varyingGoal

Variable tuning goal for gain-scheduled controllers

Syntax

VG = varyingGoal(FH,par1,par2,...)

VG = varyingGoal(___,Name,Value)

Description

When tuning fixed or gain-scheduled controllers at multiple design points (operating conditions), you might need to adjust the tuning goals as a function of operating condition, for example, to relax performance in some regions of the operating range. Use varyingGoal to construct tuning goals that depend implicitly or explicitly on the design point.

VG = varyingGoal(FH,par1,par2,...) specifies a varying goal using a template and sets of goal-parameter values. The template FH is a function handle that specifies a function, TG = FH(p1,p2,...), that evaluates to one of the TuningGoal objects. The arrays par1,par2,... specify the values of the tuning-goal parameters p1,p2,... at each design point. Use VG as you would use any TuningGoal object in an input to systune.

VG = varyingGoal(___,Name,Value)configures additional properties of the tuning goal.

Examples

Varying Tuning Goal

Create a tuning goal that specifies variable gain and phase margins across a grid of design points.

Suppose you use the following 5-by-5 grid of design points to tune your controller.

[alpha,V] = ndgrid(linspace(0,20,5),linspace(700,1300,5));

Suppose further that you have 5-by-5 arrays of target gain margins and target phase margins corresponding to each of the design points, such as the following.

[GM,PM] = ndgrid(linspace(7,20,5),linspace(45,70,5));

To enforce the specified margins at each design point, first create a template for the margins goal. The template is a function that takes gain and phase margin values and returns a TuningGoal.Margins object with those margins.

FH = @(gm,pm) TuningGoal.Margins('u',gm,pm);

Use the template and the margin arrays to create the varying goal.

VG = varyingGoal(FH,GM,PM);

To make it easier to trace which goal applies to which design point, use the SamplingGrid property to attach the design-point information to VG.

VG.SamplingGrid = struct('alpha',alpha,'V',V);

Use VG with systune as you would use any other tuning goal. Use viewGoal to visualize the tuning goal and identify design points that fail to meet the target margins.

Varying Tuning Goal With LTI Parameter

Create a tuning goal that specifies a loop shape that varies with one scheduling variable, a.

Suppose you want to specify a loop shape with a crossover frequency that varies as 2*a over three design points. To enforce this requirement, first create a template for the loop-shape goal. The template is a function that takes a numeric scalar input parameter and returns a TuningGoal.LoopShape object. The function input must be scalar, so the function constructs the LTI models representing the loop shapes.

a = [5;10;15];
s = tf('s');
FH = @(A) TuningGoal.LoopShape('u',2*A/s);

Here, 'u' is an Analysis Point in the system, the location at which you want to impose the loop-shape requirements.

Use the template and the array to create the varying goal.

VG = varyingGoal(FH,a);

Attach the design-point information to VG.

VG.SamplingGrid = struct('a',a);

Now each value of a is associated with a tuning goal that enforces the corresponding loop shape. For example, confirm that the third entry in a, a = 15, is associated with the third loop shape, 30/s.

LS3 = getGoal(VG,'index',3);
tf(LS3.LoopGain)

ans =
 
  30
  --
  s
 
Continuous-time transfer function.

Varying Tuning Goal With Frequency Focus and Loop Opening

Create a tuning goal that specifies variable gain and phase margins across a grid of design points. Configure the tuning goal to be evaluated with a loop opening at a location 'LO' and to apply only in the frequency range between 1 and 100 rad/s.

Specify the grid of design points and the corresponding grid of target gain margins and phase margins. Also, create the template function for the varying margins goal.

[alpha,V] = ndgrid(linspace(0,20,5),linspace(700,1300,5));
[GM,PM] = ndgrid(linspace(7,20,5),linspace(45,70,5));
FH = @(gm,pm) TuningGoal.Margins('u',gm,pm);

Use the template function and the margin arrays to create the varying goal. Also, use Name,Value pairs to specify:

• The location at which the loop is opened for evaluating the tuning goal (Openings property).

• The frequency range in which the varying goal applies (Focus property).

The property names and values that you specify for the underlying tuning goal are stored in the Settings property of the varying goal.

VG = varyingGoal(FH,GM,PM,'Openings','LO','Focus',[1,100])

VG = 
  varyingGoal with properties:

        Template: @(gm,pm)TuningGoal.Margins('u',gm,pm)
      Parameters: {[5x5 double]  [5x5 double]}
        Settings: {'Openings'  'LO'  'Focus'  [1 100]}
    SamplingGrid: [1x1 struct]
            Name: ''

  Variable tuning goal acting over a 5x5 grid of (gm,pm) values.

To make it easier to trace which goal applies to which design point, use the SamplingGrid property to attach the design-point information to VG.

