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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 given by \frac = \mathbf(t) + \mathbf(t) + \mathbf(t), with state vector , control vector , vector of additive disturbances, and fixed matrices can be solved by using either the classical method of solving linear differential equations or the
Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
method. The Laplace transform solution is presented in the following equations. The Laplace transform of the above equation yields s\mathbf(s) - \mathbf(0) = \mathbf(s) + \mathbf(s) + \mathbf(s) where denotes initial-state vector evaluated at . Solving for gives \mathbf(s) = (s\mathbf - \mathbf)^ \mathbf(0) + (s\mathbf - \mathbf)^ mathbf(s) + \mathbf(s) So, the state-transition equation can be obtained by taking
inverse Laplace transform In mathematics, the inverse Laplace transform of a function F(s) is a real function f(t) that is piecewise- continuous, exponentially-restricted (that is, , f(t), \leq Me^ \forall t \geq 0 for some constants M > 0 and \alpha \in \mathbb) and h ...
as \begin x(t) &= \mathcal^ \Bigl\ \mathbf(0) + \mathcal^ \Bigl\ \\ &= \mathbf(t) \mathbf(0) + \int_^ \mathbf(t - \tau) mathbf(\tau) + \mathbf(\tau)t \end where is the
state transition matrix In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical system ...
. The state-transition equation as derived above is useful only when the initial time is defined to be at . In the study of
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
s, 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 . Starting with the above equation by setting , and solving for , we get \mathbf(0) = \mathbf(-t_0) \mathbf(t_0) - \mathbf(-t_0) \int_^\mathbf(t_0 - \tau) mathbf(\tau) + \mathbf(\tau)\tau. Once the state-transition equation is determined, the output vector can be expressed as a function of the initial state.


See also

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Control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
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Control engineering Control engineering, also known as control systems engineering and, in some European countries, automation engineering, is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with d ...
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Automatic control Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machine ...
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Feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handle ...
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Process control Industrial process control (IPC) or simply process control is a system used in modern manufacturing which uses the principles of control theory and physical industrial control systems to monitor, control and optimize continuous Industrial processe ...
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PID loop PID or Pid may refer to: Medicine * Pelvic inflammatory disease or pelvic inflammatory disorder, an infection of the upper part of the female reproductive system * Primary immune deficiency, disorders in which part of the body's immune system is ...


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 {{DEFAULTSORT:Automatic Control Control theory