Mathematical Preliminaries

Abstract

In this chapter, some basic concepts concerning signals and systems are presented. From the view point of system analysis, a control system is a closed-loop system. The behaviours of the tracking error signals, the control signals and all the internal signals in the system are very important in control system design. It is, therefore, essential to have appropriate measures (or norms) for the size of these signals for analysis and controller design. From these norms, we can define deduced norms to measure the “gain” of the systems. In between, we also present the concepts of compact sets, continuous and differentiable functions, the Lipschitz condition, and Barbalat’s Lemma. Then, some basic matrix properties, the concepts of stability and Lyapunov Stability, are introduced. Finally, we present the definitions and properties/operations of (i) the stable sliding surface, (ii) Mean Value Theorems, (iii) Integral formula including integration by parts, Change of variables, Comparison theorem, and differentiation of integrals, (iv) Implicit Function Theorem, and among others for completeness. All the above concepts and formulas are essential tools used in the book.