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On Cauchy problem and stability of inversion-free feedforward control of piecewise monotonic Krasnoselskii-Pokrovskii hysteresis

Jana Kopfova, Michael Ruderman

Abstract

We consider the non-homogeneous first-order differential equation with hysteresis described by the Krasnoselskii-Pokrovskii rate-independent hysteresis operator. Existence and uniqueness of solutions as well as the boundedness of solution in response to a bounded input are proved. The global stability of the equation is also investigated. Periodic solutions and their stability are studied in addition. The differential equation under analysis constitutes the so-called inversion-free feedforward control, which was proposed for mitigating arbitrary rate-independent hysteresis effects in the actuated systems. The experimentally identified non-smooth and non-strictly monotonic hysteresis of a magnetic shape memory alloy (MSMA) actuator serves as the case study. The performed analysis is settled in a series of theorems which are illustrated by numerical examples.

On Cauchy problem and stability of inversion-free feedforward control of piecewise monotonic Krasnoselskii-Pokrovskii hysteresis

Abstract

We consider the non-homogeneous first-order differential equation with hysteresis described by the Krasnoselskii-Pokrovskii rate-independent hysteresis operator. Existence and uniqueness of solutions as well as the boundedness of solution in response to a bounded input are proved. The global stability of the equation is also investigated. Periodic solutions and their stability are studied in addition. The differential equation under analysis constitutes the so-called inversion-free feedforward control, which was proposed for mitigating arbitrary rate-independent hysteresis effects in the actuated systems. The experimentally identified non-smooth and non-strictly monotonic hysteresis of a magnetic shape memory alloy (MSMA) actuator serves as the case study. The performed analysis is settled in a series of theorems which are illustrated by numerical examples.
Paper Structure (6 sections, 6 theorems, 21 equations, 5 figures)

This paper contains 6 sections, 6 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Generalized play operator
  4. Main Results
  5. Numerical Cases
  6. Conclusions

Key Result

Theorem 4.2

Suppose that Assumption H holds. Then the solution of the Cauchy problem eq:main for any $K>0$ with initial conditions $u(0) = u_0,$$H(u)(0) = w_0,$ exists and is unique.

Figures (5)

  • Figure 1: Inversion-free feed-forward hysteresis control.
  • Figure 2: Measured (and lowpass filtered) MSMA actuator response under quasi-static operation conditions versus the identified KP-type hysteresis model with three operator elements.
  • Figure 3: Control error (a) and its logarithmic convergence (b) for a constant (step) input value $r(t) = R = 2$ applied at time $t=0.1$ sec.
  • Figure 4: Input trajectory (a) and control error (b) in case $\lim_{t \rightarrow \infty} r(t) = 2$.
  • Figure 5: Maximal steady-state absolute control error depending on the frequency $2 \pi \omega$ (a), and control error for $2 \pi \omega = 1$ Hz (b).

Theorems & Definitions (12)

  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • Theorem 4.5
  • proof
  • Theorem 4.6
  • proof
  • ...and 2 more