# New Stable Non-Vector Control Structure for Induction Motor Drive
## 2 Design of the Linear Model Reference Control Structure
In order to investigate the asymptotic stability of the system (6) according to the Lyapunov criterion, the positive definite Lyapunov function is chosen in the weighted quadratic form of the system states:  
\[V=\mathbf{e}^{T}\mathbf{P}\mathbf{e} \tag{8}\]  
The derivation of the Lyapunov function (8) after inserting (1), (2), (3), (6) and performing simple modifications is:  
\[\frac{\mathrm{d}V}{\mathrm{d}t}=\mathbf{e}^{T}\Big{(}\mathbf{A}_{M}^{T} \mathbf{P}+\mathbf{P}\mathbf{A}_{M}\Big{)}\mathbf{e}+2\Big{[}\mathbf{f}^{T} \mathbf{z}-\left(\mathbf{B}_{M}\mathbf{u}\right)^{T}\mathbf{z}\Big{]}= \mathbf{e}^{T}\mathbf{Q}\mathbf{e}+2\Big{[}\mathbf{f}^{T}\mathbf{z}-\left( \mathbf{B}_{M}\mathbf{u}\right)^{T}\mathbf{z}\Big{]} \tag{9}\]  
where the vector \(\mathbf{z}\) (\(n\times 1\)) is the weighted state deviation vector:  
\[\mathbf{z}=\mathbf{P}\mathbf{e} \tag{10}\]  
In (8)-(10) the matrix \(\mathbf{P}\) (\(n\times n\)) is a symmetric positively definite matrix which satisfies the Lyapunov matrix equation:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=(\mathbf{A}_{M}+\Delta\mathbf{A}) \mathbf{x}+(\mathbf{B}_{M}+\Delta\mathbf{B})\mathbf{u}+\mathbf{v}=\mathbf{A} _{M}\mathbf{x}+\mathbf{B}_{M}\mathbf{u}+(\Delta\mathbf{A}\mathbf{x}+\Delta \mathbf{B}\mathbf{u}+\mathbf{v}) \tag{2}\]0  
where \(\mathbf{Q}\) (\(n\times n\)) is also a symmetric positive definite matrix. Choosing the reference model according to (1) one can avoid solving (11). If the state matrix of the reference model is in the controllability form, then based on the optimal control theory [32] it is possible to determine the elements of the matrix \(\mathbf{P}\) analytically as follows:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=(\mathbf{A}_{M}+\Delta\mathbf{A}) \mathbf{x}+(\mathbf{B}_{M}+\Delta\mathbf{B})\mathbf{u}+\mathbf{v}=\mathbf{A} _{M}\mathbf{x}+\mathbf{B}_{M}\mathbf{u}+(\Delta\mathbf{A}\mathbf{x}+\Delta \mathbf{B}\mathbf{u}+\mathbf{v}) \tag{2}\]1  
where the parameter \(\alpha\) allows to set an optimal dynamics of the controlled variable satisfying the criteria of the minimum control deviation and of the minimum input energy. The model dynamics are inversely proportional to the value of the parameter \(\alpha\). The system (6) is asymptotically stable if the derivation of Lyapunov  
Then, the second expression on the right side in (14) will be negative if for each component of the vector \(\mathbf{f}\) the following inequality is met:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=(\mathbf{A}_{M}+\Delta\mathbf{A}) \mathbf{x}+(\mathbf{B}_{M}+\Delta\mathbf{B})\mathbf{u}+\mathbf{v}=\mathbf{A} _{M}\mathbf{x}+\mathbf{B}_{M}\mathbf{u}+(\Delta\mathbf{A}\mathbf{x}+\Delta \mathbf{B}\mathbf{u}+\mathbf{v}) \tag{2}\]2  
where \(\mathbf{k}_{i}^{T}\) is the \(i\)-th row of the matrix \(\mathbf{K}\). Let us note that for a single-input system instead of the matrix \(\mathbf{K}\) the row vector \(\mathbf{k}^{T}\) is used. The inequality (16) is ensured if the optional positive parameters in the matrix \(\mathbf{K}\) will have sufficiently large values. In order to achieve the zero control deviation, i.e., the difference between the output variables of the reference model and the controlled system \((y_{M}-y)\) in steady-state, the first component of the vector \(\mathbf{e}\) will be chosen as an integral of this difference:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=(\mathbf{A}_{M}+\Delta\mathbf{A}) \mathbf{x}+(\mathbf{B}_{M}+\Delta\mathbf{B})\mathbf{u}+\mathbf{v}=\mathbf{A} _{M}\mathbf{x}+\mathbf{B}_{M}\mathbf{u}+(\Delta\mathbf{A}\mathbf{x}+\Delta \mathbf{B}\mathbf{u}+\mathbf{v}) \tag{2}\]3  
By introducing this integral, the reference model (1) is extended by the new state variable \(x_{ext}\). Now, the extended deviation vector (3) \(\mathbf{e}^{*}\) will be:  
\[\mathbf{e}^{*}=\left[ \begin{array}{c}x_{ext}\\ \mathbf{e}\end{array} \right] \tag{18}\]  
The reference model extension does not affect the stability of the designed control structure provided that the conditions in Equations (10)-(12) are valid also for the extended system.