System identification

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The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. [1] System identification also includes the optimal design of experiments for efficiently generating informative data for fitting such models as well as model reduction. A common approach is 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 many details of what is actually happening inside the system; this approach is called black box system identification.



A dynamical mathematical model in this context is a mathematical description of the dynamic behavior of a system or process in either the time or frequency domain. Examples include:

One of the many possible applications of system identification is in control systems. For example, it is the basis for modern data-driven control systems, in which concepts of system identification are integrated into the controller design, and lay the foundations for formal controller optimality proofs.

Input-output vs output-only

System identification techniques can utilize both input and output data (e.g. eigensystem realization algorithm) or can include only 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 experiments. [2] In recent decades, engineers have increasingly used the theory of optimal experimental design to specify inputs that yield maximally precise estimators. [3] [4]

White- and black-box

One could build a so-called white-box model based on first principles, 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 nonlinear system identification Jin et al. [9] describe greybox modeling by assuming a model structure a priori and then estimating the model parameters. Parameter estimation is relatively easy if the model form is known but this is rarely the case. Alternatively the structure or model terms for both linear and highly complex nonlinear models can be identified using NARMAX methods. [10] This approach is completely flexible and can be used with grey box models where the algorithms are primed with the known terms, or with completely black box models where the model terms are selected as part of the identification procedure. Another advantage of this approach is that the algorithms will just select linear terms if the system under study is linear, and nonlinear terms if the system is nonlinear, which allows a great deal of flexibility in the identification.

Identification for control

In control systems applications, the objective of engineers is to obtain a good performance of the closed-loop system, which is the one comprising the physical system, the feedback loop and the controller. This performance is typically achieved by designing the control law relying on a model of the system, which needs to be identified starting from experimental data. If the model identification procedure is aimed at control purposes, what really matters is not to obtain the best possible model that fits the data, as in the classical system identification approach, but to obtain a model satisfying enough for the closed-loop performance. This more recent approach is called identification for control, or I4C in short.

The idea behind I4C can be better understood by considering the following simple example. [11] Consider a system with true transfer function :

and an identified model :

From a classical system identification perspective, is not, in general, a good model for . In fact, modulus and phase of are different from those of at low frequency. What is more, while is an asymptotically stable system, is a simply stable system. However, may still be a model good enough for control purposes. In fact, if one wants to apply a purely proportional negative feedback controller with high gain , the closed-loop transfer function from the reference to the output is, for

and for

Since is very large, one has that . Thus, the two closed-loop transfer functions are indistinguishable. In conclusion, is a perfectly acceptable identified model for the true system if such feedback control law has to be applied. Whether or not a model is appropriate for control design depends not only on the plant/model mismatch, but also on the controller that will be implemented. As such, in the I4C framework, given a control performance objective, the control engineer has to design the identification phase in such a way that the performance achieved by the model-based controller on the true system is as high as possible.

Sometimes, it is even convenient to design a controller without explicitly identifying a model of the system, but directly working on experimental data. This is the case of direct data-driven control systems.

Forward model

A common understanding in Artificial Intelligence is that the controller has to generate the next move for a robot. For example, the robot starts in the maze and then the robots decides to move forward. Model predictive control determines the next action indirectly. The term “model” is referencing to a forward model which doesn't provide the correct action but simulates a scenario. [12] A forward model is equal to a physics engine used in game programming. The model takes an input and calculates the future state of the system.

The reason why dedicated forward models are constructed, is because it allows one to divide the overall control process. The first question is how to predict future states of the system. That means, to simulate a plant over a timespan for different input values. And the second task is to search for a sequence of input values which brings the plant into a goal state. This is called predictive control.

The forward model is the most important aspect of a MPC-controller. It has to be created before the solver can be realized. If it's unclear what the behavior of a system is, it's not possible to search for meaningful actions. The workflow for creating a forward model is called system identification. The idea is to formalize a system in a set of equations which will behave like the original system. [13] The error between the real system and the forward model can be measured.

There are many techniques available to create a forward model: ordinary differential equations is the classical one which is used in physics engines like Box2d. A more recent technique is a neural network for creating the forward model. [14]

See also

Related Research Articles

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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 in the natural sciences and engineering disciplines, as well as in non-physical systems such as the social sciences. The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research. Mathematical models are also used in music, linguistics, and philosophy.

A proportional–integral–derivative controller is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an error value as the difference between a desired setpoint (SP) and a measured process variable (PV) and applies a correction based on proportional, integral, and derivative terms, hence the name.

