Recursive Least Squares and Multi-innovation Gradient Estimation Algorithms for Bilinear Stochastic 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>

Krylov subspace methods for model order reduction of bilinear control systems

A connection between multi-linear and Volterra systems

It has been a common consensus that general techniques for stabilization of nonlinear systems are available only for some special classes of nonlinear systems. Control design for nonlinear systems with uncertain components is usually carried out on a per system basis, especially when physical control constraints, and certain control performance measures such as optimum time control are imposed. Elegant adaptive control techniques are difficult to apply to this type of problems. A new neural network based control design is proposed and presented in this paper to deal with a special class of uncertain nonlinear systems with multiple inputs. The desired system dynamics are analyzed and utilized in the process of the proposed intelligent control design. The theoretical results are provided to justify the design procedures. The simulation study is conducted on a second-order bilinear system with two inputs and uncertainties on its parameters. The simulation results indicate that the proposed design approach is effective.

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.

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.

Analysis of bilinear noise models in circuits and devices

In this paper, we discuss the identification problem of multi-input output-error moving average systems. A filtering based multi-innovation stochastic gradient algorithm with forgetting factors is derived by using the filtering technique with the multi-innovation identification theory, and a multi-innovation extended stochastic gradient algorithm is given for comparison. The simulation results confirm that the proposed algorithms can generate highly accurate parameter estimates compared with the multi-innovation stochastic gradient algorithm.

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.

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.

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.

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.

In this paper we concerned with the application bilinear robust control for virotherapy model. In designing the controller, it requires a solution to the state dependent algebraic Riccati equation (SDARE). However, it is difficult to solve the SDARE. Successive method is one of the methods that can be used to solve this issue. The idea of this method is converting the bilinear systems into time-varying linear system. This method has the following step: firstly, we need to obtain the robust control for the linear system by ignoring the multiplicative term of bilinear system. Second, convert the bilinear systems into the time-varying linear systems using the previous result, and then solve the SDARE by the new performance index and the associated Hamilton-Jacobi-Isaacs equation. Last, iterate the steps until the convergence of state satisfied. The virotherapy model has been widely developed and can be considered as a model in bilinear system. There are four groups in this model: quiescent cells (Q), cancer cells (S), virus (V), and infected cells (I). Virus are injected into the human body as the control input to control the amount of the cancer cells. In this case, virus can only infect the cancer cells, and the infected cells will die when the lysis process occurs. Virus, as a control, is given with the aim of minimizing the energy used in the system. In this model we consider the body's immune response as an additive disturbance to the model. From the simulation results, it is shown that virotherapy can reduce the number of cancer cells in the body on day 50th, so the number of cancer cells in the body is only 16.6%. Based on the simulation, the next virototherapy can be done after the day 50th.

In this paper, a Hamilton-Jacobi-Bellman (HJB) equation based optimal control algorithm is proposed for a bilinear system. Utilizing the Lyapunov direct method, the controller is shown to be optimal with respect to a cost functional, which includes penalty on the control effort and the system states. In the proposed algorithm, Neural Network (NN) is used to find approximate solution of HJB equation using least squares method. Proposed algorithm has been applied on bilinear systems. Necessary theoretical and simulation results are presented to validate proposed algorithm.

Identification for multirate multi-input systems using the multi-innovation identification theory