By the end of this section, you should be able to

1. Understand benefits/drawbacks of state-space (SS) models versus transfer functions (TF)

   • Know when each may be appropriate to use

2. Create SS models for linear mechanical systems and identify state vector and matrices that describe system

   • Do this for SISO, SIMO, MISO and MIMO systems

3. Convert SS model to TF and vice-versa in MATLAB 

4. Numerically solve SS models using MATLAB (ode45)

This section introduces state-space modeling, a compact way to represent linear dynamic systems in Measurement and Dynamic Response using a state vector and the A, B, C, and D matrices. State-space form is especially useful when your model has multiple inputs or outputs, coupled equations, or when you want a representation that connects naturally to numerical simulation and control workflows. In the lessons below you will compare the benefits and drawbacks of state-space models versus transfer functions, and you will practice deciding which representation fits a given problem. You will build state-space models for linear mechanical systems by starting from free body diagrams, writing equations of motion in the time domain, and then selecting state and input variables so each equation contains only first derivatives. From there you will isolate derivatives, group coefficients, add mapping equations that relate derivatives of state variables, and assemble the matrix form needed for SISO, SIMO, MISO, and MIMO systems. You will also define outputs through the C matrix and any direct feedthrough through the D matrix, including cases where sensor definitions create relative measurements. Finally, you will use MATLAB to convert between state-space and transfer function representations and to numerically solve ordinary differential equation models using ode45 and the Runge-Kutta idea behind it. Use the table of contents below to jump to examples, workflows, and MATLAB resources.