# TuningGoal.WeightedVariance class

Package: TuningGoal

Frequency-weighted H2 norm constraint for control system tuning

## Description

Use `TuningGoal.WeightedVariance` to limit the weighted H2 norm of the transfer function from specified inputs to outputs. The H2 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 H2 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.

## Construction

```Req = TuningGoal.Variance(inputname,outputname,WL,WR)``` creates a tuning goal `Req`. This tuning goal specifies that the closed-loop transfer function H(s) from the specified input to output meets the requirement:

||WL(s)H(s)WR(s)||2 < 1.

The notation ||•||2 denotes the H2 norm.

When you are tuning a discrete-time system, `Req` imposes the following constraint:

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

The H2 norm is scaled by the square root of the sample time Ts to ensure consistent results with tuning in continuous time. To constrain the true discrete-time H2 norm, multiply either WL or WR by $\sqrt{{T}_{s}}$.

### Input Arguments

 `inputname` Input signals for the tuning goal, specified as a character vector or, for multiple-input tuning goals, a cell array of character vectors. If you are using the tuning goal to tune a Simulink® model of a control system, then `inputname` can include:Any model input.Any linear analysis point marked in the model.Any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. For example, suppose that the `slTuner` interface contains analysis points `u1` and `u2`. Use `'u1'` to designate that point as an input signal when creating tuning goals. Use `{'u1','u2'}` to designate a two-channel input. If you are using the tuning goal to tune a generalized state-space (`genss`) model of a control system, then `inputname` can include: Any input of the `genss` model Any `AnalysisPoint` location in the control system modelFor example, if you are tuning a control system model, `T`, then `inputname` can be any input name in `T.InputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_u`, then `inputname` can include `'AP_u'`. Use `getPoints` to get a list of analysis points available in a `genss` model.If `inputname` is an `AnalysisPoint` location of a generalized model, the input signal for the tuning goal is the implied input associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. `outputname` Output signals for the tuning goal, specified as a character vector or, for multiple-output tuning goals, a cell array of character vectors. If you are using the tuning goal to tune a Simulink model of a control system, then `outputname` can include:Any model output.Any linear analysis point marked in the model.Any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. For example, suppose that the `slTuner` interface contains analysis points `y1` and `y2`. Use `'y1'` to designate that point as an output signal when creating tuning goals. Use `{'y1','y2'}` to designate a two-channel output. If you are using the tuning goal to tune a generalized state-space (`genss`) model of a control system, then `outputname` can include: Any output of the `genss` model Any `AnalysisPoint` location in the control system modelFor example, if you are tuning a control system model, `T`, then `outputname` can be any output name in `T.OutputName`. Also, if `T` contains an `AnalysisPoint` block with a location named `AP_u`, then `outputname` can include `'AP_u'`. Use `getPoints` to get a list of analysis points available in a `genss` model.If `outputname` is an `AnalysisPoint` location of a generalized model, the output signal for the tuning goal is the implied output associated with the `AnalysisPoint` block: For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. `WL,WR` Frequency-weighting functions, specified as scalars, matrices, or SISO or MIMO numeric LTI models. The functions `WL` and `WR` provide the weights for the tuning goal. The tuning goal ensures that the gain H(s) from the specified input to output satisfies the inequality:||WL(s)H(s)WR(s)||2 < 1.`WL` provides the weighting for the output channels of H(s), and `WR` provides the weighting for the input channels. You can specify scalar weights or frequency-dependent weighting. To specify a frequency-dependent weighting, use a numeric LTI model. For example: ```WL = tf(1,[1 0.01]); WR = 10;``` If you specify MIMO weighting functions, then `inputname` and `outputname` must be vector signals. The dimensions of the vector signals must be such that the dimensions of H(s) are commensurate with the dimensions of `WL` and `WR`. For example, if you specify `WR = diag([1 10])`, then `inputname` must include two signals. Scalar values, however, automatically expand to any input or output dimension. If you are tuning in discrete time (that is, using a `genss` model or `slTuner` interface with nonzero `Ts`), you can specify the weighting functions as discrete-time models with the same `Ts`. If you specify the weighting functions in continuous time, the tuning software discretizes them. Specifying the weighting functions in discrete time gives you more control over the weighting functions near the Nyquist frequency. A value of `WL = []` or `WR = []` is interpreted as the identity.

## Properties

 `WL` Frequency-weighting function for the output channels of the transfer function to constrain, specified as a scalar, a matrix, or a SISO or MIMO numeric LTI model. The initial value of this property is set by the `WL` input argument when you construct the tuning goal. `WR` Frequency-weighting function for the input channels of the transfer function to constrain, specified as a scalar, a matrix, or a SISO or MIMO numeric LTI model. The initial value of this property is set by the `WR` input argument when you construct the tuning goal. `Input` Input signal names, specified as a cell array of character vectors that identify the inputs of the transfer function that the tuning goal constrains. The initial value of the `Input` property is set by the `inputname` input argument when you construct the tuning goal. `Output` Output signal names, specified as a cell array of character vectors that identify the outputs of the transfer function that the tuning goal constrains. The initial value of the `Output` property is set by the `outputname` input argument when you construct the tuning goal. `Models` Models to which the tuning goal applies, specified as a vector of indices. Use the `Models` property when tuning an array of control system models with `systune`, to enforce a tuning goal for a subset of models in the array. For example, suppose you want to apply the tuning goal, `Req`, to the second, third, and fourth models in a model array passed to `systune`. To restrict enforcement of the tuning goal, use the following command: `Req.Models = 2:4;` When `Models = NaN`, the tuning goal applies to all models. Default: `NaN` `Openings` Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loop-opening locations. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. If you are using the tuning goal to tune a Simulink model of a control system, then `Openings` can include any linear analysis point marked in the model, or any linear analysis point in an `slTuner` interface associated with the Simulink model. Use `addPoint` to add analysis points and loop openings to the `slTuner` interface. Use `getPoints` to get the list of analysis points available in an `slTuner` interface to your model. If you are using the tuning goal to tune a generalized state-space (`genss`) model of a control system, then `Openings` can include any `AnalysisPoint` location in the control system model. Use `getPoints` to get the list of analysis points available in the `genss` model. For example, if `Openings = {'u1','u2'}`, then the tuning goal is evaluated with loops open at analysis points `u1` and `u2`. Default: `{}` `Name` Name of the tuning goal, specified as a character vector. For example, if `Req` is a tuning goal: `Req.Name = 'LoopReq';` Default: `[]`

## Examples

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Create a constraint for a transfer function with one input, `r`, and two outputs, `e` and `y`, that limits the ${H}_{2}$ norm as follows:

`${‖\begin{array}{c}\frac{1}{s+0.001}{T}_{re}\\ \frac{s}{0.001s+1}{T}_{ry}\end{array}‖}_{2}<1.$`

${T}_{re}$ is the closed-loop transfer function from `r` to `e`, and ${T}_{ry}$ is the closed-loop transfer function from `r` to `y` .

```s = tf('s'); WL = blkdiag(1/(s+0.001),s/(0.001*s+1)); Req = TuningGoal.WeightedVariance('r',{'e','y'},WL,[]);```

## 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 H2 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:

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

T(s,x) is the closed-loop transfer function from `Input` to `Output`. ${‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}$ denotes the H2 norm (see `norm`).

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

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

Ts is the sample time of the discrete-time transfer function T(z,x).

## Compatibility Considerations

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Behavior changed in R2016a