The Prandtl-Ishlinskii (PI) model is a well-known mathematical model used to describe hysteresis behavior in various physical systems, particularly in smart materials, magnetic systems, and biomechanics. It is widely applied in piezoelectric actuators, ferromagnetic materials, and mechanical systems with friction.
Key Features of the Prandtl-Ishlinskii Model:
- Rate-Independent Hysteresis: It models static hysteresis effects without considering time-dependent (rate-dependent) behavior.
- Superposition of Elementary Operators: The model is constructed using a weighted sum of elementary play operators (or stop operators), which represent basic hysteresis loops.
- Analytical Simplicity: The model allows straightforward identification and inversion, making it useful for compensation and control applications.
Mathematical Formulation
The PI model is typically expressed as:
$$
y(t)=\sum_{i=1}^N w_i \Phi_{r_i}[x(t)]
$$
where:
- $y(t)$ is the output (hysteresis response),
- $x(t)$ is the input (excitation signal),
- $w_i$ are weights (scaling factors),
- $r_i$ are threshold values (hysteresis levels),
- $\Phi_{r_i}$ is the play operator with threshold $r_i$, defined as:
$$
\Phi_rx= \begin{cases}\max \left(x(t)-r, y_{prev}\right), & \text { if } x(t) \text { is increasing } \\ \min \left(x(t)+r, y_{prev}\right), & \text { if } x(t) \text { is decreasing }\end{cases}
$$
This captures the loading and unloading behavior of hysteresis.
Applications
The Prandtl-Ishlinskii model is widely used in:
- Piezoelectric actuators: Compensating hysteresis for precision positioning.