Nonlinear system System where changes of output are not proportional to changes of input

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorithms, almost all adaptive filters are digital filters. Adaptive filters are required for some applications because some parameters of the desired processing operation are not known in advance or are changing. The closed loop adaptive filter uses feedback in the form of an error signal to refine its transfer function.

Kalman filter Algorithm that estimates unknowns from a series of measurements over time

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Inverse kinematics

In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics. However, the reverse operation is, in general, much more challenging.

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Smith predictor

The Smith predictor is a type of predictive controller designed to control systems with a significant feedback time delay. The idea can be illustrated as follows.

The Eigensystem realization algorithm (ERA) is a system identification technique popular in civil engineering, in particular in structural health monitoring. ERA can be used as a modal analysis technique and generates a system realization using the time domain response (multi-)input and (multi-)output data. The ERA was proposed by Juang and Pappa and has been used for system identification of aerospace structures such as the Galileo spacecraft, turbines, civil structures and many other type of systems.

System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include any system where the inputs and outputs can be measured and include industrial processes, control systems, economic data, biology and the life sciences, medicine, social systems and many more.

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  1. Torsten, Söderström; Stoica, P. (1989). System identification. New York: Prentice Hall. ISBN   978-0138812362. OCLC   16983523.
  2. Spall, J. C. (2010), “Factorial Design for Efficient Experimentation: Generating Informative Data for System Identification,” IEEE Control Systems Magazine, vol. 30(5), pp. 38–53.
  3. Goodwin, Graham C. & Payne, Robert L. (1977). Dynamic System Identification: Experiment Design and Data Analysis. Academic Press. ISBN   978-0-12-289750-4.
  4. Walter, Éric & Pronzato, Luc (1997). Identification of Parametric Models from Experimental Data. Springer.
  5. Nielsen, Henrik Aalborg; Madsen, Henrik (December 2000). "Predicting the Heat Consumption in District Heating Systems using Meteorological Forecasts" (PDF). Lyngby: Department of Mathematical Modelling, Technical University of Denmark. S2CID   134091581. Archived from the original (PDF) on 2017-04-21.Cite journal requires |journal= (help)
  6. Nielsen, Henrik Aalborg; Madsen, Henrik (January 2006). "Modelling the heat consumption in district heating systems using a grey-box approach". Energy and Buildings. 38 (1): 63–71. doi:10.1016/j.enbuild.2005.05.002. ISSN   0378-7788.
  7. Wimpenny, J.W.T. (April 1997). "The Validity of Models". Advances in Dental Research. 11 (1): 150–159. doi:10.1177/08959374970110010601. ISSN   0895-9374. PMID   9524451. S2CID   23008333.
  8. Forssell, U.; Lindskog, P. (July 1997). "Combining Semi-Physical and Neural Network Modeling: An Example of Its Usefulness". IFAC Proceedings Volumes. 30 (11): 767–770. doi:10.1016/s1474-6670(17)42938-7. ISSN   1474-6670.
  9. Gang Jin; Sain, M.K.; Pham, K.D.; Billie, F.S.; Ramallo, J.C. (2001). Modeling MR-dampers: a nonlinear blackbox approach. Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148). IEEE. doi:10.1109/acc.2001.945582. ISBN   978-0780364950. S2CID   62730770.
  10. Billings, Stephen A (2013-07-23). Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio–Temporal Domains. doi:10.1002/9781118535561. ISBN   9781118535561.
  11. Gevers, Michel (January 2005). "Identification for Control: From the Early Achievements to the Revival of Experiment Design*". European Journal of Control. 11 (4–5): 335–352. doi:10.3166/ejc.11.335-352. ISSN   0947-3580. S2CID   13054338.
  12. Nguyen-Tuong, Duy and Peters, Jan (2011). "Model learning for robot control: a survey". Cognitive Processing. Springer. 12 (4): 319–340. doi:10.1007/s10339-011-0404-1. PMID   21487784. S2CID   8660085.CS1 maint: multiple names: authors list (link)
  13. Kopicki, Marek and Zurek, Sebastian and Stolkin, Rustam and Moerwald, Thomas and Wyatt, Jeremy L (2017). "Learning modular and transferable forward models of the motions of push manipulated objects". Autonomous Robots. Springer. 41 (5): 1061–1082. doi: 10.1007/s10514-016-9571-3 .CS1 maint: multiple names: authors list (link)
  14. Eric Wan and Antonio Baptista and Magnus Carlsson and Richard Kiebutz and Yinglong Zhang and Alexander Bogdanov (2001). Model predictive neural control of a high-fidelity helicopter model. {AIAA. American Institute of Aeronautics and Astronautics. doi:10.2514/6.2001-4164.

Further reading