CHAPTER 4 MESB 374 System Modeling and Analysis Model Reduction (Electro-mechanical Systems)

How do we determine two systems are identical? Step responses are same Responses to all inputs are the same Structures and parameters are exactly same Frequency responses are the same Bode plots are the same

Definition and Motivations Simplification of system model without losing main features of system Motivations Complicated illustration of system may be distractive Catastrophe, or Tragedy Sometimes, too detailed modeling may not be necessary $ or $2 The cost of modeling Complicated model may not be suitable for controller design Cost of implementation of controller

One Example Unit-Step Response of a system with transfer function Unit-Step Response of a system with transfer function Any detectable Differences? Poles:

One Example (continued) Bode Plots Any detectable Differences? Is there any difference between their responses if input is? When can we use the simpler one?

Model Reduction of DC Motor Original Block Diagram E i (s) - Electrical System Mechanical SystemEM Coupling - Transfer Functions

Block Diagram Reduction E i (s) + - Block diagram has been simplified. Detailed structures of system have been hidden. Is this complicated model always necessary?

Example (A)Given the following specification of a DC motor and assume there is no load, find its transfer function from input voltage to motor angular speed L A = 2 mH R A = 10  K T = 0.06 Nm/A J A = 5  Kg m 2 B = 3  Nm/(rad/sec) (B) Find the poles of the transfer function. (C)Plot the Bode diagram of the transfer function

Example Phase (deg) Magnitude (dB) Q:If we are only interested in the system response up to 400 rad/sec, can we simplify our model ? How would you simplify the model ? S1S1 S2S2 S2S2 S1S1

If frequency is smaller than 400 rad/sec, the characteristics of Bode plots is similar to that of a first-order system. Hence, we can use a first-order system to approximate it. Simplify the transfer function of whole system Simplify each block and obtain TF of the resulting system E i (s) - - TF:

Model Reduction Q:Physically, what do we mean by neglecting armature inductance ? By neglecting the armature inductance, we are assuming that it takes no time for the current to reach its steady state value when there is a step change in input voltage, i.e. a sudden change in input voltage will result in a sudden change in the armature current, which in turns will result in a sudden change in the motor torque output. This is equivalent to have direct control of the motor current.