Assignment #2
Fuzzy Controller for the Inverted Pendulum Problem
Maximum number of members per group: 3 students
Deadline for submission: October 10
Instructions
Your task is mainly to design and implement a fuzzy controller(Zero-order Sugeno Fuzzy Inference System) for balancing an inverted pendulum system. A written report detailing your system design and characterisation of its performance must accompany your program submission.
A start-up program using second-order derivative physics equations and simple graphics library are provided, simulating the complete dynamics of the cart-pendulum system. In addition, italso includes a function for collecting data points for plotting a control surface, and a fuzzy logic engine that you can utilise to implement a complete fuzzy controller. A tutorial on how to use the engine is provided in the lecture slides (Lecture - Fuzzy Logic Engine.pptx – we discussed this in the lectures).
Details of the requirements:
Part 1: Fuzzy System Design
1. Use the following inputs (combine the inputs together, as suggested in Yamakawa’s paper):
• Combined inputs (Yamakawa)
X = (A * theta) + (B * theta_dot)
Y = (C*x) + (D * x_dot)
• Definition of inputs:
x – position of the cart
x_dot – horizontal velocity of the cart
theta – angle of the pole with respect to the vertical
theta_dot – angular velocity of the pole
A, B, C and D are positive constants; they are empirically defined
2. Use the fuzzy control rules defined by Yamakawa.
• Reference: Takeshi Yamakawa’s paper, titled “A Fuzzy Inference Engine in Nonlinear Analog Mode and Its Application to a Fuzzy Logic Control”. Refer to page 517 of his paper to see what inputs were used in his design. This research paper is available for download in our Stream website.
• Yamakawa defined 13 rules to solves the control problem. Optionally, you may extend Yamakawa’s rules to 25 rules.
3. Define the rule outputs associated with each of the fuzzy rules (e.g. NL = -100, PL = 100, etc.). Note that we are implementing a Zero-Order Sugeno Fuzzy Inference System, and so the rule outputs are constants.
4. Define the fuzzy sets corresponding to the linguistic terms in your fuzzy rules.
• The fuzzy sets need to be defined according to the range of possible values for the input variables.
• Example: Input range of input variables:
o X: [-4.0 – 4.0]
o Y: [-4.0 – 4.0]
5. Implement the fuzzy sets as membership functions in your program. You may use any of the membership functions we discussed in class. (The fuzzy engine contains theimplementation of trapezoidal membership functions, if you want to use it.)
6. Define the defuzzification method for your system. (The fuzzy engine contains a centroid defuzzification method.)
7. Incorporate your fuzzy controller into the start-up program provided. (Tips on where to insert codes are provided in the start-up codes)
8. Note that in the start-up codes, there are blocks of statements that should not be modified as they are part of the implementation of the dynamics of the system. There are comments in the codes that identify these blocks of codes.
9. It is up to you to write and add any functions, classes or data structures that you may require to complete the system.
10. Your simulation system should demonstrate that the fuzzy controller is able to balance the inverted pendulum, given an initial pole angle and position.
Part 2: System Calibration
11. Calibrate your fuzzy controller by modifying the rules, shape of membership functions, etc. until it is able to balance the pendulum without exceeding the boundaries of the platform. Aim for a control solution that can balance the pendulum in a smooth fashion and can bring the cart-pole system at the centre of the platform at zero-degree angle with respect to the vertical axis.
