TuningGoal.WeightedPassivity

Frequency-weighted passivity constraint

Description

A system is passive if all its I/O trajectories (u(t),y(t)) satisfy:

$${\displaystyle {\int }_0^{T}y{\left(t\right)}^{T}u\left(t\right)dt}>0,$$

for all T > 0. TuningGoal.WeightedPassivity enforces the passivity of the transfer function:

$$H\left(s\right)=W_L\left(s\right)T\left(s\right)W_R\left(s\right),$$

where T_s is a closed-loop response in the control system being tuned. W_L and W_R are weighting functions used to emphasize particular frequency bands. Use TuningGoal.WeightedPassivity with control system tuning commands such as systune.

Creation

Syntax

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

Description

Req = TuningGoal.WeightedPassivity(inputname,outputname,WL,WR) creates a tuning goal for enforcing passivity of the transfer function:

$$H\left(s\right)=W_L\left(s\right)T\left(s\right)W_R\left(s\right),$$

where T_s is the closed-loop transfer function from the specified inputs to the specified outputs. The weights WL and WR can be matrices or LTI models.

By default, the tuning goal enforces passivity of the weighted transfer function H. You can also enforce input and output passivity indices, with a specified excess or shortage of passivity. (See getPassiveIndex for more information about passivity indices.) To do so, set the IPX and OPX properties of the tuning goal. See Weighted Passivity and Input Passivity.

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 — Input and output weighting functions

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

Input and output 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 passivity of the weighted transfer function:

$$H\left(s\right)=W_L\left(s\right)T\left(s\right)W_R\left(s\right),$$

where T(s) is the transfer function from inputname to outputname. The function WL provides the weighting for the output channels of T(s), and WR provides the weighting for the input channels. You can specify:

• Scalar weighting — use a scalar or numeric matrix.

• Frequency-dependent weighting — use a SISO or MIMO numeric LTI model. For example:

  WL = tf(1,[1 0.01]);
  WR = 10;

If WL or WR is a matrix or a MIMO model, then inputname and outputname must be vector signals. The dimensions of the vector signals must be such that the dimensions of T(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 and SISO LTI models, 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.

IPX — Target passivity at the inputs

0 (default) | scalar

Target passivity at the inputs listed in inputname, specified as a scalar value. The input passivity index is defined as the largest value of ν for which the trajectories {u(t),y(t)} of the weighted transfer function H satisfy:

$${\displaystyle {\int }_0^{T}y{\left(t\right)}^{T}u\left(t\right)dt}>\nu {\displaystyle {\int }_0^{T}u{\left(t\right)}^{T}u\left(t\right)dt},$$

for all T > 0.

By default, the tuning goal enforces strict passivity of the weighted transfer function. To enforce an input passivity index with a specified excess or shortage of passivity, set the IPX property of the tuning goal. When you do so, the tuning software:

• Ensures that the weighted response is input strictly passive when IPX > 0. The magnitude of IPX sets the required excess of passivity.

• Allows the weighted response to be not input strictly passive when IPX < 0. The magnitude of IPX sets the permitted shortage of passivity.

See Weighted Passivity and Input Passivity for an example. See getPassiveIndex for more information about passivity indices.

OPX — Target passivity at the outputs

0 (default) | scalar

Target passivity at the outputs listed in outputname, specified as a scalar value. The output passivity index is defined as the largest value of ρ for which the trajectories {u(t),y(t)} of the weighted transfer function H satisfy:

$${\displaystyle {\int }_0^{T}y{\left(t\right)}^{T}u\left(t\right)dt}>\rho {\displaystyle {\int }_0^{T}y{\left(t\right)}^{T}y\left(t\right)dt},$$

for all T > 0.

By default, the tuning goal enforces strict passivity of the weighted transfer function. To enforce an output passivity index with a specified excess or shortage of passivity, set the OPX property of the tuning goal. When you do so, the tuning software:

• Ensures that the weighted response is output strictly passive when OPX > 0. The magnitude of IPX sets the required excess of passivity.

• Allows the weighted response to be not output strictly passive when OPX < 0. The magnitude of IPX sets the permitted shortage of passivity.

See Weighted Passivity and Input Passivity for an example. See getPassiveIndex for more information about passivity indices.

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];

Input — Input signal names

cell array of character vectors

Input signal names, specified as a cell array of character vectors. The input signal names specify the input locations for determining passivity, initially populated by the inputname argument.

Output — Output signal names

cell array of character vectors

Output signal names, specified as a cell array of character vectors. The output signal names specify the output locations for determining passivity, initially populated by the outputname argument.

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 Passivity and Input Passivity

Create a tuning goal that enforces the passivity of the transfer function:

$$H\left(s\right)=\left[\begin{array}{cc}1& 0\\ 0& 10\end{array}\right]T\left(s\right)\left(\frac{1}{s}\right),$$

where $$T\left(s\right)$$ is the transfer function from an input 'd' to outputs ['y';'z'] in a control system model.

WL = tf(1,[1 0]);
WR = diag([1 10]);
TG = TuningGoal.WeightedPassivity('d',{'y','z'},WL,WR);

Use TG with systune to enforce that weighted passivity requirement.

Suppose that instead of enforcing overall passivity of the weighted transfer function H, you want to ensure that H is input strictly passive with an input feedforward passivity index of at least 0.1. To do so, set the IPX property of TG.

TG.IPX = 0.1;

Tips

• Use viewGoal to visualize this tuning goal. For enforcing passivity with IPX = 0 and OPX = 0, viewGoal plots the relative passivity indices as a function of frequency (see passiveplot). These are the singular values of $$\left(I-H\left(j\omega \right)\right){\left(I-H\left(j\omega \right)\right)}^{-1}$$. The weighted transfer function H is passive when the largest singular value is less than 1 at all frequencies.

  For nonzero IPX or OPX, viewGoal plots the relative index as described in Algorithms.

• This tuning goal imposes an implicit minimum-phase constraint on the transfer function H + I, where H is the weighted closed-loop transfer function from Input to Output, evaluated with loops opened at the points identified in Openings. The transmission zeros of H + I 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), where 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.WeightedPassivity, for a closed-loop transfer function T(s,x) from inputname to outputname, and the weighted transfer function H(s,x) = WL*T(s,x)*WR, f(x) is given by:

$$f\left(x\right)=\frac{R}{1+R/R_{\max }},R_{\max }={10}^6.$$

R is the relative sector index (see getSectorIndex) of [H(s,x);I], for the sector represented by:

$$Q=\left(\begin{array}{cc}2\rho & -I\\ -I& 2\nu \end{array}\right),$$

using the values of the OPX and IPX properties for ρ and ν, respectively. R_max is fixed at 10⁶, included to avoid numerical errors for very large R.

Version History

Introduced in R2016a