(a)Sketch the inverse function f-(g. Clearly label the axes and indicate the important numbers on the sketch (b) Express a(t)as a linear combination of y(t) and f-(y) (c)Sketch y as a function of when (c1)K1=0.5andK2=5 (c.2)K1=10andK2=0.1 (d) In order to have an approximately linear relation between the input a(t)and the output y(t), are the magnitudes of the gains K1 and K2 large Explain Problem 7 Consider the following feedback system e() t)-( 打计一 y() K( (a) Find the system function G(s) which is contained in the dashed box in the figure above. Is this system stable (b)Suppose K(s)=kp where Kp is a real constant. Can you find a range of Kp such that the closed loop system H(s)=x is stable If so, find the range of Kp (c) Suppose K(s)=Kds+kp where both Kd and Kp are real constants. Can you find ranges of Kd and Kp such that the closed loop system H(s) is stable? If so, find the ranges of Kd and K Reminder: The first 20 problems in each chapter of o& w have answers included at the end of the text. Consider using these for additional practice, either now or as you study for test(t) (a) Sketch the inverse function f −1(y). Clearly label the axes and indicate the important numbers on the sketch. (b) Express x(t) as a linear combination of y(t) and f −1(y). (c) Sketch y as a function of x when (c.1) K1 = 0.5 and K2 = 5. (c.2) K1 = 10 and K2 = 0.1. (d) In order to have an approximately linear relation between the input x(t) and the output y(t), are the magnitudes of the gains K1 and K2 large ? Explain. Problem 7 Consider the following feedback system. x(t) + e(t) 2 + 1 s −5 − 1 s y K(s) − G(s) Y (s) (a) Find the system function G(s) = E(s) which is contained in the dashed box in the figure above. Is this system stable ? (b) Suppose K(s) = Kp where Kp is a real constant. Can you find a range of Kp such that the closed loop system H(s) = Y (s) X(s) is stable ? If so, find the range of Kp. (c) Suppose K(s) = Kds + Kp where both Kd and Kp are real constants. Can you find ranges of Kd and Kp such that the closed loop system H(s) is stable ? If so, find the ranges of Kd and Kp. Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests. 4