Identification of Linear Systems Containing Unknown Gain Elements

Control of industrial plants is an important engineering field of study, where the optimization of such processes can lead to better product quality and higher profit. The present work deals with the optimization of an industrial gasifier, proposed as a benchmark challenge by ALSTOM Power Technologies in 2002. Researchers from around the globe proposed different methods for optimizing such system, using linear and nonlinear system identification techniques, different controller schemes, such as proportional-integral-derivative and model predictive controllers, and search methods for controller tuning. The authors care for the study of a methodology that can generalize the controller optimization process of real industrial systems, where linear system identification can lead to simpler systems, which are used to design more robust controllers. The proposed work deals with linear system identification and multiobjective optimization for the gasifier's controller design. To do so, the authors divide the problem in two steps, the linear system identification, where step functions are fed to the gasifier in order to detect the output response of the system, and the design of a proportional-integral-derivative controller, where spherical pruned multi-objective differential evolution algorithm is used in order to optimize the controller gains. The proposed method, unfortunately, does not provide an optimal controller, which can be explained by the complexity of the benchmark system. With such results, the authors believe a better multi-objective problem definition is necessary. Also, it is concluded that such benchmark presents a real challenge for future controller design methndologies.

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k(x>0)/k(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.

This paper deals with the identification of stable linear systems from input−output response data with low signal/noise ratios. Specifically we focus on nonparametric (finite impulse or step response, FIR or FSR) models widely used in model predictive control. A polynomial kernel representation is proposed to reduce the number of parameters needed to represent the model. This leads to parsimonious yet robust models. Linear least-squares estimation can be used with these polynomial models. The time delay and response time of the process can be explicitly included as parameters in the model; however, the nonlinear optimization method must then be used. Various properties of this model including the variance of parameter estimates are given in the paper. Both colored and white noise are considered. Simulation and experimental results are used to illustrate the advantages of this approach especially at low signal/noise ratios. The polynomials act as an adaptive filter to remove the noise.

SummaryAt the present time, there is a great deal of interest in the adaptive control of systems with rapidly varying parameters. Such problems arise in different forms and in many disciplines including finance, sociology, biology, and engineering. In general, these problems are intractable mathematically and the time variations have to be classified in some form to obtain rigorous results. In this paper, we consider the adaptive identification and control of linear systems with periodically varying parameters (referred to as linear time‐varying periodic (LTP) systems). The class of systems with known periodic parameters has been investigated widely since the pioneering work of Floquet and lends itself to rigorous mathematical analysis. However, very little was known until the early years of the last decade concerning the adaptive control of such systems. The problem was introduced by Xu and explored in detail by Narendra and Zhiling in 2006 and 2009, respectively. In 2012, a new method was introduced in adaptive control by Han and Narendra and is referred to as second‐level adaptation. In this paper, the adaptive identification and control of LTP systems using second‐level adaptation is discussed.

This paper presents a novel deterministic approach for the identification of linear continuous-time systems. The approach is based on the projection of measured signals onto the finite dimensional signal subspace whose basis is determined by the parameter structure of the target system model. In this paper, we describe the basic concept and implementation of the proposed identification framework, and develop the methodology to deal with multi-variable input and/or output case. In addition, the initial state estimation technique, which enables us to use the truncated I/O data for identification, is introduced. The important features of the proposed approach are (i) the ability to handle both open and closed loop systems in a unified way and (ii) the robustness against the measurement noise. These features are demonstrated through a numerical example.

On the optimal location of sensors for parametric identification of linear structural systems

Real-time identification and control of linear discrete systems with Gauss-Markov random parameters and conditioned quadratic cost function are considered. Algorithms for explicit calculation of the identification and control are given for systems with perfect state measurement and are illustrated by example applications. The control strategy generated by the cost function is an identitication-adaptive controller which is continually revised as new data are received. The identification equations develop an explicit model of the system which is available for other purposes if desired, and can be used apart from the control context.

