Research Projects

Mathematical models play a crucial role in understanding and analyzing control systems. Physical systems of interest in control theory are often modeled by ordinary differential equations along with some algebraic constraints. Such systems are called systems described by differential and algebraic equations (DAEs). However, it is sometimes possible to explicitly solve the algebraic constraints and transform the system model into a set of ordinary differential equations (ODEs) only. These transformations need human intervention and involve a change or elimination of system variables; thus, the resulting ODE models (also famous as state space models) do not represent many useful properties of the underlying physical phenomena. It is, therefore, necessary to work with the original DAE models. This project is devoted to studying the DAE systems in their most general form, which may be under-determined or over-determined. Any control system has three essential variables: input (control), output (measurable), and state (internal) variables. The estimation of state variables is crucial to obtaining real-time information on the system. A well-known way of estimating state variables is to design an auxiliary system called an observer, whose role is to process the information provided by the input and output variables and thereby construct a reliable estimate of the system's internal variables. Observers, which estimate only a part or linear function of internal variables without estimating the whole vector of variables, are called functional observers. The project's first aim is to design functional observers for DAEs by obtaining structural decompositions of the system coefficient matrices and nonlinearities. The main reason for choosing functional observers is their ability to estimate a combination of internal variables in less computational time than standard full-state observers. Moreover, the functional observers need less restrictive assumptions than full-state observers. On the other hand, if the state variables of a system are corrupted by noisy input/output measurement data, the state estimation becomes more interesting because we have to filtered out the noise appropriately. One of the most popular ways to deal with this problem in standard linear state space systems is the celebrated Kalman filtering approach, which generally provides an estimation for systems with Gaussian noise. In some applications, however, the noise sources may not be exactly known. In such cases, $H_\infty$ filtering techniques are well established for state space systems. In this project, the second main aim is to design filtering based functional observers for DAE control systems with Gaussian and non-Gaussian noisy measurements.