State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector $x$ at an initial time $t_0$ gives $x$ at a later time $t$. The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

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

$\dot{\mathbf{𝐱}}(t)=\mathbf{𝐀}(t)\mathbf{𝐱}(t)+\mathbf{𝐁}(t)\mathbf{𝐮}(t),\mathbf{𝐱}(t_0)={\mathbf{𝐱}}_0$,

where $\mathbf{𝐱}(t)$ are the states of the system, $\mathbf{𝐮}(t)$ is the input signal, $\mathbf{𝐀}(t)$ and $\mathbf{𝐁}(t)$ are matrix functions, and ${\mathbf{𝐱}}_0$ is the initial condition at $t_0$. Using the state-transition matrix $\mathrm{\Phi }(t,\tau )$, the solution is given by:

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

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

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

where $\mathbf{𝐈}$ 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

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

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 $\mathrm{\Phi }$ 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 ${\mathrm{\Phi }}^{-1}(t,\tau )=\mathrm{\Phi }(\tau ,t)$ and ${\mathrm{\Phi }}^{-1}(t,\tau )\mathrm{\Phi }(t,\tau )=\mathbf{𝐈}$, where $\mathbf{𝐈}$ is the identity matrix.

3. $\mathrm{\Phi }(t,t)=\mathbf{𝐈}$ for all $t$ .

4. $\mathrm{\Phi }(t_2,t_1)\mathrm{\Phi }(t_1,t_0)=\mathrm{\Phi }(t_2,t_0)$ for all $t_0\le t_1\le t_2$.

5. It satisfies the differential equation $\frac{\partial \mathrm{\Phi }(t,t_0)}{\partial t}=\mathbf{𝐀}(t)\mathrm{\Phi }(t,t_0)$ with initial conditions $\mathrm{\Phi }(t_0,t_0)=\mathbf{𝐈}$.

6. The state-transition matrix $\mathrm{\Phi }(t,\tau )$, given by

$\mathrm{\Phi }(t,\tau )\equiv \mathbf{𝐔}(t){\mathbf{𝐔}}^{-1}(\tau )$

where the $n\times n$ matrix $\mathbf{𝐔}(t)$ is the fundamental solution matrix that satisfies

$\dot{\mathbf{𝐔}}(t)=\mathbf{𝐀}(t)\mathbf{𝐔}(t)$ with initial condition $\mathbf{𝐔}(t_0)=\mathbf{𝐈}$.

7. Given the state $\mathbf{𝐱}(\tau )$ at any time $\tau$, the state at any other time $t$ is given by the mapping

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

Estimation of the state-transition matrix

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

In the time-variant case, the state-transition matrix $\mathrm{\Phi }(t,t_0)$ can be estimated from the solutions of the differential equation $\dot{\mathbf{𝐮}}(t)=\mathbf{𝐀}(t)\mathbf{𝐮}(t)$ with initial conditions $\mathbf{𝐮}(t_0)$ 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 $\mathrm{\Phi }(t,t_0)$. Now, from property 4, $\mathrm{\Phi }(t,\tau )=\mathrm{\Phi }(t,t_0)\mathrm{\Phi }(\tau ,t_0{)}^{-1}$ 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