Problem 1: The control design block diagram of a mechanical system is shown below:

Where the controller and the system are given by:

(a) Determine the transfer function of the "Controller" from E(s) to U(s).

(b) Determine the transfer function of the closed loop system from R(s) to Y(s).

(c) Assume that the closed loop transfer function for the system is of the form:

(C/s³+As²+Bs+C)

Determine the values of 'A', 'B' and 'C' in terms of 'K₁', 'K₂' and 'K₃'. Determine the steady state value of the system for a unit step input E(s) in terms of A, B, and C.

Problem 2: The following figure shows a simple model of a furnace. A packing of temperature T₁ is being heated in the furnace by an electric heater supplying heat at the rate qi. The amount of heat supplied is a function of the pressure head (liquid height), h: q_i = 2(1.5-h)². The temperature inside the furnace is T₂. An insulation layer of resistance R₁ surrounds the packing and another one of resistance R₂ surrounds the furnace. The outside wall is at temperature T₃ and the ambient temperature is T_a. The thermal capacitances of the packing, the air inside the furnace, and the outside furnace wall are C₁, C₂ and C₃, respectively.

(a) Derive the dynamic equations for this system assuming that the heat is transferred by conduction only.

(b) Draw the equivalent electric circuit.

Problem 3: The differential equation for a certain second order dynamic system is given below:

y^··(t) + y^·2(t) + 6√(y(t)) = 12

(a) Determine the equilibrium/operating point of the system, y₀.

(b) Linearize the given differential equation about the equilibrium/operating point as obtained in part (a).

(c) Represent the linearized equations in state-space form by choosing Δ_y(t) as the output of the system.

Problem 4: Consider the block diagram representation of a feedback control system given below. It is your task to select values of the real parameters K and H such that the unit step response of the closed loop system (from r to y) has all of the following characteristics:

a) the final value of y(t) is 1

b) the 1% settling time for the response is less than 3 (seconds)

c) the overshoot in the response is between 10% and 20%.

If there is a possible combination of H and K, show how you arrived at these choices. If not, explain why the objectives cannot be met.

Problem 5: Find the range of the controller gains (K, K₁) so that the feedback system is stable.

Attachment:- Assignment File.rar