Multi-input-multi-output systems

Multi-input-multi-output systems (MEMS) are systems with several inputs (u ₁ , u ₂ , ..., u _m ) and several outputs (y ₁ , y ₂ , ..., y _r ). These systems cannot be reduced to the paralleling of single-input, single-output systems, but are characterized by coupling – or interaction – phenomena insofar as the application of a signal to one input generally results in a variation of several or all outputs. We will see 2.2.1 that, as a result, a multi-input-multi-output system is not "like" a monovariable system with more complicated calculations, but is something completely different.

This article is divided into two parts.

• The first part deals with the representation of multidimensional systems, and more precisely how to move from an initial, external type of representation – i.e. input-output, either in the form of coupled differential equations, or in the form of a transfer matrix – to an internal type of representation, in the form of a state model.

  Obtaining a state model is essential, both for simulation and, even more so, for control. Unlike in the case of single-input, single-output systems, the transition from an external representation to a state representation is not necessarily straightforward, and we shall see that these difficulties are linked to the notion of "order" in multi-input, multi-output systems. Attention is drawn to these problems, as experience shows that it is at this level, prior to ordering, that the most serious mistakes are made.

• The second part deals with the control of multidimensional systems. Here again, there is no question of extending the techniques used in one-dimensional systems, and the control of a multi-input-multi-output system must be considered in a global manner. After a brief review, more historical than practical, of the first methods implemented in the frequency domain, the emphasis is placed on current techniques using state...