State-transition equation
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (January 2014) |
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by
with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, and E, can be solved by using either the classical method of solving linear
where x(0) denotes initial-state vector evaluated at . Solving for gives
So, the state-transition equation can be obtained by taking inverse Laplace transform as
The state-transition equation as derived above is useful only when the initial time is defined to be at . 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 and the corresponding initial state by , and assume that the input and the disturbance are applied at t≥0. Starting with the above equation by setting and solving for , we get
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