![]() | This article may be too technical for most readers to understand.(December 2018) |
In control theory and dynamical systems theory, the state-transition matrix is a matrix function that describes how the state of a linear system changes over time. Essentially, if the system's state is known at an initial time , the state-transition matrix allows for the calculation of the state at any future time .
The matrix is used to find the general solution to the homogeneous linear differential equation and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.
For linear time-invariant (LTI) systems, where the matrix is constant, the state-transition matrix is the matrix exponential . In the more complex time-variant case, where can change over time, there is no simple formula, and the matrix is typically found by calculating the Peano–Baker series.
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, and are matrix functions, and is the initial condition at . Using the state-transition matrix , 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 is the identity matrix. 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.
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 given by , , ..., . The corresponding solutions provide the columns of matrix . Now, from property 4, for all . The state-transition matrix must be determined before analysis on the time-varying solution can continue.