Stability of Robot Manipulators Under Saturated PID Compensation

Abstract

The joint position regulation problem for robot manipulators under a standard saturated proportional-integral differential (PID) compensator is studied in this brief. The main result states the existence of PID control gains yielding semiglobal asymptotic stability if the control torque bounds are larger than gravitational torques. Energy shaping plus damping injection methods, as well as singular perturbation analysis, are used to establish stability conditions to achieve regulation at any desired position. Some experiments are carried out to illustrate the stability results.

Saturated RISE Feedback Control for a Class of Second-Order Nonlinear Systems

“…Substituting (21), (7), (16), and (18), the expression in (26) reduces to the scalar inequality (27) where the set in (26) reduces to the scalar equality in (27) since the RHS is continuous a.e., i.e, the RHS is continuous except for the Lebesgue negligible set of times when [31], [34]. 7 Young's Inequality can be applied to select terms in (27) as (28) To facilitate the subsequent stability analysis, let be selected as , where are positive gain constants. Utilizing (28), completing the squares on and grouping terms, the expression in (27) can be upper bounded by (29) Provided the sufficient conditions in (24) are satisfied, (17) and (19) can be used to conclude that (30) where is defined as , was defined in (24), and is a continuous, positive semi-definite function for some positive constant .…”

“…Based on the definition of in (19), as . 7 The set of times is equivalent to the set of times . From (7), this set can also be represented by .…”

“…See also [3]- [5] for theoretical and practical significance of stabilization by a stable controller. Applications of stable controllers can be found in flexible structures [6], DC servo motors [7], data-communication networks [8], etc.…”

“…Results in [14] and [15] achieve global regulation of saturated nonlinear systems using a PID-like control structure and a passivity-based analysis. To compensate for uncertain dynamics and the evaluation of an unknown gravity term, the results in [7] includes an additional saturated integral term and uses energy shaping and damping injection methods to yield asymptotic tracking. More recently, a saturated PID controller was developed in [8] which uses sigmoidal functions to achieve global asymptotic regulation to a setpoint; however, it is unclear how the result can be extended to the tracking problem due to the inclusion of stationary values of an auxiliary signal.…”

Output-feedback proportional-integral-derivative-type control with multiple saturating structure for the global stabilization of robot manipulators with bounded inputs

“…For instance, semiglobal regulators with different saturating PID-type structures have been proposed in a frictionless setting by Alvarez-Ramrez J and colleagues. 10,11 The closed-loop analysis in these works was carried out using singular perturbation methodology. Through such a methodology, the authors show the existence of some suitable tuning, mainly characterized by the requirement of small enough integral action gains and sufficiently high proportional and derivative ones.…”