PART B (Analytical and numerical methods)
Remark: The remaining questions will be based on the linearised system given in Equation Eq.(6) and Equation Eq.(7).
For the remaining questions, replace the linear variable x_(1)(t) with x_(1)(t), x_(2)(t) in Eq.(4) and Eq.(5) with x_(2)(t) and f(t) with f(t) (i.e. ignoring the linearisation point). In other words, the remaining questions are based on the result of the linearisation process, which is given by:
(d^(2)x_(1)(t))/(dt^(2))+(dx_(1)(t))/(dt)+2(1+2x_(1)(t))-(dx_(2)(t))/(dt)=0
(d^(2)x_(2)(t))/(dt^(2))+(dx_(2)(t))/(dt)-(dx_(1)(t))/(dt)=f(t)
Question 4 (6 marks)
Apply Laplace transform on the linearised system in equations Eq.(6)-Eq.(7). Assume zero initial conditions, i.e. x_(1)(0)=0, x_(1)^(˙)(0)=0, x_(2)(0)=0, x_(2)^(˙)(0)=0.
Question 5 (7 marks)
Using the results from Question (4),
a) Obtain the transfer function for the linearised system (i.e. equations Eq.(6) and Eq.(7)). Note that the input of the system is f(t) and the output is x_(2)(t). (5 marks)
b) What is the order of the system? Justify your answer. (2 marks)
Question 6 (12 marks)
Using the transfer function from Question (5)
a) Obtain and clearly list all the poles and zeros of the linearised system. (Remark: there should be two zeros and four poles. Two of the poles are at s=-0.3522+-1.7214i and one at s=0. You need to find the other one real pole. You can use calculator to find these values.) (Remark 2: the location of the remaining pole is between -2 and 0). (4 marks)
b) Determine if the system is stable. Justify your answer. (3 marks)
c) Draw the "s-plane". (5 marks)
Question 7 (3 marks)
Use Final Value Theorem to predict the value of x_(2)(t) (if available) when the input is a unit impulse. Justify your answer.
