Example 1: Consider the following linear system

Design an optimal controller to minimize the following performance index.

$\displaystyle J = \sum\limits_{k = 0}^\infty(x_1^2+x_1x_2+x_2^2+0.1u^2)$

Also, find the optimal cost.

Solution: The performance index J can be rewritten as

$\displaystyle J = \sum\limits_{k = 0}^\infty(\boldsymbol{x}^T(k)\begin{bmatrix}1 & 0.5\\ 0.5 & 1 \end{bmatrix} \boldsymbol{x}(k)+0.1u^2)$

Thus, $Q=\begin{bmatrix}1 & 0.5\\ 0.5 & 1 \end{bmatrix}$ and R = 0.1.

Let us take P as

$\displaystyle P = \begin{bmatrix}p_1 & p_2\\ p_2 & p_3 \end{bmatrix}$

Then,