Linear state representations for identification of bilinear discrete-time models by interaction matrices

<div class="htmlview paragraph">As an important class of nonlinear system, polytopic bilinear system is investigated. Combined with the properties of convex polytope, the nonlinear control for polytopic bilinear system is formulated by synthesizing nonlinear <i>H</i><sub>∞</sub> controller which is designed for polytopic bilinear system at vertices. For a semi-active suspension system with controllable damping and variant stiffness elements, it is easily modeled as a polytopic bilinear system model. In this case, the desired nonlinear control properties are pursued in making effective use of the changeable damping property while the variant stiffness is taken as the affine parameter of polytopic model. Therefore, polytopic bilinear system model could be reduced to a feasible problem by polytopic convex decomposition. Then the control problem of bilinear system model is to find a solution of nonlinear <i>H</i><sub>∞</sub> control. Here, the merits in using nonlinear <i>H</i><sub>∞</sub> state feedback control are that it can confine the damping force output to 1<sup>st</sup> and 3<sup>rd</sup> phase planes where changeable damping force is produced and its nonlinear control output makes a fast state origin convergence in keeping small vibration and motion. In this paper, a practical control design demonstration for a polytopic bilinear suspension system is described after the theory description. The simulation study and experimental results demonstrate the effective control performance in improving ride comfort.</div>

Bilinear systems are a special class of nonlinear systems. Some systems can be described by using bilinear models. This paper considers the parameter identification problems of bilinear stochastic systems. The difficulty of identification is that the model structure of the bilinear systems includes the products of the states and inputs. To this point, this paper gives the input---output representation of the bilinear systems through eliminating the state variables in the model and derives a least squares algorithm and a multi-innovation stochastic gradient algorithm for identifying the parameters of bilinear systems based on the least squares principle and the multi-innovation identification theory. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

The best known model for numerically simulating the hysteretic behavior of various structural components is the bilinear hysteretic system. There are two possible mechanical formulations that correspond to the same bilinear model from a mathematical viewpoint. The first one consists of a linear elastic spring connected in series with a parallel system comprising a plastic slider and a linear elastic spring, while the second one comprises a linear elastic spring connected in parallel with an elastic-perfectly plastic system. However, the bilinear hysteretic model is unable to describe either softening or hardening effects in these components. In order to account for this, the bilinear model is extended to a trilinear one. Thus, two trilinear hysteretic models are developed and numerically tested, and the analysis shows that both exhibit three plastic phases. More specifically, the first system exhibits one elastic phase, while the second one exhibits two elastic phases according to the level of strain amplitude. Next, the change of slope between the plastic phases in unloading does not occur at the same displacement level in the two models. Furthermore, the dissipated energy per cycle in the first trilinear model, as proven mathematically and explained physically, decreases in the case of hardening and increases in the case of softening, while in the second trilinear model the dissipated energy per cycle remains unchanged, as is the case with the bilinear model. Numerical examples are presented to quantify the aforementioned observations made in reference to the mechanical behavior of the two trilinear hysteretic models. Finally, a set of cyclic shear tests over a wide range of strain amplitudes on a high damping rubber bearing is used in the parameter identification of the two different systems, namely (a) trilinear hysteretic models of the first type connected in parallel, and (b) trilinear hysteretic models of the second type also connected in parallel. The results show that the complex nonlinear shear behavior of high damping rubber bearings can be correctly simulated by a parallel system which consists of only one component, namely the trilinear hysteretic system of the first type. The second parallel system was not able to describe the enlargement of the dissipated hysteresis area for large strain amplitudes.

This paper considers the iterative parameter estimation for a dual‐rate sampled‐data bilinear system with autoregressive moving average noise. Through combining the auxiliary model identification idea with the data filtering technique, this paper derives two filtering auxiliary model gradient‐based iterative algorithms by using two different filters. The key is to construct an auxiliary model for predicting the unavailable outputs, and to transform the dual‐rate bilinear system identification model into two sub‐identification models. Finally, an auxiliary model gradient‐based iterative (AM‐GI) algorithm is presented for comparison. The simulation results indicate that the proposed algorithms are effective for identifying the dual‐rate sampled‐data bilinear systems, and can generate more accurate parameter estimates and have a higher computational efficiency than the AM‐GI algorithm.

This paper formulates optimal bilinear observers for bilinear state-space models. Observers in bilinear form, as opposed to other nonlinear forms, are required to develop an extension of observer/Kalman filter identification for simultaneous identification of a bilinear state-space model and an associated bilinear observer from noisy input–output measurements. The paper establishes the relationship between the bilinear observer gains and the interaction matrices which are used to convert the original bilinear state-space model to a form that simplifies the identification of such a model. Techniques to find the interaction matrices are developed. In the absence of noises, these matrices produce the gains of the fastest converging observer. In the presence of noises, they minimise the state estimation error in the same manner as a standard steady-state Kalman filter. Numerical examples illustrate both the theoretical and computational aspects of the proposed algorithms.

