Controller Blocks

Proportional-integral controllers. The variants differ in where the limits apply and whether they follow the IEEE formulation.

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pictl

Proportional-Integral (PI) controller.

Syntax

& pictl
name of variable x_k
name of variable x_j
data name, parameter name or math expression for K_i
data name, parameter name or math expression for K_p

Internal States

$x_i$

Discrete Variables

None.

Equations

$$\begin{aligned} \dot{x}_i &= K_i \, x_k \\ 0 &= K_p \, x_k + x_i - x_j \end{aligned}$$

Initialization

$x_i = x_j$

---

pictllim

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

Syntax

& pictllim
name of variable x_k
name of 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

$x_i$

Discrete Variables

$z \in \{-1, 0, 1\}$

Equations

$$0 = \begin{cases} \dot{x}_i = K_i x_k & \text{if } z = 0 \\ x_i - x_i^{min} & \text{if } z = -1 \\ x_i - x_i^{max} & \text{if } z = 1 \end{cases}$$

$0 = K_p x_k + x_i - x_j$

Discrete Transitions

if z = 0:
  if x_i > x_i^max:  z ← 1
  else if x_i < x_i^min:  z ← -1
else if z = 1:
  if K_i * x_k < 0:  z ← 0
else if z = -1:
  if K_i * x_k > 0:  z ← 0

Initialization

if K_i * x_k > 0:
  z ← 1;  x_i ← x_i^max
else if K_i * x_k < 0:
  z ← -1;  x_i ← x_i^min
else:
  z ← 0;  x_i ← x_j

---

pictl2lim

Proportional-Integral (PI) controller with non-windup limit on the integral term and limit on the proportional term.

Syntax

& pictl2lim
name of variable x_k
name of 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
data name, parameter name or math expression for x_p^min
data name, parameter name or math expression for x_p^max

Internal States

$x_i$ and $x_p$

Discrete Variables

$z_1 \in \{-1, 0, 1\}$ and $z_2 \in \{-1, 0, 1\}$

Equations

$$\begin{cases} 0 = K_p x_k - x_p & \text{if } z_1 = 0 \\ 0 = x_p - x_p^{min} & \text{if } z_1 = -1 \\ 0 = x_p - x_p^{max} & \text{if } z_1 = 1 \end{cases}$$ $$\begin{cases} \dot{x}_i = K_i x_k & \text{if } z_2 = 0 \\ 0 = x_i - x_i^{min} & \text{if } z_2 = -1 \\ 0 = x_i - x_i^{max} & \text{if } z_2 = 1 \end{cases}$$

$0 = x_p + x_i - x_j$

Discrete Transitions

# Proportional limiter (z_1):
if z_1 = 0:
  if x_p > x_p^max:  z_1 ← 1
  else if x_p < x_p^min:  z_1 ← -1
else if z_1 = 1:
  if K_p * x_k < x_p^max:  z_1 ← 0
else if z_1 = -1:
  if K_p * x_k > x_p^min:  z_1 ← 0

# Integral limiter (z_2):
if z_2 = 0:
  if x_i > x_i^max:  z_2 ← 1
  else if x_i < x_i^min:  z_2 ← -1
else if z_2 = 1:
  if K_i * x_k < 0:  z_2 ← 0
else if z_2 = -1:
  if K_i * x_k > 0:  z_2 ← 0

Initialization

Initialization of $x_p$:

$x_p = \min\!\left(x_p^{max},\, \max(x_p^{min},\, K_p x_k)\right)$

Initialization of $x_i$ and discrete variables:

if K_p * x_k > x_p^max:  z_1 ← 1
else if K_p * x_k < x_p^min:  z_1 ← -1
else:  z_1 ← 0

if K_i * x_k > 0:
  z_2 ← 1;  x_i ← x_i^max
else if K_i * x_k < 0:
  z_2 ← -1;  x_i ← x_i^min
else:
  z_2 ← 0;  x_i ← x_j - x_p

---

pictlieee

Proportional-Integral (PI) controller with non-windup limit on the integral term, compliant with IEEE standards.

This PI controller implements the non-windup integrator specified in:

• IEEE Std 421.5-1992, IEEE Std 421.5-2005, and IEEE Std 421.5-2016 (excitation system models for power system stability studies).

The IEEE standard specifies that the integrator state $x_1$ is frozen as soon as $x_j$ reaches its lower or upper limit. To avoid limit cycles when the integrator is released, a second “open-loop” integrator tracks what $x_1$ would be if unconstrained. The lower integrator is re-activated only when $K_p x_k + x_2$ re-enters $[x_j^{min},\, x_j^{max}]$.

Syntax

& pictlieee
name of variable x_k
name of 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_j^min
data name, parameter name or math expression for x_j^max

Internal States

$x_1$ and $x_2$

Discrete Variables

$z \in \{-2, -1, 0, 1, 2\}$

Equations

$$\text{if } z = 0: \quad \begin{cases} 0 = K_p x_k + x_1 - x_j \\ \dot{x}_1 = K_i \, x_k \\ 0 = x_2 - x_1 \end{cases}$$ $$\text{if } z = 2: \quad \begin{cases} 0 = x_j^{max} - x_j \\ \dot{x}_1 = 0 \\ 0 = x_2 - x_1 \end{cases} \qquad \text{if } z = 1: \quad \begin{cases} 0 = x_j^{max} - x_j \\ \dot{x}_1 = 0 \\ \dot{x}_2 = K_i \, x_k \end{cases}$$ $$\text{if } z = -2: \quad \begin{cases} 0 = x_j^{min} - x_j \\ \dot{x}_1 = 0 \\ 0 = x_2 - x_1 \end{cases} \qquad \text{if } z = -1: \quad \begin{cases} 0 = x_j^{min} - x_j \\ \dot{x}_1 = 0 \\ \dot{x}_2 = K_i \, x_k \end{cases}$$

Discrete Transitions

The tests marked (*) use $x_2$ (the open-loop integrator) to decide when the active integrator $x_1$ can be released.

if z = 0:
  if x_j > x_j^max:  z ← 2
  else if x_j < x_j^min:  z ← -2
else if z = 2:
  if K_p*x_k + x_1 < x_j^max:  z ← 1
else if z = 1:
  if K_p*x_k + x_1 > x_j^max:  z ← 2
  else if K_p*x_k + x_2 < x_j^max:  z ← 0   (*)
else if z = -2:
  if K_p*x_k + x_1 > x_j^min:  z ← -1
else if z = -1:
  if K_p*x_k + x_1 < x_j^min:  z ← -2
  else if K_p*x_k + x_2 > x_j^min:  z ← 0   (*)

Initialization

if x_j >= x_j^max:
  z ← 2;  x_1 ← x_j^max - K_p*x_k;  x_2 ← x_1
else if x_j <= x_j^min:
  z ← -2;  x_1 ← x_j^min - K_p*x_k;  x_2 ← x_1
else:
  z ← 0;  x_1 ← x_j;  x_2 ← x_1

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See Also

• Block Index, all CODEGEN blocks by category
• CODEGEN Model Examples, complete models built from these blocks
• User-Defined Models, the DSL file format