# TuningGoal.WeightedVariance

Frequency-weighted H_{2} norm constraint for control system
tuning

## Description

Use `TuningGoal.WeightedVariance`

to limit the
weighted *H*_{2} norm of the transfer function from
specified inputs to outputs.

The *H*_{2} norm measures:

The total energy of the impulse response, for deterministic inputs to the transfer function.

The square root of the output variance for a unit-variance white-noise input, for stochastic inputs to the transfer function. Equivalently, the

*H*_{2}norm measures the root-mean-square of the output for such input.

You can use `TuningGoal.WeightedVariance`

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

or
`looptune`

. By specifying this tuning goal, you can tune the system
response to stochastic inputs with a nonuniform spectrum such as colored noise or wind
gusts. You can also use `TuningGoal.WeightedVariance`

to
specify LQG-like performance objectives.

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

## Creation

### Description

creates a tuning goal `Req`

=
TuningGoal.Variance(`inputname`

,`outputname`

,`WL,WR`

)`Req`

. This tuning goal specifies
that the closed-loop transfer function
*H*(*s*) from the specified input to
output meets the requirement:

||W(_{L}s)H(s)W(_{R}s)||_{2}
< 1. | (1) |

_{2}denotes the

*H*

_{2}norm.

When you are tuning a discrete-time system, `Req`

imposes
the following constraint:

$$\frac{1}{\sqrt{{T}_{s}}}{\Vert {W}_{L}\left(z\right)T\left(z,x\right){W}_{R}\left(z\right)\Vert}_{2}<1.$$

The *H*_{2} norm is scaled by the
square root of the sample time *T _{s}*
to ensure consistent results with tuning in continuous time. To constrain
the true discrete-time

*H*

_{2}norm, multiply either

*W*or

_{L}*W*by $$\sqrt{{T}_{s}}$$.

_{R}### 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 weighted 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*). *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.WeightedVariance`

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

$$f\left(x\right)={\Vert {W}_{L}T\left(s,x\right){W}_{R}\Vert}_{2}.$$

*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:

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

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

*T*(

*z*,

*x*).

## Version History

**Introduced in R2016a**

## See Also

`systune`

| `looptune`

| `systune (for slTuner)`

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

(Simulink Control Design) | `TuningGoal.Gain`

| `TuningGoal.LoopShape`

| `slTuner`

(Simulink Control Design) | `norm`

| `TuningGoal.Variance`