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

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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.

Peano–Baker series

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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.

Other properties

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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   .[3]

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

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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.

See also

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References

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  1. ^ Baake, Michael; Schlaegel, Ulrike (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275: 155–159. doi:10.1134/S0081543811080098. S2CID 119133539.
  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.
  4. ^ Reyneke, Pieter V. (2012). "Polynomial Filtering: To any degree on irregularly sampled data". Automatika. 53 (4): 382–397. doi:10.7305/automatika.53-4.248. hdl:2263/21017. S2CID 40282943.

Further reading

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