VG.SamplingGrid = struct('alpha',alpha,'V',V);

Input Arguments

FH — Template for varying goal

function handle

Template for varying goal, specified as a function handle. FH specifies a function of one or more parameters that evaluates to one of the TuningGoal objects. For example, suppose you want to constrain the overshoot in the step response from an input r to an output y in your system, and you want to allow the constraint to vary across different design points. Specify the template as a function that returns a TuningGoal.Overshoot object. For example, you can specify FH as an anonymous function.

FH = @(os) TuningGoal.Overshoot('r','y',os);

Because TuningGoal.Overshoot has only one parameter besides the input and output signals, FH is a handle to a function of one argument. For other tuning goals, use more arguments. For example, TuningGoal.Margins has two parameters, the gain margin and phase margin. Therefore, for a variable margin goal, FH has two arguments.

FH = @(gm,pm) TuningGoal.Margins('u',gm,pm);

The template function allows great flexibility in constructing the design goals. For example, you can write a function, goalspec(a,b), that constructs the tuning goal specification as a nontrivial function of the parameters (a,b), and save the function in a MATLAB^® file. Your template function then calls goalspec as follows.

FH = @(a,b) TuningGoal.Margins('u',goalspec(a,b)); 

Similarly, if the tuning-goal parameters do not fit in numeric arrays, you can use the design-point index as input to FH. For example, suppose the gain and phase margin data is stored in a struct array, S, with fields GM and PM, you can use the following.

FH = @(idx) TuningGoal.Margins('u',S(idx).GM,S(idx).PM);

idx is an absolute index into the grid of design points.

par — Tuning goal parameters

numerical array

Tuning goal parameters, specified as a numerical array with the same dimensions as the model array used for gain-scheduled tuning. Provide an array for each parameter in the tuning goal, where each entry in the array is the parameter value you want to apply to the corresponding design point. For example, the TuningGoal.Overshoot goal has only one parameter, the maximum overshoot. Therefore, specify the parameters as follows.

par = osvals;

osvals is an array of overshoot values to enforce at each design point.

The TuningGoal.Margins goal has two parameters, the gain margin and phase margin. Therefore, for a variable margin goal, specify the parameters as follows:

par1 = GM;
par2 = PM;

Here, GM is an array of gain-margin values and PM is an array of phase-margin values to enforce at each design point.

To make the variable tuning goal inactive at a particular design point, set the corresponding entry of the par array (or arrays) to NaN.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Use Name,Value pairs to specify properties of the underlying tuning goal. For instance, suppose you want to create a varying gain goal that specifies a variable gain profile between points 'L' and 'V'. You also want to enforce the gain goal only in the frequency band [0 pi/Ts], with a loop opening at an analysis point labeled OuterLoop. You use Name,Value pairs to specify these properties for the gain goal.

 FH = @(w) TuningGoal.Gain('F','V',tf(w,[1 w]));
 VG = varyingGoal(FH,wdata,'Focus',[0 pi/Ts],'Openings','OuterLoop');

Which properties you can set depends on which of the TuningGoal objects your function FH evaluates to. For instance, for most varying tuning goals, you can set properties such as Openings, Models, and Focus. For a variable TuningGoal.Gain goal, you can also use Name,Value pairs to set properties such as Stabilize, InputScaling, and OutputScaling. See the individual TuningGoal object reference pages for a list of the properties of each tuning goal.

Output Arguments

VG — Variable tuning goal

varyingGoal object

Variable tuning goal, returned as a varyingGoal object. This object captures the tuning goal and its variation across design points in a MATLAB variable. Use VG in an input argument to systune just as you would use any TuningGoal object.

VG has the following properties.

Property	Description
Template	Template for the varying goal, stored as a function handle to a function of one or more parameters whose values change over the operating range for tuning. The initial value of this property is set by the FH input argument.
Parameters	Tuning goal parameters at each design point, stored as a cell array. Each entry in the cell array is a numeric array containing the parameter values at each design point. For example, for a variable margins goal with template FG = FH = @(gm,pm) TuningGoal.Margins('u',gm,pm); the value of this property is {GM,PM}, where GM and PM are arrays containing the desired gain and phase margins at each design point. The initial value of this property is set by the input arguments par1,par2,....
Settings	Property names and values to be applied to each goal instance in the varying goal, stored as a cell array. Default: {}
SamplingGrid	Design points, stored as a structure containing an array of values for each sampling variable. The design points need not lie on a rectangular grid and can be scattered throughout the operating range. The sizes of the arrays in the SamplingGrid and Parameters properties must match. For more information about sampling grids, see the SamplingGrid property description on the ss reference page. Default: struct with no fields
Name	Name for the variable goal. Set Name to a string to label the goal. Default: ''

Tips

• Use viewGoal to visualize tuning goals. For varying tuning goals, the tuning-goal plot generated by viewGoal lets you examine the tuning goal at each design point. For more information, see Validate Gain-Scheduled Control Systems.