Integrator (Discrete or Continuous)

Discrete-time or continuous-time integrator

• 

Libraries:
Simscape / Electrical / Control / General Control

Description

The Integrator (Discrete or Continuous) block implements a simple integrator in conformance with IEEE 421.5-2016^[1].

You can switch between continuous and discrete implementations of the integrator using the Sample time (-1 for inherited) parameter.

Equations

Continuous

To configure the integrator for continuous time, set the Sample time (-1 for inherited) property to 0. This representation is equivalent to the continuous transfer function:

$$G(s)=\frac{1}{s}.$$

From the preceding transfer function, the integrator defining equations are:

$$\{\begin{array}{c}\dot{x}(t)=u(t)\\ y(t)=x(t)\end{array}x(0)=x_0,$$

where:

• u is the integrator input.

• x is the integrator state.

• y is the integrator output.

• t is the simulation time.

• x₀ is the initial state of the integrator.

Discrete

To configure the integrator for discrete time, set the Sample time (-1 for inherited) property to a positive, nonzero value, or to -1 to inherit the sample time from an upstream block. The discrete representation is equivalent to the transfer function:

$$G(z)=\frac{T_s}{z-1},$$

where T_s is the sample time.

This block can integrate or accumulate a signal using a forward Euler, backward Euler, or trapezoidal method. To specify the integration or accumulation method of the block, use the Integrator method parameter. (since R2025a) If you set the Integrator method parameter to Integration: Forward Euler, the block defines the integrator equations using the forward Euler method from the discrete transfer function:

$$\{\begin{array}{c}x(n+1)=x(n)+T_{s}u(n)\\ y(n)=x(n)\end{array}x(0)=x_0,$$

where:

• u is the integrator input.

• x is the integrator state.

• y is the integrator output.

• n is the simulation time step.

• x₀ is the initial state of the integrator.

Defining Initial Conditions

You can define the state initial conditions using the input port x0. The integrator state reverts to the initial condition any time it is reset.

Limiting the Integral

You can limit the integral output using one of two methods:

• Set Limit type to Anti-windup to use the anti-windup saturation method.

  The anti-windup method limits the integrator state x between the lower saturation limit A and upper saturation limit B:

  $$A<=x<=B.$$

  Because the state is limited, the output can respond immediately to a reversal of the input sign when the integral is saturated.

• Set Limit type to Windup to use the windup saturation method.

  The windup method limits the integrator output y between the lower saturation limit A and upper saturation limit B:

  $$A<=y<=B.$$

  Because the output is limited, the state can continue to grow when the integrator is saturated. As a result, the output cannot respond to a reversal of the input sign until the state has reached the limiting saturation point.

Resetting the State

You can reset the state of the integrator by passing a nonzero signal to the Reset port of the block.

Ports

Input

u — Integrator input

vector

Integrator input.

Data Types: single | double

Reset — Revert to initial state

scalar

Integrator reset. To reset the integrator state to the value of the x0 port, pass a nonzero value to this port. Alternatively, attach a zero-valued Constant block to this port to override the external reset.

Data Types: single | double

x0 — Initial state

vector

Integrator initial state. To specify the value of the state after a reset, pass a signal to this port.

Data Types: single | double

Output

y — Integrator output

vector

Integrator output.

Data Types: single | double

Parameters

Integrator method — Option to specify integration or accumulation method

Integration: Forward Euler (default) | Integration: Backward Euler | Integration: Trapezoidal | Accumulation: Forward Euler | Accumulation: Backward Euler | Accumulation: Trapezoidal

Since R2025a

Option to specify the integration or accumulation method. For more information, see the Integration and Accumulation Methods section of the Discrete-Time Integrator block.

Dependencies

To enable this parameter, set Sample time (-1 for inherited) to -1 or a positive real number.

External reset — Reset strategy

level (default) | rising | falling | either

Select the external reset strategy for the integrator:

• Select rising to reset the state when the reset signal rises from a negative or zero value to a positive value.

• Select falling to reset the state when the reset signal falls from a positive value to a zero or negative value.

• Select either to reset the state when the reset signal changes from zero to a nonzero value, from a nonzero value to zero, or changes sign.

• Select level to reset the state when the reset signal is nonzero at the current time step or changes from nonzero at the previous time step to zero at the current time step.

Limit type — Saturation strategy

Anti-windup (default) | Windup

Select the limit type of the integrator:

• Select Anti-windup to limit the state of the integrator, preventing windup.

• Select windup to limit the output of the integrator, allowing windup of the integrator state.

Upper saturation limit — State upper limit

inf (default) | real number

Integrator upper saturation limit. Set this to inf for an unsaturated upper limit, or to a finite value to saturate the integrator using the strategy set by Limit type.

Lower saturation limit — State lower limit

-inf (default) | real number

Integrator lower saturation limit. Set this to -inf for an unsaturated lower limit, or to a finite value to saturate the integrator using the strategy set by Limit type.

Sample time (-1 for inherited) — Block sample time

-1 (default) | 0 | positive scalar

Time between consecutive block executions. During execution, the block produces outputs and, if appropriate, updates its internal state. For more information, see What Is Sample Time? and Specify Sample Time.

For inherited discrete-time operation, set this parameter to -1. For discrete-time operation, set this parameter to a positive integer. For continuous-time operation, set this parameter to 0.

If this block is in a masked subsystem or a variant subsystem that supports switching between continuous operation and discrete operation, promote this parameter to ensure correct switching between the continuous and discrete implementations of the block. For more information, see Promote Block Parameters on a Mask.

References

[1] IEEE Recommended Practice for Excitation System Models for Power System Stability Studies. IEEE Std 421.5-2016. Piscataway, NJ: IEEE-SA, 2016.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b

R2025a: Specify integration or accumulation method

You can now specify the integration or accumulation method of the block by using the Integrator method parameter. This block can integrate or accumulate a signal using a forward Euler, backward Euler, or trapezoidal method.