Spring 2015 信号与系统 Signals and Systems Chapter 11 Linear Feedback Systems

Outline 1 0. Introduction 1.Linear Feedback Systems 2. Applications and Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins

2 Introduction RECALL: Chap use time and transform domain analysis method to figure out how system responds to different kinds of input signals In many situations there are particular advantages to use the output of a system to control or modify the input based on feedback ( 反馈 ) Automobile navigation 信号与系统课程组 ©2015

3 Applications of Feedback Reducing sensitivity to disturbances and to errors 减小系统对扰动的灵敏度 信号与系统课程组 ©2015 Open-loop system Close-loop system Open-loop system: A precise characterization of the system is required Close-loop system: Insensitivity to disturbances and to imprecise knowledge of the system

+ + - F 前放 A 2 功放 A 1 A= A1A2A= A1A2 输入 输出 例：功放系统灵敏度的控制 开环时：系统增益从 1000 降至 500 ，系统性能偏差大； 闭环时：系统增益从 10 降至 9.9 ，系统性能偏差不大。 思考：如果没有前放 A 2 ，系统性能又会怎样？

5 Stabilizing a system that is inherently unstable 使一个固有的不稳定系统稳定 How to balance a broomstick in the palm of the hand? Small disturbances will cause the broom to fall over Designing a feedback system that will move the cart to keep the pendulum vertical 信号与系统课程组 ©2015

6 Outline 0. Introduction 1.Linear Feedback Systems 2. Applications and Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins

信号与系统课程组 ©2015 [Chap. 11] 2. Linear feedback systems Linear feedback systems RECALL Chap.5,9,10 Causal System Because of the typical applications in which feedback is utilized, it is natural to restrict the systems to be causal. 由于反馈最典型的应用场合都是在因果系统，因此本章 所讨论的反馈系统都局限为因果系统。

信号与系统课程组 ©2015 [Chap. 11] 2. Linear feedback systems Linear feedback systems CT feedback systemDT feedback system System function of the forward path 正向通路系统函数 System function of the feedback path 反馈通路系统函数 Recall and ： close-loop system function 闭环系统函数

9 Outline 0. Introduction 1.Linear Feedback Systems 2. Applications and Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins

11.2 Applications and consequences of feedback Inverse system design( 逆系统设计 ) 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Objective ： synthesize the inverse of a given CT system 希望综合出一个已知连续时间系统的逆系统 If the gain K is sufficiently large, then ： For example:

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback For example: The differentiation property of the capacitor is inverted to provide integration.

12 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback 11.2 Applications and consequences of feedback Compensation for nonideal elements( 非理 想元件的补偿 ) Objective ： Correct for some of the nonideal properties of the open-loop system 校正开环系统的某些非理想特性 For example: Feedback is often used in the design of amplifiers to provide constant-gain amplification in a given frequency band.

13 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback 11.2 Applications and consequences of feedback Assuming ： Then If Then 闭环频率响应为一个常数

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback is the precondition If closed-loop gain should be greater than unity. That means: must be an attenuator over the specified range of frequencies That also means: Closed-loop gain will be substantially less than the open-loop gain. 一般来说，实现一个具有近似平坦频率特性的衰减器比实现一个具有同 样频率特性的放大器来说要容易得多，因衰减器可以用无源器件组成。 然而，在总的增益上所失去的往往可以在降低总的闭环放大器的灵敏度 上得到更多的补偿。

15 信号与系统课程组 © Applications and consequences of feedback Stabilization of unstable systems( 不稳定系统的稳定 ) Objective ： Stabilize systems that are unstable without feedback 校正稳定一个原先在没有反馈时不稳定的系统 For example: Consider a simple first-order CT system with: The closed-loop system will be stable if the pole is moved into the left half of the s-plane. unstable When a>0, is this system stable?

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Then: The closed-loop system will be stable if the pole is moved into the left half of the s-plane This type of feedback system is referred to as a proportional feedback system ( 比例反馈系统 ), since the signal that is fed back is proportional to the output of the system.

