abstract
I. Basic concepts
1. control: the operation of a person or a control device to make the controlled object act according to a certain purpose.
2. Input signal: artificially given, also called given quantity.
3. Output signal: controlled quantity. It represents the state and performance of an object or process.
4. Feedback signal: the signal drawn from the output end or intermediate link and transmitted directly or after transformation to the input end comparison element, or the signal drawn from the output end and transmitted directly or after transformation to the intermediate link comparison element.
5. Deviation signal: the output of the comparison element is equal to the difference between the input signal and the main feedback signal.
6. Error signal: the difference between the expected value and the actual value of the output signal.
7. Interference signal: the signal from the inside or outside of the system that interferes with and destroys the predetermined performance and output of the system.
Second, the basic way of control
1. Open-loop control: A system whose output has no control effect on the system, or a system without feedback loop is called an open-loop control system.
2. Closed-loop control: The output control system of the system or the system with feedback loop in the system is called closed-loop control system.
Third, the basic composition of the feedback control system
1. given component: the link used to give the input signal to determine the target value (or given value) of the controlled object.
2. Measuring element: used to detect the controlled quantity, usually appearing in the feedback loop.
3. Comparison element: used to compare the actual output value detected by the measuring element with the input value given by the given element after conversion, and find out the deviation between them.
4. Amplifier: used to amplify the deviation signal given by the comparator and drive the actuator to control the controlled object with sufficient power.
5. Actuator: used to directly drive the controlled object to change the controlled quantity.
6. Correction element: also known as compensation element, it is added on the basis of the basic structure of the system, and its parameters can be flexibly adjusted to improve the performance of the system.
Four, the classification of control system
(A) according to the characteristics of the given signal classification
1. Fixed value control system
2. Servo control system
3. Program control system
(B) Classification according to the mathematical description of the system
1. linear system
2. Nonlinear system
(3) Classification according to the nature of signals transmitted by the system.
1. continuous system
2. Discrete system
(4) Classification according to the number of input and output signals of the system.
1. SISO system
2.MIMO system
(5) According to the properties of differential equations.
1. Block parameter system
2. Distributed parameter system
Verb (abbreviation for verb) controls the performance requirements of the system.
1. Stability: refers to the ability of the system to restore a stable state. Stability is a prerequisite for the normal operation of the control system.
2. Rapidity: refers to the length of time for the stable system to respond to the dynamic process.
3. Accuracy: refers to the accuracy of tracking a given signal or correcting the influence of a disturbance signal after the control system enters a steady state.
1- 1 Try to compare the advantages and disadvantages of open-loop control system and closed-loop control system.
Answer: Advantages: The open-loop control system has no feedback loop, simple structure and low cost.
Disadvantages: low control accuracy, easy to be disturbed by the outside world, and the output error can not be compensated once it appears.
1-2 explains the working principle of negative feedback and its application in automatic control system.
Answer: The measuring element detects the controlled physical quantity and gives feedback. By comparing the comparison element with a given signal, a deviation signal is generated. Then the deviation signal is amplified by the amplifier, and the actuator is driven to control the controlled object with enough power, so as to regulate the system and make the controlled quantity meet or equal to the expected value with a certain precision.
What are the basic components of 1-3 control system? What are the functions of these components?
A: The feedback control system consists of various components with different structures. A system must include the controlled object and the control device, and the control device is composed of various basic elements with certain functions. In different systems, components with completely different structures can have the same function. Therefore, the functional components that make up the system can be classified as follows according to their functions:
Given component: the link used to give the input signal to determine the target value (or given value) of the controlled object.
Measuring element: used to detect the controlled quantity, usually appearing in the feedback loop.
Comparison element: used to compare the actual output value detected by the measuring element with the input value given by the given element after conversion, and find out the deviation between them.
Amplifier: used to amplify the deviation signal given by the comparator and push the actuator to control the controlled object with sufficient power.
