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 +=-==-==+-=2

252453452312111023

2. 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 =0.5s and P.O=2%.

Figure 2

3.When the input signal of the system is t )t ()t (r +=1, the measured output response for 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 ramp input ,calculate the steady-state error respectively.

4. A negative feedback system has a loop transfer function

2

2)

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 and other roots of the characteristic equation. 5.A unity feedback system has a plant

2

11)

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.

6.Logarithm amplitude frequency characteristic curve of minimum phase systems shown below, write out the system transfer function respectively.

L(w)dB

L(w)dB 0

7. A unity feedback control system has a plant

)s )(s (s )

s ()s (G 130

12110+++=

(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 second? 2. 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 P.O and Ts.

Figure 2

3. A control system shown in figure 3.one controller is the proportional controller, Gc(s)=Kp. determine the range of Kp of stable system .

Figure 3

4. The unit-step response of a unity feedback system t 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). 5.A 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

(2)when K=10,sketch the root locus for 0

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 6

7. 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 find out the

gain margin.

Figure 7

习题三

一、A control system is shown in Figure 2.1(a)and Figure2.1(b), find the transfer

function

()()Y s R s (()0D s =), ()

()

E s R s (()0D s =), ()

()

Y s D s (()0R s =),

Figure 2.1 (a)

Figure 2.1 (b)

二 、 A unity feedback system has a plant

2

)

1()(+=

s K

s 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 4.1,

Where )(s G

C =1+

s K I , )(s G P =25(2)

s s +.

Figure 4.1

(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 percent overshoot

(P.0%) is 16.3%; (..%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 input r(t)=10+2t.

四、 A unity feedback system has a plant

2

)

4()(+=

s s K

s G (1) Sketch the root locus for 0

(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 Fig.6. (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.；

0-90-180-270Figure 6

六、The unity feedback system has open-loop transfer function ,

)15.0(100)(0+=

s s S G

The bode plot of 0()()c G s G s

is shown in Fig.7.

Fig.7

（1）Determine the transfer function of compensator Gc(s)

（2）When 9.04/g w rad s ,determine the phase margin of the system after

compensation.

习题四

1. Determine the transfer function Y(s)/R(s).

2.A 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 P.O≤ 16.3% and t s ≤4s .

(2)When ζ=0.5 and ωn =10, determine the value of K and β. (3)Determine the steady-state error for a step and a ramp input.

3.A system has the transfer function

15

815

)()()(2++==

s s s R s Y s T Determine its unity step response and its unity impulse response.

4.A 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.

5.The polar plot of the open-loop transfer function of a system is shown below. (Fig.5(2))

(1) the corresponding transfer function is the following which form? (Fig.5(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 ）

Fig.5(1)

Fig.5(2)

6.Logarithm 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.