SparseZeroPoleTruncation

Reduce sparse models using zero-pole truncation

Since R2025a

expand all in page

    Description

    The SparseZeroPoleTruncation object uses zero-pole truncation method to obtain low-order zero-pole-gain approximations of sparse state-space models. This method computes a subset of the zeros and poles of sparse models, typically in a specific low-frequency band [0 fmax]. It can yield better low-frequency approximations than modal truncation at the expense of more computation. This method also provides direct control over the roll-off slope past the frequency fmax.

    Because this method calculates zeros for each input-output pair, it is most suitable for models with small input-output sizes. Additionally, this method is applicable only to models with a valid sparss representation.

    Creation

    The reducespec function creates a SparseZeroPoleTruncation model order reduction object when you use this syntax.

    R = reducespec(sys,"zpk")

    Here, sys is a sparse LTI model. The workflow uses this object to set up MOR tasks and store results. For the full workflow, see Task-Based Model Order Reduction Workflow.

    Properties

    expand all

    This property is read-only.

    Computed zeros in the specified frequency band, returned as an Ny-by-Nu cell array, where Ny is the number of outputs and Nu is the number of inputs in the model. Here, each entry of the cell array is a vector of real-valued or complex-valued zeros corresponding to that input-output channel.

    This property is read-only.

    Computed poles in the specified frequency band, returned as an Ny-by-Nu cell array, where Ny is the number of outputs and Nu is the number of inputs in the model. Here, each entry of the cell array is a vector of real-valued or complex-valued poles of the model.

    This property is read-only.

    Computed gains in the specified frequency band, returned as an Ny-by-Nu array, where Ny is the number of outputs and Nu is the number of inputs in the model. Here, each entry of the array is the real-valued or complex valued scalar gain corresponding to that input-output channel.

    Options for zero-pole truncation, specified as a SparseZeroPoleTruncationOptions object. Use dot notation to configure options for R. For example R.Options.Focus = [0 100].

    For more information about available options, see SparseZeroPoleTruncationOptions.

    Object Functions

    view (zpk)Plot computed poles and zeros when using zero-pole truncation method
    getrom (zpk)Obtain reduced-order models when using zero-pole truncation method

    Examples

    collapse all

    Open Live Script

    This example shows how to obtain a reduced-order model of a structural beam using the zero-pole truncation method. For this example, consider a SISO sparse state-space model of a cantilever beam. This example uses the linearized model from the Linear Analysis of Cantilever Beam example.

    Load the beam model.

    load linBeam.mat
size(sys)
    Sparse second-order model with 1 outputs, 1 inputs, and 3303 degrees of freedom.

    Analyze the frequency response of the model.

    bodeplot(sys,w)

    MATLAB figure

    To perform sparse zero-pole truncation, first create a model order reduction task using reducespec with the "zpk" method.

    R = reducespec(sys,"zpk");

    For this task, set the frequency range of focus to compute modes up to 3e5 rad/s. Doing so prevents the algorithm from computing all the poles and zeros of the sparse model, which can take a long time in some cases.

    R.Options.Focus = [0 3e5];
R.Options.Display = "off";

    Run the model reduction algorithm. This computes the derived information, which are the poles, zeros, and gains of model, stored in the object R .

    R = process(R);

    You can visualize the map of computed poles and zeros using the view function. 

    view(R)

    MATLAB figure

    Obtain the reduced zero-pole-gain approximation based on the specified frequency of focus.

    rsys = getrom(R);
order(rsys)
    ans = 
38

    Compare the response of the original and reduced models.

    figure;
opt = bodeoptions;
opt.PhaseWrapping = "on";
bodeplot(sys,rsys,"--",w,opt);

    MATLAB figure

    The reduced-order model provides a good approximation for the original sparse model in the specified range of interest.

    Open Live Script

    This example shows how to obtain a reduced model using zero-pole truncation of a sparse state-space model. This example uses a sparse model obtained from linearizing a thermal model of heat distribution in a circular cylindrical rod.

    Load the model data.

    load cylindricalRod.mat
sys = sparss(A,B,C,D,E);
w = logspace(-7,-1,20);
size(sys)
    Sparse state-space model with 3 outputs, 1 inputs, and 7522 states.

    Analyze the frequency response of the model.

    sigmaplot(sys,w)

    MATLAB figure

    Create a zero-pole truncation specification object for sys and specify the frequency band of focus. For this model, you can use a frequency range from 0 rad/s to 0.01 rad/s to obtain the low-order approximation.

    R = reducespec(sys,"zpk");
R.Options.Focus = [0 1e-2];
R.Options.Display = "off";

    Run the model reduction algorithm and obtain the reduced model.

    R = process(R);
rsys = getrom(R);
sigmaplot(sys,rsys,w)

    MATLAB figure

    This thermal model has a very steep roll-off beyond 0.001 rad/s. By default, the reduced model obtained using getrom does not provide a good match for this roll-off. To mitigate this, you can use the RollOff argument of getrom and specify a minimum roll-off value beyond the frequency band of focus. Specify a roll-off slope value of -45, which corresponds to a rate of at least –900 db/decade.

    rsys2 = getrom(R,RollOff=-45);
sigmaplot(sys,rsys2,w)

    MATLAB figure

    The reduced model now provides a much better approximation of the roll-off value.

    Algorithms

    This method uses the Krylov--Schur algorithm [1] for inverse power iterations to compute poles and zeros in the specified frequency band.

    References

    [1] Stewart, G. W. “A Krylov--Schur Algorithm for Large Eigenproblems.” SIAM Journal on Matrix Analysis and Applications 23, no. 3 (January 2002): 601–14. https://doi.org/10.1137/S0895479800371529.

    Version History

    Introduced in R2025a

    See Also

    Functions

    Objects

    Topics