_2{\dot{\theta }}_3\left(\sin ({\theta }_2-{\theta }_3)\right)-m_{3s}{d}_{3}g\left(\sin {\theta }_3\right)\end{array}$$

(20)

Integrating equations from (18) ~ (20) yields the kinematic mathematical model of LLRER to be

$$\left[\underset{˜}{M}\left(\stackrel{\rightharpoonup }{\theta }\right)\right]\ddot{\stackrel{\rightharpoonup }{\theta }}+\left[\underset{˜}{C}\left(\stackrel{\rightharpoonup }{\theta },\dot{\stackrel{\rightharpoonup }{\theta }}\right)\right]\dot{\stackrel{\rightharpoonup }{\theta }}+\underset{˜}{G}\left(\stackrel{\rightharpoonup }{\theta }\right)=\left[\stackrel{\rightharpoonup }{\tau }\left(\stackrel{\rightharpoonup }{\theta },\dot{\stackrel{\rightharpoonup }{\theta }},\ddot{\stackrel{\rightharpoonup }{\theta }}\right)\right]$$

(21)

where

$$\underset{˜}{M}\left(\stackrel{\rightharpoonup }{\theta }\right)\equiv \left[\begin{array}{ccc}{D}_{11}& D_{12}& D_{13}\\ D_{21}& D_{22}& D_{23}\\ D_{31}& D_{32}& D_{33}\end{array}\right]$$

(22)

$$\underset{˜}{C}\left(\stackrel{\rightharpoonup }{\theta },\dot{\stackrel{\rightharpoonup }{\theta }}\right)\equiv \left[\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\\ H_{21}& H_{22}& H_{23}\\ H_{31}& H_{32}& H_{33}\end{array}\right]$$

(23)

$$\underset{˜}{G}\left(\stackrel{\rightharpoonup }{\theta }\right)\equiv {\left[\begin{array}{ccc}{G}_1& G_2& G_3\end{array}\right]}^T$$

(24)

$$\stackrel{\rightharpoonup }{\theta }\equiv {\left[\begin{array}{ccc}{\theta }_1& {\theta }_2& {\theta }_3\end{array}\right]}^T$$

(25)

$$\ddot{\stackrel{\rightharpoonup }{\theta }}\equiv {\left[\begin{array}{ccc}{\ddot{\theta }}_1& {\ddot{\theta }}_2& {\ddot{\theta }}_3\end{array}\right]}^T$$

(26)

$$\dot{\stackrel{\rightharpoonup }{\theta }}\equiv {\left[\begin{array}{ccc}{\dot{\theta }}_1& {\dot{\theta }}_2& {\dot{\theta }}_3\end{array}\right]}^T$$

(27)

$$\stackrel{\rightharpoonup }{\tau }\equiv {\left[\begin{array}{ccc}{\tau }_1& {\tau }_2& {\tau }_3\end{array}\right]}^T$$

(28)

$$D_{11}\equiv m_{1s}{d}_1^2+I_1+m_{2s}{l}_1^2+m_{3s}{l}_1^2$$

(29)

$$D_{12}\equiv m_{2s}{l}_1{d}_2\left(\cos ({\theta }_1-{\theta }_2)\right)+m_{3s}{l}_1{l}_2\left(\cos ({\theta }_1-{\theta }_2)\right)$$

(30)

$$D_{13}\equiv m_{3s}{l}_1{d}_3\left(\cos ({\theta }_1-{\theta }_2)\right)$$

(31)

$$D_{21}\equiv m_{2s}{l}_1{d}_2\left(\cos ({\theta }_1-{\theta }_2)\right)+m_{3s}{l}_1{l}_2\left(\cos ({\theta }_1-{\theta }_2)\right)$$

(32)

$$D_{22}\equiv m_{2s}{d}_2^2+I_2+m_{3s}{l}_2^2$$

(33)

$$D_{23}\equiv m_{3s}{l}_2{d}_3\left(\cos ({\theta }_2-{\theta }_3)\right)$$

(34)

$$D_{31}\equiv m_{3s}{l}_1{d}_3\left(\cos ({\theta }_1-{\theta }_3)\right)$$

(35)

$$D_{32}\equiv m_{3s}{l}_2{d}_3\left(\cos ({\theta }_2-{\theta }_3)\right)$$

(36)

$$D_{33}\equiv m_{3s}{d}_3^2+I_3$$

(37)

H₁₁≡0

(38)

$$H_{12}\equiv m_{2s}{l}_1{d}_2{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)+m_{3s}{l}_1{l}_2{\dot{\theta }}_2\left(\sin ({\theta }_1-{\theta }_2)\right)$$

(39)

$$H_{13}\equiv m_{3s}{l}_1{d}_3{\dot{\theta }}_3\left(\sin ({\theta }_1-{\theta }_3)\right)$$

(40)

$$H_{21}\equiv -m_{2s}{l}_1{d}_2{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_2)\right)-m_{3s}{l}_1{l}_2{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_2)\right)$$

(41)

H₂₂≡0

(42)

$$H_{23}\equiv m_{3s}{l}_2{d}_3{\dot{\theta }}_3\left(\sin ({\theta }_2-{\theta }_3)\right)$$

(43)

$$H_{31}\equiv -m_{3s}{l}_1{d}_3{\dot{\theta }}_1\left(\sin ({\theta }_1-{\theta }_3)\right)$$

(44)

$$H_{32}\equiv -m_{3s}{l}_2{d}_3{\dot{\theta }}_2\left(\sin ({\theta }_2-{\theta }_3)\right)$$

(45)

H₃₃≡0

(46)

$$G_1\equiv -m_{1s}{d}_{1}g\left(\sin {\theta }_1\right)-m_{2s}l{}_{1}g\left(\sin {\theta }_1\right)-m_{3s}l{}_{1}g\left(\sin {\theta }_1\right)$$

(47)

$$G_2\equiv -m_{2s}{d}_{2}g\left(\sin {\theta }_2\right)-m_{3s}l{}_{2}g\left(\sin {\theta }_2\right)$$

(48)

G₃≡−m_3sd₃g(sinθ₃)

(49)

Applying d₃ = l₃ = I₃ = 0 and θ₃ = 90° simplifies the 3-link LLRER to be 2-link LLRER as follows

$$\left[\begin{array}{cc}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]\left[\begin{array}{c}{\ddot{\theta }}_1\\ {\ddot{\theta }}_2\end{array}\right]+\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right]\left[\begin{array}{c}{\dot{\theta }}_1\\ {\dot{\theta }}_2\end{array}\right]+\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]=\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\end{array}\right]$$

Define the input, the output and state variables of the LLRER to be ${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}\equiv {\left[\begin{array}{cc}{\tau }_1-G_1& {\tau }_2-G_2\end{array}\right]}^T\equiv {\left[\begin{array}{cc}{I}_{\mathit{np}\_1}& I_{\mathit{np}\_2}\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}}\equiv {\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{x}}_{\mathit{st}}\equiv {\left[\begin{array}{cccc}{x}_{\mathit{st}\_1}& x_{\mathit{st}\_2}& x_{\mathit{st}\_3}& x_{\mathit{st}\_4}\end{array}\right]}^T$, x_{st_1} = θ₁, x_{st_2} = θ₂ , $x_{\mathit{st}\_3}={\dot{\theta }}_1$ and $x_{\mathit{st}\_4}={\dot{\theta }}_2$. Assume the desired tracking signals to be

$${\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}\_\mathit{track}}\equiv {\stackrel{\rightharpoonup }{y}}_{\mathit{op}\_\mathit{track}}=\left[\begin{array}{c}{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^1\\ {\mathrm{O}}_{\mathit{op}\_\mathit{track}}^2\end{array}\right]=\left[\begin{array}{c}{y}_{\mathit{op}\_\mathit{track}}^1\\ y_{\mathit{op}\_\mathit{track}}^2\end{array}\right]=\left[\begin{array}{c}\sin t\\ \sin t\end{array}\right]$$

(50)

Then the state equation and the output equation of the LLRER can be easily obtained as

$$\begin{array}{l}{\left[\begin{array}{ccc}{\dot{x}}_{\mathrm{st}\text{\_}1}& \cdots & {\dot{x}}_{\mathrm{st}\text{\_}4}\end{array}\right]}^T={\left[\begin{array}{ccc}{f}_1({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & f_4({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\\ +\left[\begin{array}{ccc}{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & {\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_\_}\mathrm{m}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]\\ {\left[\begin{array}{ccc}{I}_{\mathit{np}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & I_{\mathit{np}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\\ +{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{n\_j}+{\mathrm{\Phi }}_{\mathit{un}\_j}\right)},m=2,p=1\end{array}$$

(51)

O_{op_1} = x_{st_1}

(52)

O_{op_2} = x_{st_2}

(53)

where

f₁≡x_{st_3}

(54)

f₂≡x_{st_4}

(55)

$$f_3\equiv {\displaystyle \frac{D_{12}{H}_{21}-D_{22}{H}_{11}}{\Delta }}\left(x_{\mathit{st}\_3}\right)+{\displaystyle \frac{D_{12}{H}_{22}-D_{22}{H}_{12}}{\Delta }}\left(x_{\mathit{st}\_4}\right)$$

(56)

$$f_4\equiv {\displaystyle \frac{D_{11}{H}_{21}-D_{21}{H}_{11}}{-\Delta }}\left(x_{\mathit{st}\_3}\right)+{\displaystyle \frac{D_{11}{H}_{22}-D_{21}{H}_{12}}{-\Delta }}\left(x_{\mathit{st}\_4}\right)$$

(57)

Δ≡D₁₁D₂₂ − D₁₂D₂₁

(58)

$${\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})={\left[\begin{array}{cccc}0& 0& {\displaystyle \frac{D_{22}}{\Delta }}& {\displaystyle \frac{-D_{21}}{\Delta }}\end{array}\right]}^T$$

(59)

$${\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}2}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})={\left[\begin{array}{cccc}0& 0& {\displaystyle \frac{-D_{12}}{\Delta }}& {\displaystyle \frac{D_{11}}{\Delta }}\end{array}\right]}^T$$

(60)

$${\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}={\left[\begin{array}{cccc}0& 1& 0& 0\end{array}\right]}^T$$

(61)

Φ_{n_1} = sint,Φ_{un_1} = 0

(62)

Noting that Φ_{n_1} and Φ_{un_1} denote the noise and the uncertainty, respectively.

