Lecture 24

E12 Linear Physical Systems Analysis

Author

Prof. Emad Masroor

Published

April 16, 2026

Plan for week 14 (fill out poll on EdStem!)

  • Comprehensive Final Assigment in class on Thursday to synthesize everything we have learned

  • No HW will be assigned on week 14

  • Tuesday:

    • Instructor analyzes an electrical system on board

  • Thursday:

    • You will analyze a mechanical system in class and turn in your work, including some/all of:
  • State variables
  • Differential Equation
  • Laplace Transform
  • Block Diagram
  • Transfer Function
  • Free Response
  • Step Response
  • Impulse Response
  • Frequency Response
  • Bode Plots

Developing state-space equations

We are interested in the outputs: \(v_1\), \(i_3\) and \(i_2\).

A circuit with three outputs and one input

Note that there is a typo here. There should be no ‘dot’ on y in the second equation.

State-Space Models in MATLAB

We use the term ‘state space equations’ to refer to the set

\[\dot{\boldsymbol{x}} = \boldsymbol{A} \boldsymbol{x} + \boldsymbol{B} \boldsymbol{u} \tag{1}\]

\[{\boldsymbol{y}} = \boldsymbol{C} \boldsymbol{x} + \boldsymbol{D} \boldsymbol{u} \tag{2}\]

MATLAB has a built-in function that allows us to construct a sytem in state space form

sys = ss(A,B,C,D)

where the matrices A, B, C and D are as defined in Equation 3 and Equation 4.

Functions Available in MATLAB for state-space models

Given a ‘state-space object’ created using system1 = ss(A,B,C,D)

  1. tf(system1) – obtain the transfer functions for this system
  2. step(system1) – graph the step response
  3. impulse(system1) – graph the impulse response
  4. lsim(system1,u,t) – graph the response to an arbitrary input \(u(t)\) given as vectors u and t
  5. bodeplot(system1 – generate a Bode plot in deciBels (see Lab 5)

Multiple inputs, Multiple outputs

\[\dot{\boldsymbol{x}} = \boldsymbol{A} \boldsymbol{x} + \boldsymbol{B} \boldsymbol{u} \tag{3}\]

\[{\boldsymbol{y}} = \boldsymbol{C} \boldsymbol{x} + \boldsymbol{D} \boldsymbol{u} \tag{4}\]

If there are two inputs and two outputs, determine the size/shape of A, B, C and D. Also determine how many transfer functions are needed to describe this system. Assume there are four state variables in this system.

\[ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & -12/5 & 4/5 & 8/5 \\ 0 & 0 & 0 & 1 \\ 4/3 & 8/3 & -4/3 & -8/3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 1/3 & 0 \end{bmatrix} \]

\[ C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}, \quad D = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

Example

Use the state-space model to determine the effect on the quantity \((x_2-x_1)\) of pushing \(m_1\) with suddenly to the right, when

  1. \(m=1\), \(k=3\), and \(b = 5\)
  2. \(m=1\), \(k=3\), and \(b=0.2\)

Procedure:

  1. Determine state variables
  2. Write the state-variable equation \(\dot{\boldsymbol{x}} = \boldsymbol{A} \boldsymbol{x} + \boldsymbol{B} \boldsymbol{u}\)
  3. Write the output equation \({\boldsymbol{y}} = \boldsymbol{C} \boldsymbol{x} + \boldsymbol{D} \boldsymbol{u}\)
  4. Gather matrices A, B, C and D and use ss to make state-space model
  5. Use impulse function.

Example