Analyze Command-Line Batch Linearization Results Using Response Plots

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This example shows how to plot and analyze the step response for batch linearization results obtained at the command line. The term batch linearization results refers to the ss model array returned by the slLinearizer interface or linearize function. This array contains linearizations for varying parameter values, operating points, or both, such as illustrated in Batch Linearize Model for Parameter Variations at Single Operating Point and Vary Operating Points and Obtain Multiple Transfer Functions Using slLinearizer Interface. You can use the techniques illustrated in this example to analyze the frequency response, stability, or sensitivity for batch linearization results.

Batch Linearize Model

For this example, batch linearize the watertank Simulink® model. The following code linearize the model for four simulation snapshot times, t = [0 1 2 3]. At each snapshot time, the model parameters, A and b, are varied. The sample values for A are [10 20 30], and the sample values for b are [4 6]. The slLinearizer interface includes analysis points at the reference signal and plant output.

Open the model.

mdl = "watertank";
load_system(mdl)

Create an slLinearizer interface for the model.

sllin = slLinearizer(mdl,...
    [mdl + "/Desired  Water Level",mdl + "/Water-Tank System"]);

Specify the parameter grid.

[A_grid,b_grid] = ndgrid([10,20,30],[4 6]);
params(1).Name = 'A';
params(1).Value = A_grid;
params(2).Name = 'b';
params(2).Value = b_grid;

Set the parameters in the slLinearizer interface.

sllin.Parameters = params;

Set the operating point snapshot times in the slLinearizer interface.

sllin.OperatingPoints = [0,1,2,3];

Obtain the linearized model from the reference signal to the plant output.

linsys = getIOTransfer(sllin,...
    "Desired  Water Level","Water-Tank System");

Plot Step Responses of the Linearized Models

To plot the step responses of the linearized models, use the stepplot function.

stepplot(linsys)

MATLAB figure

The step plot shows the responses of every model in the array. This plot shows the range of step responses of the system in the operating ranges covered by the parameter grid and snapshot times.

To view the parameters associated with a particular response, click the response on the plot.

A data tip appears on the plot, providing information about the selected response and the related model. The last lines of the data tip show the parameter combination and simulation snapshot time that yielded this response. For example, in this previous plot, the selected response corresponds to the model obtained by setting A to 30 and b to 4. The software linearized the model after simulating the model for three time units.

View Step Response for Subset of Results

Suppose you want to view the responses for models linearized at a specific simulation snapshot time, such as two time units. Right-click the plot and select Array Selector. The Model Selector for LTI Arrays dialog box opens.

The Selection Criterion Setup area contains three columns, one for each model array dimension of linsys. The first column corresponds to the simulation snapshot time. The third entry of this column corresponds to the simulation snapshot time of two time units, because the snapshot time array was [0,1,2,3]. Select only this entry in the first column.

Click OK. The plot displays the responses for only the models linearized at two time units.

Plot Step Response for Specific Parameter Combination and Snapshot Time

Suppose you want to examine only the step response for the model obtained by linearizing the watertank model at t = 3, for A = 10 and b = 4. To do so, you can use the SamplingGrid property of linsys, which is specified as a structure. When you perform batch linearization, the software populates SamplingGrid with information regarding the variable values used to obtain the model. The variable values include each parameter that you vary and the simulation snapshot times at which you linearize the model. For example:

linsys(:,:,1).SamplingGrid

ans = struct with fields:
    Time: 0
       A: 10
       b: 4

Here, linsys(:,:,1) refers to the first model in linsys. This model was obtained at simulation time t = 0, for A = 10 and b = 4.

Use array indexing to extract from linsys the model obtained by linearizing the watertank model at t = 3, for A = 10 and b = 4.

sg = linsys.SamplingGrid;
sys = linsys(:,:,...
    sg.A == 10 & sg.b == 4 & sg.Time == 3);

The structure sg contains the sampling grid for all the models in linsys. The expression sg.A == 10 & sg.b == 4 & sg.Time == 3 returns a logical array. Each entry of this array contains the logical evaluation of the expression for corresponding entries in sg.A, sg.B, and sg.Time. sys is a model array that contains all the linsys models that satisfy the expression.

View the step response for sys.

stepplot(sys)