How to derive the transfer function of the state equation

　・In Japanese
premise knowledge
　・Equation of state
　・State feedback control
　・Transfer function
　・Laplace transform
　・Inverse matrix

Explain how to express state equations with transfer functions. The advantage of expressing the state equation as a transfer function is that it is possible to judge the stability of the system.

■Transfer function of a first-order system state equation

For a first-order system in its most basic form:

Laplace transform the formula (1).


Laplace transform the formula (2).


Substitute the formula (3) into the above formula.


The transfer function G(s) is as follows.



"I" is called the identity matrix and can be shown below. (The following is an identity matrix with 2 rows and 2 columns)



＜Specific calculation example＞
Calculate the transfer function when A, B, and C are as follows.


First calculate from the inverse matrix


From the above, we were able to obtain the transfer function.

■Transfer function of state equation of state feedback system

Derive the system transfer function for the following state-feedback control system.

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