Synthesis gradient observer

Question

I ask the advice of specialists on control systems. We have the following system.

Where $u(t)$ and $y(t)$ time-varying input and output. The characteristic $y(t)=f(u(t))$ is assumed to be non-inertial.

Only input and output are available for measurement.

Let me make a reservation that the variant of estimating this derivative by the Euler method, i.e.

$g(t)=\frac{y_{n}(t)-y_{n-1}(t)}{u_{n}(t)-u_{n-1}(t)}$

is not suitable, since the system does not allow elements of delays (delay by 1 time step).

Is it possible to construct an observer estimating the gradient $g(t)=\frac{dy}{du}$?

Answer 1

As you pointed out, $y=f\left(u\right)$ is a static map, hence it does not represent in any way the temporal evolution of a dynamical system.

With this in mind, resorting to an observer is fundamentally a wrong approach. An observer, in fact, provides you with an estimate of the temporal evolution of the dynamical system under subject; however, here $f\left(\cdot\right)$ has nothing to do with the time.

Thereby, you ought to fit a static map. There exist a plethora of methods to achieve this goal but you have to switch from designing an observer to a context where you deal with fitting.

You're already equipped with good knowledge to start: you essentially know the bases $b_1 = -v$, $b_2 = \exp(-v)$, and $b_3 = 1 / \left(v+1\right)$, where $v = u^2$.

You could then try to explore whether a linear combination of $b_i$ fits well $f$, or if you need to resort to polynomial fitting or even nonlinear fitting (e.g. by means of multilayer neural networks).

Once done, you can extract the gradient out of the fitted model.

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If you want to fit directly the gradient $g = df/dx$, to avoid problems related to divisions by zero, you could:

1. feed in $x_i = at_i+b$, with $a>0$ and $b<0$ (to explore the whole range);
2. read back $y_i = f\left(x_i\right)$;
3. compute $\dot{\tilde{y}}_i = \left(y_i-y_{i-1}\right)/\Delta t$;
4. compute $g_i = \dot{\tilde{y}}_i / a$;
5. fit $g$ based upon the sampled points $g_i$ exploiting the same techniques discussed above.