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

Sammanfattning: 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.

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