control systems 2 take home assignment
Post on 16-Dec-2015
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1. By Newtons second law
2. Plant transfer function =
Take Laplace transformation of
Let
Taking Laplace transformation of the above equation
Since
Therefore
3.
Here and
: This is the closed loop transfer function
3.1
Consider . Using Rouths stability criterion
To be in stable region
Therefore . But we know . Therefore for stability .
3.2
3.3
No. According to the figure only instance is acceptable when the performance specifications are taken into considered (Steady state error < 2%). In this case the rise time is less than a second. Therefore the acceleration of the vehicle is greater than. But due to power limitations of the engine and the drivetrain the maximum acceleration attainable is. Therefore we cant achieve the desired result practically with using only the proportional control.
4. Consider the diagram in part 3. Since PI control is implemented
Here and
Given that
Therefore
Consider. Using Rouths stability criterionS2
S1
S0
For stability in the system
Since the necessary performance specifications are satisfied there is no need of implementing PID control.
5.
We know
Consider
When using modified Rouths criterion Now we get
For stability
According to the above 3 equations, for, Sampling PeriodRange for
For all sampling time periods. Therefore we cant find values for such that the system is stable using Modified Rouths criterion. Therefore to obtain stable values for we have to follow trial and error (heuristics) method.
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