Control Systems Tutorials

Given a system, \begin{equation*} \dot{X}(t)=A(t)X(t)+B(t)U(t) \end{equation*} The performance measure to be minimized is \begin{equation*} J=\dfrac{1}{2}X^{T}(t_f)HX^{T}(t_f)+\dfrac{1}{2}\int^{t_f}_{t_0}[X^T(t)Q(t)X(t)+U^T(t)R(t)U(t)]dt \end{equation*}

$t_f$ is fixed.

$X(t_f)$ is free.

$H$, $Q$ are real symmetric positive semi-definite matrix.

$R$ is a real symmetric positive definite matrix.

States ($X$) and input ($U$) are not bounded.

Steps to find the optimal control law.

Step 1: Solve the following matrix differential Riccati equation \begin{equation*} \dot{K}(t)=-K(t)A(t)-A^T(t)K(t)-Q(t)+K(t)B(t)R^{-1}(t)B^T(t)K(t) \end{equation*} with the boundary condition $K(t_f)=H$. $K(t)$ is a symmetrical $n\times n$ matrix known as Kalman gain.

Step 2: : Solve the optimal state differential equation: \begin{equation*} \dot{X}^*(t)=[A(t)- B(t)R^{-1}(t)B^T(t)K(t)]X^*(t) \end{equation*} with the initial condition $X(t_0)=X_0$.

Step 3: Obtain the optimal control law $U^*(t)$ \begin{equation*} U^*(t)=-R^{-1}(t)B^T(t)K(t)X^*(t) \end{equation*}

Step 4: : Obtain the optimal performance index $J^*$

Note: For a infinite time process, $t_f \rightarrow \infty$ Kalman gain ($K$) becomes constant and $\dot{K}(t)=0$. \begin{equation*} 0=-K(t)A(t)-A^T(t)K(t)-Q(t)+K(t)B(t)R^{-1}(t)B^T(t)K(t) \end{equation*}

Engineering Tutorial         Phamacology         Blog