State-transition matrix

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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