limitations_MIMO.qmd

---
title: "Limitations for MIMO systems"
format:
    html:
        html-math-method: katex
        code-fold: true
execute:
    enabled: false
jupyter: julia-1.10
---

Multiple-input-multiple-output (MIMO) systems are subject to limitations of the same origin as single-input-single-output (SISO) systems: unstable poles, "unstable" zeros, delays, disturbances, saturation, etc. However, the vector character of inputs and outputs introduces both opportunities to mitigate those limitations, and... new limitations.

## Directions in MIMO systems

With vector inputs and vector outputs, the input-output model of an LTI MIMO system is a *matrix* (of transfer functions). As such, it can be characterized not only by various scalar quantities (like poles, zeros, etc.), but also by the associated *directions* in the input and output spaces.

::: {#exm-mimo-zeros-directions}
Consider the transfer function matrix (or matrix of transfer functions)
$$
G(s) = \frac{1}{(0.2s+1)(s+1)}\begin{bmatrix}1 & 1\\ 1+2s& 2\end{bmatrix}.
$$

Recall that a complex number $z\in\mathbb C$ is a zero of $G$ if the rank of $G(z)$ is less than the rank  of $G(s)$ for most $s$. While reliable numerical algorithms for computing zeros of MIMO systems work with state-space realizations, in this simple case we can easily verify that there is only one zero $z=1/2$.



Zeros in the RHP only exhibit themselves in some directions.

:::

## Conditioning of MIMO systems

$$\boxed{
\gamma (G) = \frac{\bar{\sigma}(G)}{\underline{\sigma}(G)}
}
$$

- Ill-conditioned for $\gamma>10$
- But depends on scaling!

Therefore minimized conditioning number
$$\boxed{
\gamma^\star(G) = \min_{D_1, D_2}\gamma(D_1GD_2)
}
$$
but difficult do compute (=upper bound on $\mu$)

RGA can be used to give a reasonable estimate.

## Relative gain array (RGA)

Relative Gain Array (RGA) as an indicator of difficulties with control
$$\boxed{\Lambda(G) = G \circ (G^{-1})^T}$$

- independent of scaling,
- sum of elements in rows and columns is 1,
- sum of absolute values of elements of RGA is very close to the minimized sensitivity number $\gamma^\star$, hence a system with large RGA entries is always ill-conditioned (but system with large $\gamma$ can have small RGA),
- RGA for a triangular system is an identity matrix,
- relative uncertainty of an element of a transfer function matrix equal to (negative) inverse of the corresponding RGA entry makes the system singular.

## Functional controllability

## Interpolation conditions for MIMO systems

## Bandwidth limitations due to unstable poles and zeros

## Limits given by presence of disturbance and/or reference

## Disturbance rejection by a plant with RHP zero

## Limits given by the input constraints (saturation)

## Limits given by uncertainty in the model: in open loop

### In open loop

### In closed loop