EECE 460 Homework, Spring 2008

B31. RG) I+ (G+ GaN Gs- Gu) B-3-5. When D(s) is zero, the closed-loop transfer function Cp(s)/R(s) is Ce) _ _ Ge ©) GL) RID It Gels) Gp @) When R(s) = 0, the closed-loop transfer function Cp(s)/D(s) is Cr) _ / DO —- 1+ Gel) Fe) When both the reference input and disturbance input are present, the output C(s) is the sum of Cp(s) and Cp(s). Hence / C= Gb)" Col = aseay [ 5269 Gol? RE) + be] B-3-6. When only the reference input R(s) is present, the output Cp(s) is given by Cee) __ 6) &O) R@) ——si«E GS) G2) For the reference.input R(s), the desired output is R(s) for the unity-feedback system such as the present system. Thus, the error Ep(s) is the difference between R(s) and the actual output Cp(s). The error Ep(s) is given by Cels) Eels) = RE) — Cel) = rey | /- EE | _ _ G,6)G26) ] = / Res ~ Ren] / 7#G,6)G2O 1? Gil) G2) @ Assuming the system to be stable, the steady-state error Eggr(t) can be given by sRe = . = her s E, =k, — Css (t) Lin Cet) = kim SEY bin 746,640 ‘ When only the disturbance input D(s) is present, the output Cp(s) is c= Cob) _ G62) D6) 1+ G,6)GE) Since the desired. output to the disturbance input D(s) is zero, the errcr can be given by £,6)=0-0)6) =-GH) Hence G25? 1+G,0 G2) For the stable system, the steady-state error @gsp(t) is given by : , -3G26) D&) e @) = hewn e, = = bo 75 F206) C6) wp) = Lox Cott) = Lin $ E56) = hits Tey The steady-state error when both the reference input R({s) and disturbance D(s) are present is the sum of @ggp(t) and eggp(t) and is given by £p6)=—Cy 5) = — Do Csslt) = Esser Zt) + Ess Ct) _ l s PG) _ | se '*G,6) G49 4+ G6 )G 25) = Lo leas [ 28)- 6,0 De] | f Define Aragr+2gau ae[2 ci), e[E], eb 2 Mw The transfer function G(s) of the system is given by Ge) = GUI-AN"B = [) 2] i I | “3 St/ > / Sti -f |lat lV a] (S#5-)(s+/) +3 ' a | ! [V2] [3 - 25459 S24 65th S34+3/ S*+6steP A MATLAB solution to this problem is given below. [uum dea] = se248A.B,C.D) 0 12 59 den = 16 8