State-transition equation

The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by $$\frac{d𝐱(t)}{dt}=𝐀𝐱(t)+𝐁𝐮(t)+𝐄𝐰(t),$$ with state vector Template:Math, control vector Template:Math, vector Template:Math of additive disturbances, and fixed matrices Template: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. The Laplace transform of the above equation yields $$s𝐗(s)-𝐱(0)=𝐀𝐗(s)+𝐁𝐔(s)+𝐄𝐖(s)$$ where Template:Math denotes initial-state vector evaluated at Template:Math. Solving for Template:Math gives $$𝐗(s)=(s𝐈-𝐀{)}^{-1}𝐱(0)+(s𝐈-𝐀{)}^{-1}[𝐁𝐔(s)+𝐄𝐖(s)].$$ So, the state-transition equation can be obtained by taking inverse Laplace transform as $$\begin{array}{cc}x(t)& ={ℒ}^{-1}\{(s𝐈-𝐀{)}^{-1}\}𝐱(0)+{ℒ}^{-1}\{(s𝐈-𝐀{)}^{-1}[𝐁𝐔(s)+𝐄𝐖(s)]\}\\ & =\mathrm{\Phi }(t)𝐱(0)+\underset{0}{\overset{t}{\int }}\mathrm{\Phi }(t-\tau )[𝐁𝐮(\tau )+𝐄𝐰(\tau )]dt\end{array}$$ where Template:Math is the state transition matrix.

The state-transition equation as derived above is useful only when the initial time is defined to be at Template:Math. In the study of control systems, specially discrete-data control systems, 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 Template:Math and the corresponding initial state by Template:Math, and assume that the input Template:Math and the disturbance Template:Math are applied at Template:Math. Starting with the above equation by setting Template:Math, and solving for Template:Math, we get $$𝐱(0)=\mathrm{\Phi }(-t_0)𝐱(t_0)-\mathrm{\Phi }(-t_0)\underset{0}{\overset{t_0}{\int }}\mathrm{\Phi }(t_0-\tau )[𝐁𝐮(\tau )+𝐄𝐰(\tau )]d\tau .$$ Once the state-transition equation is determined, the output vector can be expressed as a function of the initial state.

See also

• Control theory
• Control engineering
• Automatic control
• Feedback
• Process control
• PID loop

External links

• Control System Toolbox for design and analysis of control systems.
• http://web.mit.edu/2.14/www/Handouts/StateSpaceResponse.pdf
• Wikibooks:Control Systems/State-Space Equations
• http://planning.cs.uiuc.edu/node411.html