# Control of a nonlinear static MIMO System

I am currently writing my master thesis and trying to design a controller for my system. However, the system is somewhat unconventional.

It has a large number of inputs and outputs, is static, non-linear and time-invariant. The goal is to control the disturbance. Because it is static, conventional controllers for MIMO systems (NMPC etc.) are only of limited use.

At the moment I am looking for a similar system in another area. Does anyone have an idea in which area such a system exist?

• What do you mean by static system? May 25 '20 at 17:19
• @JoshPilipovsky That I assume that the system is so fast that I only have a nonlinear input output mapping. The problems are more the disturbances. I somehow have to control them. May 25 '20 at 18:47
• So your I/O is something like $Y(s) = \phi(s)$, where $\phi$ is the nonlinearity right? Are the disturbances delta functions then, since they are so fast? May 25 '20 at 20:46
• @JoshPilipovsky Yeah, that is how my function looks like. Actually, it looks like this $F_{x,y}^{(h)} = a^{(h)} e^{- \frac{(x-x_{a})^{2} + (y-y_{a})^{2}} {b^{(h)}} }$ Where $x_{a}$ and $y_{a}$ is my input for one h (I have ~1000 of h). $(x,y)$ is my measurement point (where I have ~1000 as well). $a$ and $b$ are just parameters. I could reduce the problem though to a size of 100x100. One disturbance is rather static. The other is dynamic and I am not even sure how to model it. May 26 '20 at 8:41
• Maybe it would be good if you included some extra information about the system. For example, a mathematical model (I suppose you have one) and if you have any info regarding the nonlinearities. May 28 '20 at 19:40

Based on your comment, it seems like you are trying to control a dynamical system to track some reference trajectory subject to Gaussian disturbances. Suppose you have some kind of model for the dynamics, then the discrete system can be written as

\begin{align} x_{k+1} &= f_k(x_k) + a_k w_k\\ y_k &= x_k \end{align}

where $$w_k$$ is a Gaussian vector with mean $$\mu_k = [x_a^{(k)}, y_a^{(k)}]^\intercal$$ and covariance $$\Sigma_{x_k} = \frac{b^{(k)}}{2} I_2$$. This is then a stochastic Control problem and there are various methods to create a stabilizing controller just check out the literature.

• Ok, thanks. I will have a look. Though, I have one disturbances which is not stochastic and very strong. It causes that the outputs of some variables h fade partly or completely. Is stochastic control still a solution then? May 27 '20 at 9:42

I think there are many systems similar to your system.

A very common way to write a system with a nonlinearty is by writing it as Luré Type System, where the nonlinearity is in the feedback loop \begin{align} \dot{x}(t) &= Ax(t) +Bw(t)\\ z(t) &= Cx + Dw(t)\\ w(t) &= \Delta(t,z) \end{align} where $$\Delta$$ is the nonlinearty.

Depending on the type of nonlinearty, you can apply all kinds of controller synthesis such as $$H_\infty$$, $$H_2$$ and QP.

For example, you can create a sector condition for the nonlinearty and use the Circle Criterion to check if the system is absolutely stable. A common nonlinearty to use in this framework is a saturation.