State-transition matrix: Difference between revisions

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{{Short description|Describes state evolution of a linear system}}

{{Technical|date=December 2018}}

In [[control theory]] and [[dynamical systems theory]], the '''state-transition matrix''' is a [[matrix whosefunction]] productthat withdescribes how the [[State space representation|state]] vectorof <math>x</math>a at[[linear ansystem]] initialchanges over time. <math>t_0</math>Essentially, givesif <math>x</math>the system's state is known at aan laterinitial time <math>tt_0</math>., Thethe state-transition matrix canallows befor usedthe tocalculation obtainof the generalstate solutionat ofany linearfuture dynamicaltime systems<math>t</math>.

The matrix is used to find the general solution to the homogeneous [[linear differential equation]] <math>\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t)</math> and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.

For [[Linear time-invariant system|linear time-invariant (LTI) systems]], where the matrix <math>\mathbf{A}</math> is constant, the state-transition matrix is the [[matrix exponential]] <math>e^{\mathbf{A}(t-t_0)}</math>. In the more complex [[Time-variant system|time-variant]] case, where <math>\mathbf{A}(t)</math> 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

: <math>\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) , \;\mathbf{x}(t_0) = \mathbf{x}_0 </math>,

where <math>\mathbf{x}(t)</math> are the states of the system, <math>\mathbf{u}(t)</math> is the input signal, <math>\mathbf{A}(t)</math> and <math>\mathbf{B}(t)</math> are [[matrix function]]s, and <math>\mathbf{x}_0</math> is the initial condition at <math>t_0</math>. Using the state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, the solution is given by:<ref name=baaschl>{{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}}</ref><ref name=rugh>{{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}}</ref>

: <math>\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</math>

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'''

:<math>\begin{align}

:<math> \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 + ...</math>

: <math>\mathbf{\Phi}(t, \tau)\equiv = \mathbf{UI}( &+ \int_\tau^t)\mathbf{U}^{-1A}(\tausigma_1)</math>\,d\sigma_1 \\

where <math>\mathbf{I}</math> is the [[identity matrix]]. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh />

&+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 \\

:<math> \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 + ...</math>\\

&+ \cdots

\end{align}</math>

where <math>\mathbf{I}</math> is the [[identity matrix]]. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh /> The series has a formal sum that can be written as

:<math>\mathbf{x\Phi}(t,\tau) = \mathbf{exp \Phimathcal{T}(t, \int_\tau)^t\mathbf{xA}(\tausigma)\,d\sigma</math>

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

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

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

4.# <math>\mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0)</math> for all <math>t_0 \leq t_1 \leq t_2</math>.

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

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

# The state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, given by <math>\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)</math> where the <math>n \times n</math> matrix <math>\mathbf{U}(t)</math> is the [[Fundamental matrix (linear differential equation)|fundamental solution matrix]] that satisfies <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math> with initial condition <math>\mathbf{U}(t_0) = \mathbf I</math>.

7.# Given the state <math>\mathbf{x}(\tau)</math> at any time <math>\tau</math>, the state at any other time <math>t</math> is given by the mapping<math>\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)</math>

4. <math>\mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0)</math> for all <math>t_0 \leq t_1 \leq t_2</math>.

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

6. The state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, given by

: <math>\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)</math>

where the <math>n \times n</math> matrix <math>\mathbf{U}(t)</math> is the [[Fundamental matrix (linear differential equation)|fundamental solution matrix]] that satisfies

: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math> with initial condition <math>\mathbf{U}(t_0) = I</math>.

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

:<math>\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)</math>

==Estimation of the state-transition matrix==

In the [[time-invariant]] case, we can define <math> \mathbf{\Phi}</math>;, using the [[matrix exponential]], as <math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>. <ref>{{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 }}</ref>

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

<math>\mathbf{\Phi}(t, \tau) = \mathbf{\Phi}(t, t_0)\mathbf{\Phi}(\tau, t_0)^{-1}</math> for all <math>t_0 \leq \tau \leq t</math>. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

== See also ==

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| volume = 275

| pages = 155–159

| doi = 10.1134/S0081543811080098

}}

| s2cid = 119133539

}}

* {{cite book

| author = Brogan, W.L.

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[[Category:Classical control theory]]

[[Category:Dynamical systems]]

[[Category:Matrix theory]]