• Dr Valeri Ougrinovski (valu@eeadfa.edu.au) Phone: 02-62688219
• Dr. Ian Petersen ( i.petersen@adfa.edu.au ) Phone: 02-62688446

Feedback control systems are widely used in manufacturing, mining, automobile and military hardware applications. In response to demands for increased efficiency and reliability, these control systems are being required to deliver more accurate and better overall performance in the face of difficult and changing operating conditions. In order to design control systems to meet the demands of improved performance and robustness when controlling complicated processes, control engineers will require new design tools and better underlying theory. In particular, a standard method of improving the performance of a control system is to add extra sensors and actuators. This necessarily leads to a multi-input multi-output control system. Thus, it is a requirement for any modern feedback control system design methodology that it be able to handle the case of multiple actuators and sensors.

Linear Quadratic Gaussian optimal control theory (LQG) is one of the major achievements of the modern control era. This controller design methodology enables a controller to be synthesized which is optimal with respect to a specified quadratic performance index. Furthermore, this theory takes into account the presence of
Gaussian white noise disturbances acting on the system. Indeed, in many practical control problems, it is straightforward to translate the required performance
objective into a problem of minimizing a quadratic cost functional. Also, in many practical control problems, the system is subject to disturbances and measurement noise which are most naturally modeled as stochastic white noise processes.

Although the LQG controller design methodology provides a systematic way of synthesizing multi-input multi-output control systems in order to optimize the control system performance, this approach suffers from a major disadvantage in that it does not take into account the issue of robustness. That is, the LQG controller design
methodology relies on the use of an accurate linear dynamical model for the process being controlled. In practice, although it is possible to obtain process models either from first principles or from experimental measurements, these models will always be subject to errors. Thus, the control system needs to be designed to be robust against these modeling errors.

Since the issue of robustness is usually critical in any control system design, the absence of any systematic way of dealing with robustness has severely limited the usefulness of the LQG control method. Indeed, the requirement for robustness in control system design has led to a great deal of research in the general area of robust control theory.

A significant new result to emerge out our previous research is the solution to a certain Minimax LQG Problem. This problem involves constructing a controller which
minimizes the worst case of a quadratic cost function in the face of a defined class of uncertainties for a given stochastic uncertain system model. The solution to this problem thus combines the optimal performance qualities of LQG control theory with the robustness properties derived from H-infinity control theory. The main aim of this project is to develop the theory of minimax LQG control to provide a practical control system design methodology which includes the previous modern control milestones of LQG control and H-infinity control. Thus, we aim to develop a grand unified controller design methodology taking into account both control system performance and control system robustness.

A key feature of the minimax LQG result developed in our previous research, is the underlying stochastic uncertain system model used. Uncertain systems are mathematical models which include representations of the plant uncertainties. As mentioned above, the use of an uncertain system to model the process being controlled enables a controller to be designed which is robust against specific types on uncertainty. The class of stochastic uncertain systems which underlies the minimax LQG method is an extension of the Integral Quadratic Constraint (IQC) uncertainty description which generalizes the IQC uncertainty description to the stochastic case by replacing the IQC by a relative entropy constraint on the uncertainty. This allows for a tractable solution to the minimax LQG problem by converting it into a risk sensitive optimal control problem. In particular, this meant that the minimax optimal control problem could be solved in the output feedback case.

Although the basic ideas behind minimax LQG control have been developed, the theory is still in its early stages and a great deal more research needs to be done in
order to develop a complete minimax LQG control theory which is widely applicable in multi-input multi-output control system design. It is the development of this theory which is the major part of this project.