PARAMETRIC IDENTIFICATION OF NONLINEAR DYNAMIC SYSTEMS USING COMBINED LEVENBERG–MARQUARDT AND GENETIC ALGORITHM

International audience

This paper presents an intelligent system identification methodology for the identification of a realistic model of an ultrasonic position estimation system that uses the difference in the time of arrivals of waves from a transmitter to various receivers. Even though a linearized formulation for the 3D system exists and is currently being used to estimate the position of the transmitter, its accuracy can still be improved further. A neural network approach is developed to train the system based on training sets obtained from the actual system, and it is proposed to use the final trained system to estimate the 3D position in real time. The weights of the neural network are obtained from an innovative procedure using genetic algorithms. Results for a simplified 1D system are presented as proof of concept. The performance of the identified 1D system using genetic algorithms is shown to be comparable to the one using the analytical model. Further, the identified system using genetic algorithms is also shown to be superior to the one using the traditional back propagation method for finding the weights for the neural networks. This work has significant applications in the identification of complex non-linear systems.

In this paper, advanced concepts for the identification of complex nonlinear systems are discussed. Three major problems are addressed: The nonlinearity of the system, noise in the data upon which the model has to be built, and the potential to incorporate qualitative and quantitative prior knowledge about the system. As an integrated solution approach, local model networks (LMNs) with appropriate parameter estimation schemes are proposed. LMNs generally offer a versatile structure for the identification of nonlinear dynamic systems. In order to account for a realistic situation when noise is present both in input and output data, an equality constrained generalised total least squares algorithm for the local model parameter estimation of the LMN is presented; the incorporation of equality constraints allows to mathematically enforce desired system properties. As an application and benchmark problem, the vertical dynamics of a vehicle is considered. After training the LMN on a rough road, excellent predictions of the behaviour of the vehicle at crossing a single obstacle are obtained, thus proving the effectiveness of the proposed algorithm. It is illustrated how both the application of a proper parameter estimation scheme and the integration of system constraints systematically improve the performance of the model.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

Identification of nonlinear systems is one of the important problems in engineering. Chaotic systems are among those nonlinear systems that have been in the center of attention of many researchers because of their complex and unpredictable behaviors. This paper proposes an algorithm for optimally estimating the parameters of system by minimizing the mean of squared errors (MSE) index. In this paper, Teaching Learning Based Optimization (TLBO) algorithm is used for solving both offline and online parameter estimation problems for chaotic systems. The validity of this algorithm in terms of convergence speed and parameter accuracy in comparison to other popular optimization algorithms such as Standard PSO (PSO), Differential Evolution (DE), Adaptive Particle Swarm Optimization (APSO) and Genetic Algorithm (GA) is shown through an illustrative example for the modeling of chaotic systems. Furthermore, in order to demonstrate the feasibility of this algorithm, it is applied to the problem of parameter identification of a well-known nonlinear Lorenz chaotic system. According to simulation results, the proposed algorithm is a very suitable algorithm for online parameter identification for a class of nonlinear chaotic systems.

Multidimensional membership functions in T–S fuzzy models for modelling and identification of nonlinear multivariable systems using genetic algorithms

Detection and classification of non-linearities based on Volterra kernels processing

The time finite element method (TFM) is employed for parametric identification of nonlinear structural dynamic systems. An advantage of TFM is the ease with which one can calculate the sensitivity of the transient response with respect to various design parameters, a key requirement for gradient-based parameter identification schemes. The method is simple, because one obtains the sensitivities of the response to system parameters by differentiating the algebraic equations, not original differential equations. These sensitivities are used in the Levenberg-Marquardt iterative direct method to identify parameters for nonlinear single- and two-degree-of-freedom systems. The measured response was simulated by integrating the example nonlinear systems using the given values of the system parameters. The accuracy and the efficiency of the present method are compared to a previously available approach that employs a multistep method to integrate nonlinear differential equations. It is seen, for the same accuracy, that the present approach requires fewer data points.

Teaching Aids for Modeling and Control of Hybrid Systems (CAMCHS)