Actuator: used to directly drive the controlled object to change the controlled quantity.
Correction element: also known as compensation element, is an element added on the basis of the basic structure of the system, and its parameters can be flexibly adjusted to improve the performance of the system.
1-4 What are the basic performance requirements for the automatic control system? What are the most important requirements?
A: Basic performance requirements: stability, speed and accuracy. The most important requirement is stability.
1-5 There are many closed-loop and open-loop control systems in daily life. Give some concrete examples to illustrate their working principles.
Answer: Open-loop control system: For example, the traditional washing machine works in the order of washing, cleaning, dewatering and drying clothes, and does not measure the output signal, that is, the cleanliness of clothes; Another example is the feed control of a simple CNC machine tool. The control device and the drive device input instructions to push the workbench to the designated position, and the position signal is no longer fed back. These are typical open-loop systems.
Closed-loop control system: Take the driving system of CNC machine tool workbench as an example. A simple control scheme is based on control.
A certain frequency and number of command pulses sent by the device drive the stepping motor to control the movement of the workbench or the tool rest without detecting the actual movement of the workbench or the tool rest. This control method is simple, but the problem is that the error of any link in the whole "transmission chain" from the driving circuit to the worktable will affect the motion accuracy or positioning accuracy of the worktable. In order to improve the control accuracy, feedback control is adopted, so that the detection device can determine the actual position of the workbench at any time (that is, its output information); Then it is fed back to the input end, compared with the control command, and then the control action is decided according to the error between the actual position and the target position of the workbench, thus eliminating the error. The detection device is the feedback link.
1-6 please describe the working principle of the automatic liquid level control system as shown in figure 1-6(a). If the structure of the system is changed as shown in figure 1-6(b), what impact will it have on the system work?
Answer: (a) In the system shown in the figure, when the outlet valve is closed, the float is in a balanced state; When the outlet valve is opened and water flows out, the water level in the water tank will drop and the float will also drop; Through leverage, the water inlet valve will open, water will flow into the water tank, and the float will rise.
(b) In the system shown in the figure, it is assumed that the float is in a balanced state when the current outlet valve is closed. When the outlet valve is opened and water flows out, the water level in the water tank will drop and the float will also drop. Through leverage, the water inlet valve will gradually close with the outflow of water until all the water in the cylinder flows out.
The principle of the warehouse gate automatic control system is shown in figure 1-7. Try to describe the working principle of automatic control of gate opening and closing, and draw the system block diagram.
A: The system block diagram is shown in the title map 1-7(a).
If you want to open the door, take out the voltage corresponding to the current state of the door, compare (subtract) it with the reference potential of the door-opening state, and then send it to the amplifier to drive the servo motor to drive the winch to open the door until the voltage corresponding to the door-opening state is equal to the reference potential of the door-opening state, and the result of comparison (subtraction) by the amplifier is zero, the actuator does not work, and the door remains open.
If you want to close the door, take out the voltage corresponding to the current state of the door, compare (subtract) it with the reference potential of the closed state, and then send it to the amplifier to drive the servo motor to drive the winch to close the door until the voltage corresponding to the state of the door is equal to the reference potential of the closed state, and the result of the comparison (subtraction) of the amplifier is zero, the actuator does not work, and the door remains closed.
Figure 1-8 is the schematic diagram of the angular velocity control system. The shaft with centrifugal speed regulation is driven by the internal combustion engine through the reduction gear, and the centrifugal force generated by the rotating flying hammer is offset by the spring force, and the required speed is adjusted through the pre-tightening force of the spring. When there is a sudden change, please explain the function of the control system.
