A linear dynamical system:
$$\dot{x} = Ax + c(t)$$Has the exact solution:
$$x(t) = e^{tA}x(0) + \int_0^t e^{(t-\tau)A}c(\tau) d\tau$$Over a small time interval, $\Delta t$, we can approximate the input, $c(t)$, as a constant. Pulling this term out of the integral we get:
$$x(t + \Delta t) = e^{\Delta t A}x(t) + c(t) \int_0^{\Delta t} e^{(\Delta t - \tau)A} d\tau$$Which evaluates to:
$$x(t + \Delta t) = e^{\Delta t A}x(t) + A^{-1} (e^{\Delta t A} - 1) c(t) $$To simplify notation, we define $\Phi = e^{\Delta t A}$, which exactly describes how the state evolves over a time step $\Delta t$ when the input is zero, $c(t) = 0$.
$$x(t + \Delta t) = \Phi x(t) + A^{-1} (\Phi - 1) c(t)$$
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length([0:0.1:10])
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function test(x)
x*2
end
test(10)
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100.0/0.1 == 1000
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expm(eye(10))
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