## 3 Model-Free Adaptive Control with Prescribed Performance Method  
### MFAC-SMC-PP Control Framework  
The principle of MFAC-SMC-PP has been extensively studied [44; 45]. The vital idea refers to finding a required function (termed error transformation function) to develop an equivalent system with several virtual variables and adopt a model-free adaptive control method to track the reference trajectory. Moreover, the SMC is used to reduce the influence of the controller under unknown disturbance [40]. In the MFAC-SMC-PP control framework, the tracking error can be limited to a small predefined boundary designed by the user without any model parameters' information. As shown in Figure 2, the prescribed performance control block can transform the tracking error into a new variable \(z(k)\) with boundary information. Then, the pseudo-partial derivative estimation block estimates the pseudo partial derivative value (i.e., \(\hat{\Phi}_{12}(k)\)) according to the historical input (i.e., \(u(k-1)\)) and output (i.e., \(y(k-1)\)) of the system for the MFAC-SMC block. Finally, the MFAC-SMC block calculates the control law (i.e., \(\tau(k)\)) and sends it into the control plant (i.e., one-DOF upper-limb exoskeleton). Assuming a discrete-time nonlinear system is considered as  
\[y(k+1)=f\big{(}u(k),\cdots,u(k-d_{u}),y(k),\cdots,y\big{(}k-d_{y}\big{)}\big{)} \tag{1}\]  
where \(u(k)\) and \(y(k)\) denote the input and output of the system at time instant \(k\), respectively, \(d_{u}\) denotes the length of the input, and \(d_{y}\) denotes the system order. Figure 2: The original MFAC-SMC-PPC control framework. There are two assumptions:  
**Assumption 1**. : _The partial differentials of the function of \(f(\cdot)\) related to the system inputs \(u(k),\cdots,u(k-d_{u})\) are successive [38]._  
**Assumption 2**. : _The system in Equation (1) satisfies the generalized Lipschitz [40]._  
By complying with the MFAC method, if the above system satisfies the following Assumptions 1 and 2, the system is expressed as Equation (2):  
\[y(k+1)=y(k)+\Delta y(k)\Phi_{1}(k)+\Delta u(k)\Phi_{2}(k) \tag{2}\]  
According to the reference, an estimation approach can be exploited to determine \(\Phi_{1}(k)\) and \(\Phi_{2}(k)\) below [45]:  
\[\hat{\[PERSON]}_{1}(k)=\hat{\[PERSON]}_{1}(k)+\frac{\eta(\Delta y(k)-g(k-1))\Delta y(k-1 )}{l(k-1)+\mu} \tag{3}\]  
\[\hat{\[PERSON]}_{2}(k)=\hat{\[PERSON]}_{2}(k)+\frac{\eta(\Delta y(k)-g(k-1))\Delta u(k-1 )}{l(k-1)+\mu} \tag{4}\]  
where \(\mu\) is adopted to prevent the generation of singular values, \(\eta>0\) is a scale constant, \(g(k-1)=\Delta y(k-1)\hat{\[PERSON]}_{1}(k-1)+\Delta u(k-1)\hat{\Phi}_{2}(k-1)\), and \(l(k-1)=\Delta y^{2}(k-1)+\Delta u^{2}(k-1)\).  
Given the modeling uncertainties and boundedness disturbance \(|f(k)|<\Gamma_{1}\) (e.g., friction force/torque) [43], Equation (2) can be rewritten as  
\[y(k+1)=y(k)+\Delta y(k)\Phi_{1}(k)+\Delta u(k)\Phi_{2}(k)+f(k) \tag{5}\]  
To address the tracking performance issue, a reference trajectory is defined as \(y_{d}(k)\). Then, the tracking error is defined as \(e(k)=y(k)-y_{d}(k)\)[40]. According to the prescribed performance method [10], a convergent bounded function--termed performance function--is defined in Equation (6). It represents the boundary of the system-tracking error. The prescribed performance can be expressed as \(-\rho(k)<e(k)<\rho(k)\):  
\[\rho(k+1)=(1-v)\rho(k)+v\rho_{\infty} \tag{6}\]  
where \(0<vs.<1\) denotes the convergence ratioand \(\rho_{\infty}>0\) is defined as a positive constant. Both of these should be predefined by the user. To solve the restricted tracking error, a strictly increasing function \(S(\cdot)\), for the error transformation function, is presented as  
\[S(z)=\frac{\exp(-z)-\exp(z)}{\exp(-z)+\exp(z)} \tag{7}\]  
where \(z\) denotes the visual state variable.