In 1948, Evans proposed a relatively simple graphic method to solve the root of closed-loop characteristic equation according to the internal relationship between open-loop and closed-loop transfer functions in feedback system. This method is called root locus method. Because the root locus method is intuitive and vivid, it has been widely used in control engineering. The root locus is the locus of the root of the closed-loop characteristic equation moving on the S plane when a parameter of the open-loop system (such as the gain of the root locus) changes from zero to infinity. The root trajectory gain K * is the coefficient corresponding to the first 1 open-loop transfer function.
Before introducing the graphic method, the significance of the root locus is illustrated by directly finding the root. The control system is shown in the above figure. The open-loop transfer function is to track the points on the S plane by using the calculation results and connect them with smooth curves to get the trajectory of the closed-loop pole moving on the S plane when K * (or k) changes from zero to infinity, that is, the root trajectory, as shown in the following figure. In the figure, the root trajectory is represented by a thick solid line, and the arrow indicates the direction in which the two root trajectories move when K * (or k) increases.
One parameter of the open-loop system changes from zero to infinity, and the trajectory of the root of the characteristic equation of the closed-loop system changes in the S plane. One of the parameters mainly refers to the root trajectory gain, but it can also be other real parameters in the system.
Root locus equation
Write the above formula as phase angle and mode condition, and according to these two conditions, we can completely determine the root locus on the S plane and the K* value on the root locus.
Root Trajectory and System Performance
1 stability
When the parameter changes from zero to infinity, the root locus will not cross the imaginary axis and enter the right half plane of S, which means that the system is stable for all k values. If the root trajectory passes through the imaginary axis and enters the right half plane of S, the system will enter an unstable state.
2 steady-state characteristics
In general, the parameter marked on the root trajectory diagram is not the open-loop gain but the root trajectory gain, and the difference between the root trajectory gain and the steady-state error coefficient is only a proportional constant.
3 dynamic characteristics
When all closed-loop poles are located on the real axis, the system is over-damped;
When the poles coincide, the system is a critical damping system;
When the closed-loop pole is a * * * yoke complex pole, the system is an underdamped system, and the unit step response is an attenuation oscillation process, and the overshoot will increase with the increase of k value.
Determine the zero pole of closed-loop system
When the gain of the root trajectory is determined, the transfer function is also determined, so that the zero-pole position of the closed-loop system is determined on the root trajectory, and the stability of the system can be judged by the position.
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Matlab calculates the root locus and gain of the system, and analyzes the root locus of the control system.
I. Concept of Root Trajectory Analysis The so-called root trajectory refers to the trajectory of the root of the characteristic equation of the closed-loop system on the S plane when a parameter of the open-loop system changes from zero to infinity. Generally speaking, this parameter is selected as the gain k of the open-loop system, and the root of the characteristic equation of the closed-loop system is the pole of the closed-loop transfer function without zero pole cancellation. Root locus analysis is a graphic method for analyzing and designing linear time-invariant control systems, which is very simple to use. It can be used to analyze various performances of the system. Stability When the open-loop gain k changes from zero to infinity, the root locus in the graph will not cross the imaginary axis and enter the right half S plane. ...
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Root Trajectory-Automatic Control Principle-Notes
Focus on mastering the changing law of tree roots. So as to design the controller/compensator. "Root"-"Pole" (same concept) first-order system, such as this system, whose transfer function is, exert influence on this system, and its Laplace transform = 1. So the output of the system is, let s+a=0, get the "root" P=-a, and get the time function of this system, where -a is its root when a >; 0, its performance on the time axis is exponential decay. For a general first-order system, if its root is represented on the complex plane (the horizontal axis is real and the vertical axis is imaginary), it must fall on the real axis, as shown by the first-order function. ...
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Expression of open-loop transfer function of first-order system _ Classical control theory (3) Root locus method (closed-loop system)
1 Root Trajectory and System Performance (1) Definition: It refers to the trajectory that the root (closed-loop pole) of the characteristic equation of the closed-loop system changes on the S plane when a parameter of the system changes from zero to infinity. Matlab command: rlocus(sys), and the default change parameter is the gain k of the open-loop transfer function. (2) Function: The root locus can directly give all the information corresponding to the closed-loop system time, and can indicate how the open-loop poles and zeros should change to meet the performance index requirements of a given closed-loop system. The root locus method can solve the roots of higher-order algebraic equations, which is actually to solve the closed-loop characteristic equations. ...
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Transfer function root locus
Root trajectory and time domain response, automatic control, automatic control in all aspects are useful, and root trajectory can be directly derived.
