控制理论基础-自控a_ch5(1)

1,Chapter 5 Root Locus Method,5.1 The Root Locus Concept5.2 The Root Locus Procedure5.3 Examples for Drawing Root Locus 5.4 Parameter Design by the Root Locus Method5.5 Relationship between Performance and the distributing of close-loop zeros and poles5.6 Compensation by Using Root Locus Method5.7 Summary 5.8 Three-term (PID) Controllers,2,Relative stability and transient performance are directly related to the location of the roots. It is necessary to adjust one or more system parameters to obtain suitable root locations.The above two facts make it worthwhile to determine how the roots migrate(移动) about the s-plane as the parameters are varied.,5.1 The Root Locus Concept,3,The root locus method was introduced by Evans in 1948.It is a graphical method for sketching the locus of roots in the s-plane as a parameter is varied.It provides an approximate sketch that can be used to obtain qualitative（定性的） information concerning the stability and performance of the system.,4,Example 5.1: A second-order system,The figure is shown the root locus as K changes from 0 to infinity.,5,Definition of Root Locus The root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter is changed.,6,5.2 The Root Locus Procedure,Magnitude and Angle RequirementProcedure for drawing root locus,7,Magnitude and Angle Requirement,Root locus (negative feedback) magnitude requirement angle requirement,8,9,10,Note: When plotting the root locus, only the angle requirement is the sufficient and necessary condition (because s is varied with K ). The magnitude requirement is used to determine K for a given root s1.The angle requirement is used to verify a test point s1 as a root location.,11,Procedure for Drawing Root Locus,Step 1：Prepare the root locus sketch.,(a) Write the characteristic equation so that the parameter of interest, K, appears as multiplier.,(b) Factor P(s) in terms of n poles and m zeros.,12,(c) Locate the open-loop poles and zeros of P(s) in the s-plane with selected symbols.The locus of the roots of 1+KP(s)=0 begins at the open-loop poles and ends at the open-loop zeros as K increases from 0 to infinity. If nm, there are n-m branches of the root locus approaching the n-m zeros at infinity.,13,Proving：,At the starting point of the root locus: K=0,14,At the end point of the root locus: K,and the characteristic equation can be written as,when,15,(d) The number of separate loci is n.,(e) The root loci are symmetrical with the respect to the horizontal real axis because complex roots must appear as pairs of complex conjugate roots.,16,Step 2：Locate the Root Locus on the Real Axis,The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros.,17,It can be ascertained by the angle requirement.,18,Step 3: The Asymptotes of the Root Locus The loci proceed to the zeros at infinity along asymptotes centered at (asymptote centroid) and with angles .,19,s时，上式可化为：为渐近线公式。有n-m个解，表示n-m条直线，经过公共点( , j0)。可得直线夹角满足：将 代入特征方程，可推导得：,20,Example 5.2,21,Step 4: Determine where the root locus crosses the imaginary axis. The point where the locus crosses the imaginary axis, we have s=jw into the characteristic equation and solving for w.,22,Example 5.3 A unity feedback system,23,Note: The actual point at which the root locus crosses the imaginary axis can also be readily evaluated by utilizing the Routh-Hurwitz criterion.,24,Step 5: Determine the breakaway point of the root locus The root locus left the real axis at a breakaway point. The breakaway point on the real axis can be evaluated graphically or analytically. The locus leaves the real axis where there are a multiplicity of roots, typically two.,25,26,Breakaway point,27,Assume the breakaway point s=d:Method 1:,28,Method 2:,29,Example 5.4 A feedback system,30,Example 5.5 Feedback systems,31,The angle between the direction of emergence (or entry) of q coincident poles (or zeros) on the real axis（根轨迹离开或进入实轴上q重极点（或零点）方向之间的夹角）,32,Example 5.6 Feedback systems,33,补充说明： 一般来说，实轴上两相邻开环极点之间若有根轨迹，则必有分离点；实轴上两相邻开环零点（包括无穷远处零点）之间若有根轨迹，则必有会合点；而实轴上开环极点和开环零点之间，或者分离点和会合点同时存在，或者都没有。