# Model-Free Tracking Control with Prescribed Performance for a Shape Memory Alloy-Based Robotic Hand
## 3 Problem Formulation  
### System Model  
A single input single output (SISO) discrete time nonlinear system can be presented as follows  
\[y(k+1)=f(y(k),y(k-1),\ldots y(k-d_{y}),u(k),u(k-1),\ldots u(k-d_{u})), \tag{1}\]  
where at time \(k\), \(y(k),u(k)\in R\) stand for the output and input, respectively, \(d_{y},d_{u}\in R\) represent the unknown order, and \(f(\cdot)\) is a nonlinear function.  
**Assumption 1**.: _The partial derivatives of \(f(\cdot)\) with respect to the \((d_{y}+2)\)th variable is continuous._  
**Assumption 2**.: _System (1) satisfies the generalized Lipschitz condition, that is, \(|y(k_{1}+1)-y(k_{2}+1)|\leq n|u(k_{1})-u(k_{2})|\) for \(u(k_{1})\
eq u(k_{2})\) and any \(k_{1}\
eq k_{2}\), \(k_{1},k_{2}\geq 0\), and \(n\) is a positive constant._  
In [32], a novel dynamic linearization model is presented as follows:  
**Theorem 1**.: _Considering system (1), which satisfies Assumptions 1 and 2, there must exist the time-varying parameters \(M(k)\) and \(B(k)\), by which system (1) can be converted into another form as follows:_  
\[y(k+1)=M(k)+\frac{\Delta u(k)}{B(k)}, \tag{2}\]  
_where \(\Delta u=u(k)-u(k-1)\)._  
The estimation of \(M(k)\) can be defined as  
\[\hat{M}(k)=y(k)-\frac{\Delta u(k-1)}{\hat{B}(k-1)}, \tag{3}\]  
where \(\hat{M}(k)\) and \(\hat{B}(k)\) are the estimations of \(M(k)\) and \(B(k)\).  
Two cost functions proposed in [32] for parameter estimation and control input are applied as follows:  
\[J(u(k))=\Big{(}y_{d}-\hat{y}(k+1)-r\hat{d}(k)\Big{)}^{2}+\theta(u(k)-u(k-1))^ {2} \tag{4}\]  
\[J(\hat{B}(k))=\big{(}\hat{B}(k)\big{(}y(k)-\hat{M}(k-1)\big{)}-\Delta u(k-1) \big{)}^{2}+\alpha\big{(}\hat{B}(k)-\hat{B}(k-1)\big{)}^{2}, \tag{5}\]  
where \(y_{d}\) is the reference output, \(\hat{d}(k)\) stands for the estimated error of the model, \(r\), \(\alpha\) and \(\theta\) are the design parameter.  
Using optimal condition \(\frac{\partial J(\hat{B}(k))}{\partial\hat{B}(k)}=0\) and \(\frac{\partial J(u(k))}{\partial u(k)}=0\), the following relationships are presented in the following:  
\[\hat{B}(k)=\hat{B}(k-1)+\frac{\big{(}y(k)-\hat{M}(k-1)\big{)}\big{(}\Delta u (k-1)-\hat{B}(k-1)\big{(}y(k)-\hat{M}(k-1)\big{)}\big{)}}{\alpha+\big{(}y(k)- \hat{M}(k-1)\big{)}^{2}} \tag{6}\]  
\[u(k)=u(k-1)+\frac{\hat{B}(k)\big{(}y_{d}(k+1)-\hat{M}(k)-r(y(k)-\hat{y}(k)) \big{)}}{1+\theta\hat{B}^{2}(k)}. \tag{7}\]  
The above control method is a new model free adaptive control (NMFAC) by [32]. For a robotic hand system, Assumption 1 can be hardly satisfied. However, Assumption 2 applies an upper limit on the the rate of system output change driven by the rate ofsystem control input. From a practical point of view, the robotic hand system can meet Assumption 2. Therefore, the SMA-based robotic hand system can be written as:  
\[\begin{array}{l}y_{a}(k+1)=y(k+1)+D(k)\\ =f(y(k),y(k-1),\ldots,y(k-d_{y}),u(k),u(k-1),\ldots,u(k-d_{u}))+D(k),\end{array} \tag{8}\]  
where \(y_{a}(k+1)\) and \(u(k)\) are the output and input of the robotic finger system, \(f(k)\) stands for the bounded disturbances with \(D(k)<|D|\), \(D>0\), and \(f(\cdot)\) represents the discrete-time SISO nonlinear system (1).