deriving a transfer fuctnion

Derive the closed loop transfer function for the given controller, and then use a second order system to approximate the 3rd-order closed loop transfer function, calculate the transient performance (rise time, time to peak, %PO, setting time), and compare with the actual performance identified from the Matlab plot.

Answer 1

[Sam Chak]

Hi @Plato

This is a homework question. However, you can use the feedback() function to compute the closed-loop transfer function if the controller K is given.

s   = tf('s');
G   = ((s + 1)*(s + 2))/((s^2 + s + 2)*(s + 3));
G   = zpk(G)

G =
 
      (s+2) (s+1)
  -------------------
  (s+3) (s^2 + s + 2)
 
Continuous-time zero/pole/gain model.

help feedback

 feedback - Feedback connection of multiple models
    This MATLAB function returns a model object sys for the negative
    feedback interconnection of model objects sys1,sys2.

    Syntax
      sys = feedback(sys1,sys2)
      sys = feedback(sys1,sys2,feedin,feedout)
      sys = feedback(sys1,sys2,'name')

      sys = feedback(___,sign)

    Input Arguments
      sys1,sys2 - Systems to connect in a feedback loop
        dynamic system models
      feedin - Subset of inputs to be used
        vector
      feedout - Subset of outputs to be used
        vector
      sign - Type of feedback
        -1 (default) | +1

    Output Arguments
      sys - Closed-loop system
        dynamic system model

    Examples
      openExample('control/InvertedPendulumAndControllerInANegativeFeedbackLoopExample')
      openExample('control/NegativeFeedbackLoopWithPlantAndControllerExample')
      openExample('control/PositiveFeedbackLoopWithPlantAndControllerExample')
      openExample('control/NegativeFeedbackLoopWithMIMOSystemsExample')
      openExample('control/FeedbackLoopBasedOnIONamesExample')
      openExample('control/ConnectSpecificInputsAndOutputsInAFeedbackLoopExample')

    See also append, connect, series, lft, sumblk, parallel

    Introduced in Control System Toolbox before R2006a
    Documentation for feedback
       doc feedback

    Other uses of feedback

       iddata/feedback    InputOutputModel/feedback