Table 1 lists all the parameters of the considered LLRER. Then, the state equation and the output equation of the LLRER can be shown by (51) based on the following physical parameters:

D₁₁≡2.3788

(63)

D₁₂≡0.514(cos(x_{st_1} − x_{st_2})) + 0.196(cos(x_{st_1} − x_{st_2}))

(64)

D₂₁≡0.514(cos(x_{st_1} − x_{st_2})) + 0.196(cos(x_{st_1} − x_{st_2}))

(65)

D₂₂≡0.418051

(66)

H₁₁≡0

(67)

$$\begin{array}{l}{H}_{12}\equiv 0.514x_{\mathit{st}\_4}\left(\sin (x_{\mathit{st}\_1}-x_{\mathit{st}\_2})\right)\\ +0.196x_4\left(\sin (x_{\mathit{st}\_1}-x_{\mathit{st}\_2})\right)\end{array}$$

(68)

$$\begin{array}{l}{H}_{21}\equiv -0.514x_{\mathit{st}\_3}\left(\sin (x_{\mathit{st}\_1}-x_{\mathit{st}\_2})\right)\\ -0.196x_3\left(\sin (x_{\mathit{st}\_1}-x_{\mathit{st}\_2})\right)\end{array}$$

(69)

H₂₂≡0

(70)

G₁≡−58.0351(sinx_{st_1})

(71)

G₂≡−14.0379(sinx_{st_2})

(72)

Define the nominal system of above nonlinear LLRER to be

$${\dot{\stackrel{\rightharpoonup }{x}}}_{\mathit{st}}(t)=\stackrel{\rightharpoonup }{f}+{\underset{˜}{g}}_{\mathrm{ip}}{I}_{\mathit{np}},{\underset{˜}{g}}_{\mathrm{ip}}\equiv \left[\begin{array}{c}{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_1}\\ {\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_2}\end{array}\right],\stackrel{\rightharpoonup }{f}\equiv \left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_5\end{array}\right]$$

(73)

$${\stackrel{\rightharpoonup }{y}}_{\mathit{op}}(t)={\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}}$$

(74)

with the relative degree {d_{rd_1},d_{rd_2}} = {2,2} meeting

<i>the following Lie equation is established:

$$L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_j}}{L}_{\stackrel{\rightharpoonup }{f}}^k{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})=0$$

(75)

for 1 ≤ i ≤ 2,1 ≤ j ≤ 2, k < d_{rd_i} − 1, where the operator L involves the Lie differentiation.

<ii>the following Lie matrix

$$\begin{array}{l}{A}_{\mathit{system}}\\ =\left[\begin{array}{cc}{L}_{g_{\mathit{ip}\_1}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& L_{g_{\mathit{ip}\_2}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\\ L_{g_{\mathit{ip}\_1}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& L_{g_{\mathit{ip}\_2}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]\end{array}$$

(76)

is a nonsingular matrix and the spanning distribution

$$\mathit{span}\left\{{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1},{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}2}\right\}$$

(77)

is involutive [33].

3. Tracking Controller Design of Nonlinear LLRER Robot

Reference [33] and reference [34] propose the theorem that under the well-defined relative degree d_rd≡d_{rd_1} + d_{rd_2} = 4 and the involutive spanning distribution, an infinitely differentiable and bijective mapping ϕ defined as follows can transform the nonlinear LLRER system into partially linear subsystems ${\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_{l\text{\_}i},i=1,2$ [33,34] as follows:

$$\begin{array}{l}{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_{\mathrm{l}\text{\_}i}\equiv {\left[\begin{array}{ccc}{\mathrm{\Pi }}_{l\_1}^i& \cdots & {\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}^T\equiv {\left[\begin{array}{ccc}{\varphi }_{l\_1}^i& \cdots & {\varphi }_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}^T\\ \equiv {\left[\begin{array}{ccc}{L}_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\_i}-1}{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T,1\le i\le 2\end{array}$$

(78)

$${\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_{l\text{\_}i}\equiv {\left[\begin{array}{ccc}{\mathrm{\Pi }}_{l\_1}^i& \cdots & {\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}^T$$

(79)

d_rd≡d_{rd_1} + d_{rd_2}

(80)

$${\mathrm{\Pi }}_{l\_1}^1={\varphi }_{l\_1}^1\equiv L_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_1}=x_{\mathit{st}\_1},$$

(81)

$${\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}1}}^1={\mathrm{\Pi }}_{l\_2}^1={\varphi }_{l\_2}^1\equiv L_{\stackrel{\rightharpoonup }{f}}^1{\mathrm{O}}_{\mathit{op}\_1}=f_1=x_{\mathit{st}\_3}$$

(82)

$${\mathrm{\Pi }}_{l\_1}^2={\varphi }_{l\_1}^2\equiv L_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_2}=x_{\mathit{st}\_2},$$

(83)

$$\begin{array}{l}{\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}2}}^2={\mathrm{\Pi }}_{l\_2}^2={\varphi }_{l\_2}^2\equiv L_{\stackrel{\rightharpoonup }{f}}^1{\mathrm{O}}_{\mathit{op}\_2}\\ =f_3={\displaystyle \frac{D_{12}{H}_{21}-D_{22}{H}_{11}}{\Delta }}\left(x_{\mathit{st}\_3}\right)+{\displaystyle \frac{D_{12}{H}_{22}-D_{22}{H}_{12}}{\Delta }}\left(x_{\mathit{st}\_4}\right)\end{array}$$

(84)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_1}^1={\displaystyle \frac{\partial {\mathrm{O}}_{\mathit{op}\_1}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\left[\stackrel{\rightharpoonup }{f}+{\underset{˜}{g}}_{\mathrm{ip}}\cdot {\stackrel{\rightharpoonup }{I}}_{\mathit{np}}+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\right]\\ ={\mathrm{\Pi }}_{l\_2}^1+{\displaystyle \sum _{j=1}^p\left(\frac{\partial {\mathrm{O}}_{\mathit{op}\_1}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\right)\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(85)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}1}}^1={\dot{\mathrm{\Pi }}}_{l\_2}^1\\ ={\displaystyle \frac{\partial L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\left[\stackrel{\rightharpoonup }{f}+{\underset{˜}{g}}_{\mathrm{ip}}\cdot {\stackrel{\rightharpoonup }{I}}_{\mathit{np}}+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\right]\\ =L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}}{\mathrm{O}}_{\mathit{op}\_1}+L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_1}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}{I}_{\mathit{np}\_1}\\ +L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_2}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}{I}_{\mathit{np}\_2}\\ +{\displaystyle \sum _{j=1}^p\left(\frac{\partial L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\right)\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(86)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_1}^2={\displaystyle \frac{\partial {\mathrm{O}}_{\mathit{op}\_2}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\left[\stackrel{\rightharpoonup }{f}+{\underset{˜}{g}}_{\mathrm{ip}}\cdot {\stackrel{\rightharpoonup }{I}}_{\mathit{np}}+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\right]\\ ={\mathrm{\Pi }}_{l\_2}^2+{\displaystyle \sum _{j=1}^p\left(\frac{\partial {\mathrm{O}}_{\mathit{op}\_2}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\end{array}$$

(87)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}2}}^2={\dot{\mathrm{\Pi }}}_{l\_2}^2\\ ={\displaystyle \frac{\partial L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\left[\stackrel{\rightharpoonup }{f}+{\underset{˜}{g}}_{\mathrm{ip}}\cdot {\stackrel{\rightharpoonup }{I}}_{\mathit{np}}+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\right]\\ =L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}}{\mathrm{O}}_{\mathit{op}\_2}+L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_1}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}{I}_{\mathit{np}\_1}+\cdots \\ +L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_2}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}{I}_{\mathit{np}\_2}\\ +{\displaystyle \sum _{j=1}^p\left(\frac{\partial L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\right)\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(88)

Since

$$I_{\mathit{np}\_\mathit{ci}}\equiv L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}}{\mathrm{O}}_{\mathit{op}\_i}$$

(89)

$$I_{\mathit{np}\_\mathit{dij}}\equiv L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_j}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i},1\le i,j\le 2$$

