# Integrator

Integrate signal

Libraries:

## Description

The Integrator block integrates an input signal with respect to time and provides the result as an output signal.

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 several different numerical integration methods to compute the output of the block, 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, set the Initial condition source parameter to `external`. An additional input port appears on the block.

Note

If you select the Limit output parameter, the initial condition must be within the saturation limits. If the initial condition is not within the block saturation limits, the block displays an error message.

### Limiting the Integral

To limit the output signal to a specified range of values, select Limit output and specify the saturation limits. When the output reaches one of the limits, the integral action is disabled to prevent integral wind up. During a simulation, you can change the limits but not whether the output is limited. The block determines output signal value using these criteria:

• When the integral value is less than or equal to the lower saturation limit, the output signal value is the lower saturation limit.

• When the integral value is between the lower saturation limit and the upper saturation limit, the output is the integral value.

• When the integral value is greater than or equal to the upper saturation limit, the output signal value is the upper saturation limit.

To generate a signal that indicates when the state is limited by the saturation limits, select Show saturation port. A second output port appears on the block.

The saturation signal has one of three values:

• `1` — State limited by upper saturation limit

• `0` — State not limited

• `–1` — State limited by lower saturation limit

When you limit the Integrator block output, the block has three zero crossing signals: one to detect when the integral value exceeds the upper saturation limit, one to detect when the integral value is less than the lower saturation limit, and one to detect when the integral value changes from being saturated to not being saturated.

Note

By default, the Limit output parameter is enabled for the Integrator Limited block, with the Upper saturation limit parameter value set to `1`, and the Lower saturation limit parameter value set to `0`.

### Wrapping Cyclic States

Several physical phenomena, such as oscillators and machinery that exhibits rotational movement, are cyclic, periodic, or rotary in nature. Modeling these phenomena in a Simulink block diagram involves integrating the rate of change for the periodic or cyclic signals to obtain the state of the movement. Over long simulation time spans, this approach can result in the states that represent periodic or cyclic signals integrating to large values. Computing trigonometric values, such as sine or cosine, for these signals takes longer as the values become larger due to angle reduction. As the signal values become larger, solver performance and accuracy degrades.

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.

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

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

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 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.

• 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 Loop Concepts). 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:

• Systems that use self-resetting integrators

• Passing states from one enabled subsystem to another

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.

### 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.

## Examples

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When you need an Integrator block to reset based on the block output value, you can use the state port to avoid creating an algebraic loop.

Open the model `SelfResettingIntegratorAlgLoop`. The Integrator block in the model is configured to reset when the reset signal changes from a positive value to a value of zero or a negative value. The reset signal is created by subtracting the current block output value from `1` so that the block resets itself each time the output value reaches `1`.

```mdl = "SelfResettingIntegratorAlgLoop"; open_system(mdl)```

Because the reset port has direct feedthrough, using the output signal to calculate the value of the reset signal creates an algebraic loop. The output signal value depends on the value of the reset signal and vice versa. Because of this mutual dependence, the software cannot calculate either value.

When you try to simulate the model, the software issues diagnostics about the algebraic loop. To see the algebraic loop in the model, use the `Simulink.BlockDiagram.getAlgebraicLoops` function. The Algebraic Loops viewer opens and shows that the model contains one real algebraic loop.

`Simulink.BlockDiagram.getAlgebraicLoops(mdl);`

The algebraic loop is highlighted in the model.

The Integrator block calculates the state value before the output value in each time step. Because the state value is available before the output value, you can use the state value to create a self-resetting Integrator block without creating an algebraic loop.

To see this solution, open the model `SelfResettingIntegrator`. The Integrator block has the state port enabled. Instead of using the output signal to calculate the value of the reset signal, the model uses the state value.

```mdl2 = "SelfResettingIntegrator"; open_system(mdl2)```

To verify that using the state instead of the output breaks the algebraic loop, call the `Simulink.BlockDiagram.getAlgebraicLoops` function again.

