I'm working on a homework problem in optimal controls and my plant model is described as: $\ddot{x}(t) = u(t)$ The performance index (cost function) is described by: $J = 1/2\int_0^5u^{2}(t)dt\,$ And the boundary conditions are $x(0) = 0,$ $x(5)=0,$ $\dot{x}(0)=2$ $\dot{x}(5)=0$ I'm not sure how to start this problem when I need $\dot{x}(t)$ to start with. Do I just integrate the plant model?
2nd Order Optimal Control Problem
1
$\begingroup$
ordinary-differential-equations
optimization
control-theory
1 Answers
2
This was a stupid question where I had a brain-fart. What confused me was that description of the system, $\ddot{x}(t)=u(t)$, looked like the state equation.
The correct states for this system would be: $x_1(t) = x(t)$ $x_2(t) = \dot{x}(t) = \dot{x}_1(t)$
After rewriting in this form, I would be able to proceed with solving for the optimal control solution using the state equation... $\dot{x}_1(t)=x_2(t)$ $\dot{x}_2(t)=u(t)$