Test for Semester 2003/2004 Principles of Automatic Control (A) 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 find its Laplace transform. 2、（10 Points） Please find the current i(t) of the system after the closure of the switch at t=0 with the initial conditions: i L (0)=0. e(t)=V and V is a constant. 3、(12 Points) Please find the transfer function between the input V i and the output V o . 4、(10 Points) For the loop transfer function )50)(1( )( ++ = sss K sGH . Please 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 sketch the root locus. 6、 (14 Points) For the loop transfer function )1( )1)(12( )( 2 + ++ = Tss ssK sGH ， 0,0 >> TK . (1) Please determine the condition, which must be satisfied for KT, , so that the closed loop system is stable. (2) For unit negative feedback system, please find the steady state errors when the reference input signals are unit step, unit ramp and unit acceleration respectively. 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. 8、(12 Points) For the Bode plot, please find the loop transfer function of the system. 9、(12 Points) For the discrete time control system, with sampling period T=1s, and a=1, k=10. Please analysis the stability of the system. C(s) )( as k + R(s) _ ZOH