Mathematical Expression for the State Transition Matrix
Mathematically , the state transition matrix can be represented as follows :
If you have a linear time – invariant system represented in state -space as :
ẋ(t) = A * x(t) + B * u(t)
where:
ẋ(t) is the derivative of the state vector x(t) with respect to time .
A is the system matrix .
B is the input matrix.
u(t) is the control input .
Then, the state transition matrix Ф(t) satisfies the following equation ẋ(t) = A * x(t) + B * u(t) , where Ф(t) is a matrix
such that x(t)= Ф(t)*x(0).
In the above equation , x(t) represents the state of the system at time t, and x(0) is the initial state at time t=0.
State Transition Matrix in Exponential Form
The below expression represents the state transition matrix in the exponential form .
Ф(t) = where :Ф(t) is the state transition matrix .A is a system matrix .
Ф(t) can be obtained by the Inverse Laplace Transform form of the Ф(s) , where Ф(s) = [sI – A]-1 (i.e., inverse of [sI-A]).
Ф(t) = L-1 { Ф(s) }
Important Properties of State Transition Matrix
A state transition matrix is a fundamental concept used to describe the Fundamental evolution of a linear time-invariant system in a state space representation. The state transition matrix is often represented by Ф(t). In this article, we will Go Through What is State Transition Matrix, What is Linear time-invariant System, the General Representation State Transition Matrix, and the Mathematical expression for the state transition matrix, and At last we will go through Solved examples of State Transition Matrix with its Application, Advantages, Disadvantages, and FAQs.
Table of Content
- State Transition Matrix
- LTI System
- General Representation
- Mathematical expression
- Steps to evaluate
- Example
- Properties
- Advantages
- Disadvantages
- Applications
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