# Integrator, Integrator Limited

Integrate signal

Continuous

## Description

The Integrator block outputs the value of the integral of its input signal with respect to time.

The Integrator Limited block is identical to the Integrator block with the exception that the output of the block is limited based on the upper and lower saturation limits. See Limiting the Integral for details.

Simulink® treats the Integrator block as a dynamic system with one state. The block dynamics are given by:

where:

• u is the block input.

• y is the block output.

• x is the block state.

• x0 is the initial condition of x.

While these equations define an exact relationship in continuous time, Simulink uses numerical approximation methods to evaluate them with finite precision. Simulink can use a number of different numerical integration methods to compute the Integrator block's output, each with advantages in particular applications. Use the Solver pane of the Configuration Parameters dialog box (see Solver Pane) to select the technique best suited to your application.

The selected solver computes the output of the Integrator block at the current time step, using the current input value and the value of the state at the previous time step. To support this computational model, the Integrator block saves its output at the current time step for use by the solver to compute its output at the next time step. The block also provides the solver with an initial condition for use in computing the block's initial state at the beginning of a simulation. The default value of the initial condition is 0. Use the block parameter dialog box to specify another value for the initial condition or create an initial value input port on the block.

Use the parameter dialog box to:

• Define upper and lower limits on the integral

• Create an input that resets the block's output (state) to its initial value, depending on how the input changes

• Create an optional state output so that the value of the block's output can trigger a block reset

Use the Discrete-Time Integrator block to create a purely discrete system.

### Defining Initial Conditions

You can define the initial conditions as a parameter on the block dialog box or input them from an external signal:

• To define the initial conditions as a block parameter, specify the Initial condition source parameter as `internal` and enter the value in the Initial condition field.

• To provide the initial conditions from an external source, specify the Initial condition source parameter as `external`. An additional input port appears under the block input.

 Note   If the integrator limits its output (see Limiting the Integral), the initial condition must fall inside the integrator's saturation limits. If the initial condition is outside the block saturation limits, the block displays an error message.

### Limiting the Integral

To prevent the output from exceeding specifiable levels, select the Limit output check box and enter the limits in the appropriate parameter fields. This action causes the block to function as a limited integrator. When the output reaches the limits, the integral action is turned off to prevent integral wind up. During a simulation, you can change the limits but you cannot change whether the output is limited. The block determines output as follows:

• When the integral is less than or equal to the Lower saturation limit, the output is held at the Lower saturation limit.

• When the integral is between the Lower saturation limit and the Upper saturation limit, the output is the integral.

• When the integral is greater than or equal to the Upper saturation limit, the output is held at the Upper saturation limit.

To generate a signal that indicates when the state is being limited, select the Show saturation port check box. A saturation port appears below the block output port.

The signal has one of three values:

• 1 indicates that the upper limit is being applied.

• 0 indicates that the integral is not limited.

• –1 indicates that the lower limit is being applied.

When you select this check box, the block has three zero crossings: one to detect when it enters the upper saturation limit, one to detect when it enters the lower saturation limit, and one to detect when it leaves saturation.

 Note:   For the Integrator Limited block, by default, Limit output is selected, Upper saturation limit is set to `1`, and Lower saturation limit is set to `0`.

### Wrapping Cyclic States

Several physical phenomena are cyclic, periodic, or rotary in nature. Objects or machinery that exhibit rotational movement and oscillators are examples of such phenomena.

Modeling these phenomena in Simulink involves integrating the rate of change of the periodic or cyclic signals to obtain the state of the movement.

The drawback with this approach, however, is that over long simulation time spans, the states representing periodic or cyclic signals integrate to large values. Further, computing the sine or cosine of these signals takes an increasingly large amount of time because of angle reduction. The large signals values also negatively impact solver performance and accuracy.

One approach for overcoming this drawback is to reset the angular state to `0` when it reaches 2π (or to –π when it reaches π, for numerical symmetry). This approach improves the accuracy of sine and cosine computations and reduces angle reduction time. But it also requires zero-crossing detection and introduces solver resets, which slow down the simulation for variable step solvers, particularly in large models.

To eliminate solver resets at wrap points, the Integrator block supports wrapped states that you can enable by checking Wrap state on the block parameter dialog box. When you enable Wrap state, the block icon changes to indicate that the block has wrapping states.

