A weighting pattern for a linear dynamical system describes the relationship between an input

u

and output

y

. Given the

time-variant system A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application. In other words, a time delay or time advance of input not only shifts the output signal in time but also change ...

described by :

\dot(t) = A(t)x(t) + B(t)u(t)

y(t) = C(t)x(t)

, then the output can be written as :

y(t) = y(t_0) + \int_^t T(t,\sigma)u(\sigma) d\sigma

, where

T(\cdot,\cdot)

is the weighting pattern for the system. For such a system, the weighting pattern is

T(t,\sigma) = C(t)\phi(t,\sigma)B(\sigma)

such that

\phi

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.

Linear time invariant system

In a LTI system then the weighting pattern is: ; Continuous :

T(t,\sigma) = C e^ B

where

e^

is the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...

. ; Discrete :

T(k,l) = C A^ B

References

{{Reflist Control theory