自动控制理论8

1、Lecture 11/12Analysis and design in the time domain using root locus North China Electric Power UniversitySun Hairong8/12/20221Topics of this class Magnitude and phase equations (Reading Module 9)Rules for plotting the root locus (Reading Module 10)System design in the complex plane using the root l

2、ocus. (Reading Module 11)21. Magnitude and phase equationsWe begin by writing the characteristic equation for the generalized feedback control system asThe value of s that satisfy the above equation may be real or complex, so rewrite the equation in the form It may be inferred that both the magnitud

3、e and phase of each are equal and write 3Given an second-order feedback system The characteristic equation2. Root-locus The roots of the equation areNow we plot the closed-loop poles on the complex plane for K=0, 0.5, 1, 2, 10, .4Firstly, for each value of K calculate s1,2： Then plot the closed-loop

4、 poles on the complex plane.53.Rules for plotting the root locusSuppose we are required to draw the root locus for the system with open loop transfer functionWe can plot the closed-loop poles on the complex plane for each value of K calculate s1,2,3The roots locus can also be drawn using the followi

5、ng rules.6Rule #1. Draw the complex plane and mark the n open-loop poles and m zeros. The locus starts at a poles for K=0, and finishes at a zero or infinity when nm, the number of segments going to infinity is therefore n-m. Rule #2. Segments of the real axis to the left of an odd number of poles a

6、nd zeros are segments of the root locus, and the complex poles and zeros have no effect.7Rule #3. The loci are symmetrical about the real axis. The angle between adjacent asymptote is and to obey the symmetry rule, the negative real axis is one asymptote when n-m is odd.Rule #4. The asymptotes inter

7、sect the real axis at8Rule #5. The angle of emergence from complex poles is given by (angles of all the vectors from all other open-loop poles to the pole in question)+ (angles of all the vectors from all other open-loop zeros to the pole in question)Rule #6. The point where the locus crosses the im

8、aginary axis may be obtained by substituting s=j into the characteristic equation and solving for .9Rule #7. The point at which the locus leaves a real-axis segment is found by determining a local maximum value of K, while the point at which the locus enters a real-axis segment is found by determini

9、ng a local minimum value of K.Rule #8. The angle between the directions of emergence (or entry) of q coincident poles (or zeros) on the real axis is given by10Rule #9. The gain at a selected point on the locus is given by Rule #10. If there are at least two more open-loop poles than open-loop zeros,

10、 the sum of the real parts of the closed-loop poles is constant, independent of K, and equal to the sum of the real parts of the open-loop poles.111Sketch the root locus based on the open-loop pole-zero map. Practice12Solution134. Performance requirement as complex-plane constraints It has to satisfy the performance requirements respectively 1. overshoot less than 10%,2. steady-state error to a unit ramp less than 10%, 3. dominant time constant less than 0.1s.4. the settlin