(90)

then the transformed system can be given by

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_1}^1(t)\\ ={\mathrm{\Pi }}_{l\_2}^1(t)+{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(91)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}1}}^1={\dot{\mathrm{\Pi }}}_{l\_2}^1=I_{\mathit{np}\_c1}+I_{\mathit{np}\_d11}{I}_{\mathit{np}\_1}+I_{\mathit{np}\_d12}{I}_{\mathit{np}\_2}\\ +{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(92)

$${\dot{\mathrm{\Pi }}}_{l\_1}^2={\mathrm{\Pi }}_{l\_2}^2+{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_2}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)$$

(93)

$$\begin{array}{l}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}2}}^2={\dot{\mathrm{\Pi }}}_{l\_2}^2=I_{\mathit{np}\_c2}+I_{\mathit{np}\_d21}{I}_{\mathit{np}\_1}+I_{\mathit{np}\_d22}{I}_{\mathit{np}\_2}\\ +{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}$$

(94)

$${\mathrm{O}}_{\mathit{op}\_i}={\mathrm{\Pi }}_{l\_1}^i,1\le i\le 2$$

(95)

Hence

$$\begin{array}{l}\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}1}}^1\\ {\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}2}}^2\end{array}\right]=\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_2}^1\\ {\dot{\mathrm{\Pi }}}_{l\_2}^2\end{array}\right]\\ =\left[\begin{array}{c}{I}_{\mathit{np}\_c1}\\ I_{\mathit{np}\_c2}\end{array}\right]+\left[\begin{array}{cc}{I}_{\mathit{np}\_d11}& I_{\mathit{np}\_d12}\\ I_{\mathit{np}\_d21}& I_{\mathit{np}\_d22}\end{array}\right]\left[\begin{array}{c}{I}_{\mathit{np}\_1}\\ I_{\mathit{np}\_2}\end{array}\right]\\ +\left[\begin{array}{c}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\\ {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}\right]\end{array}$$

(96)

To design the proposed robust tracking controller

$${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}=A_{\mathit{system}}^{-1}\{-{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}+{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\}$$

(97)

we use the vector

$$\begin{array}{l}{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}\equiv {\left[\begin{array}{cc}{I}_{\mathit{np}\_b1}& I_{\mathit{np}\_b2}\end{array}\right]}^T\equiv {\left[\begin{array}{cc}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}}{\mathrm{O}}_{\mathit{op}\_1}& L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}}{\mathrm{O}}_{\mathit{op}\_2}\end{array}\right]}^T\\ ={\left[\begin{array}{cc}{I}_{\mathit{np}\_c1}& I_{\mathit{np}\_c2}\end{array}\right]}^T\equiv {\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_c}\end{array}$$

(98)

and the virtual input [33]

$${\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\equiv {\left[\begin{array}{cc}{I}_{\mathit{np}\_v1}& I_{\mathit{np}\_v2}\end{array}\right]}^T$$

(99)

Then we can transform the nonlinear LLRER system into the following mathematical model

$$\begin{array}{l}\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}1}}^1\\ {\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}2}}^2\end{array}\right]=\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_2}^1\\ {\dot{\mathrm{\Pi }}}_{l\_2}^2\end{array}\right]\\ =\left[\begin{array}{c}{I}_{\mathit{np}\_v1}\\ I_{\mathit{np}\_v2}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\\ {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}-1}{\mathrm{O}}_{\mathit{op}\_2}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)}\end{array}\right]\end{array}$$

(100)

From (91), (93), (96) and (100), we get

$$\begin{array}{l}\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_1}^i\\ {\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\Pi }}_{l\_1}^i\\ {\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]I_{\mathit{np}\_\mathit{vi}}\\ +\left[\begin{array}{c}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\\ {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\end{array}\right]\\ d_{\mathit{rd}\text{\_}\mathrm{i}}=2,i=1,2\end{array}$$

(101)

We build the desired tracking control by ${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}=A_{\mathit{system}}^{-1}\{-{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}+{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\}$ to be [33]

$$\begin{array}{l}{I}_{\mathit{np}\_\mathit{vi}}\equiv {y_{\mathit{op}\_\mathit{track}}^i}^{(d_{\mathit{rd}\text{\_}i})}-{\epsilon }^{-d_{\mathit{rd}\text{\_}i}}{\alpha }_1^i\left[L_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-y_{\mathit{op}\_\mathit{track}}^i\right]\\ -{\epsilon }^{1-d_{\mathit{rd}\text{\_}i}}{\alpha }_2^i\left[L_{\stackrel{\rightharpoonup }{f}}^1{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-{y_{\mathit{op}\_\mathit{track}}^i}^{(1)}\right]-\cdots \\ -{\epsilon }^{-1}{\alpha }_{d_{\mathit{rd}\text{\_}i}}^i\left[L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-{y_{\mathit{op}\_\mathit{track}}^i}^{(d_{\mathit{rd}\text{\_}i}-1)}\right],1\le i\le 2\end{array}$$

(102)

where $y_{\mathit{op}\_\mathit{track}}^i$ denotes the pre-specified tracking signal and ${\alpha }_{d_{\mathit{rd}\text{\_}i}}^i$ are elements of the following Hurwitz matrix described by

$$\begin{array}{l}{A}_L^i\equiv {\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ & ⋮& & & ⋮\\ 0& 0& 0& \cdots & 1\\ -{\alpha }_1^i& -{\alpha }_2^i& -{\alpha }_3^i& \cdots & -{\alpha }_{d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}_{d_{\mathit{rd}\text{\_}i}\times d_{\mathit{rd}\text{\_}i}}\\ =\left[\begin{array}{cc}0& 1\\ -100& -100\end{array}\right],i=1,2\end{array}$$

(103)

Then we propose the desired robust control with the desired tracking signals ${\mathrm{O}}_{\mathit{op}\_\mathit{track}}^1={\mathrm{O}}_{\mathit{op}\_\mathit{track}}^2=\sin t,$ according to the feedback linearization approach as follows:

$$\begin{array}{l}{\stackrel{\rightharpoonup }{I}}_{\mathit{np}}=A_{\mathit{system}}^{-1}\{-{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}+{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\}\\ =A_{\mathit{system}}^{-1}\left(-{\left[\begin{array}{cc}{I}_{\mathit{np}\_b1}& I_{\mathit{np}\_b2}\end{array}\right]}^T+{\left[\begin{array}{cc}{I}_{\mathit{np}\_v1}& I_{\mathit{np}\_v2}\end{array}\right]}^T\right)\end{array}$$

(104)

$$A_{\mathit{system}}^{-1}=\left[\begin{array}{cc}{D}_{11}& D_{12}\\ D_{12}& D_{22}\end{array}\right]$$

(105)

$$\begin{array}{l}{I}_{\mathit{np}\_b1}\\ ={\displaystyle \frac{D_{12}{H}_{21}-D_{22}{H}_{11}}{\Delta }}\left(x_{\mathit{st}\_3}\right)+{\displaystyle \frac{D_{12}{H}_{22}-D_{22}{H}_{12}}{\Delta }}\left(x_{\mathit{st}\_4}\right)\end{array}$$

(106)

$$\begin{array}{l}{I}_{\mathit{np}\_b2}\\ ={\displaystyle \frac{D_{11}{H}_{21}-D_{21}{H}_{11}}{-\Delta }}\left(x_{\mathit{st}\_3}\right)+{\displaystyle \frac{D_{11}{H}_{22}-D_{21}{H}_{12}}{-\Delta }}\left(x_{\mathit{st}\_4}\right)\end{array}$$

(107)

$$\begin{array}{l}{I}_{\mathit{np}\_v1}=-\sin t-{\epsilon }^{-2}(100)\left(x_{\mathit{st}\_1}-\sin t\right)\\ -{\epsilon }^{-1}(100)\left(x_{\mathit{st}\_3}-\cos t\right)\end{array}$$

(108)

$$\begin{array}{l}{I}_{\mathit{np}\_v2}=-\sin t-{\epsilon }^{-2}(100)\left(x_{\mathit{st}\_2}-\sin t\right)\\ -{\epsilon }^{-1}(100)\left(x_{\mathit{st}\_4}-\cos t\right)\end{array}$$

(109)

I_{np_1}=(D₁₁)(-I_{np_b1}+I_{np_v1})+(D₁₂)(-I_{np_b2}+I_{np_v2})

(110)

I_{np_2}=(D₁₂)(-I_{np_b1}+I_{np_v1})+(D₂₂)(-I_{np_b2}+I_{np_v2})

(111)

Based on the convenience of the following discussions, let’s define

$$e_{r\_j}^i\equiv {\mathrm{\Pi }}_{l\_j}^i-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^{i(j-1)}$$

(112)

$$e_{r\_\mathit{track}}^i\equiv {\left[e_{r\_1}^i{e}_{r\_2}^i\cdots e_{r\_d_{\mathit{rd}\text{\_}i}}^i\right]}^T\in {\Re }^{d_{\mathit{rd}\text{\_}i}}$$

(113)

$$\overline{e_{r\_j}^i}\equiv {\epsilon }^{j-1}{e}_{r\_j}^i,i=1~2,j=1,2,\cdots ,d_{\mathit{rd}\text{\_}i}$$

(114)

$$\overline{e_{r\_\mathit{track}}^i}\equiv {\left[\overline{e_{r\_1}^i}\overline{e_{r\_2}^i}\cdots \overline{e_{r\_d_{\mathit{rd}\text{\_}i}}^i}(t)\right]}^T\in {\Re }^{d_{\mathit{rd}\text{\_}i}}$$

