CHAPTER3 STABILITY OF THE CONTROLSYSTEMCONTENT>3.1 TheBasics of StabilityTheory inMathematics> 3.2 Lyapunov Stability> 3.3 Lyapunov Stability Theory> 3.4 Application of Lyapunov 2nd Method tothe LTI System
CHAPTER3 STABILITY OF THE CONTROL SYSTEM • CONTENT 3.1 The Basics of Stability Theory in Mathematics 3.2 Lyapunov Stability 3.3 Lyapunov Stability Theory 3.4 Application of Lyapunov 2nd Method to the LTI System
3.1 The Basics of Stability Theory inMathematicsThe concept of stability is extremely important becausealmost every workable system is designed to be stable. If asystem is not stable, it is usually of no use in practice. The most important approach for studying thestability of control system is theLyapunovstability theory, which is introduced by theRussian mathematician Alexandr MikhailovichLyaponov in the late 19th century
3.1 The Basics of Stability Theory in Mathematics
3.1 The Basics of Stability Theory inMathematicsBefore discussing the Lyapunov stability theory, we need toreview some relevant mathematical knowledge, for examplethe norm and the quadratic form function.The norm of a vector X is a real-valuedDefinition 3.1function Xwith properties:
3.1 The Basics of Stability Theory in Mathematics
3.1 The Basics of Stability Theory inMathematicsDefinition 3.1 The norm of a vector X is a real-valuedfunction xwith properties:(1) X ≥O for all X eR" with X=o if and only ifX = 0:(2) αX=αl-Xll for all α eR and X eR";(3) Triangle inequality X +Y≤X+Y holds true forVX,YeR".There are many norms satisfy the conditions of Definition 3.1
3.1 The Basics of Stability Theory in Mathematics
3.1 The Basics of Stability Theory inMathematicsThe most commonly used norm is Euclidean norm, which isI= /x +. xdefined as.The Euclidean norm of a vector is the generalization of theidea of length and is a length measurement of a vector in thestate space. For example, X。- X.ll is used to represent thelength from the point X。 to the point Xe in the statespace
3.1 The Basics of Stability Theory in Mathematics