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 ${\displaystyle t_{0}}$, the state-transition matrix allows for the calculation of the state at any future time ${\displaystyle t}$.

The matrix is used to find the general solution to the homogeneous linear differential equation ${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)}$ 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 ${\displaystyle \mathbf {A} }$ is constant, the state-transition matrix is the matrix exponential ${\displaystyle e^{\mathbf {A} (t-t_{0})}}$. In the more complex time-variant case, where ${\displaystyle \mathbf {A} (t)}$ 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

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\;\mathbf {x} (t_{0})=\mathbf {x} _{0}}$ ,

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 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.

Other properties

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

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

See also

• Magnus expansion
• Liouville's formula

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. S2CID 119133539.
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. hdl:2263/21017. S2CID 40282943.

Further reading

• 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.

 

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