(115)

$${\overline{e}}_{r\_\mathit{track}}\equiv {\left[\begin{array}{cccc}\overline{e_{r\_\mathit{track}}^1}& \overline{e_{r\_\mathit{track}}^2}& \cdots & \overline{e_{r{\_}_{\mathit{track}}}^m}\end{array}\right]}^T\in {\Re }^{d_{\mathit{rd}}},d_{\mathit{rd}}=4$$

(116)

$${\stackrel{\rightharpoonup }{B}}^i\equiv {{\left[\begin{array}{ccccc}0& 0& \cdots & 0& 1\end{array}\right]}^T}_{d_{\mathit{rd}\text{\_}i}\times 1},1\le i\le 2$$

(117)

$$\overline{\overline{e_{r{\_}_{\mathit{track}}}^i}}\equiv {\alpha }_1^i\overline{e_{r\_1}^i}+{\alpha }_2^i\overline{e_{r\_2}^i}+\cdots +{\alpha }_{d_{\mathit{rd}\text{\_}i}}^i\overline{e_{r\_d_{\mathit{rd}\text{\_}i}}^i}$$

(118)

where the Lyapunov matrix $A_L^i$ belongs to be a Hurwitz matrix and one can apply software Matlab to find the adjointing Lyapunov matrix $K_L^i>0$ of the famous Lyapunov equation:

$${(A_L^i)}^T{K}_L^i+K_L^i{A}_L^i=-I$$

(119)

$${\lambda }_{\max }(K_L^i)\equiv \mathrm{maximum}\mathrm{eigenvalue}\mathrm{of}\mathrm{the}\mathrm{matrix}{K}_L^i$$

(120)

$${\lambda }_{\min }(K_L^i)\equiv \mathrm{minimum}\mathrm{eigenvalue}\mathrm{of}\mathrm{the}\mathrm{matrix}{K}_L^i$$

(121)

$$\begin{array}{l}{\lambda }_{\max }^{*}\equiv \max \left\{{\lambda }_{\max }(K_L^1),{\lambda }_{\max }(K_L^2)\right\}\\ \equiv \max \left\{1.005,1.005\right\}=1.005\end{array}$$

(122)

$$\begin{array}{l}{\lambda }_{\min }^{*}\equiv \min \left\{{\lambda }_{\min }(K_L^1),{\lambda }_{\min }(K_L^2)\right\}\\ \equiv \min \left\{0.05,0.05\right\}=0.05\end{array}$$

(123)

and

$$K_L^1=K_L^2=\left[\begin{array}{cc}1.0050& 0.0050\\ 0.0050& 0.0051\end{array}\right]$$

(124)

To elaborate further the detailed design of the robust controller for nonlinear LLRER sytem, let’s make the following definitions:

Definition I.

Consider a nonlinear system ${\dot{\stackrel{\rightharpoonup }{x}}}_{\mathit{st}}=\stackrel{\rightharpoonup }{f}(t,{\stackrel{\rightharpoonup }{x}}_{\mathit{st}},{\stackrel{\rightharpoonup }{I}}_{\mathit{np}})$ with the input vector ${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}$, the state vector ${\stackrel{\rightharpoonup }{x}}_{\mathit{st}}$ and the dynamic vector $a_{22}\equiv -0.0578-0.0578\left(-x_1^2+x_2^2-x_3^2\right),$. The nonlinear system is defined to have the input-to-state stable property if

$$‖{\stackrel{\rightharpoonup }{x}}_{\mathit{st}}(t)‖\le \gamma \left(‖{\stackrel{\rightharpoonup }{x}}_{\mathit{st}}(t_0)‖,t-t_0\right)+\kappa \left(\underset{t_0\le \tau \le t}{\sup }‖{\stackrel{\rightharpoonup }{I}}_{\mathit{np}}(\tau )‖\right)$$

(125)

where κ belongs to K-class function and γ belongs to KL-class function [34].

Definition II.

[34]. A nonlinear system with the external noise input ${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}$ is defined to have the almost disturbance decoupling performance if the following requirements are established:

(a)The nonlinear system has the input-to-state stable property, i.e., (125) holds.

(b)

$$\begin{array}{l}\left|{\mathrm{O}}_{\mathit{op}\_i}(t)-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^i(t)\right|\\ \le {\kappa }_1\left(‖{\stackrel{\rightharpoonup }{x}}_{\mathit{st}}(t_0)‖,t-t_0\right)+{\displaystyle \frac{1}{\sqrt{{\kappa }_2}}}{\kappa }_3\left(\underset{t_0\le \tau \le t}{\sup }‖I_{\mathit{np}}(\tau )‖\right)\end{array}$$

(126)

and

$$\begin{array}{l}{\left[{\mathrm{O}}_{\mathit{op}\_i}(\tau )-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^i(\tau )\right]}^2\mathit{d\tau }\le \\ {\displaystyle \frac{1}{{\kappa }_4}}\left[{\kappa }_5\left(‖{\stackrel{\rightharpoonup }{x}}_{\mathit{st}}{(\mathrm{t}}_0)‖\right)+{\displaystyle {\displaystyle \underset{t_0}{\overset{t}{\int }}}{\kappa }_3\left({‖I_{\mathit{np}}\left(\tau \right)‖}^2\right)\mathit{d\tau }}\right]\end{array}$$

(127)

where ${\stackrel{\rightharpoonup }{x}}_{\mathit{st}}(t_0)$ is the initial state, κ₁ belongs to KL-class function, κ₃,κ₅ belong to K-class funtions and κ₂ > 0,κ₄ > 0.

From (101), we obtain

$$\begin{array}{l}\left[\begin{array}{c}{\dot{\mathrm{\Pi }}}_{l\_1}^i-{\dot{\mathrm{O}}}_{\mathit{op}\_\mathit{track}}^{i(())}\\ {\dot{\mathrm{\Pi }}}_{l\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i-{\dot{\mathrm{O}}}_{\mathit{op}\_\mathit{track}}^{i(d_{\mathit{rd}\text{\_}\mathrm{i}}-1)}\end{array}\right]\\ =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{\Pi }}_{l\_1}^i-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^{i(())}\\ {\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^{i(d_{\mathit{rd}\text{\_}\mathrm{i}}-1)}\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]\left(I_{\mathit{np}\_\mathit{vi}}-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^{i(d_{\mathit{rd}\text{\_}\mathrm{i}})}\right)\\ +\left[\begin{array}{c}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\\ {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\end{array}\right]\\ d_{\mathit{rd}\text{\_}\mathrm{i}}=2,i=1,2\end{array}$$

(128)

Combining (112), (114) and (128) yields

$$\begin{array}{l}\left[\begin{array}{c}{\overline{e_{r\_1}^i}}^{\prime }\\ {\epsilon }^{1-d_{\mathit{rd}\text{\_}\mathrm{i}}}{\overline{e_{r\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i}}^{\prime }\end{array}\right]=\left[\begin{array}{c}{\epsilon }^{-1}\overline{e_{r\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i}\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]\left(I_{\mathit{np}\_\mathit{vi}}-{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^{i(d_{\mathit{rd}\text{\_}\mathrm{i}})}\right)\\ +\left[\begin{array}{c}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\\ {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\end{array}\right]\\ d_{\mathit{rd}\text{\_}\mathrm{i}}=2,i=1,2\end{array}$$

(129)

Substituting (102) and (103) into (129) obtains

$$\begin{array}{l}\epsilon \left[\begin{array}{c}\stackrel{.}{\overline{e_{r\_1}^i}}\\ \stackrel{.}{\overline{e_{r\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i}}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -{\alpha }_1^i& -{\alpha }_{d_{\mathit{rd}\text{\_}\mathrm{i}}}^i\end{array}\right]\left[\begin{array}{c}\overline{e_{r\_1}^i}\\ \overline{e_{r\_d_{\mathit{rd}\text{\_}\mathrm{i}}}^i}\end{array}\right]\\ +\left[\begin{array}{c}\epsilon {\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_{\stackrel{\rightharpoonup }{f}}^{1-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\\ {\epsilon }^{d_{\mathit{rd}\text{\_}i}}{\displaystyle \sum _{j=1}^p\left(\frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}{L}_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right)\left({\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right)\right)}\end{array}\right]\end{array}$$

(130)

Then, we can get

$$\epsilon \stackrel{.}{\overline{e_{r\_\mathit{track}}^i}}=A_L^i\overline{e_{r\_\mathit{track}}^i}+{\phi }_{{\mathrm{\Pi }}_l}^i\left({\mathrm{\Phi }}_{\mathit{un}\text{\_}j}+{\mathrm{\Phi }}_{n\_j}\right),i=1,2$$

(131)

$${\mathrm{O}}_{\mathit{op}\_i}={\mathrm{\Pi }}_{l\_1}^i,i=1,2$$

(132)

where

$$\begin{array}{l}{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l}^i(\epsilon )\\ \equiv \left[\begin{array}{ccc}\epsilon \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & \epsilon \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\\ ⋮& & ⋮\\ {\epsilon }^{d_{\mathit{rd}\text{\_}i}}\left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & {\epsilon }^{d_{\mathit{rd}\text{\_}i}}\left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\end{array}\right]\\ i=1,2\end{array}$$

