Hysteresis control block

Authors: Mathilde Bongrain

Diagram

hysteresis diagram detailedHysteresisDiagram

D stands for “Decreasing”
I stands for “Increasing”
B stands for “Before jumping”
A stands for “After jumping”

Syntax:

function name: hyst
input variable : \(x_i\)
output variable: \(x_j\)
data name, parameter name or math expression for \(x_I\)
data name, parameter name or math expression for \(y_{IB}\)
data name, parameter name or math expression for \(y_{IA}\)
data name, parameter name or math expression for \(x_{D}\)
data name, parameter name or math expression for \(y_{BD}\)
data name, parameter name or math expression for \(y_{DA}\)
data name, parameter name or math expression for \(z_0\)

Internal states : none

Discrete variable : \(z \in \{-1,1\}\)

Equations

\[0 = \left\{ \begin{array}{lll} x_j - y_{IA} - \frac{y_{IA} - y_{IB}}{x_I -x_D} (x_i - x_I) & if & z=1 \\ x_j - y_{DA} - \frac{y_{IB} - y_{DA}}{x_I - x_D} (x_i - x_D) & if & z=-1 \end{array} \right.\]

Discrete transitions

if z = −1 then
    if xi > xI then
        z ← 1
    end if
else
    if xi < xD then
        z ← −1
    end if
end if

Initialization of discrete variables

if xi > xI then
    z ← 1
else if xi < xD then
    z ← −1
else
    if z0 ≥ 0 then
        z ← 1
    else
        z ← 0
    end if
end if

N.B.

At \(t = 0\), if \(x_D < x_i(0) < x_I\) the initial state of the system is indeterminate, since it could operate on the ( \(y_{IA}\) , \(y_{DB}\) ) line (i.e. with an initial value of z equal to 1) as well as on the ( \(y_{DA}\), \(y_{IB}\)) line (i.e. with an initial value of z equal to −1). Hence, the user must specify \(z_0\), the initial value of z. If \(x_i(0) < x_D\) (resp. \(x_i(0) > x_I\) ) the initial value is \(z = −1\) (resp. \(z = 1\)) and \(z_0\) is not used. The data must obey \(x_D < x_I\) , otherwise the model would not correspond to hysteresis.

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