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 $\mathbf{\Phi }(t,\tau )$, the solution is given by:

$\mathbf{x}(t)=\mathbf{\Phi }(t,t_0)\mathbf{x}(t_0)+{\int }_{t_0}^t\mathbf{\Phi }(t,\tau )\mathbf{B}(\tau )\mathbf{u}(\tau )d\tau$

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 the Peano–Baker series

$\mathbf{\Phi }(t,\tau )=\mathbf{I}+{\int }_{\tau }^t\mathbf{A}({\sigma }_1)d{\sigma }_1+{\int }_{\tau }^t\mathbf{A}({\sigma }_1){\int }_{\tau }^{{\sigma }_1}\mathbf{A}({\sigma }_2)d{\sigma }_{2}d{\sigma }_1+{\int }_{\tau }^t\mathbf{A}({\sigma }_1){\int }_{\tau }^{{\sigma }_1}\mathbf{A}({\sigma }_2){\int }_{\tau }^{{\sigma }_2}\mathbf{A}({\sigma }_3)d{\sigma }_{3}d{\sigma }_{2}d{\sigma }_1+...$

where $\mathbf{I}$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.

Other properties

The state transition matrix $\mathbf{\Phi }$ satisfies the following relationships:

1. It is continuous and has continuous derivatives.

2, It is never singular; in fact ${\mathbf{\Phi }}^{-1}(t,\tau )=\mathbf{\Phi }(\tau ,t)$ and ${\mathbf{\Phi }}^{-1}(t,\tau )\mathbf{\Phi }(t,\tau )=I$, where $I$ is the identity matrix.

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

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

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

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

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

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 $\mathbf{U}(t_0)=I$.

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

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

Estimation of the state-transition matrix

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

In the time-variant case, the state-transition matrix $\mathbf{\Phi }(t,t_0)$ 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 $[1,0,\dots ,0{]}^T$, $[0,1,\dots ,0{]}^T$, ..., $[0,0,\dots ,1{]}^T$. The corresponding solutions provide the $n$ columns of matrix $\mathbf{\Phi }(t,t_0)$. Now, from property 4, $\mathbf{\Phi }(t,\tau )=\mathbf{\Phi }(t,t_0)\mathbf{\Phi }(\tau ,t_0{)}^{-1}$ 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

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. 
• Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7. https://archive.org/details/moderncontrolthe00brog.