∫

Integrator

Continuous

The Integrator block computes the time integral (accumulated sum) of an input signal. It is the inverse operation to the Derivative block and is essential for accumulating signals, computing areas, and implementing the I-term in PID controllers.

Mathematical Model

\begin{aligned} &\text{Continuous:} && y(t) = y_0 + \int_0^t u(\tau) d\tau \\ &\text{Discrete:} && y[n] = y[n-1] + \frac{\Delta t}{2}(u[n] + u[n-1]) \end{aligned}

Inputs & Outputs

Direction	ID	Label	Type	Status
→ In	`in`	Input	number	Required
→ In	`reset`	Reset	number	Optional
← Out	`out`	Output	number	Output

Parameters

Parameters	Label	Type	Default	Description
`initialCondition`	Initial Condition	number	`0`	Starting value for the integrator
`lowerLimit`	Lower Limit	number	`-1000000000000000`	Minimum output value (saturation)
`upperLimit`	Upper Limit	number	`1000000000000000`	Maximum output value (saturation)

Usage Examples

Constant Integration (1 second)

Constant input (value=2) integrated over 1 second produces linearly increasing output (0 to 2).

Velocity from Acceleration

Accelerometer signal (constant 5 m/s²) integrated to produce velocity (0 to 5 m/s over 1 sec).

Remarks & Best Practices

Inverse of Derivative: Complementary operation (Integrator + Derivative ≈ identity with delay)
Trapezoidal Rule: More accurate than Euler for smooth signals
I-term in PID: Essential for eliminating steady-state error
Saturation: Prevents integrator windup when output hits limits. Once saturated, the accumulated value is clamped; it can still decrease (or increase) when the input reverses.
Initial Condition: Set non-zero to start accumulation from different point

Related Components

Derivative: Inverse operation (time differentiation)
PIDController: Uses Integrator for I-term
Gain: Scale the integrated signal
Sum: Combine with other signals