`Simulink.BlockDiagram.getAlgebraicLoops(mdl2);`
```No algebraic loops were found. ```

To see the self-resetting Integrator in action, simulate the model.

`simOut = sim(mdl2);`

To view the output and reset signals in the Scope, double-click the Scope block. The Integrator block output signal oscillates between a value of `0` and `1`. The output value resets to `0` each time the reset signal value is `0`.

You can use the state port to prevent creating an algebraic loop when you need an Integrator block in one enabled subsystem to use the state of an Integrator block in another enabled subsystem.

Open the model `EnabledSubsystemStatesAlgLoop`. The model contains two enabled subsystems, `A` and `B`. A Constant block provides the first input to each subsystem. The output from subsystem `A` is the other input for subsystem `B`. The output from subsystem `B` is the other input for subsystem `A`. A Pulse block provides the enable signal for both subsystems. The Pulse block enables subsystem `B` directly. The Pulse block output is inverted using a Logical Operator block to create the enable signal for subsystem `A`. As a result, the execution of the enabled subsystems alternates between subsystem `A` and subsystem `B` as the value of the Pulse block output changes.

```mdl = "EnabledSubsystemStatesAlgLoop"; open_system(mdl);```

The enabled subsystems each contain an Integrator block that uses an input port to provide the initial condition for the block. The Enable block in each subsystem is configured to reset the states each time the subsystem executes. When the state of the Integrator block resets, the initial value comes from the initial condition input port, which is connected to the output of the Integrator block in the other subsystem. Passing the state from one subsystem to another allows for continual integration of the signal while alternating execution between the subsystems.

However, to compute the output value from subsystem `B`, the solver needs the output from subsystem `A` and vice versa. Connecting the output of one Integrator block to the input of the other creates an algebraic loop. To see the loop in the model, use the `Simulink.BlockDiagram.getAlgebraicLoops` function. The Algebraic Loops viewer opens and shows that the model contains one real algebraic loop.

`Simulink.BlockDiagram.getAlgebraicLoops(mdl);`

The algebraic loop is highlighted in the model.

To avoid creating this algebraic loop, you can use the state port instead of the output port to pass the state of the Integrator block in one subsystem to the initial condition of the Integrator block in the other. The solver computes the block state value at an earlier point in each time step, so the output for subsystem `B` no longer depends on the output from subsystem `A` and vice versa.

To see this solution, open the model `EnabledSubsystemStates`. The input, enable, and output signals are the same, but the output from subsystem `A` no longer acts as a second input for subsystem `B` and vice versa.

```mdl2 = "EnabledSubsystemStates"; open_system(mdl2)```

The software does not support connecting the state port of the Integrator block to an output port on an enabled subsystem. The state port of the Integrator block in subsystem `A` is connected to the initial condition port for the Integrator block in subsystem `B` using GoTo and From blocks with global visibility. The state port of the Integrator block in subsystem `B` connects to the initial condition port for the Integrator block in subsystem `B` using the same strategy.

To verify that using the state port resolved the algebraic loop, use the `Simulink.BlockDiagram.getAlgebraicLoops` function again.

`Simulink.BlockDiagram.getAlgebraicLoops(mdl2);`
```No algebraic loops were found. ```

Simulate the model. To view the results, double-click the Scope block. The Scope window displays the enable signal and the output signals from each subsystem.

`out = sim(mdl2);`

For another example of a system that passes state values between enabled subsystems, see Building a Clutch Lock-Up Model.

## Ports

### Input

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The input signal is a real-valued scalar, vector, or matrix that this block integrates.

This port does not have direct feedthrough.

Data Types: `double`

The external reset signal is a Boolean scalar, vector, or matrix that resets the block state to the initial condition. See Resetting the State.

This port has direct feedthrough.

#### Dependencies

To enable this port, select External Reset.

Data Types: `Boolean`

The initial condition signal is a real-valued scalar, vector, or matrix that sets the initial condition for the block.

This port has direct feedthrough.

#### Dependencies

To enable this port, set the Initial Conditions parameter to `external`.

Data Types: `double`

### Output

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The output signal is a scalar, vector, or matrix with the value of the integrated input signal. The dimensions of the integrated signal match the dimensions of the input signal.

Data Types: `double`

The output saturation signal is a scalar, vector, or matrix that indicates when each element in the block state is limited by the upper or lower saturation limit. The signal value indicates whether the value is limited and which limit is applied.