A: Working principle: When the engine drives the load to rotate, a pair of flying hammers are driven by gears to rotate horizontally. The flying hammer can drive the sleeve to slide up and down through the hinge, and a balance spring is installed in the sleeve. When the sleeve rolls up and down, the opening of the oil supply valve is adjusted through the connecting rod. When the generator is running normally, the centrifugal force generated by the rotation of the flying hammer is balanced with the resilience of the spring, and the sleeve keeps a certain height, so that the valve is in a balanced position. If the generator speed decreases due to the increase of load, the flying hammer will slide down the sleeve due to the decrease of centrifugal force, and the opening of the oil supply valve will be increased through the connecting rod to increase the generator speed. Similarly, if the generator speed increases due to the decrease of load, the flying hammer will slide the sleeve upward due to the increase of centrifugal force, and reduce the opening of the oil supply valve through the connecting rod, forcing the generator speed to fall back. In this way, the centrifugal governor can automatically resist the influence of load change on the speed and keep the speed of the generator near the expected value.
The schematic diagram of 1-9 angular position servo system is shown in figure 1-9. The task of the system is to control the angular position of the working machine and track the handle angle at any time. Try to analyze its working principle and draw the system block diagram.
A: (1) Working principle: closed-loop control.
As long as the rotation angle of the working machine is consistent with the rotation angle of the handle, the bridge circuit composed of two ring potentiometers is in a balanced state and has no voltage output. This means that there is no deviation in tracking, the motor does not move, and the system is static.
If the handle angle changes, the bridge outputs a deviation voltage and drives the motor to rotate through the amplifier. Drag the working machine to make it deflect in the required direction through the reducer. At this time, the system reaches a new equilibrium state, and the motor stops running, thus achieving the purpose of angular position tracking.
(2) The controlled object of the system is the working machine, and the controlled quantity is the angular displacement of the working machine. The given quantity is the angular displacement of the handle. The functional elements of each part of the control device are: the handle completes the setting, the bridge completes the detection and comparison, and the motor and reducer complete the execution function.
The block diagram of the system is shown in the topic 1-9(2).
1- 10 is the schematic diagram of the electric furnace temperature control system. Try to analyze the working process of the system to keep the temperature of the electric furnace constant, point out the controlled object, the controlled quantity and the functions of each component of the system, and finally draw the system block diagram.
(1) Working principle: closed-loop control.
As long as the thermocouple measures the temperature of the electric furnace, the output voltage is consistent with the given voltage, there is no deviation voltage, the voltage amplifier and power amplifier have no voltage output, the motor does not move, the resistance wire does not heat up, and the system is static.
If the temperature of the electric furnace changes, the output voltage of the thermocouple measuring the temperature of the electric furnace also changes, which is inconsistent with the given voltage and produces a deviation voltage. The amplifier drives the motor to rotate, and after the reducer slows down, it drags the sliding rheostat pointer to move, so that the heating power of the resistance wire changes. When the voltage corresponding to the furnace temperature is equal to the given voltage, the system reaches a new equilibrium state and the motor stops running, thus achieving the purpose of constant temperature control.
(2) The controlled object of the system is the electric furnace, and the controlled quantity is the furnace temperature. The given reference quantity is a given voltage. The functional elements of each part of the control device are: the sliding rheostat completes the comparison, the thermocouple completes the detection, and the amplifier, motor and reducer complete the execution function.
The block diagram of the system is shown in figure 1- 10(a).
The second chapter is the mathematical method of Laplace transform.
abstract
First, the definition of Laplace transform
Let the time function ≥0, then the Laplace transform of is defined as:
Second, Laplace transform of typical time function
1. unit pulse function,
2. Unit step function,
3. unit slope function,
4. Unit acceleration function,
5. Exponential function,
6. Sine function,
7. Cosine function,
8. Power function,
Third, the nature of Laplace transform
1. Linear attribute
If it is, it is a constant. rule
2. Delay theorem
If it is.