Derivation of closed-loop transfer function of simulink model and frequency domain analysis of Simulink model by Matlab (including implementation program)
The closed-loop transfer function of simulink model is derived, and the frequency domain analysis of Simulink model is carried out by Matlab function, and the implementation program is given.
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Control course-Introduction: 4. Design of Root Trajectory Controller
In this tutorial, we will introduce the root trajectory, show how to create the root trajectory using MATLAB, and demonstrate how to design a feedback controller that meets some performance indicators using the root trajectory. The main MATLAB commands used in this tutorial include: feedback, rlocus, step, controlsystemdesigner. The root locus of the closed-loop pole open-loop transfer function H(s)H(s)H(s) is usually the locus curve of the closed-loop pole formed when the proportional gain KKK changes from 0 to infinity. ...
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Root locus and frequency response
First, let's understand the definition of root. In a closed-loop first-order system, there is a formula in which the transfer function G(s) is S, so it may have zero points and poles. Speaking of Poles, everyone must be very familiar with them. Here, we can understand it as "root" = "pole". How to draw the root locus? Let me briefly mention that the root locus has an open-loop pole pointing to the open-loop zero, which corresponds to the left and right. If there is no corresponding pole, use infinite pole/zero instead (for example, if there is only one pole and there is no zero, then the root locus points from this pole to negative infinity). In addition, the root locus is symmetric about the real axis, and its convergence point and separation point are not important. Interested students can search by themselves. Finally, the stability of the system can be judged by the amplitude of the gain. .
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Determination of Root Trajectory and System Parameters
1, the root locus mentioned above can judge the stability of the system through the pole distribution of the closed-loop transfer function. Of course, there are some simpler discriminant methods, such as Routh stability criterion and leonid hurwicz stability criterion. But they all judge whether the system is stable, so how to judge the degree of stability (stability margin) of the system? Or when system parameters ...
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Root Trajectory of matlab Closed-loop Transfer Function, Design of Automatic Control System —— Series Correction Design Based on Root Trajectory
Similar to the frequency domain method, the root locus method has two methods to design the system: 1) conventional method; 2)Matlab method. The root locus method of Matlab allows visual design, which has the advantages of simple operation, intuitive interface, good interaction and high design efficiency. At present, the commonly used Matlab design methods are: 1) direct programming method; 2) The powerful Rltool tool provided by 2)Matlab control toolbox; 3) applications provided by third parties, such as CTRLLAB. Based on the design idea of root trajectory, this section will focus on the first and second methods. ...
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202005 18 how to draw the root locus of closed-loop characteristic equation quickly?
Q: What is the root locus method? A: Given a closed-loop characteristic equation, with the change of a certain parameter, the root of the closed-loop characteristic equation (that is, the pole of the closed-loop transfer function) changes constantly, and the graph describing the changing trajectory of the root is called the root trajectory graph. Q: Is it an open-loop system or a closed-loop system? Answer: The poles (roots) of the closed-loop system are studied by using the zero poles of the open-loop system. Q: What is the difference between an open-loop transfer function and a closed-loop transfer function? A: The closed-loop transfer function is g1+GH \ frac {g} {1+GH}1+ghg? , the open-loop transfer function is GHGHGH. Q: How to draw a root trajectory quickly and approximately? A: First, introduce the unknown theory of opposites attract (fiction). ......
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Draw the root locus diagram and zero-pole distribution diagram in the textbook
Signals and Systems, Volume II, Third Edition-Zheng Example 1 1-8 The structure of the known feedback system is shown in figure 1 1-2 1. Try to draw its root trajectory diagram. This picture shows the root trajectory of A(s)F(s), not the root trajectory of the whole closed-loop system, nor the root trajectory of A(s). The analytical part of this problem is actually to calculate the intersection point on the horizontal axis. -
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System Performance Analysis Based on Root Trajectory
This section briefly introduces the application of root locus method in system analysis and design through several examples. 5.3. 1 single parameter design example 5.4: the block diagram of the known system is as follows: try to determine the appropriate feedback coefficient k by root locus method, so that the system has a closed-loop * * * yoke complex pole with damping ratio of 0.4. Solution: The open-loop transfer function of the system can be obtained from the block diagram as follows:
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Root locus analysis
Root Trajectory Analysis Based on MATLAB
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How to calculate the root trajectory of matlab closed-loop transfer function, and the intersection of open-loop transfer function with trajectory and imaginary axis
According to the principle of university automatic control, it is not easy to calculate the step of finding the closed-loop root locus diagram by the open-loop transfer function (finding the intersection of the root locus and the imaginary axis). A complex number equals 0, and the result is that the real part equals 0 and the imaginary part equals 0. This is too simple. Ask again: -w 3j-6w 2+9wj+k = 0, (-w 3+9w) -6W 2 How to draw the Nyquist diagram of the open-loop transfer function with matlab% k =10k =10; d=conv( ...
