tailieunhanh - Advanced Robotics - Control of Interactive Robotic Interfaces Volume 29 Part 4

Tham khảo tài liệu 'advanced robotics - control of interactive robotic interfaces volume 29 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 46 2 Control of Port-Hamiltonian Systems where k is a positive constant then the control law u fifix V is such that the energy balance Hd x tỴ - Hd x 0 i VT r y r dr - d t Jo is satisfied with Hd x H x Ha x Proof. By replacing the control law u fifix V into Eq. we get Hfitfit H x 0 y fiTficfir y r dr ịị VT r y r dr dfi Substituting Eq. into Eq. the following equation holds H x t H x 0 Ha x t k J VT r y r dr d t from which it follows that H x t Hafixfi - H x 0 k j VT -T y r dr d t From Eq. it follows that necessarily Ha x o k and thus Eq. can be rewritten as H fix ft Hafixft - H x 0 - Hafix 0 Ị VT -T y r dr dft Finally setting Hd x t H xft Hfixft the balance of Eq. follows. Remark . The term fg fi1 xfir yfr dr can be interpreted as the energy supplied by the controller to the plant. Therefore the condition of Eq. expresses the fact that the energy supplied by the controller can be expressed as a function of the state. The energy of the closed loop system is the difference between the energy stored by the plant and the energy supplied by the controller therefore this energy shaping strategy is called energy balancing passivity based control energy balancing PBC . If the closed loop energy Hd x has a strict minimum in the desired configuration X then setting V 0 X is stable and the Lyapunov function is represented by the difference between the energy stored by the system and the energy supplied by the controller. Example . Consider an n-DOF fully-actuated mechanical system with generalized coordinates q e Q. The port-Hamiltonian model of this system has been obtained in Example and is Energy Shaping of Port-Hamiltonian Systems 47 AA 0 In 0 0 M 0 p -In 0 VO D q p l V L b 72 .r7 J ijp j VI y 0 BT g D y dp u with H q p - pTM 1 q p V q where V q is the potential energy and where D q is a positive definite matrix representing the viscous friction present in the system. Suppose that qd e Q is