Test for Semester 2003/2004 Principles of Automatic Control Institute of Automation, College of Information Engineering, Zhejiang University of Technology Name: , Student Number: , Scores: . 1、 (8 Points) Suppose that a signal can be expressed as following: ? ? ? ? ? ≤≤ ≤≤ = ? otherwise TtTe Tt tf at ,0 2 , 0 , 0 )( Please try to find its Laplace transform. 2、 （ 10 Points） suppose that the unit impulse response function of a system with zero initial conditions can be written as ))(()( 1 2 1 12 1 2 T t e T TT t T T Ktk ? ? ?= δ 。 （1）try to find the transfer function of the system。 （2）try to find the step response of the system with zero initia l conditions. 3、 (12 Points) Please try to find the transfer function between the input V i and the output V o . - + C 0 R 0 R 1 C 1 U i R U o 4、 (10 Points) For the loop transfer function )15.0)(11.0( )( ++ = sss K sGH . Please try to use the Nyquist criterion to find the range of gain K, so that the closed loop system is stable. 5、 (8 Points) For the loop transfer function )20)(2( )1( )( 2 ++ + = sss sK sGH . Please try to sketch the root locus. 6、 (14 Points) For the loop transfer function )1( )1)(12( )()( 2 + ++ = Tss ssK sHsG ， 0,0 >> TK Please try to determine the condition, which must be satisfied for KT, , so that the closed loop system is stable. 7、 Please try to find d τ , so that the steady state error is zero when the reference input signal is ttr =)( .（ suppose that TK, are known.） )(sE )1( Tss K + )(sR s d τ )(sC 7、 (14 Points) Please use the Nyquist criterion to analysis the stability of the closed loop system with the loop transfer function 0. 0,T , )1( )1( )( 2 >> + + = τ τ Tss sK sGH . And then check with the Routh criterion. 7、 (12 Points) For the Bode plot (Magnitude~w) for a minimum phase system, try to find the loop transfer function of the system. 0.4 10 1 -20dB/dec -40dB/dec -20dB/dec -60dB/dec L( )ω ω 8、 (12 Points) For the discrete time control system, with sampling period T=1s, and a=1, k=10. Please try to analysis the stability of the system. C(s) )( as k + R(s) _ ZOH