Analysis of bilinear noise models in circuits and devices

Ch. 21. On the theory of discrete and continuous bilinear time series models

To date, significant efforts have been made to study the theory of bilinear time series models, especially simple bilinear models. Much less efforts, however, have been made to identify optimal models. Focused on optimal model identification, this study attempts to fill this gap. Full and subset one-dimensional bilinear models are proposed and shown to be robust in achieving stationarity for all non-linear series. The parameters of the models are estimated using robust nonlinear least-square method and Newton-Raphson iterative method, and statistical properties of the derived estimates are investigated. An algorithm is proposed to eliminate redundant parameters from full order bilinear models.

When a homogeneous bilinear system is dyadic, it permits the use of the so-called division controller for global stabilization. However, due to singularities in the control law, a dead-zone must be cascaded with the division controller in practical implementation. The contribution of this note is to show that the division controller, when cascaded with a dead-zone, can asymptotically drive the bilinear system state into a bounded set around the origin with the size of the bounded set proportional to the size of dead-zone. This note further proposes a C/sup 0/ modification and a C/sup /spl infin// modification of the conventional discontinuous dead-zone so that the control signals from the modified division controllers are continuous and smooth, respectively.

The paper develops a bilinear perturbation fuzzy model of a DC-DC converter. A methodology for developing the discrete-time bilinear model around a nominal operating point is presented first. The multi-zonal bilinear model is then combined to obtain a fuzzy model using the Takagi-Sugeno method (1985). It is shown that this model predicts the behavior of the system fairly accurately. The fuzzy model is then used in a feedback-linearized control to track a given reference. The controller is obtained through the minimization of a Lyapunov function.

The identification of nonlinear systems with artificial neural networks models has been successfully used in many applications. Most processes in industry are characterized by nonlinear and time-varying behavior. In this context, the identification of mathematical models for nonlinear systems is vital in many fields of engineering. The Radial Basis Function Neural Network (RBF-NN) is a powerful approach for nonlinear identification and can be improved using Particle Swarm Optimization (PSO) approaches. This paper presents a multivariable nonlinear system identification using RBF-NN combined with standard PSO and Constriction Factor PSO (CFPSO) approaches in order to determine the RBF-NN parameters. RBF-NN is considered to be a good choice for black-box modeling problems due to its rapid learning capacity and, therefore, has been applied successfully to nonlinear time series modeling and nonlinear identification. On the other hand, PSO was inspired by the choreography of bird flocks and fish schools and can be seen as a distributed behavior algorithm that performs multidimensional search. Furthermore, promising simulation results from performance analysis of the proposed RBF-NN with PSO training approaches are presented and discussed in this paper showing promising results.

We study some model control problems which arise in connection with optimal manipulation of dissipative quantum dynamics. It is shown that the problem of optimal control of quantum mechanical phenomenon in presence of dissipation can be reduced to the study of optimization problems associated with a class of constrained bilinear control systems. These bilinear systems x/spl dot/=(A+/spl Sigma//sub i//sup n/u/sub i/B/sub i/)x are characterized by the fact that the controls can be expressed as polynomial functions of fewer parameters, i.e. u/sub i/=g/sub i/ (/spl upsi//sub 1/, /spl upsi//sub 2/,...,/spl upsi//sub k/) where g/sub i/ are polynomials and k<n. A general study of these systems is expected to find immediate applications in coherent control of quantum mechanical phenomenon.

Identification of bilinear systems subject to white inputs is studied. Assuming bilinearity we use the crosscovariances between the output and the higher order Hermite polynomials of the input for estimating the coefficients of the Hermite series expansion of the output. The parameters of the bilinear model are obtained via a balanced factorization of the appropriately constructed Hankel matrix. The skewness of the singular values of the estimated Hankel matrix is applied for testing the order selection for the bilinear model.

SummaryThis article studies the parameter identification problem for bilinear state space models with time‐varying time delays. Considering the correlation of time delays, the Markov chain switching mechanism is adopted to model the delay sequence. Based on the observer canonical form, the bilinear state space model is transformed into a regressive form. A bilinear state observer is designed to estimate the state variables. Under the variational Bayesian scheme, the system parameters, discrete delays, and the Markov transition probabilities are identified by using the measurement data. A numerical example and a continuous stirred tank reactor simulation are employed to validate the effectiveness of the proposed algorithm.

Piezoelectric elements (PEMs) are used in a variety of applications. In this paper, we developed a full analytical model and a simple system identification (SI) method of a piezoelectric actuator, which includes piezostack elements and a three-stage amplification mechanism. The model was derived separately for each unit of the system. Next, the units were combined, while taking into account their coupling. The hysteresis phenomenon, which is significant in piezoelectric materials, is described extensively. The theoretical model was verified in a laboratory setup. This setup includes a piezoelectric actuator, measuring devices and an acquisition system. The measured results were compared to the theoretical results. Some of the most well-known forms of system identification are shown briefly, while a new and simple algorithm is described systematically and verified by the model. The main advantage of this work is to provide a solid background and domain knowledge of modelling and system identification methods for further investigations in the field of piezoelectric actuators. Due to their simplicity, both the model and the system identification method can be easily modified in order to be applied to other PEMs or other amplification mechanism methods. The main novelty of this work lies in applying a simple system identification algorithm while using the system-level approach for piezoelectric actuators. Lastly, this review work is concluded and some recommendations for researchers working in this area are presented.