Fixed-time nonsingular terminal sliding mode control for a class of nonlinear systems with mismatched disturbances and its applications

Abstract

This article investigates a new fixed-time control scheme for a general class of nonlinear systems with high-order mismatched disturbances. An integral sliding mode surface is designed based on the estimations of the mismatched disturbances, which enables the control scheme to reject the mismatched disturbances. A continuous terminal sliding mode control law is then proposed, which guarantees that the convergence time is finite and independent of the initial conditions. Furthermore, the rigorous analysis for the fixed-time stability is presented with the bi-limit homogeneous property and Lyapunov theory. Finally, both the numerical simulations and the experimental tests are given to verify the superiority of the proposed fixed-time control scheme. The core of the proposed control scheme, which is developed with the nonsingular terminal sliding mode control (NTSMC) technique and fixed-time disturbance observer, ensures the fixed-time convergence and strong robustness to mismatched disturbances. The practical significance of the proposed control design is demonstrated by a real application to a crane system.

Appendix

Proof of Theorem 1

Define the estimation errors $e_i^0 = z_i^0 - x_i$ and $e_i^j = z_i^j - d_i^{(j-1)}$ for $i=1,\cdots ,n$ and $j=1,\cdots ,\ell _i$.

The error dynamics of the disturbance observer (2) is formulated by

$$\begin{aligned} \left\{ \begin{array}{l} \dot{e}_i^0 = e_i^1 - \lambda _i^0 {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^0}} - \gamma _i^0 \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^0}, \\ \dot{e}_i^1 = e_i^2 - \lambda _i^1 {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^1}} - \gamma _i^1 \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^1}, \\ \vdots \\ \dot{e}_i^{\ell _i} = - \lambda _i^{\ell _i} {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^{\ell _i} }} - \gamma _i^{\ell _i} \big \lceil {e_i^0} \big \rfloor ^{\beta _i^{\ell _i} } - d_i^{(\ell _i)}. \\ \end{array} \right. \end{aligned}$$

(23)

The compact expression of the error system (23) can be rewritten by

$$\begin{aligned} \dot{e}_i = f_i(e_i) + D_i, \end{aligned}$$

(24)

where the error vector $e_i=[e_i^0,\ldots ,e_i^{\ell _i}]^T$, the functional vector $f_i(e_i) =[f_i^0(e_i),\ldots ,f_i^{\ell _i}(e_i)]^T$ with $f_i^{j}(e_i) = e_i^{j+1} - \lambda _i^j {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^j}} - \gamma _i^j \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^j}$ for $j=0,\cdots ,\ell _i-1$, and $f_i^{\ell _i}(e_i) = - \lambda _i^{\ell _i} {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^{\ell _i}}} - \gamma _i^{\ell _i} \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^{\ell _i}}$, and the disturbance vector $D_i = [0,\ldots ,0,- d_i^{(\ell _i)}]^T$.

The subsequent proof will be presented with two steps. First of all, the stability of undisturbed error system (24) will be discussed, i.e., $D_i=0$. Then, the fixed-time convergence of complete error dynamic system (24) will be investigated, i.e., $D_i\ne 0$.

Step 1: It follows from [14] that the vector field $f_i(e_i)$ is bi-limit homogeneous with associated triples $(r_{i,0},k_{i,0},f_{i,0})$ and $(r_{i,\infty },k_{i,\infty },f_{i,\infty })$, where $r_{i,0}=[1,\alpha _i,\cdots ,\ell _i\alpha _i-(\ell _i-1)]$, $r_{i,\infty }=[1,\beta _i,\cdots ,\ell _i\beta _i-(\ell _i-1)]$, $k_{i,0}=\alpha _i-1$, $k_{i,\infty }=\beta _i-1$, and the approximating vector fields are selected as

