# New Stable Non-Vector Control Structure for Induction Motor Drive
## 5 Simulation of the IM Drive in Basic Operation States  
The verification of the control structure properties was carried out by simulation in MATLAB with the motor parameters specified in Table 1. The desired value of the mechanical angular speed of the IM was set in the reference model to \(\omega_{mrf}=150\) rad/s and the stator current vector components were \(i_{1x}=0\) A, \(i_{1y}=20\) A. The proposed controller structure is based on [PERSON]'s second method, which defines the range of its optional parameters, i.e., values of the elements of the matrix \(\mathbf{P}\) and the positive optional vector of parameters \(\mathbf{k}^{T}\) for which the controlled system is stable. The elements of the matrix \(\mathbf{P}\) are computed from the Lyapunov matrix Equation (11), where the positive definite matrix \(\mathbf{Q}\) must be chosen. The values of elements of the matrix \(\mathbf{Q}\) (and thus of the matrix \(\mathbf{P}\)) influence the speed of decreasing the Lyapunov function (Equation (8)), which means a deceleration rate of the control deviation \(e\). The system (6) will be stable for any arbitrarily selected elements of the matrix \(\mathbf{Q}\) when fulfilling the condition of its positive definiteness. However, if we choose the reference model which ensures the optimal dynamic properties of the controlled system according to the criteria of minimal control deviation and minimum input energy, then it is possible to avoid solving the Lyapunov matrix Equation (11), because the elements of the matrix \(\mathbf{P}\) can be determined analytically, based on Equation (12) from the matrix (27) (see [32]):  
\[\mathbf{P}=\left[ \begin{array}{ccc}1562.5&625&62.5\\ 625&312.5&37.5\\ 62.5&37.5&7.5\end{array} \right] \tag{32}\]  
The designed controlled structure also includes the controller gain, which is represented by optional elements of the vector \(\mathbf{k}^{T}\) in Equation (28) for calculating the input \(u\). These elements must be positive and their value must be chosen in such a way that during the operation the IM must not reach the current limit. To fulfil all requirements, the vector \(\mathbf{k}^{T}\) in (28) has been set to the value \(\mathbf{k}^{T}=\) [0.0031 0.0019 0.00038]. These values were obtained by a gradual increasing the values of individual elements of the vector \(\mathbf{k}^{T}\) so that the  
Figure 4: IM drive block diagram. dynamics of decreasing the Lyapunov function would be the largest one at considering physical limits of the control system. An advantage of the described settings of parameters against the classical approaches consists of the fact they can be determined from measurements of responses and form the knowledge of the system limits without knowledge of its precise parameters (e.g., of the rotor resistance, moment of inertia, etc.). The dynamics of the reference model setting (using the parameter \(\alpha\)) must take into account the physical properties of the drive, which in our case are presented by the current (torque) overloading of the IM. The dynamic properties of the IM are described by unit step characteristics shown in Figure 2. The value of the parameter \(\alpha\) is inversely proportional to the model dynamics. This means that based on the Shannon-Kotelnik theorem for stabilizing the angular speed of the IM within 1 s, the value of the optional parameter is chosen as \(\alpha=5\). The operation cycle of the IM drive consists of three phases: starting the IM drive, running at a constant speed, and stopping (Figure 5a). An external (additive) disturbance of the IM drive--the load torque \(T_{L}=T_{N}=20\) Nm--was also introduced in time \(t=3\) s. This is valid for all simulation experiments.