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−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 V_c  as the first state variable and the current through the inductor i_L 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.