Articles of optimal control

Inverse of State-space representation (control)

Ask two questions from a paper (2012 ACC): Consider the plant: Let X be the stabilizing solution of the Riccati equation: where . Define the LQR gain by . The transfer matrix has a left spectral factorization , where WL is given by Questions: If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom […]

Robust Control VS Optimal Control

What’s the diffrents between Optimal Control and Robust Control? I know that Optimal Control have the controllers: LQR – State feedback controller LQG – State feedback observer controller LQGI – State feedback observer integrator controller LQGI/LTR – State feedback observer integrator loop transfer recovery controller (for increase robustness) And Robust Control have: $H_{2}$ controller $H_{\infty}$ […]

Optimal control

Consider the growth equation: $ \dot{x} = tu $, with $x(0)=0$ and $x(1)=1$, and with the cost function: $ J= \int_0^1 u^2 dt $. Show that $u^*=3t$ is a successful control, with $x^*=t^3$ and $J^*=3$ the corresponding trajectory and cost. If $u=u^* + v $ is another successful control, show that $\int_0^1 vt dt = […]

Fastest curve from $p_0$ to $p_1$

I’m trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = v_1$ which minimizes $T$ (or $p^{-1}(x_1)$) given the constraint $|p”(t)| \le 1$. From what I’ve read, I think this […]

Time-optimal control to the origin for two first order ODES – Trying to take control as we speak!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it with all I have got: $\begin{pmatrix} \dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}3 & 1 \\ 4 & 3\end{pmatrix}\begin{pmatrix}x_1 […]

A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Let $\gamma \in(0,1)$, ${\bf F},{\bf Q} \in \mathbb R^{n\times n}$, ${\bf H}\in \mathbb R^{n\times r}$, and ${\bf R}\in \mathbb R^{r\times r}$ be given and suppose that ${\bf P}$,${\bf W}$,${\bf X}\in \mathbb R^{n\times n}$, and ${\bf K}$,${\bf L}\in \mathbb R^{n\times r}$ satisfy \begin{align} {\bf P} &={\bf F}({\bf I}_{n}-{\bf K} {\bf H}^\top){\bf P}{\bf F}^\top+{\bf Q}, \;\;\;\;\;\;\;\;\text{where}\;\;\;\;{\bf […]

Is there any guaranteed stability margins for Extended Kalman Filter (EKF)?

LQR controllers have guaranteed stability margins, but LQG controllers has not guaranteed stability margins, due to the linear kalman filter. But what will happen if I replace the linear kalman filter with the Extended Kalman Filter(EKF), which is a nonlinear kalman filter? Do I receive guaranteed stability margins then?

Selection problem: how to solve?

Here is what we have done. We started from having a system subject to $\dot x(t)=f(x(t),u(t))$ (dropping explicit dependence of $f$ from $t$ for simplicity), with $u(t)\in\mathcal{U}$ for all $t$, and our problem was to minimize the time taken to go from $x(0)=x^0$ to $x(T_f)=x^f$, $T_f$ being the final time at which $x^f$ is reached. […]

A good reference on optimal control theory

Ok, so I am reading about decision making and I came across this subject. Fortunately it has a Wiki, but the point is I want to see some examples, and learn to solve regular problems of this field. Of course I can go for the book by Pontryagin but then I don’t think that would […]