# DDC Control Techniques for Three-Phase BLDC Motor Position Control
## 3 DDC Control System Design  
### Design of MFAC  
Consider the Discrete time nonlinear system having Single Input and Single Output (SISO);  
\[y(\kappa+1)\ =f\big{(}y(\kappa),y(\kappa-1),\ldots,y(\kappa-n_{y}),u(\kappa),u( \kappa-1),\ldots,u(\kappa-n_{u})\big{)} \tag{8}\]  
where \(y(\kappa)\in R\), \(u(\kappa)\in R\) are the input and output, respectively, at time instant \(\kappa\), \(n_{y}\) and \(n_{u}\) is the unknown order, and \(f(\ldots)\) is an unknown time-varying nonlinear function. Detailed information on the succeeding assumptions is given in reference [19]. **Assumption 1**: _The proposed system model is observable and controllable, and a bounded control input signal should exist that converts the system response identically to the expected output for the expected bounded system output signal \(y^{*}(\kappa+1)\)._  
**Assumption 2**: \(f(\ldots)\) _is a Partial derivative and is continuous with respect to the control input \(u(\kappa)\)._  
**Assumption 3**: _The general Lipschitz condition should be satisfied by the proposed system._  
\[|\Delta y(\kappa+1)|\leq c|\Delta u(\kappa)|\;\text{For any}\;\kappa\;\text{and} \;\Delta u(\kappa)\
eq 0 \tag{9}\]  
_where \(\Delta y(\kappa+1)\ =y(\kappa+1)-y(\kappa),\;\Delta u(\kappa)=u(\kappa)-u( \kappa-1)\), and positive constant \(c\)._  
Nonlinearity is present in the system, which restricts the rate of change of the system. The purpose of control is to design an appropriate control input \(u(\kappa)\) to make the system output \(y(\kappa+1)\to 0\). \(\lambda\in(0,1)\) is introduced into the criterion function where the system is to linearize in an acceptable range. The CFDL data model of the control system is described as  
\[\Delta y(\kappa+1)=\phi(\kappa)\Delta u(\kappa). \tag{10}\]  
The system should satisfy the Assumptions 1-3 for the CFDL method. First, the system is a continuous moving system whose dynamics satisfy the smoothness condition. Second, a bounded change of system input must cause a bounded change of the system when the control input is not out of the permitted range, namely, that the generalized Lipschitz condition is satisfied. The generalized PPD estimation and MFAC control law can be written as,  
\[u(\kappa)=u(\kappa-1)+\frac{\rho_{\kappa}\hat{\phi}(\kappa)}{\lambda+|\hat{ \phi}(\kappa)|^{2}}(y^{*}(\kappa+1)-y(\kappa)) \tag{11}\]  
\[\hat{\phi}(\kappa)=\hat{\phi}(\kappa-1)+\frac{\eta_{\kappa}\Delta u(\kappa-1) }{\mu+|\Delta u(\kappa-1)|^{2}}\times\big{(}\Delta y(\kappa)-\hat{\phi}( \kappa-1)\Delta u(\kappa-1)\big{)} \tag{12}\]  
\[\text{If}\;\hat{\phi}(\kappa)\leq\varepsilon\,\text{or}\,\Delta u(\kappa-1) \leq\varepsilon\,\text{then}\,\hat{\phi}(\kappa)=\hat{\phi}(1) \tag{13}\]  
where \(\eta_{\kappa},\rho_{\kappa}\in(0,2)\) are the step-size constants, \(\varepsilon\) is a positive small number, \(\mu\) and \(\lambda\) are weighting factors, and \(\hat{\phi}(1)\) and \(\hat{\phi}(2)\) are the initial values. The sampling instant \(\Delta u(\kappa)\) values are too small now considering \(\phi(\kappa)\) that is a time-varying parameter and can be ignored by \(u(\kappa)\). The adaptive based controller is proposed by the integration of Equations (10), (12) and (13). By combining these equations, it will make the Equation (14). \[y(\kappa+1)=y(\kappa)+\big{[}\big{(}\hat{\phi}(\kappa)+\delta_{\phi}(\kappa) \big{)}\Delta u(\kappa)\big{]} \tag{14}\]The parameter estimation error is written as \(\delta_{\phi}(\kappa)=\left[\phi(\kappa)-\hat{\phi}(\kappa)\right]\). The PPD of the proposed system is \(\phi(\kappa)\), and it is not used in practice.