## 2 Control Algorithm
### PISMC for a Weighted-Control System  
Based on the work in [19], we extend our work to a nonlinear system of weighted control, which is usually described by the differential equations  
\[\bar{x}^{(n)} = f\left(\bar{x},\hat{x},\cdots,\bar{x}^{(n-1)},t\right) \tag{41}\] \[+\hat{g}\left(\bar{x},\hat{x},\cdots,\bar{x}^{(n-1)},t\right)\, \hat{u}(t),\]  
where \(\tilde{u},\tilde{x}\in\mathbb{R}\) are the input and output, respectively, and \(f,\tilde{g}:\mathbb{R}^{n}\rightarrow\mathbb{R}\) are smooth functions. The gain \(\tilde{g}\) is unknown, yet assumed to be uniformly bounded.  
**Assumption 2**.: _Assume \(0<\beta_{min}\leq\tilde{g}(\tilde{\mathbf{X}},t)\leq\beta_{max}\), where \(\beta_{min}\) and \(\beta_{max}\) are given constants, time-varying or state-dependent functions [27]._  
We assume that the control input may enter multiplicatively in the dynamics, and it is natural to choose our estimate \(\hat{g}\) of gain \(\tilde{g}\) as the geometric mean of the above bounds.  
\[\hat{g} = \sqrt{\beta_{\min}\beta_{\max}}. \tag{42}\]  
Assumption 2 can then be written in the form of  
\[\beta^{-1}\leq\frac{\hat{g}}{\hat{g}}\leq\beta, \tag{43}\]  
where  
\[\beta=\sqrt{\frac{\beta_{\max}}{\beta_{\min}}}>0. \tag{44}\]  
Since the PISMC is designed to be robust to the bounded multiplicative uncertainty, as in (43), we call \(\beta\) the gain margin of our design, by analogy to the terminology used in linear control. Let  
\[\hat{G} = \frac{1}{\sqrt{\beta_{\max}\beta_{\min}}}. \tag{45}\]  
The PISMC \(\tilde{u}^{*}\) is modified from the aforementioned \(\tilde{u}\) and it has the form:  
\[\tilde{u}^{*} = \hat{G}\left(\hat{u}_{eq}+v_{sw}\right), \tag{46}\]  
where the \(\hat{u}_{eq}\) and \(v_{sw}\) are defined in (13) and (14), respectively. We summarize the controller design in Theorem 2.  
Figure 6: Comparison of parameter effectiveness. (**a**) Outputs \(\bar{x}\); (**b**) control input \(\tilde{u}\); (**c**) cost of controls \(J\).  
**Theorem 2**: _Consider the system (41), which satisfies Assumptions 1 and 2. Let the function \(\tilde{u}^{*}\) be specified in (46) with \(\hat{u}_{aq}\), \(v_{sw}\), and \(\hat{w}\) defined in (13), (14), and (15), respectively. Let the gain \(k^{*}\) be designed to satisfy (5). If there exits a \(k^{*}\) such that_  
\[k^{*} = \left|\left(\beta-1\right)\left(\hat{f}+\hat{w}\right)\right|+ \beta(M+\eta), \tag{47}\]  
_then the tracking error as defined in (3) will tend to the neighborhood of zero within a finite time._  
Proof: To prove that PISMC \(\hat{u}^{*}\) satisfies the sliding conditions, the following Lyapunov function is employed:  
\[V = \frac{1}{2}s^{2}.\]  
If we choose \(\hat{u}^{*}\) as in (46), then follow similar steps as those in (18):  
\[\hat{V} = s\hat{s}\] \[= (\hat{f}+\Delta f+\hat{g}\hat{u}^{*}+\hat{w})s\] \[\leq -\left[\hat{g}\hat{G}k^{*}-\left|\left(\hat{f}+\hat{w}\right) \left(1-\hat{g}\hat{G}\right)\right|-\hat{F}\right]|s|.\]  
Finally, let  
\[k^{*} \geq \left|\left(\beta-1\right)(\hat{f}+\hat{w})\right|+\beta(\hat{F}+ \eta). \tag{48}\]  
We obtain  
\[\hat{g}\hat{G}k^{*}-\left|\left(\hat{f}+\hat{w}\right)\left(1- \hat{g}\hat{G}\right)\right|-\hat{F} \geq 0,\]  
and hence \(\hat{V}<0\). This ensures that any trajectory \(e(t)\) converges to the neighborhood around of origin in the phase plane.