Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators

We present a simple global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators. Nonlinear control structures based on the classical PID controller, which assure global asymptotic stability of the closed-loop system, have emerged. Some works that deal with global nonlinear PID regulators based on Lyapunov theory have been reported by (Arimoto, 1995a), (Kelly, 1998) and (Santibáñez & Kelly, 1998). Recently, a particular case of the class of nonlinear PID global regulators originally proposed in (Santibáñez & Kelly, 1998) was presented by (Sun et al., 2009). Few saturated PID controllers (that is, bounded PID controllers taking into account the actuator torque constraints) have been reported: for the case of semiglobal asymptotic stability, a saturated linear PID controller was presented in (Alvarez et al., 2003) and (Alvarez et al., 2008); for the case of global asymptotic stability, saturated nonlinear PID controllers were introduced in (Gorez, 1999), (Meza et al., 2005), (Santibáñez et al., 2008). The work introduced by (Gorez, 1999) was the first bounded PID-like controller in assuring global regulation; the latter works, introduced in (Meza et al., 2005) and (Santibáñez et al., 2008), also guarantee global regulation, but with the advantage of a controller structure which is simpler than that presented in (Gorez, 1999). A local adaptive bounded regulator was presented by (Laib, 2000). Recently a new saturated nonlinear PID regulator for robots has been proposed in (Santibáñez et al., 2010), the controller structure considers the saturation phenomena of the control computer, the velocity servo-drivers and the torque constraints of the actuators. The work in (Orrante et al., 2010) presents a variant of the work presented by (Santibáñez et al., 2010), where now the controller is composed by a saturated velocity proportional (P) inner loop, provided by the servo-driver, and a saturated position proportional-integral (PI) outer loop, supplied by the control computer. In this chapter we use a passivity based approach to explain the results of global regulation of a class of nonlinear PID controllers proposed by (Santibáñez & Kelly, 1998), that include the particular cases reported by (Arimoto, 1995a) and (Kelly, 1998). At the end of the 80’s, it was established in (Kelly & Ortega, 1988) and (Landau & Horowitz, 1988) that the nonlinear Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators