System identification
Encyclopedia
In control engineering
, the field of system identification uses statistical methods to build mathematical model
s of dynamical system
s from measured data. System identification also includes the optimal design of experiments
for efficiently generating informative data for fitting
such models as well as model reduction.
or process in either the time or frequency domain. Examples include:
, e.g. a model for a physical process from the Newton equations
, but in many cases such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many systems and processes.
A much more common approach is therefore to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into the details of what is actually happening inside the system. This approach is called system identification. Two types of models are common in the field of system identification:
In the context of non-linear model identification Jin et al. describe greybox modeling as assuming a model structure a priori and then estimating the model parameters. This model structure can be specialized or more general so that it is applicable to a larger range of systems or devices. The parameter estimation is the tricky part and Jin et al. point out that the search for a good fit to experimental data tend to lead to an increasingly complex model. They then define a black-box model as a model which is very general and thus containing little a priori information on the problem at hand and at the same time being combined with an efficient method for parameter estimation. But as Nielsen and Madsen point out, the choice of parameter estimation can itself be problem-dependent.
) or can only include the output data (e.g. frequency domain decomposition). Typically an input-output technique would be more accurate, but the input data is not always available.
. In recent decades, engineers have increasingly used the theory of optimal experimental design
to specify inputs that yield maximally precise estimator
s.
Control engineering
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...
, the field of system identification uses statistical methods to build mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s of dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s from measured data. System identification also includes the optimal design of experiments
Design of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
for efficiently generating informative data for fitting
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
such models as well as model reduction.
Overview
A dynamical mathematical model in this context is a mathematical description of the dynamic behavior of a systemSystem
System is a set of interacting or interdependent components forming an integrated whole....
or process in either the time or frequency domain. Examples include:
- physicalPhysical systemIn physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
processes such as the movement of a falling body under the influence of gravity; - economicEconomic systemAn economic system is the combination of the various agencies, entities that provide the economic structure that defines the social community. These agencies are joined by lines of trade and exchange along which goods, money etc. are continuously flowing. An example of such a system for a closed...
processes such as stock marketStock marketA stock market or equity market is a public entity for the trading of company stock and derivatives at an agreed price; these are securities listed on a stock exchange as well as those only traded privately.The size of the world stock market was estimated at about $36.6 trillion...
s that react to external influences.
White and Black-Box
One could build a so-called white-box model based on first principlesFirst principles
In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption. In mathematics, first principles are referred to as axioms or postulates...
, e.g. a model for a physical process from the Newton equations
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
, but in many cases such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many systems and processes.
A much more common approach is therefore to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into the details of what is actually happening inside the system. This approach is called system identification. Two types of models are common in the field of system identification:
- grey box model: although the peculiarities of what is going on inside the system are not entirely known, a certain model based on both insight into the system and experimental data is constructed. This model does however still have a number of unknown free parameterParameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s which can be estimated using system identification.. One example, uses the monod saturation model for microbial growth. The model contains a simple hyperbolic relationship between substrate concentration and growth rate, but this can be justified by molecules binding to a substrate without going into detail on the types of molecules or types of binding. Grey box modeling is also known as semi-physical modeling.
- black box model: No prior model is available. Most system identification algorithms are of this type.
In the context of non-linear model identification Jin et al. describe greybox modeling as assuming a model structure a priori and then estimating the model parameters. This model structure can be specialized or more general so that it is applicable to a larger range of systems or devices. The parameter estimation is the tricky part and Jin et al. point out that the search for a good fit to experimental data tend to lead to an increasingly complex model. They then define a black-box model as a model which is very general and thus containing little a priori information on the problem at hand and at the same time being combined with an efficient method for parameter estimation. But as Nielsen and Madsen point out, the choice of parameter estimation can itself be problem-dependent.
Input-Output vs Output-Only
System identification techniques can utilize both input and output data (e.g. eigensystem realization algorithmEigensystem realization algorithm
The Eigensystem realization algorithm is a system identification technique popular in civil engineering, in particular in structural health monitoring...
) or can only include the output data (e.g. frequency domain decomposition). Typically an input-output technique would be more accurate, but the input data is not always available.
Optimal design of experiments
The quality of system identification depends on the quality of the inputs, which are under the control of the systems engineer. Therefore, systems engineers have long used the principles of the design of experimentsDesign of experiments
In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...
. In recent decades, engineers have increasingly used the theory of optimal experimental design
Optimal design
Optimal designs are a class of experimental designs that are optimal with respect to some statistical criterion.In the design of experiments for estimating statistical models, optimal designs allow parameters to be estimated without bias and with minimum-variance...
to specify inputs that yield maximally precise estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
s.
See also
- Mathematical modelMathematical modelA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
- System realization
- Parameter estimation
- Linear time-invariant system theoryLTI system theoryLinear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
- Nonlinear autoregressive exogenous modelNonlinear autoregressive exogenous modelIn time series modeling, a nonlinear autoregressive exogenous model is a nonlinear autoregressive model which has exogenous inputs. This means that the model relates the current value of a time series which one would like to explain or predict to both:...
- System dynamicsSystem dynamicsSystem dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use...
- Systems theorySystems theorySystems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can be applied to all types of systems at all nesting levels in all fields of research...
Further reading
- Daniel Graupe: Identification of Systems, Van Nostrand Reinhold, New York, 1972 (2nd ed., Krieger Publ. Co., Malabar, FL, 1976)
- Eykhoff, Pieter: System Identification - Parameter and System Estimation, John Wiley & Sons, New York, 1974. ISBN 0-471-24980-7
- Lennart LjungLennart LjungLennart Ljung is a Swedish Professor in the Chair of Control Theory at Linköping University since 1976. He is known for his pioneering research in system identification, and is regarded as a leading researcher in control theory.- Education :...
: System Identification — Theory For the User, 2nd ed, PTR Prentice HallPrentice HallPrentice Hall is a major educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 6-12 and higher-education market. Prentice Hall distributes its technical titles through the Safari...
, Upper Saddle River, N.J., 1999. - Jer-Nan Juang: Applied System Identification, Prentice Hall, Upper Saddle River, N.J., 1994.
- Oliver Nelles: Nonlinear System Identification, Springer, 2001. ISBN 3-540-67369-5
- T. Söderström, P. Stoica, System Identification, Prentice Hall, Upper Saddle River, N.J., 1989. ISBN 0-13-881236-5
- R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, IEEE Press, New York, 2001. ISBN 978-0-7803-6000-6