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:{{cite journal|last1=Baake|first1=Michael|last2=Schlaegel|first2=Ulrike|title=The Peano Baker Series|journal=Proceedings of the Steklov Institute of Mathematics|year=2011|volume=275|pages=155–159|doi=10.1134/S0081543811080098|s2cid=119133539}}{{cite book|last1=Rugh|first1=Wilson|title=Linear System Theory|date=1996|publisher=Prentice Hall|location=Upper Saddle River, NJ | isbn = 0-13-441205-2}}

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

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 a product integral, referred to as the Peano–Baker series

: $\begin{align}
\mathbf{\Phi}(t,\tau) = \mathbf{I} &+ \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 \\
&+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 \\
&+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^{\sigma_2}\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 \\
&+ \cdots
\end{align}$

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

: $\mathbf{\Phi}(t,\tau) = \exp \mathcal{T}\int_\tau^t\mathbf{A}(\sigma)\,d\sigma$

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

3. $\mathbf{\Phi}(t, t) = \mathbf I$ for all $t$ .{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}

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

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 $\mathbf{\Phi}(t_0, t_0) = \mathbf I$ .

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

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

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

Estimation of the state-transition matrix

In the time-invariant case, we can define $\mathbf{\Phi}$ , using the matrix exponential, as $\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}$ . {{cite journal |last1=Reyneke |first1=Pieter V. |title=Polynomial Filtering: To any degree on irregularly sampled data |journal=Automatika |date=2012 |volume=53 |issue=4 |pages=382–397|doi=10.7305/automatika.53-4.248 |s2cid=40282943 |url=http://hrcak.srce.hr/file/138435 |doi-access=free |hdl=2263/21017 |hdl-access=free }}

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 $[1,\ 0,\ \ldots,\ 0]^\mathrm{T}$ , $[0,\ 1,\ \ldots,\ 0]^\mathrm{T}$ , ..., $[0,\ 0,\ \ldots,\ 1]^\mathrm{T}$ . 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 $t_0 \leq \tau \leq t$ . The state-transition matrix must be determined before analysis on the time-varying solution can continue.

State-transition matrix

Linear systems solutions

Peano–Baker series

Other properties

Estimation of the state-transition matrix

See also

References

Further reading