TuningGoal.WeightedVariance

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.

Creation

Syntax

Req = TuningGoal.Variance(inputname,outputname,WL,WR)

Description

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.

(1)

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

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

$$\frac{1}{\sqrt{T_s}}{\Vert W_L\left(z\right)T\left(z,x\right)W_R\left(z\right)\Vert }_2<1.$$

The H₂ 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 H₂ norm, multiply either W_L or W_R by $$\sqrt{T_s}$$.

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.

WL,WR — Frequency-weighting functions

[] (default) | scalar | matrix | SISO numeric LTI model | MIMO numeric LTI model

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.

(2)

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 output channels of transfer function to constrain

scalar | matrix | SISO numeric LTI model | MIMO numeric LTI model

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 input channels of transfer function to constrain

scalar | matrix | SISO numeric LTI model | MIMO numeric LTI model

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

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

Weighted Constraint on H2 Norm

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:

$${\Vert \begin{array}{c}{\displaystyle \frac{1}{s+0.001}{T}_{re}}\\ {\displaystyle \frac{s}{0.001s+1}{T}_{ry}}\end{array}\Vert }_2<1.$$

$$T_{re}$$ 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:

$$f\left(x\right)={\Vert W_{L}T\left(s,x\right)W_R\Vert }_2.$$

T(s,x) is the closed-loop transfer function from Input to Output. $${\Vert \cdot \Vert }_2$$ denotes the H₂ norm (see norm).

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

$$f\left(x\right)=\frac{1}{\sqrt{T_s}}{\Vert W_L\left(z\right)T\left(z,x\right)W_R\left(z\right)\Vert }_2.$$

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

Version History

Introduced in R2016a

R2016a: Functionality moved from Robust Control Toolbox

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