Computation of nonlinear balanced realization and model reduction based on Taylor series expansion

Model reduction of two-dimensional (2-D) Separable-in-Denominator Digital Filters (SDDF) using frequency domain balanced realization is proposed. The frequency domain controllability and observability Grammians are introduced and their appropriate Lyapunov equations are developed. The approach could be viewed as the generalization of the existing balanced structure approach in a specific frequency range of operation. Various properties of the proposed frequency domain balanced structure are investigated. A comparison study of the proposed method with the available techniques is presented using numerical examples

Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

An effective nonlinear model reduction approach, empirical Gramians balanced reduction approach, is studied, in order to reduce the computation complexity in nonlinear power system model application. The realization procedure is: firstly, computing the empirical controllable and observable Gramians matrices, secondly, computing the balance transformation matrix to obtain the balanced system model of the original model, then, and obtaining the diagonal Hankel singular values of balance transformation matrix. Finally, deciding the lower-order subspace to obtain the reduced model. A 15-machine nonlinear power system model is taken as an example to perform the reduction simulation analysis. Simulation result shows that, under the condition to maintain the stability and input-output dynamic behavior of the original system, the order of the reduced system model could be as low as 1/5 of that of the original system model, verifying the effectiveness of the proposed approach.

Due to the serious drift phenomenon of underactuated air cushion vehicle, the actual trajectory is determined by the total speed and course angle. In this paper, the course angle and total speed model are derived from the general four degrees of freedom air cushion vehicle model and named nonlinear vector model. Nonlinear vector model can be used to directly design the course and total speed controllers for underactuated air cushion vehicle. Adaptive radial basis function neural network is introduced to deal with the strong nonlinearity and uncertainty of air cushion vehicle’s complex dynamics. However, the adaptive weights to be calculated and updated may be too many in each sampling period. For the relief of the burden caused by the online computing, parameter reduction algorithm is designed in this paper. It gives us a power to choose the number of online update parameters freely. Then the new trajectory tracking control method with independent total speed and course controller is designed based on nonlinear vector model and parameter reduction algorithm. The designed controller ensures that the tracking errors are uniformly ultimately bounded. Also, only a few weights need to be updated online. The effectiveness and superiority of the designed controller is verified by simulation results.

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

This paper presents a new method for the design of separable-denominator 2-D IIR filters with nearly linear phase in the passband. The design method is based on a balanced realization model reduction technique. The nearly linear-phase 2-D IIR filter is designed using 2-D model reduction from a linear-phase 2-D FIR filter, which serves as the initial filter. The structured controllability and observability Gramians Ps and Qs serve as the foundation for this technique. onal positive-definite matrices that satisfy 2-D Lyapunov equations. An efficient method is used to compute these Gramians by minimizing the traces of Ps and Qs under linear matrix inequality (LMI) constraints. The use of these Gramians ensures that the resulting 2-D IIR filter preserves stability and can be implemented using a separable-denominator 2-D filter with fewer coefficients than the original 2-D FIR filter. Numerical examples show that the proposed method compares favorably with existing techniques.

In this paper the standard optimal Hankel-norm model reduction and balanced model reduction are generalized to optimal Hankel-norm model reduction and balanced model reduction over a disc in the complex plane. Important properties of the new model-reduction concept such as the monotone property of the generalized Hankel norm with respect to the domain, and the L ∞ error bound over a disc D are given. Applications like the approximation of unstable systems and model reduction for specific working signals are discussed.

The pneumatic drive as a thermodynamic system is described based on the principles of "thermodynamics of a body of variable mass". In describing thermodynamic and gas-dynamic processes in pneumatic drives, two scientific schools have developed: researchers of the first school represent processes in the cavities of the drive as polytropic processes with a variable polytropic index, and the second consider processes based on the equation of energy (heat) balance of the gas-pneumatic drive in an open space. The authors investigate a nonlinear mathematical model obtained on the concepts of the second school. In the cycles of reducing the number of independent parameters that determine the nature of the transient process in the drive, a transition to a dimensionless form of equations is carried out based on the principle of minimizing dimensionless complexes (dynamic similarity criteria) that determine the nature of the transient process. The task of linearization of the nonlinear model was set with the purpose of obtaining analytical expressions for all state variables based on the linear model, which will allow avoiding the numerical step methods of integration of the original nonlinear model in calculations. It is shown that the replacement of nonlinear dependencies by the first terms of their expansion in a Taylor series (the tangent method), which is practiced for servo hydraulic pneumatic drives, leads to large errors in relation to discrete drives. The proposed linearization by the secant method with the choice of the optimal form of the sowing allowed to significantly increase the calculated accuracy of the linear mathematical model. and obtain the state variables of the pneumatic drive in analytical form. The calculations performed using the second-order linear model convincingly indicate that it is quite adequate to the calculated accuracy of the nonlinear mathematical model. In the entire range of the most probable parameters, the calculated accuracy of the second-order mathematical model is quite sufficient for practical use. Thus, the need to use step-by-step numerical methods for integrating the equations of a nonlinear model and organizing the computational process on a computer is eliminated.

