Chapter7.1: Theoretical Basis of Frequency Domain Analysis

This series of blogs mainly describes the application of Matlab software in automatic control. If you have no theoretical basis for automatic control, please learn the series of blog posts on automatic control first. This series of blogs will not explain the theoretical knowledge of automatic control in detail.
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1. Theoretical basis of frequency domain analysis

1.1 Basic concept of frequency characteristics

1.1.1 Overview of frequency characteristics

• The frequency characteristic of the control system reflects the response performance of the system to the sinusoidal input signal;
• The frequency domain analysis method can directly obtain the frequency characteristics through experiments to analyze the quality of the system, and the application of the frequency characteristic analysis system can draw qualitative and quantitative conclusions, and has obvious physical meanings;
• The method of analyzing and designing a system using frequency characteristics as a mathematical model is called the frequency characteristic method, also known as the frequency response method;
• The basic idea of ​​the frequency response method: treat each variable in the control system as some signals, and these signals are synthesized by many sinusoidal signals of different frequencies, and the movement of each variable is the sum of the system's responses to signals of different frequencies;

1.1.2 Definition of Frequency Characteristics

Frequency characteristics: refers to the relationship characteristics of the ratio of steady-state output to input relative to frequency under the action of a sinusoidal signal;

Frequency characteristic function:
$$G(\mathrm{j}\omega )=\frac{X_o(j}{X_i\left(j\right)\_}=A\left(\omega \right)e^{j\phi \left(\omega \right)}$$
among them：$A\left(\omega \right)=\frac{X_o(oh}{X_i\left(oh\right)}$Called the amplitude-frequency characteristic, $\phi \left(\omega \right)={Phi}_o\left(oh\right)-{Phi}_i\left(\omega \right)$ is called the phase-frequency characteristic;

The frequency characteristic is also expressed as:
$$G(\mathrm{j}\omega )=\frac{X_o(j}{X_i\left(j\right)\_}=p\left(\omega \right)+j\theta \left(\omega \right)$$
where:$p\left(\omega \right)$ is$The\; real\; part\; of\; G\left(j\omega \right)$ , called the real frequency characteristic,$\theta \left(\omega \right)$ is$The\; imaginary\; part\; of\; G\left(j\omega \right)$ is called the imaginary frequency characteristic;

Given the function:
$$\{\begin{array}{llc}& p\left(\omega \right)=A\left(\omega \right)\cos \phi \left(\omega \right)，& i\left(o\right)=A\left(\omega \right)\sin \phi \left(\omega \right)\\ & & \\ & A\left(\omega \right)=\sqrt{p^2\left(oh\right)+i^2\left(oh\right)}，& \phi \left(\omega \right)=\arctan \frac{\theta (\omega }{p\left(\omega \right)}\end{array}$$
When the input is a non-sinusoidal periodic signal, its input can be expanded into a superposition of sine waves using Fourier series, and its output is the superposition of corresponding sine waves; at this time, the system frequency characteristic is defined as the Fourier transform of the system output and the input The ratio of the Fourier transform of the quantity;

1.1.3 Features of frequency domain analysis method

• It is suitable for performance analysis of each link, open loop and closed loop system.

  Using the Nyquist stability criterion, the stability and performance of the closed-loop system can be analyzed according to the open-loop frequency characteristics of the system through the drawing method, without having to solve the characteristic roots of the system, thereby avoiding the difficulty of directly solving the differential equation;

• Frequency characteristics have a clear physical meaning.

  The frequency characteristics of many components can be determined by experiments. For components or systems whose mechanisms are complex or unclear and difficult to write differential equations, the frequency characteristics can be measured by using signal generators and precision measuring instruments in the laboratory;

• There is a definite correspondence between the performance index in the frequency domain and the performance index in the time domain.

  For the second-order system, there is a definite correspondence between the frequency characteristic and the performance index of the time-domain transition process. For the high-order system, there is an approximate correspondence by linking the changes of system parameters and structures with the time-domain transition process index;

• The frequency design can take into account the requirements of both dynamic response and noise suppression.

  When the system has severe noise in certain frequency ranges, the frequency analysis method can be used to design a system that can satisfactorily suppress these noises;

• Among the correction methods, the frequency domain analysis method is very convenient for correction.

  When the performance index of the system is given in the form of amplitude margin, phase margin and error coefficient, it is very convenient to use the frequency analysis method to analyze and design the system;

• The frequency method cannot fully analyze nonlinear systems.