The work presented in this dissertation was conducted in an effort to achieve improvement in conventional and digital control methods applied to chemical processes. Emphasis was placed on simplicity and effectiveness in an attempt to provide control improvements which are easily implemented and yet efficient in their tasks. In these studies digital simulation was used exclusively utilizing the IBM 7040 and 360 and the XDS Sigma V computers. Initially the inaccuracies involved in obtaining a firstorder plus dead time model graphically from an open loop response curve are discussed. It is pointed out that fine tuning of two and three mode controllers based on these inaccurate models is necessary and presently must be done by trial and error. Modifications of existing controller tuning relations are presented which guide fine tuning attempts and eliminate much of the trial and error procedure. Second, the subject of linear discrete models applied to sampled-data systems is considered. A multiple linear least squares technique is applied to sampled system input/output records to identify parameters of a linear discrete model. The method is demonstrated to give good results when applied xii to linear and non-linear systems with and without dead time. The simplicity and flexibility of implementing this modeling technique are discussed. Application of this method to systems with measurement noise are shown to be improved when the data are filtered. Third, the implementation of linear discrete models in Deadbeat, Dahlin and Kalman controller synthesis methods are presented. It is demonstrated that the form of the models fits well into each of these controller design philosophies. The Dahlin algorithms which incorporate a tuning parameter give the most consistent performance for model orders and sampling times considered. In general, the performance of controllers based on low order models is seen to compare well with that of higher order controllers. Fourth, utilizing the speed of the multiple linear regression identification, the application of this technique in an adaptive control scheme is presented. Applying a recursive solution of the regression equation enables the updating of the linear discrete model at each sample instant and subsequently the adaptation of the control strategy. This scheme is demonstrated to provide closer control of a non linear, time-variant system than is possible with a stationary model. xiii CHAPTER I The form of the transfer function, HG(z), comprised of the digital hold device and the process itself, is such that multiple linear regression techniques can be applied to identify the model constants (parameters). Model identification is then based only on a linear regression of the sampled-data records of the system input and output. A modeling technique based on multiple linear regression only requires the solution of a set of normal equa tions and therefore is rapid enough to be applied on-line. Model order can also be changed with relative ease requiring only an other pass through the system input/output response data already collected. The effectiveness of this identification method is demonstrated on linear systems as well as on a simulated non linear plug flow reactor system. The effect of measurement noise on modeling accuracy is also considered, and comparison of identification using both filtered and unfiltered data are presented. The advantage of filtering data under these operating conditions is illustrated. Linear discrete models obtained for the simulated reactor were demdnstrated in their capacities as both one step ahead predictor models and as free running models. In these studies the higher order models provided significantly better results. Digital controller synthesis methods are the subject of Chapter IV. The discrete form of the model fits well into these design methods. Flexibility in varying model order is carried over to allow: for flexibility in controller order. The advantages of this flexibility are illustrated with the Deadbeat, Dahlin and Kalman controller design algorithms. Controllers based on each of these design methods are implemented in the plug flow reactor system and compared as to overall control performance and manipulate variable action. Comparisons are made also for both fast and slow sampling rates. Finally, Chapter V incorprates the multiple linear regres sion modeling technique with a direct controller synthesis method in an attempt to achieve an adaptive control scheme. The intent here is to update the process model with each new sample taken, and therefore adapt the controller, which is di rectly based on this model to any variation in the operating levels of the system. Adaptive control in this fashion is demonstrated in application to the non-linear plug flow reactor system with a slowly changing feed rate. The result is a demonstration of an adaptive control strategy which has the capability to function in many possible process applications.

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 1996

In recent years, extensive works on genetic algorithms have been reported covering various applications. Genetic algorithms (GAs) have received significant interest from researchers and have been applied to various optimization problems. They offer many advantages such as global search characteristics, and this has led to the idea of using this programming method in modelling dynamic non-linear systems. In this paper, a methodology for model structure selection based on a genetic algorithm was developed and applied to non-linear discrete-time dynamic systems. First the effect of different combinations of GA operators on the performance of the model developed is studied. A proposed algorithm called modified GA, or MGA, is presented and a comparison between a simple GA and a modified GA is carried out. The performance of the proposed algorithm is also compared to the model developed using the orthogonal least squares (OLS) algorithm. The adequacy of the developed models is tested using one-step-ahead prediction and correlation-based model validation tests. The results show that the proposed algorithm can be employed as an algorithm to select the structure of the proposed model.

Our thermodynamic approach to the study and design of robust optimal control processes in nonlinear (in general global unstable) dynamic systems used soft computing based on genetic algorithms with a fitness function as minimum entropy production. Control objects were nonlinear dynamic systems involving essentially nonlinear stochastic differential equations. An algorithm was developed for calculating entropy production rate in control object motion and in control systems. Part 1 discusses relation of the Lyapunov function (measure of stability) and the entropy production rate (physical measure of controllability). This relation was used to describe the following qualitative properties and important relations: dynamic stability motion (Lyapunov function), Lyapunov exponent and Kolmogorov-Sinai entropy, physical entropy production rates, and symmetries group representation in essentially nonlinear systems as coupled oscillator models. Results of computer simulation are presented for entropy-like dynamic behavior for typical benchmarks of dynamic systems such as Van der Pol, Duffing, and Holmes-Rand, and coupled oscillators. Parts 2 and 3 discuss the application of this approach to simulation of dynamic entropy-like behavior and optimal benchmark control as a 2-link manipulator in a robot for service use and nonlinear systems under stochastic excitation.

A strategy to nonlinear dynamic system identification based on Fuzzy Kalman Filter, is proposed. A mathematical formulation based on fuzzy Takagi-Sugeno structure is presented: the algorithm FCM estimates the fuzzy sets; from the data input and output of a nonlinear dynamic system, the ERA/DC algorithm based on clustering, estimated by FCM algorithm, is applied to obtain the matrices A, B, C, and D (state matrix, input influence matrix, output influence matrix, and direct transmission matrix, respectively) to each rule of the fuzzy model consequent. A Fuzzy Kalman Filter is applied to estimate states and output of a nonlinear dynamic system. Computational results show the efficiency of the proposed methodology, once that the TS Fuzzy Model Based on Fuzzy Kalman Filter follow the behavior relative to output and states of a nonlinear dynamic system, and compared it to other methodology it shows a better performance related to follow the behavior of states.

This article presents a theoretical framework for the identification of nonlinear dynamic MISO systems with orthonormal base function models on fundamental basics of the Volterra theory. In the past the Volterra theory was used for the identification of nonlinear dynamic SISO systems (e.g. Hammerstein and Wiener models). In principle it is possible to extend these approaches to systems with more than one input (i.e. MISO systems). In former times this failed due to a lack of computational performance. In this paper an approach is presented which allows the identification of arbitrary coupled Hammerstein and Wiener models with multiple inputs. Some fundamental considerations for the identification of MISO systems based on arbitrary coupled Hammerstein models are made and extended to MISO systems based on arbitrary coupled Wiener models. This extended Volterra theory results in equations where the unknown parameters can be separated of the input values in a linear manner. In order to approximate the truncated impulse responses of the linear dynamic systems and to reduce the number of unknown parameters orthonormal base functions (OBFs) are introduced. As an example the proposed identification method is applied to a MISO system based on forward and backward coupled Hammerstein and Wiener models.