---
date: '2024-03-18'
description: assignment on open-loop systems with root locus plots, unity feedback systems, damping ratio calculation, marginal stability using routh-hurwitz, and pd controller design.
id: A3
modified: 2026-06-05 15:08:42 GMT-04:00
seealso:
  - '[[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/assignment3.pdf|problem]]'
tags:
  - sfwr3dx4
title: Open-loop system
created: '2024-03-18'
published: '2024-03-18'
pageLayout: default
slug: thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/A3
permalink: https://aarnphm.xyz/thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/A3.md
generator:
  quartz: v4.6.0
  hostedProvider: Cloudflare
  baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
## Problemè 1

A unity feedback system has transfer function

$$
G(s) = \frac{K}{s(s^2+4s+13)}
$$

> \[!question\] 1.a
>
> Plot the root locus for this problem

We need to find the poles and zeros of the open-loop transfer function $G(s)$. The poles are given by the roots of the denominator polynomial:

$$
s(s^2+4s+13) = 0
\implies s = 0, -2 \pm 3j
$$

![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/p1a.webp]]
And the following code to draw the plot:

```python
import matplotlib.pyplot as plt
import control as ctl

numerator = [1]
denominator = [1, 4, 13, 0]
G_s = ctl.TransferFunction(numerator, denominator)
rlist, klist = ctl.root_locus(G_s, Plot=True, grid=True)

# Show the plot
plt.show()
```

> \[!question\] 1.b
>
> Find the value of $K$ that gives a damping ration of 0.2588

K is approximately $10.000$.

The following code is used:

```python
# Find the gain K for a damping ratio of 0.2588
K_range = np.linspace(0, 10, 1000)
for K in K_range:
  poles = np.roots(G.den[0][0] + K * G.num[0][0])
  zeta_actual = -np.cos(np.angle(poles[0]))
  if np.abs(zeta_actual - zeta) < 0.001:
    break

# Print the gain value
print(f'The gain K for a damping ratio of 0.2588 is approximately: {K:.3f}')
```

> \[!question\] 1.c
>
> Find the location of the roots for the value of $K$ found in 1.b

With $K=10$, the poles are `[-1.5+2.78388218j -1.5-2.78388218j -1. +0.j        ]`.

The code is:

```python
K = 10
print(np.roots([1, 4, 13, K]))
```

> \[!question\] 1.d
>
> Plot the step response of your closed-loop system, along with the step response of an ideal second order system with damping ratio 0.2588 and poles that correspond to the two poles with imaginary parts.

![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/p1d.webp]]
Here is the code for that:

```python
# Extract the imaginary part of the complex poles
wn = np.abs(poles[0].imag)

# Create the closed-loop transfer function with the found gain K
G_cl = ctl.feedback(K * G, 1)

# Create an ideal second-order system with the same damping ratio and natural frequency
G_ideal = ctl.tf([wn**2], [1, 2 * zeta * wn, wn**2])

# Generate time vector for simulation
t = np.linspace(0, 10, 1000)

# Simulate the step response of the closed-loop system and the ideal system
_, y_cl = ctl.step_response(G_cl, t)
_, y_ideal = ctl.step_response(G_ideal, t)

# Plot the step responses
plt.figure()
plt.plot(t, y_cl, label='Closed-loop System')
plt.plot(t, y_ideal, '--', label='Ideal Second-Order System')
plt.xlabel('Time')
plt.ylabel('Output')
plt.title('Step Response Comparison')
plt.legend()
plt.grid()
plt.show()
```

> \[!question\] 1.e
>
> Find the value of K that leads to a marginally stable system.

The characteristic equation is given by:

$$
1 + G(s) = 0
$$

or

$$
s^3 + 4s^2 + 13s + K = 0
$$

Let’s setup the [[thoughts/Routh-Hurwitz criterion|Routh-Hurwitz table]]:

| $s^3$ | 1                    | 13 |
| ----- | -------------------- | -- |
| $s^2$ | 4                    | K  |
| $s^1$ | $b=13 - \frac{K}{4}$ | 0  |
| $s^0$ | K                    | -  |

For marginal stability, the system must have poles on the imaginary axis. This occurs when the first element of any row in the Routh array is zero.

Let $b=0$, then $13 - \frac{K}{4} = 0 \implies K = 52$.

Therefore, the system is marginally stable for $K = 52$. This is the critical gain $K_{cr}$.
For $K < 52$, all elements in the first column of the Routh array are positive, indicating stability.
For $K > 52$, there is a sign change in the first column, indicating instability

For frequency oscillation at marginal stability, solve characteristic equation for $s$ with $K=52$:

$$
(s^2+4)(s+13) = 0
$$

imaginary roots are $\pm 2j$, thus frequency of oscillation is $2$ rad/s.

---

## Problemè 2

Consider the open-loop system

$$
G(s) = \frac{(s+10)}{(s+1)(s+2)(s+12)}
$$

> \[!question\] 2.a
>
> Suppose that design specifications are that the $\%OS$ is 20% and the settling time is 1 second. Use the root-locus approach to design a PD controller for this system.

Given $\%OS$ is 20% and settling time is 1 second, we can find $\zeta$, $\sigma$, $\omega_{n}$ as:

$$
\begin{align*}
\zeta &= -\ln(\%OS/100) / \sqrt{\pi^2 + \ln^2(\%OS/100)} \approx 0.456 \\
\sigma &= \frac{4}{\zeta T_s} \approx 8.77 \\
\omega_{n} &= 19.24
\end{align*}
$$

The code to find:

```python
import numpy as np
import matplotlib.pyplot as plt
import control as ctl

# Desired specifications
OS = 0.20  # 20% overshoot
Ts = 1  # 1 second settling time

# Calculations for desired pole locations
zeta = -np.log(OS) / np.sqrt(np.pi**2 + np.log(OS) ** 2)  # damping ratio
sigma = -4 / (zeta * Ts)  # real part of poles
wd = sigma / zeta  # imaginary part of poles

print(zeta, sigma, wd)
```

So desired dominant poles are:

$$
s_{1,2} = -\zeta \omega_n \pm j\omega_{n}\sqrt{1-\zeta^2} = -4 \pm 7.66j
$$

Propose a PD controller $D(s) = K(s+z)$ where $K$ is the gain and $z$ is the zero introduced by the PD controller.

![[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/p2a.webp]]

The code can be found [[thoughts/university/twenty-three-twenty-four/sfwr-3dx4/a3/p2.py|here]]