126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closed-loop system with unity feedback, (c) estimate the bandwidth of the closed-loop system, and (d) the setting time (2% criterion) of the system. Ans: (b) 7.73 % (c) 5857 rad/sec (d) t s = 1.26 sec RE4.3 The output and input of a position control system is related by: 500( s 100) Y(s) = 2 R(s). s 60s 500 (a) If r(t) is a unit step input, find the output y(t). (b) What is the final value of y(t)? Ans: (a) y(t) = 100 112.5e 10t 12.5e 50t, (b) y ss = 100 RBES 4.1 The pure time delay e st may be approximated by a transfer function as: e st ( 1 T s / 2 ) ( 1 T s / 2) for 0 < ω < 2/T. btain the Bode diagram for the actual transfer function and the approximation for T = 2 for 0 < ω < 1. Hints: ATAB command < g = zpk([ ],[ ],1, inputdelay, 2)> will produce transfer function g = e 2s. 4.2 The pneumatic actuator of a tendon-operated robotic hand can be represented by 2500 ( s 45)( s 340) (a) lot the frequency response of G( jω) and show that the magnitudes of G(jω) are 16 db and 34.8 db respectively at ω = 10 and ω = 300. Also show that the phase is 141 at ω = 500. 4.3 The dynamics of a vertical takeoff aircraft are approximately represented by the transfer function 2 8 ( s 025. ) The controller transfer function is represented by (see Fig. 4.3) G c ( s 6 ) and H(s) = s ( s 2) (a) btain the Bode plot of the loop transfer function (s) = G c (s)g(s)h(s) with K 1 = 1.5. (b) Compute the steady-state error for the closed loop system for a wind disturbance of D(s) = 1/s. (c) btain the frequency response ( jω) and the peak amplitude of the resonant peak along with the resonance frequency (d) Find out the gain and phase margins from (jω) plot (e) Estimate the damping ratio of the system from the phase margin. Ans: (b) e ss = 1(c) r = 324 db, ω r = 0.5, (d) G =, = 74.04 at ω cp = 13.07 rad/s (e) δ = 0.74 D(s) R(s) G (s) c G(s) altitude (s) H(s) Fig. 4.3
ANAYSIS F INEAR SYSTES 127 4.4 A system is described by a set of differential equations as shown below: yt &( ) = 2y(t) a 1 x(t) = 2u(t) &x (t) a 2 y(t) = 4u(t) where u(t) is an input. (a) Select a suitable set of state variables and obtain the state variable representation (b) Find the characteristic roots of the system in terms of the parameters a 1 and a 2. Ans: (b) s = 1 ± 1 aa 1 2 4.5 The simplified state variable vector representation for depth control of a submarine is given by : N 0 1 0 0 x & = 0075 1115 0. 125 x 01. u(t) 0 0. 075 265 0075. where u(t) is the deflection of the stern plane. (a) Examine the stability of the system. (b) btain the discrete-time approximation with sampling period of 0.25 sec as well as 2.5 sec. btain and compare the responses for both the sampling periods. Ans: (b) T = 0.25, x(k 1) = T = 2.5, x(k 1) = 0. 99977 0. 24655 0. 00379 00185 0. 97256 0. 02982 xk ( ) 00002 0. 01789 0. 93677 N 0. 97859 2. 18290 0. 28817 01637 0. 75681 0. 19649 xk ( ) 00130 0. 11790 0. 53281 00307 02437 0. 01792 26605 19667 0. 12105 uk ( ) uk ( ) 4.6 The forward path transfer function, with unity negative feedback, is given by : 10( s 4) s( 01. s 1)( 2s 1) (a) Find the steady state error due to ramp input. (b) Find the dominant roots of the closed loop system and estimate the settling time (2% criteria) to a step input. (c) Compute the step response and find over shoot and settling time and compare with results in part (b). 4.7 achine tools are automatically controlled as shown in Fig. 4.7. These automatic systems are called numerical machine controls. Considering one axis, the desired position of the machine tool is compared with the actual position and is used to actuate a solenoid coil and the shaft of a hydraulic actuator. The transfer function of the actuator is given by: G 1 (s) = X( s ) Y( s ) = 1 s(. 05s 1) The output voltage of the difference amplifier is E 0 (s) = A[X(s) R d (s)], where r d (t) is the desired position input. The force on the shaft is proportional to the current i so that F = K 2 i(t), where K 2 = 2.0. The force is balanced against the spring. F = Ky(t), where K is the spring constant and is numerically equal to 1.2, and R = 10 ohm, and = 0.5 henry. (a) Determine the forward path transfer function and the gain A such that the phase margin is 50. (b) For the gain A of part (a), determine the resonance peak r, resonance frequency ω r and the bandwidth of the closed-loop system. (c) Estimate the percent overshoot of the transient response for a unity step of the desired position and the settling time to within 2% of the final value. Chapter 4
128 INTRDUCTIN T CNTR ENGINEERING Ans: (a) 24. A, A = 6.95 s(. 05s 10)(. 05s 1) (b) r = 1.23 db, ω r = 1.29 rad/sec, BW = 2.31 rad/sec (c) 16.2%,t s = 4.49 sec osition feedback x r d Difference amplifier A E o i Cutting tool R Spring, K Work piece y (a) Fluid supply R (s) d A E (s) o 1 Rs I(s) K 2 K G(s) F(s) Y(s) X(s) Tool position (b) Fig. 4.7 (a) Tool position control, (b) Block diagram 4.8 A closed loop system for controlling chemical concentration is shown in Fig. 4.8. The feed is granular in nature and is of varying composition. It is desired to maintain a constant composition of the output mixture by adjusting the feed-flow valve. The transfer function of the tank and output valve is given by 4 4s 1 and that of the controller is represented as : G c K 2 s The transport of the feed along the conveyor introduces a delay time of 2 seconds. (a) Draw the Bode diagram when K 1 = 1, K 2 = 0.2, and investigate the stability of the system. (b) Also draw the Bode diagram when K 1 = 0.1 and K 2 = 0.02, and investigate the stability of the system. Ans.: (a) unstable (b) stable, G = 20.5 db = 84.7.