(133)

$${\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\equiv {\left[\begin{array}{ccc}{\mathrm{\Phi }}_{\mathit{un}\_1}& \cdots & {\mathrm{\Phi }}_{\mathit{un}\_p}\end{array}\right]}^T$$

(134)

$${\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n\equiv {\left[\begin{array}{ccc}{\mathrm{\Phi }}_{n\_1}(t)& \cdots & {\mathrm{\Phi }}_{n\_p}(t)\end{array}\right]}^T$$

(135)

Appropriately choose L_f to play the role of the Lyapunov functions for transformed system (131) by

$$L_f\equiv k\left[L_{f\_l}^1+L_{f\_l}^2\right]\equiv k{L}_{f\_l},L_{f\_l}\equiv L_{f\_l}^1+L_{f\_l}^2$$

(136)

and

$$L_{f\_l}^i\left(\overline{e_{r\_\mathit{track}}^i}\right)\equiv {\displaystyle \frac{1}{2}}{\overline{e_{r\_\mathit{track}}^i}}^T{K}_L^i\overline{e_{r\_\mathit{track}}^i},i=1.2$$

(137)

Then, we can obtain the derivative of the overall Lyapunov function L_f for the transformed system by

$$\begin{array}{c}{\displaystyle \raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.}\left(L_f\right)\\ ={\displaystyle \frac{k}{2}}\left[{\left(\stackrel{•}{\overline{e_{r\_\mathit{track}}^1}}\right)}^T{K}_L^1\overline{e_{r{\_}_{\mathit{track}}}^1}+{\left(\overline{e_{r\_\mathit{track}}^1}\right)}^T{K}_L^1\left(\stackrel{•}{\overline{e_{r\_\mathit{track}}^1}}\right)\right.\\ \left.+{\left(\stackrel{•}{\overline{e_{r\_\mathit{track}}^2}}\right)}^T{K}_L^2\overline{e_{r\_\mathit{track}}^2}+{\left(\overline{e_{r\_\mathit{track}}^2}\right)}^T{K}_L^2\left(\stackrel{•}{\overline{e_{r\_\mathit{track}}^2}}\right)\right]\end{array}$$

$$={\displaystyle \frac{k}{2\epsilon }}{\left(\overline{e_{r\_\mathit{track}}^1}\right)}^T\left[K_L^1\left(A_L^1\right)+{\left(A_L^1\right)}^T{K}_L^1\right]\overline{e_{r\_\mathit{track}}^1}$$

$$+{\displaystyle \frac{k}{2\epsilon }}{\left(\overline{e_{r\_\mathit{track}}^2}\right)}^T\left[K_L^2\left(A_L^2\right)+{\left(A_L^2\right)}^T{K}_L^2\right]\overline{e_{r\_\mathit{track}}^2}$$

$$+{\displaystyle \frac{k}{\epsilon }}\left\{{\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)}^T\left[{\left({\phi }_{{\mathrm{\Pi }}_l}^1\right)}^T{K}_L^1\overline{e_{r\_\mathit{track}}^1}+{\left({\phi }_{{\mathrm{\Pi }}_l}^2\right)}^T{K}_L^2\overline{e_{r\_\mathit{track}}^2}\right]\right\}$$

$$\begin{array}{l}=-{\displaystyle \frac{k}{2\epsilon }}\left[{‖\overline{e_{r\_\mathit{track}}^1}‖}^2+{‖\overline{e_{r\_\mathit{track}}^2}‖}^2\right]\\ +{\displaystyle \frac{k}{\epsilon }}\left\{{\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)}^T\left[{\left({\phi }_{{\mathrm{\Pi }}_l}^1\right)}^T{K}_L^1\overline{e_{r\_\mathit{track}}^1}+{\left({\phi }_{{\mathrm{\Pi }}_l}^2\right)}^T{K}_L^2\overline{e_{r\_\mathit{track}}^2}\right]\right\}\end{array}$$

$$\begin{array}{l}\le -{\displaystyle \frac{k}{2\epsilon }}\left[{‖\overline{e_{r\_\mathit{track}}^1}‖}^2+{‖\overline{e_{r\_\mathit{track}}^2}‖}^2\right]+{\displaystyle \frac{k}{\epsilon }}\left[‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖\right.\\ \left.\left(‖{\phi }_{{\mathrm{\Pi }}_l}^1‖‖K_L^1‖‖\overline{e_{r\_\mathit{track}}^1}‖+‖{\phi }_{{\mathrm{\Pi }}_l}^2‖‖K_L^2‖‖\overline{e_{r\_\mathit{track}}^2}‖\right)\right]\end{array}$$

$$\le -{\displaystyle \frac{k}{\epsilon }}\left[{\displaystyle \frac{2L_{f\_l}^i}{{\lambda }_{\max }(K_L^1)}}+{\displaystyle \frac{2L_{f\_l}^2}{{\lambda }_{\max }(K_L^2)}}\right]$$

$$\begin{array}{l}+{\displaystyle \frac{225}{30}}{\displaystyle \frac{k^2}{{\epsilon }^2}}{‖{\phi }_{{\mathrm{\Pi }}_l}^1‖}^2{‖K_L^1‖}^2{‖\overline{e_{r\_\mathit{track}}^1}‖}^2+{\displaystyle \frac{1}{30}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\\ +{\displaystyle \frac{225}{30}}{\displaystyle \frac{k^2}{{\epsilon }^2}}{‖{\phi }_{{\mathrm{\Pi }}_l}^2‖}^2{‖K_L^2‖}^2{‖\overline{e_{r\_\mathit{track}}^2}‖}^2+{\displaystyle \frac{1}{30}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\end{array}$$

$$\begin{array}{l}\le -{\displaystyle \frac{k}{\epsilon }}{\displaystyle \frac{1}{{\lambda }_{\max }^{*}}}{L}_{f\_l}\\ +{\displaystyle \frac{225}{30}}{\displaystyle \frac{k^2}{{\epsilon }^2}}{‖{\phi }_{{\mathrm{\Pi }}_l}^1‖}^2{‖K_L^1‖}^2{‖\overline{e_{r\_\mathit{track}}^1}‖}^2+{\displaystyle \frac{1}{30}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\\ +{\displaystyle \frac{225}{30}}{\displaystyle \frac{k^2}{{\epsilon }^2}}{‖{\phi }_{{\mathrm{\Pi }}_l}^2‖}^2{‖K_L^2‖}^2{‖\overline{e_{r\_\mathit{track}}^2}‖}^2+{\displaystyle \frac{1}{30}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\end{array}$$

$$\begin{array}{l}\le -\left({\displaystyle \frac{1}{\epsilon {\lambda }_{\max }^{*}}}-{\displaystyle \frac{225}{30}}{\displaystyle \frac{k{‖{\phi }_{{\mathrm{\Pi }}_l}^1‖}^2{‖K_L^1‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^1)}}-{\displaystyle \frac{225}{30}}{\displaystyle \frac{k{‖{\phi }_{{\mathrm{\Pi }}_l}^2‖}^2{‖K_L^2‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^2)}}\right)\\ {\left(\sqrt{k\left(L_{f\_l}\right)}\right)}^2+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\end{array}$$

$$=-Q{\left(\sqrt{k\left(L_{f\_l}\right)}\right)}^2+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2$$

(138)

where

$$Q\equiv {\displaystyle \frac{1}{\epsilon {\lambda }_{\max }^{*}}}-{\displaystyle \frac{225}{30}}{\displaystyle \frac{k{‖{\phi }_{{\mathrm{\Pi }}_l}^1‖}^2{‖K_L^1‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^1)}}-{\displaystyle \frac{225}{30}}{\displaystyle \frac{k{‖{\phi }_{{\mathrm{\Pi }}_l}^2‖}^2{‖K_L^2‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^2)}}>0$$

(139)

i.e.,

$${\displaystyle \frac{d}{\mathit{dt}}}\left(L_f\right)\le -\mathit{Qk}{L}_{f\_l}+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2$$

(140)

Define

$${\alpha }_s(\epsilon )\equiv {\displaystyle \frac{Q}{2}}$$

(141)

F≡2α_s(ε)

(142)

$$F_1\equiv {\displaystyle \frac{1}{15}}{\left(\underset{t_0\le \tau \le t}{\sup }‖{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n‖\right)}^2$$

(143)

$$F_2\equiv {\displaystyle \frac{k(\epsilon )}{2}}{\lambda }_{\min }^{*}$$

(144)

Then

$$\begin{array}{l}{\displaystyle \raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.}\left(L_f\right)\le -2{\alpha }_{s}k{L}_{f\_l}+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\\ \le -F\cdot F_2\left({‖\overline{e_{r\_\mathit{track}}}‖}^2\right)+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\end{array}$$

(145)

Define the error vector $\overline{e_{r\_\mathit{track}}}$ of the transformed system by

$$\begin{array}{l}\overline{e_{r\_\mathit{track}}}\equiv {\left[\begin{array}{cc}\overline{e_{r\_\mathit{track}}^1}& \overline{e_{r\_\mathit{track}}^2}\end{array}\right]}^T\\ \equiv {\left[\begin{array}{cc}\overline{e_{r\_1}^1}& \overline{e_{r\_\mathit{rem}}^1}\end{array}\right]}^T,\overline{e_{r\_\mathit{rem}}^1}\in {\Re }^{d_{\mathit{rd}}-1}\end{array}$$

(146)

Then we can get

$$\begin{array}{l}{\displaystyle \raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.}\left(L_f\right)\\ \le -F\cdot F_2\left({‖\overline{e_{r\_1}^1}‖}^2+{‖\overline{e_{r\_\mathit{rem}}^1}‖}^2\right)+{\displaystyle \frac{1}{15}}{‖\left({\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\right)‖}^2\end{array}$$

(147)

Next, we will prove the fact that the robust tracking control in Figure 4 possesses the almost disturbance decoupling performance, and the globally exponential stability of the LLRER system in Appendix A.

