In designing a controller, it is very important to know the system dynamics well since the knowledge on the system is directly related to the performance of the designed control system. In general, more knowledge on the dynamics of systems can result in better performance of the control systems.
In modelling the dynamics of the systems, there are two representative methods:i.e., the state equation in the time domain and the transfer function in the frequency domain. In the modelling, it is very important to express the dynamics of the system as perfectly as possible since the design of controller is based on the model. But the perfect model is often very high order and/or includes nonlinearity in general, and so it is very difficult to design a controller for a complicated system model using the well-known linear system controller design techniques.
Therefore the trade-off is essential between the exact system model and the easiness in designing a controller, and the linearization and the approximation is frequently employed. Also, the physical system is operated under noise and disturbances which is often ignored in designing a controller. We shall call these differences between the physical system and the obtained linear model as uncertainty, i.e., the terminology "uncertainty" includes the system model error, the system disturbances and noise. The uncertainty is classified as unstructured uncertainty and structured uncertainty based on its knowledge. The unstructured uncertainty is the case when only its upper norm bound is known while the structured uncertainty is the case with its structure known as when it can be expressed as a linear combination of known matrices.
In designing a controller one must consider this uncertainty to satisfy the desired system stability and performance. The robust stability means the capability of maintaning the stability against the uncertainty.
Also, one must note that the controller output is connected to the actuators such as valves and/or motors. But these actuators have the typical nonlinearity, called saturation. The saturating actuator is usually a source of the degradation of system performance and it can be a source of instability of the system.
In this thesis, we first consider the robust stability of uncertain linear systems, which are time invariant system and either continuous-time or discrete-time. Also, the unstructured uncertainty as well as the structured uncertainty are considered. In each case, we review representative previous works and present the new robust stability condition. The obtained results are compared with the previous representative works by examples.
Next we study the robust stability of uncertain linear systems with saturated actuator, and presents some robust stability conditions to guarantee the stability. Both symmetrical and non-symmetrical types of actuators are considered.
Finally, we have treated the robust stability issues of LQ optimal control which operates under uncertainty, and takes the suboptimality beta problem.
