1 Introduction
Differential output response of the solution of the system is the system of equations by the expression, analyze the dynamic characteristics of the system.
If the output response curve is plotted, it can directly reflect the dynamic process, but the process is very complicated to solve.
For linear time-invariant systems, the transfer function is commonly used in a mathematical model is established based on the Laplace transform.
The system transfer function described can avoid the trouble solving differential equations, indirect Relationship between structure and system parameters and system performance.
The dynamic performance and may transfer function in the complex plane shape directly determining system, find ways to improve the quality of the system.
The transfer function 2
2.1 Concepts and Definitions
In the zero initial conditions , the linear system given constant output of the Laplace transform of the output causes the ratio of the Laplace transform of the input quantity .
Zero initial conditions :
1. t <0, the input and its first derivative are both 0.
2. Before the system is applied to the input, the system is in steady state operation, i.e., t <0, and its output is also the first derivative are zero.
Setting the linear time-invariant differential equation for the system:
Under zero initial conditions Laplace transform to obtain
The transfer function of the system
It has the following characteristics:
1. Differential simple than by Laplace transform, the real domain complex calculus has been transformed into simple algebra.
2. Typically the input signal, the output of the transfer function there is a certain correspondence relationship, when the input is a unit impulse function, the function as an input, the output of which is the same as the function and the transfer function.
= Jω, 3. Analysis of the system may make s in the transfer function in the frequency domain.
4. G (s) determined the distribution of zeros and poles system dynamics.
Equivalent elastic stiffness :
Equivalent complex impedance :
2.2 Examples of the transfer function
2.2.1 mass - spring - damper system transfer function
All initial conditions are zero, its Laplace transform as follows:
By definition, the system transfer function is:
2.2.2 RLC passive transfer function circuit network
2.3 Some Conclusions
1. The transfer function is the mathematical model of the complex s domain, the parameters depends only on the structure and parameters of the system itself , regardless of the form of input systems .
2. If the input is given, the system is completely output characteristic (s) determined by the transfer function G , i.e., the transfer function of the system is characterized by the inherent dynamic characteristics inherent .
3. The transfer function of the inherent characteristics of the system described by the relationship between the system input and the output, i.e. to the external system input - output characteristic of the internal characteristics of the system will be described .
The general form of the transfer function of 2.4
Linear System:
When the all-zero initial condition, the Laplace transform of the equation system can be obtained in the form of a general transfer function:
2.4.1 characteristic equation zeros and poles
make:
then:
D (s) = 0 is called the system characteristic equation , which is known as the root system characteristic root .
The characteristic equation of the system determines the dynamic characteristics , D (s) the s is equal to the highest order of the order of the system .
When s = 0, there is
Where, K is called the system static amplification factor or static gain .
From the perspective of differential equations to see, at this time corresponds to derivative terms are all zero. Thus the reaction K ratio in static system, the output and input.
2.4.2 zeros and poles
The G (s) written in the following form:
The system transfer function characterized in that the pole root system , values of zeros and poles depends entirely on the configuration parameters of the system .
2.4.3 poles and zeros profile
The zero transfer function poles in the complex plane represents the graph is called a zero of the transfer function, the pole profile . FIG zero point with "O" represents the pole by "×" indicates.
Some explanations 2.4.4 Transfer Function
1. The transfer function is a linear system to a constant system parameter indicates the relationship between the input and the output , the transfer function of the concept generally applies only to linear time-invariant systems .
2. The transfer function is a complex function of s, transfer coefficient as a function of the coefficients and the respective corresponding differential equation equal , depends entirely on the system configuration parameters .
3. The transfer function is defined under zero initial conditions , i.e. before time zero, the system for a given operating point of equilibrium in a relatively quiescent state. Thus, the transfer function does not reflect the movement of the entire system in a non-zero initial conditions .
4 shows a relationship between the transfer function of only the input and output system, can not describe changes of the system's internal intermediate variable .
The transfer function of a relationship can only represent one input and one output, adapted to describe a single-input single-output system , for the use of multiple input multiple output transfer function matrix .
2.4.5 impulse response function
The initial condition is 0, the Laplace transform of the output response system under the action of the pulse input unit is
Laplace transformed back
g (t) referred to the system impulse response function ( weighting function ).
Impulse response function and transfer function that contains the same information about the dynamics of the system .
Noting the complex domain is equivalent to multiplying the time domain convolution , and therefore, the
Linear systems known at any input action, the time domain output
Wherein, when t <time 0, g (t) = x (t) = 0.
Its transfer function is typically part 3
3.1 links
A portion having a certain element, the element group or information element to determine the relationship is referred to as a transmission link , the link is called frequently encountered typical link .
Any complex system consisting of the total can be attributed by some typical aspects.
3.2 Classification link
Suppose the system has real zeros to b, c to the complex zeros, d nonzero real pole, e v pole pair of complex poles and a zero.
Linear system transfer function poles and zeros of the expression
1. For the real zeros Z I = [alpha I and real pole P J = [beta] -D J , which can be transformed into the formula because of the following forms:
2. For the complex zeros of the Z ℓ = [alpha ℓ + j [omega] ℓ and Z ℓ +. 1 = [alpha ℓ -jω ℓ , which can be transformed into the formula because of the following forms:
In the formula,
3. For complex pole pair P K = [alpha K + j [omega] K and P K +. 1 = [alpha K -jω K , which can be transformed into the formula because of the following forms:
In the formula,
Thus, the transfer function can be written:
In the formula,
For the system static magnification .
From the above equation, the transfer function expression that contains six different factors:
In general, any linear system can be viewed as a series combination of a typical segment represented by the above six factors. Typical of the six sectors are called:
Note: the presence of pure practical system time delays, the output of the input exactly reproduced, but the delay time [tau], i.e.:
at this time
Namely the last part of the above.
3.3 A typical example links
3.3.1 proportional element
Output without distortion, without inertially follow the input, both the proportional relationship.
Proportional element transfer function:
3.3.2 The first-order inertia
Any equation of motion for the following first-order differential equations:
Form part called first-order inertia, its transfer function is:
Such as: spring - damper link
3.3.3 Differential link
Output proportional to the differential input.
Passive Differential Network :
Clearly, a passive network including differential and differential inertia link, called the inertia differential part .
Only when | Ts | << 1 only when approximately differentiating element.
The transfer function is:
When called first-order differential link .
The differential part of output is the derivative of the input that the output reflects the changing trend of the input signal , so that the relevant input to the system to changes in the situation notice.
Therefore, the differential part is often used to improve the dynamic performance of the control system .
3.3.4 integral part
Output is proportional to the time integral of the amount of input .
1. The output depending on the input amount of time the accumulation process.
2. has a significant hysteresis:
Such as when the input is a constant value A, since the
Output to achieve the elapsed time T must be input at t = 0 value A.
Therefore, the integral element used to improve steady-state accuracy of the system .
For example, active integrator network:
3.3.5 order Oscillation link
Containing two separate energy storage element and the memory can be interchangeable, with the resulting properties of the output oscillation equations of motion:
The transfer function is:
Another common standard second-order transfer function of the oscillating link (K = 1):
[omega] n- called undamped natural angular frequency .
The mass - spring - damper system:
Transfer Function:
3.3.6 Delayed link
Delay and inertia of the difference link:
1. inertia has had an output from the input start time, only due to inertia, the output with a lag output was close to the required value.
2. From the beginning part of the input start delay within 0 ~ τ time, there is no output, but after t = τ, the output time [tau] equal to the previous input.
3.4 Summary
