Simple implementation of Almost Passivity in PID controlled systems

The theory, use and implementation of robust adaptive control algorithms require the understanding of Passivity and Almost Passivity. For LTI system these are the Strictly Positive Realness (SPR) and Almost Strictly Positive Realness (ASPR) concepts. Although these concepts have been defined in the existing literature, their grasp is not straightforward for the practicing control engineer that deals with real-world plants. In an attempt to present the interpretation and meaning of these concepts in a more intuitive way, in this paper we use various frequency domains illustrations that may help the control engineer better grasp their implications. An important result that is formalized and proved is that any system controlled by PID controller can be made to become ASPR and thus, robust adaptive control can be used towards improvement of performance. An example is also given in order to demonstrate these concepts.