## 2 Control Algorithm
### Proportional-Integral Sliding Mode Control  
To improve the performance of a SMC, this paper adopts a PISMC as our algorithm. Consider a dynamic system described by an \(n\)th-order nonlinear differential equation  
\[\tilde{x}^{(n)}(t) = f\left(\tilde{x},\dot{x},\cdots,\tilde{x}^{(n-1)},t\right)+ \tilde{n}(t), \tag{6}\]  
where \(\tilde{x}\in\mathbb{R}\) is the output of the dynamic system, \(f\) is the system function and must be at least once differentiable, \(\tilde{n}\in\mathbb{R}\) is the system input, and \(t\) is time. We define the state vector, \(\tilde{\mathbf{X}}=\left[\tilde{x},\dot{\tilde{x}},\cdots,\tilde{x}^{(n-1)} \right]^{T}\), for future derivations. **Assumption 1**. : _Consider (6) and assume that there exists an uncertain parameter \(\Delta f\) in the dynamic function \(f(\tilde{\mathbf{X}},t)\), satisfying_  
\[f = \dot{f}+\Delta f, \tag{7}\]_where \(\hat{f}\) is the estimation and \(\Delta f\) is the variation of function \(f\). Assume_  
\[|\Delta f| \leq \hat{F}(\tilde{\mathbf{X}},t), \tag{8}\]  
_given \(\tilde{F}\) the boundary of system uncertainty [27]._  
The goal of the PISMC proposed in this paper is to design a switching control law to let the output \(\tilde{x}(t)\) approach a given \(\tilde{x}_{d}(t)\), as shown in (2). To design the control system, the system states are assumed to be attainable via a state observer, which sometimes requires measurement of an output or output error, as defined in (3). The sliding surface of the PISMC consists of two sliding surfaces, \(s_{1}\) and \(s_{2}\), the proportional sliding surface and the integral sliding surface, respectively. Here,  
\[s_{1} = \left(\frac{d}{dt}+\lambda\right)^{n-1}e(t),\]  
as defined in (4), and  
\[s_{2} = \left(\frac{d}{dt}+\alpha\right)^{n}\tilde{\varsigma}_{e}, \tag{9}\]  
where \(\lambda>0\), \(\alpha>0\), \(\tilde{\varsigma}_{e}(t)=\int_{0}^{t}e(\tau)d\tau\). The operator \((d/dt+\gamma)^{n}\) is defined as  
\[\left(\frac{d}{dt}+\gamma\right)^{n} = \frac{d^{n}}{dt^{n}}+\left( \begin{array}{c}n\\ 1\end{array} \right)\gamma\frac{d^{n-1}}{dt^{n-1}}+\cdots+\left( \begin{array}[] {c}n\\ n-1\end{array} \right)\gamma^{n-1}\frac{d}{dt}+\gamma^{n}, \tag{10}\]  
where \(\gamma\in\mathbb{R}\) is a parameter to tune, and \(\left( \begin{array}{c}n\\ m\end{array} \right)\) denotes the \(k\)-combinations of a set having \(n\) elements. Applying the operator to a function \(h(t)\) yields  
\[\left(\frac{d}{dt}+\gamma\right)^{n}h = h^{(n)}+\left( \begin{array}{c}n\\ 1\end{array} \right)\gamma h^{(n-1)}+\cdots+\left( \begin{array}{c}n\\ n-1\end{array} \right)\gamma^{n-1}h+\gamma^{n}h, \tag{11}\]  
where the superscript with parentheses denotes the order of derivative, while the superscript without parentheses denotes the power of the parameter \(\gamma\). In our case, \(\gamma=\lambda\) or \(\gamma=\alpha\). They are two complementary parameters, which provide the capability of tuning the gains of controller. We note that the expansion of (9) is different from the conventional PSMC or PID sliding surface, since \(\lambda\) and \(\alpha\) are generally distinct. The PISMC is designed as [19]:  
\[\tilde{u} = \hat{u}_{eq}+v_{sw}, \tag{12}\]  
where, by letting \(s=s_{1}+s_{2}\),  
\[\hat{u}_{eq} = -\hat{f}-\hat{w}, \tag{13}\] \[v_{sw} = -ks\text{gn}(s), \tag{14}\]with  
\[\hat{w} = -\hat{x}_{d}^{(n)}+\left[\sum_{m=0}^{n-2}\left( \begin{array}{c}n-1 \\ m\end{array} \right)e^{(m+1)}\lambda^{n-m-1}\right] \tag{15}\] \[+\left[\sum_{m=0}^{n-1}\left( \begin{array}{c}n-1\\ m\end{array} \right)e^{(m+1)}\alpha^{n-m-1}\right]\] \[+\alpha\left[\sum_{m=0}^{n-1}\left( \begin{array}{c}n-1\\ m\end{array} \right)e^{(m)}\alpha^{n-m-1}\right],\]  
and  
\[\mathrm{sgn}(s) = \left\{ \begin{array}{rl}1&\mathrm{if}\,s>0\\ 0&\mathrm{if}\,s=0\\ -1&\mathrm{if}\,s<0\end{array} \right..\]  
The controller design is summarized in Theorem 1. Theorem 1: [19] _Consider the system of (6) which satisfies Assumption 1.