Simulink allows wrapping states that are bounded by upper and lower values parameters of the wrapped state. The algorithm for determining wrapping states is given by:

`$y=\left\{\begin{array}{l}x\\ x-\left({x}_{u}-{x}_{{}_{l}}\right)⌊\frac{x-{x}_{l}}{{x}_{u}-{x}_{l}}⌋\end{array}x\in \left[{x}_{l},{x}_{u}\right)\text{otherwise}$`

where:

• xl is the lower value of the wrapped state.

• xu is the upper value of the wrapped state.

• y is the output.

The support for wrapping states provides these advantages.

• It eliminates simulation instability when your model approaches large angles and large state values.

• It reduces the number of solver resets during simulation and eliminates the need for zero-crossing detection, improving simulation time.

• It eliminates large angle values, speeding up computation of trigonometric functions on angular states.

• It improves solver accuracy and performance and enables unlimited simulation time.

### Resetting the State

The block can reset its state to the specified initial condition based on an external signal. To cause the block to reset its state, select one of the External reset choices. A trigger port appears below the block's input port and indicates the trigger type.

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

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

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

• Select `level hold` to reset the state when the reset signal is nonzero at the current time step.

The reset port has direct feedthrough. If the block output feeds back into this port, either directly or through a series of blocks with direct feedthrough, an algebraic loop results (see Algebraic Loops). Use the Integrator block's state port to feed back the block's output without creating an algebraic loop.

 Note:   To be compliant with the Motor Industry Software Reliability Association (MISRA®) software standard, your model must use Boolean signals to drive the external reset ports of Integrator blocks.

Selecting the Show state port check box on the Integrator block's parameter dialog box causes an additional output port, the state port, to appear at the top of the Integrator block.

The output of the state port is the same as the output of the block's standard output port except for the following case. If the block is reset in the current time step, the output of the state port is the value that would have appeared at the block's standard output if the block had not been reset. The state port's output appears earlier in the time step than the output of the Integrator block's output port. Use the state port to avoid creating algebraic loops in these modeling scenarios:

• Self-resetting integrators (see Creating Self-Resetting Integrators)

• Handing off a state from one enabled subsystem to another (see Handing Off States Between Enabled Subsystems)

 Note   When updating a model, Simulink checks that the state port applies to one of these two scenarios. If not, an error message appears. Also, you cannot log the output of this port in a referenced model that executes in Accelerator mode. If logging is enabled for the port, Simulink generates a "signal not found" warning during execution of the referenced model.

### Creating Self-Resetting Integrators

The Integrator block's state port helps you avoid an algebraic loop when creating an integrator that resets itself based on the value of its output. Consider, for example, the following model.

This model tries to create a self-resetting integrator by feeding the integrator's output, subtracted from 1, back into the integrator's reset port. However, the model creates an algebraic loop. To compute the integrator block's output, Simulink software needs to know the value of the block's reset signal, and vice versa. Because the two values are mutually dependent, Simulink software cannot determine either. Therefore, an error message appears if you try to simulate or update this model.

The following model uses the integrator's state port to avoid the algebraic loop.

In this version, the value of the reset signal depends on the value of the state port. The value of the state port is available earlier in the current time step than the value of the integrator block's output port. Therefore, Simulink can determine whether the block needs to be reset before computing the block's output, thereby avoiding the algebraic loop.

### Handing Off States Between Enabled Subsystems

The state port helps you avoid an algebraic loop when passing a state between two enabled subsystems. Consider, for example, the following model.

The enabled subsystems, A and B, contain the following blocks:

Subsystem ASubsystem B

In this model, a constant input signal drives two enabled subsystems that integrate the signal. A pulse generator generates an enabling signal that causes execution to alternate between the two subsystems. The enable port of each subsystem is set to reset, which causes the subsystem to reset its integrator when it becomes active. Resetting the integrator causes the integrator to read the value of its initial condition port. The initial condition port of the integrator in each subsystem is connected to the output port of the integrator in the other subsystem.

This connection is intended to enable continuous integration of the input signal as execution alternates between two subsystems. However, the connection creates an algebraic loop. To compute the output of A, Simulink needs to know the output of B, and vice versa. Because the outputs are mutually dependent, Simulink cannot compute the output values. Therefore, an error message appears if you try to simulate or update this model.

The following version of the same model uses the integrator state port to avoid creating an algebraic loop when handing off the state.

The enabled subsystems, A and B, contain the following blocks:

Subsystem ASubsystem B

In this model, the initial condition of the integrator in A depends on the value of the state port of the integrator in B, and vice versa. The values of the state ports are updated earlier in the simulation time step than the values of the integrator output ports. Therefore, Simulink can compute the initial condition of either integrator without knowing the final output value of the other integrator. For another example of using the state port to hand off states between conditionally executed subsystems, see the `sldemo_clutch` model.

