System Identification Lecture 1 Lazareva P.A. 2021/2022 Course structure • 8 lectures (16 academic hours) • 4 labs (16 hours) • Independent work (76 hours) • Pass/Fail grade To get a credit • Lecture attendance • Timely completion of lab work and lab reports • Be prepared to explain general issues of system identification Course content 1 General topics of system identification. Topic 1.1 The problem of identification of dynamical systems, basic concepts, classification. Topic 1.2 Mathematical models of systems and test inputs. 2 Identification Methods for linear systems. Topic 2.1 Parametric identification methods. Topic 2.2 Non-parametric identification methods. 3 Practical aspects of identification. Topic 3.1 Identification of closed-loop, nonlinear and discrete systems. Topic 3.2 Identification under a priori uncertainty. Topic 3.3 Theoretical and experimental verification of mathematical models. Introduction • A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. • Their relationship is reflected in the observed input and output signals. • System examples? In a broad sense: • A model is a description of the essential aspects of a real system, presenting information about the system in a convenient form. • The model is a simplified representation of reality. Systems Models Complex Approximate , idealization Exploring a real system The model helps to is expensive, not study the interesting convenient properties of the system Introduction • Imagine that we are studying an unknown dynamical system. • To study its properties and develop a control systemwe need to build a mathematical model. • A model is a description of the essential aspects of a real system that provides information about it in a convenient form. Examples of model types: Disturbances • block diagrams, w(t) • operator equations, Input Output • algebraic equations, signal signal System • differential equations, • • • • Markov chains, transfer functions, frequency characteristics, weight functions, etc. u(t) y(t) Introduction Models are being built Analytically (white box models) • • • • • Physical laws, known analytical dependences are used. The object has a simple structure and is comprehensively studied. Physical interpretation required. Estimate parameters of a physical model from data. Example: aircraft flight model Experimental (black box models) • • • • Easy to build and use. The measurement data of the inputs and outputs of the system are used. Less general models. Linear (linearized). Example: security pricing models for stock market Combined (experimental-analytical, gray box models) • Given generic model structure estimate parameters from data. • Example: neural network model of an engine I d e n t i f i c a t i o n Industrial Use of System ID • Process control - most developed approaches – all plants and processes are different – need to do identification, cannot spend too much time on each – industrial identification tools • Aerospace – white-box identification, specially designed programs of tests • Automotive – white-box, significant effort on model development and calibration • Disk drives – used to do thorough identification, shorter cycle time • Embedded systems – simplified models, short cycle time Identification task • System identification is a set of methods for constructing mathematical models of a dynamic system based on measurement data. • The task of identification is reduced, in general case, to the definition of a model operator that converts the input signals of the object into output signals. • Mathematically: y (t ) A( f )u (t ) Where A( f ) is a mathematical operator that is not known in advance and is to be determined; y (t ) - vector of outputs; u (t ) - control vector (input). Classification of identification methods 1. Classification according to the amount of initial information about the object under study: • Identification in a broad sense: the establishment of mathematical relationships between measured inputs and outputs given their measurements in time (structural identification). • Identification in the narrow sense: determination of the parameters of a given mathematical model based on the results of "input-output" measurements (parametric and non-parametric identification). Classification of identification methods 2. Classification by type of experiment: • Methods of active experiment. It is possible to purposefully form input signals for the object under study. • Methods of passive experiment. In this case, the researcher can observe and process the input and output signals of the object, but cannot interfere with its functioning. • A passive experiment is almost always possible, while an active experiment cannot be carried out for many objects and processes under study. Classification of identification methods 3. By the nature of the signals used: • deterministic methods - when conducting active identification based on deterministic signals; • statistical methods - during passive identification, as well as in real conditions, the signals are subject to disturbances and very noisy. 4. On the basis of time costs: • operational - current tracking of the changing characteristics of the object is provided based on recurrent estimation algorithms; • retrospective - first, the entire data array is collected, and parameter estimates are obtained after processing it. Identification procedure 1. Data collection • Measurement data requirements • Choice of input signal u(t) • Setting up an experiment 2. Selection of a set of candidate models (model structures) • A priori knowledge about the identified object, engineering experience, intuition are used • Choice of model structure (structural identification) 3. Choice of identification method and loss function • Choosing an accuracy criterion for the obtained model 4. Identification of model parameters • Determining the numerical values of model parameters from measurement data 5. Verification of the resulting model • Comparison of the output signals of the object and the model when applying the same input (new data set that wasn’t used for identification) Common Questions • What is the purpose of the model? • How to design an experiment? How much measurement data to use? • What to do with measurement noise? What data preprocessing should be done? • What to do with non-linear effects and changing parameters over time? • How to choose a model structure? • What identification method to use? • How to evaluate the quality of the resulting model? Model class definition Models are classified on the basis of: • Static • Dynamic • Linear • Nonlinear • Stationary • Non-stationary • Continuous • Discrete • One-dimensional • Multidimensional • Deterministic • Stochastic • Lumped Parameters • With distributed parameters Model class definition • Static - the behavior of the output depends only on the values of the input at the current time. • Dynamic - the behavior of the output depends on the previous values of the input. • Linear - the response to two different input signals is equivalent to the sum of the responses to each of these signals separately (superposition principle). • Nonlinear - the response to two different input signals is not equivalent to the sum of the reactions to each of these signals separately, the change in output is not proportional to the change in input. Model class definition • Stationary - the response to the same inputs does not depend on the time of application of these inputs, i.e. the parameters of such an object do not depend on time. • Non-stationary - the values of the object's parameters change in time. • Continuous - the state of the input and output variables changes or is measured continuously for a certain period of time. • Discrete - the state of outputs and inputs is determined only at discrete times. • To describe discrete systems, lattice functions are used, which are analogs of continuous functions, and difference equations , which are analogs of differential equations. Model class definition • One-dimensional - objects that have single input and one output (SISO). • Multidimensional (multi-connected) - have multiple inputs and multiple outputs (MIMO). • Deterministic - the behavior of the output does not depend on random factors. • Stochastic - the behavior of the output depends on random factors. • With lumped parameters - input and output variables depend only on time (only on one variable), described by ODEs. • With distributed parameters - the output depends on several variables: on time and on spatial coordinates. Described by partial differential equations. Structural identification • Selection of an object from its environment and the environment interacting with it. • Ranking the inputs and outputs of the object according to the degree of their influence on the behavior of the object. • Determination of the optimal number of inputs and outputs of the object, taken into account in the model. • Determining the nature of the connection between the inputs and outputs of the object model. Identification procedure System Evaluation of the Identification Process Model Identification Algorithm Measurement postprocessing Y Active Identification • The object of study is derived from normal conditions (nominal mode of operation, nominal parameters, etc.). • Research is carried out in specialized laboratory conditions. • Test signals of a special type are fed to the inputs of the object. It can be: step and pulse time signals, harmonic signals, random signals with given parameters. They are used in the development of new technologies in relation to existing industrial facilities, in the study of new phenomena, in the initial development of a mathematical model. Laboratory Object Object Passive identification • The object functions in the control loop, is in the process of normal operation. • Only natural control signals are received at its inputs. • Used to refine the mathematical model, to track changes in the object. • The information is operationally used in the control system. Controller Object Identification Mixed type identification • The object is not taken out of normal operation. • Test inputs are added to the control signals, which make it possible to identify the object without degrading the quality of the main control process. Experiment data post-processing • Measurement data is not suitable for direct use in identification algorithms. • The data may be flawed: 1. High frequency disturbances, above frequencies that are of interest in relation to the dynamic properties of the system. 2. Rare picks and overshoots. 3. Drift and bias, low-frequency disturbances, possibly of a periodic nature. • In data-based identification problems, one should always first plot the data to view these defects. • After post-processing, the array of measurement data is used in the selected identification method. Experiment data post-processing High frequency disturbances. • Indicate that the selection of the sampling interval was not successful enough. • If the sampling interval turned out to be unnecessarily small, it is always possible to thin out the data sequence by selecting each s-th observation from the original array. • If a digital filter is used, it is applied before the data is decimated. Experiment data preprocessing Slow disturbances. • Low frequency disturbances, bias, trends, drift and periodic (seasonal) changes are not uncommon. • They usually come from external sources that may or may not be included in the model. • There are basically two different approaches: 1. Removal of disturbances through special data pre-processing (by direct subtraction). 2. Introduction of a noise model to account for disturbances (noise models with poles on or near a circle of unit radius). Classification of identification methods • Nonparametric identification methods Determination of the transfer function by the time responces of the object Determination of the transfer function from the frequency characteristics of the object Correlation methods Spectral methods • Parametric identification methods Least square method Prediction error method Instrumental Variable Method Stochastic Approximation Method Accuracy estimation of the resulting model • The most important identification stage is verification of the obtained model. • In general case, the parameters estimation of a model with a given structure is carried out by minimizing the selected model quality criterion - the loss function (residual function) . • Example – mean squared error e y (t ) yˆ (t ) N where N is the number of measurements. • The smaller the criterion value, the higher the quality of identification. • The model can never be accepted as a "real" or "true" description of an object due to its innate approximation. Identification procedure as a closed system • The identification procedure has a natural logical order: first we collect data, then we form a set of models, and then we choose the best model. • It is common for the first chosen model to fail the test for compliance with the experimental data. • Then we should go back and select another model or change the search criteria. The model may be unsatisfactory for the following reasons: The numerical method cannot find a model suitable for the selected criterion. Incorrectly chosen criterion. An ill-formed set of models may not contain a high-quality model at all. The data collected is not informative.