自动控制理论第七章新.docVIP

E7.1 Let us consider a device that consists of a ball rolling on the inside rim of a hoop [11]. This model is similar to the problem of liquid fuel sloshing in a rocket. The hoop is free to rotate about its horizontal principle axis as shown in Figure E7.1. The angular position of the hoop may be controlled via the torque T applied to the hoop from a torque motor attached to the hoop drive shaft. If negative feedback is used, the system characteristic equation is =0. (a) Sketch the root locus. (b) Find the gain when the roots are both equal. (c) Find these equal roots. (d) Find the settling time of the system when the roots are equal. E7.2 A tape recorder has a speed control system so that H(s)=1 with negative feedback an . (a) Sketch a root locus for K, and show that the dominant toots ate s=-0.35±j0.08 when K=6.5. (b) For the dominant roots of part (a), calculate the settling time and overshoot for a step input. E7.3 A control system for an automobile suspension tester has negative unity feedback and a process [12] . We desire the dominant roots to have a ζ equal to 0.5. Using the root locus, show that K=7.35 is required and the dominant roots are s=-1.3±j2.2. E7.4 Consider a unity feedback system with. (a) Find the angle of departure of the root locus from the complex poles. (b) Find the entry point for the root locus as it enters the real axis. Answers: ±;-2.4 E7.6 One version of a space station is shown in Figure E6.3 [30].It is critical to keep this station in the proper orientation toward the sun and the Earth for generating power and communications. The orientation controlling may be represented by a unity feedback system with an actuator and controller, such as . Sketch the root locus of the system as K increases. Find the value of K that results in an unstable system. Answers: K=96 E7.9 The world’s largest telescope is located in Hawaii. The primary mirror has a diameter of 10 m and consists of a mosaic of 36 hexagonal segments with the orientation