A Dual-Frequency Approach Based on the Subspace Method for Identifying Nonlinear Vibration Structures

Linear regressive realizations of LTI state space models

Ahlstrom Corp, Finland

In this paper, a least-mean-squares (LMS) finite memory (FM) estimator for a stochastic discrete-time state space model is obtained by taking the conditional expectation of the estimated state given a finite number of inputs and outputs measured on the recent finite horizon. Any a priori state information is not involved and any arbitrary constraints are not imposed. For a general discrete-time state space model with both system and measurement noises, the LMS FM estimator is represented in a closed-form. It turns out that the proposed LMS FM estimator has the unbiased property and the linear structure with respect to inputs and outputs on the recent finite horizon.

In this paper, we propose a subspace state space system identification method using prior knowledge. In this method, we give an algorithm to identify the partial state which corresponds to unknown modes of the system. This scheme is effective when some system poles cannot be identified because of the measurement noise or a high gain feedback controller. The effectiveness of the proposed method is evaluated by numerical examples including the comparison with the conventional 4SID method

This paper addresses identification of Hammerstein systems using the sampled input-output data in frequency domain. When the static memoryless nonlinear part of Hammerstein model can be considered as a polynomial with a known order, the following linear part can be estimated as a multi-input single-output (MISO) systems with the numerical algorithm for subspace state space system identification method. Simulation results illustrate this approach is effective.

The chapter concerns numerical issues encountered when the pipeline flow process is modeled as a discrete-time state-space model . In particular, issues related to computational complexity and computability are discussed, i.e., simulation feasibility which is connected to the notions of singularity and stability of the model. These properties are critical if a diagnostic system is based on a discrete mathematical model of the flow process. The starting point of the study is determined by the partial differential equations obtained from the momentum and mass conservation laws by using physical principles. A realizable computational model is developed by approximation of the principal equations using the finite difference method. This model is expressed in terms of the recombination matrix A which is the key of the analysis by taking into account its possible singularity and stability . The nonsingularity of the matrix A for nonzero and finite, time and spatial steps is proven by the Lower–Upper decomposition. A feature of the discrete model allows the derivation of a nonsingular aggregated model, whose stability can be analyzed. By considering the Courant–Friedrichs–Lewy condition and data from experimental studies, numerical stability conditions are derived and limitations for the feasible discretized grid are obtained. Moreover, the optimal relationship between the time and space steps which ensures a maximum stability margin is derived. Because the inverse of matrix A, composed of four tridiagonal matrices, is required for the main diagnosis methods, two analytical methods for the inversion are discussed which reduce the system’s initialization time and allow designing an accurate and fast diagnosis algorithm. By considering that each inversion method generates its particular structure, two different flow models are generated: one based on auxiliary variables and the other suitable if the stability condition of A is satisfied. The applicability of the two models is shown by considering the norm of the difference between their behaviors for a finer discretization grid . A similarity measure is proposed which considers the number of pipeline segments as well as the ratio between the time and spatial steps . Thus, the system’s computational efficiency is improved and satisfactory results are shown with respect to the base model, if a highly dimensional model with the approximated diagonal matrix is considered.

The conventional method for identifying the transfer function of an unknown linear system consists of a least squares fit of its input to its output. It is equivalent to identifying the frequency response of the system by calculating the empirical cross-spectrum between the system's input and output, divided by the empirical auto-spectrum of the input process. However, if the additive noise at the system's output is correlated with the input process, e.g., in case of environmental noise that affects both system's input and output, the method may suffer from a severe bias effect. We present a modification of the cross-spectral method that exploits nonstationary features in the data in order to circumvent bias effects caused by correlated stationary noise. The proposed method is particularly attractive to problems of multichannel signal enhancement and noise cancellation, when the desired signal is nonstationary in nature, e.g., speech or image.

In this paper, we treat the problem of decentralized implicit adaptive regulation for large-scale stochastic systems composed into a set of interconnected systems that are described by discrete-time state-space mathematical models with unknown parameters. The key idea in the decentralized regulation method is to design local regulator using only local information such that the state of each interconnected system is regulated to a certain constant reference signal. The main contribution is the proposition of a decentralized implicit adaptive regulator based on state-feedback strategy that can be applied to stochastic interconnected systems with unknown parameters. Furthermore, the practical implementation of the proposed decentralized implicit adaptive regulator can be made easily (low-cost implementation of the electronic components, short computation of the decentralized control law, etc.). A theorem is established and proved which gives sufficient stability conditions of the resulting closed-loop interconnected systems by using the Lyapunov method. An example of numerical simulation is treated to test the performance of the proposed decentralized implicit adaptive regulator.

