# 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. By static, I mean that the sampling time is so high that all dynamics are faded during that time.

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?

Formally, the system is

$$F_{x,y}^{(h)} = a^{(h)} e^{- \frac{(x-x_{a})^{2} + (y-y_{a})^{2}} {b^{(h)}} }$$ The state space representation of the system is

$$\dot F_{x,y}^{(h)} = 2\frac{a^{(h)}}{b^{(h)}} \left[(x-x_{a}) \dot x_{a} + (y-y_{a}) \dot y_{a}\right] e^{ \frac{(x-x_{a})^{2} + (y-y_{a})^{2}} {b^{(h)}} }$$

with

\begin{align} \dot x_{a} &= u_{1}^{(h)}\\ \dot y_{a} &= u_{2}^{(h)}\\ \text{output}_{1} &= \sum_{h=1}^{h_\text{end}} F^{(h)}\text. \end{align}

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). I would have one ouput for every measurement point. $$a$$ 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.

• By static, I mean that the sampling time is so high that all dynamics are faded during that time. Does that imply that phase information is erased through that channel? Or does it imply the channel changes so much, and your receiver averages across that, that the phase is averaged, too? – Marcus Müller May 26 at 13:40
• (Channel=System, by the way) Could also be pretty helpful if you told us the nature of the nonlinearity, i.e. tell us whether how you're mathematically modelling it! (I'm really a bit confused by how you'd get any information across this channel – a mathematically stricter description of it would probably help me understand what kind of system you're looking at. – Marcus Müller May 26 at 13:44
• The formulation would look like this: $F_{x,y}^{(h)} = a^{(h)} e^{- \frac{(x-x_{a})^{2} + (y-y_{a})^{2}} {b^{(h)}} }$ In a state space it would look like this: $\dot F_{x,y}^{(h)} = 2a^{(h)}/b^{(h)} \cdot [(x-x_{a}) \dot x_{a} + (y-y_{a}) \dot y_{a}] e^{ \frac{(x-x_{a})^{2} + (y-y_{a})^{2}} {b^{(h)}} }$ $\dot x_{a} = u_{1}^{(h)}$ $\dot y_{a} = u_{2}^{(h)}$ $output_{1} = \sum_{h=1}^{h_{end}} F^{(h)}$ – David Zanger May 26 at 13:54
• 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). I would have one ouput for every measurement point. $a$ 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. – David Zanger May 26 at 14:01
• ah, that gets a tiny bit small on my screen; since it's important to the question, I'll go ahead and add it to it – Marcus Müller May 26 at 14:17