Discrete system means that the change of system state only occurs at some time points, and it is often random. For example, the daily traffic volume of an intersection has no practical significance to the computer simulation of discrete models, but only statistical significance, so no model in simpowersystem belongs to discrete systems. But when choosing models and simulation algorithms, what do discrete models, discrete solvers, discrete simulation types and so on often refer to? In fact, it refers to discrete time, that is, discrete time model.
The continuity mentioned below refers to the continuity in time, and the continuous model refers to the continuous time model. Discretization refers to the discreteness in time, and discrete model refers to discrete time model, but in the physical world, they all belong to continuous systems. Why discretize the continuous model? Mainly from the mathematical model of the system, the former is modeled by differential equation, and the latter is modeled by difference equation, which is more suitable for computer calculation. The former adopts numerical integration method for simulation solver, and the latter adopts state updating discrete algorithm for difference equation.
In simpowersystem library, for some physical equipment, both its continuous model and its discrete model are given, for example:
A very important parameter of discrete model is sampling time. How to discretize a continuous model from the perspective of mathematical modeling will be introduced later. Powergui is often used in simpowersystem to discretize the continuous model in the system, so that the discretization algorithm can be used to facilitate computer calculation.
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2. Mathematical modeling of continuous model and discrete model.
Note: Continuity and discreteness here refer to continuity and discreteness in time, and have nothing to do with continuous systems and discrete systems in the real world. The so-called mathematical modeling is what kind of mathematical language is used to describe the model.
The mathematical model of continuous system can usually be expressed in the following forms: differential equation, transfer function and state space expression, which can be transformed into each other, and the state space expression is most beneficial to computer calculation.
① Differential equation:
A continuous system can be expressed as a higher order differential equation, i.e.
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② transfer function
Laplace transform is performed on both sides of the above formula. Assuming that the initial values of the derivatives of y and u (including zero order) are all zero, there are
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Then the transfer function of the differential equation is described as follows:
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③ State space expression
The state space expression of linear time-invariant system includes the following two matrix equations:
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(7- 1)
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(7-2)
Equation (7- 1) consists of n first-order differential equations, which are called state equations. Equation (7-2) is called the output equation by L linear algebraic equations.
Therefore, the following state equation and output equation are obtained (let a0= 1):
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The discrete model assumes that the input, output and internal state of the system are discrete functions of time, that is, time series:
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Where t is a discrete time interval, in fact, t is the sample time in the above article.
Note: The discrete model here still refers to the discrete time model, which has nothing to do with the discrete event model in the real world. The dispersion in simpowersystem refers to the dispersion in time, which has nothing to do with the concept of dispersion we learned in signal school.
Discrete time model includes difference equation, discrete transfer function, weight sequence and discrete state space model.
① difference equation
The general expression of the difference equation is:
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Similarly, the difference equation can be transformed into the following expression.
3. Discretization of continuous model
Such as the screenshot in 7. 1. Continuous system vs discrete system explains how to get its discrete model from continuous model, (RMS? Discrete RMS value), and how powergui discretizes the continuous model, that is, how simulator transforms the differential equation into the difference equation.
Assume that the state equation of a continuous system is
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At present, the input and output terminals of the system are artificially added with sampling switches, and at the same time, in order to restore the input signal to the original signal, a keeper is added at the input terminal, as shown in the figure. Now assume that it is a zero-order holder, that is, assume that all components of the input vector are constant at any two consecutive sampling moments, for example, for the nth sampling period u(t)=u(nt), where t is the sampling interval.
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According to the sampling theorem, when the sampling frequency WS and the maximum signal frequency wmax satisfy ws >; 2 wmax, the original signal can be uniquely determined from the sampled signal. The signal can be reconstructed by passing the sampled discrete signal through a low-pass filter. It is worth noting that the sampler and holder shown in the figure do not actually exist, but are fictional for the purpose of discrete equations.
Solve the following equation and perform Laplace transform on both sides of another program, so that
that is
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Through a series of Rasmussen inverse transformation and convolution, the difference equation is finally obtained (the specific process need not be concerned)
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Collectively referred to as the discrete coefficient matrix of the system.
An important parameter T, namely sampling interval, namely sampling time, is introduced in the conversion process. Whether it is powergui or other discrete models, as long as it involves discretization, it will inevitably involve sample time, as shown in the following figure.
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So how long is the sampling time, as long as the sampling theorem is satisfied, that is, the sampling frequency of the signal is more than twice the maximum frequency of the signal itself.