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Another example ： consider the second-order system a)How to choose a to make this system unstable? b)How to make this system stable by using feedback? Check yourself…

Applications and consequences of feedback Step 1.Pole point analysis This system is unstable in each case! 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback

Applications and consequences of feedback Step 2.Consider the use of proportional feedback Recall the discussion of second-order systems in Chap. 6: Obviously, with proportional feedback we can only influence that value of, and consequently, we can't stabilize the system because we can’t introduce any damping. 故单靠比例反馈不能使该二阶系统稳定 For such a system to be stable, must be real and positive, and must be positive 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback

Applications and consequences of feedback When we choose K1 and K2 to guarantee that Then the pole is moved into the left half of the s-plane, and this system are stable. 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Step 3.Consider proportional-plus-derivative feedback 比例加微分的反馈类型

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback 离散时间不稳定系统的稳定反馈应用 例：考虑某种动物繁殖模型, 自然繁殖使每代总数加倍,e[n] 代 表外界影响对总数引起的增减. Is this system stable? 繁殖模型不稳定

Applications and consequences of feedback Adding feedback to make this system stable 减慢增长的一些控制因素 动物迁徙 \ 疾病 \ 自然灾害等外部因素 Suppose that the regulative influences account for the depletion is a fixed proportion of the population 若 ，则系统不稳定 若 ，则系统稳定 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback

23 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback 11.2 Applications and consequences of feedback Sampled-data feedback systems Objective ： DT feedback techniques are of great importance in a wide variety of applications involving CT systems 离散时间反馈技术在涉及连续时间系统的各种应用中也很重要 反馈系统因果性对离散时间反馈信号转换为连续时间信号的过程 施加了限制（理想低通滤波器或任何非因果近似都不允许应用）

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Tracking systems( 跟踪系统 ) Objective ： To have the output track or follow the input 使输出跟踪输入 Example: Consider the DT feedback system denotes the system function of the system whose output is to be controlled represents a compensator, which is the element to be designed Let

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Specializing to, we obtain Specifically, for good tracking performance, we would like to be small. That is, However, if the gain is too large, the closed-loop system may have undesirable characteristics (such as too little damping) or might in fact become unstable. 由此可得反馈系统设计的一个基本原则： 好的跟踪特性要求大的增益 Consequently, for that range of frequencies for which is nonzero, we would like to be large.

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback There are other reasons to limit the gain in a tracking system Because any measuring device used will have inaccuracies and error sources, and we could rewrite Y(z) We would like to be small to minimize the influence of on Good tracking performance requires a large gain or large 跟踪的要求与减小测 量误差的影响是互为 矛盾的 干扰误差源

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Destabilization caused by feedback Feedback can have undesirable effects and can cause instability 反馈也会有一些不希望的后果，引起不稳定 Feedback in audio systems Note that in addition to other audio inputs, the sound coming from the speaker itself may be sensed by the microphone

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback is the amplifier gain represents the attenuation is the propagation delay Note that the output from the feedback path is added to the external input. Could be equivalent to If This system is unstable, which will lead to an excessive amplification and distortion of audio signals. 正反馈 Positive Feedback 例 11.7

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback Example: considering a DT feedback system How to choose b to make this system stable? Check yourself… Step 1.Write the closed-loop system function Step 2.Use the stability criterion to choose b

Applications and consequences of feedback 信号与系统课程组 ©2015 [Chap. 11] 3. Consequences of feedback If the pole is inside the unit circle, the system is stable Step 1.Write the closed-loop system function Step 2.Use the stability criterion to choose b

31 Outline 0. Introduction 1.Linear Feedback Systems 2. Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins 以下不讲 !

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems System stability Depend on Poles distribution Other performancePoles an zeros distribution Feedback can be used to relocate the poles to improve system performance With improper choice of feedback a stable system can be destabilized How to check whether the feedback is appropriate?

33 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems Root-locus analysis is a graphical technique for plotting the closed-loop poles of a rational system function Q(s) or Q(z) as a function of the value of the gain ( 随着可调增益的变化，闭环极点在复平面内的轨迹 ) Could be used both in CT and DT systems Analyze the system performance Determine the system structure and parameters Correct device performance 根轨迹分析法可被用来

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems An introductory example Considering a linear feedback system 根轨迹即为讨论随 的变化，闭环系统函数 的运动轨迹

Root-locus analysis of linear feedback systems Unit circle Pole of closed-loop system Pole of open-loop system 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems  >0 ， the pole of Q(z) moves to the left of z=2;, the pole lies inside the unit circle, and thus the system is stable ;  <0 ， the pole of Q(z) moves to the right of z=2.

