2-1 (a),(b) (pp.50) Problem: Prove that the systems shown in Fig. (a) and Fig. (b) are similar.(that is, the format of differential equation is similar). Where electric pressure u1 and displacement x1 are inputs; voltage u2 and displacement x2 are outputs, k1,k2 and k3 are elastic coefficient of the spring, f is damping coefficient of the friction

Fig. (a) Fig. (b)

2-3 (pp. 51) In pipeline,the flux q through the valve is proportional to the square root of the pressure difference p, that is, q=K. suppose that the system changes slightly around initial value of flux q 0. Problem: Linearize the flux equation.

2-5 (pp. 51) Suppose that the system’s output is under the step input r(t)=1(t) and zero initial condition. Problems: 1. Determine the system’s transfer function and the output response c(t) when r(t)=t, 2. Sketch system response curve.

2-6 (pp. 51) Suppose that the system’s transfer function is, and the initial condition is Problem: Determine the system’s unit step response c(t).

2-13 (a), (d) (pp. 53) Problem: Determine the close loop transfer function of the system shown in the following figures, using Mason Formula. Fig. (a) Fig. (b)

3-2 (pp. 83) Suppose that the thermometer can be characterized by transfer function. Now measure the temperature of water in the container by thermometer. It needs one minute to show 98% of the actual temperature of water. Problem: Determine the time constant of thermometer.

3-4 (pp. 83) Suppose that the system’s unity step response is Problem: (1) Solve the system’s close-loop transfer function. (2)Determine damp ratio and un-damped frequency.

3-5 (pp. 83) Suppose that the system’s unity step response is Problem: Determine the system’s overshoot, peak time and setting time

 3-8 (pp. 83) Suppose that unity step response of a second –order system is shown as follows. Problem: If the system is a unity feedback, try to determine the system’s open loop transfer function.

3-11 (pp. 84) Problem: Determine the stability of the systems described by the following characteristic equations,using Routh stability criterion. (1) (2) (3)

3-16 (pp. 16) Suppose that the open loop transfer function of the unity feedback system is described as follows. Problem: Determine the system’s steady- state error when r(t)=1(t), t, respectively (1) (2) (3)

3-19 (a) (pp. 85) Problem: Determine the system’ steady- state error which is shown as follows.

4-2 (pp.108) The system’s open-loop transfer function is Problem: Prove that the point s1=-1+j3 is in the root locus of this system, and determine the corresponding K.

4-4 (pp.109) A open-loop transfer function of unity feedback system is described as Problems : (1) Draw root locus of the system (2) Determine the value K when the system is critically stable. (3) Determine the value K when the system is critically damped.

4-7 (pp. 109) Consider a systems shown as follows: Problems: 1.Determine the range of K when the system has no overshoot, using locus method. 2. Analysis the effect of K on system’s dynamic performance. Where

4-10 (pp. 110) The open-loop transfer functions of unity feedback system are described as: Problem: Draw root locus with varying parameters being a and T respectively.

5-2 (1) (pp.166) A unity feedback system is shown as follows. Problem: Determine the system’s steady- state output when input signal is

5-7 （ 3 ） (pp.167) Problem: Draw logarithm amplitude frequency asymptotic characteristics and logarithm phase- frequency characteristic of the following transfer function 。

5-8 (pp. 167) The logarithm amplitude frequency asymptotic characteristics of a minimum phase angle system is shown as follows. Problem: Determine the system’s open loop transfer function 。 5-8 （ a ）

5-8 （ b ） 5-8 （ c ） 5-8 （ d ）

5-10 (pp. 168) The system’s open loop amplitude-phase curve is shown as follows ， where P is the number of poles in right semi-plane of G （ s ） H （ s ）. Problem: Determine the stability of the close- loop system 。

5-10 （ a ） 5-10 （ b ） 5-10 （ c ）

5-12 （ 1 ）, （ 2 ） (pp ) The open loop transfer function of the unity feedback system is shown below ： Problem: Determine the system’s stability using logarithm frequency stability criterion, the phase angle margin and amplitude margin of the steady system 。