SPR and ASPR Untangled

The theory, proofs, use and implementation of robust adaptive control algorithms require the understanding of the concepts of Strictly Positive Real (SPR) and Almost Strictly Positive Real (ASPR) plants. Although these concepts are defined in the existing literature, both in time and frequency domains, their grasp is not straight forward for the practicing control engineer that deals with real-world plants. Here, we attempt to present the interpretation and meaning of these concepts in a more intuitive way for the practicing control engineer. That is, we use the Bode, Nyquist, Nichols and Root-Locus domains that may help the control engineer better grasp their implications. The paper is oriented towards the practitioner and therefore any proof that already appears in the literature is only cited. An important result that is formalized and proved is that any stable system can be made ASPR. An example is also given in order to demonstrate these concepts.