
State-transition matrix
Encyclopedia
In control theory
, the state-transition matrix is a matrix whose product with the state vector
at an initial time
gives
at a later time
. The state-transition matrix can be used to obtain the general solution of linear dynamical systems. It is also known as the matrix exponential
.
model
The general solution is given by
The state-transition matrix
, given by
where
is the fundamental solution matrix that satisfies
is a
matrix that is a linear mapping onto itself, i.e., with
, given the state
at any time
, the state at any other time
is given by the mapping
While the state transition matrix φ is not completely unknown, it must always satisfy the following relationships:
and
for all
and where
is the identity matrix
.
And φ also must have the following properties:
If the system is time-invariant, we can define φ as:

In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, the state-transition matrix is a matrix whose product with the state vector




Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
.
Overview
Consider the general linear state spaceState space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
model
The general solution is given by
The state-transition matrix

where

is a






While the state transition matrix φ is not completely unknown, it must always satisfy the following relationships:




Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
And φ also must have the following properties:
1. | ![]() |
2. | ![]() |
3. | ![]() |
4. | ![]() |
If the system is time-invariant, we can define φ as:

In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.