Documentation

# 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 $\sqrt{{T}_{s}}$.

## 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