1.1 Lecture notes layout

The course aims at providing the students with all necessary tools for being able to apply the theory to practical optimal control design problems.

Optimal control design is based on the designer’s familiarity with the engineering situation and understanding of the underlying physics. These allow to formulate a relevant mathematical model based on which the optimal control is designed.

There are four essential steps in an optimal control design:

• full-state feedback design where it is assumed that all states are measured and available for feedback;
• state estimator design where the concern is to estimate values of the states when they cannot all be measured directly, but certain measurements are available;
• controller reduction where the concern is to approximate a complicated state estimate feedback controller obtained from the above two steps by a simpler one - complication usually being measured by the state dimension;
• implementation of the controller where the controller has to be approximated in discrete-time to be implemented on physical hardware.

The first part of the course assumes precise knowledge of the structure and parameters of the dynamic systems investigated and precise measurements of the states for feedback control. In practice, such precise knowledge is seldom available. Thus, it is important to be able to deal with random fluctuations in the environment and in the measurement system. The second part of the course thus is concerned with this topic, starting with a review of the fundamentals of probability and random processes and proceeding to the design of the ”best” control system, on the average, using noisy measurements and taking into account random perturbations of the system by the environment.