Title: Flexible Manipulator Control Accommodating Passivity Violations: Theory and Experiments (½ hour)
Abstract: The passivity and small gain theorems are fundamental results of input-output stability theory; both have been extensively used in robotics and aerospace control, for example. The passivity theorem states that the negative feedback interconnection of a passive system and a very strictly passive system (i.e., input strictly passive with finite gain) is stable. The small gain theorem states that the negative feedback interconnection of two systems is stable provided the loop gain is strictly less than one.
Although the passivity theorem is extremely powerful, the passive nature of the nominal plant to be controlled is often destroyed via unmodeled sensors and actuators which, for example, add time and/or phase delay. These passivity violations disallow the use of the passivity theorem nominally guaranteeing closed-loop stability. Utilizing the small gain theorem is possible provided the plant has finite gain (which is probable), but the controllers may be overly conservative owing to the small gain requirement. This is especially true in the context of flexible manipulator control where the modes of the structure are modestly damped, resulting in very large gain.
To recover some of the high gain feedback characteristics of passivity-based control, various authors have investigated "mixing" or "blending" the passivity and small gain properties of systems [1,2,3]. Essentially, the passive nature of the plant being controlled will be exploited when passive properties hold, and the finite gain characteristics of the plant will be exploited when passivity is violated. This allows high gain feedback within the passive bandwidth of the plant, and gain limited feedback when passivity is violated. This control structure naturally fits with physically realizable systems because all real systems roll-off at high frequency (i.e, mass-spring-damper systems, RLC circuits, etc.).
In this talk stability and control of systems which possess hybrid properties, that is those that possess both passive and finite gain properties, will be discussed. Two practical and challenging problems will be addressed: the tip control of a one-link flexible manipulator, and the joint control of a two-flexible manipulator. Both of theses systems are nominally passive, but passivity is destroyed by sensor dynamics. Synthesis of hybrid controllers via frequency weighting will be presented, along with experimental results.
References
[1] Griggs, W.M., Anderson, B.D.O., Lanzon, A. "A mixed small gain and
passivity theorem in the frequency domain." Systems & Control Letters, Volume
56, Issues 9-10, September-October 2007, Pages 596-602.
[2] Griggs W.M., Anderson, B.D.O., Lanzon, A., Rotkowitz,
M.C. "Interconnections of nonlinear systems with mixed small gain and
passivity properties and associated input-output stability results." Systems
& Control Letters, Volume 58, Issue 4, April 2009, Pages 289-295.
[3] Forbes, J.R., Damaren, C.J., "A Hybrid Passivity and Finite Gain
Stability Theorem: Stability and Control of Systems Possessing Passivity
Violations." IET Control Theory & Applications, 2010 (to appear).