Linear Time-invariant Systems Research Articles

Stability Analysis Using Quadratic Constraints for Systems With Neural Network Controllers

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal innerapproximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. This in turn allows for off-by-one IQCs to refine the description of activation functions by capturing their slope restrictions. Both results rely on semidefinite programming to approximate the ROA. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty.

Broadband Time-Modulated Absorber beyond the Bode-Fano Limit for Short Pulses by Energy Trapping

Wideband absorption is a popular topic in microwave engineering to protect sensitive devices against broadband sources. However, the Bode-Fano criterion defines the trade-off between bandwidth and efficiency for all passive linear time-invariant systems. In this paper, we propose a broadband absorber beyond the Bode-Fano limit by creating an energy trap using time-modulated switches or diodes. This work starts with an ideal-circuit model to prove the concept, followed by two electromagnetic realizations---a frequency-selective-surface approach for general bandwidth broadening and a low-profile printed-circuit-board design. The prototype of the latter is built and measured, demonstrating a Bode-Fano integral larger than 1. This approach paves the way to many practical ultrawideband-absorber designs.

An Ellipsoidal Predictor–Corrector State Estimation Scheme for Linear Continuous-Time Systems With Bounded Parameters and Bounded Measurement Errors

For linear time-invariant dynamic systems with exactly known coefficients of their system matrices for which measurements with bounded errors are available at discrete time instants, an optimal polygonal state estimation scheme was recently published. This scheme allows for tightly enclosing all possible state trajectories in presence of uncertain, but bounded, system inputs which may be varying arbitrarily within in their bounds. Moreover, this approach is also capable of accounting for uncertainty related to the measurement time instants. However, the drawback of this polygonal technique is its rapidly increasing complexity for larger system dimensions. For that reason, the polygonal state enclosures are replaced by a computationally less expensive, but nearly optimal, ellipsoidal enclosure technique in this paper. Numerical simulations for representative benchmark examples focusing both on applications with precisely known and uncertain parameters conclude this contribution.

An overview of structural systems theory

This paper provides an overview of the research conducted in the context of structural (or structured) systems. These are parametrized models used to assess and design system theoretical properties without considering a specific realization of the parameters (which could be uncertain or unknown). The research in structural systems led to a principled approach to a variety of problems, into what is known as structural systems theory. Hereafter, we perform a systematic overview of the problems and methodologies used in structural systems theory since the latest survey by Dion et al. in 2003. During this period, most of the focus seems to be on structural system’s properties related to controllability/observability and decentralized control, in the context of linear time-invariant systems, under the classic assumption that the parameters are independent and belonging to infinite fields. Notwithstanding, it is notable an increase in research in topics that go beyond such scope and underlying assumptions, as well as applications in a variety of domains. Lastly, we provide a compilation of open questions on several settings and we discuss future directions in this field.

Event‐triggered MPC for linear systems with bounded disturbances: An accumulated error based approach

A novel accumulated error based event-triggered model predictive control (MPC) algorithm is put forward on the basis of the dual-mode control strategy and state-feedback control for constrained continuous-time linear time-invariant (LTI) systems with constraints and bounded disturbances. First, a new event-triggered mechanism (ETM) is constructed based on the accumulated error (AE) between the optimal state trajectory and the real state trajectory. Next, in order to stabilize the system, the dual-mode event-triggered MPC strategy is established. Then, sufficient conditions are established to guarantee the algorithm feasibility, system stability, and to avoid the Zeno behavior. Finally, the numerical simulation shows the validity of the algorithm.

A linear algorithm for the minimal realization problem in physical coordinates with a non-invertible output matrix

In this paper we present a linear algorithm that estimates some physical parameters of a continuous-time system, described by an analytical mathematical model, when not all the state variables can be measured. The algorithm starts from the well-known subspace methods and applies some linear transformations to recover, at least partially, the estimated model in physical coordinates. Some analytical investigations and numerical experiments are shown for this method, which has general application within linear time-invariant (LTI) dynamical systems.

Analytical solution of impulse response function of finite-depth water waves

Short-term deterministic water wave prediction is of great importance to the operations of ships and offshore structures. The use of the convolution integral of input wave profiles and an impulse response function is one of the candidates to predict water waves on the basis of the linear time-invariant system. However, this approach is not often used today because of the non-causality and the absence of the analytical solution of the impulse response function of finite-depth water; only the deep water impulse response function is available. In the paper, we present the analytical solution of the impulse response function using the dispersion relation of finite-depth water waves. In addition, the prediction abilities in space and time are investigated. It is shown that non-causal influence decreases exponentially and the predictable time increases linearly as the distance between measuring and prediction points. We conducted a tank experiment to predict irregular waves based on the JONSWAP spectrum. The prediction accuracy by the impulse response function of finite-depth water is higher than that by the deep water impulse response function. Besides, we find that the errors of the use of the analytical solution and that of the numerical solution obtained by the inverse discrete Fourier transform are comparable.

