State-transition matrix

In control theory and dynamical systems theory, the state-transition matrix is a matrix function that describes how the state of a linear system changes over time. Essentially, if the system's state is known at an initial time ${\displaystyle t_{0}}${\displaystyle t_{0}}, the state-transition matrix allows for the calculation of the state at any future time ${\displaystyle t}${\displaystyle t}.

The matrix is used to find the general solution to the homogeneous linear differential equation ${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)}${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)} and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.

For linear time-invariant (LTI) systems, where the matrix ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} } is constant, the state-transition matrix is the matrix exponential ${\displaystyle e^{\mathbf {A} (t-t_{0})}}${\displaystyle e^{\mathbf {A} (t-t_{0})}}. In the more complex time-variant case, where ${\displaystyle \mathbf {A} (t)}${\displaystyle \mathbf {A} (t)} can change over time, there is no simple formula, and the matrix is typically found by calculating the Peano–Baker series.

The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\;\mathbf {x} (t_{0})=\mathbf {x} _{0}}${\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 ${\displaystyle \mathbf {x} (t)}${\displaystyle \mathbf {x} (t)} are the states of the system, ${\displaystyle \mathbf {u} (t)}${\displaystyle \mathbf {u} (t)} is the input signal, ${\displaystyle \mathbf {A} (t)}${\displaystyle \mathbf {A} (t)} and ${\displaystyle \mathbf {B} (t)}${\displaystyle \mathbf {B} (t)} are matrix functions, and ${\displaystyle \mathbf {x} _{0}}${\displaystyle \mathbf {x} _{0}} is the initial condition at ${\displaystyle t_{0}}${\displaystyle t_{0}}. Using the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}${\displaystyle \mathbf {\Phi } (t,\tau )}, the solution is given by:^{[1] [2]}

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,t_{0})\mathbf {x} (t_{0})+\int _{t_{0}}^{t}\mathbf {\Phi } (t,\tau )\mathbf {B} (\tau )\mathbf {u} (\tau )d\tau }${\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

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

where ${\displaystyle \mathbf {I} }${\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

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

where ${\displaystyle {\mathcal {T}}}${\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 ${\displaystyle \mathbf {\Phi } }${\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 ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)} and ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=\mathbf {I} }${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=\mathbf {I} }, where ${\displaystyle \mathbf {I} }${\displaystyle \mathbf {I} } is the identity matrix.
3. ${\displaystyle \mathbf {\Phi } (t,t)=\mathbf {I} }${\displaystyle \mathbf {\Phi } (t,t)=\mathbf {I} } for all ${\displaystyle t}${\displaystyle t} .^[3]
4. ${\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})}${\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})} for all ${\displaystyle t_{0}\leq t_{1}\leq t_{2}}${\displaystyle t_{0}\leq t_{1}\leq t_{2}}.
5. It satisfies the differential equation ${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})} with initial conditions ${\displaystyle \mathbf {\Phi } (t_{0},t_{0})=\mathbf {I} }${\displaystyle \mathbf {\Phi } (t_{0},t_{0})=\mathbf {I} }.
6. The state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}${\displaystyle \mathbf {\Phi } (t,\tau )}, given by ${\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )}${\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )} where the ${\displaystyle n\times n}${\displaystyle n\times n} matrix ${\displaystyle \mathbf {U} (t)}${\displaystyle \mathbf {U} (t)} is the fundamental solution matrix that satisfies ${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)}${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)} with initial condition ${\displaystyle \mathbf {U} (t_{0})=\mathbf {I} }${\displaystyle \mathbf {U} (t_{0})=\mathbf {I} }.
7. Given the state ${\displaystyle \mathbf {x} (\tau )}${\displaystyle \mathbf {x} (\tau )} at any time ${\displaystyle \tau }${\displaystyle \tau }, the state at any other time ${\displaystyle t}${\displaystyle t} is given by the mapping${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}

In the time-invariant case, we can define ${\displaystyle \mathbf {\Phi } }${\displaystyle \mathbf {\Phi } }, using the matrix exponential, as ${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}. ^[4]

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

• Baake, M.; Schlaegel, U. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275: 155–159. doi:10.1134/S0081543811080098. S2CID 119133539.
• Brogan, W.L. (1991). Modern Control Theory . Prentice Hall. ISBN 0-13-589763-7.

• Classical control theory
• Dynamical systems
• Matrix theory

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