The Team

Jaimin Vaidya, Abhishek Malhotra, Sibhi Srinivasan, Vaishnav Srivaths, Brinda Mehta , Dylan Dhakhwa, Grace Cloonan, Krinal Doma,Travis Miller,Gavin Chan

Introduction

This project examines a blimp with longitudinal and lateral motion each controlled by two sets of propellers. The overall blimp system, as well as its propeller control hardware, are shown below: With this control, the blimp can either act as a chaser or an evader; the chaser has a control system that tells it to pursue the evader while minimizing time, while the evader has a controller to help it avoid the chaser by maximizing the relative distance between the evader and the chaser.

This specific analysis will look at the evader’s control system and use the system’s performance to design a controller that can help the evader blimp achieve its goal. The process of this analysis will be to analyze the blimp’s open-loop performance, determine a preliminary control design, and to apply that design to the evader to meet certain performance requirements.

Open-Loop System Performance

Section A

The open-loop system characteristics have to assume a second-order system. The transfer function for the linearized longitudinal dynamics is:

$\frac{y_a(s)}{u_a(s)} = \frac{3.6083}{s^3+0.8229s^2+0.1294s}$

Because the given linearized longitudinal dynamics is represented by a third-order system, we must use second-order dominance to approximate the performance. We used the denominator as the characteristic equation and set it equal to a third-order expansion:

$s^3+0.8229s^2+0.1394s = (s+p)(s^2+2\zeta \omega_ns+\omega_n^2)$

$s^3+0.8229s^2+0.1394s =s^3+(2n+p)s^2+(\omega_n^2+2pw_n)s+pn2$

$n=0.37336 rad/s ,\quad \zeta =1.102$

This led to an overdamped system, for which there is no settling time or overshoot.

The transfer function for the linearized rotational dynamics is:

$\frac{y_h(s)}{u_h(s)} = \frac{0.1257}{s^3+2.542s^2+0.6254s+1.5901}$

Because the given linearized rotational dynamics is represented by a third-order system, we must use second-order dominance to approximate the performance. We use the denominator as the characteristic equation and set it equal to the third-order expansion:

$s^3+2.542s^2+0.6254s+1.5901 = (s+p)(s^2+2\zeta \omega_ns+\omega_n^2)$

$s^3+2.542s^2+0.6254s+1.5901 =s^3+(2n+p)s^2+(\omega_n^2+2pw_n)s+pn2$

$n=0.37336 rad/s ,\quad \zeta =0$

This led to an undamped system, for which there is no settling time or overshoot.

The transfer function for the linearized forward dynamics is represented by a second-order system:

$\frac{y_f(s)}{u_f(s)} = \frac{6.2339s^2-9.5876s+292.43}{s^4+3.7028s^3+28.6058s^2+88.9349s+71.1363}$

Using MATLAB’s stepinfo(SYS) command, we were able to calculate the open-loop performance of the forward dynamics system as follows:
Settling time: 9.9984 s
Overshoot: 2.7575% Which yielded a value of 1 for the linearized longitudinal dynamics system, a value of 0.927 for the linearized rotational dynamics system, and a value of 0.196 for the linearized forward dynamics system.

Section B

Controller Design

Summary

LTAR is the most recent innovation in the historied field of lighter-than-air flight. Using modern control engineering, a blimp equipped with 2 sets of propellers and a control system can successfully perform several functions including chasing or evading an independant object. Given linearized transfer functions to describe the aircraft’s flight, the systems were evaluated for damping and steady state error of a step input. Additionally, further examination of the forward mode of lateral motion provided opportunity for root locus discovery. Through the determination of poles and zeros as well as the break-in and break-out points and their corresponding angles of arrival or departure allowed for accurate plotting.

Closed loop feedback controllers were then created using simulations of closed loop models for each mode of motion. Utilizing SIMULINK’s various tools and design values for the controllers, PID controllers were developed and tuned to provide acceptable overshoot, settling time, and steady state error.

Beginning with altitude control and heading angle, PID controllers were designed to minimize response times and overshoot. These were consequently used in the creation of the circle path and evasion codes. The first model has 3 main subsystems which individually achieves portions of the end goal including circle generation, PID post processing, and signal generation. The evasion code is instead divided into 2 main subsystems, one concerning object avoidance in the horizontal plane, the other for the same purpose in the vertical axis. These systems work individually utilizing the separate propellers in tandem to space the system from the chaser. Although we found the chaser to catch our evading system through 100% efficient simulations, we believe our evader to be successful due to correctly attempting its main directive of increasing relative distance between it and the chaser.

References

Robots. Lofaro Labs. (n.d.). Retrieved April 22, 2023, from http://lofarolabs.com/index.php/robots/
Lighter-than-air flight. National Museum of the United States Air Force™. (n.d.). Retrieved April 22, 2023, from https://www.nationalmuseum.af.mil/Visit/Museum-Exhibits/Fact-Sheets/Display/Article/196758/lighter-than-air-flight/#:~:text=There%20are%20two%20basic%20types,vehicles%20that%20can%20be%20steered.