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems Consider a CT feedback system 22 The locus of the poles is determined by the algebraic expressions for the system functions of and is not inherently associated with whether the system is a CT or DT system

37 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems Equation for the closed-loop poles If we could not find closed-form expressions for the closed-loop poles ①Plot the location of the pole ②Change the gain ③Plot the root locus The poles are the solutions of the equation 根轨迹方程 可直接由此式勾画根轨迹

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 当 就得出系统闭环极点即为 的极点上 The poles of closed-loop system are the solutions of the equation:

39 当 就得出系统闭环极点即为 的零点上 11.3 Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems The right-hand side of this equation is real, a point s 0 can be a closed-loop pole only if the left-hand side is also real The Angle Criterion （角判据）

41 That is, for s 0 to be a closed-loop pole, we must have Angle criterion 11.3 Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems Then If Then If

43 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems 角判据是确定根轨迹的充分必要条件！！

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems Example 11.1 ： Consider a linear feedback system Please sketch the root locus According to angle criterion 回顾 9.4 节：有理拉普拉斯变换 在复平面上某一点 的相位等于从每个零点到 的零点向量的 相角之和，减去从每个极点到 的极点向量的相角之和

45 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 只有平行于虚轴，且平分极点 -1 和 -2 的那根直线上的点才满足条件 It is same as points in the upper half of the s-plane

48 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems 闭环极点即为开环极点 闭环极点即为开环零点

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems Properties of the root locus To make the sketching of a locus far less tedious, we will discuss some properties of the root locus Assuming zeros poles Regarded as the overall gain

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 根轨迹的起点（ K=0 ） ： G(S)H(S) 极点 终点（ K=  ） ： G(S)H(S) 零点

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 实轴上某一区域右边的零、极点个数之和为奇数，则该区域为 K>0 的根轨迹 ; 实轴上某一区域右边的零、极点个数之和为偶数，则该区域为 K<0 的根轨迹 ; 判断实轴上各段是 否位于根轨迹上

52 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems ①From property 1 the root locus begins at the poles ②From property 3 the entire portion of the real axis between the two poles will lie on the root locus for a positive or negative range of values of K

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

补充：根轨迹的渐近线 性质 4 根据性质 4 ，当开环传递函数中 m < n 时，将有 n  m 条 根轨迹分支沿着与实轴夹角为  a ，交点为  a 的一组渐近 线趋于无穷远处，且有： n  m  1 (k = 0,1, …, n  m  1)

例 设某负反馈系统的开环传递函数为 试确定系统根轨迹条数、起点和终点、渐近线及根轨迹 在实轴上的分布。 解： 解：开环极点 p 1 = 0 、 p 2 =  1 、 p 3 =  5 。 系统的根轨迹有三条分支，分别起始于系统的三个 有限的开环极点，由于不存在有限的开环零点，当 K g  时，沿着三条渐近线趋向无穷远处；三条渐近线在实 轴上的交点

 0 j  实轴上 K>0 的根轨迹分布在 (0 ，  1) 和 (  5 ，   ) 的实轴段上。 60  三条渐近线与正实轴上间的夹角： -2

补充：根轨迹分离点和会合点 两条或两条以上的根轨迹在 s 平面上相遇后立即分开的 点，称为根轨迹的分离点 ( 会合点 ) 。  0 j  z1z1 j1 A p1p1 p2p2 Kg  Kg =0

分离点的性质： 1 ）分离点是系统闭环重根； 2 ）由于根轨迹是对称的，所以分离点或位于实轴上， 或以共轭形式成对出现在复平面上； 实轴上相邻两个开环零（极）点之间（其中之一可 为无穷零（极）点）若为根轨迹，则必有一个分离点； 3 ）实轴上相邻两个开环零（极）点之间（其中之一可 为无穷零（极）点）若为根轨迹，则必有一个分离点； 4 ）在一个开环零点和一个开环极点之间若有根轨迹，该段无 分离点或分离点成对出现。 j   0

确定分离点位置的方法： 式中， z i 、 p j 是系统 G(s)H(s) 的零点和极点。 法一 法一：重根法（极值法） 法二 法二：公式法 设分离点的坐标为 d ，则 d 满足如下公式： k 为分离点处根轨迹的分支数 分离点上，根轨迹的切线与正实轴的夹角称为根轨迹的分离角， 用下式计算： k 为分离点处根轨迹的分支数。

例 求系统根轨迹的分离点。 解：根据上例，系统实轴上的根轨 迹段（  1 ， 0 ），位于两个开环极点之间， 该轨迹段上必然存在根轨迹的分离点。设 分离点的坐标为 d ，则 3d d + 5 = 0 d 1 =  d 2 =  3.53( 不在 K>0 根轨迹 上，舍去 ) 分离点上根轨迹的分离角为 ±90° 。  0 jj d 1 =  0.472

