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.