State-transition equation

The State-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation is given by

$dx(t)/dt=Ax(t)+Bu(t)+Ew(t)$

can be solved by using either the classical method of solving linear differential equations or the Laplace transform method. The Laplace transforme solution is presented in the following equations. Laplace transforme of the above equation yeilds $sX(s)-x(0)=AX(s)+BU(s)+EW(s)$ where x(0) denotes Initial-state vector evaluated at $t=0$ . Solving for $X(s)$

$X(s)=(sI-A)^{-}1x(0)+(sI-A)^{-}1[BU(s)+EW(s)]$

So, the state-transition equation can be obtained by taking inverse Laplace transforme as

$x(0)=''L''^{-}1[(sI-A)^{-}1]x(0)+''L''^{-}1{(sI-A)^{-}1[BU(s)+EW(s)]}=\phi (t)x(0)+\int _{0}^{t}\phi (t-\tau )[Bu(\tau )+Ew(\tau )dt$

The state-transition equation as derived above is useful only when the initial time is defined to be at $t=0$ . 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 $t_{0}$ and the corresponding intial state by $x(t_{0})$ , and assume that the input $u(t)$ and the distrubance $w(t)$ are applied at t≥0. We start with the above equation by setting $t=t_{0}$ and solving for $x(0)$ , we get

$x(0)=\phi (-t_{0})x(t_{0})-\phi (-t_{0})\int _{0}^{t_{0}}\phi (t_{0}-\tau )[Bu(\tau )+Ew(\tau )d\tau$

Once the state-transition equation is determined, the output vector can be expressed as a function of initial state.

External links

• 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