I have the following 2nd order linear system with appropriate initial conditions.
$$\textbf{X}''(t)+\textbf{A}(t)\textbf{X}'(t)+\textbf{B}(t)\textbf{X}(t)=\textbf{F}(t)+\textbf{C}(t)\textbf{U}(t)$$
$\textbf{X}$ is a unknown vector.
$\textbf{A, B, C, F}$ are time varying matrices.
$\textbf{U}(t)$ is a control input matrix.
I wonder what is the cost function (performance index) for the following 2nd order linear system for finding optimal control function $\textbf{U}(t)$?
But I didn' t find enough reference books, articles etc. (Do you have good references?)
Is it the following functional, right?
$$J=\frac{1}{2}\int_{0}^{t_f}(\textbf{X}^T\textbf{QX}+\dot{\textbf{X}}^T\textbf{R}\dot{\textbf{X}}+\textbf{U}^T\textbf{ZU})dt$$
where $\textbf{Q,R,Z}$ are the semipositive matrices.
Best Answer
You can also write the second order differential equation as a first order using
$$ \dot{z} = \begin{bmatrix} 0 & I \\ -B(t) & -A(t) \end{bmatrix} z + \begin{bmatrix} 0 \\ F(t) \end{bmatrix} + \begin{bmatrix} 0 \\ C(t) \end{bmatrix} u $$
where $z=\begin{bmatrix}x^\top & \dot{x}^\top \end{bmatrix}^\top$.
A quadratic cost function can also be generalized to
$$ J =\frac{1}{2}\int_0^{t_f} \begin{bmatrix}z \\ u\end{bmatrix}^\top \!M \begin{bmatrix}z \\ u\end{bmatrix} dt $$
with $M=M^\top\succeq0$ plus come extra constraints similar for LQR.