# DDC Control Techniques for Three-Phase BLDC Motor Position Control
## 3 DDC Control System Design
### Design of Data-Driven SMC  
Consider the general discrete time single input, single output (SISO) nonlinear system in Equation (8). The system should satisfy the assumptions in Section 3.1 for the PFDL method. The basic theory and a stability analysis of MFA-SMC is present in Reference [19]. Figure 15: Step response of the BLDC Motor using MFLAC. The discrete sliding mode function is given below:  
\[s(\kappa)=e(\kappa)+c\cdot e(\kappa-1) \tag{21}\]  
where \(c\) must satisfy the Hurwitz condition \(c>0\). Additionally, the tracking error is \(e(\kappa)\), which is well-defined as:  
\[e(\kappa)=y^{*}(\kappa)-y(\kappa) \tag{22}\]  
where \(y^{*}(\kappa)\) is the desired tracking trajectory, and \(y(\kappa)\) is the system output. By using the discrete exponential reaching law,  
\[s(\kappa+1)\;=\;(1-q)s(\kappa)-\;\varepsilon\cdot sgn(s(\kappa)),\; \varepsilon>0,\;0<q<1. \tag{23}\]  
From Equations (21) and (23), the equivalent derived equation is:  
\[e(\kappa+1)\;=\;(1-q)\cdot s(\kappa)-\;\varepsilon\cdot sgn(s(\kappa))-\;c \cdot e(\kappa). \tag{24}\]  
Let  
\[\beta 1=e(\kappa+1)\;=\;(1-q)\cdot s(\kappa)-\;\varepsilon\cdot sgn(s(\kappa))- \;c\cdot e(\kappa).\]  
By using Equation (24), the tracking error becomes  
\[(y^{*}(\kappa+1)-y(\kappa+1))\;=\;\beta 1. \tag{25}\]  
Solving Equations (15) and (25) produces the following result:  
\[\Delta u(\kappa)=\Bigg{(}\Bigg{(}\frac{1}{\hat{\phi}(\kappa)+\big{(}\hat{\phi }(\kappa-1)-\hat{\phi}(\kappa-2)\big{)}}\Bigg{)}\cdot\left(y^{*}(\kappa+1)\;- \;y(\kappa)-\beta 1\right)\Bigg{)}. \tag{26}\]  
Combining \(\Theta(k)=\bigg{(}\frac{1}{\hat{\phi}(\kappa)+\big{(}\hat{\phi}(\kappa-1)- \hat{\phi}(\kappa-2)\big{)}}\bigg{)}\). Put the value of \(\beta 1\), and then Equations (21) and (26) establish the following control law:  
\[u(\kappa)=\big{(}u(\kappa-1)+\Theta(\kappa)\times\left(y^{*}(\kappa+1)-y( \kappa)-\beta 2+\varepsilon\cdot sgn(e(\kappa)+c\cdot e(\kappa-1))\right) \tag{27}\]  
where \(\beta 2=(1-c-q)\cdot e(\kappa)-c(1+q)\cdot e(\kappa-1)\). Since \(\Theta(\kappa)\) is linear convergence, a slow varying parameter may not be able to satisfy the necessity of the controlled system; therefore, Equation (27) can be improved in the following expression:  
\[u(\kappa)=\Big{(}u(\kappa-1)+\widetilde{\Theta}(\kappa)\times\left[y^{*}( \kappa+1)\;-y(\kappa)-\beta 2+\varepsilon\cdot sgn(e(\kappa)+c\cdot e(\kappa-1))\right]\Big{)} \tag{28}\]  
where  
\[s(\kappa)=e(\kappa)+c\cdot e(\kappa-1) \tag{21}\]0  
Equation (27) is the MFA-based SMC control law that converges exponentially and will result in a reduction of time and will not cause problems with stability, where \(\lambda\) is the weighting factor, \(\rho_{\kappa}\) is the step sequence, and \(\rho_{\kappa}\subset(0,2)\). Solving Equations (21) and (27) results in stable polynomials \(c\) that range from \(0<c<\min\left\{\frac{1-q}{2-q},\frac{-1+\sqrt{4q^{2}-8q+5}}{2(1-q)}\right\}\). An analysis shows that \(\widetilde{\Theta}(\kappa)\) is quadratic convergence and that the \(\hat{\phi}(\kappa-1)\) and \(\hat{\phi}(\kappa-2)\) values are present in the control law; thus, the MFA-based adaptive SMC is greatly improved. It can be seen that Equation (27) has a discontinuous \(\mathrm{sgn}(. )\) function. This function will establish a high frequency chattering at the sliding mode surface. This high frequency will reduce the performance of the control system, and cannot be applied to an actual control system. To eliminate the high frequency chattering, replace \(f(s(\kappa))=\frac{s(\kappa)}{|s(\kappa)|+\Delta}\) with the \(\text{sgn}(. )\) function in Equation (27). By replacing the above function, it will convert the discontinuous function of the continuous signal that will result in a reduction of the chattering effect. The new control algorithm becomes the following equation:  
\[s(\kappa)=e(\kappa)+c\cdot e(\kappa-1) \tag{21}\]1  
where \(f(e(\kappa)+c\cdot e(\kappa-1))=\left(\frac{e(\kappa)+c\cdot e(\kappa-1)}{| \varepsilon(\kappa)+c\cdot e(\kappa-1)|+\Delta}\right)\).