# TuningGoal.Variance

Noise amplification constraint for control system tuning

## Description

Use `TuningGoal.Variance`

to specify a tuning goal
that limits the noise amplification from specified inputs to outputs.

The noise amplification is defined as either:

The square root of the output variance, for a unit-variance white-noise input

The root-mean-square of the output, for a unit-variance white-noise input

The

*H*_{2}norm of the transfer function from the specified inputs to outputs, which equals the total energy of the impulse response

These definitions are different interpretations of the same quantity. `TuningGoal.Variance`

imposes the same limit on these quantities.

You can use `TuningGoal.Variance`

for control system
tuning with tuning commands, such as `systune`

or
`looptune`

. Specifying this tuning goal allows you to tune the system
response to white-noise inputs. For stochastic inputs with a nonuniform spectrum (colored
noise), use `TuningGoal.WeightedVariance`

instead.

After you create a tuning goal, you can further configure the tuning goal by setting Properties of the object.

## Creation

### Description

creates a tuning goal that limits the noise amplification of the transfer function from
`Req`

= TuningGoal.Variance(`inputname`

,`outputname`

,`maxamp`

)`inputname`

to `outputname`

to the scalar value
`maxamp`

.

When you tune a control system in discrete time, this tuning goal assumes that the
physical plant and noise process are continuous. To ensure that continuous-time and
discrete-time tuning give consistent results, `maxamp`

is interpreted
as a constraint on the continuous-time *H*_{2}
norm. If the plant and noise processes are truly discrete and you want to constrain the
discrete-time *H*_{2} norm to the value
`maxamp`

, set the third input argument to
`maxamp`

`/sqrt(Ts)`

, where `Ts`

is the sample time of the model you are tuning.

### Input Arguments

## Properties

## Examples

## Tips

When you use this tuning goal to tune a continuous-time control system,

`systune`

attempts to enforce zero feedthrough (*D*= 0) on the transfer that the tuning goal constrains. Zero feedthrough is imposed because the*H*_{2}norm, and therefore the value of the tuning goal (see Algorithms), is infinite for continuous-time systems with nonzero feedthrough.`systune`

enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term.`systune`

returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the tuning goal or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term.When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice.

To fix parameters of tunable blocks to specified values, use the

`Value`

and`Free`

properties of the block parametrization. For example, consider a tuned state-space block:`C = tunableSS('C',1,2,3);`

To enforce zero feedthrough on this block, set its

*D*matrix value to zero, and fix the parameter.C.D.Value = 0; C.D.Free = false;

For more information on fixing parameter values, see the Control Design Block reference pages, such as

`tunableSS`

.This tuning goal imposes an implicit stability constraint on the closed-loop transfer function from

`Input`

to`Output`

, evaluated with loops opened at the points identified in`Openings`

. The dynamics affected by this implicit constraint are the*stabilized dynamics*for this tuning goal. The`MinDecay`

and`MaxRadius`

options of`systuneOptions`

control the bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, use`systuneOptions`

to change these defaults.

## Algorithms

When you tune a control system using a `TuningGoal`

, the software
converts the tuning goal into a normalized scalar value
*f*(*x*). The vector *x* is the vector of
free (tunable) parameters in the control system. The software then adjusts the parameter
values to minimize *f*(*x*) or to drive
*f*(*x*) below 1 if the tuning goal is a hard
constraint.

For `TuningGoal.Variance`

, *f*(*x*) is
given by:

$$f\left(x\right)={\Vert \frac{1}{\text{MaxAmplification}}T\left(s,x\right)\Vert}_{2}.$$

*T*(*s*,*x*) is the closed-loop
transfer function from `Input`

to `Output`

. $${\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert}_{2}$$ denotes the *H*_{2} norm (see
`norm`

).

For tuning discrete-time control systems, *f*(*x*) is
given by:

$$f\left(x\right)={\Vert \frac{1}{\text{MaxAmplification}\sqrt{{T}_{s}}}T\left(z,x\right)\Vert}_{2}.$$

*T _{s}* is the sample time of the discrete-time
transfer function

*T*(

*z*,

*x*).

## Version History

**Introduced in R2016a**

## See Also

`looptune`

| `systune`

| `looptune (for slTuner)`

(Simulink Control Design) | `systune (for slTuner)`

(Simulink Control Design) | `slTuner`

(Simulink Control Design) | `viewGoal`

| `evalGoal`

| `norm`

| `TuningGoal.WeightedVariance`