 Note   Simulink does not permit three or more enabled subsystems to hand off a model state. If Simulink detects that a model is handing off a state among more than two enabled subsystems, it generates an error.

### Specifying the Absolute Tolerance for the Block Outputs

By default Simulink software uses the absolute tolerance value specified in the Configuration Parameters dialog box (see Error Tolerances for Variable-Step Solvers) to compute the output of the Integrator block. If this value does not provide sufficient error control, specify a more appropriate value in the Absolute tolerance field of the Integrator block dialog box. The value that you specify is used to compute all the block outputs.

### Selecting All Options

When you select all options, the block icon looks like this.

## Data Type Support

The Integrator block accepts and outputs signals of type `double` on its data ports. The external reset port accepts signals of type `double` or `Boolean`.

## Parameters and Dialog Box

### External reset

Reset the states to their initial conditions when a trigger event occurs in the reset signal.

#### Settings

Default: `none`

`none`

Do not reset the state to initial conditions.

`rising`

Reset the state when the reset signal rises from a zero to a positive value or from a negative to a positive value.

`falling`

Reset the state when the reset signal falls from a positive value to zero or from a positive to a negative value.

`either`

Reset the state when the reset signal changes from a zero to a nonzero value or changes sign.

`level`

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.

`level hold`

Reset the state when the reset signal is nonzero at the current time step.

#### Command-Line Information

 Parameter: `ExternalReset` Type: string Value: `'none'` | `'rising'` | `'falling'` | `'either'` | `'level'` | ```'level hold'``` Default: `'none'`

### Initial condition source

Get the initial conditions of the states.

#### Settings

Default: `internal`

`internal`

Get the initial conditions of the states from the Initial condition parameter.

`external`

Get the initial conditions of the states from an external block.

#### Tips

Simulink software does not allow the initial condition of this block to be `inf` or `NaN`.

#### Dependencies

Selecting `internal` enables the Initial condition parameter.

Selecting `external` disables the Initial condition parameter.

#### Command-Line Information

 Parameter: `InitialConditionSource` Type: string Value: `'internal'` | `'external'` Default: `'internal'`

### Initial condition

Specify the states' initial conditions.

#### Settings

Default: `0`

#### Tips

Simulink software does not allow the initial condition of this block to be `inf` or `NaN`.

#### Dependencies

Setting Initial condition source to `internal` enables this parameter.

Setting Initial condition source to `external` disables this parameter.

#### Command-Line Information

 Parameter: `InitialCondition` Type: scalar or vector Value: `'0'` Default: `'0'`

### Limit output

Limit the block's output to a value between the Lower saturation limit and Upper saturation limit parameters.

#### Settings

Default: Off

On

Limit the block's output to a value between the Lower saturation limit and Upper saturation limit parameters.

Off

Do not limit the block's output to a value between the Lower saturation limit and Upper saturation limit parameters.

#### Dependencies

This parameter enables Upper saturation limit.

This parameter enables Lower saturation limit.

#### Command-Line Information

 Parameter: `LimitOutput` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Upper saturation limit

Specify the upper limit for the integral.

#### Settings

Default: `inf`

Minimum: value of Output minimum parameter

Maximum: value of Output maximum parameter

#### Dependencies

Limit output enables this parameter.

#### Command-Line Information

 Parameter: `UpperSaturationLimit` Type: scalar or vector Value: `'inf'` Default: `'inf'`

### Lower saturation limit

Specify the lower limit for the integral.

#### Settings

Default: `-inf`

Minimum: value of Output minimum parameter

Maximum: value of Output maximum parameter

#### Dependencies

Limit output enables this parameter.

#### Command-Line Information

 Parameter: `LowerSaturationLimit` Type: scalar or vector Value: `'-inf'` Default: `'-inf'`

### Wrap state

Enable wrapping of states between the Wrapped state upper value and Wrapped state lower value parameters. Enabling wrap states eliminates the need for zero-crossing detection, reduces solver resets, improves solver performance and accuracy, and increases simulation time span when modeling rotary and cyclic state trajectories.

#### Settings

Default: `off`

On

Enable wrap states between the Wrapped state upper value and Wrapped state lower value parameters.

If you specify Wrapped state upper value as `inf` and Wrapped state lower value as `-inf`, wrapping will never occur.

Off

Do not enable wrap states.

