Discrete-Time Integrator (Simulink Reference)

Simulink Reference

Discrete-Time Integrator

Perform discrete-time integration or accumulation of a signal

Library

Discrete

Description

You can use the Discrete-Time Integrator block in place of the Integrator block to create a purely discrete system.

The Discrete-Time Integrator block allows you to

Define initial conditions on the block dialog box or as input to the block.
Define an input gain (K) value.
Output the block state.
Define upper and lower limits on the integral.
Reset the state depending on an additional reset input.

These features are described below.

Integration and Accumulation Methods

The block can integrate or accumulate using the Forward Euler, Backward Euler, and Trapezoidal methods. For a given step n, Simulink updates y(n) and x(n+1). In integration mode, T is the block's sample time (delta T in the case of triggered sample time). In accumulation mode, T = 1; the block's sample time determines when the block's output is computed but not the output's value. K is the gain value. Values are clipped according to upper or lower limits.

Forward Euler method (the default), also known as Forward Rectangular, or left-hand approximation.

For this method, 1/s is approximated by T/(z-1). The resulting expression for the output of the block at step n is

```
y(n) = y(n-1) + K*T*u(n-1)
```

Let x(n+1) = x(n) + K*T*u(n). The block uses the following steps to compute its output:

Step 0:     y(0)   = x(0) = IC (clip if necessary)
            x(1)   = y(0) + K*T*u(0)

Step 1:     y(1)   = x(1)
            x(2)   = x(1) + K*T*u(1)

Step n:     y(n)   = x(n)
            x(n+1) = x(n) + K*T*u(n) (clip if necessary)

With this method, input port 1 does not have direct feedthrough.

Backward Euler method, also known as Backward Rectangular or right-hand approximation.

For this method, 1/s is approximated by T*z/(z-1). The resulting expression for the output of the block at step n is

```
y(n) = y(n-1) + K*T*u(n)
```

Let x(n) = y(n-1). The block uses the following steps to compute its output

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0)

or, depending on Use initial condition as initial and reset value for 
parameter:

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0) = x(0) + K*T*u(0)

Step 1:     y(1)   = x(1) + K*T*u(1)
            x(2)   = y(1)

Step n:     y(n)   = x(n) + K*T*u(n)
            x(n+1) = y(n)

With this method, input port 1 has direct feedthrough.

Trapezoidal method. For this method, 1/s is approximated by

```
T/2*(z+1)/(z-1)
```

When T is fixed (equal to the sampling period), let

```
x(n) = y(n-1) + K*T/2 * u(n-1)
```

The block uses the following steps to compute its output

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0) + K*T/2 * u(0)

or, depending on Use initial condition as initial and reset value for 
parameter:

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0) = x(0) + K*T/2*u(0)

Step 1:     y(1)   = x(1) + K*T/2 * u(1)
            x(2)   = y(1) + K*T/2 * u(1)

Step n:     y(n)   = x(n) + K*T/2 * u(n)
            x(n+1) = y(n) + K*T/2 * u(n)

Here, x(n+1) is the best estimate of the next output. It isn't quite the state, in the sense that x(n) != y(n).

If T is variable (i.e. obtained from the triggering times), the block uses the following algorithm to compute its outputs

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0)

or, depending on Use initial condition as initial and reset value for 
parameter:

Step 0:     y(0)   = x(0) = IC (clipped if necessary)
            x(1)   = y(0) = x(0) + K*T/2*u(0)

Step 1:     y(1)   = x(1) + T/2 * (u(1) + u(0))
            x(2)   = y(1)

Step n:     y(n)   = x(n) + T/2 * (u(n) + u(n-1))
            x(n+1) = y(n)

With this method, input port 1 has direct feedthrough.

The block reflects the selected integration or accumulation method, as this figure shows.

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 parameter 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, as shown in this figure.

Using the State Port

In two situations, you must use the state port instead of the output port:

When the output of the block is fed back into the block through the reset port or the initial condition port, causing an algebraic loop. For an example of this situation, see the bounce model.
When you want to pass the state from one conditionally executed subsystem to another, which can cause timing problems. For an example of this situation, see the clutch model.

You can correct these problems by passing the state through the state port rather than the output port. Although the values are the same, Simulink generates them at slightly different times, which protects your model from these problems. You output the block state by selecting the Show state port check box.

By default, the state port appears on the top of the block, as shown in this figure.

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. Doing so 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 output is determined as follows:

When the integral is less than or equal to the Lower saturation limit and the input is negative, 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 and the input is positive, 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, as shown in this figure.

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.

