# State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector $\displaystyle{ x }$ at an initial time $\displaystyle{ t_0 }$ gives $\displaystyle{ x }$ at a later time $\displaystyle{ t }$. The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

## 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

$\displaystyle{ \dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) , \;\mathbf{x}(t_0) = \mathbf{x}_0 }$,

where $\displaystyle{ \mathbf{x}(t) }$ are the states of the system, $\displaystyle{ \mathbf{u}(t) }$ is the input signal, $\displaystyle{ \mathbf{A}(t) }$ and $\displaystyle{ \mathbf{B}(t) }$ are matrix functions, and $\displaystyle{ \mathbf{x}_0 }$ is the initial condition at $\displaystyle{ t_0 }$. Using the state-transition matrix $\displaystyle{ \mathbf{\Phi}(t, \tau) }$, the solution is given by:[1][2]

$\displaystyle{ \mathbf{x}(t)= \mathbf{\Phi} (t, t_0)\mathbf{x}(t_0)+\int_{t_0}^t \mathbf{\Phi}(t, \tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau }$

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 the Peano–Baker series

$\displaystyle{ \mathbf{\Phi}(t,\tau) = \mathbf{I} + \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^{\sigma_2}\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 + ... }$

where $\displaystyle{ \mathbf{I} }$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.[2]

## Other properties

The state transition matrix $\displaystyle{ \mathbf{\Phi} }$ satisfies the following relationships:

1. It is continuous and has continuous derivatives.

2, It is never singular; in fact $\displaystyle{ \mathbf{\Phi}^{-1}(t, \tau) = \mathbf{ \Phi}(\tau, t) }$ and $\displaystyle{ \mathbf{\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, \tau) = I }$, where $\displaystyle{ I }$ is the identity matrix.

3. $\displaystyle{ \mathbf{\Phi}(t, t) = I }$ for all $\displaystyle{ t }$ .[3]

4. $\displaystyle{ \mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0) }$ for all $\displaystyle{ t_0 \leq t_1 \leq t_2 }$.

5. It satisfies the differential equation $\displaystyle{ \frac{\partial \mathbf{\Phi}(t, t_0)}{\partial t} = \mathbf{A}(t)\mathbf{\Phi}(t, t_0) }$ with initial conditions $\displaystyle{ \mathbf{\Phi}(t_0, t_0) = I }$.

6. The state-transition matrix $\displaystyle{ \mathbf{\Phi}(t, \tau) }$, given by

$\displaystyle{ \mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau) }$

where the $\displaystyle{ n \times n }$ matrix $\displaystyle{ \mathbf{U}(t) }$ is the fundamental solution matrix that satisfies

$\displaystyle{ \dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t) }$ with initial condition $\displaystyle{ \mathbf{U}(t_0) = I }$.

7. Given the state $\displaystyle{ \mathbf{x}(\tau) }$ at any time $\displaystyle{ \tau }$, the state at any other time $\displaystyle{ t }$ is given by the mapping

$\displaystyle{ \mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau) }$

## Estimation of the state-transition matrix

In the time-invariant case, we can define $\displaystyle{ \mathbf{\Phi} }$, using the matrix exponential, as $\displaystyle{ \mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)} }$. [4]

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

## References

1. Baake, Michael; Schlaegel, Ulrike (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics 275: 155–159. doi:10.1134/S0081543811080098.
2. 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.