# New Stable Non-Vector Control Structure for Induction Motor Drive
## 2 Design of the Linear Model Reference Control Structure  
The desired system dynamics of a controlled system are very often described by a reference model. When this is a linear system, it can be optimally designed using standard methods of the optimal control theory [33]. The state-space reference model for the controlled system with \(n\) state variables and \(p\) inputs is described by the state equation:  
\[\frac{\mathrm{d}\mathbf{x}_{M}}{\mathrm{d}t}=\mathbf{A}_{M}\mathbf{x}_{M}+ \mathbf{B}_{M}\mathbf{w} \tag{1}\]  
where \(\mathbf{x}_{M}\) (\(n\times 1\)) is the state vector of the reference model, \(\mathbf{A}_{M}\) (\(n\times n\)) is the state matrix of the reference model, \(\mathbf{B}_{M}\) (\(n\times p\)) is the input matrix of the model and \(\mathbf{w}\) (\(p\times 1\)) is the vector of the desired values. The controlled system is described in state space as a nonlinear continuous system with parametric and with additive disturbances (or deviations from the reference model) in the form:  
\[\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}=(\mathbf{A}_{M}+\Delta\mathbf{A}) \mathbf{x}+(\mathbf{B}_{M}+\Delta\mathbf{B})\mathbf{u}+\mathbf{v}=\mathbf{A} _{M}\mathbf{x}+\mathbf{B}_{M}\mathbf{u}+(\Delta\mathbf{A}\mathbf{x}+\Delta \mathbf{B}\mathbf{u}+\mathbf{v}) \tag{2}\]  
where \(\mathbf{x}\) (\(n\times 1\)) is the state vector of the controlled structure, \(\mathbf{u}\) (\(p\times 1\)) the vector of input variables, \(\Delta\mathbf{A}\) (\(n\times n\)), \(\Delta\mathbf{B}\) (\(n\times p\))--the matrices of the parametric disturbances and \(\mathbf{v}\) (\(n\times 1\))--the vector of additive disturbances. In this case, the goal of the electric drive control is twofold:  
1. To reach the zero state of the state vector \(\mathbf{x}\). 2. To reach the zero state of all deviation of state variables from the desired values. For this reason, the deviations of the state vector components from the desired values are suitable to be chosen as the state variables of the controlled system. The system's stability is investigated with regard to these deviations. Let us calculate the deviation between the reference model and controlled system:  
\[\mathbf{e}=\mathbf{x}_{M}-\mathbf{x} \tag{3}\]  
where \(\mathbf{e}\) (\(n\times 1\)) is the vector of deviations between the state variables of the model \(\mathbf{x}_{M}\) according to (1) and of the system \(\mathbf{x}\), defined in Equation (2). By differentiating this vector of deviations one gets:  
\[\frac{\mathrm{d}\mathbf{e}}{\mathrm{d}t}=\frac{\mathrm{d}\mathbf{x}_{M}}{ \mathrm{d}t}-\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} \tag{4}\]  
After inserting Equations (1) and (2), the expanded system is:  
\[\frac{\mathrm{d}\mathbf{e}}{\mathrm{d}t}=\mathbf{A}_{M}(\mathbf{x}_{M}- \mathbf{x})+\mathbf{B}_{M}\mathbf{w}-\mathbf{B}_{M}\mathbf{u}-\Delta\mathbf{A }\mathbf{x}-\Delta\mathbf{B}\mathbf{u}-\mathbf{v} \tag{5}\]  
\[\frac{\mathrm{d}\mathbf{e}}{\mathrm{d}t}=\mathbf{A}_{M}\mathbf{e}-\mathbf{B}_{M }\mathbf{u}+\mathbf{f} \tag{6}\]where the vector \(\mathbf{f}\left(n\times 1\right)\) presents a generalized disturbance vector comprising all parametrical and additive disturbances affecting the system with regard to its reference model:  
\[\mathbf{f}=-\Delta\mathbf{A}\mathbf{x}-\Delta\mathbf{B}\mathbf{u}-\mathbf{v}+ \mathbf{B}_{M}\mathbf{w} \tag{7}\]  
The goal of the controller design is to find such mathematical formulation for determining the input vector \(\mathbf{u}\) for which the zero solution of the system (6) is asymptotically stable, i.e., \(\lim\limits_{l\rightarrow\infty}\mathbf{e}=0\).