Stability Analysis for Digital PD Control of Flexible Systems

Abstract

In this work we extent our recent results on the stability of rigid systems to cases involving flexible ones. We present closed form analytical expressions that describe the boundaries of the stability regions for digital PD control systems. This is obtained using a newly adopted approach based on the critical stability constraints of Jury test. The considered system consists of a single rigid and a single flexible mode. This simulates many practical systems such as antenna, space shuttle, and robot arm. The obtained closed-form stability criterion shows that the stability region is almost a right triangle in the gain space. The right-angle sides of the proposed triangle coincide exactly with two boundaries of the stability region. The hypotenuse of the triangle is found to well approximate the third boundary. An analytical expression is derived for the slope of the hypotenuse. This approximation results in defining the stability region using only two parameters which is convenient for practical applications. Case study is presented to show the practical usefulness of the proposed close-form stability criteria through investigating the control design of high speed flexible robot arm.

Stability Analysis for Digital PD Control of Flexible Systems Including Damping

“…The results show that for practical values of damping ratio, ξ = 0.001 -0.02 (21) , and if the system belongs to the first class, ξ < ξ 1 , the stability region is almost a right triangle similar to the case of zero damping (16) . The stability criteria obtained previously for zero-damping system can well approximate this case of lightly damped systems.…”

“…(17). The three solutions can be defined as: (14) - (16). For D > 0, one can obtain the real root of the cubic equation (10) by substitution for z by the one that has real value from the three solutions (z 1 ,z 2 ,z 3 ) in Eq.…”

“…In Ref. (16), the results of Ref. (14) is extended to cases involving flexible systems without damping, where the stability region for single-rigid/singleflexible mode system is defined analytically in a closedform that is suitable for design purposes.…”