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−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.
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.
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.
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.
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's and 0'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.
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