• `-1` — Value limited by lower saturation limit

• `0` — Value not limited by either saturation limit

• `1` — Value limited by upper saturation limit

#### Dependencies

To enable this port, select Show saturation port.

Data Types: `double`

The block state signal is a real-valued scalar, vector, or matrix with a value that matches the block state value.

#### Dependencies

To enable this port, select Show state port.

Data Types: `double`

## Parameters

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Specify the type of trigger to use for the external reset signal.

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

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

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

#### Programmatic Use

 Block Parameter: `ExternalReset` Type: character vector , string Values: `'none'` | `'rising'` | `'falling'` | `'either'` | `'level'` | `'level hold'` Default: `'none'`

Select source of initial condition:

• `internal` — Get the initial conditions of the states from the Initial condition parameter.

• `external` — Get the initial conditions of the states from an external signal. When you select this option, an input port appears on the block.

#### Programmatic Use

 Block Parameter: `InitialConditionSource` Type: character vector, string Values: `'internal'` | `'external'` Default: `'internal'`

Set the initial state of the Integrator block.

#### 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.

#### Programmatic Use

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

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

• Selecting this check box limits the block output to a value between the Lower saturation limit and Upper saturation limit parameters.

• Clearing this check box does not limit the block output values.

#### Dependencies

Selecting this parameter enables the Lower saturation limit and Upper saturation limit parameters.

#### Programmatic Use

 Block Parameter: `LimitOutput` Type: character vector , string Values: `'off'` | `'on'` Default: `'off'`

Specify the upper limit for the integral as a scalar, vector, or matrix. You must specify a value between the Output minimum and Output maximum parameter values.

#### Dependencies

To enable this parameter, select the Limit output check box.

#### Programmatic Use

 Block Parameter: `UpperSaturationLimit` Type: character vector, string Values: scalar | vector | matrix Default: `'inf'`

Specify the lower limit for the integral as a scalar, vector, or matrix. You must specify a value between the Output minimum and Output maximum parameter values.

#### Dependencies

To enable this parameter, select the Limit output check box.

#### Programmatic Use

 Block Parameter: `LowerSaturationLimit` Type: character vector , string Values: scalar | vector | matrix Default: `'-inf'`

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.

If you specify Wrapped state upper value as `inf` and Wrapped state lower value as `-inf`, wrapping does not occur.

#### Dependencies

Selecting this parameter enables Wrapped state upper value and Wrapped state lower value parameters.

#### Programmatic Use

 Block Parameter: `WrapState` Type: character vector, string Values: `'off'` | `'on'` Default: `'off'`

Upper limit of the block output.

#### Dependencies

Selecting Wrap state enables this parameter.

#### Programmatic Use

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

Lower limit of the block output.

#### Dependencies

Selecting Wrap state enables this parameter.

#### Programmatic Use

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

Select this check box to add a saturation output port to the block. When you clear this check box, the block does not have a saturation output port.

#### Dependencies

Selecting this parameter enables a saturation output port.

#### Programmatic Use

 Block Parameter: `ShowSaturationPort` Type: character vector , string Values: `'off'` | `'on'` Default: `'off'`

Select this check box to add a state output port to the block. When you clear this check box, the block does not have a state output port.

#### Dependencies

Selecting this parameter enables a state output port.

#### Programmatic Use

 Block Parameter: `ShowStatePort` Type: character vector , string Values: `'off'` | `'on'` Default: `'off'`
• 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.

#### Programmatic Use

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

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

#### Tip

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

#### Programmatic Use

 Block Parameter: `IgnoreLimit` Type: character vector, string Values: `'off'` | `'on'` Default: `'off'`

#### Programmatic Use

 Block Parameter: `ZeroCross` Type: character vector | string Values: `'off'` | `'on'` Default: `'on'`
• 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 character vector, string, cell array, or structure.

#### Programmatic Use

 Block Parameter: `ContinuousStateAttributes` Type: character vector, string Values: `' '` | user-defined Default: `' '`

## Block Characteristics

 Data Types `double` Direct Feedthrough `yesa` Multidimensional Signals `no` Variable-Size Signals `no` Zero-Crossing Detection `yes` a Ports of this block have different direct feedthrough characteristics. See port descriptions for details.

## Version History

Introduced before R2006a