  The frequency method is mainly used in the analysis and research of single-input and single-output linear steady-state systems, and it is also applied in multi-input multi-output linear steady-state systems; in nonlinear, there are only some local typical applications, and it cannot be comprehensively analyzed for nonlinear systems. analysis;

1.1.4 Frequency Domain Performance Indicators

• Resonant frequency ${oh}_r$: Indicates the amplitude-frequency characteristic A ( ω ) A(\omega)The frequency corresponding to the maximum value of$A$$($$\omega$$)\; ;$
• Resonant peak value $M_r$: Indicates the maximum value of the amplitude-frequency characteristic, $M_r$A large value indicates that the system responds strongly to the sinusoidal signal of the frequency, that is, the system has poor stability and the overshoot of the step response is large;
• frequency band ${oh}_b$: Indicates the amplitude-frequency characteristic $The\; amplitude\; of\; A\left(\omega \right)$ decays to$0.707$ times the corresponding frequency;${oh}_b$Large indicates that the system has a strong ability to reproduce fast-changing signals, and the distortion is small, that is, the system is fast, the step response rise time is short, and the adjustment time is short;
• Zero frequency $A\left(0\right)$ : Indicates frequency$oh=$Amplitude at$0$$A\left(0\right)$ represents the final value of the system step response,$A\left(0\right)$ and1 1The difference between$1$$A\left(0\right)$ is closer to$1$ , the higher the system accuracy;

1.2 Expression method of frequency characteristics

1.2.1 Polar plot (Nyquist plot)

• System frequency characteristic representation:
  $$G(\mathrm{j}\omega )=A\left(\omega \right)e^{j\phi \left(\omega \right)}$$

• Use a vector to represent a certain frequency ${oh}_i$Under $G(\mathrm{j}{\omega }_i)$ The length of the vector$A\left({oh}_i\right)$ , the vector polar coordinate angle is$f\left(o_i\right)$，$f\left(o_i\right)$ positive direction is taken counterclockwise, the polar coordinates coincide with the rectangular coordinates, and the vertex of the polar coordinates is at the origin of the coordinates, as shown in the figure below:

  

• Frequency characteristic $G\left(j\omega \right)$ is the input frequencyω \omegaThe complex variable function of$\omega$$\omega Yu$ 0$0\to \infty$ ,$The\; curve\; of\; G\left(j\omega \right)$ change, that is, the trajectory of the vector endpoint, is called the polar coordinate diagram;

• In a polar plot, when $oh={oh}_i$, the projection on the real axis is the real frequency characteristic $p\left(o_i\right)$ , the projection on the imaginary axis is the imaginary frequency characteristic;

1.2.2 Logarithmic coordinate diagram (Bode diagram)

• $The\; Bode$ diagram consists of logarithmic amplitude-frequency characteristics and logarithmic phase-frequency characteristics, as shown in the following figure:

  

• The logarithmic amplitude-frequency characteristic is the logarithmic value of the frequency characteristic $L\left(\omega \right)=20\lg A\left(\omega \right)$ and frequency$The\; relationship\; curve\; of\; \omega$ , the logarithmic phase-frequency characteristic is the phase angle$\phi \left(\omega \right)$ and frequency$The\; relationship\; curve\; of\; \omega$ ;

• The vertical axis of the logarithmic amplitude-frequency characteristic is $L\left(\omega \right)=20\lg A\left(\omega \right)$ in$dB$ (decibel), using a linear scale,$A\left(\omega \right)$ increases by$10$ times,$L\left(\omega \right)$ increases by$20dB$ , the abscissa adopts logarithmic division, that is,ω \omegaAfter taking the logarithm of$\omega \; ,\; it\; is\; the\; equal\; point;$

• The horizontal axis of the logarithmic phase-frequency characteristic adopts logarithmic graduation, and the vertical axis adopts linear graduation, and the unit is ° (degree);

• Log plot ( $Bode$ diagram) advantages:

  • The multiplication of amplitudes is transformed into logarithmic addition operation, which greatly simplifies the drawing work of system frequency characteristics;
  • The horizontal axis adopts logarithmic division, which reduces the scale and expands the frequency field of view, and can represent the system frequency characteristics in a larger frequency range; in a $On\; the\; Bode$ diagram, not only the middle and high frequency bands of the frequency characteristics can be drawn, but also the low frequency band can be drawn, which is beneficial to the analysis and design of the system;
  • Asymptotic logarithmic amplitude-frequency characteristics can be drawn, and standard templates can be made to draw accurate logarithmic frequency characteristics;

1.2.3 Logarithmic phase diagram (Nichols diagram)

• The logarithmic phase diagram is also called $The\; Nichols$ graph is a graph that combines the logarithmic amplitude-frequency characteristics and phase-frequency characteristics when the angular frequency is the parameter variable;

