Order Selection for Stochastic Bilinear Systems*

<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.

Analysis of bilinear noise models in circuits and devices

Bilinear systems can be viewed as a bridge between linear and nonlinear systems, providing a promising approach to handle various nonlinear identification and control problems. This paper provides a formal justification for the extension of interaction matrices to bilinear systems and uses them to express the bilinear state as a linear function of input–output data. Multiple representations of this kind are derived, making it possible to develop an intersection subspace algorithm for the identification of discrete-time bilinear models. The technique first recovers the bilinear state by intersecting two vector spaces that are defined solely in terms of input–output data. The new input–output-to-state relationships are also used to extend the equivalent linear model method for bilinear system identification. Among the benefits of the proposed approach, it does not require data from multiple experiments, and it does not impose specific restrictions on the form of input excitation.

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 presents two fast least-squares lattice algorithms for adaptive nonlinear filters equipped with bilinear system models. Bilinear models are attractive for adaptive filtering applications because they can approximate a large class of nonlinear systems adequately and usually with considerable parsimony in the number of coefficients required. The lattice filter formulation transforms the nonlinear filtering problem into an equivalent multichannel linear filtering problem and then uses multichannel lattice filtering algorithms to solve the nonlinear filtering problem. The first of the two approaches is an equation-error algorithm that uses the measured desired response signal directly to compute the adaptive filter outputs. This method is conceptually very simple however it will result in biased system models in the presence of measurement noise. The second approach is an approximate least-squares output-error solution. In this case the past samples of the output of the adaptive system itself are used to produce the filter output at the current time. This approach is expected to reduce the effect of measurement noise on the behavior of the system. Results of several experiments that demonstrate and compare the properties of the adaptive bilinear filters are also presented in the paper.

Recursive Identification of Continuous Bilinear Systems

This paper is concerned with the design of reduced-order state estimators for bilinear stochastic discrete-time systems subjected to estimation error covariance assignment. The purpose of the problem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such that the steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisation of estimation error covariances that certain bilinear error dynamic processes may possess is presented, (2) the characterisation of all reduced-order state estimators that assign such error covariances is explicitly derived, and (3) the solvability of the assignability conditions is discussed. Furthermore, an illustrative example is used to demonstrate the effectiveness of the proposed design procedure.

Investigated herein are the dynamic responses of three different single degree of freedom (SDOF) bilinear dynamical models of a cracked concrete beam subjected to the El Centro earthquake. The first model is an equivalent unilinear model while the other two models have different stiffness values but differ from each other in the dependence of their damping coefficients on the sense of amplitude. All three models have the same natural period. The three models have been found to predict different response spectra for the same bilinear dynamical system. The effect of constant loading on the dynamical response of the concrete structures subjected to sinusoidal and random loading has been found to be significant for more flexible structures. Implications of these theoretical predictions for seismic design of concrete structures are discussed.

This paper tackles a 3D inverted pendulum stabilization problem, where the pendulum is attached on a quadrotor. We first derive the mathematical model of the quadrotor-pendulum system based on the Euler-Lagrange equation. Then, with hope for better control performance than traditional linear quadratic optimal control by efficiently using the vertical input, the bilinear approximation model is built by the second order Taylor expansion. We then propose an inverse optimal stabilization law for the bilinear system, conduct the convergence analysis and give the interpretation of the optimality. Finally, the validity of the present approach is demonstrated via simulations, where we explicitly show the effectiveness of the vertical input by comparing with the linear quadratic control case.

The optimal control of the hydrogen evolution reactions is attempted for the regulation and change of set‐point problems, taking into account that model parameters are uncertain and I/O signals are corrupted by noise. Bilinear approximations are constructed, and their dimension eventually increased to meet accuracy requirements with respect to the trajectories of the original plant. The current approximate model is used to evaluate the optimal feedback through integration of the Hamiltonian equations. The initial value for the costate is found by solving a state‐dependent algebraic Riccati equation, and the resulting control is then suboptimal for the electrochemical process. The bilinear model allows for an optimal Kalman–Bucy filter application to reduce external noise. The filtered output is reprocessed through a non‐linear observer in order to obtain a state‐estimation as independent as possible from the bilinear model. Uncertainties on parameters are attenuated through an adaptive control strategy that exploits sensitivity functions in a novel fashion. The whole approach to this control problem can be applied to a fairly general class of non‐linear continuous systems subject to analogous stochastic perturbations. All calculations can be handled on‐line by standard ordinary differential equations integration software. Copyright © 2007 John Wiley & Sons, Ltd.

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy and in the absence of it. This model also contains a control function responsible for the transition from the stage of such therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the problem of minimizing the cancer cells load both during the entire period of melanoma treatment and at its final moment is stated. Such a minimization problem has a nonconvex control set, which can lead to the absence of an optimal solution to the stated minimization problem in the classes of admissible modes traditional for applications. To avoid this problem, the control set is imposed to be convex. As a result, a relaxed minimization problem arises, in which the optimal solution exists. An analytical study of this minimization problem is carried out using the Pontryagin maximum principle. The corresponding optimal solution is found in the form of synthesis and may contain a singular arc. It shows that there are values of the parameters of the bilinear control model, its initial conditions, and the time interval for which the original minimization problem does not have an optimal solution, because it has a sliding mode. Then for such values it is possible to find an approximate optimal solution to the original minimization problem in the class of piecewise constant controls with a predetermined number of switchings. This research presents the results of the analysis of the connection between such an approximate solution of the original minimization problem and the optimal solution of the relaxed minimization problem based on numerical calculations performed in the Maple environment for the specific values of the parameters of the bilinear control model, its initial conditions, and the time interval.

This paper studies the realizability property of bilinear input-output (i/o) models in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The complete list of 2nd and 3rd order realizable input-output bilinear models together with the corresponding state equations is given. In the general case some subclasses of realizable bilinear models together with their state-space realizations are presented, including the diagonal bilinear model and the special subclass of the superdiagonal bilinear model.

The modeling of nonlinear effects in bilinear systems using Wiener-Hammerstein models is considered. Such models are chosen based on the block-oriented structure of bilinear systems and the shapes of their Volterra kernels. Theoretical analysis is given for first-order bilinear systems, and simulation results are presented for first- and second-order systems. While the nonlinearity is only approximately Wiener-Hammerstein, the models are able to capture a significant part of the nonlinear dynamics. In order to reduce the complexity of optimizing the model parameters, the linear subsystems are estimated using the technique of linear interpolation in the frequency domain.