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...

As to control systems, transient performance is as important as steady-state performance. For some special dynamic systems, transient performance is a more prior index in comparison with the ste...

Writing a New Post YAML --- title: TITLE date: YYYY-MM-DD HH:MM:SS +0900 #Tokyo categories: [TOP_CATEGORIE, SUB_CATEGORIE] tags: [TAG1, TAG2] author: <author_id> math: true #Mathematical i...