• The logarithmic phase diagram looks like this:

  

• $Nichols$ diagram features: the vertical axis is$L\left(\omega \right)=20\lg A\left(\omega \right)$ in$dB$ (decibel), using linear scale; abscissa using logarithmic scale, unit is ° (degree), frequency$\omega$ is the parameter variable;

1.3 Frequency characteristics of typical links

1.3.1 Typical links

Typical links are divided into: minimum phase link and non-minimum phase link;

Minimum phase link:

• Proportional link: $K，(K>0)$；
• Inertial link: $1/(Ts+1)，(T>0)$；
• First-order differential link: $Ts+1，(T>0)$；
• Solution: $1/(s^2/o_n^2+2zs/{\omega }_n+1)，(o_n>0，0\le g<1)$；
• Given the function: $s^2/o_n^2+2zs/{\omega }_n+1，({oh}_n>0，0\le g<1)$；
• Integral link: $1/s$；
• Differentiation link: $s$；

Non-minimum phase link:

• Proportional link: $K，(K<0)$；
• Inertial link: $1/(-Ts+1)，(T>0)$；
• First-order differential link: $-Ts+1，(T>0)$；
• Solution: $1/(s^2/o_n^2-2zs/{\omega }_n+1)，(o_n>0，0<g<1)$；
• Given the function: $s^2/o_n^2-2zs/{\omega }_n+1，({oh}_n>0，0<g<1)$；

The typical link decomposition of the open-loop transfer function can express the open-loop system as a series of several typical links:
$$G(s)H(s)=\prod _{i=1}^N{G}_i(s)$$
设典型环节的频率特性为：
$$G_i(j\omega )=A_i(\omega ){\mathrm{e}}^{j{\phi }_i(\omega )}$$
则系统开环频率特性为：
$$G(j\omega )H(j\omega )=\left[\prod _{i=1}^N{A}_i(\omega )\right]{\mathrm{e}}^{j\left[{\sum }_{i=1}^N{\phi }_i\left(\omega \right)\right]}$$
system open-loop amplitude-frequency characteristics and open-loop phase-frequency characteristics are:
$$A\left(\omega \right)=\prod _{i=1}^N{A}_i\left(\omega \right)，\phi \left(\omega \right)=\sum _{i=1}^N{Phi}_i\left(\omega \right)$$
system open loop characteristic:
$$L\left(\omega \right)=20\lg A\left(\omega \right)=\sum _{i=1}^{N}20\lg A_i\left(oh\right)=\sum _{i=1}^N{L}_i\left(oh\right)$$

• The open-loop frequency characteristics of the system are represented by the synthesis of the frequency characteristics of the typical links that make up the open-loop system;
• The open-loop logarithmic frequency characteristic of the system is expressed as the superposition of logarithmic frequency characteristics of typical links;

1.3.2 Frequency characteristics of typical links

Some characteristics of the typical link frequency characteristic curve:

1. Non-minimum phase and corresponding minimum phase links

   The proportional link of minimum phase $G(s)=K(K>0)$ , referred to as the proportional link, the amplitude-frequency and phase-frequency characteristics are as follows:
   $$A\left(\omega \right)=K，\phi \left(\omega \right)=0°$$
   non-minimum phase proportional link$G\left(s\right)=-K(K>0)$ , the amplitude-frequency and phase-frequency characteristics are as follows:
   $$A\left(\omega \right)=K，\phi \left(\omega \right)=-$$ The inertia link G ( s ) of the minimum phase of 180° = 1 T s + 1 , ( T > 0 ) G(s)=\displaystyle\frac{1}{Ts+1}, (T>
   0$$)$$$G(s)=\frac{1}{Ts+1}，(T>0)$ , the amplitude-frequency and phase-frequency characteristics are:
   $$A\left(\omega \right)=\frac{1}{\sqrt{1+T^2{o}^2}},\phi \left(\omega \right)=-\arctan T\omega$$
   non-minimum phase inertia link$G\left(s\right)=\frac{1}{-Ts+1}，(T>0)$ , the amplitude-frequency and phase-frequency characteristics are:
   $$A\left(\omega \right)=\frac{1}{\sqrt{1+T^2{o}^2}},\phi \left(\omega \right)=\arctan T\omega$$

