State-transition matrix

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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.

Linear systems solutions[edit]

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 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 the second term is known as the zero-state response.

Peano-Baker series[edit]

The most general transition matrix is given by the Peano-Baker series

where is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.[2]

Other properties[edit]

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

The state transition matrix must always satisfy the following relationships:

for all and where is the identity matrix.[3]

And also must have the following properties:


If the system is time-invariant, we can define ; 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.


  • Baake, M.; Schlaegel, U. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275. pp. 155–159.
  • Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7.


  1. ^ Baake, Michael; Schlaegel, Ulrike (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275: 155–159.
  2. ^ a b Rugh, Wilson (1996). Linear System Theory. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-441205-2.
  3. ^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.

See also[edit]