Name: phase-variable state equations from transfer functions
Uploaded: 2024-11-21T14:54:05.000Z
Duration: 1 min 23 s
Channel: JoVE

状态空间表示法用于在数字计算机上对物理系统进行仿真。为此，必须先将传递函数转换为状态空间形式。
考虑一个 nth常系数线性微分方程
选择输出及其前 n-1 阶导数作为状态变量。对这一系列方程求导，并将结果代回原方程，即可得到状态方程。
每个后续的状态变量都定义为前一个状态变量的导数。
然后将得到的方程表示为向量-矩阵形式，在微分方程的负系数基础上形成由1和0构成的特定模式。这种独特结构即为状态方程的相变量形式。
考虑一个传递函数。将方程两边交叉相乘，然后在假设初始条件为零的情况下，通过取拉普拉斯逆变换得到相应的微分方程。
状态变量选择为逐阶导数。
然后对等式两边进行微分，得到状态方程和输出方程。
然后将这些方程表示为向量-矩阵形式。

传递函数到状态空间

Transfer Function to State Space

状态空间表示是在数字计算机上模拟物理系统的强大工具，需要将传递函数转换为状态空间形式。考虑一个具有常数系数的 n 阶线性微分方程，就像 RLC 电路中遇到的方程一样。状态变量被选为输出及其 n-1 个导数。对这些变量进行微分并将它们代回原始方程可得出状态方程。

在 RLC 电路中，电容器两端的电压和流过电感器的电流可用作状态变量。例如，在二阶系统（如串联 RLC 电路）中，我们可以将电容器两端的电压 V_c 定义为第一个状态变量，将流过电感器的电流 i_L 定义为第二个状态变量。首先将 RLC 电路的传递函数交叉相乘，得到微分方程。通过应用拉普拉斯逆变换并假设初始条件为零，可以推导出方程的时域形式。

<img alt="Equation1" src="/files/ftp_upload/16101/16101_Equation_001.svg" style="height: 140px;">
<img alt="Equation2" src="/files/ftp_upload/16101/16101_Equation_002.svg" style="height: 60px;">

然后将状态变量选为输出的连续导数，并对方程的两边应用微分来生成状态方程。

<img alt="Equation3" src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;">

然后将系统的微分方程以向量矩阵形式表示，产生 1 和 0 的独特模式，以及原始微分方程的负系数。

<img alt="Equation4" src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;">

这被称为状态方程的相变量形式。矩阵结构为模拟和分析系统的动态行为提供了一种清晰简洁的方法。

状态空间表示法用于在数字计算机上模拟物理系统。为此，必须首先将传递函数转换为……

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Electrical Engineering

电气工程

Unit 1

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Modeling in Time and Frequency Domain

时域与频域建模

KEY TERMS AND CONCEPTS

关键术语与概念

phase-variable state equations from transfer functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer.

The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is first cross-multiplied to form the differential equation. Then the inverse Laplace transform is applied, with zero initial conditions, to get the time-domain form.

Next, the state variables are chosen from the output and its successive derivatives. For a second-order system such as a series RLC circuit, the voltage across the capacitor, Vc, can be used as the first state variable. The current through the inductor, iL, can be used as the second state variable.

After the variables are selected, each one is differentiated and substituted back into the original equation. This process generates the state equations. The equations are then written in vector-matrix form.

In phase-variable form, the matrix has a clear pattern of 1s and 0s. It also contains the negative coefficients from the original differential equation. This structure gives a compact way to simulate and analyze the system's dynamic behavior.

Phase-Variable State Equations from Transfer Functions

phase-variable-state-equations-from-transfer-functions

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Learn how transfer functions are converted into phase-variable state equations using state variables, inverse Laplace transforms, and matrix form for RLC circuits.

Phase-Variable State Equations from Transfer Functions | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Phase-Variable State Equations from Transfer Functions

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