Tham khảo tài liệu 'advances in pid control part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 50 Advances in PID Control where Kp Kv and Ki are positive definite diagonal n X n matrices and q qd q denotes the position error vector. Even though the PID controller for robot manipulators has been very used in industrial robots Arimoto 1995a there still exist open problems that make interesting its study. A open problem is the lack of a proof of global asymptotic stability Arimoto 1994 . The stability proofs shown until now are only valid in a local sense Arimoto 1994 Arimoto et al. 1990 Arimoto Miyazaki 1983 Arimoto 1996 Dorsey 1991 Kelly 1995 Kelly et al. 2005 Rocco 1996 Wen 1990 or in the best of the cases in a semiglobal sense Alvarez et al. 2000 Meza et al. 2007 . In Ortega et al. 1995a a so-called PI2D controller is introduced which is based on a PID structure but uses a filter of the position in order to estimate the velocity of the joints and adds a term which is the integral of such an estimate of the velocity this added term motivates the name PI2D for this controller semiglobal asymptotic stability was proved. To solve the global positioning problem some globally asymptotically stable PID-like regulators have also been proposed Arimoto 1995a Gorez 1999 Kelly 1998 Santibanez Kelly 1998 such controllers however are nonlinear versions of the classical linear PID. We propose a new global asymptotic stability analysis by using passivity theory for a class of nonlinear PID regulators for robot manipulators. For the purpose of this chapter it is convenient to recall the following definition presented in Kelly 1998 . Definition 1. F m e x with 1 m 0 e 0 and x E IRn denotes the set of all continuous differentiable increasing functions sat x sat xj sat 2 sat xn T such that x sat x m x Vx E IR x e e sat x me V x E IR x e 1 dx sat x 0 V x E IR where stands for the absolute value. For instance the nonlinear vector function sat q sat r 1 sat j2 considered in Arimoto Arimoto 1995a whose entries are given by sat qn T sin x if x n 2 sat x Sin x 1 if x n 2 1 if x n 2
