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 $e^{𝐀(t-t_0)}$. 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 $𝜱(t,\tau )$, the solution is given by:

$𝐱(t)=𝜱(t,t_0)𝐱(t_0)+\underset{t_0}{\overset{t}{\int }}𝜱(t,\tau )𝐁(\tau )𝐮(\tau )d\tau$

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

$\begin{array}{cc}𝜱(t,\tau )=𝐈& +\underset{\tau }{\overset{t}{\int }}𝐀({\sigma }_1)d{\sigma }_1\\ & +\underset{\tau }{\overset{t}{\int }}𝐀({\sigma }_1)\underset{\tau }{\overset{{\sigma }_1}{\int }}𝐀({\sigma }_2)d{\sigma }_{2}d{\sigma }_1\\ & +\underset{\tau }{\overset{t}{\int }}𝐀({\sigma }_1)\underset{\tau }{\overset{{\sigma }_1}{\int }}𝐀({\sigma }_2)\underset{\tau }{\overset{{\sigma }_2}{\int }}𝐀({\sigma }_3)d{\sigma }_{3}d{\sigma }_{2}d{\sigma }_1\\ & +\cdots \end{array}$

where $𝐈$ 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

$𝜱(t,\tau )=\exp 𝒯\underset{\tau }{\overset{t}{\int }}𝐀(\sigma )d\sigma$

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.

1. It is continuous and has continuous derivatives.
2. It is never singular; in fact ${𝜱}^{-1}(t,\tau )=𝜱(\tau ,t)$ and ${𝜱}^{-1}(t,\tau )𝜱(t,\tau )=𝐈$, where $𝐈$ is the identity matrix.
3. $𝜱(t,t)=𝐈$ for all $t$ .
4. $𝜱(t_2,t_1)𝜱(t_1,t_0)=𝜱(t_2,t_0)$ for all $t_0\le t_1\le t_2$.
5. It satisfies the differential equation $\frac{\partial 𝜱(t,t_0)}{\partial t}=𝐀(t)𝜱(t,t_0)$ with initial conditions $𝜱(t_0,t_0)=𝐈$.
6. The state-transition matrix $𝜱(t,\tau )$, given by $𝜱(t,\tau )\equiv 𝐔(t){𝐔}^{-1}(\tau )$ 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 $𝐔(t_0)=𝐈$.
7. Given the state $𝐱(\tau )$ at any time $\tau$, the state at any other time $t$ is given by the mapping$𝐱(t)=𝜱(t,\tau )𝐱(\tau )$

Estimation of the state-transition matrix

In the time-invariant case, we can define $𝜱$, using the matrix exponential, as $𝜱(t,t_0)=e^{𝐀(t-t_0)}$.

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 $[1,0,\dots ,0{]}^{\mathrm{T}}$, $[0,1,\dots ,0{]}^{\mathrm{T}}$, ..., $[0,0,\dots ,1{]}^{\mathrm{T}}$. The corresponding solutions provide the $n$ columns of matrix $𝜱(t,t_0)$. Now, from property 4, $𝜱(t,\tau )=𝜱(t,t_0)𝜱(\tau ,t_0{)}^{-1}$ for all $t_0\le \tau \le t$. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

See also

• Magnus expansion
• Liouville's formula