2. a) In this question you have to determine the closed loop transfer function and then use the characteristic equation to determine the closed loop poles (roots) of the system. Prove that the closed loop transfer function is given by:

C(s)/R(s) = G(s)/(I + G(s)H(s))

b) What is the significance of the characteristic equation? i.e. 1 + G(s)H(s) = 0.

c) A control system has a feed forwad loop of G(s) = (K_p)/(s^2 + 10s) and a feed back Loop of H(s) = 1. Determine the roots fo the characteristic equation when K_p = 16, 25 and 50 and draw the position of the roots on the s plane. For each root pattern sketch its associated time domain response to a step input. If possible, us SIMULINK to verify which gain gives the over-damped, critically damped and under-damped response.

d) Briefly explain the advantages and disadvantages associated with closed loop control systems. Cite an example of an open and closed loop control system.

3. a) In this question you are being asked to select values for K_p and K_t to satisfy some design criteria. Prior to this you much explain the following terms using sketches to support your answers:
- Damping ration
- maximum overshoot
- peak time
- rise time
- settling time

b) For the system show in the figure attached, determine its closed loop transfer function and subsequently explain what the inner loop with the feedback gain L, does.

c) When J = I Kg-m^2 and B = 1 N/ms, determine the values of feed forward gain K_p and feedback gain K_t to satisfy the following design requirements.
Maximum overshoot M_p = 0.2
Peak time t_p = 1 sec

d) What will be the system's 2% settling time where,
The maximum overshoot is given by M_p = e^-x and x = (damping ratio * pi)/sqrt(1 - damping ratio^2)
The peak time is given by t_p = pie/[(w_n)sqrt(1-damping ratio^2)]
The 2% settling time is given by t_p = 4/(damping ration * w_n)
Standard form of the second order system C(s)/R(s) = (w_n)^2 / [(s^2) + 2(damping ration * w_n * s) + (w_n)^2]

This solution includes a graph of (1) Root positions in s plane, (2) step responses, and (3) Illustration of performance indexes. The simulink file is attached as Q37030.mdl and the related matlab file is Q370302c.m, which can be run directly. The discussion is 744 words. All calculations are formatted in the attached Word document.

A system described in the attachment is under feedback control of the form u = Kx + r where r is the reference input.
(i) Show that (A,C) is observable.
(ii) Compute a K of the form {see attachment} so that (A - BK, C) is unobservable. (I.e., the closedloop system is unobservable)
(iii) Find the transfer function of the open

Question 1:
a) Construct the root-locus for the K > 0 for the transfer function
GH = K / [s(s+1)(s^2+7s+12)
b) If the design value for the gain is K = 6, calculate the gain margin.
c) Determine the value of the gain factor K for which the system with the above open loop transfer has closedloop poles with a damping ra

Consider a unity-feedback control system with the closedloop transfer function:
C(s)/R(s) = Ks+b/s^2+as+b
Determine the open loop transfer function G(s).
Show that the steady-state error in the unit-ramp response is given by:
e_ss = 1/K_v = a-K/b
See attached file for equations in equation editor. Please use Micros

Derive an expression for the closedloop transfer function:
h(s) = Y(s)
----
R(s) for a gain k = 10.
Work i have done so far:
Ea(s) = R(s) - B(s) = R(s) - H(s)Y(s)
Because the output is related to the actuating signal by G(s), we have
Y(s) = G(s) Ea(s)
Therefore:
Y(s) = G(

1. a. Draw the Bode diagram for the attached system.
b. Is the system stable?
2. Robots can be used in manufacturing and assembly operations where accurate, fast and versatile manipulation is required. The open-loop transfer function of a direct-drive arm may be approximated by:
G H (s) = K(s+2) / s(s+5)

Consider a block diagram describing a system under proportional-integral control (as show in figure in attachment):
Find the constraints and determine the range (using the Routh-Herwitz criterion) of Kp and Ki. Also, find the closedloop system transfer function assuming the controller gains are set to a specific value.
(Ple

Problem 1:
Consider the differential equation: d2y/dt2 + (3)dy/dt +2y = u
where y(0) = dy(0)/dt = 0 and u(t) is a unit step.
1. Determine the solution y(t) analytically.
I have the following(using Laplace transform):
[s2Y(s) - sy(0)] + 3[sY(s) - y(0)] + 2Y(s) = 1/(s)
s2Y(s) + 3sY(s) + 2Y(s) = 1/(s)
Y(s)[s2 + 3s

1) G(s) =
Use Matlab and obtain Nyquist Plot
2) G(s) =
Kv1=0.325
Kv2=0.45
Note: K does not equal Kv
Determine the phase margin, gain margin, and closed-loop bandwidth for each case. Determine these both graphically, from the appropriate bode plots, and using the Matlab functions margin and bandwidth

Consider the paper machine control in the attached file. Plot the bandwidth of the closed-loop system as K varies in the interval 1 ≤ K ≤ 50. What gain is required to provide a bandwidth of 10 rad/sec?
See attached file for full problem description.

Consider the block diagram attached, describing a process under Proportional-Integral-Derivative control.
1) Is the system open loop stable? Justify your answer
2) Let Ki = 10. Use the Routh-Hurwitz criterion to find the range of Kd, and Kp in terms of Kd, so that closedloop stability is achieved.
3) Suppose that Ki =