$$\begin{aligned} f_{i,0}(e_i)= & {} \left[ {\begin{array}{*{20}{c}} e_i^1 - \lambda _i^0 {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^0}}\\ e_i^2 - \lambda _i^1 {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^1}}\\ \vdots \\ - \lambda _i^{\ell _i} {\big \lceil {e_i^0} \big \rfloor ^{\alpha _i^{\ell _i} }} \end{array}} \right] ,\nonumber \\ f_{i,\infty }(e_i)= & {} \left[ {\begin{array}{*{20}{c}} e_i^1 - \gamma _i^0 \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^0}\\ e_i^2 - \gamma _i^1 \big \lceil {e_i^0 } \big \rfloor ^{\beta _i^1}\\ \vdots \\ - \gamma _i^{\ell _i} \big \lceil {e_i^0} \big \rfloor ^{\beta _i^{\ell _i} } \end{array}} \right] . \end{aligned}$$

(25)

According to [43], we can obtain that $\dot{e}_i =f_{i,0}(e_i)$, $\dot{e}_i =f_{i,\infty }(e_i)$, and $\dot{e}_i =f_{i}(e_i)$ are globally asymptotically stable. Furthermore, it follows from [14] that the states of the system $\dot{e}_i =f_{i}(e_i)$ will converge to the origin within a fixed time. Additionally, according to [44], there exists a well-defined Lyapunov function $V_i(e_i)$ and a real constant $p>0$, such that the time derivative along the dynamics $\dot{e}_i =f_{i}(e_i)$ satisfies

$$\begin{aligned}{} & {} \frac{{\partial V_i(e_i)}}{{\partial {e_i}}} f_i(e_i) \nonumber \\{} & {} \quad \le - p \left( V_i(e_i)^\frac{k_{i,V_0}+k_{i,0}}{k_{i,V_0}} + V_i(e_i)^\frac{k_{i,V_\infty }+k_{i,\infty }}{k_{i,V_\infty }} \right) , \end{aligned}$$

(26)

for real numbers $k_{i,V_0}\ge \textrm{max}\{1,\alpha _i,\cdots ,\ell _i\alpha _i-(\ell _i-1)\}$ and $k_{i,V_\infty }\ge \textrm{max} \{1,\beta _i,\cdots , \ell _i\beta _i-(\ell _i-1)\}$.

Step 2: Let us reconsider the disturbance $D_i$. The time derivative of $V_i(e_i)$ along the error dynamics (24) can be presented by

$$\begin{aligned} \dot{V}_i(e_i)= \frac{{\partial V_i(e_i)}}{{\partial {e_i}}} f_i(e_i) - \frac{{\partial V_i(e_i)}}{{\partial {e_i^{\ell _i}}}} d_i^{(\ell _i)}. \end{aligned}$$

(27)

According to the Theorem 1 of [44], we can find a real constant $q>0$ to satisfy

$$\begin{aligned}{} & {} \left| - \frac{{\partial V_i(e_i)}}{{\partial {e_i^{r_i}}}} d_i^{(\ell _i-1)} \right| \nonumber \\{} & {} \quad \le \left\{ \begin{array}{l} 2q { V_i(e_i)^\frac{k_{i,V_0} - [r_{i,0}]_{\ell _i+1}}{k_{i,V_0}}} L_i, ~~~~\textrm{for}~V_i(e_i)\le 1, \\ 2q { V_i(e_i)^\frac{k_{i,V_\infty } - [r_{i,\infty }]_{\ell _i+1}}{k_{i,V_\infty }}} L_i,~~\textrm{for}~V_i(e_i) \ge 1, \end{array} \right. \nonumber \\ \end{aligned}$$

(28)

where $[r_{i,0}]_{\ell _i+1}$ and $[r_{i,\infty }]_{\ell _i+1}$ denote the $(\ell _i+1)$-th element of $r_{i,0}$ and $r_{i,\infty }$, respectively.