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so called singular value functions of a system describe the relative importance of each state component from an input-output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, it appears that the singular value functions as currently defined are dependent on a particular factorization of the observability function. It is shown by example that in a fixed coordinate frame this factorization is not unique, and thus other distinct sets of the singular value functions and balanced realizations are possible.

A method for order reduction of nonlinear differential-algebraic models of arbitrary index is presented. The approach is a direct generalization of a method suggested by Pallaske (1987) for the reduction of explicit differential equation models. A rigorous development as well as a theoretical and numerical evaluation of the reduction technique is provided. Strong emphasis is on implementational issues such as the choice of tuning parameters for a particular problem. The case studies discussed include the reduction of a strongly nonlinear catalytic fixed bed reactor model.

Nonlinear dynamic modeling and model reduction strategy are studied in this paper for a rotating thin cylindrical shell. The nonlinear dynamic model is first established in terms of ordinary differential equations, in which the effects of Coriolis and centrifugal forces are considered, as well as the initial hoop tension due to rotation. This model describes both the in‐plane vibrations and the flexural vibration and reflects the coupling effects of those deformations. Based on this original model, a novel model reduction strategy is proposed to reduce the degrees of freedom by neglecting vibration modes predominated by in‐plane vibrations. Meanwhile, for the reduced‐order model, the in‐plane vibrations’ contributions to the rotating shell’s response are still preserved. To validate the dynamic model and the model reduction strategy, comparisons and simulations are carried out. Subsequently, nonlinear dynamic behaviors are investigated preliminarily by analyzing the rotating cylindrical shell’s amplitude‐frequency responses under different excitation levels.

A self-organizing filter and smoother for the general nonlinear non-Gaussian state-space model is proposed. An expanded state-space model is defined by augmenting the state vector with the unknown parameters of the original state-space model. The state of the augmented state-space model, and hence the state and the parameters of the original state-space model, are estimated simultaneously by either a non-Gaussian filter/smoother or a Monte Carlo filter/smoother. In contrast to maximum likelihood estimation of model parameters in ordinary state-space modeling, for which the recursive filter computation has to be done many times, model parameter estimation in the proposed self-organizing filter/smoother is achieved with only two passes of the recursive filter and smoother operations. Examples such as automatic tuning of dispersion and the shape parameters, adaptation to changes of the amplitude of a signal in seismic data, state estimation for a nonlinear state space model with unknown parameters, and seasonal adjustment with a nonlinear model with changing variance parameters are shown to exemplify the usefulness of the proposed method.

Nonlinear model predictive control has become increasingly popular in the chemical process industry. However, computational requirements grow with the complexity of the models. Many rigorous dynamic models require too much computation time to be useful for real-time model based controllers. This presents a need for model reduction techniques. The method introduced here reduces nonlinear systems, while retaining most of the input-output properties of the original system. The reduction itself is based on empirical gramians which capture the nonlinear behavior of the system in a region around an operating point. The gramians are then balanced and the less important states reduced. A Galerkin projection is performed onto the remaining states. This method has the advantage that it only requires linear matrix computations while being applicable to nonlinear systems.

A new filtering method is proposed to accurately estimate target state via decreasing the nonlinearity between radar polar measurements (or spherical measurements in three-dimensional (3D) radar) and target position in Cartesian coordinate. The degree of linearity is quantified here by utilizing correlation coefficient and Taylor series expansion. With the proposed method, the original measurements are converted from polar or spherical coordinate to a carefully chosen Cartesian coordinate system that is obtained by coordinate rotation transformation to maximize the linearity degree of the conversion function from polar/spherical to Cartesian coordinate. Then the target state is filtered along each axis of the chosen Cartesian coordinate. This method is compared with extended Kalman filter (EKF), Converted Measurement Kalman filter (CMKF), unscented Kalman filter (UKF) as well as Decoupled Converted Measurement Kalman filter (DECMKF). This new method provides highly accurate position and velocity with consistent estimation.