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System performance analysis of root locus method based on automatic control principle
This paper mainly describes the principle of analyzing system stability by root locus method and the method steps of designing parameter Kg according to root locus. Finally, the influence of dominant pole, dipole pair and open-loop zero on the system performance is given.
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Matlab to find system root locus and system gain, experimental report of automatic control principle (root locus of linear system) ...
Experiment 4 Root Trajectory of Linear System 1. Experimental purpose 1. Familiar with some basic programming statements and formats used by MATLAB in control system. 2. Draw the root trajectory of the system with MATLAB statements. 3. Master the graphic method of analyzing system performance with root trajectory. 4. Grasp the influence of system parameter change on the position of characteristic root. Basic knowledge The root locus of MATLAB function refers to the change locus of the root of the characteristic equation on the S plane when a parameter of the system changes from zero to infinity. This parameter is usually selected as the gain k of the open-loop system. The method of manual drawing of root trajectory introduced in textbook ...
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Newly released 8 17 automatic control principle-1- open-loop transfer function and closed-loop transfer function
First of all, the open-loop transfer function is aimed at the closed-loop system, not the open-loop system. For example: open-loop transfer function under input: g (s) = g1H2H3/(1-g2h1) Error transfer function under input: r (s)/e (s) = (1-g2h2)/(.
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Automatic control principle of hot recommendation Chapter 4 Root locus method
This paper lists the basic concepts of root locus method. B-root trajectory equation c is the rule of drawing root trajectory. The root trajectory rule is introduced. C.b Root Trajectory Rule (Conventional Root Trajectory) D Introduction by Root Trajectory Analysis System. The stability and performance index of closed-loop control system are mainly determined by the position of closed-loop system poles on the complex plane. When analyzing and designing the system, it is very meaningful to determine the position of the closed-loop pole (i.e. the characteristic root) in the complex plane: the position of the closed-loop pole in the complex plane determines the performance index of the system stability system, and the position of the closed-loop pole also determines it. ...
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Matlab- automatic control principle draws focus root trajectory according to transfer function.
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The concept of root locus
One parameter of the open-loop system changes from zero to infinity, and the trajectory of the root of the characteristic equation of the closed-loop system changes in the S plane. One of the parameters mainly refers to the root trajectory gain, but it can also be other real parameters in the system.
Root locus equation
Write the above formula as phase angle and mode condition, and according to these two conditions, we can completely determine the root locus on the S plane and the K* value on the root locus.
Root Trajectory and System Performance
1 stability
When the parameter changes from zero to infinity, the root locus will not cross the imaginary axis and enter the right half plane of S, which means that the system is stable for all k values. If the root trajectory passes through the imaginary axis and enters the right half plane of S, the system will enter an unstable state.
2 steady-state characteristics
In general, the parameter marked on the root trajectory diagram is not the open-loop gain but the root trajectory gain, and the difference between the root trajectory gain and the steady-state error coefficient is only a proportional constant.
3 dynamic characteristics
When all closed-loop poles are located on the real axis, the system is over-damped;
When the poles coincide, the system is a critical damping system;
When the closed-loop pole is a * * * yoke complex pole, the system is an underdamped system, and the unit step response is an attenuation oscillation process, and the overshoot will increase with the increase of k value.
Determine the zero pole of closed-loop system
When the gain of the root trajectory is determined, the transfer function is also determined, so that the zero-pole position of the closed-loop system is determined on the root trajectory, and the stability of the system can be judged by the position.
Open CSDN for a better reading experience.
Matlab calculates the root locus and gain of the system, and analyzes the root locus of the control system.
I. Concept of Root Trajectory Analysis The so-called root trajectory refers to the trajectory of the root of the characteristic equation of the closed-loop system on the S plane when a parameter of the open-loop system changes from zero to infinity. Generally speaking, this parameter is selected as the gain k of the open-loop system, and the root of the characteristic equation of the closed-loop system is the pole of the closed-loop transfer function without zero pole cancellation. Root locus analysis is a graphic method for analyzing and designing linear time-invariant control systems, which is very simple to use. It can be used to analyze various performances of the system. Stability When the open-loop gain k changes from zero to infinity, the root locus in the graph will not cross the imaginary axis and enter the right half S plane. ...
