Question

Question 4 (a) A feedback control system with a proportional controller is shown in Figure Q4 (a). (i) Sketch the root locus of the system, (ii) Design the proportional controller (choose the value of K) such that the damping ratio does not exceed 0.5 and the time constant is less than 1 second. [All necessary steps of root locus construction and controller design must be shown). C(s) R(S) + s(s+4)(s + 10) Figure Q4 (a). A feedback control system [11 + 4 marks] (b) Consider the system described by the differential equation, Y + 3y + 2y = u (i) Derive a state space representation of the system, (ii) Construct its state diagram. [6 +4 marks]

Answer 1

to givom system, open loop Transfer function is: G63) = S(574) (5+10). RO: No. of branches of Root Locus = No. of open loop poles (P) = 3, towards 2utas o . - P-2-3-0 = 3 No. of branches . Re: Real axis terminating . Root low: R,L N.R.L . R.L -10 -4 ° R®: Angle of Asymptores: No. of Asymptotes - P-2-3 Angle of Asymptores love (2971) 1809 - 1909, 2X180°, Sk180' P-Z 3 3 where q=0, 1, 2, ... P-2-1 s0,2 = 60°, 1200, +80°. 300° Centroido < real part of I - real rea pe part of 2 edes - zeros - 14-0 - -4.66 P-2 3 RA: R☺: Broak away point: I+ K → 35 +235 + 40 = 0, Yoots Suza -28 Vi28) 4x3x40 6 s(s++)(5+10) 5(5+4)('5+10)+ K = 0. S(57145+ 40)+K 20. s 1434+ 405 +K = 0. Ka - [8 +14.547 405) 6 = -23 17.435 = -1,760, -7.572

by substituting above yoots in eyno, K is tue for s=-1,760 @ i it is a valid B,A point. Ke-16-1.760+ 14(-1.76) + 401-1,76)) : -1-5.451 + 43.36-70.4] - 32,491 = +ve RO: axis: Intersection Of Root locus with w C.E: 53+14.52 + 403 +K=0, by constructing Roult Array: 5214 K 560-K 20 19 560= Kmor 560.K Auxillary Eqr: 145²7 Kon=0. 145 +560 20 Sa+16.324 -560 = -40, Ich 5 <0,5 Time constant (1) - Śwn < Iseci Let {=0.530 = costs & O= cos" (0.5) = 60° Since % 1/3 <wn of was wizás > -5, 3-1,123 let 3-0,5 →wn = 1,123 = 2.246 ka product of length of vectors from all potes to point "sd' . prod, of leaghts of vectors from all zeros to point si 2.5X 3.5% = 78.65 79

domsontho 1 A 00 13 6. An COM

Let y = x X = tu - 2x2 - 3X3 hot n = x 13.1 po 1 0141 - c ][3] [!] - state Equation, [ 23 ] [° Output Equatim: y = x Y=loo 1 1 state diagram: Cimstructed based on abou equations 2