tailieunhanh - Control of Robot Manipulators in Joint Space - R. Kelly, V. Santibanez and A. Loria Part 3

Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Lyapunov Stability 41 A continuous function w IRn Iflis said to be globally positive definite or simply positive definite if 1. VK O 0 2. w x 0 V X J 0 . According to this definition it should be clear that a positive definite function is also locally positive definite. Also according to the definition of positive definite matrix a quadratic function f IR IR i. e. of the form fix xTPx PeIR x is positive definite if and only if p 0. The function VK ir is said to be locally negative definite if VK ir is I ora c ontinuousfunction V Ib x II ll . o . which also depends on time we say that V t x is resp. locally positive definite if dd f 0 0 rr . 2. V t x W x V t 0 V X e IR resp. for small ir h d w .d h rtp. kicsily 0 deMSe tahUon. . . . T . c . Definition . Radially unbounded function and decrescent function A continuous function w IRn Iflis said to be radially unbounded if at I o -goo. d os we say that V i I rad ally aalmmded fVlt x t z E -fE ZEE h . -. ffi z zso A continuous function V 1 X JRn JR is locally decrescent if there existt a floml 0asitive definite function w JRn JR such that V i x W x vt 0 V. e JRn Jar small - . If V f x is independent of t . if V f x V x then V x is decrescent- The following examples illustrate the concepts presented above. Example . Consider the whs oDhe fimctions Vi x with i 1 . 4 as depicted in Figure . It is apparent from these graphs that 42 2 Mathematical Preliminaries Figure . Examples Vi x is locally positive definite but is not globally positive definite 16 is locally positive definite globally positive definite and radially unbounded Vz x is locally positive definite and globally positive definite but it is not radially unbounded v4 x iff posf fi ve defimte and radially unb ounded. Example . The function ĨV xi X2 x4 x is positive definite and radially w is independent of t it follows immediately that it is also decrescent. 0 Lyapunov Stability 43 Example . The function V t X1 X2 t l