Write the governing differential equations of the system in

Question

Topic: Dynamics and Controls

[1] Reduce the block diagram given in Fig. 2 and find the transfer function from X(s) to Y(s).

Bonus: Compute the transfer function from X(s) to E(s)

[2] The mast in Fig. 3 rotates around point O. Ignore the pulley's friction and mass.

(a) Write the equations of motion of the mast (Output = θ, Input=F) in terms of the parameters shown on the graph ONLY.

(b) Linearize the equation(s) around equilibrium point(s).

(c) Find the transfer function representing the mast angle (θ) as a function of the motor pulling force (F).

[3] Consider the motor used to raise the mast in Problem 3. The cable force F is tangential to the shaft of the motor. The motor bearing has viscous damping coefficient b. The schematic diagram of the motor electrical and mechanical components is shown in Fig. 4. The rotation of the motor is designated by θ_m and the current in the motor winding is designated by i_a. The applied voltage to the motor is v_a. Write the equations for the motor dynamics using the parameters shown on the Fig. 4 ONLY.

Bonus: Write the state space equations of the system.

[4] (a) Draw the line magnitude plot of the system shown in Fig. 5 for K=10 (use provided graph sheet).

(b) For which value(s) of K the system is unstable.

Bonus: Draw the magnitude plot for K=100 (1 point. No credit is given if the magnitude plot is drawn following the same steps as in (a)).

[5] For each of the root loci shown in Fig. 6, tell whether or not the sketch can be a root locus. If the sketch cannot be a root locus, explain why. List all reasons on the graph itself.

[6] Sketch the general shape of the root locus for each of the open-loop pole-zero plots shown in Fig. 7.

[7] The root locus for the system shown in Fig. 8a is shown in Fig 8b. (a) Using the root locus, find the complex poles which will lead to the closed-loop system having a natural frequency equal to ω_n= 4.0 rad/s and a damping ratio ζ = 0.36. (b) What is the value of K for your choice?

[8] What type of controller C(s) will lead to perfect tracking of unit step input for the system in Fig. 9? P or PI? Explain why.

Bonus:

What is gain margin? What is phase margin?

How do you choose P, I, or D or a combination of these to use for controlling your system?

What is Routh Hurwitz array and what is it used for?

Can you state one problem with the Routh Hurwitz method?

What is aliasing?

What is Nyquist Frequency?

State one advantage and one disadvantage of open-loop control.

State one advantage and one disadvantage of Closed-loop control.

Why do you need to linearize a system?

How do you linearize a system?

[9] A linear model of a DC motor is shown in the figure below. The definitions of the system parameters and variables are as follows:

θ_m: motor-rotor rotational displacement

J_m: motor-rotor inertia

B_m: motor viscous-damping coefficient

v_a: armature voltage

i_a: armature current

R_a: armature resistance

L_a: armature inductance

v_bemf:  back emf voltage

a) Write the governing differential equations of the system. In this linear model, you may assume that the motor's back end constant is equal to its torque constant (k_bemf = k_t).

b) Consider that there are an input v_a()t and an output θ_m (t)  in the system. Derive the transfer function from the input to the output.

[10] You may consider this a continuation of the previous problem. With a given set of parameters, a DC motor transfer function can be written as 0.2/(s² + 0.4s + 0.8).

a) Can a PD controller D(s) = k_p+ k_ds make the steady-state error to a unit step input be zero? If so, prove it. If not, find the steady-state error to a unit step input (as a function of k_p and k_d).

b) Design a PD controller D(s) = k_p + k_ds that satisfies the following closed-loop transient specifications: t_s < 4.6 s and t_r < 1.8 s. Neglect the effect of additional zeros.

c) Can a PID controller D(s)= k_p + k_is^-1 + k_ds make the steady-state error to a unit step be zero? If so, prove it. If not, find the steady-state error to a unit step input (as a function of k_p, k_i, and k_d).

[11] Fig. 3 shows a typical control system block diagram. The plant transfer function is given by

G(s) = 1/s(s+10)(s+1000)

(a) Sketch the root locus for K > 0. Calculate the approximate location of the breakaway pole by neglecting the open-loop pole at s = -1000. You do not need to calculate the jω-axis crossing point.

(b) Show that the proportional controller K cannot make the closed-loop system satisfy the following performance criteria.

• The step response has a settling time T_s ≤ 0.575 sec. (Ts = 4.6/σ for 1% criterion).
• All closed-loop poles have a damping ratio ζ ≥ 0.707.

[12] The system described by the following block diagram must be designed so that its closed-loop poles are placed within the shaded regions as shown in the diagram given below.

a) Determine the range of values the natural frequency and the damping coefficient could take for the closed loop poles to lie in the shaded region, where θ₁ = 32^o, θ₂ = 29^o.

b) For the closed loop poles to be in the shaded region, determine the range of the constant, 'k'.

c) For ω_n = 50; ζ = 0.67, what are the values of the constant, ' k₁' and 'k'?

[13] The following figure shows a simple model of an automatic egg hatcher. The egg has a temperature of T₁ and being heated in the hatcher by an electric heater supplying heat at the rate Q_i(t). Q_i(t) is supplied by a resistance R as shown in the figure. A temperature control system is used in the hatcher to control the amount of heat going into the system. The angle θ of the lever of the controller is proportional to the temperature T₂ (constant of proportionality is α). The heat extraction variable resistor is proportional to the force in the spring k (constant of proportionality is β). The temperature inside the hatcher is T₂, the walls are at temperature T₃ and the ambient temperature is T_a. The thermal capacitances of the egg, the air inside the hatcher and the hatcher walls are C_h1, C_h2 and C_h3 respectively. Assume small angles and mass less lever.

(a) Derive the equations of motion for this system assuming that the heat is transferred by conduction inside the hatcher and by convection outside it (convective heat transfer coefficients at the wall is h_c3). Use variables shown on the diagram only.

(b) Derive the state-space equations assuming temperature T₁ and T₃ as the outputs.

[14] (a) Complete the magnitude plot of the open-loop system given by:

G(s)H(s) = (100s+7000)/(s³ + 34s² + 320s + 800)

(The phase plot is already drawn for you)

(a) Determine the phase and gain margins of the system.

(b) If an input u(t)=10sin(10t+π/2) is applied to the open loop,

what will be the steady state y(t)?

Attachment:- Assignment File.rar