Noting that we can extend aforementioned complete control-design process to get two more general theorems for general nonlinear uncertain systems with external disturbances and uncertainties as follows:

$$\begin{array}{l}{\left[\begin{array}{ccc}{\dot{x}}_{\mathrm{st}\text{\_}1}& \cdots & {\dot{x}}_{\mathrm{st}\text{\_}n}\end{array}\right]}^T={\left[\begin{array}{ccc}{f}_1({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & f_n({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\\ +\left[\begin{array}{ccc}{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & {\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_\_}\mathrm{m}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]\\ {\left[\begin{array}{ccc}{I}_{\mathit{np}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & I_{\mathit{np}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\\ +{\left[\begin{array}{ccc}\delta f_{\mathit{un}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & \delta f_{\mathit{un}\_n}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{n\_j}+{\mathrm{\Phi }}_{\mathit{un}\_j}\right)}\end{array}$$

(148)

$$\begin{array}{l}{\left[\begin{array}{ccc}{y}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & y_{\mathit{op}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\\ ={\left[\begin{array}{ccc}{\mathrm{O}}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & {\mathrm{O}}_{\mathit{op}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T\end{array}$$

(149)

i.e.,

$${\dot{\stackrel{\rightharpoonup }{x}}}_{\mathit{st}}(t)=\stackrel{\rightharpoonup }{f}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})+{\underset{˜}{g}}_{\mathrm{ip}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}}){\stackrel{\rightharpoonup }{I}}_{\mathit{np}}+\delta {\stackrel{\rightharpoonup }{f}}_{\mathit{un}}+{\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}\left({\mathrm{\Phi }}_{n\_j}+{\mathrm{\Phi }}_{\mathit{un}\_j}\right)}$$

(150)

$${\stackrel{\rightharpoonup }{y}}_{\mathit{op}}(t)={\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})$$

(151)

where ${\stackrel{\rightharpoonup }{x}}_{\mathit{st}}(t)\equiv {\left[\begin{array}{ccc}{x}_{\mathit{st}\_1}(t)& \cdots & x_{\mathit{st}\_n}(t)\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}\equiv {\left[\begin{array}{ccc}{I}_{\mathit{np}\_1}& \cdots & I_{\mathit{np}\_m}\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n\equiv {\left[\begin{array}{ccc}{\mathrm{\Phi }}_{n\_1}(t)& \cdots & {\mathrm{\Phi }}_{n\_p}(t)\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}\equiv {\left[\begin{array}{ccc}{\mathrm{\Phi }}_{\mathit{un}\_1}& \cdots & {\mathrm{\Phi }}_{\mathit{un}\_p}\end{array}\right]}^T$, ${\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}$ denote the system states vector, the input vector, the output vector, the noise vector, the uncertain vector, the noise-adjointing vector of the nonlinear system, respectively. Consider the system relating vectors $\stackrel{\rightharpoonup }{f}\equiv {\left[\begin{array}{ccc}{f}_1& \dots & f_n\end{array}\right]}^T$, ${\underset{˜}{g}}_{\mathrm{ip}}\equiv {\left[\begin{array}{ccc}{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1}& \dots & {\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}\mathrm{m}}\end{array}\right]}^T$ and ${\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}}\equiv {\left[\begin{array}{ccc}{\mathrm{O}}_{\mathit{op}\_1}& \dots & {\mathrm{O}}_{\mathit{op}\_m}\end{array}\right]}^T$ to be differentially continuous and the uncertain vector $\delta {\stackrel{\rightharpoonup }{f}}_{\mathit{un}}$ to be $\delta {\stackrel{\rightharpoonup }{f}}_{\mathit{un}}\equiv {\displaystyle \sum _{j=1}^p{\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_j}^{*}{\mathrm{\Phi }}_{\mathit{un}\_j}}$.

Assumption I.

The inequality is established:

$$‖{\stackrel{\rightharpoonup }{\beta }}_{n\_t}(t,{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n,\overline{e_{r\_\mathit{track}}})-{\stackrel{\rightharpoonup }{\beta }}_{n\_t}(t,{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n,0)‖\le M_n\left(‖\overline{e_{r\_\mathit{track}}}‖\right)$$

(152)

where M_n > 0, ${\stackrel{\rightharpoonup }{\beta }}_{n\_t}(t,{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n,\overline{e_{r\_\mathit{track}}})\equiv {\stackrel{\rightharpoonup }{\beta }}_n({\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l,{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n)$.

Define the nominal system of above nonlinear system to be

$${\dot{\stackrel{\rightharpoonup }{x}}}_{\mathit{st}}(t)=\stackrel{\rightharpoonup }{f}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})+{\underset{˜}{g}}_{\mathrm{ip}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}}){\stackrel{\rightharpoonup }{I}}_{\mathit{np}}$$

(153)

$${\stackrel{\rightharpoonup }{y}}_{\mathit{op}}(t)={\stackrel{\rightharpoonup }{\mathrm{O}}}_{\mathit{op}}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})$$

(154)

with the relative degree {d_{rd_1},d_{rd_2},…,d_{rd_m}} meeting

<i>the following Lie equation is established:

$$L_{{\stackrel{\rightharpoonup }{g}}_{\mathit{ip}\_j}}{L}_{\stackrel{\rightharpoonup }{f}}^k{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})=0$$

(155)

for 1 ≤ i ≤ m,1 ≤ j ≤ m, k < d_{rd_i} − 1, where m is the input number and the operator L involves the Lie differentiation.

<ii>the following Lie matrix

$$\begin{array}{l}{A}_{\mathit{system}}\\ \equiv \left[\begin{array}{ccc}{L}_{g_{\mathit{ip}\_1}}{L}_f^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & L_{g_{\mathit{ip}\_m}}{L}_f^{d_{\mathit{rd}\text{\_}1}-1}{\mathrm{O}}_{\mathit{op}\_1}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\\ ⋮& & ⋮\\ L_{g_{\mathit{ip}\_1}}{L}_f^{d_{\mathit{rd}\text{\_}m}-1}{\mathrm{O}}_{\mathit{op}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & L_{g_{\mathit{ip}\_m}}{L}_f^{d_{\mathit{rd}\text{\_}m}-1}{\mathrm{O}}_{\mathit{op}\_m}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]\end{array}$$

(156)

is a nonsingular matrix and the spanning distribution

$$\mathit{span}\{{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}1},{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}2},\cdots ,{\stackrel{\rightharpoonup }{g}}_{\mathrm{ip}\text{\_}\mathrm{m}}\}$$

(157)

is involutive [33].

Assume that the considered nonlinear system possesses the well-defined relative degree and the involutive spanning distribution and then the following mapping ϕ described as

ϕ:ℜⁿ → ℜⁿ

(158)

$$\begin{array}{l}{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_{\mathrm{l}\text{\_}i}\equiv {\left[\begin{array}{ccc}{\mathrm{\Pi }}_{l\_1}^i& \cdots & {\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}^T\equiv {\left[\begin{array}{ccc}{\varphi }_{l\_1}^i& \cdots & {\varphi }_{l\_d_{\mathit{rd}\text{\_}i}}^i\end{array}\right]}^T\\ \equiv {\left[\begin{array}{ccc}{L}_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})& \cdots & L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\_i}-1}{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\end{array}\right]}^T,\\ d_{\mathit{rd}}\equiv d_{\mathit{rd}\text{\_}1}+d_{\mathit{rd}\text{\_}2}+\cdots +d_{\mathit{rd}\text{\_}m}\end{array}$$

(159)

$${\phi }_{n\_k}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})\equiv {\mathrm{\Pi }}_{n\_k}(t),k=d_{\mathit{rd}}+1,d_{\mathit{rd}}+2,\cdots ,n$$

(160)

and

$$L_{g_{\mathit{ip}\_j}}{\varphi }_{n\_k}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})=0,k=d_{\mathit{rd}}+1,d_{\mathit{rd}}+2,\cdots ,n,1\le j\le m$$

(161)

is an infinitely differentiable and bijective mapping that transforms the complicated nonlinear system to be both subsystems including the partially nonlinear Π_{n_k} subsystem and partially linear subsystem ${\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_{\mathrm{l}\text{\_}i},i=1,\mathrm{...},m$, respectively [33].

