**Block Category:** Integration

**Input:** Real scalar

**Description:** The limitedIntegrator block integrates
the input value and limits the internal state to specified upper and lower
limits. If the integral state reaches its limit, it backs off the limit as soon
as the derivative changes sign. You set the integration algorithm with the
System > System Properties command. Available algorithms are Euler,
trapezoidal, Runge Kutta 2nd and 4th orders, adaptive Runge Kutta 5th order,
adaptive Bulirsh-Stoer, and backward Euler (Stiff). You can reset the integrator
block to 0 using the System
> Reset States command.

The inputs to the block are *x*_{1}, the
derivative; *x*_{2} (U), the upper limit; and *x*_{3}
(L), the lower limit.

The limitedIntegrator block is used in the prevention of wind-up in PI and PID controllers in control applications. It is also used in kinematics, electrical circuits, process control, and fluid dynamics.

**Code Generation:** When generating code for C2000 or
ARM Cortex targets and you include one or more limitedIntegrator blocks in your
diagram and you activate Check for Performance Issues in the Code Generation
dialog box, Embed warns you that large RAM blocks are not supported for the
target. You can continue with code generation; however, the generated code may
not fit in the target RAM and the code will run slower.

When generating code for Arduino or MSP430 targets, you cannot include limitedIntegrator blocks in your diagram. Embed halts code generation and issues a message to replace the block.

**Checkpoint State:** Contains the value of the
integrator state at the checkpoint. If you have not checkpointed your simulation
using the **System > System Properties** command, the value is 0. You can
enter a value as a C
expression.

**ID:** Represents an identification number for the
block, which holds the state number that Embed assigns to the integrator. The
number of states in any block diagram equals the number of integrators. The
default is 0.

**Initial Condition:** Indicates the initial value of
the integrator. The default is 0. You can enter a value as a C expression.

**Label:** Indicates a user-defined block label that appears when
**View > Block Labels** is activated.

**1. Integration with constant limits**

Consider a system whose dynamics are given by the differential equation:

Furthermore, assume that *x *must lie in the limits 5
≤ *x* ≤ 6 and that *x*(0) = 5. This system can be realized as shown
below.

During simulation, the limitedIntegrator block limits the output to be within the upper and lower limits, namely 6 and 5, respectively.

**2. Integration with time-varying limits**

Consider a system whose dynamics are given by the differential equation:

Furthermore, assume that *x *must lie in the limits
0.2*t* ≤ *x* ≤ 2*t* and that *x*(0) = 0. This system
can be realized as:

A ramp block is used to access simulation time, t; simulation time is then used to feed the sin block, and two gain blocks, set to 2 and 0.2, to generate the time-varying upper and lower limits. During simulation, the time-varying limits and the output of the limitedIntegrator block are displayed in the plot blocks.