自控原理习题

习题一1. The differential equation of a system shown below, try to establish the dynamic structure diagram of the system and determine the transfer function Y(s)/R(s),Y(s)/N 1(s), Y(s)/N 2(s).dt)t (dy dt )t (y d )t (x )t (n )t (x )t (x )t (x dt)t (dx )t (x )t (x )t (x )t (x )t (x )t (n )t (y )t (r )t (x +=-==-==+-=22524534523121110232. A Feedback control system has the structure shown in figure 2. Determine the K 1 and K 2 so that the unit step response y(t) has T p = and =2%.Figure 2the input signal of the system is t )t ()t (r +=1, themeasuredoutputresponsefor t e .).t ()t (y 109090--+=. (1)Determine the system transfer function)s (R )s (Y . (2) For a unit step input and a unit rampinput ,calculate the steady-state error respectively.4. A negative feedback system has a loop transfer function 22)s (s K)s (GH +=.(1) Using the Routh-Hurwitz criterion, find the value of the gain K when the closed-loop system has two poles on the imaginary axis. (2)Find the value of the gain K when s=-3 andother roots of the characteristic equation. unity feedback system has a plant211)s ()s (K )s (G -+=(1)Sketch the root locus of the closed-loop system. (2)Determine the range of K for stability. (3)Find the gain K at the breakaway point. amplitude frequency characteristic curve ofminimum phase systems shown below, write out the system transfer function respectively.L(w)dBL(w)dB 07. A unity feedback control system has a plant)s)(s (s )s ()s (G 13012110+++=(1) Plot the Bode diagram of G(s).(2) Determine the phase margin PM and gain margin GM.(3) Plot the polar plot and Mark GM, PM on the this map.习题二1. A unity negative feedback control system has the plant)2()(K s s K s G +=(1) Determine the percent overshoot due to a unit step input.(2) For what range of K is the settling time less than 1 second2. A control system is shown in figure 2. (1)when r(t)=t and n(t)=t, determine all steady-state error of the system.(2)when r(t)=1(t) and n(t)=0,determine the and Ts.Figure 23. A control system shown in figure controller is the proportional controller, Gc(s)=Kp. determine the range of Kp of stable system .Figure 34. The unit-step response of a unity feedback systemt t e e t y 10602.12.01)(---+=(1)Find the transfer function of the system)()(s R s Y . (2)Find the open-loop transfer function G(s). unity feedback system has a plant )0,0(,)10(1()()(>>-+=T K s T K s H s G s s ）(1)when T=1,Sketch the root locus for 0<K<∞. Find the range of K when all roots are negative real parts. (2)whenK=10,sketchtherootlocusfor0<T<∞.Determine the range of T of the stable system. 6. the following open-loop transfer function G(s)H(s) have no poles on the right half s-plane , their polar plots shown in figure 6. Please analyze the stability of the corresponding system.Figure 67. A closed-loop control system shown in Figure 7. (1)Draw the Bode plot of the open-loop transfer function G(s).(2)Calculate the phase margin.(3)Prove the phase is -180°when 14.0=ω and findout the gain margin.Figure 7习题三一、A control system is shown in Figure (a)and (b), find the transferfunction()()Y s R s (()0D s =), ()()E s R s (()0D s =), ()()Y s D s (()0R s =),Figure (a)Figure (b)二 、 A unity feedback system has a plant2)1()(+=s Ks G .(1) Plot the polar plot for )(s G when 4=K ； (2) Calculate the phase and magnitude at 1=ω；(3) Determine the range of K for which the system is stable; (4) Calculate the steady-state error ss e for input ()sin r t t =.三、A control system is shown in Figure ,Where )(s G C =1+sK I, )(s G P =25(2)s s +.Figure(1)To assure the system is stable, determine the condition that I K ,t K should satisfy ;(2)When I K =0, t K ≠0, determine the value of t K when the percentovershoot %) is %; (..%P O =)(3)For the condition of (2), find the time response y(t) for a step input r(t)=A, for t>0;(4)When I K ≠0, t K ≠0, find the steady-state error ss e for inputr(t)=10+2t.四、 A unity feedback system has a plant2)4()(+=s s Ks G(1) Sketch the root locus for 0<K<∞;(2) Determine the value of the roots on the axis j -ω and the gain K required for those roots;(3) Determine the range of K , that result in a system that satisfy step response with no overshoot.五、The open-loop bode plot of a unity feedback system are shown in .(1)Find the open-loop transfer function G(s)；. (2)Determine the phase margin and gain margin ；. (3)When K=20,whether the system is stable.；12.24100.1-20dB/dec-40dB/dec-60dB/decωωdBL )(ω)(ωΦ0o -90o -180o -270o-6.2dBFigure 614六、The unity feedback system has open-loop transfer function ,)15.0(100)(0+=s s S GThe bode plot of 0()()c G s G s is shown in .（1）Determine the transfer function of compensator Gc(s)（2）When 9.04/g w rad s ,determine the phase margin of the system aftercompensation.习题四1. Determine the transfer function Y(s)/R(s).Feedback control system has the structure shown in figure 2.Figure 2(1)Show the permissible area for the poles of Y(s)/R(s) in order to achieve ≤ % and t s ≤4s . (2)When ζ= and ωn =10, determine the value of K and β.(3)Determine the steady-state error for a step and a ramp input.system has the transfer function15815)()()(2++==s s s R s Y s T Determine its unity step response and its unity impulse response.unity negative feedback system has a loop transfer function)1()1()5.0()(2-++=s s s K s G .(1)Sketch the root locus of q(s)=1+G(s)=0. (2)Determine the range of K for stability. (3)Find the value of the gain K when s=0 and other roots of the characteristic equation.polar plot of the open-loop transfer function of a system is shown below. (2))(1) the corresponding transfer function is the following which form (1))(2) determine the stability of the system using Nyquist stability criteria.)Ti ,K (,)s T )(s T )(s T (K )s (H )s (G )b ()T ,K (,)s T )(s T (T (K )s (H )s (G )a (i 0011100111321213>>+++=>>+++=s s ）(1)(2)amplitude frequency characteristic curve of minimum phase systems as shown below, write out the system transfer function respectively.7. A unity feedback control system has a plant)s )(s (s )s (G 3212++= (1) Plot the Bode diagram of G(s).(2) Determine the phase margin PM and gain margin GM.(3) Plot the polar plot and Mark GM, PM on the map.。