Inverted Pendulum

The purpose of this project is to apply controls theory to balance and stabilize an inverted pendulum. The apparatus used is a rotary inverted pendulum manufactured by Quanser. The following parameters are determined in the system:

Controllability and Observability of the system.

The gain if the system has state-feedback based on a time costraint to stabalize.

Gain margin - point where the LTI system loses its asymptotic stability.

Designing a state-feedback and output-feedback system which follows a constant and variable command.

In this project task, we focus on the command following problem for two types of systems.

1) STATE-FEEDBACK SYSTEM

In particular, assume that the state vector 𝑥(𝑡) is available for feedback. Design a state-feedback control structure of your choice in order to make 𝜃(𝑡) to follow the time-varying command 𝑐 𝑡 = cos (𝛽𝑡). Here, consider that the initial condition is 𝑥: = [0 0 0 0]^T. In your design, you must satisfy the following requirements:

a. 𝜃(𝑡) needs to converge to command 𝑐 around 𝑇 seconds, where 1.5 < 𝑇 < 3. 𝜃(𝑡) can make some acceptable level of oscillations, but not excessive (i.e., try your best to avoid oscillations as much as you can, significant oscillations are not acceptable for 𝜃(𝑡)).

b. 𝛼(𝑡) needs to converge to zero around 𝑇 seconds, where 0.5 < 𝑇 < 4.

c. Control signal 𝑢(𝑡) needs to stay between -20 and 20 volts (i.e., −20 ≤ 𝑢 𝑡 ≤ 20).

2) OUTPUT-FEEDBACK SYSTEM

Specifically, design an output-feedback control structure of your choice in order to make 𝜃(𝑡) to follow the constant command 𝑐 = 1.5 radians. Here, consider that the initial condition is 𝑥: = [0.02 0.04 − 0.02 − 0.04]^T. In your design, you must satisfy the following requirements:

a. 𝜃(𝑡) needs to converge to command 𝑐 around 𝑇 seconds, where 1.5 < 𝑇 < 3. 𝜃(𝑡) can make some acceptable level of oscillations, but not excessive (i.e., try your best to avoid oscillations as much as you can, significant oscillations are not acceptable for 𝜃(𝑡)).

b. 𝛼(𝑡) needs to converge to zero around 𝑇 seconds, where 0.5 < 𝑇 < 4.

c. Control signal 𝑢(𝑡) needs to stay between -20 and 20 volts (i.e., −20 ≤ 𝑢 𝑡 ≤ 20).