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 parameter choices. A trigger port appears below the block's input port and indicates the trigger type, as shown in this figure.

Select rising to trigger the state reset when the reset signal has a rising edge.
Select falling to trigger the state reset when the reset signal has a falling edge.
Select either to trigger the reset when either a rising or falling signal occurs.
Select level to trigger the reset and hold the output to the initial condition while the reset signal is nonzero.

The reset port has direct feedthrough. If the block output is fed back into this port, either directly or through a series of blocks with direct feedthrough, an algebraic loop results. To resolve this loop, feed the output of the block's state port into the reset port instead. To access the block's state, select the Show state port check box.

Choosing All Options

When all options are selected, the icon looks like this.

Data Type Support

The Discrete-Time Integrator block accepts real signals of any data type supported by Simulink, including fixed-point data types.

Parameters and Dialog Box

The Main pane of the Discrete-Time Integrator block dialog appears as follows:

Integrator method: Specify the integration or accumulation method.
Gain value: Specify a value by which to multiply the integrator input. Specifying a value other than 1.0 (the default) is semantically equivalent to connecting a signal to the input of the integrator via a Gain block, i.e., to; Using this parameter to specify the input gain eliminates a multiplication operation in the generated code. Realizing this benefit, however, requires that this parameter be nontunable. Accordingly, the Real-Time Workshop generates a warning during code generation if the Model Parameter Configuration dialog box for this model declares this parameter to be tunable. If you want to tune the input gain, set this parameter to 1.0 and use an external Gain block to specify the input gain.
External reset: Resets the states to their initial conditions when a trigger event occurs in the reset signal.
Initial condition source: Gets the states' initial conditions from the Initial condition parameter (internal) or from an external block (external). Simulink does not allow the initial condition of this block to be inf or NaN.
Initial condition: The states' initial conditions. This parameter is only available if the Initial condition source parameter is set to internal. Simulink does not allow the initial condition of this block to be inf or NaN.
Use initial condition as initial and reset value for: When you set this parameter to State and output,; y(0) = IC; x(0) = IC; or at reset; y(n) = IC; x(n) = IC; When you set this parameter to State only (most efficient),; x(0) = IC; or at reset; x(n) = IC
Sample time: The time interval between samples. The default is 1. In accumulation mode, the sample time specifies when the block's output is computed. See Specifying Sample Time in the online documentation for information on how to specify the sample time.
Limit output: If selected, limits the block's output to a value between the Lower saturation limit and Upper saturation limit parameters.
Upper saturation limit: The upper limit for the integral. This parameter is only available if you select the Limit output parameter.
Lower saturation limit: The lower limit for the integral. This parameter is only available if you select the Limit output parameter.
Show saturation port: If selected, adds a saturation output port to the block.
Show state port: If selected, adds an output port to the block for the block's state.
Ignore limit and reset when linearizing: Select this option to 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. This allows you to linearize a model around an operating point that causes the integrator to reset or saturate.

The Signal data types pane of the Discrete-Time Integrator block dialog appears as follows:

Output data type mode: Specify the output data type and scaling via the dialog box, or inherit the data type and scaling from the driving block or by backpropagation.
Output data type: Specify any data type, including fixed-point data types. This parameter is only visible if you select Specify via dialog for the Output data type and scaling parameter.
Output scaling value: Set the output scaling using either binary point-only or [Slope Bias] scaling. This parameter is only visible if you select Specify via dialog for the Output data type and scaling parameter.
Lock output scaling against changes by the autoscaling tool: Select to lock scaling of outputs. This parameter is only visible if you select Specify via dialog for the Output data type mode parameter.
Round integer calculations toward: Select the rounding mode for fixed-point operations.
Saturate on integer overflow: Select to have overflows saturate.

The State properties pane of this block pertains to code generation and has no effect on model simulation. See "Block States: Storing and Interfacing" in the Real-Time Workshop documentation for more information.

Characteristics

Direct Feedthrough
Yes, of the reset and external initial condition source ports. The input has direct feedthrough for every integration method except forward Euler and accumulation forward Euler.

Sample Time
Specified in the Sample time parameter

Scalar Expansion
Yes, of parameters

States
Inherited from driving block and parameter

Dimensionalized
Yes

Zero Crossing
No

Direct Feedthrough	Yes, of the reset and external initial condition source ports. The input has direct feedthrough for every integration method except forward Euler and accumulation forward Euler.
Sample Time	Specified in the Sample time parameter
Scalar Expansion	Yes, of parameters
States	Inherited from driving block and parameter
Dimensionalized	Yes
Zero Crossing	No

Discrete State-Space Discrete Transfer Fcn