## 3 Suboptimally Controlled Induction Motor
### Synthesis of the Suboptimal Control for the AC Motor  
In order to obtain the suboptimal control that regulates the discrete state \(\bar{x}_{m}(k)\) of the nonlinear discrete system (4), first observe that the nonlinear map \(f_{m}(. )\) is well defined with respect to its arguments. The origin is a fixed point of the system given by (4), which means \(f_{m}(0)=0\) when the control vector \(\bar{u}_{m}(k)\in\mathbb{R}^{2}\) is equal to zero and \(k=0,1,\ldots,N\) for some \(N\in\mathbb{N}\). Consider the following definition about controllability for discrete systems. **Definition 1**. : _The pair \((\bar{x}_{m0},\bar{x}_{m1})\) is controllable if there exists an admissible control sequence \(\bar{u}_{m}(k)\) such that the system defined by (4) is transferred from \(\bar{x}_{m0}\) to \(\bar{x}_{m1}\) in \(N\) finite number of steps._  
It is not difficult to demonstrate that the system given by (4) is locally controllable. In fact, one can linearize the system given by (4) around the fixed point zero and verify that the linear discrete system is controllable. Therefore, the system given by (4) is controllable according to Definition 1, at least in the local sense (see, for example, ref. [8]). The main ideas of the dynamic programming approach exposed in the following were originally exposed in ref. [19]. Here, the algorithm is adapted to be applied to regulate all the states of the nonlinear model given by (4). Next, the performance index is defined as a quadratic form to be locally minimized with respect to the control sequence \(\bar{u}_{m}(k)\) subject to the trajectories of the system given by (4). In fact, the following quadratic index is considered:  
\[J=\frac{1}{2}\bar{x}_{m}^{T}(N)H\bar{x}_{m}(N)+\frac{1}{2}\sum_{k=0}^{N-1} \Bigl{\{}\bar{x}_{m}^{T}(k)Q\bar{x}_{m}(k)+\bar{u}_{m}^{T}(k)R\bar{u}_{m}(k) \Bigr{\}}, \tag{7}\]  
where \(H,\ Q\geq 0\) and \(R>0\) are square matrices of appropriate dimensions, and \(t_{f}=T_{s}N\) defines the horizon. As in the linear quadratic regulator (LQR), matrices \(H\) and \(Q\) penalize the state convergence, and \(R\) penalizes the energy applied to the system. The optimization problem could be established as follows:  
**Problem 1**. : _To find a suboptimal sequence \(\bar{u}_{m}(k)\) that minimizes (in the local sense) to the performance index given by (7), **along the trajectories** of the system given by (4)._  
The problem of finding a global optimal control sequence that minimizes the performance index given by (7) involves the solution of the Hamilto-Jacobi-Bellman (HJB) equation in its discrete version. However, finding the solution of the HJB equation for nonlinear systems represents a very difficult problem [32]. The following proposition gives a solution to Problem 1. **Proposition 1**. : _Consider the system given by (4) and the quadratic performance index given by (7). A suboptimal sequence \(\bar{u}_{m}(k)\), which causes the performance index (7) to reach a minimum in the local sense, is described by the following entries:_  
\[v_{\text{sat}}^{*}(N-k) =-\sigma T_{\text{s}}\Big{(}\frac{f_{1}(\bar{x}_{m}(N-k))s_{41}+f_{ 2}(\bar{x}_{m}(N-k))s_{42}+f_{4}(\bar{x}_{m}(N-k))s_{44}}{r_{11}r^{2}+s_{44}T_{ 2}^{2}}\Big{)}, \tag{8}\] \[v_{\text{sb}}^{*}(N-k) =-\sigma T_{\text{s}}\Big{(}\frac{f_{1}(\bar{x}_{m}(N-k))s_{12}+f_ {2}(\bar{x}_{m}(N-k))s_{23}+f_{5}(\bar{x}_{m}(N-k))s_{56}}{r_{22}r^{2}+s_{56}T_{ 2}^{2}}\Big{)},\]  
_where \(s_{i\bar{j}}=h_{i\bar{j}}\) if \(k=1\), \(s_{i\bar{j}}=q_{i\bar{j}}\) if \(k=2,\ldots,N\), and the matrices \(H\), \(Q\), and \(R\) are given as follows:_  
\[H =\left[ \begin{array}{cccc}h_{11}&0&0&h_{14}&h_{15}\\ 0&h_{22}&0&h_{24}&h_{25}\\ 0&0&h_{33}&0&0\\ h_{41}&h_{42}&0&h_{44}&0\\ h_{51}&h_{52}&0&0&h_{55},\end{array} \right], \tag{9}\] \[Q =\left[ \begin{array}{cccc}q_{11}&0&0&q_{14}&q_{15}\\ 0&q_{22}&0&q_{24}&q_{25}\\ 0&0&q_{33}&0&0\\ q_{41}&q_{42}&0&q_{44}&0\\ q_{51}&q_{52}&0&0&q_{55}\end{array} \right],\]  
_and_  
\[R=\left[ \begin{array}{cc}r_{11}&0\\ 0&r_{22}\end{array} \right].