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.
Model structures
LPVcore
package pre-fixSimilar 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:
with , , , , and polynomials similar to and from LPVcore.lpvio
. All polynomials except are monic. 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
.
The expected format for the polynomials (a cell array of pmatrix
objects) differs from the convention used by idpoly
. For example, idpoly
expects an -by- cell array of row vectors for the specification of , 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:
Furthermore, the following model structure with general noise model is also supported:
where the white noise processes and 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:
- Gradient-based PEM for the identification of LPV-IO and LPV-SS model structures.
- 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:
- The output equation (i.e., the and matrices) is allowed to be scheduling dependent.
- 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:
- A cell array of LTI models.
- A cell array of values of the frozen extended scheduling signal corresponding to the operating points of the provided LTI models.
- An
LPVcore.lpvlfr
model object providing the structure of the scheduling dependence and the initial values of the coefficients. - Optionally, an
hinfstructOptions
options set, since thehinfid
method relies on thehinfstruct
command to perform the optimization. This option set is directly passed to thehinfstruct
command.
A typical workflow for identifying an LPV-LFR model from local data is as follows:
- Identify the local LTI models using, e.g., the System Identification Toolbox.
- Define the model structure in terms of the scheduling dependence and construct an
LPVcore.lpvlfr
model based on this. - Pass the identified local LTI models, along with the operating points and
LPVcore.lpvlfr
model, tohinfid
.
A model structure for LPV-LFR system representation is not yet available. Thus, setting model parameters non-free is not possible.
References
Model type | Data type | Algoritm | Reference | Status |
---|---|---|---|---|
IO | Global | Gradient-based PEM (ARX, ARMAX, OE, BJ) | 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 | (with kernelization) | link | ✔️ |
LFR | Local | An optimization based approach for behavioral interpolation | link | ✔️ |