3. Laplace transform of periodic function
If the function is a periodic function with periodicity, there is
4. Displacement Theorem in Complex Number Field
If, for any constant (real number or complex number), there is
5. Time scales change naturally
If it is an arbitrary constant, then
6. Differential properties
If, then
7. Integral properties
If, then
8. Initial value theorem
If so, then
9. Final value theorem
If and exist, then
10. Complex differential theorem
If, then
1 1. Complex integral theorem
If, then
12. Convolution theorem
2- 1 Try to find the Laplace transform of the following function
( 1)
Solution:
(2)
Solution:
(3)
Solution:
(4)
Solution:
(5)
Solution:
(6)
Solution:
(7)
Solution:
(8)
Solution:
2-2 Known
(1) Use the final value theorem to find the time value.
Solution:
(2) The time value is obtained by inverse Laplace transform.
Solution:
Two or three are known.
(1) is evaluated by the initial value theorem.
Solution:
(2) Find by taking the Laplace inverse transformation, and then find.
Solution:
2-4 Find the Laplace transform of the function shown in the figure below.
Solution: (1) According to the Laplace transform property of periodic signals, we can get
Solution:
2-5 Try to find the inverse Laplace transform of the following function.
( 1)
Solution:
(2)
Solution:
(3)
Solution:
(4)
Solution:
(5)
Solution:
(6)
Solution:
(7)
Solution:
(8)
Solution:
2-6 Find the following convolution
( 1)
Solution:
(2)
(3)
(4)
2-7 Solve the following differential equation with Laplace transform.
( 1)
Solution:
(2)
The third chapter is the mathematical model of the system
abstract
I. Basic concepts
1. linear system
When the mathematical model of a system can be described by a linear differential equation, the system is called a linear system. The general expression of differential equation of linear system is
2. Nonlinear system
Systems whose dynamic characteristics are described by nonlinear differential equations are called nonlinear systems.
Second, the steps of establishing the differential equation of the system
1. Determine the input and output of the system or component. The input or disturbance input of the system is the input of the system, while the controlled quantity is the output. For a link or element, the input and output should be determined according to the signal transmission of the system.
2. According to the order of signal transmission, from the input end of the system, according to the laws of motion followed by each variable (such as Kirchhoff's law in the circuit, Newton's law in mechanics, thermodynamics law in the thermodynamic system and energy conservation law), the dynamic differential equations of each link in the motion process are listed. When writing columns, some secondary factors are ignored according to the working conditions, and whether there is load effect between adjacent members is considered.
3. The differential equation describing the input and output of the system is obtained by eliminating the intermediate variables listed in the differential equation.
4. The differential equation obtained by sorting. Generally, items related to output are placed on the left side of the equal sign, and items related to input are placed on the right side of the equal sign, in descending order.
Third, the transfer function
Definition of transfer function: The ratio of Laplace transform of output to Laplace transform of input of single-input single-output linear time-invariant system under zero initial condition is the transfer function of linear time-invariant system.
Fourth, the transfer function of typical links
1. proportional link
2. Inertial connecting rod
3. Overall link
4. Differential link
5. Swing the connecting rod
6. Delay link
Verb (abbreviation of verb) signal flow chart
Signal flow graph is a graph representing a set of simultaneous linear algebraic equations. From the point of view of describing the system, describe the flow of signals from one point to another in the system, and show the relationship between signals, including all the information contained in the structure diagram, which corresponds to the structure diagram one by one.
Mason formula:
Where-total transfer function;
The transfer function of the first forward path;
-Characteristic formula of signal flow graph.
Where-the transfer function of the first circuit;
-sum of all loop transfer functions in the system;
-the product of the transfer functions of two non-contact loops;
-the sum of transfer function products of every two non-contact loops in the system;
-the product of the transfer functions of three non-contact circuits;
-sum of transfer function products of every three non-contact loops in the system;
-is the cofactor of the characteristic formula of the first forward path, that is, in the characteristic formula of the signal flow graph, the loop transfer function in contact with the first forward path is replaced by zero, that is.
State space description of intransitive verb system
(A) the state equation description of univariate system
Equation of state.
