TuningGoal.WeightedGain

Frequency-weighted gain constraint for control system tuning

Description

Use TuningGoal.WeightedGain to limit the weighted gain from specified inputs to outputs. The weighted gain is the maximum across frequency of the gain from input to output, multiplied by weighting functions that you specify. You can use the TuningGoal.WeightedGain tuning goal for control system tuning with tuning commands such as systune or looptune.

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

Creation

Syntax

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

Description

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

||WL(s)H(s)WR(s)||∞ < 1.

(1)

The notation ||•||_∞ denotes the maximum gain across frequency (the H_∞ norm).

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

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)||∞ < 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.

Focus — Frequency band in which tuning goal is enforced

[0,Inf] (default) | [0,pi/Ts] | row vector of two elements

Frequency band in which tuning goal is enforced, specified as a row vector of the form [min,max]. For continuous time, the default value is equal to [0,Inf]. For discrete time, the default value is equal to [0,pi/Ts], where Ts is the model sample time.

Set the Focus property to limit enforcement of the tuning goal to a particular frequency band. Express this value in the frequency units of the control system model you are tuning (rad/TimeUnit). For example, suppose Req is a tuning goal that you want to apply only between 1 and 100 rad/s. To restrict the tuning goal to this band, use the following command:

Req.Focus = [1,100];

Stabilize — Stability requirement on closed-loop dynamics

1 (true) (default) | 0 (false)

Stability requirement on closed-loop dynamics, specified as 1 (true) or 0 (false).

By default, TuningGoal.Gain imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain requirement. If stability is not required or cannot be achieved, set Stabilize to false to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, set Stabilize to false.

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 Weighted Gain of Closed-Loop System

Create a tuning goal requirement that constrains the gain of a closed-loop SISO system from its input, r, to its output, y. Weight the gain at its input by a factor of 10 and at its output by the frequency-dependent weight $$1/(s+0.01)$$.

WL = tf(1,[1 0.01]);
WR = 10;
Req = TuningGoal.WeightedGain('r','y',WL,WR);

You can use the requirement Req with systune to tune the free parameters of any control system model that has an input signal named 'r' and an output signal named 'y'.

You can then use viewGoal to validate the tuned control system against the requirement.

Constrain Weighted Gain Evaluated with a Loop Opening

Create a requirement that constrains the gain of the outer loop of the following control system, evaluated with the inner loop open.

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

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);  
T.InputName = 'r';
T.OutputName = 'y';

Create a tuning requirement that constrains the gain of this system from r to y. Weight the gain at the output by $$s/(s+0.5)$$.

WL = tf([1 0],[1 0.5]);
Req = TuningGoal.WeightedGain('r','y',WL,[]);

This requirement is equivalent to Req = TuningGoal.Gain('r','y',1/WL). However, for MIMO systems, you can use TuningGoal.WeightedGain to create channel-specific weightings that cannot be expressed as TuningGoal.Gain requirements.

Specify that the transfer function from r to y be evaluated with the outer loop open for the purpose of tuning to this constraint.

Req.Openings = 'AP1';

By default, tuning using TuningGoal.WeightedGain imposes a stability requirement as well as the gain requirement. Practically, in some control systems it is not possible to achieve a stable inner loop. When this occurs, remove the stability requirement for the inner loop by setting the Stabilize property to false.

Req.Stabilize = false;

The tuning algorithm still imposes a stability requirement on the overall tuned control system, but not on the inner loop alone.

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 the command viewGoal(Req,T).

Tips

• 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.WeightedGain, f(x) is given by:

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

T(s,x) is the closed-loop transfer function from Input to Output. $${\Vert \cdot \Vert }_{\infty }$$ denotes the H_∞ norm (see getPeakGain).

Version History

Introduced in R2012b

R2016a: Functionality moved from Robust Control Toolbox

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