   • The minimum phase inertia link and the non-minimum phase inertia link have the same amplitude-frequency characteristics, but the phase-frequency characteristics have opposite signs, and the amplitude-phase curves are symmetrical about the real axis;
   • The minimum phase inertia link and the non-minimum phase inertia link have the same logarithmic amplitude-frequency curve, and the logarithmic phase-frequency curve is symmetrical about the 0° line;
   • The above two features are applicable to the oscillation link and the non-minimum phase oscillation link, the first-order differential link and the non-minimum phase first-order differential link, the second-order differential link and the non-minimum phase second-order differential link;

2. Typical links where the transfer functions are reciprocals of each other

   In the typical link of the minimum phase, the transfer functions of the integral link and the differential link, the inertial link and the first-order differential link, the oscillation link and the second-order differential link are reciprocals of each other, and the relationship is as follows: G 1 ( s ) = 1 G 2 ( s
   $$G_1(s)=\frac{1}{G_2\left(s\right)}$$
   任$G_1\left(j\omega \right)=A_1\left(o\right)e^{j{\phi }_1\left(\omega \right)}$ ,then:
   $$\{\begin{array}{ll}& {Phi}_2\left(oh\right)=-f_1\left(oh\right)\\ & \\ & L_2\left(oh\right)=20\lg A_2\left(oh\right)=20\lg \frac{1}{A_1\left(oh\right)}=-L_1(oh\end{array}$$
   In a typical link where the transfer functions are reciprocals, the logarithmic amplitude-frequency curve is symmetrical about the 0dB line, and the logarithmic phase-frequency curve is symmetrical about the 0° line; this conclusion is also applicable in the non-minimum phase link;

3. Oscillation link and second order differential link

   The transfer function of the oscillation link is:
   $$G(s)=\frac{1}{(s/o_n{)}^2+2z(s/{\omega }_n)+1};{oh}_n>0，0<g<1$$
   Frequency characteristics of the oscillation link:
   $$A\left(\omega \right)=\frac{1}{\sqrt{{\left(\begin{array}{c}1-\frac{{oh}^2}{{oh}_n^2}\end{array}\right)}^2+4g^2\frac{{oh}^2}{{oh}_n^2}}}$$

   $$\phi \left(\omega \right)=-\arctan \frac{2g\frac{}{{oh}_n}}{1-\frac{{oh}^2}{{oh}_n^2}}=\{\begin{array}{ll}& -\arctan \frac{2g\frac{}{{oh}_n}}{1-\frac{{oh}^2}{{oh}_n^2}},oh\le {oh}_n\\ & \\ & -\left(\begin{array}{c}180-\arctan \frac{2g\frac{}{{oh}_n}}{\frac{{oh}^2}{{oh}_n^2}-1}\end{array}\right),oh>{oh}_n\end{array}$$

   $\phi \left(0\right)=0°，\phi (\infty )=-180°$ , the phase-frequency characteristic curve decreases monotonously from 0° to -180°; when$oh={oh}_n$时，$f(o_n)=-90°$，$A\left({oh}_n\right)=\frac{1}{2g},$ the intersection point of the oscillation link and the imaginary axis is$-\mathrm{j}\frac{1}{2g}$；

   $A(0)=1，A(\infty )=0$，求$A\left(\omega \right)$的极值，
   $$\frac{dA(\omega }{\mathrm{d}\omega }=\frac{-\left[-\frac{2\omega }{{\omega }_n^2}\left(1-\frac{{\omega }^2}{{\omega }_n^2}\right)+4{\zeta }^2\frac{\omega }{{\omega }_n^2}\right]}{{\left[{\left(1-\frac{{oh}^2}{{oh}_n^2}\right)}^2+4{\zeta }^2\frac{{\omega }^2}{{\omega }_n^2}\right]}^{\frac{3}{2}}}=0$$
   得谐振频率：
   $${\omega }_r={\omega }_n\sqrt{1-2{\zeta }^2}，0<\zeta \le \sqrt{2}/2$$
   谐振峰值：
   $$M_r=A({\omega }_r)=\frac{1}{2\zeta \sqrt{1-{\zeta }^2}}，0<\zeta \le \sqrt{2}/2$$
   当$0<\zeta <\frac{\sqrt{2}}{2}$时，
   $$\frac{\mathrm{d}{M}_r}{\mathrm{d}\zeta }=\frac{-(1-2g^2)}{g^2(1-g^2{)}^{\frac{3}{2}}}<0$$
   ${oh}_r，M_r$Both are the damping ratio ζ \zetaDecreasing function of$\zeta$$(0<g\le \frac{\sqrt{2}}{2})$；当$0<g<\frac{\sqrt{2}}{2}$时，且$\omega \in (0,{\omega }_r)$时，$A(\omega )$单调增；$\omega \in ({\omega }_r,\infty )$时，$A(\omega )$单调减；当$\frac{\sqrt{2}}{2}<\zeta <1$时，$A(\omega )$单调减；

