"Principles of Automatic Control" study notes-personal arrangement

1. Basic concept of automatic control

Automatic control : refers to the use of additional equipment or devices to make a certain working state or parameter of a machine, equipment or production process automatically operate according to a predetermined law without direct participation.

Open-loop control : An open-loop control system refers to a system without feedback of the controlled quantity, that is, the flow of control information in the system does not form a closed loop.
Closed loop control : Control with feedback of the controlled quantity, that is, the output signal of the system returns to the input end of the system along the feedback channel, forming a closed channel, also called feedback control.

The task of the automatic control system : the controlled quantity and the given value are equal or maintain a fixed proportional relationship at all times, without any deviation, and not affected by interference.

The dynamic process of the system : also called the transition process, it refers to the whole process of the controlled quantity changing with time after the system is affected by the external signal (given value or interference).

Automatic control performance indicators : indicators that reflect the pros and cons of system control performance, which are often evaluated in terms of stability, speed, and accuracy in engineering .

2. Mathematical model foundation

2.1 Concept of control system mathematical model

The mathematical expression describing the relationship between the input and output variables of the control system and the internal variables is called the mathematical model of the system.

2.2 Methods of establishing mathematical models

One is the mechanism analysis modeling method, called the analytical method;

The second is the experimental modeling method, called system identification.

Common models:

1. External description model-differential equation, transfer function

2. Internal description model-state space method

3. Signal flow graph model

3. Time domain mathematical model of linear system

3.1 Differential equation

It is the most basic mathematical model of the control system. To study the movement of the system, the differential equations of the system must be listed.

A control system is composed of several components with different functions. First, according to the physical laws of each component, the differential equations of each component must be listed to obtain a differential equation group, and then the intermediate variables will be eliminated to obtain the total input and output of the control system. The differential equation.

3.2 Transfer function

The differential equation of the control system is a mathematical model that describes the dynamic performance of the system in the time domain . Under the given external effects and initial conditions, the output response of the system can be obtained by solving the differential equation. This method is relatively intuitive.

Laplace transform is a powerful tool for solving linear differential equation, the differential equation can be converted into his domain into complex frequency domain algebraic equations, and the mathematical model can be obtained in the complex domain control systems - the transfer function.

Transfer function : The ratio of the Laplace transform of the output of the system to the Laplace transform of the input of the linear time-invariant system under the zero initial condition.

A few explanations of the transfer function :

1. As a mathematical model, the transfer function is only suitable for linear time-invariant systems. This is because the transfer function is derived through Laplace transformation, which is a linear integral operation.

2. The linear time-invariant differential equations of linear time-invariable systems or components correspond one-to-one with the transfer functions, which are descriptions of the same system or component in different domains.

3. The transfer function characterizes the inherent characteristics of the linear time-invariant system or the component itself. It has nothing to do with the form of its input signal, but is related to the position of the input signal and the location of the output signal. So when it comes to the transfer function, the input and output must be specified.

4, the transfer function is a complex variable $S$ rational fraction, and molecules, the denominator polynomial coefficients are real numbers, the denominator polynomial is $\tiny N$ greater than or equal to the number of the numerator polynomial $\tiny M$ , $\tiny N\geqslant M$ .

5. The transfer function is defined under zero initial conditions. The zero initial condition of the control system has two meanings:
one means that the input $\tiny t\geqslant 0$ only takes effect at time ;

The second means that the system is in a stable working state before the input is added to the system.

6. The transfer function only represents the relationship between single input and single output (SISO). For multiple input multiple output (MIMO) systems, it can be represented by a transfer function matrix.

7. The transfer function can be expressed as

$\tiny G\left ( s \right )=K\frac{\left ( s-z_{1} \right )\left ( s-z_{2} \right )\cdots \left ( s-z_{m} \right )}{\left ( s-p_{1} \right )\left ( s-p_{2} \right )\cdots \left ( s-p_{n} \right )}$

In the formula, $\tiny p_{1},p_{2}\cdots \cdots p_{n}$ is the root of the denominator polynomial, called the pole of the transfer function; $\tiny z_{1},z_{2}\cdots \cdots z_{m}$ is the root of the numerator polynomial, called the zero of the $\tiny K$ transfer function ; called the gain of the transfer function.

8. The denominator polynomial of the transfer function is called the characteristic polynomial, which is recorded as

$\tiny D\left ( s \right )=a_{0}s^{n}+a_{1}s^{n-1}+\cdots +a_{n-1}s+a_{n}$

It is $\tiny D\left ( s \right )=0$ called the characteristic equation.

