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

The matrix is used to find the general solution to the homogeneous linear differential equation $\dot{𝐱} (t) = 𝐀 (t) 𝐱 (t)$ 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 $𝐀$ is constant, the state-transition matrix is the matrix exponential ⁠ $\exp (𝐀 (t - t_{0}))$ ⁠. In the more complex time-variant case, where $𝐀 (t)$ 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 $\dot{𝐱} (t) = 𝐀 (t) 𝐱 (t) + 𝐁 (t) 𝐮 (t), 𝐱 (t_{0}) = 𝐱_{0},$ where $𝐱 (t)$ are the states of the system; $𝐮 (t)$ is the input signal; $𝐀 (t)$ and $𝐁 (t)$ are matrix functions; and $𝐱_{0}$ is the initial condition at ⁠ $t_{0}$ ⁠. Using the state-transition matrix ⁠ $𝜱 (t, τ)$ ⁠, the solution is given by:^[1]^[2] $𝐱 (t) = 𝜱 (t, t_{0}) 𝐱 (t_{0}) + \int_{t_{0}}^{t} 𝜱 (t, τ) 𝐁 (τ) 𝐮 (τ) d τ .$

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: $\begin{matrix} 𝜱 (t, τ) = 𝐈 & + \int_{τ}^{t} 𝐀 (σ_{1}) d σ_{1} \\ + \int_{τ}^{t} 𝐀 (σ_{1}) \int_{τ}^{σ_{1}} 𝐀 (σ_{2}) d σ_{2} d σ_{1} \\ + \int_{τ}^{t} 𝐀 (σ_{1}) \int_{τ}^{σ_{1}} 𝐀 (σ_{2}) \int_{τ}^{σ_{2}} 𝐀 (σ_{3}) d σ_{3} d σ_{2} d σ_{1} \\ + \dots, \end{matrix}$ where $𝐈$ 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 $𝜱 (t, τ) = \exp 𝒯 \int_{τ}^{t} 𝐀 (σ) d σ,$ where $𝒯$ 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 $𝜱$ 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 $𝜱^{- 1} (t, τ) = 𝜱 (τ, t)$ and ⁠ $𝜱^{- 1} (t, τ) 𝜱 (t, τ) = 𝐈$ ⁠, where $𝐈$ is the identity matrix.
$𝜱 (t, t) = 𝐈$ for all ⁠ $t$ ⁠.^[3]
$𝜱 (t_{2}, t_{1}) 𝜱 (t_{1}, t_{0}) = 𝜱 (t_{2}, t_{0})$ for all ⁠ $t_{0} \leq t_{1} \leq t_{2}$ ⁠.
It satisfies the differential equation $\frac{\partial}{\partial t} 𝜱 (t, t_{0}) = 𝐀 (t) 𝜱 (t, t_{0})$ with initial conditions ⁠ $1$ ⁠.
The state-transition matrix ⁠ $𝜱 (t, τ)$ ⁠, given by $𝜱 (t, τ) \equiv 𝐔 (t) 𝐔^{- 1} (τ)$ where the $n \times n$ matrix $𝐔 (t)$ is the fundamental solution matrix that satisfies $\dot{𝐔} (t) = 𝐀 (t) 𝐔 (t)$ with initial condition ⁠ $𝐔 (t_{0}) = 𝐈$ ⁠.
Given the state $𝐱 (τ)$ at any time ⁠ $τ$ ⁠, the state at any other time $t$ is given by the mapping ⁠ $𝐱 (t) = 𝜱 (t, τ) 𝐱 (τ)$ ⁠.

Estimation of the state-transition matrix

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

In the time-variant case, the state-transition matrix $𝜱 (t, t_{0})$ can be estimated from the solutions of the differential equation $\dot{𝐮} (t) = 𝐀 (t) 𝐮 (t)$ with initial conditions $𝐮 (t_{0})$ given by ⁠ $[1, 0, \dots, 0]^{T}$ ⁠, ⁠ $[0, 1, \dots, 0]^{T}$ ⁠, ..., ⁠ $[0, 0, \dots, 1]^{T}$ ⁠. The corresponding solutions provide the $n$ columns of matrix ⁠ $𝜱 (t, t_{0})$ ⁠. Now, from property 4, $𝜱 (t, τ) = 𝜱 (t, t_{0}) 𝜱 (τ, t_{0})^{- 1}$ for all ⁠ $t_{0} \leq τ \leq t$ ⁠. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

References

↑ 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.
1 2 Rugh, Wilson (1996). Linear System Theory. Upper Saddle River, New Jersey: Prentice Hall. ISBN 0-13-441205-2.
↑ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
↑ 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.

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