   The transfer function of the second-order differential link is the reciprocal of the transfer function of the oscillation link. According to the symmetry, the logarithmic frequency curve of the second-order differential link can be obtained: { A ( 0 ) = 1 φ ( 0 )
   $$\{\begin{array}{ll}& A(0)=1\\ & \phi \left(0\right)=0°\end{array}，\{\begin{array}{ll}& A({oh}_n)=2g\\ & f\left(o_n\right)=90°\end{array}，\{\begin{array}{ll}& A(\infty )=\infty \\ & \phi (\infty )=180°\end{array}$$
   When the damping ratio $\frac{\sqrt{2}}{2}<g<1$ o'clock，$A\left(\omega \right)$ increases monotonically from 1 to$\infty$ ; when the damping ratio$0<g<\frac{\sqrt{2}}{2}$,from $oh\in (0,{oh}_r)$ ,$Let\; A\left(\omega \right)$ be the 1st set
   $$\{\begin{array}{ll}& A({oh}_r)=2g\sqrt{1-g^2}<1\\ & {oh}_r=oh\sqrt{1-2g^2}\end{array}$$
   在$oh\in ({oh}_r,\infty )$ ,$A\left(\omega \right)$ increases monotonously;

4. Logarithmic amplitude-frequency asymptotic characteristic curve

   In order to simplify the plotting of the logarithmic amplitude-frequency curves of the inertial link, the first-order differential link, the oscillation link and the second-order differential link, the low-frequency and high-frequency asymptotes are commonly used to approximate the logarithmic amplitude-frequency curve, which is called the logarithmic amplitude-frequency curve Near characteristic curve;

   The logarithmic amplitude-frequency asymptotic characteristic of the inertial link is:
   $$L_a\left(oh\right)=\{\begin{array}{ll}0,& oh<\frac{1}{T}\\ -20\lg T\_& oh>\frac{1}{T}\end{array}$$
   

   The low frequency part is a zero decibel line, and the high frequency part is a slope of $-20dB/dec$ straight line, two straight lines intersect at$oh=\frac{1}{T}$, said frequency $\frac{1}{T}$is the handover frequency of the inertial link; using the asymptotic characteristics to approximate the logarithmic amplitude-frequency characteristics, there is an error: $\Delta L(\omega )=L\left(\omega \right)-L_a\left(\omega \right)$ , the error is largest at the handover frequency, about$-3dB$ ; the logarithmic amplitude-frequency asymptotic characteristic curves of the first-order differential link and the non-minimum phase first-order differential link and the inertial link are equal to$The\; 0dB$ lines are mirror images of each other;

   Determine the quantity of the solution:
   $$L\left(\omega \right)=-20\lg \sqrt{{\left(1-\frac{{oh}^2}{{oh}_n^2}\right)}^2+4g^2\frac{{oh}^2}{{oh}_n^2}}$$
   当$oh<<{oh}_n$When , $L\left(\omega \right)\approx 0$ , the low frequency asymptote is the 0dB line; when$oh>>{oh}_n$时，$L\left(\omega \right)=-40\lg \frac{}{{oh}_n}$, the high frequency asymptote is over $({oh}_n,0)$ , a straight line with a slope of -40dB/dec, and the handover frequency of the oscillation link is${oh}_n$,Let us define the following:
   $$L_a\left(oh\right)=\{\begin{array}{ll}0,& oh<{oh}_n\\ -40\lg \frac{}{{oh}_n},& oh>{oh}_n\end{array}$$
   The logarithmic amplitude-frequency asymptotic characteristic curves of the non-minimum phase oscillation link and the oscillation link are the same, and the logarithmic amplitude-frequency asymptotic characteristic curves of the second-order differential link and the non-minimum phase second-order differential link and the oscillation link are symmetrical about the 0dB line;

   The equation of a line in semi-logarithmic coordinates is:
   $$k=\frac{L_a({oh}_2)-L_a({oh}_1)}{\lg {oh}_2-\lg {oh}_1}$$
   Excludes: $[{oh}_1,lg(o_1)]、[{oh}_2,lg(o_2)]$ are two points on the line,$k(dB/dec)$ is the slope of the line;

1.3.3 Open-loop amplitude-phase characteristic curve

The method of drawing a rough open-loop amplitude-phase characteristic curve:

• The starting point of the open-loop amplitude-phase characteristic curve $(oh=0_{+})$ and end point$(oh=\infty )$；

