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 H₂ 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

Syntax

Req = TuningGoal.Variance(inputname,outputname,maxamp)

Description

Req = TuningGoal.Variance(inputname,outputname,maxamp) creates a tuning goal that limits the noise amplification of the transfer function from 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 H₂ norm. If the plant and noise processes are truly discrete and you want to constrain the discrete-time H₂ 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

inputname — Input signals for tuning goal

character vector | cell array of character vectors

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 (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points to the slTuner interface. Use getPoints (Simulink Control Design) 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 model

  For 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 tuning goal

character vector | cell array of character vectors

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 (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points to the slTuner interface. Use getPoints (Simulink Control Design) 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 model

  For 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.

maxamp — Maximum noise amplification from inputname to outputname

positive scalar

Maximum noise amplification from inputname to outputname, specified as a positive scalar value. This value specifies the maximum value of the output variance at the signals specified in outputname, for unit-variance white noise signal at inputname. This value corresponds to the maximum H₂ norm from inputname to outputname.

When you tune a control system in discrete time, this tuning goal assumes that the physical plant and noise process are continuous, and interprets maxamp as a bound on the continuous-time H₂ norm. This ensures that continuous-time and discrete-time tuning give consistent results. If the plant and noise processes are truly discrete, and you want to bound the discrete-time H₂ norm instead, specify the value maxamp/$$\sqrt{T_s}$$. T_s is the sample time of the model you are tuning.

Properties

MaxAmplification — Maximum noise amplification

positive scalar

Maximum noise amplification, specified as a positive scalar value. This property specifies the maximum value of the output variance at the signals specified in Output, for unit-variance white noise signal at Input. This value corresponds to the maximum H₂ norm from Input to Output. The initial value of MaxAmplification is set by the maxamp input argument when you construct the tuning goal.

InputScaling — Input signal scaling

[ ] (default) | vector of positive real elements

Input signal scaling, specified as a vector of positive real elements.

Use this property to specify the relative amplitude of each entry in vector-valued input signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from Input to Output when the tuning goal is evaluated.

Suppose T(s) is the closed-loop transfer function from Input to Output. The tuning goal is evaluated for the scaled transfer function D_o^–1T(s)D_i. The diagonal matrices D_o and D_i have the OutputScaling and InputScaling values on the diagonal, respectively.

The default value, [] , means no scaling.

OutputScaling — Output signal scaling

[ ] (default) | vector of positive real elements

Output signal scaling, specified as a vector of positive real values.

Use this property to specify the relative amplitude of each entry in vector-valued output signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from Input to Output when the tuning goal is evaluated.

Suppose T(s) is the closed-loop transfer function from Input to Output. The tuning goal is evaluated for the scaled transfer function D_o^–1T(s)D_i. The diagonal matrices D_o and D_i have the OutputScaling and InputScaling values on the diagonal, respectively.

The default value, [] , means no scaling.

Input — Input signal names

cell array of character vectors

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

cell array of character vectors

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 tuning goal applies

NaN (default) | vector of scalars

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.

Openings — Feedback loops to open when evaluating the tuning goal

{} (default) | cell array of character vectors

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 (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points and loop openings to the slTuner interface. Use getPoints (Simulink Control Design) 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.

Name — Name of tuning goal

[] (default) | character vector

Name of the tuning goal, specified as a character vector.

For example, if Req is a tuning goal:

Req.Name = 'LoopReq';

Examples

Constrain Noise Amplification Evaluated with a Loop Opening

Create a requirement that constrains the amplification of the variance from the analysis point AP2 to the output y of the following control system, measured with the outer loop open.

Create a model of the system. To do so, specify and connect the numeric plant models G1 and G2, and the tunable controllers C1 and C2. Also specify and connect the AnalysisPoint blocks AP1 and AP2 that mark points of interest for analysis and tuning.

G1 = tf(10,[1 10]);
G2 = tf([1 2],[1 0.2 10]);
C1 = tunablePID('C','pi');
C2 = tunableGain('G',1);
AP1 = AnalysisPoint('AP1');
AP2 = AnalysisPoint('AP2');
T = feedback(G1*feedback(G2*C2,AP2)*C1,AP1);

Create a tuning requirement that constrains the noise amplification from the implicit input associated with the analysis point, AP2, to the output y.

Req = TuningGoal.Variance('AP2','y',0.1);

This constraint limits the amplification to a factor of 0.1.

Specify that the transfer function from AP2 to y is evaluated with the outer loop open when tuning to this constraint.

Req.Openings = {'AP1'};

Use systune to tune the free parameters of T to meet the tuning requirement specified by Req. You can then validate the tuned control system against the requirement using viewGoal(Req,T).

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 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. $${\Vert \cdot \Vert }_2$$ denotes the H₂ 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

R2016a: Functionality moved from Robust Control Toolbox

Prior to R2016a, this functionality required a Robust Control Toolbox™ license.