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State-Space Averaging Model

A technique for deriving averaged, continuous-time models from switched-mode converters. It approximates the system dynamics by averaging the switching function over the switching period, leading to equations like \frac{d\vec{x}}{dt} = \vec{A}_{avg}\vec{x} + \vec{B}_{avg}\vec{u}.
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The statement of the theorem

Let x⃗(t)\vec{x}(t) be the state vector and u⃗(t)\vec{u}(t) be the input vector. The averaged model is derived by approximating the time-varying dynamics dx⃗dt=f(x⃗,u⃗,s(t))\frac{d\vec{x}}{dt} = f(\vec{x}, \vec{u}, s(t)) using the average operator ⟨⋅⟩\langle \cdot \rangle: dx⃗dt≈A⃗avgx⃗+B⃗avgu⃗\frac{d\vec{x}}{dt} \approx \vec{A}_{avg}\vec{x} + \vec{B}_{avg}\vec{u} where A⃗avg=⟨∂f∂x⃗⟩\vec{A}_{avg} = \langle \frac{\partial f}{\partial \vec{x}} \rangle and B⃗avg=⟨∂f∂u⃗⟩\vec{B}_{avg} = \langle \frac{\partial f}{\partial \vec{u}} \rangle, representing the time average over the switching period TsT_s.