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System identification

Overview

LPVcore contains state-of-the-art identification algorithms for several model structures. On this page, a brief overview is given of the classes and functions that can be used to perform identification. First, the classes representing model structures with identifiable parameters is given. Then, The available algorithms and corresponding functions for the identification of these model structures are presented. Finalldfy, a table containing references to each algorithm is given.

Suggested examples

Model structures

Use the LPVcore package pre-fix

Similar to the core system representation objects, all system representation objects with identifiable parameters must be prefixed with LPVcore. upon instantiation.

LPVcore.lpvidpoly

The LPVcore.lpvidpoly class represents a DT or CT polynomial model structure similar to idpoly of the System Identification Toolbox. More specifically, the following model structure is supported:

A(ρ,ξ)y^=y+v,F(ρ,ξ)y=B(ρ,ξ)u,D(ρ,ξ)v=C(ρ,ξ)e,A(\rho, \xi) \hat{y} = y + v, \\[4pt] F(\rho, \xi) y = B(\rho, \xi) u, \\[4pt] D(\rho, \xi) v = C(\rho, \xi) e,

with AA, BB, CC, DD, and FF polynomials similar to AA and BB from LPVcore.lpvio. All polynomials except BB are monic. ee is a white noise signal with a given variance. Based on the general model structure, ARX, ARMAX, OE, and BJ model structures can be constructed by setting certain polynomials to identity. For example, the following code snippet shows the construction of a DT LPV-OE model structure with a noise variance of 0.1 and a sampling time of 1 second:

rho = preal('rho', 'dt');
NoiseVariance = 1E-1;
Ts = 1; % seconds
B = {1 - rho, rho^2};
F = {1, rho, rho^2};
model = LPVcore.lpvidpoly([], B, [], [], F, NoiseVariance, Ts);

Many methods that work with idpoly objects have been extended to function properly for LPVcore.lpvidpoly objects. For example, the nparams function can be used to find the number of (free) model parameters:

model = LPVcore.lpvidpoly(...);
nparams(model);
nparams(model, 'free')

For a complete overview of the available functions, type doc LPVcore.lpvidpoly.

info

The expected format for the polynomials (a cell array of pmatrix objects) differs from the convention used by idpoly. For example, idpoly expects an NyN_y-by-NyN_y cell array of row vectors for the specification of AA, where each element of the row vector corresponds to a coefficient. In contrast, LPVcore.lpvidpoly does not expect the size of the cell array to reflect the number of inputs or outputs. Instead, the size of each cell array entry (pmatrix object) should be consistent with the number of inputs or outputs.

LPVcore.lpvidss

The LPVcore.lpvidss class represents an LPV-SS representation with innovation noise model similar to the idss class of the System Identification Toolbox:

ξx=A(ρ)x+B(ρ)u+K(ρ)e,y=C(ρ)x+D(ρ)u+e.\xi x = A(\rho) x + B(\rho) u + K(\rho) e, \\[4pt] y = C(\rho) x + D(\rho) u + e.

Furthermore, the following model structure with general noise model is also supported:

ξx=A(ρ)x+B(ρ)u+G(ρ)v,y=C(ρ)x+D(ρ)u+H(ρ)e,\xi x = A(\rho) x + B(\rho) u + G(\rho) v, \\[4pt] y = C(\rho) x + D(\rho)u + H(\rho) e,

where the white noise processes ee and vv are normally distributed according to a specified variance. The following code snippet constructs a DT LPVcore.lpvidss model structure with innovation noise model with variance 0.1 and sampling time 1 second:

rho = preal('rho', 'dt');
NoiseVariance = 1E-1;
Ts = 1; % seconds
A = randn(3) * rho;
B = ones(3, 1) + ones(3, 1) * rho^2;
K = ones(3, 1);
C = ones(1, 3);
D = zeros(1);
model = LPVcore.lpvidss(A, B, C, D, 'innovation', K, [], NoiseVariance, Ts);

Algorithms

Template system representations

All system identification methods in LPVcore rely on the specification of a template system representation object. The purpose of such an object is to specify information such as the scheduling dependence and non-free model parameters (for gray-box identification). The template system representation objects are standard LPVcore.lpvrep objects: algorithms estimating IO models often require a template system representation in the form of an LPVcore.lpvidpoly object, while algorithms estimating SS models often require an LPVcore.lpvidss object.

The values of the model parameters of the template system representation may or may not be used in the algorithm itself; this depends on the chosen initialization method. The documentation for a particular algorithm indicates under which conditions the value of the model parameters is considered as an initial guess.

Prediction Error Method (PEM)

Two types of approaches for PEM are available in LPVcore:

  1. Gradient-based PEM for the identification of LPV-IO and LPV-SS model structures.
  2. Pseudo-linear least squares PEM for the identification of LPV-IO model structures.

For LPVcore.lpvidpoly model structures, a PEM identification can be invoked using the lpvpolyest command. The estimation is configured through the use of the lpvpolyestOptions object. Both aforementioned commands function in a similar way to polyest and polyestOptions as found in the System Identification Toolbox.

For LPVcore.lpvidss model structures, a PEM identification can be invoked using the lpvssest command.

Subspace

LPVcore contains a work-in-progress (WIP) version of a subspace algorithm for the estimation of an LPVcore.lpvidss model structure with an innovation noise model. The algorithm is called LPV-PBSIDopt (original paper: link). Compared to the original paper, the version in LPVcore offers more flexibility:

  1. The output equation (i.e., the CC and DD matrices) is allowed to be scheduling dependent.
  2. Each scheduling dependent matrix can have its own number and type of basis functions.

To accomplish the additional flexibility, the implementation requires additional derivations not contained in the original paper. Therefore, a PDF file containing these derivations is included here for the interested reader:

Extensions of LPV-PBSIDopt for LPVcore (PDF)

Local LPV-LFR identification

This section is still a work-in-progress!

The local LPV-LFR identification approach is available under the function hinfid. It requires the following inputs:

  1. A cell array of LTI models.
  2. A cell array of values of the frozen extended scheduling signal corresponding to the operating points of the provided LTI models.
  3. An LPVcore.lpvlfr model object providing the structure of the scheduling dependence and the initial values of the coefficients.
  4. Optionally, an hinfstructOptions options set, since the hinfid method relies on the hinfstruct command to perform the optimization. This option set is directly passed to the hinfstruct command.

A typical workflow for identifying an LPV-LFR model from local data is as follows:

  1. Identify the local LTI models using, e.g., the System Identification Toolbox.
  2. Define the model structure in terms of the scheduling dependence and construct an LPVcore.lpvlfr model based on this.
  3. Pass the identified local LTI models, along with the operating points and LPVcore.lpvlfr model, to hinfid.
info

A model structure for LPV-LFR system representation is not yet available. Thus, setting model parameters non-free is not possible.

References

Model typeData typeAlgoritmReferenceStatus
IOGlobalGradient-based PEM (ARX, ARMAX, OE, BJ)link✔️
IOGlobalPseudo-Linear Least Squares PEM (ARX, ARMAX, OE, BJ)link✔️
SSGlobalGradient-based search for DDLC parametrized LPV-SS identificationlink (Algorithm 7.1)✔️
SSGlobalLPVPBSIDopt\mathrm{LPV-PBSID_{opt}} (with kernelization)link✔️
LFRLocalAn HH_{\infty} optimization based approach for behavioral interpolationlink✔️