When analyzing signals in time domain, sometimes the time domain parameters of some signals are the same, but it does not mean that the signals are exactly the same. Because the signal not only changes with time, but also is related to information such as frequency and phase, it is necessary to further analyze the frequency structure of the signal and describe the signal in frequency domain. The transformation of dynamic signal from time domain to frequency domain is mainly realized by Fourier series and Fourier transform. Periodic signals depend on Fourier series, and aperiodic signals depend on Fourier transform. A simple example of frequency domain analysis can be illustrated by figure 1: children's toys in a simple linear process. The linear system consists of a spring-suspended weight mounted on a handle. Children control the position of heavy objects by moving the handle up and down.
Anyone who has played this game knows that if the handle moves in a more or less sinusoidal way, the weight will start to oscillate at the same frequency, although the oscillation of the weight is not synchronized with the movement of the handle. Only when the spring can't fully extend, the weight and the spring will move synchronously, and the frequency is relatively low.
With the increasing frequency, the swing phase of the weight may be earlier or later than that of the handle. At the natural frequency point of the process object, the height of the weight swing will reach the highest. The natural frequency of the process object is determined by the mass of the weight and the strength coefficient of the spring.
When the input frequency is larger and larger than the natural frequency of the process object, the amplitude of the weight oscillation will tend to decrease and the phase will be more delayed (in other words, the amplitude of the weight oscillation will be smaller and smaller, and its phase will be more and more delayed). In the case of extremely high frequency, the weight only moves slightly, just opposite to the direction of the handle. All linear process objects show similar characteristics. These process objects are all about converting sine wave input into sine wave output with the same frequency. The difference is that the amplitude and phase of the output and input are changed. The change of amplitude and phase depends on the phase lag and gain of the process object. The gain can be defined as "the proportional coefficient between the amplitude of the output sine wave and the amplitude of the input sine wave amplified by the process object", while the phase lag can be defined as "the lag degree of the output signal relative to the input sine wave".
Different from the steady-state gain k value, "gain and phase lag of process object" will change according to the frequency of input sine wave signal. In the above example, the spring weight does not significantly change the amplitude of the low-frequency sine wave input signal. In other words, the object has only one low-frequency gain coefficient. When the signal frequency is close to the natural frequency of the process object, because the amplitude of its output signal is greater than that of its input signal, its gain coefficient is greater than that at the above-mentioned low frequency. However, when the toy in the above example is shaken quickly, it can be considered that the high-frequency gain of the processing object is zero because the heavy object can hardly vibrate.
The phase lag of process objects is an exception. Because the weight oscillates synchronously with the handle when the handle moves very slowly, in the above example, the phase lag starts from the low-frequency input signal close to zero. When high-frequency signals are input, the phase lag is "-180 degrees", that is, the weight and the handle move in the opposite direction (therefore, we often use'180 degrees' to describe this reverse movement).
Porter spectrum shows the complete spectrum of system gain and phase lag of spring-weighted objects in the frequency range of 0.0 1- 100 radian/second. This is an example of Porter's Atlas, which is a graphic analysis tool invented by Hendrick Porter of Bell Laboratories in AD 1940. Using this tool, we can judge the vibration amplitude and phase of the corresponding output signal driven by a sine wave input signal with a certain frequency. To get the amplitude of the output signal, just multiply the amplitude of the input signal by "the gain coefficient corresponding to this frequency in Bode diagram". To get the phase of the output signal, just add the phase of the input signal to the "phase lag value corresponding to this frequency in Bode diagram". The gain coefficient and phase lag value shown in the Bode diagram of the process object reflect the very definite characteristics of the system. For an experienced control engineer, the diagram tells him all the characteristics of the process object that he needs to know accurately. Therefore, using this tool, control engineers can not only predict "system response caused by future control of sine wave", but also know "system response caused by any control action".
Fourier theorem makes the above analysis possible. This theorem shows that any continuously measured time series or signal can be expressed as infinite superposition of sine wave signals with different frequencies. Mathematician Fourier proved this famous theorem in 1822, and created a famous algorithm called Fourier transform, which uses the directly measured original signal to calculate the frequency, amplitude and phase of different sine wave signals through accumulation.
