State-transition matrix


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 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.

Linear systems solutions

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:
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.

Peano–Baker series

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. 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.

Other properties

The state transition matrix satisfies the following relationships. These relationships are generic to the product integral.
  1. It is continuous and has continuous derivatives.
  2. It is never singular; in fact and, where is the identity matrix.
  3. for all .
  4. for all.
  5. It satisfies the differential equation with initial conditions.
  6. The state-transition matrix, given by where the matrix is the fundamental solution matrix that satisfies with initial condition.
  7. Given the state at any time, the state at any other time is given by the mapping

Estimation of the state-transition matrix

In the time-invariant case, we can define, using the matrix exponential, as.
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.