In this paper, a reduced-order modeling approach based on computational fluid dynamics is presented for an elastic wing with control surfaces in the transonic regime. To treat the computational fluid dynamics grid around the geometrical discontinuities, due to the deflection of control surfaces, the constant volume tetrahedron method and the transpiration method are combined together without deforming the grid. Based on the input–output data from the computational fluid dynamics solver, one multiple-input/multiple-output discrete-time state-space model for the wing is identified via a robust subspace algorithm. For each control surface, one one-input/multiple-output discrete-time state-space model is identified using the same algorithm. With the precomputed state-space models for a few flight parameters, the generalized aerodynamic forces over a range of flight parameters can be computed by interpolating the output data. The methodology is applied to an elastic wing model with two control surfaces...

Subspace-based identification for continuous-time stochastic systems via distribution-theoretic approach

Discrete-time state-space models have been extensively used in simulation-based design of dynamical systems. These prediction models may not accurately represent the true physics of a dynamical system due to potentially flawed understanding of the system, missing physics, and/or numerical approximations. To improve the validity of these models at new design locations, this paper proposes a novel dynamic model discrepancy quantification (DMDQ) framework. Time-instantaneous prediction models are constructed for the model discrepancies of “hidden” state variables, and are used to correct the discrete-time prediction models at each time-step. For discrete-time models, the hidden state variables and their discrepancies are coupled over two adjacent time steps. Also, the state variables cannot be directly measured. These factors complicate the construction of the model discrepancy prediction models. The proposed DMDQ framework overcomes these challenges by proposing two discrepancy modeling approaches: an estimation-modeling approach and a modeling-estimation approach. The former first estimates the model discrepancy and then builds a nonparametric prediction model of the model discrepancy; the latter builds a parametric prediction model of the model discrepancy first and then estimates the parameters of the prediction model. A subsampling method is developed to reduce the computational effort in building the two types of prediction models. A mathematical example and an electrical circuit dynamical system demonstrate the effectiveness of the proposed DMDQ framework and highlight the advantages and disadvantages of the proposed approaches.

Discrete-time models with state-dependent parameters are derived for non-linear sampled-data systems. Velocity-based linearization of the continuous-time system followed by integration over the sampling interval is used to represent the sampled-data system by a discrete-time state-space model with state-dependent parameters. By reconstructing the state a quasi-ARX model is obtained, the parameters of which are functions of past inputs and outputs. As the velocity-based linearization procedure does not involve suppression of higher-order terms, the state-dependent state-space model and the quasi-ARX description give exact representations of the non-linear system.

This paper proposes a new fixed-lag smoother for state estimation in discrete-time state space models, which is called the unbiased, fixed-lag, and optimal (UFO) smoother. The proposed UFO smoother is obtained by directly solving an optimization problem with the unbiasedness constraint using only the most recent finite observations. Therefore, the proposed UFO smoother is both unbiased and optimal by design. The proposed UFO smoother also has inherent properties such as deadbeat, time-invariance, and robustness. Via numerical simulations, the performance of the proposed UFO smoother is evaluated by the comparison with that of the existing fixed-lag Kalman smoother, which shows that the proposed UFO smoother could be appropriate for fast estimation of signals that vary relatively quickly.

An adaptive LQG control with no control cost is considered. In such case the loop transfer recovery (LTR) effect can be obtained. The control problem is handled using discrete-time state-space model and the parameter estimation is performed for corresponding ARMAX model which can be represented in innovation state-space form. Thus the direct estimation of model parameters is possible by means of standard ERLS procedure and the adaptive control is implemented through certainty equivalence principle. In such a situation the problem of solution continuity of Riccati equation can arise for nonminimum-phase systems. Computer simulations of third-order systems modeled by a second-order minimum-phase and nonminimum-phase models are given to illustrate the robustness and performance properties of the adaptive controller, particularly with respect to the modelling error parameter η.

LCL filters installed at converter outputs offer a higher harmonic attenuation than L filters. However, as a three order resonant circuit, it is difficult to stabilize and has a risk of oscillating with the power grid. Therefore, careful design is required to damp LCL resonance. Compared to a passive damping method, an active damping method is a more attractive solution for this problem, since it avoids extra power losses. In this paper, the damping capabilities of capacitor current, capacitor voltage, and grid-side current feedback methods, are analyzed under the discrete-time state-space model. Theoretical analysis shows that the grid-side current feedback method is more suitable for use in active power filters, because it can damp LCL resonance more effectively than the other two methods when the ratio of the resonance and the control frequency is between 0.225 and 0.325. Furthermore, since there is no need for extra sensors for additional states measurements, this method provides a cost-efficient solution. To support the theoretical analysis, the proposed method is tested on a 7-kVA single-phase shunt active power filter.