I'm in need of help finding a third-order or higher system in which I can derive a transfer function. We have a class project in which we need to find a real-life example of the system that equates to a 3rd order system or higher.

The problem that I have is that I do not know what a third-order system looks like in real life. I can find a lot of examples of 2nd order systems (acceleration, velocity, and displacement). But no 3rd order! Help!

Project guidelines:

Step 1: Find a system of interest to you, discuss why this system is of specific interest to you and why this is a good topic for the class project;

Step 2: Model the system in three forms: differential equations, transfer function, and state-space representation. Note if the system is nonlinear, please linearize it first;

Step 3: Find and plot system output under step input and another input of your choice. Discuss the physical implications;

Step 4: Study the system stability, discuss the physical meaning of instability in your case

  • $\begingroup$ Pneumatic linear rails fromm festo come to mind… $\endgroup$ Jan 17, 2022 at 9:14
  • $\begingroup$ 3 (connected) water tanks. each will contribute one state (water level in that tank). Or, cascade a second order system and a first order system to get a third order system; $\endgroup$
    – AJN
    Jan 17, 2022 at 11:58
  • $\begingroup$ Newton's second law always gives you two, so anything where a force acts to move an object gets you 2/3 of the way there. Use your imagination for the last one... $\endgroup$
    – Pete W
    Jan 17, 2022 at 14:15
  • $\begingroup$ One that pops in my mind is the transfer function of the voltage supplied to a DC motor and the angle of the output. Here, the torque applied to the mass is a function of the current through the circuit. However, as a DC motor is also a coil, the current is equal to the difference of the voltage. This adds (albeit a very large) pole. In mechanical terms, this means the change in angular acceleration (angular jerk) is limited by the voltage you can supply. $\endgroup$
    – Petrus1904
    Jan 18, 2022 at 10:55

1 Answer 1


If you know second-order systems you can always add a time delay to obtain the third order as all real systems have them. Often a Pade approximation is used for a time delay. For the Laplace transform {e^-st} the first order Pade approximation for time delay, t, is (1-t/(2s))/(1+t/(2s)). Time delays always cause degradation of system performance due to delay of system information. One can determine how much delay the system can tolerate. MATLAB and SIMULINK have embedded time delay and high order Pede approximations and also there are numerous examples on the internet.


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