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

State-space representation is used for simulating physical systems on digital computers. For this, the transfer function must first be converted into state space.
Consider an nth-order, linear differential equation with constant coefficients.
The output and its n-1 derivatives are chosen as the state variables. Differentiating these series of equations and substituting them back into the original equations, provides the state equations.
Each subsequent state variable is defined as the derivative of the previous one.
The resulting equations are then represented in vector-matrix form, creating a distinct pattern of 1&#39;s and 0&#39;s along with the negative coefficients of the differential equation. This unique structure is the phase-variable form of the state equations.
Consider a transfer function. The equation is cross-multiplied, and the corresponding differential equation is then found by taking the inverse Laplace transform, assuming zero initial conditions.
The state variables are chosen as successive derivatives.
Differentiation is then applied to both sides of the equation, yielding the state equations and the output equation.
These equations are then presented in vector-matrix form.

transfer function to state space

State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n&#8722;1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC circuit, the voltage across the capacitor and the current through the inductor can be used as the state variables. For example, in a second-order system, such as a series RLC circuit, we can define the voltage across the capacitor Vc as the first state variable and the current through the inductor iL as the second state variable. The transfer function for an RLC circuit is first cross-multiplied to obtain the differential equation. By applying the inverse Laplace transform and assuming zero initial conditions, one can derive the time-domain form of the equation.
<img alt="Linear circuit system equations and transfer function diagram for electrical analysis, H(s) equation." src="/files/ftp_upload/16101/16101_Equation_001.svg" style="height: 140px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
<img alt="RLC circuit differential equation, Ld&#178;Vc/dt&#178; + RdVc/dt + (1/C)Vc = Vin, mathematical formula." src="/files/ftp_upload/16101/16101_Equation_002.svg" style="height: 60px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
State variables are then chosen as successive derivatives of the output, and differentiation is applied to both sides of the equation to generate the state equations.
<img alt="Electrical circuit differential equation; LC, RC components; voltage function analysis equation." src="/files/ftp_upload/16101/16101_Equation_003.svg" style="height: 25px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
The system's differential equations are then represented in vector-matrix form, yielding a distinct pattern of 1's and 0's, along with the negative coefficients of the original differential equation.
<img alt="Natural frequency and damping ratio equations; dynamic system analysis; mechanical resonance." src="/files/ftp_upload/16101/16101_Equation_004.svg" style="height: 90px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
This is known as the phase-variable form of the state equations. The matrix structure provides a clear and concise method for simulating and analyzing the dynamic behavior of the system.

Transfer Function to State Space

State-space representation is used for simulating physical systems on digital computers. For this, the transfer function must first be converted into ...

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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

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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.

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Phase-Variable State Equations from Transfer Functions

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