Combining (26) and (28), and considering the dominant scopes of two power functions, we have

$$\begin{aligned} \dot{V}_i(e_i)\le & {} - p V_i(e_i)^\frac{k_{i,V_\infty }+k_{i,\infty }}{k_{i,V_\infty }} - p V_i(e_i)^\frac{k_{i,V_0}+k_{i,0}}{k_{i,V_0}} \nonumber \\{} & {} + 2q { V_i(e_i)^\frac{k_{i,V_\infty } - [r_{i,\infty }]_{\ell _i+1}}{k_{i,V_\infty }}} L_i \end{aligned}$$

(29)

for $V_i(e_i)\ge 1$, and

$$\begin{aligned} \dot{V}_i(e_i)\le & {} - p V_i(e_i)^\frac{k_{i,V_0}+k_{i,0}}{k_{i,V_0}} - p V_i(e_i)^\frac{k_{i,V_\infty }+k_{i,\infty }}{k_{i,V_\infty }} \nonumber \\{} & {} + 2q { V_i(e_i)^\frac{k_{i,V_0} - [r_{i,0}]_{\ell _i+1}}{k_{i,V_0}}} L_i \end{aligned}$$

(30)

for $V_i(e_i)\le 1$.

Define two constants $c_1= (\frac{2qL_i}{p})^\frac{k_{i,V_\infty } k_{i,V_0}}{k_{i,V_\infty }k_{i,0} +k_{i,V_0}[r_{i,\infty }]_{\ell _i+1}}$ and $c_2= (\frac{2qL_i}{p})^\frac{k_{i,V_0} k_{i,V_\infty }}{k_{i,V_0}k_{i,\infty } +k_{i,V_\infty }[r_{i,0}]_{\ell _i+1}}$.

For the clarity of presentation, the following analysis will be given with two cases.

Case 1: If the Lipschitz constant $L_i$ satisfies $\frac{2qL_i}{p}\ge 1$, then $c_1\ge 1$ and $c_2\ge 1$. According to the inequality (30), we have

$$\begin{aligned} \begin{array}{l} \dot{V}_i(e_i) \le - p V_i(e_i)^\frac{k_{i,V_\infty }+k_{i,\infty }}{k_{i,V_\infty }},~~\textrm{for}~V_i(e_i)\ge c_1\ge 1. \end{array}\nonumber \\ \end{aligned}$$

(31)

Define a set

$$\begin{aligned} \begin{array}{l} S_1 = \{e_i | V_i(e_i)<c_1 \}. \end{array} \end{aligned}$$

(32)

Therefore, if the initial value of estimation error $e_i$ is at the outside of the set $S_1$, then $e_i$ will converge into $S_1$ in a finite settling time $T_{o1} = \frac{1}{p}\frac{k_{i,V_\infty }}{k_{i,\infty }} c_1 ^{-\frac{k_{i,\infty }}{k_{i,V_\infty }}}$, which is independent of initial conditions.

Case 2: If the Lipschitz constant $L_i$ satisfies $\frac{2qL_i}{p} < 1$, then we have

$$\begin{aligned} \dot{V}_i(e_i) \le \left\{ \begin{array}{l} - p V_i(e_i)^\frac{k_{i,V_0}+k_{i,0}}{k_{i,V_0}}, ~~~~\textrm{for}~c_2 \le V_i(e_i)\le 1, \\ - p V_i(e_i)^\frac{k_{i,V_\infty }+k_{i,\infty }}{k_{i,V_\infty }},~~\textrm{for}~V_i(e_i) \ge 1. \end{array} \right. \end{aligned}$$

(33)

Define a set

$$\begin{aligned} \begin{array}{l} S_2 = \{e_i | V_i(e_i)<c_2 \}. \end{array} \end{aligned}$$

(34)

Therefore, if the initial value of estimation error $e_i$ is at the outside of the set $S_2$, then $e_i$ will converge into $S_2$ in a finite settling time $T_{o2} = \frac{1}{p} (\frac{k_{i,V_\infty }}{k_{i,\infty }} + \frac{k_{i,V_0}}{|k_{i,0}|})$, which is also independent of initial conditions.

To sum up, the disturbance estimation error $e_i$ can converge to the small neighborhood of the origin in a finite settling time $T_o= \textrm{max} \{T_{o1},T_{o2} \}$. $\square$