Solutions Manual to Accompany Homework For Linear Control System Engineering Liu Sijiu HIT 651 Copyright 2006 P1-2 Redesign the turbine speed control system discussed in Sample Problem 1.1, but replace the fly-ball governor with the tachometer shown in Fig.1.2. A tachometer consists basically of a small DC motor operated in reverse as a generator, the shaft being rotated continuously, producing a DC voltage proportional to shaft speed. Solution 2 P1-4. Shown in Fig.P1.4 is a water-level control system comprising a tank, inlet pipe, slide valve, and float. Details of the operation of the flow valve are also shown in the figure. Draw a block diagram of the feedback control system and identify the main elements, writing down the mathematical transfer functions where appropriate. Solution P1-5 A method of producing a displacement proportional to an input displacement but with a much larger output force is shown in Fig.P1.5. The input displacement x causes the movement of the spool valve to produce a differential flow to the hydraulic actuator. Draw a block diagram of the system, label the principal parts of the control loop, and identify as many of the transfer functions as possible. State any assumptions made in the analysis. Solution 3 P2-1 A system unknown transfer function is shown in Fig.P2.1. If a unit impulse applied at the input produces at the output a signal described by the time function , determine the unknown transfer function. t 3 e2) t ( c = Solution t 3 e2) t ( c = = 3s 2 R C + = P2-2. Find the solution of the differential equation z3 dt dz x8 dt dx dt xd 2 2 +=+ When and all other initial conditions are zero. t2 e) t (z = Solution 10 1 4 31 ) 2 1 s ( 11s 2s 1 C 8ss Cs 2s 1 8ss 3s 2s 1 ) s (Z 2 1 2 2 2 + + = + + + = + + + = = t 2 31 sin 31 23 t 2 31 cose1 . 0e1 . 0) t (y t 2 1 t2 P2-3 For the system shown in Fig.P2.3., determine the relationship between voltage and current, express this relationship in the form of a transfer function and determine the current as a function of time when the voltage is a step change from zero to 10V. Solution 1RCsLCs Cs Cs/1sLR 1 2 + = + , 632326 6 2 10s10s 10 1s10s10 1010 1RCsLCs Cs s 10 ) s (Y + = + = + = ) t866sin(e01154. 0) t1000 2 3 sin(e 3 2 10) t (y t500t5002 = P2-8 For the system shown in Fig2.8 determine the closed loop transfer function C/R. Solution HG1 GG G G 1 HG1 G R C 1 21 1 2 1 1 + + = + + = P2-9 For the single input system shown in Fig2.9, find the transfer function of output to input C/R. Solution 232 21 HGG1 GG + = 121232 321 HGGHGG1 GGG + 3321121232 321 HGGGHGGHGG1 GGG R C + = 4 P3.2 Determine the output of the open-loop system G(s)=a/(1+sT) to the input r(t)=t. Sketch both input and output as function of time, and determine the steady-state error between the input and output. Compare the result with that given by Fig.3.7. Solution: )sT1 (s a )sT1 ( a ) s (R) s ( c 2 + = + =)TeTt (a) t ( c T/ t += )TeTt (atcre T/ t += As , t = = 1afor 1aforT aTattess P3.5 An open-loop first-order system is characterized b the transfer function s1 1 )S(G + =, where the time constant is. Calculate the steady-state error when the system input is r(t)=1+6t. Confirm the result by using the final-value theorem. s5= Solution: By the superposition theorem, the system output c(t) could be considered as a sum of a step response and a ramp response. That is 5/ t5/ t5/ t e530)e1 ()e5t (6) 1t6(cr) t ( e =+=, yields 30)( e= By the final value theorem for Laplaces transform, ) 1s5(s 5) s6( 1s5 1 1) s 1 s 6 () s (G1) s (R) s (C) s (R) s (E 2 + + = + += 30 ) 1s5(s 5) s6( sLim)S(EsLim) t ( eLim s0st = + + = P3.7 One definition of the bandwidth of a system is the frequency range over which the amplitude of the output signal is greater than 70% of the input signal amplitude when a system is subjected to a harmonic input. Find a relationship between the bandwidth and time constant of a first-order system. What is the phase angle at the bandwidth frequency? Solution: 2 2 707. 0 1 1 s1 1 22 b = + = + or 21 22 b =+ i.e. , phase lag is at the bandwidth frequency. =/1 b = 45) 1 (tan)(tan)j (G 11 45 P3.8 Figure P3.8 shows the experimentally obtained voltage output of an unknown system subjected to a step input of +10V. Determine the transfer function of the system and locate its pole on the complex plane. Solution: System appears to be first order. So suppose the transfer function is as follow: s1 K R C ) s (G + =, then )e1 (K10) t ( c / t = From final value theory V5 . 2K10 1s K s 10 sLim) s (CsLim) t ( cLim 0s0st = + = , hence K=0.25 Now consider time constant， = ) t ( c K10 1 1ln 1 t =1.4983 Matlab: K=0.25, c=1.22, 1.84, 2.16, 2.33, 2.41, t=1:5, tao=t./(-log(ones(1,5)-C/10/K)*ones(5,1)/5 5 P4.1P4.1 Figure P4.1 shows a