Analysis of disturbance-generated bifurcation in an adaptive system with e1-modification law

A modified adaptation law, the so-called e₁-modification law, has been proposed by K.S. Narendra and A.M. Annaswamy (1987) for robust model-reference adaptive control in the presence of bounded disturbances. Using this law, the authors show analytically that in the case of regulation, a prototype adaptive control system, with a single unknown pole and an unknown constant disturbance at its input, possesses multiple equilibria and undergoes various bifurcations. The analysis begins by transforming the augmented system equations so that there are only two independent parameters. Then explicit expressions for the equilibria and their bifurcations are given, followed by the proof of the existence of the Hopf bifurcation. As the constant disturbance is varied, saddle-node bifurcation, (subcritical) Hopf bifurcation, and saddle-connection bifurcation occur in that order. Consequently, in this case, the e ₁-modification law is qualitatively equivalent to the σ-modification law