View source for Bose-Einstein condensation of excitons

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{{Orphan|date=January 2014}}
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The '''State-transition equation''' is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation is given by
:<math>\frac {dx(t)}{dt}=Ax(t)+Bu(t)+Ew(t) </math>
can be solved by using either the classical method of solving linear [[differential equations]] or the [[Laplace transform]] method. The Laplace transform solution is presented in the following equations.
Laplace transform of the above equation yields 
:<math>sX(s)-x(0)=AX(s)+BU(s)+EW(s) </math>
where x(0) denotes '''Initial-state vector''' evaluated at <math>t=0</math> . Solving for <math>X(s)</math>
:<math>X(s)=(sI-A)^{-1} x(0) + (sI-A)^{-1}[BU(s)+EW(s)] </math>
So, the state-transition equation can be obtained by taking inverse Laplace transform as 
:<math>x(t)=L^{-1}[(sI-A)^{-1}] x(0) + L^{-1} {(sI-A)^{-1} [BU(s) + EW(s)]}= \phi(t)x(0)+\int_{0}^{t} \phi(t-\tau)[Bu(\tau)+Ew(\tau)]dt </math>

The state-transition equation as derived above is useful only when the initial time is defined to be at <math>t=0</math> . In the study of [[control system]], specially discrete-data control system, it is often desirable to break up a state-transition process into a sequence of transitions, so a more flexible initial time must be chosen. Let the initial time be represented by <math>t_0</math> and the corresponding initial state by <math>x(t_0)</math>, and assume that the input <math> u(t)</math> and the disturbance <math> w(t)</math> are applied at t≥0. 
We start with the above equation by setting <math> t=t_0</math> and solving for <math>x(0)</math>, we get 
:<math> x(0)= \phi (-t_0)x(t_0)-\phi(-t_0)\int_{0}^{t_0}\phi(t_0 - \tau)[Bu(\tau)+Ew(\tau)]d\tau</math>
Once the state-transition equation is determined, the output vector can be expressed as a function of initial state.

==External links==
* [http://www.mathworks.com/products/control/ Control System Toolbox] for design and analysis of control systems.

==Further reading==
* http://web.mit.edu/2.14/www/Handouts/StateSpaceResponse.pdf
* https://en.wikibooks.org/wiki/Control_Systems/State-Space_Equations
* http://planning.cs.uiuc.edu/node411.html

==See also==
* [[Control theory]]
* [[Control engineering]]
* [[Automatic control]]
*[[Feedback]]
*[[Process control]]
*[[PID loop]]

{{DEFAULTSORT:Automatic Control}}
[[Category:Control theory]]