Integrator
Integrate signal
 Library:
Simulink / Commonly Used Blocks
Simulink / Continuous
Description
The Integrator block outputs the value of the integral of its input signal with respect to time.
Simulink^{®} treats the Integrator block as a dynamic system with one state. The block dynamics are given by:
$$\{\begin{array}{c}\dot{x}(t)=u(t)\\ y(t)=x(t)\end{array}\text{}x({t}_{0})={x}_{0}$$
where:
u is the block input.
y is the block output.
x is the block state.
x_{0} 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 DiscreteTime 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 zerocrossing 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:
$$y=\{\begin{array}{cc}x& x\in [{x}_{l},{x}_{u})\\ x({x}_{u}{x}_{l})\lfloor \frac{x{x}_{l}}{{x}_{u}{x}_{l}}\rfloor & \text{otherwise}\end{array}$$
where:
x_{l} is the lower value of the wrapped state.
x_{u} 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 zerocrossing 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.
About the State Port
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:
Selfresetting integrators (see Creating SelfResetting 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 SelfResetting 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 selfresetting 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 A  Subsystem 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 A  Subsystem 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 Building a Clutch LockUp 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 VariableStep 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.
Ports
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
.
Input
Output
Parameters
Model Examples
Block Characteristics
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 