The order of the denominator polynomial of the transfer function is always greater than or equal to the order of the numerator polynomial, ie $\tiny n\geqslant m$ . This is due to the inertia of the actual system.

9. In actual engineering, many different physical systems have exactly the same transfer function, so the transfer function only describes the relationship between output and input, and does not provide any physical structure of the system.

10. A transfer function is only suitable for single-input and single-output systems, so the intermediate variables in the signal transmission process cannot be reflected.

11. For the unknown transfer function of the system, the system transfer function can be obtained by adding inputs with known characteristics to the system and then studying its output, and a complete description of its dynamic characteristics can be given.

12. The inverse pull transformation of the transfer function is the impulse response of the system.

4. Structure diagram

Mathematical models such as differential equations and transfer functions. They all use pure mathematical expressions to describe the system characteristics, which cannot reflect the influence of various components in the system on the performance of the entire system. Although the system schematic diagram and functional block diagram reflect the physical structure of the system, they lack the relationship between the variables in the system. Quantitative Relationship.
The structure diagram can not only describe the quantitative relationship between the variables in the system, but also clearly show the influence of the system components on the system performance.
The control system is composed of some components. According to different functions, the system is divided into several links or subsystems. The function of each subsystem can be represented by a one-way function.
According to the transmission direction of the information in the system, the function blocks of the various subsystems are sequentially connected with signal lines to form the structure of the system, which is also called the block diagram of the system.

Series-parallel connection and feedback connection of structure diagram:

Typical structure of automatic control system:

The mathematical model of the system has three forms: differential equation, transfer function and dynamic structure diagram. The three can be easily converted to each other through pull conversion. In the analysis of automatic control systems, transfer functions and dynamic structure diagrams are the most common.
The transfer function of the system can be easily obtained by simplifying the structure diagram; the equivalent transformation of the dynamic structure diagram and Mason's formula are effective tools to find the system transfer function.
The transfer function of the system can be divided into open-loop transfer function, closed-loop transfer function and error transfer function. The closed-loop transfer function and error transfer function are divided into the given input and the interference input, and the system can be The total output under quantification and interference, and the total error of the system.

5. Signal flow diagram

The structure diagram of a more complex control system is often multi-loop and cross. In this case, it is troublesome to simplify the structure diagram , and it is easy to make mistakes. It is more convenient if the structure diagram is transformed into a signal flow diagram, and then the Mason formula is used to find the transfer function of the system.
The signal flow graph is a signal transmission network composed of nodes and branches .
In the signal flow diagram, small circles represent variables or signals, called nodes.
The line segment connecting the two nodes is called a branch, and the signal can only be transmitted in the direction of the arrow of the branch.
The mathematical operator marked next to the branch is called the transfer function or transfer gain. The transfer gain can be a constant or a complex function. When the transfer function is 1, it can be omitted.
The signal is transmitted in one direction along the arrow on the branch.
The branch is equivalent to a multiplier. When the signal flows through the branch, it is multiplied by the branch gain and becomes another signal.
For a given system, the signal flow graph is not unique.
The basic rule of using a signal flow diagram to express a system of equations is: the end signal of a branch is equal to the start point signal multiplied by the branch transfer function.

For example, the algebraic equation $\tiny x_{2}=ax_{1}$ can be expressed as the signal flow diagram shown in the figure below.

Terms commonly used in signal flow diagrams:

Source node (input node): On the source node, there are only signal output branches but no signal input branches. It generally represents the input variables of the system.
Trap node (output node): On the trap node, there are only branches for signal input and no signal output. It generally represents the input variables of the system.
Hybrid node: On the hybrid node, there are both signal output branches and signal input branches.
Pathway: A path through each connected branch in the direction of the branch arrow. If the path intersects with any node not more than once, it is called an open path.
Forward path: When the signal passes from the input node to the output node, each node passes only once, called the forward path.
The total gain of the forward path : the product of the gain of each branch on the forward path is called the total gain of the forward same path, which is generally $\tiny P_{k}$ expressed in terms of the total gain of the forward path .
Loop: A closed path in which the start and end points are at the same node and the signal passes through each node no more than once is called a loop.
Loop gain: The product of the gain of each branch on the loop is called loop gain, which is generally $\tiny L_{a}$ expressed in terms of.
Non-contact circuits: When there are no common nodes between the circuits, they are called non-contact circuits.