The state equation of the first-order linear time-invariant SISO system is
Nano; Fiber (/kloc-negative ninth power of 0/0)
2. Output equation
If the output is specified, the matrix form of the system output equation is
Or abbreviated as
3. State space expression
(2) State equation description of multivariable system.
Or rewritten as a matrix equation.
3- 1 Solve the differential equation of the system shown in Figure 3- 1(a) and (b).
Solution: (1) input f(t) and output y(t)
(2) mass m:
(3) finishing:
(b) Solution: (1) Input f(t) and output y(t).
(2) introducing an intermediate variable x(t) as the right displacement of the connection point, (y >；; x)
(3) ①
②
(4) Eliminating intermediate variables from ① and ② leads to:
3-2 Find the transfer functions of the three mechanical systems shown in Figure 3-2(a), (b) and (c). The figure shows the input displacement and the output displacement. It is assumed that the load effect at the output can be ignored.
(a) Solution: (1) Input and output
(2) mass m:
(3) finishing:
(4) Laplace transform on both sides:
(5) Transfer function:
(b) Solution: (1) Input and output
(2) The intermediate variable X is introduced as the displacement of the connection point with C..
(3) ①
②
(4) eliminating the intermediate variable x to obtain:
(5) Laplace transform on both sides:
(6) Transfer function:
(c) Solution: (1) Input and output
(2)
(3) Laplace transform on both sides:
(4) Transfer function:
3-3 Solve the differential equation of the mechanical system shown in Figure 3-3. The figure shows the input torque, circumferential damping and moment of inertia.
Solution: Let the input of the system be (i.e.) and the output be (i.e.), and make dynamic analysis of the disc and mass respectively, and list some dynamic equations as follows:
By eliminating intermediate variables, the system dynamics equation can be obtained.
3-4 The pulley transmission system is shown in Figure 3-4, where the radii of 1 wheel and wheel 2 are sum, the moment of inertia is sum, and the viscous friction coefficient is sum. If the belt drive does not slip and the belt quality is ignored, try to find the transfer function of the pulley drive system. Where are input torque and output rotation angle.
Solution: The torque equation of wheel 1 is
The torque equation of wheel 2 is
For the above two types of Delaunay transformation, and set the initial conditions to zero, there are
Because wheel 1 and wheel 2 do equal work, there is
rule
Substituting the relationship between sum into the Laplace transform equation and eliminating the sum of intermediate variables, we can get
Available after completion
One of them is.
3-5 proves that the systems shown in Figure 3-5(a) and (b) are similar systems.
Solution: (a)( 1) input, output
(2) Transfer function of the system:
(b)( 1) input, output
(2) The intermediate variable X is introduced as the displacement from the connection point of c 1.
(3) ① ②
(4) Laplace transform on both sides: ①
②
(5) Eliminate the intermediate variables to obtain:
(6) Transfer function:
The two systems (a) and (b) have the same mathematical model, so they are similar systems.
3-6 In the passive network shown in Figure 3-6, it is known, try to find the transfer function of the network, and explain whether the network is equivalent to RC network series?
The solution is shown in Figure 3-6. Using the complex impedance method, the transfer function of the network can be obtained as follows
Because the transfer function of two series RC networks is
Therefore, this network is not equivalent to a network composed of two RC networks in series.
3-7 The relationship between output and input is
(a) Find the corresponding steady-state output values when working points are found respectively.
(b) Make a small deviation linearization model at these working points, define the sum with the deviation from the working points, and write a new linearization model.
Solution: (a) Substitute them, that is, when the working point is, the corresponding steady-state output values are respectively.
(b) According to the linearization method of nonlinear system, the nonlinear function is expanded into Taylor series near the working point, and the higher-order term is omitted.
therefore
If so, yes.
When the working point is,
When the working point is,
When the working point is,
3-8 If the system transfer function block diagram is shown in Figure 3-8, find:
(1) is the input, and it was the closed-loop transfer function of the output.
