# New Stable Non-Vector Control Structure for Induction Motor Drive
## 4 Design of the IM Drive Control  
The objective of the IM drive control consists of control of its angular speed where the motor dynamics are prescribed by a linear reference model to be designed. For the control structure design according to Section 2, it is not necessary to know an exact model of the controlled system. The electromagnetic phenomena within the motor are considerably faster than the mechanical phenomena on the motor shaft and the electrical circuits comprise all nonlinearities. The electromagnetic phenomena dynamics are replaced by the 1 st order proportional system and the mechanical phenomena are similarly replaced by the 1 st order system. The simplified model of the IM presents a nonlinear dynamic system of the 2 nd order as shown in Figure 3.  
The static nonlinearity in generating the mechanical torque of the \(T_{mech}\) motor is usually described by the Kloss formula with the 1 st order dynamics and the electromechanical linear subsystem is described by the equation of the torque equilibrium on the drive shaft. Let us select the state variables of the IM as shown in Figure 3: \(x_{1}=y=\omega_{m}\) (the rotor angular speed) and \(x_{2}=\mathrm{d}x_{1}/\mathrm{d}t=\mathrm{d}\omega_{m}/\mathrm{d}t\) (the rotor acceleration corresponding to its motor dynamic torque \(T_{D}\)). As the IM is considered as a nonlinear 2 nd order system, according to [32] its required dynamics are prescribed by the 2 nd order linear reference model:  
\[\left[ \begin{array}{c}\dot{x}_{1M}\\ \dot{x}_{2M}\end{array} \right]=\left[ \begin{array}{cc}0&1\\ -\frac{\alpha^{2}}{2}&-\alpha\end{array} \right]\left[ \begin{array}{c}x_{1M} \\ x_{2M}\end{array} \right]+\left[ \begin{array}{c}0\\ \frac{\alpha^{2}}{2}\end{array} \right]w \tag{23}\]  
Figure 3: Simplified model of the IM presenting a 2 nd order nonlinear system. Figure 2: Time courses of the IM model torque and speed at motor starting when supplied by the current vector. where the state variables \(x_{1M}\) and \(x_{2M}\) prescribe the dynamic behavior of the corresponding state variables of the controlled system, i.e., \(x_{1M}=y_{M}=\omega_{mM}\) and \(x_{2m}=\mathrm{d}x_{1M}/\mathrm{d}t=\mathrm{d}\omega_{mM}/\mathrm{d}t\). The optimal dynamics of the controlled variable can be set by the optional positive parameter \(\alpha\) in the reference model (23). To ensure the zero control deviation of the angular speed of the controlled system and the reference model at steady-state, the reference model is extended by a new state variable \(x_{ext}\). To create a deviation between the state variable of the reference model and the system according to (17), the following relation is used:  
\[x_{ext}=\int\;(x_{1M}-x_{1})\mathrm{d}t=\int\;(y_{M}-y)\mathrm{d}t \tag{24}\]  
Now the extended vector of the deviation will be in the form:  
\[\mathbf{e}^{*}=\left[ \begin{array}{c}x_{ext}\\ e_{1}\\ e_{2}\end{array} \right] \tag{25}\]  
This means the extended reference model of the IM drive presents the 3 rd order system. According to the optimization theory [32], its state matrices are in the form:  
\[\mathbf{A}_{M}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ -\frac{\alpha^{3}}{2}&-\frac{3\alpha^{2}}{2}&-\frac{3\alpha}{2}\end{array}  \right];\;\;\mathbf{b}_{M}=\left[ \begin{array}{c}0\\ 0\\ \frac{\alpha^{2}}{2}\end{array} \right] \tag{26}\]  
where \(\mathbf{b}_{M}\) is the input vector in the case of the one-input model. The optimal matrices \(\mathbf{P}\) and \(\mathbf{Q}\), that satisfy the Lyapunov matrix Equation (11), are:  
\[\mathbf{P}=\left[ \begin{array}{ccc}\frac{\alpha^{5}}{2}&\alpha^{4}&\frac{ \alpha^{3}}{2}\\ \alpha^{4}&\frac{5\alpha^{3}}{2}&\frac{3\alpha^{2}}{2}\\ \frac{\alpha^{3}}{2}&\frac{3\alpha^{2}}{2}&\frac{3\alpha}{2}\end{array}  \right];\;\;\mathbf{Q}=-\alpha\mathbf{P} \tag{27}\]  
The IM as a controlled system will follow the extended reference model with \(\lim\limits_{t\rightarrow\infty}\mathbf{e}^{*}=\mathbf{0}\), i.e., the controlled system will be asymptotically stable, if the input \(u\) is calculated using (15):  
\[u=\left[