# Integrator with Wrapped State (Discrete or Continuous)

Discrete-time or continuous-time integrator with wrapped state

• Library:
• Simscape / Power Systems / Simscape Components / Control / General Control

## Description

The Integrator with Wrapped State (Discrete or Continuous) block implements a wrapped state integrator in conformance with IEEE 421.5-2016[1].

Use this block to generate periodic signals such as angles or to represent a voltage-controlled oscillator. You can switch between continuous and discrete implementations of the integrator using the Sample time parameter.

### Equations

#### Continuous

To configure the integrator for continuous time, set the Sample time property to `0`. This representation is equivalent to the continuous transfer function:

`$G\left(s\right)=\frac{1}{s}.$`
From the preceeding transfer function, the integrator defining equations are:
`$\left\{\begin{array}{c}\stackrel{˙}{x}\left(t\right)=u\left(t\right)\\ y\left(t\right)=x\left(t\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\left(0\right)={x}_{0},$`
where:

• u is the integrator input.

• x is the integrator state.

• y is the integrator output.

• t is the simulation time.

• x0 is the initial state of the integrator.

#### Discrete

To configure the integrator for discrete time, set the Sample time property to a positive, nonzero value, or to `-1` to inherit the sample time from an upstream block. The discrete representation is equivalent to the transfer function:

`$G\left(z\right)=\frac{{T}_{s}}{z-1},$`
where Ts is the sample time. From the discrete transfer function, the integrator equations are defined using the forward Euler method:
`$\left\{\begin{array}{c}x\left(n+1\right)=x\left(n\right)+{T}_{s}u\left(n\right)\\ y\left(n\right)=x\left(n\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\left(0\right)={x}_{0},$`
where:

• u is the integrator input.

• x is the integrator state.

• y is the integrator output.

• n is the simulation time step.

• x0 is the initial state of the integrator.

### Defining Initial Conditions

You can define the state initial conditions using Initial condition parameter.

### Wrapping Cyclic States

The integrator wraps its state between the specified lower and upper values. This diagram shows the outputs of a wrapped and nonwrapped state integrator for a constant input.

In the diagram, the lower and upper limits are 0 and , respectively.

## Ports

### Input

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Integrator input.

Data Types: `single` | `double`

### Output

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Integrator output.

Data Types: `single` | `double`

## Parameters

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Integrator upper limit.

Integrator lower limit.

Integrator initial state.

Integrator sample time. Set this to `0` to implement a continuous integrator. To implement a discrete integrator, set this to `-1` or a positive number.

## References

[1] IEEE Recommended Practice for Excitation System Models for Power System Stability Studies. IEEE Std 421.5-2016. Piscataway, NJ: IEEE-SA, 2016.