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 […]

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 $H_{\infty}$ […]

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 = […]

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 […]

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 […]

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 […]

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?

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. […]

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 […]

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