Max-plus linear inverse problems: 2-norm regression and system identification of max-plus linear dynamical systems with Gaussian noise

In many real-time systems we need to know the exact values of system parameters. Parameters identification for linear and stable systems can be done by taking special measurements. Problems can occur if the tested system is nonlinear and its parameters change due to certain variations or they depend on time. In such cases special procedures have to be applied. These procedures should operate quickly, and their computational time delay cannot affect on the entire control process. In this paper, determination of the frequency of cantilever beam with embedded piezoelectric actuator is studied. A short overview of Model Based Methods is described. A nonlinear equation of motion of the beam is rewritten as an equation with step wise changes of its natural frequency. Proposed mathematical identification methods use some input and output variables for identifying lack of information about certain parameters in a mathematical model. We take into account Decoupling Observers and Model Reference Adaptive System (MRAS). Due to the structure of the mathematical description, MRAS system found out to be more adequate option. Algorithms based on Model Identification Techniques (MIT), as well as an algorithm satisfying Lyapunov stability condition, were tested. The latter was robust on certain residual effects. Therefore, we chose it for the final test. Obtained results confirmed this choice. Presented figures show characteristics of natural frequency that have been identified. Therefore, the implemented adjusting algorithm satisfying Lyapunov stability found out to be suitable for an application in DSP. However, reducing the processing time is still a problem for future studies.

ABSTRACTA method for the identification of single input single output linear systems is presented. The method employs a Bayesian approach to compute the posterior distribution of the model parameters given the data-set. Since this distribution is often unavailable in closed form, a Metropolis Hastings algorithm is implemented to draw samples from it. To implement the sampler, the inclusion of prior information regarding the model order of the identified system is discussed. As one of the main contributions of this work, a prior over the Hankel singular values of the model is imposed. Numerical examples illustrate the method.

The problem of recursive robust identification of linear discrete-time single-input single-output dynamic systems with correlated disturbances is considered. Problems related to the construction of optimal robust stochastic approximation algorithms in the min-max sense are demonstrated. Since the optimal solution cannot be achieved in practice, several robustified stochastic approximation algorithms are derived on the basis of a suitable non-linear transformation of normalized residuals, as well as step-by-step optimization with respect to the weighting matrix of the algorithm. The convergence of the developed algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms. The results indicate the most suitable algorithms for applications in engineering practice.

Recently, a new regularized least squares approach to linear system identification has been introduced where the penalty term on the impulse response is defined by so called stable spline kernels. They encode information on regularity and BIBO stability, and depend on a small number of parameters that can be estimated from data. In this paper, we provide new nonsmooth formulations of the stable spline estimator. In particular, we consider linear system identification problems in a very broad context, where regularization functionals and data misfits can come from a rich set of piecewise linear quadratic functions. Moreover, our analysis includes polyhedral inequality constraints on the unknown impulse response. For any formulation in this class, we show that interior point methods can be used to solve the system identification problem, with complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> )+O(mn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) in each iteration, where n and m are the number of impulse response coefficients and measurements, respectively. The usefulness of the framework is illustrated via a numerical experiment where output measurements are contaminated by outliers.

The identification of linear dynamic systems in the delta domain has been proposed in this paper with the help of a hybrid metaheuristic algorithm combining chaotic firefly algorithm (CFA) and grey wolf optimiser (GWO). GWO performs the global search, while CFA fine-tunes the solutions through its local search abilities, thereby balancing exploration and exploitation features. Linear systems with static nonlinearities at the input are termed as the Hammerstein model, whereas linear systems with static nonlinearities at the output are known as the Wiener model. A test case with continuous polynomial nonlinearities has been taken up for Hammerstein and Wiener system identification in the delta domain. Delta operator parameterisation unifies identification of continuous-time systems with the discrete domain at a higher sampling rate. Pseudo-random binary sequence (PRBS), polluted with white Gaussian noise of fixed signal-to-noise ratio (SNR), has been considered as the input signal to estimate the unknown model parameters as well as static nonlinear coefficients. The hybrid algorithm not only supersedes the parent heuristics of which it is constituted but also proves better in comparison with some standard and latest heuristic approaches reported in the literature. Nonparametric statistical tests are performed to validate the results. The plots of fitness function (normalised value) against the number of iterations also support the convergence speed and accuracy of the results.

Identification of Linear Systems with Nonlinear Distortions

An operational matrix for the integration of Chebyshev vector whose elements are Chebyshev polynomials is first derived in this paper. Using the operational matrix of integration, a time‐invariant linear system can be represented by a set of simultaneous linear algebraic equations with the approximation of time functions by Chebyshev series. Analysis and parameter identification of linear system can thus be made by solving the algebraic equations.