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, including the specific methods, that can be used to perform identification. First, the classes representing model structures with identifiable parameters are given. Then, The available algorithms and corresponding functions for the identification of these model structures are presented. Finally, a table containing references to each algorithm is given.

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 rather general model structure is supported:

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

with $\xi$ being either differentiation $\frac{d}{dt}$ in CT or backward time-shift $\q^{-1}$ in DT, while $A$, $B$, $C$, $D$, and $F$ are polynomials similar to $A$ and $B$ from LPVcore.lpvio. All polynomials except $B$ are required to be monic. $e$ is a white noise signal with a variance either specified by the user or estiamted from data. Based on the general model structure, ARX, ARMAX, OE, and BJ model structures can be constructed by setting certain polynomials to identity.

• give overview of the various configurations

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'); % scheduling variable
NoiseVariance = 1E-1; % noise variance spec.
Ts = 1;  % sampling time in seconds
B = {1 - rho, rho^2}; % Creation of the coefficients of a 2nd order B-polynomial 
F = {1, rho, rho^2}; % Creation of the coefficients of a 2nd order F-polynomial
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.

MIMO

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 $N_y$-by-$N_y$ cell array of row vectors for the specification of $A$, 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.

Suggested example

• examples/T2_tutorial_sysID_IO.m

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:

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

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

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

where the white noise processes $e$ and $v$ 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);

MIMO

Suggested example

• examples/T3_tutorial_sysID_SS.m

LPVcore.lpvobf

OBF based model structures involve a fitler bank of orthonormal basis functions generated by a set of pole locations and parameter-varying expansion coefficient blocks preceeding or following the OBF bank or they are in feedback connection with it, see link for detials. In LPVcore only the first two options are implemented currently, called the Wiener and Hammerstein configurations.

• Introduce general form
• Give example

MIMO

Suggested example

• examples/T4_tutorial_advanced_sysID.m

Data generation

The defined model structures can be also used to define a data-generating system or their predictied or simulated responses with respect to given input trajectories can be easily computed.

• data generation (lsim)
• data set creation, plotting etc. (iddata)

Algorithms

Template-based model specification

As in the LPV case, not only the order of the polynomials or state-dimension of various model structure specifications can vary, but also specificaiton of the type of dependency of each coefficent, hence a flexible, yet simple way of specifying orders and coefficent depedencies via templates have been introduce. Hence, all system identification methods in LPVcore rely on the specification of a model template object. Such an object specifies involved dimensions, orders, scheduling dependence, and non-free model parameters (for graybox identification). The model template objects are standard LPVcore.lpvrep objects: algorithms estimating IO models often require a template representation in the form of an LPVcore.lpvidpoly object, while algorithms estimating SS models often require an LPVcore.lpvidss object.

The primary function of the templates is to specify structure and the values of the model parameters of the template representation are by default not used in the estiamtion algorithms. However, it can be specified by the user for the algorithm to use the template parameters for (i) intitializaiton or (ii) as fixed parameter values in case of graybox identification. The documentation for a particular algorithm indicates under which conditions the value of the model parameters is considered as an initial guess or kept fixed during estimation.

Prediction Error Methods (PEM)

Besides of linear least-squares estimation of ARX models, two types of approaches for PEM are available in LPVcore for the estimation of general LPVcore.lpvidpoly models:

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 each specific variant of the general polynomial models there is a specilised estiamtion approach:

• ARX

• ARMAX

• ARMA

• OE

• BJ

• EXAMPLES

For PEM identification of SS model, i.e., LPVcore.lpvidss model structures, the lpvssest command can be used.

• EXAMPLE

Estiamtion options

Instrumental Variable Methods (IB)

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 $C$ and $D$ 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.

Model (In)Validation

• examplain compare

References

Model type	Data type	Algoritm	Reference	Status
IO	Global	Gradient-based PEM (ARX, ARMAX, OE, BJ)	link, link	✔️
IO	Global	Pseudo-Linear Least Squares PEM (ARX, ARMAX, OE, BJ)	link	✔️
SS	Global	Gradient-based search for DDLC parametrized LPV-SS identification	link (Algorithm 7.1)	✔️
SS	Global	$\mathrm{LPV-PBSID_{opt}}$ (with kernelization)	link	✔️
LFR	Local	An $H_{\infty}$ optimization based approach for behavioral interpolation	link	✔️