Theoretically, Fourier transform and Bode diagram can be used together to predict the reaction of linear process objects when they are affected by the time sequence of control actions. See below for details:
1) The control function provided to the process object is theoretically decomposed into different sine wave signal components or frequency spectra by using the mathematical method of Fourier transform.
2) Using the Porter diagram, we can judge what happens when each sine wave signal passes through the process object. In other words, the amplitude and phase changes of sine wave at each frequency can be found on the graph.
3) On the contrary, by using the inverse Fourier transform method, each sine wave signal can be converted into a signal.
Because the inverse Fourier transform is essentially an accumulation process, the linear characteristics of the process object will ensure that a group of individual effects produced by "various theoretical sine waves calculated in the first step" should be equivalent to those produced by "accumulation groups of different sine waves". Therefore, the total signal calculated in the third step can represent "the actual value of the process object when the provided control action is input to the process object".
Please note that in the above steps, there is no point that is not composed of a single sine wave generated by the controller drawn in the figure. All these frequency domain analysis techniques are conceptual. This is a convenient mathematical method. Fourier transform (or closely related Laplace transform) is used to convert time domain signals into frequency domain signals, and then Bode diagram or other frequency domain analysis tools are used to solve some problems at hand. Finally, the inverse Fourier transform is used to convert the frequency domain signal into the time domain signal.
Most control design problems that can be solved by this method can also be solved by direct manipulation in time domain, but for calculation, the frequency domain method is usually simpler. In the above example, the frequency spectrum of the actual value of the process is calculated by multiplication and subtraction, and the actual value of the process is obtained by Fourier transform of the given control function, and then analyzed by comparing it with Bode diagram.
If all sine waves are accumulated correctly, a signal with the predicted shape of Fourier transform will be generated. When this phenomenon is sometimes not intuitive, it may help to give an example.
Please think again about the children's heavy objects in the above example-spring toys, seesaws on the playground, boats on the ocean. Suppose the ship undulates in the form of sine wave with frequency W and amplitude A on the sea surface, and we also assume that the seesaw is swinging in the form of sine wave with frequency 3w and amplitude A/3, and the child is shaking the toy in the form of sine wave with frequency 5w and amplitude A/5. Three independent sine waves show that if we observe three different sine wave motions respectively, each sine wave motion will take on a new form.
Now suppose a child is sitting on a seesaw, which is fixed on the deck of the ship in turn. If the sine waves of the three are arranged correctly, then the overall movement of the toy is about a square wave-as shown in Figure 4: sine waves of the three.
The above is not a very accurate practical example, but it is certain that the sum of the fundamental sine wave, the third harmonic with amplitude and the fifth harmonic with amplitude is approximately equal to the square wave with frequency w and amplitude a. Even if we add the seventh harmonic with one-seventh amplitude and the ninth harmonic with one-ninth amplitude, the total waveform is more like a square wave. In fact, Fourier theorem has been explained. When sine waves with different frequencies are infinitely accumulated into an infinite series, the total superimposed signal thus generated is a square wave with amplitude A in a strict sense. Fourier theorem can also be used to decompose aperiodic signals into infinite superposition of sine wave signals.
It is very useful to analyze the time domain performance by solving differential equations, but it is more troublesome for more complex systems. Because the calculation workload of solving differential equations will increase with the increase of the order of differential equations. In addition, when the equation has been solved and the response of the system can not meet the technical requirements, it is not easy to determine how to adjust the system to obtain the expected results. From the engineering point of view, we hope to find a method to predict the system performance without solving differential equations. At the same time, it can also point out how to adjust the technical indicators of system performance. Frequency domain analysis method has the above characteristics, which is a classic method to study control system and an engineering method to evaluate system performance through graphic analysis in frequency domain. This method is a method to study the system performance in frequency domain with the frequency of input signal as a variable. Frequency characteristics can be obtained from differential equations or transfer functions, and can also be measured by experimental methods. Frequency domain analysis method does not need to directly solve the differential equation of the system, but indirectly reveals the time domain performance of the system, which can conveniently show the influence of system parameters on the system performance and further indicate how to design and correct it. This analysis method is beneficial to system design and can estimate the frequency range that affects system performance. Especially when there are some components in the system that are difficult to be described by mathematical models, the frequency characteristics of the system can be obtained by experimental methods, so that the system and components can be analyzed accurately and effectively.