Continue to visit
Root Trajectory-Automatic Control Principle-Notes
Focus on mastering the changing law of tree roots. So as to design the controller/compensator. "Root"-"Pole" (same concept) first-order system, such as this system, whose transfer function is, exert influence on this system, and its Laplace transform = 1. So the output of the system is, let s+a=0, get the "root" P=-a, and get the time function of this system, where -a is its root when a >; 0, its performance on the time axis is exponential decay. For a general first-order system, if its root is represented on the complex plane (the horizontal axis is real and the vertical axis is imaginary), it must fall on the real axis, as shown by the first-order function. ...
Continue to visit
Expression of open-loop transfer function of first-order system _ Classical control theory (3) Root locus method (closed-loop system)
1 Root Trajectory and System Performance (1) Definition: It refers to the trajectory that the root (closed-loop pole) of the characteristic equation of the closed-loop system changes on the S plane when a parameter of the system changes from zero to infinity. Matlab command: rlocus(sys), and the default change parameter is the gain k of the open-loop transfer function. (2) Function: The root locus can directly give all the information corresponding to the closed-loop system time, and can indicate how the open-loop poles and zeros should change to meet the performance index requirements of a given closed-loop system. The root locus method can solve the roots of higher-order algebraic equations, which is actually to solve the closed-loop characteristic equations. ...
Continue to visit
Transfer function root locus
Root trajectory and time domain response, automatic control, automatic control in all aspects are useful, and root trajectory can be directly derived.
Derivation of closed-loop transfer function of simulink model and frequency domain analysis of Simulink model by Matlab (including implementation program)
The closed-loop transfer function of simulink model is derived, and the frequency domain analysis of Simulink model is carried out by Matlab function, and the implementation program is given.
Continue to visit
Control course-Introduction: 4. Design of Root Trajectory Controller
In this tutorial, we will introduce the root trajectory, show how to create the root trajectory using MATLAB, and demonstrate how to design a feedback controller that meets some performance indicators using the root trajectory. The main MATLAB commands used in this tutorial include: feedback, rlocus, step, controlsystemdesigner. The root locus of the closed-loop pole open-loop transfer function H(s)H(s)H(s) is usually the locus curve of the closed-loop pole formed when the proportional gain KKK changes from 0 to infinity. ...
Continue to visit
Root locus and frequency response
First, let's understand the definition of root. In a closed-loop first-order system, there is a formula in which the transfer function G(s) is S, so it may have zero points and poles. Speaking of Poles, everyone must be very familiar with them. Here, we can understand it as "root" = "pole". How to draw the root locus? Let me briefly mention that the root locus has an open-loop pole pointing to the open-loop zero, which corresponds to the left and right. If there is no corresponding pole, use infinite pole/zero instead (for example, if there is only one pole and there is no zero, then the root locus points from this pole to negative infinity). In addition, the root locus is symmetric about the real axis, and its convergence point and separation point are not important. Interested students can search by themselves. Finally, the stability of the system can be judged by the amplitude of the gain. .
Continue to visit
Determination of Root Trajectory and System Parameters
1, the root locus mentioned above can judge the stability of the system through the pole distribution of the closed-loop transfer function. Of course, there are some simpler discriminant methods, such as Routh stability criterion and leonid hurwicz stability criterion. But they all judge whether the system is stable, so how to judge the degree of stability (stability margin) of the system? Or when system parameters ...
Continue to visit
Root Trajectory of matlab Closed-loop Transfer Function, Design of Automatic Control System —— Series Correction Design Based on Root Trajectory
Similar to the frequency domain method, the root locus method has two methods to design the system: 1) conventional method; 2)Matlab method. The root locus method of Matlab allows visual design, which has the advantages of simple operation, intuitive interface, good interaction and high design efficiency. At present, the commonly used Matlab design methods are: 1) direct programming method; 2) The powerful Rltool tool provided by 2)Matlab control toolbox; 3) applications provided by third parties, such as CTRLLAB. Based on the design idea of root trajectory, this section will focus on the first and second methods. ...
Continue to visit
202005 18 how to draw the root locus of closed-loop characteristic equation quickly?
