In control theory, the state-transition matrix is a matrix whose product with the state vector
at an initial time
The state-transition matrix can be used to obtain the general solution of linear dynamical systems.
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form where
are the states of the system,
is the input signal,
is the initial condition at
, the solution is given by:[1][2] The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input.
The second term is known as the zero-state response and defines how the inputs impact the system.
The most general transition matrix is given by a product integral, referred to as the Peano–Baker series where
This matrix converges uniformly and absolutely to a solution that exists and is unique.
[2] The series has a formal sum that can be written as where
is the time-ordering operator, used to ensure that the repeated product integral is in proper order.
The Magnus expansion provides a means for evaluating this product.
The state transition matrix
satisfies the following relationships.
These relationships are generic to the product integral.
2, It is never singular; in fact
It satisfies the differential equation
with initial conditions
is the fundamental solution matrix that satisfies 7.
, the state at any other time
is given by the mapping In the time-invariant case, we can define
, using the matrix exponential, as
[4] In the time-variant case, the state-transition matrix
can be estimated from the solutions of the differential equation
with initial conditions
The corresponding solutions provide the
columns of matrix
The state-transition matrix must be determined before analysis on the time-varying solution can continue.