# Control of a nonlinear static MIMO System

Engineering Asked by David Zanger on August 5, 2020

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?

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.

Answered by Josh Pilipovsky on August 5, 2020

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.

Answered by useless-machine on August 5, 2020

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