# 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.

## 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
    ${\begin{aligned}\mathbf {\Phi } (t,\tau )=\mathbf {I} &+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\int _{\tau }^{\sigma _{2}}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}\\&+\cdots \end{aligned}}$
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.