#### Dependencies

This parameter enables Wrapped state upper value.

This parameter enables Wrapped state lower value.

#### Command-Line Information

 Parameter: `WrapState` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Wrapped state upper value

Specify the upper value for the wrap state.

#### Settings

Default: `'pi'`

#### Dependencies

Wrap state enables this parameter.

#### Command-Line Information

 Parameter: `WrappedStateUpperValue` Type: scalar or vector Value: `'2*pi'` Default: `'pi'`

### Wrapped state lower value

Specify the lower value for the wrap state.

#### Settings

Default: `-pi`

#### Dependencies

Wrap state enables this parameter.

#### Command-Line Information

 Parameter: `WrappedStateLowerValue` Type: scalar or vector Value: `'0'` Default: `'-pi'`

### Show saturation port

Add a saturation output port to the block.

#### Settings

Default: Off

On

Add a saturation output port to the block.

Off

Do not add a saturation output port to the block.

#### Command-Line Information

 Parameter: `ShowSaturationPort` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Show state port

Add an output port to the block for the block's state.

#### Settings

Default: Off

On

Add an output port to the block for the block's state.

Off

Do not add an output port to the block for the block's state.

#### Command-Line Information

 Parameter: `ShowStatePort` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Absolute tolerance

Specify the absolute tolerance for computing block states.

#### Settings

Default: `auto`

• You can enter `auto`, –1, a positive real scalar or vector.

• If you enter `auto` or –1, then Simulink uses the absolute tolerance value in the Configuration Parameters dialog box (see Solver Pane) to compute block states.

• If you enter a real scalar, then that value overrides the absolute tolerance in the Configuration Parameters dialog box for computing all block states.

• If you enter a real vector, then the dimension of that vector must match the dimension of the continuous states in the block. These values override the absolute tolerance in the Configuration Parameters dialog box.

#### Command-Line Information

 Parameter: ` AbsoluteTolerance` Type: string, scalar, or vector Value: `'auto'` | `'-1'` | any positive real scalar or vector Default: ` 'auto'`

### Ignore limit and reset when linearizing

Cause Simulink linearization commands to treat this block as unresettable and as having no limits on its output, regardless of the settings of the block's reset and output limitation options.

#### Settings

Default: Off

On

Cause Simulink linearization commands to treat this block as unresettable and as having no limits on its output, regardless of the settings of the block's reset and output limitation options.

Off

Do not cause Simulink linearization commands to treat this block as unresettable and as having no limits on its output, regardless of the settings of the block's reset and output limitation options.

#### Tip

Use this check box to linearize a model around an operating point that causes the integrator to reset or saturate.

#### Command-Line Information

 Parameter: `IgnoreLimit` Type: string Value: `'off'` | `'on'` Default: `'off'`

### Enable zero-crossing detection

#### Settings

Default: On

On

Use zero crossings to detect and take a time step at any of the following events: reset, entering or leaving an upper saturation state, entering or leaving a lower saturation state.

Off

Do not use zero crossings to detect and take a time step at any of the following events: reset, entering or leaving an upper saturation state, entering or leaving a lower saturation state.

If you select this check box, Limit output, and zero-crossing detection for the model as a whole, the Integrator block uses zero crossings as described.

#### Command-Line Information

 Parameter: `ZeroCross` Type: string Value: `'off'` | `'on'` Default: `'on'`

### State Name (e.g., 'position')

Assign a unique name to each state.

#### Settings

Default: `' '`

If this field is blank, no name assignment occurs.

#### Tips

• To assign a name to a single state, enter the name between quotes, for example, `'velocity'`.

• To assign names to multiple states, enter a comma-delimited list surrounded by braces, for example, `{'a', 'b', 'c'}`. Each name must be unique.

• The state names apply only to the selected block.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB® workspace, enter the variable without quotes. A variable can be a string, cell array, or structure.

#### Command-Line Information

 Parameter: `ContinuousStateAttributes` Type: string Value: `' '` | user-defined Default: `' '`

## Examples

The following example models show how to use the Integrator block:

• `sldemo_hardstop`

• `sldemo_suspn`

• `sldemo_wheelspeed_absbrake`

## Characteristics

 Data Types Double Sample Time Continuous Direct Feedthrough Yes, of the reset and external initial condition source ports Multidimensional Signals No Variable-Size Signals No Zero-Crossing Detection Yes, if enabled and you select the Limit output check box, one for detecting reset, one each to detect upper and lower saturation limits, and one when leaving saturation Code Generation Yes