Stability of A Class of Relay Feedback Systems Arising in Controlled Mechanical Systems with Ideal Coulomb Friction ⋆

Coulomb friction is inevitable in every mechanical system with contact motion. When the mechanical system with Coulomb friction is under feedback control, it can destabilize the system by generating limit cycles. Controlled mechanical systems with ideal Coulomb friction can be viewed as a particular class of relay feedback systems characterized by the zero DC gain property and the positivity of the first Markov parameter. This paper elaborates recent results on sufficient conditions to guarantee the global pointwise stability of such systems. The scope of analysis has been kept broad so that the results apply to systems with multiple inertia elements and multiple Coulomb friction sources. To employ the recent advances in the absolute stability theory, the limiting arguments are adopted to approximate the relay elements to continuous functions. As a result, a new sufficient condition on the global pointwise stability of the systems with multiple Coulomb friction sources is derived by extending the existing result with a single Coulomb friction source when the stiction level is larger than the Coulomb friction level. Also, it has been shown that the describing function criterion is indeed an exact condition when the order of the closed-loop system is 3. Simulation results are presented with a flexible joint mechanism to illustrate the main points.