Transfer Function Blocks

Linear transfer functions in the Laplace domain, several with non-windup output or rate limits.

---

tf1p

Transfer function between input and output: one time constant.

Syntax

& tf1p
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T

Internal States

None.

Discrete Variables

None.

Equations

$T \, \dot{x}_j = -x_j + G \, x_i$

Notes

The time constant $T$ can be zero, in which case the block behaves as a simple gain $x_j = G \, x_i$.

---

tf1plim

Transfer function between input and output: one time constant with non-windup limits on output.

Syntax

& tf1plim
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T
data name, parameter name or math expression for x_min
data name, parameter name or math expression for x_max

Internal States

None.

Discrete Variables

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

Equations

$$\begin{cases} T \, \dot{x}_j = G \, x_i - x_j & \text{if } z = 0 \\ 0 = x_j - x_{max} & \text{if } z = 1 \\ 0 = x_j - x_{min} & \text{if } z = -1 \end{cases}$$

Discrete Transitions

if z = 0:
  if x_j > x_max:  z ← 1
  else if x_j < x_min:  z ← -1
else if z = 1:
  if G*x_i - x_j < 0:  z ← 0
else if z = -1:
  if G*x_i - x_j > 0:  z ← 0

Initialization

if x_j >= x_max:  z ← 1
else if x_j <= x_min:  z ← -1
else:  z ← 0

Notes

The time constant $T$ can be zero. In this case the block behaves as a gain ($x_j = G x_i$) but the limits $x_{min}$ and $x_{max}$ remain in effect.

---

tf1pvlim

Transfer function between input and output: one time constant with non-windup limits on output. The limits are variables.

This block is identical in behaviour to tf1plim except that $x_{min}$ and $x_{max}$ are variable states rather than fixed data parameters.

Syntax

& tf1pvlim
name of variable x_i
name of variable x_j
name of variable x_min
name of variable x_max
data name, parameter name or math expression for G
data name, parameter name or math expression for T

Internal States

None.

Discrete Variables

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

Equations

$$\begin{cases} T \, \dot{x}_j = G \, x_i - x_j & \text{if } z = 0 \\ 0 = x_j - x_{max} & \text{if } z = 1 \\ 0 = x_j - x_{min} & \text{if } z = -1 \end{cases}$$

Discrete Transitions

if z = 0:
  if x_j > x_max:  z ← 1
  else if x_j < x_min:  z ← -1
else if z = 1:
  if G*x_i - x_j < 0:  z ← 0
else if z = -1:
  if G*x_i - x_j > 0:  z ← 0

Initialization

if x_j >= x_max:  z ← 1
else if x_j <= x_min:  z ← -1
else:  z ← 0

Notes

The time constant $T$ can be zero. In this case the block behaves as a gain ($x_j = G x_i$) but the variable limits remain in effect.

---

tf1p2lim

Transfer function between input and output: one time constant, with limits on rate of change of output and non-windup limits on output.

Syntax

& tf1p2lim
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T
data name, parameter name or math expression for x_min
data name, parameter name or math expression for x_max
data name, parameter name or math expression for dx_min/dt
data name, parameter name or math expression for dx_max/dt

Internal States

$x_1$, initialized at $\min\!\left[\max\!\left(0,\, T \, \dot{x}_{min}\right),\, T \, \dot{x}_{max}\right]$

Discrete Variables

$z_1 \in \{-1, 0, 1\}$ (rate limiter) and $z_2 \in \{-1, 0, 1\}$ (output limiter)

Equations

$$\begin{cases} 0 = x_1 - G \, x_i + x_j & \text{if } z_1 = 0 \\ 0 = x_1 - T \, \dot{x}_{max} & \text{if } z_1 = 1 \\ 0 = x_1 - T \, \dot{x}_{min} & \text{if } z_1 = -1 \end{cases}$$ $$\begin{cases} T \, \dot{x}_j = x_1 & \text{if } z_2 = 0 \\ 0 = x_j - x_{max} & \text{if } z_2 = 1 \\ 0 = x_j - x_{min} & \text{if } z_2 = -1 \end{cases}$$

$\dot{x}_{max}$ (resp. $\dot{x}_{min}$) is the maximum (resp. Minimum) rate of change of $x_j$ with time.

