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 t 0 {\displaystyle t_{0}} , the state-transition matrix allows for the calculation of the state at any future time t {\displaystyle t} . The matrix is used to find the general solution to the homogeneous linear differential equation x ˙ ( t ) = A ( t ) 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 A {\displaystyle \mathbf {A} } is constant, the state-transition matrix is the matrix exponential exp ( A ( t − t 0 ) ) {\displaystyle \textstyle \exp {(\mathbf {A} (t-t_{0}))}} . In the more complex time-variant case, where 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.
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
x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , x ( t 0 ) = 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 x ( t ) {\displaystyle \mathbf {x} (t)} are the states of the system; u ( t ) {\displaystyle \mathbf {u} (t)} is the input signal; A ( t ) {\displaystyle \mathbf {A} (t)} and B ( t ) {\displaystyle \mathbf {B} (t)} are matrix functions; and x 0 {\displaystyle \mathbf {x} _{0}} is the initial condition at t 0 {\displaystyle t_{0}} . Using the state-transition matrix Φ ( t , τ ) {\displaystyle \mathbf {\Phi } (t,\tau )} , the solution is given by:
x ( t ) = Φ ( t , t 0 ) x ( t 0 ) + ∫ t 0 t Φ ( t , τ ) B ( τ ) u ( τ ) d τ . {\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 a product integral, referred to as the Peano–Baker series:
Φ ( t , τ ) = I + ∫ τ t A ( σ 1 ) d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) d σ 2 d σ 1 + ∫ τ t A ( σ 1 ) ∫ τ σ 1 A ( σ 2 ) ∫ τ σ 2 A ( σ 3 ) d σ 3 d σ 2 d σ 1 + ⋯ , {\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 I {\displaystyle \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
Φ ( t , τ ) = exp T ∫ τ t A ( σ ) d σ , {\displaystyle \mathbf {\Phi } (t,\tau )=\exp {\mathcal {T}}\int _{\tau }^{t}\mathbf {A} (\sigma )\,d\sigma ,}
where 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.
Other properties The state transition matrix Φ {\displaystyle \mathbf {\Phi } } satisfies the following relationships. These relationships are generic to the product integral.
It is continuous and has continuous derivatives. It is never singular; in fact Φ − 1 ( t , τ ) = Φ ( τ , t ) {\textstyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)} and Φ − 1 ( t , τ ) Φ ( t , τ ) = I {\displaystyle \textstyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=\mathbf {I} } , where I {\displaystyle \mathbf {I} } is the identity matrix.
Φ ( t , t ) = I {\displaystyle \mathbf {\Phi } (t,t)=\mathbf {I} } for all t {\displaystyle t} .
Φ ( t 2 , t 1 ) Φ ( t 1 , t 0 ) = Φ ( 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 t 0 ≤ t 1 ≤ t 2 {\displaystyle t_{0}\leq t_{1}\leq t_{2}} . It satisfies the differential equation ∂ ∂ t Φ ( t , t 0 ) = A ( t ) Φ ( t , t 0 ) {\textstyle {\frac {\partial }{\partial t}}\mathbf {\Phi } (t,t_{0})=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})} with initial conditions 1 {\displaystyle {1}} . The state-transition matrix Φ ( t , τ ) {\displaystyle \mathbf {\Phi } (t,\tau )} , given by Φ ( t , τ ) ≡ U ( t ) U − 1 ( τ ) {\textstyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )} where the n × n {\displaystyle n\times n} matrix U ( t ) {\displaystyle \mathbf {U} (t)} is the fundamental solution matrix that satisfies U ˙ ( t ) = A ( t ) U ( t ) {\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)} with initial condition U ( t 0 ) = I {\displaystyle \mathbf {U} (t_{0})=\mathbf {I} } . Given the state x ( τ ) {\displaystyle \mathbf {x} (\tau )} at any time τ {\displaystyle \tau } , the state at any other time t {\displaystyle t} is given by the mapping x ( t ) = Φ ( t , τ ) x ( τ ) {\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 Φ {\displaystyle \mathbf {\Phi } } , using the matrix exponential, as Φ ( t , t 0 ) = exp ( A ( t − t 0 ) ) {\displaystyle \textstyle \mathbf {\Phi } (t,t_{0})=\exp {(\mathbf {A} (t-t_{0}))}} . In the time-variant case, the state-transition matrix Φ ( t , t 0 ) {\displaystyle \mathbf {\Phi } (t,t_{0})} can be estimated from the solutions of the differential equation u ˙ ( t ) = A ( t ) u ( t ) {\displaystyle {\dot {\mathbf {u} }}(t)=\mathbf {A} (t)\mathbf {u} (t)} with initial conditions u ( t 0 ) {\displaystyle \mathbf {u} (t_{0})} given by [ 1 , 0 , … , 0 ] T {\displaystyle \textstyle [1,\ 0,\ \ldots ,\ 0]^{\mathrm {T} }} , [ 0 , 1 , … , 0 ] T {\displaystyle \textstyle [0,\ 1,\ \ldots ,\ 0]^{\mathrm {T} }} , ..., [ 0 , 0 , … , 1 ] T {\displaystyle \textstyle [0,\ 0,\ \ldots ,\ 1]^{\mathrm {T} }} . The corresponding solutions provide the n {\displaystyle n} columns of matrix Φ ( t , t 0 ) {\displaystyle \mathbf {\Phi } (t,t_{0})} . Now, from property 4, Φ ( t , τ ) = Φ ( t , t 0 ) Φ ( τ , t 0 ) − 1 {\textstyle \mathbf {\Phi } (t,\tau )=\mathbf {\Phi } (t,t_{0})\mathbf {\Phi } (\tau ,t_{0})^{-1}} for all t 0 ≤ τ ≤ t {\displaystyle t_{0}\leq \tau \leq t} . The state-transition matrix must be determined before analysis on the time-varying solution can continue.
See also Magnus expansion Liouville's formula
References
Further reading Brogan, W. L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7.