[For personal use] Automatic Control Chapter 4 Root Locus

Q stands for questions that help understand concepts
K stands for fragmented knowledge points in the combing process
U stands for unsolved problems

Basic Concepts of Traditional Root Locus

Q1: What is the root locus

根轨迹是系统所有闭环极点的集合It reflects the root locus gain $K^{\ast }$ varies from 0 to$+\infty$ ,the closed-loop polesin the complex domain

K1.1 What is stability related to

1. Stability refers to the ability of the system to return to the original equilibrium state after the disturbance disappears. That is, for an ideal pulse $\delta \left(t\right)$ , system impulse response$c\left(t\right)$ should satisfy$$\underset{t\to \infty }{\lim }c(t)=0$$
2. It can be proved that the necessary and sufficient condition for the stability of the linear system is: all roots of the characteristic equation of the closed-loop system have negative real parts, or in other words, the poles of the closed-loop transfer function are all located in the left half plane of s

K1.2 root locus gain $K^{\ast }$ and open loop gainKKWhat's the matter with$K$

Open loop gain $K$ is tail 1, root locus gain$K^{\ast }$ first 1

Q2: What is the known quantity of the question form/question?

Generally speaking, the title will give a system, and the open-loop transfer function is $G\left(s\right)$ , feedback transfer$H\left(s\right)$ are known [sometimes unit negative feedback,$H(s)=1$ ]
According to the open-loop transfer function, some information of the closed-loop poles can be determined

K2.1 The relationship between closed-loop zero-pole and open-loop zero-pole✨

1. Open loop root locus gain $K^{\ast }=K_G^{\ast }{K}_H^{\ast }$. For a unity negative feedback system, the closed-loop gain is simply the open-loop gain.
2. Closed-loop transfer function $The\; zero\; point\; of\; \Phi \left(s\right)$ is KG ∗ K_G^*ofthe openandthe pole point of the feedback transfer function$K_G^{\ast }$times
3. Closed-loop transfer function $\Phi \left(s\right)$ pole andopen-loop zero,open-loop pole,root locus gain$K^{\ast }$ All related
   $$\Phi \left(s\right)=\frac{K_G^{\ast }{\prod }_{i=1}^{f(open\; loop}\left(s-z_i\right){\prod }_{j=1}^{h(feedback}\left(s-p_j\right)}{{\prod }_{i=1}^{n(total}\left(s-p_i\right)+K^{\ast }{\prod }_{j=1}^{m(total}\left(s-z_j\right)}$$

Q3: How to understand the amplitude/phase angle condition?

The pole of the closed-loop transfer function should satisfy $$1+G(s)H(s)=0$$
幅值条件和相角条件如下
$$\begin{array}{l}|G(s)H(s)|=\frac{K^{\ast }|s-z_1|\cdots |s-z_m|}{|s-p_1||s-p_2|\cdots |s-p_n|}=K^{\ast }\frac{{\prod }_{i=1}^m|\left(s-z_i\right)|}{{\prod }_{j=1}^n|\left(s-p_j\right)|}=1\\ \\ \angle G(s)H(s)={\sum }_{i=1}^m\angle \left(s-z_i\right)-{\sum }_{j=1}^n\angle \left(s-p_j\right)=(2k+1)\end{array}$$
How to understand the phase angle condition?
The root trajectory equation is actually a vector equation, so it can be considered geometrically.
$s$ points to all zero points$z_i$The sum of the angles between the vector of and the direction of the positive real axis - $s$ points to all poles$p_j$The sum of the angles between the vector of and the positive real axis direction = negative real axis direction angle $(2k+1)p$

K3.1 $\frac{{\prod }_{i=1}^n(s-z_i)}{{\prod }_{j=1}^m(s-p_j)}$Angles of such fractions

After simplification, the included angle of the fraction is " the included angle of the numerator - the included angle of the denominator "

K3.2 The phase angle condition is to determine ssSufficient and Necessary Conditions for the Root Locus of$the\; S-\; Plane$

K3.3 Need to determine the K ∗ K^* corresponding to a point on the root locus$K^{\ast }$ , the modulus condition is required

System Performance Analysis

Q4A: Increase the effect of open-loop zero-pole

In fact, the concept of this block is relatively abstract and general. In the later stage, you can consider combining the phase angle margin to learn the
open-loop zero point:
to a certain extent, increasing the open-loop zero point can improve the dynamic performance and stability of the closed-loop system. Left [at the zero point] curved [because the root locus will point to the zero point]

K4A.1 Open loop zero point is equivalent to KD proportional differential control

K4A.2 The influence of open-loop zero point on system stability and dynamic performance is sometimes contradictory✨

As shown in the figure, the stability of figure (b) is better, but the dynamic performance of figure (a) is better than that of figure (b)
[because the dominant pole of figure (a) is a conjugate pole, the response is fast and the adjustment time is short; (b) The dominant pole is on the negative real axis, which is equivalent to a first-order system, and the response speed is slow and the adjustment time is long. 】

Open-loop poles:
Open-loop poles will degrade the dynamic performance of the closed-loop system

Q4B: Increase the effect of closed-loop zero-pole

In fact, the concept of this piece is relatively abstract and general, and it can be considered in combination with phase angle margin learning in the later stage

Closed loop zero:

Stability:
Adding closed-loop zeros does not change the root locus and does not affect stability

Dynamic performance:
Increasing the closed-loop zero point is equivalent to reducing the damping of the system , so that the transition process of the system has a tendency to overshoot, and this effect will increase as the closed-loop pole approaches the coordinate origin.

closed loop pole

The pole close to the imaginary axis is called the dominant pole [refer to K4A.1 analysis ]
closed loopreal numberThe role of the dominant pole is equivalent to increasing the damping of the system, so that the peak time lags and the overshoot decreases.

K4B.1 Adding closed-loop zeros does not change system stability and root locus

The reason is simple, the two are only related to the closed-loop poles of the system.

K4B.2 Closed-loop real poles imply smoothing, overshoot reduction complex poles imply oscillations

K4B.3 The closed-loop real number zero point advances the peak time, increases the overshoot, reduces the system damping, and the closer to the origin, the stronger the effect

K4B.4 The closed-loop real pole makes the peak time lag, the overshoot decreases, and the system damping increases. The closer to the origin, the stronger the effect

UQ❓ How to understand the role of closed-loop zero-pole?

UQ2 What is a smoothing filter

couple

If the closed-loop poles and closed-loop zeros are close together, they are called dipoles .
As long as the dipoles are not close to the origin of the coordinates, their effect on the dynamic performance of the system is minimal.

Q5 Is there a situation where the conjugate (second-order) closed-loop pole is not the dominant pole [as shown in the figure]

Yes, exist. See K4A.1 Analysis

K5.1 The pole of the input signal is not within the selection range of the dominant pole

Q6