例 已知系统开环传函为 试绘制系统的根轨迹。 解：  0 jj d =  2.47 渐近线与实 轴交点 渐近线与实轴 夹角

64 如何推演根轨迹特性？ 1. 图示出 的极点和零点 2. 利用根轨迹的 4 个性质勾画根轨迹 3. 利用根轨迹其他特性，得到更准确的根轨迹图 11.3 Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

Root-locus analysis of linear feedback systems Example 11.2 ①From property 1 and 2, the root locus begins at the poles one branch terminates at the zeros and the other at infinity 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

66 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems

s =  1.95 渐近线与 实轴交点 渐近线与 实轴夹角 分离点 由两个极点（实数或复数）和一个有限零点组成的开环系统，只要 有限零点没有位于两个实数极点之间，当 K 从零变到无穷时，闭环 根轨迹的复数部分，是以该零点为圆心、零点到分离点为半径的一 个圆或圆的一部分。（可在数学上严格证明）

 0 j  -2 j1 d =  0.59( 舍去 ) d =  3.41 d

70 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems 11.3 Root-locus analysis of linear feedback systems Example 11.3 The root locus analysis method of a DT feedback system are identical to those used in the CT feedback system ZZ

Root-locus analysis of linear feedback systems 信号与系统课程组 ©2015 [Chap. 11] 4. Root-Locus Analysis of Linear Feedback Systems Z Z

72 Outline 0. Introduction 1.Linear Feedback Systems 2. Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion 根轨迹法需要正向通路和反馈通路系统函数的解析表达式， 且需这些变换是有理函数才适用 Nyquist criterion 奈奎斯特判据 Do not provide detailed information concerning the location of the closed-loop poles as a function of K Can be applied to nonrational system functions and in situations in which no analytic description of forward and feedback path system functions

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion Objective ： Outline the basic ideas behind the Nyquist criterion for both CT and DT systems

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion The Encirclement Property 围线性质 P plane W plane

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion More generally

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

《自动控制原理》

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion Example 11.4 P plane W plane P plane W plane

The Nyquist stability criterion P plane Check yourself… 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion The Nyquist criterion for CT LTI feedback systems s plane

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion Example 11.5

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion There are no right-half-plane open-loop poles, and the Nyquist criterion requires that there be no net encirclements of the point -1/K

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion Example 11.6

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion The Nyquist criterion for DT LTI feedback systems Nyquist stability criterion is based on the difference in the number of poles and zeros inside a contour The choice of the contour Similarity between DT and CT Difference between DT and CT

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion In order to make use of encirclement property 例 已证明

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion 单位圆上，

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion

The Nyquist stability criterion 信号与系统课程组 ©2015 [Chap. 11] 5. Nyquist Stability Criterion Example 11.8

93 Outline 0. Introduction 1.Linear Feedback Systems 2. Consequences of Feedback 3. Root-Locus Analysis of Linear Feedback Systems 4. Nyquist Stability Criterion 5.Gain and Phase Margins

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Whether a feedback system is stable Determine how much the gain can be perturbed and how much additional phase shift can be added before it becomes unstable We want to know We introduce and examine the concept of the margin of stability in a feedback system

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins In the actual system How much variation can be tolerated without losing system stability?

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Example 11.9

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins As an alternative, we can identify the gain and phase margins from a log magnitude-phase diagram The phase margin can be read off by locating the intersection of the log magnitude-phase plot with the 0-dB line

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Example P plane

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Example If unavoidable time delay is introduced into this feedback path: How small this delay must be to ensure the stability of the closed- loop system?

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Every point on the curve is shifted to the left

Gain and Phase Margins 信号与系统课程组 ©2015 [Chap. 11] 6. Gain and Phase Margins Example 11.13

103 Chapter 11 Linear Feedback Systems A Summary Review

104 CT and DT Linear feedback 线性反馈 Changes in the parameters in a feedback control system lead to changes in the behavior of the system Root-locus analysis 根轨迹分析法 Plot the poles of the closed-loop system as a function of a gain parameter 根轨迹性质可以在不必进行复杂计算的条件下， 得到一个相当准确的根轨迹图 Nyquist criterion 奈奎斯特判据 其是在用不着得到一个详细的闭环极点位置的条件下， 确定反馈系统稳定性的一种方法 Gain and phase margins 增益和相位裕度 这两个量提供了反馈系统稳定性裕度的一种度量， 检验系统有多大的稳健程度