Asymptotic Output Tracking: Code Issues

I ask for help from specialists in differential equations, dynamical systems, optimal control and

general control theory;

I have the following system of differential equations:

begin{cases} frac{dx(t)}{dt}=G(t) \ frac{dz(t)}{dt}+z(t)=frac{df}{dt} \ frac{dG(t)}{dt}+G(t)=z(t) cdot alpha sin(omega t) \ frac{dH(t)}{dt}+H(t)=z(t) cdot (frac{16}{alpha^2}(sin(omega t)-frac{1}{2})) \ frac{dX(t)}{dt}+X(t)=frac{dx(t)}{dt} end{cases}

where, $x,z,G,H,X$ – variables; $f=-(x(t)+alpha sin(omega t)-x_e)^2$; $alpha, omega$ – parameters.

As an output $y$, I assign:

$y=tanh(k cdot H(t))$

As an reference signal $r_1$, I assign:

$r_1=-1$

As an constant time $p_1$, I assign:

$p_1=-1$

Well, I tried to program this in the Mathematica program and ran into a difficulty that I can’t get over yet. **Question:** *in which of the equations should the control signal $u(t)$ be placed?*

I chose the first equation, then the original system of equations will look like this:

begin{cases} frac{dx(t)}{dt}=G(t)+u(t) \ frac{dz(t)}{dt}+z(t)=frac{df}{dt} \ frac{dG(t)}{dt}+G(t)=z(t) cdot alpha sin(omega t) \ frac{dH(t)}{dt}+H(t)=z(t) cdot (frac{16}{alpha^2}(sin(omega t)-frac{1}{2})) \ frac{dX(t)}{dt}+X(t)=frac{dx(t)}{dt} end{cases}

(***)

```
Clear("Derivative")
ClearAll("Global`*")
Needs("Parallel`Developer`")
S(t) = (Alpha) Sin((Omega) t)
M(t) = 16/(Alpha)^2 (Sin((Omega) t) - 1/2)
f = -(x(t) + S(t) - xe)^2
Parallelize(
asys = AffineStateSpaceModel({x'(t) == G(t) + u(t),
z'(t) + z(t) == D(f, t), G'(t) + G(t) == z(t) S(t),
H'(t) + H(t) == z(t) M(t),
1/k X'(t) + X(t) == D(x(t), t)}, {{x(t), xs}, {z(t), 0.1}, {G(t),
0}, {H(t), 0}, {X(t), 0}}, {u(t)}, {Tanh(k H(t))}, t) //
Simplify)
pars1 = {Subscript(r, 1) -> -1, Subscript(p, 1) -> -1}
Parallelize(
fb = AsymptoticOutputTracker(asys, {-1}, {-1, -1}) // Simplify)
pars = {xs = -1, xe = 1, (Alpha) = 0.3, (Omega) = 2 Pi*1/2/Pi,
k = 100, (Mu) = 1}
Parallelize(
csys = SystemsModelStateFeedbackConnect(asys, fb) /. pars1 //
Simplify // Chop)
plots = {OutputResponse({csys}, {0, 0}, {t, 0, 1})}
```

At the end, I get an error.

```
At t == 0.005418556209176463`, step size is effectively zero;
singularity or stiff system suspected
```

It seems to me that this is due to the fact that either in the system there is a ksk somewhere, or I have put the control input signal in the wrong equation. I need the support of a theorist who can help me choose the right sequence of actions to solve the problem.

I would be glad to any advice and help.