Q: What is the root locus method? A: Given a closed-loop characteristic equation, with the change of a certain parameter, the root of the closed-loop characteristic equation (that is, the pole of the closed-loop transfer function) changes constantly, and the graph describing the changing trajectory of the root is called the root trajectory graph. Q: Is it an open-loop system or a closed-loop system? Answer: The poles (roots) of the closed-loop system are studied by using the zero poles of the open-loop system. Q: What is the difference between an open-loop transfer function and a closed-loop transfer function? A: The closed-loop transfer function is g1+GH \ frac {g} {1+GH}1+ghg? , the open-loop transfer function is GHGHGH. Q: How to draw a root trajectory quickly and approximately? A: First, introduce the unknown theory of opposites attract (fiction). ......
Continue to visit
Draw the root locus diagram and zero-pole distribution diagram in the textbook
Signals and Systems, Volume II, Third Edition-Zheng Example 1 1-8 The structure of the known feedback system is shown in figure 1 1-2 1. Try to draw its root trajectory diagram. This picture shows the root trajectory of A(s)F(s), not the root trajectory of the whole closed-loop system, nor the root trajectory of A(s). The analytical part of this problem is actually to calculate the intersection point on the horizontal axis. -
Continue to visit
System Performance Analysis Based on Root Trajectory
This section briefly introduces the application of root locus method in system analysis and design through several examples. 5.3. 1 single parameter design example 5.4: the block diagram of the known system is as follows: try to determine the appropriate feedback coefficient k by root locus method, so that the system has a closed-loop * * * yoke complex pole with damping ratio of 0.4. Solution: The open-loop transfer function of the system can be obtained from the block diagram as follows:
Continue to visit
Root locus analysis
Root Trajectory Analysis Based on MATLAB
Continue to visit
How to calculate the root trajectory of matlab closed-loop transfer function, and the intersection of open-loop transfer function with trajectory and imaginary axis
According to the principle of university automatic control, it is not easy to calculate the step of finding the closed-loop root locus diagram by the open-loop transfer function (finding the intersection of the root locus and the imaginary axis). A complex number equals 0, and the result is that the real part equals 0 and the imaginary part equals 0. This is too simple. Ask again: -w 3j-6w 2+9wj+k = 0, (-w 3+9w) -6W 2 How to draw the Nyquist diagram of the open-loop transfer function with matlab% k =10k =10; d=conv( ...
Continue to visit
System performance analysis of root locus method based on automatic control principle
This paper mainly describes the principle of analyzing system stability by root locus method and the method steps of designing parameter Kg according to root locus. Finally, the influence of dominant pole, dipole pair and open-loop zero on the system performance is given.
Continue to visit
Matlab to find system root locus and system gain, experimental report of automatic control principle (root locus of linear system) ...
Experiment 4 Root Trajectory of Linear System 1. Experimental purpose 1. Familiar with some basic programming statements and formats used by MATLAB in control system. 2. Draw the root trajectory of the system with MATLAB statements. 3. Master the graphic method of analyzing system performance with root trajectory. 4. Grasp the influence of system parameter change on the position of characteristic root. Basic knowledge The root locus of MATLAB function refers to the change locus of the root of the characteristic equation on the S plane when a parameter of the system changes from zero to infinity. This parameter is usually selected as the gain k of the open-loop system. The method of manual drawing of root trajectory introduced in textbook ...
Continue to visit
Newly released 8 17 automatic control principle-1- open-loop transfer function and closed-loop transfer function
First of all, the open-loop transfer function is aimed at the closed-loop system, not the open-loop system. For example: open-loop transfer function under input: g (s) = g1H2H3/(1-g2h1) Error transfer function under input: r (s)/e (s) = (1-g2h2)/(.
Continue to visit
Automatic control principle of hot recommendation Chapter 4 Root locus method
This paper lists the basic concepts of root locus method. B-root trajectory equation c is the rule of drawing root trajectory. The root trajectory rule is introduced. C.b Root Trajectory Rule (Conventional Root Trajectory) D Introduction by Root Trajectory Analysis System. The stability and performance index of closed-loop control system are mainly determined by the position of closed-loop system poles on the complex plane. When analyzing and designing the system, it is very meaningful to determine the position of the closed-loop pole (i.e. the characteristic root) in the complex plane: the position of the closed-loop pole in the complex plane determines the performance index of the system stability system, and the position of the closed-loop pole mainly determines it. ...
Continue to visit
Matlab- automatic control principle draws focus root trajectory according to transfer function.
20 19 Python engineer recruitment standard for unicorn enterprises >; & gt& gt ...
Continue to visit
other
Write comments
comment
1
Thumbs up
step on
share
Go to CSDN APP to read the full text.
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CSDN
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