---
date: '2024-12-16'
description: final exam review covering laplace and z-transforms, poles and zeros, pid control, and sampled-data systems.
id: finals
modified: 2026-06-05 15:08:42 GMT-04:00
tags:
- sfwr4aa4
title: Real-time control systems, and scheduling
created: '2024-12-16'
published: '2024-12-16'
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---
see also [[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/midterm|some OS-related for real-time system]], [[thoughts/university/twenty-three-twenty-four/sfwr-3dx4|Control systems]], or [[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/Frequency Domain]]
## time domain versus frequency domain
_use Laplace transform_ to convert from time domain to frequency domain
Consider the following circuit:
$$
\begin{aligned}
\text{RC} \frac{d v_o(t)}{dt} + v_o(t) &= v_i(t) \\
v_i(t) &= 1
\end{aligned}
$$
![[thoughts/Laplace transform]]
> \[!tip\] derivatives and integral
>
> if $\mathcal{L}[f(t)] = F(s)$ then we have
>
> $$
> \begin{aligned}
> \mathcal{L}[f^{'}(t)] &= sF(s) -f(0) \\
> \mathcal{L}[\int_0^{t}f(t) dt] &= \frac{F(s)}{s}
> \end{aligned}
> $$
For higher derivatives we have $\mathcal{L}[f^{''}(t)] = s^{2} F(s) - sf(0) - f^{'}(0)$
## another form of system model
we can replace $s$ with $jw$
ex: $G(jw) = \frac{1}{1+jw \text{RC}}$, so $|G(jw)| = |\frac{1}{1+jw \text{RC}}| = \frac{1}{\sqrt{1+(w \text{RC}^2)}}$
reasoning: we substitute Laplace transform with [[thoughts/Fourier transform]] with $s=jw$
example for a first-order system
$$
\begin{aligned}
Y(s) &= \frac{s+2}{s(s+5)} = \frac{2}{5s} + \frac{3}{5(s+5)} \\
y(t) &= \frac{2}{5} + \frac{3}{5} e^{-5t} \\[8pt]
\because \text{total response} &= \text{forced} + \text{natural}
\end{aligned}
$$
stability: total response = natural response + forced response
> \[!tip\] the output response of a system
>
> 1. natural (transient) response: $\frac{3}{5} e^{-5t}$
> 2. forced response (steady-state) response: $\frac{2}{5}$
### poles and zeros
_zeros and poles generate the amplitude for both forced and natural response_
$$
Y(s) = \frac{s+2}{s(s+5)}
$$
$s=0,-5$ are poles and $s=-2$ are zeros
> \[!note\] poles
>
> - at origin, generated **step function**
> - at -5 generate transient response $e^{-5t}$
![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/system response]]
![[thoughts/Root locus]]
| Control Type | Transfer function $T(s)$ | Key Characteristics | Effects |
| ---------------- | ----------------------------------------- | -------------------------- | ---------------------------------------------------------------------------- |
| Proportional (P) | $\frac{K_p G_p}{1 + K_p G_p}$ | Basic control action | - Affects speed of response
- Cannot eliminate steady-state error |
| Integral (I) | $\frac{K_I}{s^2 + s + K_I}$ | Integrates error over time | - Eliminates steady-state error
- Output reaches 1 at steady state |
| PI | $\frac{K_I + sK_p}{s^2 + (1+K_p)s + K_I}$ | Combines P and I | - P impacts response speed
- I forces zero steady-state error |
| Derivative (D) | $\frac{K_D s}{(1+K_D)s + 1}$ | Based on rate of change | - Adds open-loop zero
- Can affect stability
- Provides damping effect |
![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/PID controller#PID control]]
## S and Z-transform table
| Time Domain $x(t)$ | Laplace Transform $X(s)$ | Z Transform $X(z)$ |
| ------------------------------------------------------------------------- | ------------------------------------- | ----------------------------------------------------------------------------- |
| $\delta(t) = \begin{cases} 1 & t = 0 \\ 0 & t = kT, k \neq 0 \end{cases}$ | $1$ | $1$ |
| $\delta(t - kT) = \begin{cases} 1 & t = kT \\ 0 & t \neq kT \end{cases}$ | $e^{-kTs}$ | $z^{-k}$ |
| $u(t)$ (unit step) | $\frac{1}{s}$ | $\frac{z}{z - 1}$ |
| $t$ | $\frac{1}{s^2}$ | $\frac{Tz}{(z - 1)^2}$ |
| $t^2$ | $\frac{2}{s^3}$ | $\frac{T^2 z(z + 1)}{(z - 1)^3}$ |
| $e^{-at}$ | $\frac{1}{s + a}$ | $\frac{z}{z - e^{-aT}}$ |
| $1 - e^{-at}$ | $\frac{a}{s(s + a)}$ | $\frac{(1 - e^{-aT})z}{(z - 1)(z - e^{-aT})}$ |
| $te^{-at}$ | $\frac{1}{(s + a)^2}$ | $\frac{Tz e^{-aT}}{(z - e^{-aT})^2}$ |
| $t^2 e^{-at}$ | $\frac{2}{(s + a)^3}$ | $\frac{T^2 e^{-aT} z(z + e^{-aT})}{(z - e^{-aT})^3}$ |
| $b e^{-bt} - a e^{-at}$ | $\frac{(b - a)s}{(s + a)(s + b)}$ | $\frac{z [ z(b - a) - (b e^{-aT} - a e^{-bT}) ]}{(z - e^{-aT})(z - e^{-bT})}$ |
| $\sin \omega t$ | $\frac{\omega}{s^2 + \omega^2}$ | $\frac{z \sin \omega T}{z^2 - 2z \cos \omega T + 1}$ |
| $\cos \omega t$ | $\frac{s}{s^2 + \omega^2}$ | $\frac{z(z - \cos \omega T)}{z^2 - 2z \cos \omega T + 1}$ |
| $e^{-at} \sin \omega t$ | $\frac{\omega}{(s + a)^2 + \omega^2}$ | $\frac{z e^{-aT} \sin \omega T}{z^2 - 2z e^{-aT} \cos \omega T + e^{-2aT}}$ |
![[thoughts/Z-transform]]
![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/closed loop system]]
| System Type | Transfer Function | Diagram |
| ------------------------- | ------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------- |
| Basis | $\frac{C(z)}{R(z)} = \frac{G(z)}{1+Z[G(s)H(s)]}$ | ![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/closed-loop-sampled-data-system.webp]] |
| w/ digital sensing device | $C(z) = \frac{Z[G(s)R(s)]}{1+Z(G(s)H(s))}$ | ![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/closed-loop-tf-sensing-device.webp]] |
| w/ digital controller | $\frac{C(z)}{R(z)} = \frac{G_{1}(z)G_{1}(z)}{1+G_{1}(z)Z(G_{2}(s)H(s))}$ | ![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/closed-loop-tf-digital-controller.webp]] |
![[thoughts/university/twenty-four-twenty-five/sfwr-4aa4/CCS to DCS]]