Pattern in control theory
A weighting pattern for a linear dynamical system describes the relationship between an input
and output
. Given the time-variant system described by
![{\displaystyle {\dot {x))(t)=A(t)x(t)+B(t)u(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e0dbfe5779f0d9e4824944d90a6d7d6ffd7386)
,
then the output can be written as
,
where
is the weighting pattern for the system. For such a system, the weighting pattern is
such that
is the state transition matrix.
The weighting pattern will determine a system, but if there exists a realization for this weighting pattern then there exist many that do so.[1]
Linear time invariant system
In a LTI system then the weighting pattern is:
- Continuous
![{\displaystyle T(t,\sigma )=Ce^{A(t-\sigma )}B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51069b5a9aca0794d1b64b422c89766ad74ea31b)
where
is the matrix exponential.
- Discrete
.