Proportional integral (PI) controller with non-windup limit on the integral term

Diagram

pictlim diagram

Proportional integral (PI) controller with non-windup limit on the integral term diagram

Syntax:

• function name: pictllim
• input variable : \(x_k\)
• output variable: \(x_j\)
• data name, parameter name or math expression for \(K_I\)
• data name, parameter name or math expression for \(K_p\)
• data name, parameter name or math expression for \(x_i^{min}\)
• data name, parameter name or math expression for \(x_i^{max}\)

Internal states : variable \(x_i\)

Discrete variable : \(z \in \{-1,0,1\}\)

Equations

\[ \begin{align} \left\{ \begin{array}{lll} \dot{x_i} = K_i x_k & if & z=0 \\ 0= x_i - x_i^{min} & if & z=-1 \\ 0 = x_i - x_i^{max} & if & z=1 \end{array} \right. \end{align} \]

\[ 0 = K_p x_k + x_i - x_j \]

Discrete transitions

if z = 0 then
    if xi > xmaxi then
        z ← 1
    else if xi < xmini then
        z ← −1
    end if
else if z = 1 then
    if Ki*xk < 0 then
        z ← 0
    end if
else if z = −1 then
    if Ki*xk > 0 then
        z ← 0
    end if
end if

Initialization of internal state variables and discrete variables

if Ki*xk > 0 then
    z ← 1
    xi ← xmaxi
else if Ki*xk < 0 then
    z ← −1
    xi ← xmini
else
    z ← 0
    xi ← xj
end if