Example: State Space Solver
Use the solver statespace to solve a state space representation of a system of first-order ordinary differential equations (ODEs).
Specifically, find the solution to the case of unforced harmonic oscillator in which the right-hand side of the harmonic oscillation equation is 0:
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There are four cases for the solution - overdamped, critically damped, underdamped and full-state feedback.
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Overdamped Solution
1. Write the mathematical equation for the overdamped solution:
2. Define the initial conditions, the mass of the object, the damping constant, the start and end of the integration interval, and the number of points:
3. Set the natural, or resonant, frequency of the system.
4. Verify that the condition of overdampness exists:
5. Write the ODE in matrix form:
6. Call the statespace function:
7. Plot the solution:
Critically Damped Solution
1. Set the natural, or resonant, frequency of the system.
2. Verify that the condition of critical dampness exists:
3. Write the ODE in matrix form:
4. Call the statespace function:
5. Plot the solution:
Underdamped Solution
1. Set the natural, or resonant, frequency of the system.
2. Verify that the condition of underdampness exists
3. Write the ODE in matrix form:
4. Call the statespace function:
5. Plot the solution:
Full State Feedback Solution
1. Set initial conditions.
2. Write the ODE in matrix form.
3. Define the additional functions.
4. Call the statespace function.
5. Plot the solution.