(2) Closed-loop transfer function of input and output.
(3) Comparing the denominators of the above transfer functions, what conclusions can be drawn.
Solution: (1) as input, at that time:
If you think the output, there is
If you think the output, there is
If you think the output, there is
If you think the output, there is
(2) As input, at that time:
If you think the output, there is
If you think the output, there is
If you think the output, there is
If you think the output, there is
(3) From the above, we can know that for the same closed-loop system, when the input values are different, the transfer function of the forward channel is different, the transfer function of the feedback loop is different, and the transfer function of the system is also different, but the denominator of the system transfer function remains unchanged, because this denominator reflects the inherent characteristics of the system and has nothing to do with the outside world.
3-9 It is known that a system consists of the following equations. Try to draw the system structure diagram and find the closed-loop transfer function C(s)/R(s).
Solution: According to the system equation, the system structure diagram can be drawn, as shown in Figure 3-9.
pass by
Available:
replace
get
because
therefore
that is
Another solution: (1) The forward channel transfer function of the system is obtained by simplifying the structure and moving the exit point backward.
The closed-loop transfer function of the system is
(2) Using signal flow chart, the system has a forward channel, three independent loops and no non-contact loop.
The transfer function of the system can be obtained from Mason formula, as shown below
3- 10 try to simplify the system structure diagram shown in figure 3- 10, and find the corresponding transfer function sum.
Solution: When only the effect is considered, the simplified structure diagram (topic 3- 10(a)) can be obtained by feedback connection equivalence, and the transfer function of the system is
Only considering the function, the system structure is shown in Figure 3- 10(b). The system moves backward and sums after comparing the points.
Series connection and parallel connection are equivalent, and a simplified structure diagram can be obtained, as shown in Figure 3- 10(c). The system transfer function is
Another solution: signal flow graph method can be used to verify the results.
The signal flow chart of the system in theme figure 3- 10 is shown in theme figure 3- 10(d).
As can be seen from the figure, when only the action is considered, the system has a forward channel and two independent loops, and there is no non-contact loop, namely
The transfer function of the system can be obtained from Mason formula, as shown below
Considering only the action, it can be seen from the figure that this system has two forward channels and two independent circuits, and there is no connection between them.
Contact circuit, i.e.
The transfer function of the system can be obtained from Mason formula, as shown below
.
3- 1 1 It is known that the transfer function block of a system is shown in Figure 3- 1 1, where R(s) is the input, C(s) is the output and N(s) is the interference. Try to find out what the value of G(s) is, and the system can eliminate the influence of interference.
Solution:
If so,
Then, that is
3- 12 Solve the transfer function of the system shown in Figure 3- 12.
Solution:
3- 13 Solve the transfer function of the system shown in Figure 3- 13.
Solution:
3- 14 Solve the transfer function of the system shown in Figure 3- 14.
Solution:
The transfer function of 3- 15 problem solving system is shown in figure 3- 15.
Solution: (1)
(2)
The signal flow chart of the known system is shown in Figure 3- 16. Try to find the transfer function of the system.
Solution: As shown in Figure 3- 16, the system has a forward channel, three independent loops and no non-contact loop, namely
The transfer function obtained from the Mei Sen gain formula is
Let the differential equation of the system be
Find the state space expression of the system.
Solution: If yes, the state equation and the output equation can be derived.
The transfer function of a given system is
Try to write its state space expression.
Solution:
3- 19 Let the system transfer function be
Try to transform the mathematical model of MATLAB into the system state equation.
Solution: Enter the following MATLAB mathematical model conversion program in the MATLAB command window, and the matrices A, B, C and D will be generated.
num =；
b =；
d =[0]；
[num，den]=ss2tf(A，B，C，D)
Quantity =
0 1.0000 4.0000 3.0000
den =
1 8 16 0
In other words, the transfer function of the system is