Order reduction of nonlinear quasi-periodic systems subjected to external excitations

This paper presents order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov–Perron (L–P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P Transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L–P Transformation. This approach is similar to the Lyapunov–Floquet (L–F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems’ essential dynamics. This work presents reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations. The methods proposed here use the L–P Transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The ‘quasi-periodic invariant manifold’ based technique yields ‘reducibility conditions’. These conditions (referred to in the perturbation literature as resonance conditions) help us understand the system’s various types of resonant interactions. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one must preserve all these ‘resonant’ states in the reduced-order model. Thus, if the ‘reducibility conditions’ are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.

SYNTHESIS AND ANALYSIS OF LINEAR, DISCRETE AND TIMEINVARIANT SYSTEMS, USED IN THE FIELD OF COMMUNICATIONS, USING MATLAB AND SIGNAL PROCESSING TOOLBOX

Synthesis and Analysis of Linear Discrete Time-Invariant Systems Applied in Communications Using MATLAB and Signal Processing Toolbox: The paper presents the main steps in the process of synthesizing and analyzing linear discrete time-invariant systems applied in the field of communications. The simulation results of the systems implementing pre-emphasis and de-emphasis and removing DC offset are given in the paper such as impulse responses, frequency responses and pole-zero plots using MATLAB and Signal Processing Toolbox.

Event-triggered control for linear time-varying systems using a positive systems approach

This paper presents a new design for the event-triggered output feedback control of uncertain linear time-varying systems. Departing from the traditional treatments of event-triggered linear time-invariant systems, it is shown that factoring of fundamental solutions as products of solutions of matrix differential equations, together with small-gain arguments and the notions of interval observers and positive systems, lead to a new class of robust event-triggered output-feedback controllers. A key innovation is our use of new event triggers, which use vectors of absolute values instead of the usual Euclidean 2-norms. An illustrative example of a curve tracking system from marine robotics validates the efficacy of the proposed design scheme.

Stochastic model predictive control for linear systems with unbounded additive uncertainties

This paper presents two stochastic model predictive control methods for linear time-invariant systems subject to unbounded additive uncertainties. The new methods are developed by formulating the chance constraints into deterministic form, which are treated in analogy with robust constraints, by using the probabilistic reachable set. The first one is the time-varying tube-based stochastic model predictive control algorithm, which is designed by employing the time-varying probabilistic reachable sets as tubes. The second one is the constant tube-based stochastic model predictive control algorithm, which is developed by enforcing a constant tightened constraint in the entire prediction horizon. In addition, the soft constraints are proposed to associate with the state initialization in the algorithms to enhance the feasibility. The algorithm feasibility and closed-loop stability results are provided. The efficacy of the approaches is demonstrated by means of numerical simulations.

On the prescribed-time attractivity and frozen-time eigenvalues of linear time-varying systems

A system is called prescribed-time attractive if its solution converges at an arbitrary user-defined finite time. In this note, necessary and sufficient conditions are developed for the prescribed-time attractivity of linear time-varying (LTV) systems. It is proved that the frozen-time eigenvalues of a prescribed-time attractive LTV system have negative real parts when the time is sufficiently close to the convergence moment. This result shows that the ubiquitous singularity problem of prescribed-time attractive LTV systems can be avoided without instability effects by switching to the corresponding frozen-time system at an appropriate time. Consequently, it is proved that the time-varying state-feedback gain of a prescribed-time controller, designed for an unknown linear time-invariant system, approaches the set of stabilizing constant state-feedback gains.

A Power-Efficient Multichannel Low-Pass Filter Based on the Cascaded Multiple Accumulate Finite Impulse Response (CMFIR) Structure for Digital Image Processing

The author offers a power-efficient multichannel low-pass filter for digital image processing based on the cascade multiple accumulate finite impulse response (CMFIR) structure in this study. The CMFIR filter was created using the outputs of a linear time-invariant system (LTI), which was built using a cascaded integrator comb (CIC) and a MAC low-pass filter. The sample rate convertor based on CIC filters effectively conducts decimation or interpolation. The sample rate convertor with the CIC filter can only accommodate narrowband transmissions and so cannot be utilized for wideband signals. The MAC architecture-based sample rate convertor is a good solution for high-bandwidth signals, but it uses more resources like registers and flip-flops, which increases power consumption. Here, the CMFIR low-pass filter acts as an interpolator, introducing a sample to boost the image's resolution. CMFIR is a useful tool for addressing the issue of aliasing during sampling. In addition, the genetic algorithm was used to increase the filter's resource utilization and power consumption efficiency.

