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
: $\dot(t) = \mathbf(t) \mathbf(t) + \mathbf(t) \mathbf(t) , \;\mathbf(t_0) = \mathbf_0$,
where $\mathbf(t)$ are the states of the system, $\mathbf(t)$ is the input signal, $\mathbf(t)$ and $\mathbf(t)$ are matrix functions, and $\mathbf_0$ is the initial condition at $t_0$. Using the state-transition matrix $\mathbf(t, \tau)$, the solution is given by:
: $\mathbf(t)= \mathbf (t, t_0)\mathbf(t_0)+\int_^t \mathbf(t, \tau)\mathbf(\tau)\mathbf(\tau)d\tau$

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
:$\mathbf(t,\tau) = \mathbf + \int_\tau^t\mathbf(\sigma_1)\,d\sigma_1 + \int_\tau^t\mathbf(\sigma_1)\int_\tau^\mathbf(\sigma_2)\,d\sigma_2\,d\sigma_1 + \int_\tau^t\mathbf(\sigma_1)\int_\tau^\mathbf(\sigma_2)\int_\tau^\mathbf(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 + ...$
where $\mathbf$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.

Other properties

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

1. It is continuous and has continuous derivatives.

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

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

4. $\mathbf(t_2, t_1)\mathbf(t_1, t_0) = \mathbf(t_2, t_0)$ for all $t_0 \leq t_1 \leq t_2$.

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

6. The state-transition matrix $\mathbf(t, \tau)$, given by
: $\mathbf(t, \tau)\equiv\mathbf(t)\mathbf^(\tau)$
where the $n \times n$ matrix $\mathbf(t)$ is the fundamental solution matrix that satisfies
: $\dot(t)=\mathbf(t)\mathbf(t)$ with initial condition $\mathbf(t_0) = I$.

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

Estimation of the state-transition matrix

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

In the time-variant case, the state-transition matrix $\mathbf(t, t_0)$ can be estimated from the solutions of the differential equation $\dot(t)=\mathbf(t)\mathbf(t)$ with initial conditions $\mathbf(t_0)$ given by ,\ 0,\ \ldots,\ 0T, ,\ 1,\ \ldots,\ 0T, ..., ,\ 0,\ \ldots,\ 1T. The corresponding solutions provide the $n$ columns of matrix $\mathbf(t, t_0)$. Now, from property 4,
$\mathbf(t, \tau) = \mathbf(t, t_0)\mathbf(\tau, t_0)^$ for all $t_0 \leq \tau \leq t$. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

See also

* Magnus expansion
* Liouville's formula

References

Further reading

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{{Matrix classes

Classical control theory