Weighting pattern
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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.
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 .
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