discrete LQR controller based on C++

• This project gives an example, which can be modified by users.
• Every block during option start and option end should be modified by users according to the actual situation, where contains necessary parameters and printing options.
• Need to pre install dependency Eigen:
  sudo apt-get install libeigen3-dev
sudo ln -s /usr/include/eigen3/Eigen /usr/include/Eigen
• Provide simple parameter type checking function & analysis of calculation results.
• Users can set parameter SYSTEM_TYPE in CONTINUOUS or DISCRETE according to the actual situation known by users. If users input the information of a continuous system, this component can trans that into discrete.
• Midpoint Euler method & Backward Euler method are used to trans system discretization if necessary.
• In main function, a loop is used to simulate the normal operation of a classical 1-dim position-velocity-acceleration system.

$$\begin{bmatrix}p_{k+1}\\v_{k+1}\end{bmatrix} = \begin{bmatrix}1 & T\\0 & 1\end{bmatrix} \cdot \begin{bmatrix}p_{k}\\v_{k}\end{bmatrix} + \begin{bmatrix}\frac{1}{2} \cdot T^2 \\T\end{bmatrix} \cdot u_k$$

• The test result is as the follow figure, as the time going, the state variable of the system converge to 0. the number of iterations is 20:

• Reference:
  连续方程的离散化与离散LQR原理
  迭代法求黎卡提（Riccati）方程的解