TuningGoal.Passivity

Passivity constraint for control system tuning

Description

Use TuningGoal.Passivity to enforce passivity of the response between specified inputs and outputs, when using a control system tuning command such as systune. You can also use TuningGoal.Passivity to ensure a particular excess or shortage of passivity (see getPassiveIndex).

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. Equivalently, a system is passive if its frequency response is positive real, which means that for all ω > 0,

$$G\left(j\omega \right)+G{\left(j\omega \right)}^H>0$$

Creation

Syntax

Req = TuningGoal.Passivity(inputname,outputname)

Req = TuningGoal.Passivity(inputname,outputname,nu,rho)

Description

Req = TuningGoal.Passivity(inputname,outputname) creates a tuning goal for enforcing passivity of the response from the specified inputs to the specified outputs.

Req = TuningGoal.Passivity(inputname,outputname,nu,rho) creates a tuning goal for enforcing:

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

for all T > 0. This tuning goal enforces an excess of passivity at the inputs or outputs when nu > 0 or rho > 0, respectively. The tuning goal allows for a shortage of input passivity when nu < 0. See getPassiveIndex for more information about these indices.

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.

nu — Target passivity at 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:

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

for all T > 0. Equivalently, nu is the largest ν for which:

$$G\left(j\omega \right)+G{\left(j\omega \right)}^H>2\nu I$$

When you set a target nu in a TuningGoal.Passivity goal, the tuning software:

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

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

rho — Target passivity at 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:

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

for all T > 0.

When you set a target rho in a TuningGoal.Passivity goal, the tuning software:

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

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

Properties

IPX — Target passivity at inputs

scalar

Target passivity at the inputs, specified as a scalar value. This value specifies the required amount of passivity at the inputs listed in inputname. The initial value of this property is set by the input argument nu when you create the TuningGoal.Passivity goal.

OPX — Target passivity at outputs

scalar

Target passivity at the outputs, stored as a scalar value. This value specifies the required amount of passivity at the outputs listed in outputname The initial value of this property is set by the input argument rho when you create the TuningGoal.Passivity 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];

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

Passivity Requirement

Create a requirement that ensures passivity in the response from an input or analysis point 'u' to an output or analysis point 'y' in a control system.

TG = TuningGoal.Passivity('u','y');

Use viewGoal to visualize the tuning goal.

viewGoal(TG)

The requirement is satisfied when the relative passivity index R < 1 at all frequencies. The shaded area represents the region where the requirement is violated. When you use this requirement to tune a control system CL, viewGoal(TG,CL) shows R for the specified inputs and outputs on this plot, enabling you to identify frequency ranges in which the passivity requirement is violated.

Input Passivity in Specified Frequency Range

Create a requirement that ensures that the response from an input 'u' to an output 'y' is input strictly passive, with an excess of passivity of 2.

TGi = TuningGoal.Passivity('u','y',2,0);

Restrict the requirement to apply only within the frequency range between 0 and 10 rad/s.

TGi.Focus = [0 10];

Tips

• Use viewGoal to visualize this tuning goal. For enforcing passivity with nu = 0 and rho = 0, viewGoal plots the relative passivity indices as a function of frequency (see passiveplot). These are the singular values of $$\left(I-G\left(j\omega \right)\right){\left(I-G\left(j\omega \right)\right)}^{-1}$$. The transfer function G from inputname to outputname (evaluated with loops open as specified in Openings) is passive when the largest singular value is less than 1 at all frequencies.

  For nonzero nu or rho, viewGoal plots the relative index as described in Algorithms.

• This tuning goal imposes an implicit minimum-phase constraint on the transfer function G + I. The transmission zeros of G + 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 the TuningGoal.Passivity goal, for a closed-loop transfer function G(s,x) from inputname to outputname, 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 [G(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.

Version History

Introduced in R2016a