• The intersection point of the open-loop amplitude-phase characteristic curve and the real axis. Let $oh={oh}_x$时，$G(\mathrm{j}{\omega }_x)H(\mathrm{j}{\omega }_x)$ is:
  $$In\left[G\right(j{\omega }_x\left)H\right(j{\omega }_x)]=0或\phi ({\omega }_x)=\angle [G(\mathrm{j}{\omega }_x\left)H\right(j{\omega }_x)]=k\pi ，k=0,\pm 1，\pm 2，\dots$$
  其中：${\omega }_x$称为穿越频率；

  开环频率特性曲线与实轴交点坐标值为：
  $$\mathrm{Re}[G(\mathrm{j}{\omega }_x)H(\mathrm{j}{\omega }_x)]=G(\mathrm{j}{\omega }_x)H(\mathrm{j}{\omega }_x)$$

• 开环幅相特性曲线的变化范围(象限、单调性)；

绘制概略开环幅相特性曲线规律小结：

• 开环幅相特性曲线的起点，取决于比例环节$K$和系统积分或微分环节的个数$\nu$(系统型别)；

  • $\nu <0$，起点为原点；
  • $\nu =0$，起点为实轴上的点$K$处，$K$为系统开环增益，$K$有正负之分；
  • $n>0$，设$n=4k+i(k=0,1,2,\dots ;i=1,2,3,4),$ then$K>0$ is$i\times (-90°)$ at infinity,$K<0$ is$i\times (-90°)-180°$ to infinity;

• The end point of the open-loop amplitude-phase characteristic curve depends on the order sum of the minimum phase link and the non-minimum phase link in the numerator and denominator polynomial of the open-loop transfer function;

  Let the order of the numerator and denominator polynomials of the system open-loop transfer function be $m$ and$n$ , write down$In\; addition\; to\; K$ , the order sum of the minimum phase link in the numerator polynomial is$m_1$, the order sum of the non-minimum phase link is $m_2$, the order sum of the minimum phase link in the denominator polynomial is $n_1$, the order sum of the non-minimum phase link is $n_2$, then:
  $$m=m_1+m_2,n=n_1+n_2$$

  $$\phi (\infty )=\{\begin{array}{ll}[(m_1-m_2)-(n_1-n_2)]\times 90°，& K>0\\ & \\ [(m_1-m_2)-(n_1-n_2)]\times 90°-180°,& K<0\end{array}$$

  When the open-loop system is the minimum phase system, if:
  $$\begin{array}{rlrl}& n=m,& & G(\mathrm{j}\infty )H(\mathrm{j}\infty )=K^{\ast }\\ & & & \\ & n>m,& & G(j\infty )H(j\infty )=0\angle [(n-m)\times (-90°)]\end{array}$$
  where: $K^{\ast }$ is the open-loop root locus gain of the system;

• If there is a constant amplitude oscillation link in the open-loop system, the multiplicity $l$为正整数，即开环传递函数具有如下形式：
  $$G(s)H(s)=\frac{1}{(\frac{s^2}{{\omega }_n^2}+1{)}^l}{G}_1(s)H_1(s)$$
  $G_1(s)H_1(s)$不含$\pm \mathrm{j}{\omega }_n$的极点，则当$\omega$趋于${\omega }_n$时，$A\left(\omega \right)$早于无百，而：
  $$\begin{array}{rl}& f(o_{n^{-}})\approx {Phi}_1\left({oh}_n\right)=\angle [G_1(j{\_}_n)H_1(j{\_}_n)]\\ & \\ & f\left(o_{n^{+}}\right)\approx {Phi}_1\left({oh}_n\right)-l\times 180°\end{array}$$
  即$\phi \left(\omega \right)$在$oh={oh}_n$Nearby, phase angle abrupt change $-l\times 180°$；

1.3.4 Open-loop logarithmic frequency characteristic curve

Open-loop logarithmic amplitude-frequency asymptotic characteristics of the system:
$$L_a\left(oh\right)=\sum _{i=1}^N{L}_{a_i}\left(\omega \right)$$
For any open-loop transfer function, it is decomposed according to typical links, and the typical links that make up the system are divided into three parts:

• $\frac{K}{s^n}$或$\frac{-K}{s^n}(K>0)$；
• The first-order link, including the inertial link, the first-order differential link and the corresponding non-minimum phase link, the handover frequency is $\frac{1}{T}$；
• The second-order link, including the oscillation link, the second-order differential link and the corresponding non-minimum phase link, the handover frequency is ${oh}_n$；

