Weighting pattern
Pattern in control theory
title: "Weighting pattern" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["control-theory"] description: "Pattern in control theory" topic_path: "general/control-theory" source: "https://en.wikipedia.org/wiki/Weighting_pattern" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Pattern in control theory ::
A weighting pattern for a linear dynamical system describes the relationship between an input u and output y. Given the time-variant system described by : \dot{x}(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_0}^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^{A(t-\sigma)} B where e^{A(t-\sigma)} is the matrix exponential.
; Discrete : T(k,l) = C A^{k-l-1} B.
References
References
- Brockett, Roger W.. (1970). "Finite Dimensional Linear Systems". John Wiley & Sons.
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