Markov transition matrix

State-transition matrix

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.

Overview

Consider the general linear state space model
dot{mathbf{x}}(t) = mathbf{A}(t) mathbf{x}(t) + mathbf{B}(t) mathbf{u}(t)
mathbf{y}(t) = mathbf{C}(t) mathbf{x}(t) + mathbf{D}(t) mathbf{u}(t)
The general solution is given by
mathbf{x}(t)= mathbf{Phi} (t, t_0)mathbf{x}(t_0)+int_{t_0}^t mathbf{Phi}(t, tau)mathbf{B}(tau)mathbf{u}(tau)dtau
The state-transition matrix mathbf{Phi}(t, tau), given by
mathbf{Phi}(t, tau)equivmathbf{U}(t)mathbf{U}^{-1}(tau)
where mathbf{U}(t) is the fundamental solution matrix that satisfies
dot{mathbf{U}}(t)=mathbf{A}(t)mathbf{U}(t)
is a n times n matrix that is a linear mapping onto itself, i.e., with mathbf{u}(t)=0, given the state mathbf{x}(tau) at any time tau, the state at any other time t is given by the maping
mathbf{x}(t)=mathbf{Phi}(t, tau)mathbf{x}(tau)

While the state transtion matrix φ is not completely unknown, it must always satisfy the following relationships:

frac{partial phi(t, t_0)}{partial t} = A(t)phi(t, t_0)

phi(tau, tau) = I

And φ also must have the following properties:

1. phi(t_2, t_1)phi(t_1, t_0) = phi(t_2, t_0)
2. phi^{-1}(t, tau) = phi(tau, t)
3. phi^{-1}(t, tau)phi(t, tau) = I
4. frac{dphi(t, t_0)}{dt} = A(t)

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

phi(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 dependant on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

References

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