Home Page:Youkoutaku import numpy as np import scipy as sp import pandas as pd from pandas import Series, DataFrame import matplotlib.pyplot as plt import matplotlib as mpl import seaborn as sns %...
Problem Consider a simple system: [\begin{bmatrix} \dot x_1 \ \dot x_2 \end{bmatrix}=\begin{bmatrix} 0 &1\ 0.5 &0 \end{bmatrix}\begin{bmatrix} x_1 \ x_2 \end{bmatrix}+\begin{bmatrix}0\ 1 \...
By the HJB equation, we discuss the linear quadratic regulators (LQR) for continuous systems. Continuous systems Consider a continuous system as the following equation. [\dot{x}{(t)}=f(x{(t)},u_{...
We discuss a typical problem: for the linear system, the performance function is in quadratic form, and the control goal is to stabilize the state at 0 (regulation problem). Then, its’ controllers ...
Continuous System [\dot x(t)=f\left(x(t),u(t),t\right)] $x(t):$ system states $u(t):$ input $f():$ linear or nonlinear function Performance function The performance function from anytime...
Discrete-time system [x_{[k+1]}=f(x_{[k]},u_{[k]})] $x_{[k]}:$ system states $u_{[k]}:$ input $f():$ linear or nonlinear function Performance function [J=h(x_{[N]})+\sum_{k=0}^{N-1}g(x_...
Bellman optimal theory An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the stat...
Unicycle Model State Space [x(t)=\begin{bmatrix} x_1(t) x_2(t) x_3(t) x_4(t) \end{bmatrix}=\begin{bmatrix} p_x(t) p_y(t) v(t) \theta(t) \end{bmatrix}] $p_x(t)$: position in x direction ...
1. The derivatives of scalar function by vector Consider a scalar function by a variable as [f(u)=u^2-2u-1] where $f,u\in\mathbb{R}$. [\frac{df(u)}{du}=2u-2] The extremum is at $u=1$ in $f(u)$...
State-space representation [\dot{x}(t)=Ax(t)+Bu(t)] [y(t)=Cx(t)+Du(t)] Exponential of matrix [\dot{x}=ax \implies x=x(0)e^{at}] where $a$ is a scale constant. [\dot{x}=Ax \implies x=x(0)e^{At...