# Solution of a system of nonlinear ODE in the form of a state space with nonzero initial conditions

I have a system of differential equations like this:

$$\begin{cases} \frac{dx}{dt} = hpf_1 \cdot \alpha \cdot sin(\omega t) + \frac{d}{dt}(\alpha \cdot sin(\omega t)) \\ \frac{dhpf_1}{dt} + hpf_1 = \frac{d}{dt}(extr(t)) \end{cases}$$

where: $$extr(t)$$ - Any function that has one extreme (minimum or maximum). For example $$extr(t) = e^{-(x(t))^2}$$

$$x$$ and $$hpf_1$$ - variables of the system of differential equations.

I'm trying to get solution, but transforming the system into a state-space form. How to set nonzero initial conditions?

pars = {\[Alpha] = 0.1, h = 1, \[Omega] = 2 Pi 0.5, \[Beta] = 1}

extr = Exp[-(x[t])^2]

AffineStateSpaceModel[{x'[t] ==
hpf1[t] \[Alpha] Sin[\[Omega] t] + u[t],
hpf1'[t] + h hpf1[t] == D[extr, t]}, {x[t], hpf1[t]}, u[t], x[t], t]

Plot[OutputResponse[%, D[\[Alpha] Sin[\[Omega] t], t], {t, 0, 3}] //
Evaluate, {t, 0, 3}, PlotRange -> Full]


## 1 Answer

See this answer for the topic:

https://math.stackexchange.com/questions/3923222/changing-the-quality-of-the-transient-process-in-a-nonlinear-system-part-ii

A notation in the form of an affine state space can be obtained in the form:

extr = Exp[-(x[t])^2]

AffineStateSpaceModel[{x'[t] ==
hpf1[t] \[Alpha] Sin[\[Omega] t] + u[t],
hpf1'[t] + h hpf1[t] == D[extr, t]}, {x[t], hpf1[t]}, u[t], x[t], t]