Iterative parameter estimation methods for dual‐rate sampled‐data bilinear systems by means of the data filtering technique

ABSTRACTThis article considers the parameter estimation problems of multivariable autoregressive output‐error autoregressive moving‐average systems. To alleviate the identification difficulty of systems, we decompose the multivariable autoregressive output‐error autoregressive moving‐average system into several subsystems. To reduce the impact of colored noises on parameter estimation accuracy, we propose an auxiliary model maximum likelihood generalized extended gradient‐based iterative algorithm using the auxiliary model idea and the maximum likelihood principle. In addition, we use the multi‐innovation identification theory to introduce a moving data window and propose an auxiliary model maximum likelihood moving‐data‐window generalized extended gradient‐based iterative algorithm. These two proposed algorithms produce more accurate parameter estimates than the existing auxiliary model generalized extended gradient‐based iterative algorithm. Simulation examples illustrate the effectiveness of the proposed algorithms.

Maximum likelihood methods are based on the probability and statistics theory, and significant for parameter estimation and system modeling. This paper combines the maximum likelihood principle with the data filtering technique for parameter estimation of a class of bilinear systems. The input-output representation of a bilinear system is derived through eliminating the state variables in the model. Then, a filtering based maximum likelihood iterative least squares algorithm is proposed for identifying the parameters of bilinear systems with colored noises by filtering the input-output data with a filter. A least squares based iterative algorithm is given for comparison. The simulation results indicate that the proposed algorithm is effective for identifying bilinear systems. The filtering based maximum likelihood iterative least squares algorithm is more accurate under different noise variance, and has higher computational efficiency.

In this paper, we will discuss the problem of optimal model order reduction of bilinear control systems with respect to the generalization of the well-known ${\cal H}_2$-norm for linear systems. We revisit existing first order necessary conditions for ${\cal H}_2$-optimality based on the solutions of generalized Lyapunov equations arising in bilinear system theory and present an iterative algorithm which, upon convergence, will yield a reduced system fulfilling these conditions. While this approach relies on the solution of certain generalized Sylvester equations, we will establish a connection to another method based on generalized rational interpolation. This will lead to another way of computing the ${\cal H}_2$-norm of a bilinear system and will extend the pole-residue optimality conditions for linear systems, also allowing for an adaption of the successful iterative rational Krylov algorithm to bilinear systems. By means of several numerical examples, we will then demonstrate that the new techniques outperform the method of balanced truncation for bilinear systems with regard to the relative ${\cal H}_2$-error.

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.

Time-series measurements of metabolite concentration have become increasingly more common, providing data for building kinetic models of metabolic networks using ordinary differential equations (ODEs). In practice, however, such time-course data are usually incomplete and noisy, and the estimation of kinetic parameters from these data is challenging. Practical limitations due to data and computational aspects, such as solving stiff ODEs and finding global optimal solution to the estimation problem, give motivations to develop a new estimation procedure that can circumvent some of these constraints. In this work, an incremental and iterative parameter estimation method is proposed that combines and iterates between two estimation phases. One phase involves a decoupling method, in which a subset of model parameters that are associated with measured metabolites, are estimated using the minimization of slope errors. Another phase follows, in which the ODE model is solved one equation at a time and the remaining model parameters are obtained by minimizing concentration errors. The performance of this two-phase method was tested on a generic branched metabolic pathway and the glycolytic pathway of Lactococcus lactis. The results showed that the method is efficient in getting accurate parameter estimates, even when some information is missing.

This paper considers the parameter estimation for multi-input output error autoregressive (OEAR) systems. The main contribution of this paper is to propose iterative algorithms for multi-input OEAR systems using the data filtering technique. Firstly, a least squares iterative (LSI) identification algorithm is given for multi-input OEAR systems as a comparison. Secondly, we transform an OEAR system into two identification models and present a data filtering based least squares iterative (F-LSI) identification algorithm. Finally, an illustrative simulation example is provided to test the proposed algorithms, the simulation results indicate that the F-LSI identification algorithm can produce more accurate parameter estimates than the LSI identification algorithm.

For parameter estimation of output error moving average (OEMA) systems, this study combines the auxiliary model identification idea with the filtering theory, transforms an OEMA system into two identification models and presents a filtering and auxiliary model-based recursive least squares (F-AM-RLS) identification algorithm. Compared with the auxiliary model-based recursive extended least squares algorithm, the proposed F-AM-RLS algorithm has a high computational efficiency. Moreover, a filtering and auxiliary model-based least squares iterative (F-AM-LSI) identification algorithm is derived for OEMA systems with finite measurement input-output data. Compared with the F-AM-RLS approach, the proposed F-AM-LSI algorithm updates the parameter estimation using all the available data at each iteration, and thus can generate highly accurate parameter estimates.

<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>

SummaryThis study investigates the parameter identification issues of bilinear‐in‐parameter systems through fractional adaptive algorithms. An auxiliary model based modified fractional least mean square algorithm is proposed for accelerating the parameter estimation accuracy based on the auxiliary model identification idea and the introduced convergence index, a normalized modified hierarchical fractional least mean square algorithm is presented for improving the computational efficiency based on the hierarchical identification principle. The proposed normalized fractional adaptive strategies are effective and could provide more accurate parameter estimates comparing with conventional counterparts for bilinear‐in‐parameter identification model based on the mean square error metrics and the average predicted output error. The effectiveness and accuracy of the proposed algorithms are further verified and validated through numerical simulations for different noise variances, fractional orders and gain parameters.

Parameter estimation for Non-uniformly Sampled Wiener Systems with Dead-zone Nonlinearities

Identification of Input-Output Bilinear Systems

SummaryThis paper focuses on the parameter estimation issues of multivariate equation‐error autoregressive moving average systems. By applying the gradient search and the multi‐innovation theory, we derive a multi‐innovation gradient based iterative (MI‐GI) algorithm. In order to improve the computational efficiency and the parameter estimation accuracy, a filtering and decomposition based gradient iterative (F‐D‐GI) algorithm is presented by using the data filtering technique and the decomposition technique. The key is to choose an appropriate filter to filter the input‐output data and to transform an original system into several subsystems. Compared with the MI‐GI algorithm, the F‐D‐GI algorithm can generate more accurate parameter estimates. Finally, an illustrative example is provided to indicate the effectiveness of the proposed algorithms.

Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems

Maximum likelihood LM identification based on particle filtering for scarce measurement-data MIMO Hammerstein Box-Jenkins systems

This article considers the iterative identification problems for a class of feedback nonlinear systems with moving average noise. The model contains both the dynamic linear module and the static nonlinear module, which brings challenges to the identification. By utilizing the key term separation technique, the unknown parameters from both linear and nonlinear modules are included in a parameter vector. Furthermore, an auxiliary model maximum likelihood gradient‐based iterative algorithm is derived to estimate the unknown parameters. In addition, an auxiliary model maximum likelihood stochastic gradient algorithm is derived as a comparison. The numerical simulation results indicate that the auxiliary model maximum likelihood gradient‐based iterative algorithm can effectively estimate the parameters of the feedback nonlinear systems and get more accurate parameter estimates than the auxiliary model maximum likelihood stochastic gradient algorithm.