## 3 Model-Free Adaptive Control with Prescribed Performance Method
### Stability Analysis  
According to reference [14], the Lyapunov function was selected to be \(V=s^{2}/2\). After differentiation and linearization, it yields:  
\[\dot{V}\rightarrow\frac{(s(k)-s(k-1))}{T}s(k)=\frac{1}{T}s(k)\Delta s(k) \tag{31}\]  
Given \(T>0\), \(\dot{V}\) and \(s(k)\Delta s(k)\) are of same sign. Since \(\Delta s(k){=}s(k{+}1)-s(k)\), it yields:  
\[\Delta s(k)=z(k{+}1)-(1-K)z(k) \tag{32}\]  
By substituting Equations (25) and (29) into (32), it also yields:  
\[\Delta s(k)=\frac{1}{2}\ln\biggl{(}\frac{\rho+\varphi(k{+}1)e(k{+}1)}{\varsigma (k)\rho-\varsigma(k)\varphi(k{+}1)e(k{+}1)}\biggr{)} \tag{33}\]  
Using Equations (16), (30) and (33), \(\Delta s(k)\) is denoted by  
Figure 4: MFAC-SMC-PPC control framework with a tuning function. \[\Delta s(k){=}\frac{1}{2}\ln\left(\frac{\frac{2\varsigma(k)}{1+\varsigma(k)}\rho+ \left(\Delta f(k)-\Gamma_{1}\left(sign(s(k))+\Gamma_{2}|s(k)|^{0.5}sign(s(k)) \right)\right)}{\varsigma(k)\left(\frac{2}{1+\varsigma(k)}\rho-\left(\Delta f(k )-\Gamma_{1}\left(sign(s(k))+\Gamma_{2}|s(k)|^{0.5}sign(s(k))\right)\right) \right)}\right) \tag{34}\]  
From references [38; 45], it yields:  
\[ \begin{array}{l}v_{a}{=}\ 2\rho\\ v_{b,k}\frac{1+\varsigma(k)}{\varsigma(k)}\\ v_{c,k}=1+\varsigma(k)\end{array}  \tag{35}\]  
Since \(\varsigma(k)\) refers to an exponential function (i.e., \(\varsigma(k)>0\)), \(v_{b,k}\) and \(v_{c,k}\) are positive. Moreover, \(v_{a}>0\) is a positive constant. Then, the stability analysis is expressed below. Step 1. If \(s(k)>0\), \(sign(s(k))=1\). \(\Delta s(k)\) can be denoted by  
\[\Delta s(k){=}\frac{1}{2}\ln\left(\frac{v_{a}+v_{b,k}\left(\Delta f(k)-\Gamma _{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)\right)}{v_{a}-v_{c,k}\left(\Delta f( k)-\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)\right)}\right) \tag{36}\]  
Since \(|\Delta f(k)|\leq\Gamma_{1}\) and \(\Gamma_{2}>0\), it yields \(\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)>\Gamma_{1}\geq|\Delta f(k)|\). Then, \(v_{a}+v_{b,k}\left(\Delta f(k)-\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right) \right)<v_{a}\). \(v_{a}<v_{a}-v_{c,k}\left(\Delta f(k)-\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5} \right)\right)\). This can yield \(\Delta s(k)<0\). Step 2. If \(s(k)<0\), \(sign(s(k))=-1\). \(\Delta s(k)\) can be denoted by  
\[\Delta s(k){=}\frac{1}{2}\ln\left(\frac{v_{a,k}+v_{b,k}\left(\Delta f(k)+ \Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)\right)}{v_{a,k}-v_{c,k}\left( \Delta f(k)+\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)\right)}\right) \tag{37}\]  
This yields that \(v_{a,k}-v_{c,k}\left(\Delta f(k)+\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5} \right)\right)<v_{a,k}\). Then, we can obtain that \(v_{a,k}<v_{a,k}+v_{b,k}\left(\Delta f(k)+\Gamma_{1}\left(1+\Gamma_{2}|s(k)|^{0.5}\right)\right)\), which means \(\Delta s(k)>0\). Accordingly, whether \(s(k)>0\) or \(s(k)<0\), we can obtain that \(s(k)\Delta s(k)<0\) and that the proposed controller is stable. Finally, the convergence analysis of the tracking error is presented below. Since the tracking error does not enter the predesigned boundary before \(k_{in}\), the convergence analysis was discussed for \(t>k_{in}\). Selecting a time instant of \(k_{t}\) in the control process and defining \(k_{t}\geq k_{in}\), the convergence analysis process can be expressed step by step. Step 1. Assuming \(s(k_{t}-1)>0\). In the stable domain, it yields that \(s(k_{t}-1)\Delta s(k_{t}-1)\) < 0, and then \(s(k_{t})<s(k_{t}-1)\). According to Equation (32), it yields:  
\[s(k_{t})-s(k_{t}-1)=z(k_{t})+(K-1)z(k_{t}-1)\leq 0 \tag{38}\]  
As impacted by \(z(k_{t}-1)=0.5\ln\left((\rho+e(k_{t}-Step 2. Assume that \(s(k_{t}-1)<0\). In the stable domain, this yields that \((s_{k_{t}}-1)-(\Delta sk_{t-1}-1)>0\) and \(s^{2}(k_{t})/2<s^{2}(k_{t-1})/2\). Then, we can obtain \(s(k_{t}-1)<s(k_{t})<-s(k_{t}-1))\). A similar deduction is presented as in Equation (40). As impacted by \(z(k_{t})=0.5 ln((\rho+e(k))/(\rho-e(k)))\), it yields \(-\rho<e(k)<\rho\). Thus, also indicates that the tracking error converged:  
\[