Weighting pattern
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.[1]
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].
References
- ↑ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
Original source: https://en.wikipedia.org/wiki/Weighting pattern.
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