Weighting pattern

Short description: Pattern in control theory

A weighting pattern for a linear dynamical system describes the relationship between an input [math]\displaystyle{ u }[/math] and output [math]\displaystyle{ y }[/math]. Given the time-variant system described by

[math]\displaystyle{ \dot{x}(t) = A(t)x(t) + B(t)u(t) }[/math]

[math]\displaystyle{ y(t) = C(t)x(t) }[/math],

then the output can be written as

[math]\displaystyle{ y(t) = y(t_0) + \int_{t_0}^t T(t,\sigma)u(\sigma) d\sigma }[/math],

where [math]\displaystyle{ T(\cdot,\cdot) }[/math] is the weighting pattern for the system. For such a system, the weighting pattern is [math]\displaystyle{ T(t,\sigma) = C(t)\phi(t,\sigma)B(\sigma) }[/math] such that [math]\displaystyle{ \phi }[/math] 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

[math]\displaystyle{ T(t,\sigma) = C e^{A(t-\sigma)} B }[/math]

where [math]\displaystyle{ e^{A(t-\sigma)} }[/math] is the matrix exponential.

Discrete

[math]\displaystyle{ T(k,l) = C A^{k-l-1} B }[/math].