ANAYSIS F INEAR SYSTES 129 Feed Stirrer G (s) c Concentration set point Concentration feed back Conveyor utput mixture Fig. 4.8 Chemical concentration control 4.9 In an automatic ship-steering system the deviation of the heading of the ship from the straight course is measured by radar and is used to generate the error signal. The block diagram representation shown in Fig. 4.9. This error signal is used to control the rudder angle δ(s). The transfer function of the ship-steering system is given by: E( s ) 016. ( s 018. ) ( s 03. ) = δ() s 2 s ( s 024). ( s 03. ) where E(s) is the aplace transform of the deviation of the ship from the desired heading and δ(s) is the transfer function of deflection of the steering rudder. btain the frequency response with k 1 = 0. (a) Is this system stable with k 1 = 0.? Ans: No (b) With k 1 = 0, is it possible to stabilize this system by lowering the forward path gain of the transfer function G(s)? Ans: No (c) Repeat parts (a) when k 1 = 0.01 and k 2 = 1. Ans: yes Chapter 4 Desired heading E(s) Heading G(s) (s) k 1 ks 2 Fig. 4.9 Automatic ship steering 4.10 A typical chemical reactor control scheme is shown in Fig. 4.10. The chemical process is represented by G 3 and G 4 and disturbance by D(s). The controller and the actuator valve are represented by G 1 and G 2 respectively and the feedback sensor is represented by H(s). We will assume that G 2, G 3, and G 4 are all of the form: G i (s) = K i 1 τ i s where τ 3 = τ 4 = 5 seconds, and K 3 = K 4 = 0.2 and H(s) = 1. The valve constants are K 2 = 10 and τ 2 = 0.4, second. The close loop system is required to maintain a prescribed steady-state error. (a) With G 1, find the proportional gain such that steady state error is less than 5% of the step input. For this value of K 1, find the overshoot to a step change in the reference signal r(t). (b) Now with a I controller G 1 (1 1/s), find K 1 so as to get an overshoot less than 25% but greater than 5%. For these calculations D(s) is assumed to be zero.
130 INTRDUCTIN T CNTR ENGINEERING (c) Now with r(t) = 0, find the settling time of the output within 2% of the steady state value for cases (a) and (b) when subjected to a step disturbance. Ans: (a) K 1 47.5, with K 1 = 50, over shoot 78.9% (b) K 1 = 0.2, over shoot = 14.7%. (c) t s = 73.4 sec, and 92.1 sec D(s) R(s) G (s) 1 G 2(s) G 3(s) G 4(s) Y(s) H(s) Fig. 4.10 Chemical reactor control 4.11 Consider a system with a closed-loop transfer function (s) = Y( s ) 144. = 2 2 R( s) ( s 0. 6s 0. 36) ( s 0. 16s 4) (a) lot the frequency response and step response. 4.12 The Bode magnitude plot of transfer function K( 02. s 1) ( τ1s 1) s( 01. s 1) ( 005. s 1) ( τ2s 1) is shown in Fig. 4.12. Determine K, τ 1 and τ 2 from the plot. Ans: K = 4, τ 1 = 1, τ 2 = 1/50 40 Bode magnitude diagram agnitude (db) 20 0 20 40 10 1 10 0 10 1 10 2 10 3 Frequency (rad/sec) Fig. 4.12 Bode plot of G(s)