Development and evaluation of methods for control of multiple-input multiple output systems

Licentiate thesis, 2018

In control, the most common type of system is the multiple-input multiple-output
(MIMO) system, where the same input may affect multiple outputs, or
conversely, the same output is affected by multiple inputs. In this
thesis two methods for controlling MIMO systems are examined, namely
linear quadratic Gaussian (LQG) control and decentralized control,
and some of the difficulties associated with them.

One difficulty when implementing decentralized control is to decide
which inputs should control which outputs, that is the input-output
pairing problem. There are multiple ways to solve this problem, among
them using gramian based measures, which include the Hankel interaction
index array, the participation matrix and the Sigma2 method.
These methods take into account system dynamics as opposed to many
other methods which only consider the steady-state system. However,
the gramian based methods have issues with input and output scaling.
Generally, this is resolved by scaling all inputs and outputs to have
equal range. However, in this thesis it is demonstrated how this can
result in an incorrect pairing. Furthermore this thesis examines other
methods of scaling the gramian based measures, using either row or
column sums, or by utilizing the Sinkhorn-Knopp algorithm. This thesis
shows that there are considerable benefits to be gained from the alternative
scaling of the gramian based measures, especially when using the Sinkhorn-Knopp
algorithm. The use of this method also has the advantage that the
results are completely independent of the original scaling of the
inputs and outputs.

An alternative way to control a MIMO system is to implement an LQG
controller, which yields a single control structure for the entire
system using a state based controller. It has been proposed that LQG
control can be an effective control scheme to be used on networked
control systems with wireless channels. These channels have a tendency
to be unreliable with package delays and package losses. This licentiate
thesis examines how to implement an LQG controller over such unreliable
communication channels, and proposes an optimal controller which minimizes
the cost function.

When new methods of control system design and analysis are introduced
in the control engineering field, it is important to compare the new
results with existing methods. Often this requires application of
the methods on examples, and for this purpose benchmark processes
are introduced. However, in many areas of control engineering research
the number of examples are relatively few, in particular when MIMO
systems are considered. For a thorough assessment of a method, however,
as large number of relevant models as possible should be used. As
a remedy, a framework has been developed for generating linear MIMO
models based on predefined system properties, such as model type,
size, stability, time constants, delays etc. This MIMO generator,
which is presented in this thesis, is demonstrated by using it to
evaluate the previously described scaling methods for the gramian
based pairing methods.