1、E7.1Let 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 contr
2、olled 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. 2(4)1Ks+(a) Sketch the root locus. (b) Find the gain when the roots are both equal. (c) Find these equal roots. (d) Find the settli
3、ng time of the system when the roots are equal. E7.2A tape recorder has a speed control system so that H(s)=1 with negative feedback an . 2()(45)KGss=+(a) Sketch a root locus for K, and show that the dominant toots ate s=-0.35j0.08 when K=6.5. (b) For the dominant roots of part (a), calculate the se
4、ttling time and overshoot for a step input.E7.3A control system for an automobile suspension tester has negative unity feedback and a process 12 . We desire the 2(48)KsG+=dominant roots to have a equal to 0.5. Using the root locus, show that K=7.35 is required and the dominant roots are s=-1.3j2.2.E
5、7.4Consider a unity feedback system with . (a) Find the 2(1)()45KsG+=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: ;-2.425oE7.6One version of a space station is shown in Figure E6.3 30.It is critical to ke
6、ep 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 . 25()410)KGss=+Sketch the root locus of the system as K increases. Find
7、the value of K that results in an unstable system.Answers: K=96 E7.9The worlds 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 of each segment actively controlled. This unity feedback system for
8、the mirror segments has . 2()5)KGss=+(a) Find the asymptotes and draw them in the s-plane. (b) Find the angle of departure from the complex poles. (c) Determine the gain when two roots lie on the imaginary axis.(d) Sketch the root locus.E7.25A closed-loop feedback system is show in Figure E7.25. For
9、 what range of values of the parameters K is the system stable? Sketch the root locus as 02 is larger ()cGs nwthan 0.15 (seek a maximum ). (b) Plot the response, q(t), for a step input r(t)with K as in (a). (c) A designer suggests an anticipatory controller with . Sketch the root locus for this syst
10、em as K 12()()csKs=+varies and determine a K so that the of all the closed-loop roots is 0.8. (d) Plot the response, q(t), for a step input r(t) with K as in (c). DP7.2A larger helicopter uses two tandem rotors rotating in opposite directions, as shown in Figure P7.33 (a). The controller adjusts the
11、 tilt angle of the main rotor and thus the forward motion as shown in Figure DP7.2. The helicopter dynamics are represented by , and the controller is selected as 210()4.59Gss=+. 1(1)cK(a) Sketch the root locus of the system and determine K when of the complex roots is equal to 0.6. (b) Plot the res
12、ponse of the system to a step input r(t) and find the settling time (with a 2% criterion) and over-shoot for the system of part(a). What is the steady-state for error for a step input? (c) Repeat parts (a) and (b) when the of the complex roots is 0.41. Compare the results with those obtained in part
13、s (a) and (b).DP7.4A welding torch is remotely controlled to achieve high accuracy while operating in changing and hazardous environments 21. A model of the welding arm position control is shown in Figure DP7.4, with the disturbance representing the environmental changes. (a) With D(s)=0, select and
14、 K to provide high-quality performance of the position 1control system. Select a set of performance criteria and examine the results of your design. (b) For the system in part (a), let R(s)=0 and determine the effect of a unit step D(s)=1/s by obtaining y(t). DP7.5A high-performance jet aircraft wit
15、h an autopilot control system has a unity feedback and control system, as shown in Figure DP7.5. Sketch the root locus, and predict the step response of the system, and compare it to the predicted response.DP7.10A pilot crane control is shown in Figure DP7.10 (a). The trolley is moved by an input F(
16、t) in order to control x(t) and (t)13. The fmodel of the order to control is shown in Figure DP7.10 (b). Design a controller that will achieve control of the desired variables when =K. ()cGsDP7.13The automatic control of an air plane is one example that requires multiple-variable feedback methods. I
17、n this system, the attitude of an aircraft is controlled by three sets of surfaces: elevators, a rudder, and ailerons, as shown in Figure DP7.13 (a). By manipulating these surfaces, a pilot can set the aircraft on a desired flight path 20.An autopilot, which will be considered here, is an automatic
18、control system that controls the roll angle by adjusting aileron fsurfaces. The deflection of the aileron surfaces by an angle qgenerates a torque due to air pressure on these surfaces. This causes a rolling motion of the aircraft. The aileron surfaces are controlled by a hydraulic actuator with a t
19、ransfer function 1/s.The actual roll angle is measured and compared with the input. fThe difference between the desired roll angle will drive the dfhydraulic actuator, which in turn adjusts the deflection of the aileron surface.A simplified model where the rolling motion can be considered independen
20、t of other motion is assumed, and its block diagram is shown in Figure DP7.13 (b). Assume that =1 and that the roll rate 1Kis fed back using a rate gyro. The step response desired has an fovershoot less than 10% and a settling time (with a 2% criterion) less than 9 seconds. Select the parameters and
21、 .a2MP7.4A unity negative feedback system has the open loop transfer function . Using MATLAB, obtain the root locus as p varies; 2(1)36psGs+-=0p. For what values of p is the closed-loop stable?MP7.8Consider the feedback control system in Figure MP7.8. Using MATLAB, plot the root locus for 0K. Find t
22、he value of K resulting in a damping ratio of the closed-loop poles equal to 0.707. MP7.9Consider the system represented in state variable from = Ax + Buxy= Cx + Du,where C= 1 -9 12, and D= 0. (a) 011,0.524ABk=-Determine the characteristic equation. (b) Using Routh-Hurwitz criterion, determine the values of k for which the system is stable. (c) Using MATLAB, plot the root locus and compare the results to those obtained in (b).