记${oh}_{\min }$is the minimum handover frequency, which means $oh<{oh}_{\min }$The frequency range is the low frequency band;

Open-loop logarithmic amplitude-frequency asymptotic characteristic curve drawing steps:

1. Typical link decomposition of open-loop transfer function;

2. Determine the handover frequency of the first-order link and the second-order link, and mark each handover frequency on the $On\; the\; \omega$ axis;

3. Draw the asymptotic characteristic line in the low frequency band: at $oh<{oh}_{\min }$In the frequency band, the slope of the amplitude-frequency asymptotic characteristic of the open-loop system depends on $\frac{K}{{oh}^n}$, so the slope of the line is $-20\nu dB/dec$ ;to obtain the low frequency asymptote, it is necessary to determine a point on the line, the method is as follows:

   • Method 1: At $oh<{oh}_{\min }$In the range, choose a point ${oh}_0$,Determine $L_a({oh}_0)=20\lg K-20n\lg {oh}_0$；
   • Method 2: Take the frequency as a specific value ${oh}_0=1$ , then$L_a(1)=20\lg K$；
   • Method 3: Take $L_a\left({oh}_0\right)$ is the special value 0, there is$\frac{K}{{oh}_0^{}}=1$ ,则${oh}_0=K^{\frac{1}{n}}$；

   过点$({oh}_0,L_a({oh}_0))$在$oh<{oh}_{\min }$The range can be used as a slope of -20 $\nu$ dB/dec straight line;

4. 作$oh\ge {oh}_{\min }$Frequency band asymptotic characteristic line; at each handover frequency point, the slope changes, and the change rule depends on the type of typical link corresponding to the handover frequency. When multiple links in the system have the same handover frequency, the slope at the handover frequency point The change of should be the algebraic sum of the slope change values ​​corresponding to each link;

The change table of the slope at the handover frequency point:

1.3.5 Delay link and delay system

The link in which the output quantity reproduces the change of the input quantity without distortion after a constant delay is called the delay link; the system containing the delay link is called the delay system;

The time-domain expression of the input and output of the delay link is:
$$c(t)=1(t-t)r(t-\tau )$$
where:$\tau$ is the delay time;

The transfer function of the delay link is:
$$G(s)=\frac{C(s)}{R(s)}={\mathrm{e}}^{The\; frequency\; characteristic\; of\; -\tau s}$$
delay link is:
$$G(j\omega )=e^{-jto}=1\cdot \angle (-57.3t\omega )$$

1.4 Performance analysis of open-loop and closed-loop frequency characteristics

1.4.1 Relationship between open-loop logarithmic frequency characteristics and time domain response

1. Low frequency band analysis.

   • Low frequency band: refers to $L\left(\omega \right)=20\lg The\; asymptote\; of\; |G\left(j\omega \right)|$ is in the frequency band before the first corner frequency, and the characteristics of this frequency band are completely determined by the integral link and the open-loop magnification;

   • Low-frequency characteristics of low frequency:
     $$L_d\left(oh\right)=20\lg K-20n\lg \omega$$
     where:$K$ is the open-loop magnification,$\nu$ is the number of integral links in the open-loop transfer function;

   • The frequency characteristics of the low frequency band determine the steady-state performance of the system;

2. Mid-band analysis.

   • Middle frequency band: refers to the open-loop logarithmic amplitude-frequency characteristic curve at the open-loop cut-off frequency ${oh}_c$near ( $Near\; 0dB$ ), the characteristics of this frequency band reflect the stability and rapidity of the dynamic response of the closed-loop system;
   • The dynamic characteristics of the time domain response mainly depend on the shape of the mid-frequency band;
   • Three parameters that reflect the shape of the mid-band: open-loop cut-off frequency ${oh}_c$, the slope of the mid-frequency band, and the width of the mid-frequency band;
   • The slope of the frequency band in the open-loop logarithmic amplitude-frequency characteristic is preferably $-20dB/dec$ , and the length is desired to be as long as possible to ensure that the system has sufficient phase angle margin; the slope of the middle frequency band is$-\; At40dB/dec$ , the frequency range occupied by the mid-frequency band should not be too long, otherwise the phase angle margin is very small; if the slope of the mid-frequency band is smaller, the system will be difficult to stabilize;
   • Cutoff frequency ${oh}_c$The higher the value, the stronger the ability of the system to reproduce the signal, and the better the rapidity of the system;

