Multivariable (MIMO) Control

Multivariable systems have coupled inputs and outputs, so each actuator affects several variables, demanding coordinated design.

When Everything Affects Everything

A multivariable or MIMO (multiple-input multiple-output) system has several inputs and several outputs that interact: moving one actuator changes more than one measured variable. Treating each input-output pair as an independent single loop then fails, because the loops fight one another through the coupling. MIMO control coordinates all the loops together.

The coupling problem

• Each output depends on several inputs, so tuning one loop disturbs the others.
• Strong cross-coupling can make independently stable single loops unstable when closed together.
• The interaction is captured by a transfer-function matrix, whose off-diagonal entries measure the coupling.
• The relative gain array is a common tool for deciding which input to pair with which output when decentralized control is attempted.

Two design philosophies

One approach is decoupling: design a compensator that cancels the cross-coupling so the system behaves like independent single loops, which can then be tuned separately. This works when the model is accurate but is sensitive to error. The other approach is fully centralized design, treating the system as one coupled entity and designing a single multivariable controller, which is the natural setting for state-space, LQR, LQG, and H-infinity methods.

Why state-space fits

State-space methods handle MIMO systems as naturally as single loops: the matrices A, B, C, and D simply carry more columns and rows, and controllability, observability, pole placement, and optimal control all apply unchanged. This is a major reason modern control shifted from transfer functions to state-space as systems grew more coupled.

Directionality and conditioning

MIMO systems have a property with no single-loop analogue: directionality. The same-sized input in different directions can produce very different-sized outputs, measured by the singular values of the transfer-function matrix. An ill-conditioned system, with a large ratio between its largest and smallest singular values, is hard to control in its weak direction and demands careful robust design.

Fusion devices are strongly multivariable: many magnetic coils jointly determine plasma position, shape, and current, so their control is a coordinated MIMO problem rather than a set of independent loops.