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
3.1 The Basics of Stability Theory inMathematicsDefinition 3.2Definiteness of a scalar function V(X)A scalar function V(X) is calledpositivedefiniteasV(X)>O for all X eR" with V(X)=o if and only ifX=0.tA scalar function V(X) is called positive semi-definiteasV(X)≥0 for all X R" with V(X)=o if and only ifX=0
3.1 The Basics of Stability Theory in Mathematics
3.1 The Basics of Stability Theory inMathematicsDefinition 3.2Definiteness of a scalar function V(X)A scalar function V(X) is callednegativedefiniteasV(X)<Ofor all X eR" with V(X)=O if and only ifX=0.A scalar function V(X) is called negative semi-definiteasV(X)≤O for all X Rn with V(X)=o if and only ifX=0..A scalar function V(X) is called indefinite when it presentsboth positive and negative values for X + 0
3.1 The Basics of Stability Theory in Mathematics
3.1 The Basics of Stability Theory inMathematicsDefinition 3.3 TheV(X)quadratic form functionis athein.realreal homogeneousvariablespolynomialXi, X2,".., Xn of the form.nnV(X)=ZZp;x,x;where Pi are real.The quadratic form function can also be written with thevector X =[xi x, -.. x,jT and the matrix P as. V(X)= XI PXP11 P12Pinis a real symmetric matrix and it...P22P12P2nwhere P=is called the matrix of the...:.quadratic form function V(X) ..PinP2nPmm
3.1 The Basics of Stability Theory in Mathematics
Example 3.1 Rewrite the following scalar function in theform such as V(X) = XT PXV(X) = x - 3x2 + x3 - 4x2 - 2xix2 + 4xix3 - 8xix4 - 4xgx4Letting X=[xi x22 x， xJ ， the scalarSolutionfunction V(X) can be written as.21-1Axi0-10-3X2V(X)=X'PX=x x2 x33X4201-2X310-4-2-4X4where. P is a real symmetric matrix and is called the matrixof the quadratic form function V(X)
3.1 The Basics of Stability Theory inMathematicsDefinition 3.4Given a nxn matrix P such asP11P12pinP22P12P2np..PinPmm.P2nlet P denote the matrix formed by deleting the lastn-krows and columns of P , where k =l,2,...,n . Thedeterminant △ =Pkis called theleading principal minor determinant about Pof order k
3.1 The Basics of Stability Theory in Mathematics