1,Chapter 5 Root Locus Method,5.1 The Root Locus Concept 5.2 The Root Locus Procedure 5.3 Examples for Drawing Root Locus 5.4 Parameter Design by the Root Locus Method 5.5 Relationship between Performance and the distributing of close-loop zeros and poles 5.6 Compensation by Using Root Locus Method 5.7 Summary 5.8 Three-term (PID) Controllers,2,5.6 Compensation by Using Root Locus Method,1. Introduction,The design of a control system is concerned with the arrangement, or the plan, of the system structure and the selection of suitable components and parameters. The alteration or adjustment of a control system in order to provide a suitable performance is called compensation.,3,A compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance.,4,2. Phase-lead design using the root locus,Adding a single zero moves root locus to the left and achieves the higher stability.,5,unstable,理硕教育专注于北理工考研辅导,本资料由理硕教育整理，理硕教育是全国唯一专注于北理工考研辅导的学校，相对于其它机构理硕教育有得天独厚的优势。丰富的理工内部资料资源与人力资源确保每个学员都受益匪浅，确保理硕教育的学员初试通过率89%以上，复试通过率接近100%，理硕教育现开设初试专业课VIP一对一，初试专业课网络小班，假期集训营，复试VIP一对一辅导，复试网络小班，考前专业课网络小班，满足学员不同的需求。因为专一所以专业，理硕教育助您圆北理之梦。详情请查阅理硕教育官网,6,7,stable,8,Shortcomings It is difficult to realize The noise is amplified, especially, the higher frequency noise.,9,Phase-lead network,10,List the system specification and translate them into a desired root location for the dominant roots. Sketch the uncompensated root locus, and determine whether the desired root location can be realized. If a compensator is necessary, place the zero of the phase-lead network directly below the desired root location (or to the left of the first two real poles). Determine the pole location so that the total angle at the desired root location is 180 and therefore is on the compensated root locus. Evaluate the total system gain at the desired root location and then calculate the error constant. If the error constant is not satisfactory, a Phase-lag compensator is needed.,11,Example 5.15 Lead compensator using root locus,The uncompensated loop transfer function is,The specifications for the system are,Therefore,12,Choose a desired dominant root location as,Place the zero of the compensator directly below the desired root location at s=-1.,13,Therefore, the compensator is,14,The compensated loop transfer function is,and at the root location,the acceleration constant is,15,A computer simulation of the compensated system is,These values compare moderately well with the specified values of 35% and 4s.,16,Example 5.16 Lead compensator for a type-one system,The uncompensated loop transfer function is,The specifications for the system are,The gain of the uncompensated system must be K=40, and the roots are,17,Select the real part of the desired roots as zwn=4, where wn=8.89 for z=0.45.,If the natural frequency of the roots is large, the velocity constant should be reasonably large.,The desired closed-loop system is,Its open-loop transfer function is,If z be constant,18,Place the zero of the compensator at s=-4. We get,The compensated loop transfer function is,and at the root location,the velocity constant is,19,The velocity constant is less than 20, so we should repeat the design procedure with a new wn=10. We have,20,3. Phase-lag design using the root locus,The compensator is,If the compensator pole and zero appear relatively close together and near the origin of the s-plane compared to wn, it can increase the error constant by the factor a while altering the root location very slightly.,21,Obtain the root locus of the uncompensated system. Determine the transient performance for the system and locate suitable dominant root locations on the uncompensated root locus that will satisfy the specifications. Calculate the loop gain at the desired root location and the system error constant. Compare the uncompensated error constant with the desired error constant, and calculate the necessary increase that must result from the pole-zero ratio a of the compensator. Locate the compensator pole and zero near the origin of the s-plane in comparison to wn.,22,Example 5.17 Design of a phase-lag compensator,The uncompensated loop transfer function is,The specifications for the system are,The gain of the uncompensated system must be K=2000, and at the desired location of z=0.707 on the root locus, K=236.,23,The necessary ratio of the zero to the pole of the compensator is,We choose z=0.1 and p=0.1/9 in order to allow a small margin of safity. The compensated system is,24,5.7 Summary,The relationship between performance and roots location The concept of Root Locus The Root Locus drawing procedure The generalized Root Locus (zero-degree and parameter RL) A powerful tool for the analysis and design of control system,25,The relative stability and the transient response performance of a closed-loop control system are directly related to the location of the closed-loop roots of the characteristic equation. The root lo