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Weighting Pattern

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Given an LTVC system {x˙(t)=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ y(t)=C(t)x(t) \end{cases}A weighting pattern, T(t,τ)T(t,\tau) defined as T(t,τ)=C(t)ΦA(t,τ)B(τ)T(t,\tau)=C(t)\Phi_{A}(t,\tau)B(\tau)describes the relationship between input u(t)u(t) and output y(t)y(t) in the following form: y(t)=t0t1T(t,τ)u(τ)dτy(t)=\int\limits _{t_{0}}^{t_{1}}T(t,\tau)u(\tau) \, d\tau

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