Optimal and Robust Control

Knowledge (I memorize and understand)

1. Formulate the general problem of calculus of variations. Explain the difference between the variation and differential.
2. Write down the Euler-Lagrange equation and explain that it constitutes the first-order necessary condition of optimality for the calculus of variations problem \(\min_{y(x)} \int_a^b L(x,y(x),y'(x))\).
3. Give the first-order necessary conditions of optimality for a general (possibly nonlinear) optimal control problem on a fixed and finite time interval. Highlight that it comes in the form of a set of differential and algebraic equations together with the boundary conditions that reflect the type of the problem.
4. Give the first-order necessary conditions for an optimal control problem on a fixed and finite time interval with a continous LTI system and a quadratic cost - so-called LQ problem. Discuss how the boundary conditions change if the final state if regarded fixed or free.
5. Characterize qualitatively the solution to the LQ-optimal control problem on a fixed and finite time interval with a fixed final state. Namely, you should emphasize that it is an open-loop control.
6. Characterize qualitatively the solution to the LQ-optimal control problem on a fixed and finite time interval with a free final state. Namely, you should emphasize that it is a time-varying state-feedback control and that the time-varying feedback gains are computed from the solution to the differential Riccati equation.
7. Explain the basic facts about LQ-optimal control on an infinite time interval with a free final state. Namely, you should explain that it comes in the form of a state feedback and that the feedback gain can be computed either as the limiting solution to the differential Riccati equation or (and this is preferrable) as a solution to Algebraic Riccati Equation (ARE). The latter option brings in some issues related to existence and uniqueness of a stabilizing controller, which you should discuss.

Skills (I can use the knowledge to solve a problem)

1. Solve the continuous-time LQ-optimal control problem using solvers available in Matlab.