Construct the Root-Locus of the closed-loop control system with the following open-loop transfer function, applying the 7 rules for Root-Locus definition: GOL (s) = K(s+1) 2

Consider the control system shown in Figure 1. Copy it into your answer book, labelling the actual output, control signal, desired output, plant, proportional action and derivative action. Also write down the control algorithm. Q1 (a) + s+1 kas I+ s+ 5 kp Figure 1 Determine the open-loop transfer function for the system shown in Figure 1 and hence obtain an expression for the closed-loop transfer function. (b) Find the state-space representation in phase-variable form for the above closed- loop system. Assume kp = 2 and ka = 1. (c) (d) Draw a signal flow diagram for the above system, assuming phase-variable form.

2 parts (A-B) SOLVE CAREFULLY!! Please Write Clearly and Box the final Answer(s) label them

Q2 Figure Q2a shows the block diagram of a control system. Using the block diagram reduction method, determine the closed loop transfer function of the system shown in Figure Q2a. Show all intermediate steps in the reduction process including block diagrams and working. (а) The transfer functions of the blocks in Q2a are as follows: G1(s) = ! G2(s) = s G3(s) = s² H1(s) = s H2(s) = 2s R(s), C(s) GI(s) G2(s) G3(s) H1(s) H2(s) Figure Q2a Continued overleaf Page 3 of 12 (b) Determine the range of K for which the system shown in figure Q2b is stable. R(s) E(s) C(s) 1 s2 + 10s + 10 Figure Q2b

3. The block diagram given here shows a closed-loop control of a third order plant. Comment on the stability of (i) the open-loop transfer function and (ii) closed-loop transfer function using Routh-Hurwitz stability criterion. What should be the appropriate value(s) of K to have stability in closed-loop control? R(s) - K s+1 s3 + s² + 2s+8 C(s)

Simplify the block diagram shown in Figure 1 and obtain the closed-loop transfer function ?(?)/?(?).

3. A robot force control system with unity feedback has a loop transfer function K(s +2.5) (s² +2s+2)(s² + 4s+4) KL(s) = KG(s) =- Find break-in and breakaway points (if any). b. Find the jw-axis crossing and the gain, K, at the crossing. c. Determine the range of K stability.

Please explain each step clearly and include a proper image of what the block diagram and plot must look like. thank you

The previous solutions are posted on screenshot. please refer to them for gain value.  solve :  An integrator is added to the PD controller, leading to a PID controller C(s) = kp +ki/s + kds. Keep the PD controllers gains kp and kd from above, and plot the closed-loop poles as the gain ki varies from 0 ! 1 (Hint: The closed-loop poles satisfy 1 + P (s)C(s) = 0, translate this to Evan’s form 1 + kiG(s) = 0). Determine gains such that a load disturbance is rejected in about 10 time units, while keeping the amount of overshoot (from r to y) under 20%. Why can ki not be too large?

Q2 A PID controller is designed to control the robot leg, and the control system is shown in Figure 2 where K is a positive gain. (a) (b) R(S) Find the closed-loop transfer function (s)/R (s). Use the Routh-Hurwitz stability criterion to determine the range of the parameter K such that the closed-loop system is stable. K 1+-+s S PID Controller 2 s² + 3s +2 Robot Leg dynamics Figure 2 e(s)