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State transition matrix

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In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t 0 gives x at a later time t . The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

Contents

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

x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , x ( t 0 ) = x 0 ,

where x ( t ) are the states of the system, u ( t ) is the input signal, and x 0 is the initial condition at t 0 . Using the state-transition matrix Φ ( t , τ ) , the solution is given by:

x ( t ) = Φ ( t , t 0 ) x ( t 0 ) + ∫ t 0 t Φ ( t , τ ) B ( τ ) u ( τ ) d τ

Peano-Baker series

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

Φ ( t , τ ) = I + ∫ τ t A ( σ 1 ) d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) d σ 2 d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) ∫ τ σ 2 A ( σ 3 ) d σ 3 d σ 2 d σ 1 + . . .

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

Other properties

The state-transition matrix Φ ( t , τ ) , given by

Φ ( t , τ ) ≡ U ( t ) U − 1 ( τ )

where U ( t ) is the fundamental solution matrix that satisfies

U ˙ ( t ) = A ( t ) U ( t )

is a n × n matrix that is a linear mapping onto itself, i.e., with u ( t ) = 0 , given the state x ( τ ) at any time τ , the state at any other time t is given by the mapping

x ( t ) = Φ ( t , τ ) x ( τ )

The state transition matrix must always satisfy the following relationships:

∂ Φ ( t , t 0 ) ∂ t = A ( t ) Φ ( t , t 0 ) and Φ ( τ , τ ) = I for all τ and where I is the identity matrix.

And Φ ; also must have the following properties:

If the system is time-invariant, we can define Φ ; as:

Φ ( t , t 0 ) = e A ( t − t 0 )

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.

References

State-transition matrix Wikipedia
(Text) CC BY-SA


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