Limitations for MIMO systems
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.
Example 1 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.