Discrete Transitions

# Rate limiter (z_1):
if z_1 = 0:
  if x_1 > T*dx_max:  z_1 ← 1
  else if x_1 < T*dx_min:  z_1 ← -1
else if z_1 = 1:
  if G*x_i - x_j < T*dx_max:  z_1 ← 0
else if z_1 = -1:
  if G*x_i - x_j > T*dx_min:  z_1 ← 0

# Output limiter (z_2):
if z_2 = 0:
  if x_j > x_max:  z_2 ← 1
  else if x_j < x_min:  z_2 ← -1
else if z_2 = 1:
  if x_1 < 0:  z_2 ← 0
else if z_2 = -1:
  if x_1 > 0:  z_2 ← 0

Initialization

if G*x_i - x_j > T*dx_max:  z_1 ← 1
else if G*x_i - x_j < T*dx_min:  z_1 ← -1
else:  z_1 ← 0

if x_j > x_max:  z_2 ← 1
else if x_j < x_min:  z_2 ← -1
else:  z_2 ← 0

Notes

The time constant $T$ can be zero. In this case $x_1 = 0$ throughout the simulation, the block behaves as a gain $x_j = G x_i$, and both limiters are ignored ($x_{max} \to \infty$, $x_{min} \to -\infty$, $T\dot{x}_{max} \to \infty$, $T\dot{x}_{min} \to -\infty$).

---

tf1p2limr

As tf1p2lim, with one time constant, rate limits and non-windup output limits, plus an integrator reset.

The block behaves exactly like tf1p2lim until the reset condition is met. When the parameter $p_{reset}$ becomes equal to the flag $f_{reset}$, the internal state is driven to zero and the block passes its input straight through ($x_j = x_i$) until the condition clears. The flag is intended to be set during the simulation with a CHGPRM disturbance, which makes the reset controllable from the disturbance file.

Uses the same block diagram as tf1p2lim.

Syntax

& tf1p2limr
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T
data name, parameter name or math expression for x_min
data name, parameter name or math expression for x_max
data name, parameter name or math expression for dx_min/dt
data name, parameter name or math expression for dx_max/dt
data name, parameter name or math expression for the reset parameter
data name, parameter name or math expression for the reset flag

The two trailing arguments are the only difference from tf1p2lim.

Internal States

As tf1p2lim.

Discrete Variables

As tf1p2lim, plus the reset state: the block switches to the pass-through equation $0 = x_j - x_i$ while the reset is active.

---

tf1p1z

Transfer function between input and output: one zero and one pole.

The transfer function is $\displaystyle x_j = G \frac{1 + s T_z}{1 + s T_p} x_i$.

Syntax

& tf1p1z
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T_z
data name, parameter name or math expression for T_p

Internal States

$x_1$

Discrete Variables

None.

Equations

$$\begin{aligned} \dot{x}_1 &= G \, x_i - x_j \\ 0 &= T_p \, x_j - G \, T_z \, x_i - x_1 \end{aligned}$$

Initialization

$x_1 = G \, (T_p - T_z) \, x_i$

Notes

• $T_z = 0$ is allowed: the block reduces to $T_p \dot{x}_j = -x_j + G x_i$ (i.e., tf1p).
• $T_p = 0$ is formally allowed but produces a pure differentiator $G(1 + sT_z)$; undesired transients may occur at discontinuities of $x_i$ or $\dot{x}_i$.
• $T_p = T_z = T$: $x_1$ is initialized to zero and remains zero, giving $x_j = G x_i$.

---

tf2p2z

Transfer function between input and output: two real zeros and two real poles.

The transfer function is $\displaystyle x_j = G \frac{(1 + n_1 s)(1 + n_2 s)}{(1 + d_1 s)(1 + d_2 s)} x_i$.

Syntax

& tf2p2z
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for n_1
data name, parameter name or math expression for n_2
data name, parameter name or math expression for d_1
data name, parameter name or math expression for d_2

Internal States

$x_1$ and $x_2$

Discrete Variables

None.

Equations

State-space controllable canonical form:

$$\begin{aligned} \dot{x}_1 &= x_2 \\ d_2 \, \dot{x}_2 &= -x_1 - d_1 x_2 + d_2 x_i \\ 0 &= G(d_2 - n_2) x_1 + G(n_1 d_2 - d_1 n_2) x_2 + G n_2 d_2 x_i - d_2^2 x_j \end{aligned}$$

Initialization

$x_2 = 0, \qquad x_1 = d_2 \, x_i$

Notes

Exception. If $d_2 < 0.005$, both $d_2$ and $n_2$ are set to zero and the transfer function reduces to:

$G \frac{1 + n_1 s}{1 + d_1 s}$

as in the tf1p1z block. If additionally $d_1 < 0.005$, both $d_1$ and $n_1$ are set to zero and the block becomes a simple gain $x_j = G x_i$.

---

tfder1p

Transfer function: derivative with one time constant.

The transfer function is $\displaystyle x_j = \frac{G s}{1 + sT} x_i$.

Syntax

& tfder1p
name of variable x_i
name of variable x_j
data name, parameter name or math expression for G
data name, parameter name or math expression for T

Internal States

$x_1$

Discrete Variables

None.

Equations

$$\begin{aligned} T \, \dot{x}_1 &= x_j \\ 0 &= G \, x_i - x_1 - x_j \end{aligned}$$

Initialization

$x_1 = G \, x_i$

Notes

The model allows $T = 0$. In this case $x_j = 0$ as expected; the second equation is formally retained but becomes redundant.

---

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