State-transition matrix

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.

The most general transition matrix is given by a product integral, referred to as the Peano–Baker series

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

where ${\displaystyle \mathbf {I} }$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.^[2] The series has a formal sum that can be written as

${\displaystyle \mathbf {\Phi } (t,\tau )=\exp {\mathcal {T}}\int _{\tau }^{t}\mathbf {A} (\sigma )\,d\sigma }$

where ${\displaystyle {\mathcal {T}}}$ 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.

The state transition matrix ${\displaystyle \mathbf {\Phi } }$ 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 ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}$ and ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=\mathbf {I} }$, where ${\displaystyle \mathbf {I} }$ is the identity matrix.
3. ${\displaystyle \mathbf {\Phi } (t,t)=\mathbf {I} }$ 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 ${\displaystyle \mathbf {\Phi } (t_{0},t_{0})=\mathbf {I} }$.
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 ${\displaystyle \mathbf {U} (t_{0})=\mathbf {I} }$.
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${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}$

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 ${\displaystyle [1,\ 0,\ \ldots ,\ 0]^{\mathrm {T} }}$, ${\displaystyle [0,\ 1,\ \ldots ,\ 0]^{\mathrm {T} }}$, ..., ${\displaystyle [0,\ 0,\ \ldots ,\ 1]^{\mathrm {T} }}$. 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.

• Magnus expansion
• Liouville's formula