,34,Step 6：Determine the angle of departure of the locus from complex poles and the angle of arrival at complex zeros, using the phase criterion. The angle of departure from complex poles is given by (2k+1)(angles of the vectors from all other open-loop poles to the poles in question) + (angles of the vectors from the open-loop zeros to the complex pole in question).,35,Proving：,36,The angle of arrival at a complex zero may be found from the same rule and then the sign changed to produce the final result.,37,Proving：,38,Example 5.7 Feedback systems,39,40,Step 7: Complete the root locus sktechRule 1：The sum of the closed-loop poles If there are at least two more open-loop poles than open-loop zeros, the sum of the closed-loop poles is constant, independent of K, and equal to the sum of the open-loop poles. (如果开环极点比开环零点至少多2个，闭环极点的和为一不依赖于K的常数，且等于开环极点的和。）,41,闭环极点的和与积设闭环系统特征方程：则有以下特征根与多项式系数的关系：对于稳定系统，有：,42,例4.3 例4.1系统根轨迹与虚轴交于：由于闭环特征方程：则：因此：,43,可利用相角条件计算闭环极点位置；例：求闭环特种根的实部为-4时的值。,44,Rule 2：The gain at a selected root location sx The gain Kx at a selected root sx, x=1,2,n, on the locus is obtained by joining the point to all open-loop poles and zeros and measuring the length of each line . The gain is given by,45,Example 5.8,46,At the breakaway point s=-2.6, Gain K is,47,Summary of the Root Locus plotting: Table 7.2 at page 424,48,5.3 Examples for drawing root locus,Example 5.9: Plot the root locus of the following system,49,Step 1: find the poles and zeros of GH and plot them.,50,Step 2: draw locus on the real axis,51,Step 3: calculate asymptote angles and center of asymptotes:,Since n-m=1, there is only one asymptote at 180, corresponding to the real axis.,52,Step 4: Not necessary.Step 5: Determine the breakaway point on the real axis,53,54,Example 5.10:,Drawing root locus,55,5. The points crossing the imaginary axis,56,57,58,Example 5.11:,Drawing root locus,59,5. The points crossing the imaginary axis,60,61,62,Generalized root locus1. Zero-degree root locusSystem contains inner positive-feedback-loopOr for K=0 -2. Parameter root locusEquivalent unity feedback transform,63,1. Zero-degree root locusThe close-loop characteristic equation:,magnitude requirement angle requirement,64,Rules for drawing zero-degree root locus,1. Number of root locus branches is the number of characteristic roots.2. The symmetry of root locus.3. The start point and end point.4. The locus on the real axis has even number of real poles and zeros on its right.5. The Asymptotes of the Root Locus .,65,6. The breakaway point of the root locus7. The angle of departure of the locus from complex poles and the angle of arrival at complex zeros.8. The root locus crossing the imaginary axis.,66,Example 5.12: A positive-feedback system:,Drawing the zero-degree root locus,67,68,Example 5.13: Feedback system,Drawing the parameter root locus.,2. Parameter root locus,69,70,71,5.4 Parameter Design by the Root Locus Method,The root locus method can be readily extended to the investigation of two or more parameters.,Example 5.14: The closed-loop characteristic equation is with two parameters:,72,The effect of varying b is determined from the root locus equation:,First of all, let b=0, and therefore we evaluate the effect of varying a by the equation:,Rewritten as:,73,The sketch of the root locus is shown in the figure.,Let a=a1, the root locus equation of the closed-loop system becomes:,74,The root locus as b varies is shown in the figure.,75,As a be selected to different value, the root locus of b is changed as shown in the figure.,76,5.5 Relationship between Performance and the Distributing of Close-loop Zeros and Poles,2. If there is a closed-loop zero close to a pole, then the residue at this po