Appropriately choose $L_{f\_n}({\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n)$ and $L_{f\_l}\left(\overline{e_{r\_\mathit{track}}}\right)$ to play the role of the Lyapunov functions for the transformed nonlinear subsystem and the transformed linear subsystem, respectively and then compositely design an overall Lyapunov function $L_{f\_l+n}\left(\overline{e_{r\_\mathit{track}}},{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n\right)$ of the transformed system by

$$L_{f\_l+n}\left(\overline{e_{r\_\mathit{track}}},{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n\right)\equiv L_{f\_n}({\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n)+k(\epsilon )L_{f\_l}(\overline{e_{r\_\mathit{track}}})$$

(162)

$$L_{f\_l}(\overline{e_{r\_\mathit{track}}})=L_{f\_l}^1\left(\overline{e_{r\_\mathit{track}}^1}\right)+\cdots +L_{f\_l}^m\left(\overline{e_{r\_\mathit{track}}^m}\right)$$

(163)

and

$$L_{f\_l}^i\left(\overline{e_{r\_\mathit{track}}^i}\right)\equiv {\displaystyle \frac{1}{2}}{\overline{e_{r\_\mathit{track}}^i}}^T{K}_L^i\overline{e_{r\_\mathit{track}}^i}$$

(164)

Theorem I.

Exist an infinitely differentiable and bijective function $L_{f\_n}:{\Re }^{n-r}\to {\Re }^{+}$for the transformed system (Π_{n_k}, ${\mathrm{\Pi }}_{l\_d_{\mathit{rd}\text{\_}m}}^m$) such that the following criteria hold:

$$(\mathrm{a}){\eta }_{n\text{\_}1}{‖{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n‖}^2\le L_{f\_n}({\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n)\le {\eta }_{n\text{\_}2}{‖{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n‖}^2$$

(165)

$$(\mathrm{b}){\nabla }_t{L}_{f\_n}+{({\nabla }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n}{L}_{f\_n})}^T{\stackrel{\rightharpoonup }{\beta }}_n(t,{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n,0)\le -27{\alpha }_x{L}_{f\_n}$$

(166)

$$(\mathrm{c})‖{\nabla }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n}{L}_{f\_n}‖\le {\eta }_{n\text{\_}3}‖{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n‖,{\eta }_{n\text{\_}3}>0$$

(167)

and the proposed robust tracking controller is designed by

$${\stackrel{\rightharpoonup }{I}}_{\mathit{np}}=A_{\mathit{system}}^{-1}\{-{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}+{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\}$$

(168)

$$\begin{array}{l}{\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_b}\equiv {\left[\begin{array}{cccc}{I}_{\mathit{np}\_b1}& I_{\mathit{np}\_b2}& \cdots & I_{\mathit{np}\_\mathit{bm}}\end{array}\right]}^T\\ \equiv {\left[\begin{array}{cccc}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}1}}{\mathrm{O}}_{\mathit{op}\_1}& L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}2}}{\mathrm{O}}_{\mathit{op}\_2}& \cdots & L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}m}}{\mathrm{O}}_{\mathit{op}\_m}\end{array}\right]}^T\end{array}$$

(169)

$${\stackrel{\rightharpoonup }{I}}_{\mathit{np}\_v}\equiv {\left[\begin{array}{cccc}{I}_{\mathit{np}\_v1}& I_{\mathit{np}\_v2}& \cdots & I_{\mathit{np}\_\mathit{vm}}\end{array}\right]}^T$$

(170)

$$\begin{array}{l}{I}_{\mathit{np}\_\mathit{vi}}\equiv {y_{\mathit{op}\_\mathit{track}}^i}^{(d_{\mathit{rd}\text{\_}i})}-{\epsilon }^{-d_{\mathit{rd}\text{\_}i}}{\alpha }_1^i\left[L_{\stackrel{\rightharpoonup }{f}}^0{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-y_{\mathit{op}\_\mathit{track}}^i\right]\\ -{\epsilon }^{1-d_{\mathit{rd}\text{\_}i}}{\alpha }_2^i\left[L_{\stackrel{\rightharpoonup }{f}}^1{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-{y_{\mathit{op}\_\mathit{track}}^i}^{(1)}\right]-\cdots \\ -{\epsilon }^{-1}{\alpha }_{d_{\mathit{rd}\text{\_}i}}^i\left[L_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}({\stackrel{\rightharpoonup }{x}}_{\mathit{st}})-{y_{\mathit{op}\_\mathit{track}}^i}^{(d_{\mathit{rd}\text{\_}i}-1)}\right]\end{array}$$

(171)

$$Q=\left[\begin{array}{cc}{Q}_{11}& Q_{12}\\ Q_{12}& Q_{22}\end{array}\right]$$

(172)

$$Q_{11}=27{\alpha }_x-{\displaystyle \frac{225}{30}}{\displaystyle \frac{{\eta }_{n\_3}^2}{{\eta }_{n\_1}}}{‖{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n}‖}^2$$

(173)

$$Q_{12}=-\left[{\displaystyle \frac{{\eta }_{n\text{\_}3}{M}_n}{\sqrt{2k(\epsilon ){\eta }_{n\text{\_}1}{\lambda }_{\min }^{*}}}}\right]$$

(174)

$$\begin{array}{l}{Q}_{22}={\displaystyle \frac{1}{\epsilon {\lambda }_{\max }^{*}}}-\left({\displaystyle \frac{225}{30}}\right){\displaystyle \frac{k(\epsilon ){‖{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l}^1‖}^2{‖K_L^1‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^1)}}-\cdots \\ -\left({\displaystyle \frac{225}{30}}\right){\displaystyle \frac{k(\epsilon ){‖{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l}^m‖}^2{‖K_L^m‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^m)}}\end{array}$$

(175)

$${\alpha }_s(\epsilon )\equiv {\displaystyle \frac{Q_{11}+Q_{22}-{\left[{(Q_{11}-Q_{22})}^2+4Q_{12}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{4}}$$

(176)

F≡2α_s(ε)

(177)

$$F_1\equiv {\displaystyle \frac{m+1}{30}}{\left(\underset{t_0\le \tau \le t}{\sup }‖{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n‖\right)}^2$$

(178)

$$F_2\equiv \min \left\{{\eta }_{n\text{\_}1},{\displaystyle \frac{k(\epsilon )}{2}}{\lambda }_{\min }^{*}\right\}$$

(179)

$$\begin{array}{l}{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l}^i(\epsilon )\\ \equiv \left[\begin{array}{ccc}\epsilon \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & \epsilon \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\\ ⋮& & ⋮\\ {\epsilon }^{d_{\mathit{rd}\text{\_}i}}\left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & {\epsilon }^{d_{\mathit{rd}\text{\_}i}}\left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{L}_{\stackrel{\rightharpoonup }{f}}^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\end{array}\right]\end{array}$$

(180)

$$\begin{array}{l}{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_n}(\epsilon )\\ \equiv \left[\begin{array}{ccc}\left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\phi }_{\mathrm{n}\text{\_}{d}_{\mathit{rd}}+1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\phi }_{n\_d_{\mathit{rd}}+1}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\\ ⋮& & ⋮\\ \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\phi }_{n\_n}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & \left({\displaystyle \frac{\partial }{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}{\phi }_{n\_n}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\end{array}\right]\end{array}$$

(181)

where the dynamic matrix Q > 0 belongs to be positive definite matrix and the parameter k(ε) passes through the original point and the ε/k(ε) approach to zero, i.e.,

$$\underset{\epsilon \to 0}{\lim }\epsilon /k(\epsilon )\to 0$$

(182)

Then the nonlinear system based on the proposed robust tracking controller (168) has the almost disturbance decoupling performance. Moreover, the desired tracking errors will be exponentially reduced by the criterion F ⋅ F₂ > 1 with the exponential convergence rate

$${\displaystyle \frac{F\cdot F_2}{{\xi }_{\max }}}$$

(183)

$${\xi }_{\max }\equiv \max \left\{{\eta }_{n\text{\_}2},{\displaystyle \frac{k}{2}}{\lambda }_{\max }^{*}\right\}$$

(184)

and the exponential convergence radius $\underset{¯}{r}$ of the global final attractor is equal to be

$$\sqrt{{\displaystyle \frac{F_1}{F\cdot F_2}}}$$

(185)

If the dimension of the nonlinear system is same to the relative degree, i.e., the considered system belongs to be complete feedback linearizable, then the simplified version of Theorem I can be summarized as follows.

Theorem II.

The robust tracking problem of the nonlinear system with almost decoupling disturbance performance can be well achieved via the proposed controller (248) by tuning the following variables F ⋅ F₂ > 1 with

$$\begin{array}{l}Q={\displaystyle \frac{1}{\epsilon {\lambda }_{\max }^{*}}}-\left({\displaystyle \frac{225}{30}}\right){\displaystyle \frac{k(\epsilon ){‖{\phi }_{{\mathrm{\Pi }}_l}^1‖}^2{‖K_L^1‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^1)}}-\cdots \\ -\left({\displaystyle \frac{225}{30}}\right){\displaystyle \frac{k(\epsilon ){‖{\phi }_{{\mathrm{\Pi }}_l}^m‖}^2{‖K_L^m‖}^2}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\epsilon }^2{\lambda }_{\min }(K_L^m)}}>0\end{array}$$

(186)

$${\alpha }_s(\epsilon )\equiv {\displaystyle \frac{Q}{2}}$$

(187)

F≡2α_s(ε)

(188)

$$F_1\equiv {\displaystyle \frac{m}{30}}{\left(\underset{t_0\le \tau \le t}{\sup }‖{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_n+{\stackrel{\rightharpoonup }{\mathrm{\Phi }}}_{\mathit{un}}‖\right)}^2$$

