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Chapter 26
26. MIMO Controller Parameterisations
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An idealized requirement in MIMO control system design is that of
decoupling. As discussed in section §20.4, decoupling can take various
forms ranging from static (where decoupling is only demanded for
constant reference signals) up to full dynamic decoupling (where
decoupling is demanded at all frequencies). Clearly full dynamic
decoupling is a stringent demand. Thus, in practice, it is more usual to
seek dynamic decoupling over some desired bandwidth. If a plant is
dynamically decoupled, then changes in the setpoint of one process
variable leads to a response in that process variable but all other
process variables remain constant. The advantages of such a design are
intuitively clear; i.e. a temperature may be required to be changed but
it may be undesirable for other variables (e.g. pressure) to suffer any
associated transient.
This chapter describes the design procedures necessary to achieve
dynamic decoupling. In particular, we discuss
 dynamic decoupling for stable minimum phase systems
 dynamic decoupling for stable nonminimum phase systems
 dynamic decoupling for open loop unstable systems.
As might be expected, full dynamic decoupling is a strong requirement
and is generally not cost free. We will thus also quantify the
costs of decoupling using frequency domain procedures. These allow a
designer to assess a priori whether the cost associated with
decoupling is acceptable in a given application.
Of course, some form of decoupling is a very common requirement. For
example, static decoupling is almost always called for. The question
then becomes, over what bandwidth will decoupling (approximately) hold.
All forms of decoupling come at a cost, and this can be evaluated using
the techniques presented here. It will turn out that the additional cost
of decoupling is a function of open loop poles and zeros in the right
half plane. Thus, if one is interested in restricting decoupling to some
bandwidth then by restricting attention to those open loop poles and
zeros that fall within this bandwidth one can get a feel for the cost of
decoupling over that bandwidth. In this sense, the results presented in
this chapter are applicable to almost all MIMO design problems since
some form of decoupling over a limited bandwidth (usually around dc) is
almost always required.
We will also examine the impact of actuator saturation on decoupling.
In the case of static decoupling, it is necessary to avoid integrator
windup. This can be achieved using methods that are analogous to the
SISO case treated in Chapter11. In the case of full dynamic decoupling,
special precautions are necessary to maintain decoupling in the face of
actuator limits. We show that this is indeed possible by appropriate use
of MIMO anti windup mechanisms.
Summary
 Recall that key closed loop specifications shared by SISO and MIMO
design include
 continued compensation of disturbances
 continued compensation of model uncertainty
 stabilization of openloop unstable systems
whilst not
 becoming too sensitive to measurement noise
 generating excessive control signals
and accepting inherent limitations due to
 unstable zeros
 unstable poles
 modeling error
 frequency and time domain integral constraints
 Generally, MIMO systems also exhibit additional complexities due
to
 directionality (several inputs acting on one output)
 dispersion (one input acting on several outputs)
 and the resulting phenomenon of coupling
 Designing a controller for closed loop compensation of this MIMO
coupling phenomenon is called decoupling.
 Recall (Chapter 20) that there are different degrees of decoupling,
including
 static (i.e.,
is diagonal)
 triangular (i.e.,
is triangular)
 dynamic (i.e.,
is diagonal)
 Due to the fundamental law of
,
if
exhibits any of these decoupling properties, so does
.
 The severity and type of tradeoffs associated with decoupling
depend on
 whether the system is minimum phase
 the directionality and cardinality of nonminimum phase zeros
 unstable poles.
 If all of a systems unstable zeros are canonical (their
directionality affects one output only), then their adverse effect
is not spread to other channels by decoupling provided that the
direction of decoupling is congruent with the direction of the
unstable zeros.
 The price for dynamically decoupling a system with noncanonical
nonminimum phase zeros of simple multiplicity is that
 the effect of the nonminimum phase zeros is potentially
spread across several loops
 therefore, although the loops are decoupled, each of the
affected loops needs to observe the bandwidth and sensitivity
limitations imposed by the unstable zero dynamics.
 If one accepts the less stringent triangular decoupling, the
effect of dispersing limitations due to nonminimum phase zeros can
be minimized.
 Depending on the case, a higher cardinality of nonminimum phase
zeros can either enforce or mitigate the adverse effects.
 If a system is also openloop unstable, there may not be any way
at all for achieving full dynamic decoupling by a one d.o.f.
controller although it is always possible by a two d.o.f.
architecture for reference signal changes.
 If a system is essentially linear but exhibits actuator
nonlinearities such as input or slew rate saturations, then the
controller design must reflect this appropriately.
 Otherwise, the MIMO generalization of the SISO windup phenomenon
may occur.
 MIMO windup manifests itself in two aspects of performance
degradation:
 transients due to growing controller states
 transients due to the nonlinearity impacting on directionality
 The first of these two phenomenon
 is in analogy to the SISO case
 is due to the saturated control signal not being able to
annihilate the control errors sufficiently fast compared to the
controller dynamics; therefore the control states continue to
grow in response to the nondecreasing control. These wound
up states produce the transients when the loop emerges from
saturation.
 can be compensated by a direct generalization of the SISO anti
windup implementation.
 The second phenomenon
 is specific to MIMO systems
 is due to uncompensated interactions arising from the input
vector loosing its original design direction.
 In analogy to the SISO case, there can be regions in state space,
from which an open loop unstable MIMO system with input saturation
cannot be stabilized by any control.
 More severely than in the SISO case, MIMO systems are difficult to
control in the presence of input saturation, even if the linear loop
is stable and the controller is implemented with anti windup. This
is due to saturation changing the directionality of the input
vector.
 This problem of preserving decoupling in the presence of input
saturation can be addressed by anti windup schemes that scale the
control error rather than the control signal.
