# Documentation

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

## Description

The Lead-Lag (Discrete or Continuous) block implements a lead-lag compensator in conformance with IEEE 421.5-2016[1].

You can switch between continuous and discrete implementations of the block using the Sample time parameter.

### Equations

#### Continuous

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

`$G\left(s\right)=\frac{{T}_{1}s+1}{{T}_{2}s+1},$`
where:

• T1 is the lead time constant.

• T2 is the lag time constant.

From the preceeding transfer function, the compensator defining equations are:

`$\left\{\begin{array}{c}\stackrel{˙}{x}\left(t\right)=\frac{1}{{T}_{2}}\left(u\left(t\right)-x\left(t\right)\right)\\ y\left(t\right)=\frac{{T}_{1}}{{T}_{2}}u\left(t\right)+\left(1-\frac{{T}_{1}}{{T}_{2}}\right)x\left(t\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(0\right)=x\left(0\right)={u}_{0},$`
where:

• u is the block input.

• x is the block state.

• y is the block output.

• t is the simulation time.

• u0 is the initial input to the block.

#### Discrete

To configure the compensator 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:

`$\frac{{T}_{1}z+\left({T}_{s}-{T}_{1}\right)}{{T}_{2}z+\left({T}_{s}-{T}_{2}\right)},$`
where:

• T1 is the lead time constant.

• T2 is the lag time constant.

• Ts is the compensator sample time.

From the discrete transfer function, the compensator equations are defined using the forward Euler method:

`$\left\{\begin{array}{c}x\left(n+1\right)=\left(1-\frac{{T}_{s}}{{T}_{2}}\right)x\left(n\right)+\left(\frac{{T}_{s}}{{T}_{2}}\right)u\left(n\right)\\ y\left(n\right)=\left(1-\frac{{T}_{1}}{{T}_{2}}\right)x\left(n\right)+\left(\frac{{T}_{1}}{{T}_{2}}\right)u\left(n\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(0\right)=x\left(0\right)={u}_{0},$`
where:

• u is the block input.

• x is the state.

• y is the block output.

• n is the simulation time step.

• u0 is the initial input to the block.

### Initial Conditions

The block sets the state and output initial conditions to the initial input.

### Limiting the Integral

Set the Upper saturation limit and Lower saturation limit parameters to use the anti-windup saturation method.

The anti-windup method limits the compensator state between the lower saturation limit A and upper saturation limit B:

`$A<=x<=B\text{\hspace{0.17em}}.$`
Because the state is limited, the output can respond immediately to a reversal of the input sign when the integral is saturated.

This block does not provide a windup saturation method. To use the windup saturation method, set the Upper saturation limit parameter to `inf`, the Lower saturation limit parameter to `-inf`, and attach a Saturation block to the output.

### Bypass Compensator Dynamics

Set the lag time constant to zero or to a value equal to that of the lead time constant to ignore the dynamics of the compensator. When bypassed, the block feeds the input directly to the output:

`$\begin{array}{c}{T}_{1}=0\\ {T}_{2}=0\\ {T}_{1}={T}_{2}\end{array}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=u\text{\hspace{0.17em}}.$`
In the continuous case, both the sample time and at least one time constant must be zero.

## Ports

### Input

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Lead-lag compensator input signal. The block uses the input initial value to determine the state initial value.

Data Types: `single` | `double`

### Output

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Data Types: `single` | `double`

## Parameters

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Compensator lead time constant. To bypass the dynamics of the compensator. set this value to `0` or to the value of the Lag time constant, T2 parameter.

Compensator lag time constant. To bypass the dynamics of the compensator. set this value to `0` or to the value of the Lead time constant, T1 parameter.

Compensator upper state limit. Set this to `inf` for an unsaturated upper limit, or to a finite value to prevent upper windup of the system's integrator.

Compensator lower state limit. Set this to `-inf` for an unsaturated lower limit, or to a finite value to prevent lower windup of the system's integrator.

Compensator sample time. Set this to `0` to implement a continuous lead-lag compensator. Set this to `-1` or a positive number to implement a discrete lead-lag compensator.

## References

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