(189)

$$F_2\equiv {\displaystyle \frac{k(\epsilon )}{2}}{\lambda }_{\min }^{*}$$

(190)

$$\begin{array}{l}{\phi }_{{\stackrel{\rightharpoonup }{\mathrm{\Pi }}}_l}^i(\epsilon )\\ \equiv \left[\begin{array}{ccc}\epsilon \left({\displaystyle \frac{\partial {\mathrm{O}}_{\mathit{op}\_i}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & \epsilon \left({\displaystyle \frac{\partial {\mathrm{O}}_{\mathit{op}\_i}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\\ ⋮& & ⋮\\ {\epsilon }^{d_{\mathit{rd}\text{\_}i}}\left({\displaystyle \frac{\partial L_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_1}^{*}& \cdots & {\epsilon }^{d_{\mathit{rd}\text{\_}\mathrm{i}}}\left({\displaystyle \frac{\partial L_f^{d_{\mathit{rd}\text{\_}i}-1}{\mathrm{O}}_{\mathit{op}\_i}}{\partial {\stackrel{\rightharpoonup }{x}}_{\mathit{st}}}}\right){\stackrel{\rightharpoonup }{q}}_{\mathit{noise}\_p}^{*}\end{array}\right]\end{array}$$

(191)

Moreover, the tracking errors of the transformed subsystem can be reduced by adjusting the variable F ⋅ F₂ > 1 with the exponential convergence rate

$${\displaystyle \frac{F\cdot F_2}{{\xi }_{\max }}}$$

(192)

where

$${\xi }_{\max }\equiv {\displaystyle \frac{k}{2}}{\lambda }_{\max }^{*}$$

(193)

and the exponential convergence radius of the global final attractor

$$\underset{¯}{r}=\sqrt{{\displaystyle \frac{F_1}{F\cdot F_2}}}$$

(194)

To systematically design the proposed robust tracking controller for LLRER, an useful algorithm of Theorem I is depicted as follows. An effective AI Python program is applied to construct the proposed robust tracking controller in section III.

(Step i)According to the given outputs for LLRER, one firstly obtain the relative degree d_{rd_1},…,d_{rd_m}.

(Step ii)Applying (158) gets the infinitely differentiable and bijective mapping.

(Step iii)Properly choose parameters ${\alpha }_{d_{\mathit{rd}\text{\_}i}}^i$ such that $A_L^i$ is Hurwitz via (103) and (119) and apply software Matlab to calculate the adjointing Lyapunov matrix $K_L^i>0$.

(Step iv)Appropriately design the Lyapunov function L_{f_n} of the nonlinear transformed subsystem to establish the requirements (152) and (165) ~ (167). It is worth noting that if the dimension of the nonlinear system is same to the relative degree, i.e., the considered system belongs to be complete feedback linearizable [34], then immediately jump to (Step v).

(Step v)According to (172) ~ (181), (182) and (184), properly select parameters k,α_s(ε),ε such that F ⋅ F₂ > 1.

(Step vi)Immediately construct the proposed robust tracking controller by (168).

4. Simulation of Nonlinear LLRER Robot

Designing ε = 1, $k=50\sqrt{\epsilon }$, d_{rd_1} = 2, d_{rd_2} = 2, α_s = 0.495625, Q = 0.99125, F = 0.99125, F₁ = 0.067 and F₂ = 1.25 yields the fact that all the requirements of Theorem II are established. The output trajectories and the input trajectories (ε = 1) of the LLRER system are shown by software Matlab in Figure 5 and Figure 6, respectively. Noting that the proposed rotational inertias of link1~link2 indeed drive the outputs to robustly track the pre-specified angles ${\mathrm{O}}_{\mathit{op}\_\mathit{track}}^1=\sin t$, ${\mathrm{O}}_{\mathit{op}\_\mathit{track}}^2=\sin t$ for link1 and link2, respectively. Based on the observations of (192) ~ (193) , Figure 5 and Figure 7, it is obvious to see the fact that the exponentially convergent rates of system tracking errors for the LLRER system with smaller ε (ε = 0.4) is better than larger ε (ε = 1).

Figure 5. The state trajectories for ε = 1.

Figure 5. The state trajectories for ε = 1.

Figure 6. The input trajectories for ε = 1.

Figure 6. The input trajectories for ε = 1.

Figure 7. The state trajectories for ε = 0.4.

Figure 7. The state trajectories for ε = 0.4.

Figure 8. The input trajectories for ε = 0.4.

Figure 8. The input trajectories for ε = 0.4.

In order to show the outstanding features that allow engineers to systematically build proposed robust tracking control, we use the famous AI language “Python” to generate an effective system in this study, where this effective controller-design human–machine interaction package includes inputs: (i)the state equation and output equation of the nonlinear LLRER system; (ii)the items of the state number, the output number, the input number and the desired pre-specified tracking signals ${\mathrm{O}}_{\mathit{op}\_\mathit{track}}^1=\sin t,{\mathrm{O}}_{\mathit{op}\_\mathit{track}}^2=\sin t$; (iii)the disturbances, the Lyapunov functions for transformed nonlinear and linear subsystems. The controller-design human–machine interaction package will exert its excellent symbol-operation function of AI “Python” to generate both Matlab working files (cytquadff_new1.m and cytquadsimulation_new1.m) as shown in Figure 9 for the LLRER system. Then, we can implement these both working Matlab files to depict the complete trajectories before, on and after entering the exponential convergent radius symbolized by blue circles as shown in Figure 10, Figure 11 and Figure 12, respectively, with the exponential convergent radius $\underset{¯}{r}\approx 0.284$.

5. Comparison to Traditional Approach

In this section, we will compare the proposed feedback linearization method with the singular perturbation approach with high feedback-gain [30,31] that “firstly” address the almost disturbance decoupling problem. The severe drawback of the singular perturbation method with high feedback-gain needs to establish the sufficient condition that the considered dynamics multiplied by the disturbance should meet the famous “structural triangle condition” for the almost disturbance decoupling problem. [30,31] had pointed out the fact that the following nonlinear system cannot solve the almost disturbance decoupling problem:

$$\left[\begin{array}{c}{\dot{x}}_{\mathit{st}\_1}(t)\\ {\dot{x}}_{\mathit{st}\_2}(t)\end{array}\right]=\left[\begin{array}{c}{\tan }^{-1}\left(x_{\mathit{st}\_2}\right)\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]I_{\mathit{np}}+\left[\begin{array}{c}1\\ 0\end{array}\right]{\mathrm{\Phi }}_n$$

(195)

O_{op_1} = x_{st_1},Φ_n(t) = 0.6sint

(196)

From (195) and (196), it is easy to obtain the following results: $L_f^0{\mathrm{O}}_{\mathit{op}\_1}={\mathrm{O}}_{\mathit{op}\_1}=x_{\mathit{st}\_1}$, $L_f^1{\mathrm{O}}_{\mathit{op}\_1}={\tan }^{-1}(x_{\mathit{st}\_2})$, $L_{g_{\mathit{ip}}}{L}_f^0{\mathrm{O}}_{\mathit{op}\_1}=0$, $L_{g_{\mathit{ip}}}{L}_f^1{\mathrm{O}}_{\mathit{op}\_1}={\displaystyle \frac{1}{1+x_{\mathit{st}\_2}^2}}$ and

$${\tilde{g}}_{\mathrm{ip}}\equiv \left[\begin{array}{c}0\\ {\displaystyle \frac{1}{L_{g_{\mathit{ip}}}{L}_f^1{\mathrm{O}}_{\mathit{op}\_1}}}\end{array}\right]=\left[\begin{array}{c}0\\ 1+x_{\mathit{st}\_2}^2\end{array}\right]$$

(197)

Therefore, the main sufficient condition of [30,31] is not well established since the distribution ${\tilde{g}}_{\mathrm{ip}}$ is not complete. In contrast, the following proposed robust tracking controller can well address the almost disturbance decoupling problem:

$$\begin{array}{l}{I}_{\mathit{np}}=\left(x_{\mathit{st}\_2}^2+1\right)\\ [(-\cos t)-256(x_{\mathit{st}\_1}-(\cos t))-16({\tan }^{-1}{x}_{\mathit{st}\_2}-(-\sin \mathrm{t}))]\end{array}$$

(198)

The system output trajectory of the considered noniliear system shown by (195) with the proposed controller (198) is depicted in Figure 13. From the observation of Figure 13, one can conclude the fact that the presented robust tracking controller described by (198) can indeed make the system output trajectory track the pre-specified signal cost with the almost disturbance decoupling performance.

6. Conclusions

This study presents the well robust tracking control design of the highly nonlinear lower limb rehabilitation exoskeleton robot system via the feedback linearization method to replace the singular perturbation approach with high feedback-gain that needs to satisfy the difficult sufficient condition that some distributions should be complete, and proposes the global exponential stability without addressing the complex Hamilton-Jacobin equation. Since the disturbance plays important role for the lower limb rehabilitation exoskeleton robot system, this article uses more stricter restriction of the disturbance with simultaneous absolute-value sense, integration-value sense and input-to-state stable sense.

To show the outstanding superiority of current research and let one easily design the robust tracking control, we apply the powerful AI “Python” to design a powerful controller-design human–machine interaction package. This current research firstly proposes the significant formula of nonlinear exponential convergence rate and the important formula of exponential convergent radius for the highly nonlinear LLRER system. At last, based on the proposed model, the effectiveness of proposed theorem is rigorously validated for the nonlinear LLRER by carrying out some simulations.