4. Simulator continuous model simulation algorithm (also translated as simulatesolver) and the concept of step size.
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The computer simulation algorithm of continuous system is numerical integration method, that is, the computer uses numerical integration to solve the differential equation, so as to get its approximate solution. The specific method is as follows
(1) Euler method and improved Euler method;
The existing differential equations are as follows:
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The right-hand integral of the above formula cannot be solved by computer, and its geometric meaning is the area of the curve f(t, y) on the interval (ti, ti+ 1). When (ti, ti+ 1) is small enough, it can be approximately replaced by a rectangular region:
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Where h is the integration step size.
Note: In the simulator simulation calculation, H is actually the simulation time interval.
Therefore, the following formula can be obtained:
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So as long as we know the current state and step size, we can get the next state. Its geometric significance is as follows:
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Analyze its error characteristics:
According to Taylor expansion:
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The truncation error is known.
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It is proportional to the step size h2, so if the computer wants to make the approximate integral more accurate, it will reduce the step size, but it will increase the truncation error.
② Improved Euler method (prediction-correction method)
Calculate the right-hand integral of the integral formula (3. 1.2) by using the trapezoidal area formula, and get
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Write the above formula in recursive difference format:
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It can be seen from the above formula that when calculating y n+ 1, we need to know that fn+ 1, FN+ 1 = F (Tn+ 1, FN+ 1) depends on yn+ 1 itself. Therefore, it is necessary to calculate each estimated ypn+ 1 by Euler method, and then substitute this value into the original equation to calculate fpn+ 1. Finally, the modified ypn+ 1 is obtained by using the following formula. So the improved Euler method can be described as
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③ Runge-Kutta method
Euler's method is to combine
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The first-order one-step method is obtained by Taylor series expansion and truncation of h2, so the accuracy is low. If the expansion is truncated after taking multiple terms, a higher-order numerical solution can be obtained, but it is difficult to directly calculate the higher-order derivative of the function by Taylor series expansion. Runge-Kutta method uses the idea of indirectly using Taylor series expansion, that is, the linear combination of function value f at n points replaces the derivative of f, and then the coefficient is determined according to Taylor series expansion to improve the order of the algorithm. This can not only avoid calculating the derivative of the function, but also ensure the calculation accuracy. Runge-Kutta method is one of the most basic algorithms in many simulation software packages because of its many advantages.
④ Linear multi-step method
The numerical solutions mentioned above are all one-step methods. As long as you know that in the calculation
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. That is to say, the Y value of continuous moments can be calculated recursively according to the initial conditions, so this method can be started automatically. Here is another algorithm, multi-step method.
When using this algorithm to solve, you may need to
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The value of every moment. Obviously, the multi-step calculation formula can't be started by itself, which takes up a lot of memory in the calculation process, but it can improve the calculation accuracy and speed. For example, Adams-Beshihos explicit multi-step method.
⑤ Rigid system solution
The so-called rigid system is used to describe the solution of the differential equation of this kind of system, which is often affected by multiple time constants * * *. Some small time constants have little influence on the solution, but they are indeed indispensable. For example, the following equation is the solution of a simple differential equation of a rigid system:
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When the time is long, the characteristic solution-1000 has little influence on the equation, but at the beginning, the influence of e-1000t can not be ignored. Therefore, if the computer numerical solution is introduced earlier, in order to ensure the stability of the solution, when choosing the H step, it must be ensured that the H step is very small, which will inevitably increase the number of calculations and increase the calculation time. Moreover, because e-1000t hardly works under certain conditions, this increase will not greatly improve the calculation accuracy, that is to say, it is useless to calculate the rigid system by conventional solution.
So far, many numerical methods have been proposed to solve rigid equations, which are basically divided into explicit formulas, implicit formulas and predictive correction types.
Renard method is often used to display formulas.
Implicit equations are all stable, so they are all suitable for solving equations describing rigid systems, such as implicit Runge-Kutta method. However, this method needs iteration every step of calculation, so it is difficult to use in engineering because of its large amount of calculation. Therefore, the semi-implicit Runge-Kutta method proposed by Rosenbrock is often used to solve rigid equations.
Gear algorithm is often used to solve the rigid equation in predictive correction model.