3. High frequency analysis.

   • High frequency band: refers to the frequency band where the open-loop logarithmic amplitude-frequency characteristic curve is after the middle frequency band. The shape of the high frequency band mainly affects the initial section of the time domain response;
   • When performing system analysis, the high-frequency band can be approximated, that is, a small inertial link is used to equivalent multiple small inertial links, and the time constant of the equivalent small inertial link is equal to the time constant of the replaced multiple small inertial links. and;
   • The amplitude of the open-loop logarithmic amplitude-frequency characteristic of the system in the high-frequency band directly reflects the system's ability to suppress high-frequency interference signals; the lower the amplitude of the high-frequency part, the stronger the anti-interference ability of the system;

4. The shape of each frequency band of a good system.

   • The low frequency band must have a certain height and slope;
   • The slope of the mid-band is preferably $-20dB/dec$ with sufficient width;
   • The high-frequency band adopts the characteristics of rapid attenuation to suppress unnecessary high-frequency interference;

1.4.2 Relationship between open-loop frequency characteristics and time-domain response

The mutual conversion formula of the frequency domain index and time domain index of a typical second-order system:
$$\begin{array}{rl}& {oh}_c=\sqrt{\sqrt{1+4g^4}-2g^2}{oh}_n,c(o_c)=\arctan \frac{2}{\sqrt{\sqrt{1+4g^4}-2g^2}}\end{array}$$
高阶系统的频域指标和时域指标相互转换公式：
$$\begin{array}{rl}& {\sigma }_p=0.16+0.4\left(\frac{1}{\sin \gamma }-1\right)，35°\le \gamma \le 90°\\ & \\ & t_s=\frac{\pi }{{\omega }_c}\left[2+1.5\left(\frac{1}{\sin \gamma }-1\right)+2.5{\left(\frac{1}{\sin \gamma }-1\right)}^2\right]\end{array}$$

1.4.3 闭环频率特性与时域响应的关系

A typical second-order system closed-loop transfer function is:
$$\Phi \left(s\right)=\frac{{oh}_n^2}{s^2+2z{o}_{n}s+{oh}_n^2}$$
The closed-loop frequency characteristics of the system can be obtained as follows:
$$\Phi \left(j\omega \right)=\frac{{oh}_n^2}{\left(j\right)\_{}^2+j2z{o}_{n}oh+{oh}_n^2}$$
The closed-loop amplitude-frequency characteristic of the system is:
$$M\left(\omega \right)=\frac{{oh}_n^2}{\sqrt{({oh}_n^2-{oh}^2{)}^2+(2z{o}_{n}oh{)}^2}}$$
电视题环相镇characteristics：
$$\phi \left(\omega \right)=-\arctan \frac{2z{o}_n}{{oh}_n^2-{oh}^2}$$
The resonance peak M r M_r of the second order system$M_r$and time-domain overshoot $p_p$Solution:
$$p_p={\mathrm{e}}^{-zp/\sqrt{1-g^2}}\times 100\%，M_r=\frac{1}{2g\sqrt{1-g^2}}$$

• Resonant peak value $M_r$Only with the damping ratio $\zeta$ related, overshoot$p_p$It also depends only on the damping ratio $g$ ;

• $The\; smaller\; \zeta$ ,$M_r$The faster it increases, the overshoot $p_p$Also very large, more than $40\%$ , at this time the system generally does not meet the requirements of the transient response index;

• 当$0.4<g<0.707$ ,$M_r$One $p_p$The changing trend is basically the same, at this time the resonance peak value $M_r=1.2～1.5$ ,interpretation$p_p=20\%to30\%$ , the system response is ideal;

• ζ $g>0.707$ , no resonance peak,$M_r$ฎ$\sigma \%$ correspondence no longer exists, so in design generally$The\; value\; of\; \zeta$ is between0.4 and 0.7 0.4 and 0.7Between$0.4$$and$$0.7\; ;$

• The resonant frequency of the second order system ${oh}_r$vs peak time $t_p$Functional:
  $$t_p{oh}_r=\frac{Pi\sqrt{1-2g^2}}{\sqrt{1-g^2}}$$

• ζ \zetaWhen$\zeta$${oh}_r$vs peak time $t_p$Inversely proportional to ${oh}_r$The larger the value, $t_p$The smaller the value, the faster the system time response;

• 二阶系统闭环截止频率${\omega }_b$与过渡过程时间$t_s$间的关系：
  $${\omega }_b{t}_s=\frac{3～4}{\zeta }\sqrt{1-2{\zeta }^2+\sqrt{2-4{\zeta }^2+4{\zeta }^4}}$$

• 当阻尼比$\zeta$给定后，闭环截止频率${\omega }_b$与过渡过程时间$t_s$成反比，${\omega }_b$越大，系统响应速度越快；