Control Systems Tutorials

# EE 351 Advanced Control Systems

## Summary: Linear Tracking Problem

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_f)-r(t_f)]^{T}H[X(t_f)-r(t_f)]+\dfrac{1}{2}\int^{t_f}_{t_0}([X(t)-r(t)]^TQ(t)[X(t)-r(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*} \begin{equation*} \dot{S}(t)=-[A^T(t)-K(t)B(t)R^{-1}(t)B^T(t)]S(t)+Q(t)r(t) \end{equation*} with the boundary condition $K(t_f)=H$,and $S(t_f)=-Hr(t_f)$. $K(t)$ is a symmetrical $n\times n$ matrix known as Kalman gain. $S$ is a $n \times 1$ vector.

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)-B(t)R^{-1}B^T(t)S(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)-R^{-1}(t)B^T(t)S(t) \end{equation*}

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

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