CMPSC 200 10 D State Space Example 1

Описание: Convert a 2nd-Order ODE to State Space Form for MATLAB (ODE Solvers) | 4y'' - 2y' + 7y = 3 sin(t)

In this example, you will learn how to rewrite a second-order ordinary differential equation (ODE) as a first-order state space system so you can solve it in MATLAB using standard ODE solvers (like ODE45). This is a core workflow in system dynamics, controls, and numerical simulation.

ODE given:
4*y''(t) - 2*y'(t) + 7*y(t) = 3*sin(t)

Step 1: Choose state variables (one per order)
x1 = y
x2 = y'

Step 2: Write the state derivatives
x1' = x2
Start from 4*x2' - 2*x2 + 7*x1 = 3*sin(t)
Solve for x2':
x2' = (-7/4)*x1 + (1/2)*x2 + (3/4)*sin(t)

State space form (matrix form you can code directly):
x' = A*x + B*u(t)
where x = [x1; x2], u(t) = sin(t)
A = [0 1; -7/4 1/2]
B = [0; 3/4]
and y = x1, y' = x2

More ODE and MATLAB resources (course page):
(https://sites.google.com/view/jmm-mat...)

Related videos from this channel:
Analytical Solution to an ODE I: (   • CMPSC 200 10 A Analytical Solution to an O...  )
Analytical Solution to an ODE II: (   • CMPSC 200 10 B Analytical Solution to an O...  )
Solving ODEs Numerically: (   • CMPSC 200 10 C Solving ODEs numerically  )
State Space Example 1: (   • CMPSC 200 10 D State Space Example 1  )
State Space Example 2: (   • CMPSC 200 10 E State Space Example 2  )
Using ODE45 in MATLAB: (   • CMPSC 200 10 F Using ODE45: how to use ODE...  )
ODE45 Example I: (   • CMPSC 200 10 G ODE45 Example I: solving ou...  )
ME 357 Intro to State Space Modeling: (   • ME 357 09 A Introduction to State Space Mo...  )

#MATLAB #StateSpace #DifferentialEquations #ODE45 #SystemDynamics #NumericalMethods

Timestamps
0:00 Goal: rewrite an ODE for MATLAB solving
0:11 Present the ODE and identify variables
0:32 Define state variables x1 and x2
1:18 Differentiate to relate x1' and x2'
1:46 Rewrite the ODE using state variables
2:57 Isolate x2' and simplify coefficients
3:54 Mapping equation x1' = x2
4:19 Final state equations ready to implement
4:44 What MATLAB will output: y and y' over time