5. The simulation algorithm of the simulator discrete model and the concept of step size.
The mathematical modeling of discrete model generally adopts the way of difference equation, and the simulation algorithm in matlab adopts discrete algorithm, that is, the discrete module is updated according to the simulation step (that is, the difference equation is calculated at regular intervals).
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As for the concept of its step size, it is similar to the concept of H in the continuous model, but the selection of its size is closely related to the sample time, which will be explained below.
6.6. Simulation/configuration parameters in Simulink.
With the above knowledge, we can introduce the setting of simulink simulation parameters.
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The solver (simulation solver) in the above figure is programmed by computer language with various algorithms introduced above.
The continuous solver is a numerical integration method and the discrete solver is a discrete solution method.
Step size has variable step size and fixed step size. The step size in the continuous solver is h, that is, the integration time interval. For the discrete solver, the step size is closely related to the sample time in the model to be simulated, so it cannot be taken casually.
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① Variable step size (variable step size)
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That is to say, the variable step size will adjust the step size appropriately according to the speed of model state change, that is, the time interval between adjacent simulation calculations, which not only ensures a certain accuracy, but also reduces the number of simulations, thus reducing the simulation time.
For the continuous solver, the maximum step length and minimum step length can be set artificially, and then the computer automatically selects the integration step length h for numerical integration. Here is its analog solver (ODE stands for ordinary differential method).
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② Fixed step size (fixed step size)
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That is, the simulation uses the same step size from beginning to end. Note: For the continuous solver, the fixed step size can be considered as arbitrary; For dicretesolver, the fixed step size can be auto (that is, simulation can help you get it). If people choose it, they must abide by a certain relationship with the sample time, which will be introduced below.
Note: Some peripheral modules, such as DSP and fpga, are built in simulink, and the code is automatically generated after the simulation, which can run on the actual equipment. In this case, the simulation step size must be fixed. See the figure below for specific instructions:
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③ Discrete solver
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Solver is a discrete algorithm, that is, the state of discrete blocks is constantly updated at each discrete point, and the step size is closely related to sampletime in the model.
As can be seen from the above difference equation, the t sampling time in the difference equation is fixed. Regardless of variable step size or fixed step size, the simulation step size must appear in all integer multiples of the sampling time, that is, the simulation step size must be set so that the simulator can calculate the simulation model at 1T, 2T, 3T, so as not to miss the transformation of the principal state.
If a discrete simulation model has multiple sampling times, then to ensure that each model can be simulated within its adoption time of 1T, 2T and 3T, the minimum step size can only be the common divisor of each simulation time, and the maximum common divisor is also called the basic sampling time. Examples are as follows.
Assuming that there are two sampling times T 1=2e-6 and T2=4e-6 in the simulated discrete model, the common divisors are 1e-6 and 2e-6, while the basic sampling time is 2e-6.
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If the fixed step size is adopted, in order not to miss the state change of the model at each sampling moment, the simulation time of the simulator must include an integer multiple of each sampling moment, so its fixed step size must take the common divisor of each sampletime, which can be 1e-6 or 2e-6; If auto is written, the basic sampling time is = 2e-6; If you write other steps, the simulation will prompt an error.
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The above simulation process is as follows:
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Arrows represent simulation steps, that is, the simulator will simulate calculation once at each arrow; The circle represents the sampling time of the model. In fact, only at this moment can the state of the discrete model change, that is, the solution of the difference equation can change. As can be seen from the above figure, setting the step size in this way ensures that the simulator is simulated at every sampling time.
If variable step size is adopted, the simulator will automatically adjust the step size according to each sample time in the model, so that the simulation time is equal to the sample time.
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At this time, there is also a maximum step length limit. If auto is written in the above picture, the above simulation process is as follows:
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It can be seen that the simulator only performs simulation calculation at the sampling time, which reduces the number of simulations and saves time.
If the maximum step length =0.7e-6, what about the simulation process? As shown in the figure below:
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When the variable step is visible, the simulator will always track the sampletime, even if someone is limited by the size of maxstep. Generally, you can choose automatic.
⑥ About the function of powergui
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In simpowersystem simulation, Powergui has two main functions:
I: discretize some continuous models in the system so that the simulator can use discrete algorithm to calculate. Note: It has no influence on the existing discrete model, as shown in the following figure:
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The discrete sampling time of powergui is 2e-6, while the sampling time of discrete modules in the system is 4e-6. The discrete function of powergui has no effect on it.
Ⅱ: Provide various graphical user interface tools for analyzing signals and data in the simulation process (especially FFT analysis).
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