1.2. Classification of Identification Methods

Identification methods are a set of mathematical and computational tools used to determine the parameters of dynamic system models based on experimental data. There are several key identification methods that are applied depending on the type of system, the availability of data, and the requirements placed on the model.

Different identification methods depend substantially on the different forms in which mathematical models are represented: ordinary differential equations, difference equations, convolution equations, and so on. At the same time, none of the identification methods is universal for identifying all types of mathematical models; each is instead used in specific areas of application.

Identification methods can be classified according to various criteria [6, 8, 9]:

1. By the design of the experiment:

• •Active identification methods.
• •Passive identification methods.

2. By the availability of prior information about the control object:

• •Structural identification methods.
• •Parametric identification methods.

3. Depending on the identification criterion used:

• •The least squares method.
• •Markov estimates (or the generalized least squares method).
• •The maximum likelihood method.

4. By the implementation scheme of the identification algorithm:

• •Explicit schemes.
• •Schemes with a tunable (adjustable) model.

5. By the main systemic characteristics of the mathematical model:

• •Physical (real-world) and mathematical (symbolic).
• •One-dimensional and multidimensional.
• •Static or dynamic.
• •Linear or nonlinear.
• •Stationary or non-stationary.
• •Discrete or continuous.
• •Deterministic or stochastic.
• •Lumped and distributed.
• •Input–output characteristics and state-space description.
• •Structured and aggregated.
• •Parametric and non-parametric.

6. By the type of incoming data

• Online identification: parameters are updated in real time as data arrives (for example, adaptive methods).
• Offline identification: all data is collected in advance, and identification is carried out in a static manner over the entire data set.

7. By the type of model.

• Parametric identification: assumes that the structure of the model is known, and the task consists of finding specific values of the parameters (for example, linear regression).

• Non-parametric identification: the model is built without an explicit assumption about its form, for example, methods that use neural networks or time series.

Main identification methods:

1. The least squares method (LSM)

This is one of the most popular methods for estimating model parameters. The basic idea is to minimize the sum of squared deviations between the observed output values of the system and the values calculated by the model.

• Linear LSM: Used for linear models, where the output variable depends linearly on the parameters.
• Nonlinear LSM: Used for nonlinear systems, where the dependency is more complex and iterative optimization is required.

2. The maximum likelihood method

This method is based on a probabilistic interpretation of the data. The goal is to find the model parameters that maximize the probability of observing the available data, based on the chosen probabilistic model. This method is often used when there is noise in the data.

3. Bayesian identification methods

These methods use prior information about the system's parameters and combine it with the observed data. The Bayesian approach makes it possible to account for uncertainty in both the parameters and the data, which can be useful when there are few observations.

4. The method of moments

This method is based on equating the empirical moments (mean values, variance, etc.) of the observed data to the theoretical moments of the model. This method is widely used for simple systems and situations where the parameters can be expressed in terms of distribution moments.

5. Adaptive identification methods

These are applied in real time for systems with changing parameters. Adaptive methods continuously update the model parameters based on newly incoming data. Examples include the recursive least squares (RLS) method and adaptive filters.

6. Frequency-domain methods

This method is used to identify systems in the frequency domain. The system's frequency characteristics (for example, transfer functions) are estimated based on the system's responses to harmonic or random signals. An example is the Bode method, which constructs frequency characteristics for linear systems.

7. Identification from impulse and transient response characteristics

For systems for which data on the response to impulse or step input signals are available, special methods are used, such as the Z-transform method. This is especially useful for linear systems.

8. Methods based on neural networks and machine learning

These are used for complex and nonlinear systems, when the structure of the system is unknown or difficult to model using traditional methods. Neural networks can be trained on data and predict the system's behavior, thereby performing the identification task.

9. The subspace method

This method is applied to multidimensional systems described by states. It is based on decomposing the data into subspaces of the system's states and outputs. An example is the N4SID algorithm (Numerical Subspace State Space System Identification), which makes it possible to obtain a system model in state space.

The choice of identification method depends on the tasks at hand, the data, and the characteristics of the system. For example, for linear systems with good data, simple methods such as LSM are used, while for nonlinear or heavily noisy systems, more complex algorithms may be used, such as machine learning methods or Bayesian approaches.