(a) Find the damping ratio and natural frequency for each second-order system of Problem 4.17.6 and show that the value of the damping ratio conforms to the type of response (underdamped, overdamped, and so on) predicted in that problem. [Section: 4.5]

Derive the equation of motion and transfer function of the mechanical system.

) Consider a system with the following block-flow diagram: a. (10 points) What manner of control would you describe this as having? b. (10 points) Derive the overall transfer function for the system. C/R. c. (10 points) Using the result from Part b, is there an offset for this system? To make this simpler, you may consider R to be a unit step change. d. (15 points) Derive the response by hand for the response to a unit-step change in R. To make comparison simpler, evaluate C(2).

A closed-loop control system is shown in Figure 4, where G(s) is the transfer function of the system,G.(s) is the transfer function of the controller, D(s) is a disturbance, R(s) and Y(s) are the input1and output, respectively. The open-loop system is unstable with a transfer function of G (s)(1/s-Y)'= where y is 6 The objectives of the controller, G.(s), are to make the closed-loop system stable and at the same time to minimize the effect of the disturbance. Ks+yConsider a proportional-integral (PI) controller with a transfer function G.(s)Answer theSfollowing questions: Determine a suitable value of K so that the closed-loop system is stable. Note: you need to justify the choice of your designed gain K. Find the steady-state error for the case whereR(s) =1and D(s) = 0.S Find the steady-state output of the system when R(s)= 0 and D(s) = -S Base on the above analysis, explain whether the objectives of the controller have been met or not.

\text { 1. A control system has the forward path transfer function } G(s)=\frac{1}{(s+1)(2 s+1)}, \text { and a } unity feedback gain, H=1. Find the steady-state error for a step input with amplitude K. (ii)Find the steady-state error for a ramp (linear) input with slope L. (iii)Discuss / describe the reasons for results obtained in (i) and (ii). If H is not 1, and if the input to the system is R(s), derive the expression for the error E(s), in terms of H and R(s). (Note: the error signal is the output of the summing junction.)(iv)

5.) The system shown in Figure P4.6 has a unit step input.Find the output response as a function of time. Assume the system is under damped. Notice that the result will be Eq. (4.28). [Section: 4.6]

Consider a multiple inputs control system described by a block diagram shown in Fig. 2. Find the complete output for the system when both inputs act simultaneously.

For each of the transfer function shown below, find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. For 1(a) alone, compute the time constant, rise time, and 2% settling time. Use the formula sheet. For full credit, do the following: • Clearly mark the poles with X and zeros with 0 Write the expression for the output in the frequency domain • Write an expression for the general form of the step response without solving for the inverse Laplace transform Compute the required performance specifications using the formula (problem 1(a) alone).

A system is described by the following differential equation: with the initial conditions x(0) = 1, x(0) = -1. Show a block diagram of the system, giving its transfer function and all pertinent inputs and outputs. (Hint: the initial conditions will show up as added inputs to an effective system with zero initial conditions.)

(a) Assuming an uncharged capacitor in the figure below, use Laplace tranform to find an expression for the voltage across the capacitor after the switch closes at t = 0. Find the time constant, rise time, and settling time for the calculated voltage. [Sections: 4.2, 4.3]