Inverted Pendulum

https://github.com/jpaine126/InvertedPendulumControl_Demo

The model used for this project is taken from the "Controls Tutorials for Matlab and Simulink" course on inverted pendulums, found here. All images shown are also taken from this site.

Overview

This section will give a brief overview about the Inverted Pendulum Dynamic System, and will discuss the different control actions and options in the simulation.

The Inverted Pendulum

The inverted pendulum is a classic example of an unstable dynamic system. The goal of this project is to simulate this system, and to design and implement a control scheme that balances the pendulum. More info on the dynamics can be found here.

Running the Simulation

Run the simulation by running python -m Inverted_Pendulum_Control_Demo from the directory root. Listed in the next few sections are the different parameters that can be changed in the simulation. After the simulation is done running, the states and the control force/error are plotted.

Physical Paramters

M_Cart = 0.5 % kg M_Arm = 0.2 % kg length = 0.3 % m b = 0.1 % N/m/s g = 9.8 % m/s^2

Controller Parameters

PID Controller

The PID Controller has three gains for the proportional, integral, and derivative of the error signal.

K_P = 10 % Proportional Gain K_I = 1 % Integral Gain K_D = 1 % Derivative Gain

On line 124 there are a couple other parameters that can be changed pertaining to the antiwind up on the integrator term (antiwindupmode, control_limit). This shouldn't need to be changed at all, but you can experiment with making the control action weaker with the antiwindup turned off, to see why it is necessary in the first place.

Linear Quadratic Regulator

The LQR has three parameters that can be tuned to change the controller response. q11 and q33 represent the relative cost of the associated state in the system, and r represents the cost of the control force, meaning that a higher value causes the controller to react more strongly to that state. Individual gains can be manually set to zero after calculation to see the effect of eliminating the cost of that state.

``` q11 = 5000 % Cost on position error q33 = 100 % Cost on angle error

r = 1 % Cost on control force ```

On line 138, there are two parameters for setting a maximum control force for the LQR (max_input, max_input_on). By default it is turned off. This may need to be turned on when using a lot of noise to ensure a reasonable response.

Kalman Filter

The Kalman Filter has two matrices used in tuning, each associated with noise in the simulation. R is used for the Measurement Noise, and Q is used for the Process Noise. Q is calculated using a basic model with the equation:

``` Q = [[(dt^4) / 4, (dt^3) / 2], [(dt^3) / 2, (dt^2)]]

``` where dt is the timestep for the estimator. This matrix can be changed to also represent the expected noise from the sensors, where each sensors noise value would be on the diagonal.

The process noise is set to an arbitrary small value, since the model used in the filter is exact to the one used in the simulation. Process noise can represent different things, including unmodeled dynamics of the real system, unknown excitation of the system, etc.

```

R - Measurement Noise

kalmannoise = 0.001 Rkalman = np.array([[(kalmannoise**4)/4, (kalmannoise3)/2], [(kalman_noise3)/2, kalman_noise**2]])

Q - Process Noise

Q_kalman = np.array([[.01, 0, 0, 0], [0, .01, 0, 0], [0, 0, .01, 0], [0, 0, 0, .01]]) ```

Initial Conditions

x = 0 % Initial Cart Position - m x_dot = 0 % Initial Cart Velocity - m/s theta = 0 % Initial Pendulum Angle - rad theta_dot = 2 % Initial Pendulum Angular V - rad/s

Time and Simulation Parameters

The simulation has three different switches to select. Two different controllers can be chosen: PID and LQR. Niose in the system can be turned on or off, and a Kalman Filter can be used in place of full state feedback. Any combination of these options can be used to run the simulation.

The variable noise_val is the noise used in the model.

The dt_control should be chosen so that dt_plant is a multiple of dt_control.

```

Selected Control Scheme

control_scheme = 2

1 - PID

2 - LQR

Noise

measure_noise = 1

0 - No Noise

1 - Noise

noise_val = 0.002

Observers

observer = 1

0 - None

1 - Kalman Filter

...

tfinal = 10 % Length of Simulation - s dtplant = 0.001 % Size of Time Step for Plant Dynamics - s dt_control = 0.01 % Size of Time Step for Controller Update - s ```

Planned Features

• [x] Multiple Control Algorithms, including Linear Quadratic Regulator.
• [x] Adding noise and multiple types of observers to the system.
• [x] Changing the plant simulation to use the full non-linear equations of motion.
• [x] A graphical interface in panel that will allow the user to select the control algorithm from a drop down menu, and will have text input for changing gains, parameters, sample time, and initial conditions.