Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

Abstract

Abstract. This paper is concerned with integral control of systems with hysteresis. Using an inputoutput approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L 2 -stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2 -stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system. Mathematics Subject Classification. 34G20, 47J40, 47N70, 93C23, 93C25, 93D10, 93D25.

PID control of second-order systems with hysteresis

“…In order to provide an appropriate analytical framework for the main investigation in Section 4 below, here we assemble some results from [12,13].…”

“…Furthermore, observe that system (17) has the same form as the system in [13, (4.1)], with h = 0, ϑ = 0 and ψ = id in the latter and with the roles of ρ and g in the latter being played, respectively, by G(0)r 1 + r 2 and r 1 (G(0) − g ⋆ θ) − q ∈ L 2 (R + ) in the present context. Therefore, Theorem 4.1 in [13] may be applied to establish that (17) has a unique solution y ∈ W 1,1 loc (R + ), and lim t→∞ẏ (t) = 0, (Φ(y)) ′ ∈ L 2 (R + ) and lim t→∞ (Φ(y))(t) =: Φ ∞ exists. Moreover, Theorem 4.1 in [13] also shows that y is bounded, provided that r 1 + r 2 /G(0) is an interior point of NVS Φ.…”

“…Therefore, Theorem 4.1 in [13] may be applied to establish that (17) has a unique solution y ∈ W 1,1 loc (R + ), and lim t→∞ẏ (t) = 0, (Φ(y)) ′ ∈ L 2 (R + ) and lim t→∞ (Φ(y))(t) =: Φ ∞ exists. Moreover, Theorem 4.1 in [13] also shows that y is bounded, provided that r 1 + r 2 /G(0) is an interior point of NVS Φ. It remains to show that Φ ∞ = r 1 + r 2 /G(0).…”

“…The analytical framework of the present paper is based on frequency-domain conditions developed recently in [12,13] which ensure existence, regularity and certain asymptotic properties of solutions of a feedback interconnection of a linear (possibly infinite dimensional) system and a hysteresis operator Φ. Within this framework, the main contribution of the paper is as follows.…”

Stability and robust position control of hysteretic systems

“…In [11][12][13], control of various systems that include a hysteretic component is studied using techniques for nonlinear dynamical systems. In [11], pure integral control with a time-varying gain is studied, with additional dynamics included in the loop.…”