State-transition matrix

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 : \dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) , ;\mathbf{x}(t_0) = \mathbf{x}_0 , where \mathbf{x}(t) are the states of the system, \mathbf{u}(t) is the input signal, \mathbf{A}(t) and \mathbf{B}(t) are matrix functions, and \mathbf{x}0 is the initial condition at t_0. Using the state-transition matrix \mathbf{\Phi}(t, \tau), the 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)d\tau

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

The most general transition matrix is given by a product integral, referred to as the Peano–Baker series :\begin{align} \mathbf{\Phi}(t,\tau) = \mathbf{I} &+ \int_\tau^t\mathbf{A}(\sigma_1),d\sigma_1 \ &+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2),d\sigma_2,d\sigma_1 \ &+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^{\sigma_2}\mathbf{A}(\sigma_3),d\sigma_3,d\sigma_2,d\sigma_1 \ &+ \cdots \end{align} where \mathbf{I} is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique. The series has a formal sum that can be written as :\mathbf{\Phi}(t,\tau) = \exp \mathcal{T}\int_\tau^t\mathbf{A}(\sigma),d\sigma where \mathcal{T} 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

The state transition matrix \mathbf{\Phi} 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 \mathbf{\Phi}^{-1}(t, \tau) = \mathbf{ \Phi}(\tau, t) and \mathbf{\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, \tau) = \mathbf I, where \mathbf I is the identity matrix.
3. \mathbf{\Phi}(t, t) = \mathbf I for all t .
4. \mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0) for all t_0 \leq t_1 \leq t_2.
5. It satisfies the differential equation \frac{\partial \mathbf{\Phi}(t, t_0)}{\partial t} = \mathbf{A}(t)\mathbf{\Phi}(t, t_0) with initial conditions \mathbf{\Phi}(t_0, t_0) = \mathbf I.
6. The state-transition matrix \mathbf{\Phi}(t, \tau), given by \mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau) where the n \times n matrix \mathbf{U}(t) is the fundamental solution matrix that satisfies \dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t) with initial condition \mathbf{U}(t_0) = \mathbf I.
7. Given the state \mathbf{x}(\tau) at any time \tau, the state at any other time t is given by the mapping\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)

Estimation of the state-transition matrix

In the time-invariant case, we can define \mathbf{\Phi}, using the matrix exponential, as \mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}.

In the time-variant case, the state-transition matrix \mathbf{\Phi}(t, t_0) can be estimated from the solutions of the differential equation \dot{\mathbf{u}}(t)=\mathbf{A}(t)\mathbf{u}(t) with initial conditions \mathbf{u}(t_0) given by [1,\ 0,\ \ldots,\ 0]^\mathrm{T}, [0,\ 1,\ \ldots,\ 0]^\mathrm{T}, ..., [0,\ 0,\ \ldots,\ 1]^\mathrm{T}. The corresponding solutions provide the n columns of matrix \mathbf{\Phi}(t, t_0). Now, from property 4, \mathbf{\Phi}(t, \tau) = \mathbf{\Phi}(t, t_0)\mathbf{\Phi}(\tau, t_0)^{-1} for all t_0 \leq \tau \leq t. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

References

References

1. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics.
2. (1996). "Linear System Theory". Prentice Hall.
3. Brockett, Roger W.. (1970). "Finite Dimensional Linear Systems". John Wiley & Sons.
4. (2012). "Polynomial Filtering: To any degree on irregularly sampled data". Automatika.