On noise covariance estimation for Kalman filter-based damage localization

In Structural Health Monitoring, Kalman filters can be used as prognosis models, and for damage detection and localization. For a proper functioning, it is necessary to tune these filters with noise covariance matrices for process and measurement noise, which are unknown in practice. Therefore, in the presented work, we apply an autocovariance least-squares method with semidefinite constraints solely based on model parameters. We facilitate this novel approach by formulating the considered innovations covariance function in infinite horizon, which follows inherently from the assumption of linear time-invariant systems. For damage analysis, we adapt a framework based on state-projection estimation errors that was recently established, and yet only applied using H∞ filters. These estimators represent an alternative to Kalman filters, and are considered robust. Because of this property, the necessity of filter tuning is relaxed, and a naive design is often considered. Based on the damage analysis framework, we derive a new damage indicator that features a high sensitivity towards localized damage. We demonstrate the efficacy of the proposed schemes for noise covariance estimation and damage analysis in a series of simulations inspired by a preceding laboratory test. We finally offer experimental validation, based on vibration test data of a cantilever beam featuring damages at multiple positions, where high sensitivity towards small local stiffness changes is achieved. In our investigations, we compare the damage detection and localization performance of Kalman and H∞ filters as well as differences in mode shape curvatures (MSC). In the simulation studies, the proposed Kalman filter-based approach outperforms the alternative strategy using H∞ estimators. The experimental investigations demonstrate a significantly higher sensitivity of the filters towards localized damage compared to differences in MSCs. Considering the totality of investigations, the combined application of both estimators can lead to an increased robustness and sensitivity regarding damage detection and localization.

A GMM-Based Secure State Estimation Approach against Dynamic Malicious Adversaries

We consider the secure state estimation of linear time-invariant Gaussian systems subject to dynamic malicious attacks. An error compensator is proposed to reduce the impact of local error data on state estimation. Based on that, a new estimation algorithm based on the Gaussian mixture model (GMM) aiming at dynamic attacks is proposed, which can cluster the local state estimates autonomously and improve the remote estimation accuracy effectively. The superiority of the proposed algorithm is verified by numerical simulations.

Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.

Fast actuator and sensor fault estimation based on adaptive unknown input observer

This study evaluates the robust fault estimation problem of systems with actuator and sensor faults though the simultaneous use of unknown input disturbances and measurement noise. Specifically, an augmented descriptor system is preliminarily developed by creating an augmented state consisting of system states and sensor faults. Next, a novel fast adaptive unknown input observer (FAUIO) is proposed for the system to enhance its fault estimation performance. The existence condition of the novel FAUIO is then introduced for linear time-invariant systems with unknown input disturbances. Furthermore, the proposed FAUIO is extended to a class of Lipschitz nonlinear systems with unknown input disturbances and measurement noise to investigate the robust fault estimation problem. Accordingly, an H∞ performance index is employed to attenuate the influence of disturbances on fault estimation. Moreover, the linear matrix inequality (LMI) technique is applied to solve the designed FAUIO. Finally, the effectiveness of the developed FAUIO is validated via the simulation of two examples.

Prediction-based controller for linear systems with stochastic input delay

We study the stabilization problem of a linear time-invariant system with an unknown stochastic input delay. We propose to robustly compensate for the stochastic delay with a constant time horizon prediction-based controller. We prove the mean-square exponential stabilization of the closed-loop system under a sufficient condition, which requires the range of the delay values to be sufficiently narrow and the constant delay used in the prediction-based controller to be chosen in this range. Numerical simulations illustrate the relevance of this condition and the merits of our control design.

A unified non-linear system model view of hyperelasticity, viscoelasticity and hysteresis exhibited by rubber

Rubber modeling is an old subject and so many models exist that it is difficult to have a clear vision of what exists and is more appropriate. Rather than attempting a standard review, this paper proposes classification using the traditional system modeling strategy, where raw measurements are either processed to obtain non-parametric models, or used to identify parametric models, whose accuracy can be controlled by order selection or by numerical implementation considerations. A full test campaign, including multi-step relaxation, low speed triangular and sine tests, on a large deformation compression sample is used to illustrate the need to model and combine the base behaviors known as hyperelasticity, viscoelasticity, and rate independent hysteresis. The equivalence between linear viscoelasticity and linear time invariant systems is used to clarify the link between order selection and accuracy of a generalized Maxwell model. Rate independent hysteresis is analyzed using a convolution product like the one used for viscoelastic transients by introducing a relaxation modulus. Measurements of the hysteretic relaxation modulus are used to propose strategies to measure the asymptotic hyperelastic modulus and discriminate between different hysteretic model forms. A parallel between Iwan and Maxwell models is detailed, and non-parametric models are used to show that the two overlap in the low frequency small deformation regime. Regularized rate independent hysteresis and non-linear viscoelasticity are finally shown to lead to a similar view allowing a transition between the rate independent and linear relaxation models. The instantaneous ratio analytic force and displacement signals, or instant complex modulus, is introduced as novel non-parametric estimation of sine measurements and shown to be a powerful tool to analyze and validate the fact that a force rate relaxation with non-linear relaxation frequencies is most appropriate to represent the non-linear coupling of all three effects.

Phase Preserving Balanced Truncation for Order Reduction of Positive Real Systems

This paper presents a new passivity-preserving order reduction method for linear time-invariant passive systems, which are also called positive real (PR) systems, with the aid of the balanced truncation (BT) method. The proposed method stems from the conic positive real balanced truncation (CPRBT) method, which is a modification of the BT method for PR systems. CPRBT presents an algorithm in which the reduced models are obtained from some Riccati equations in which the phase angle of the transfer function has been taken into consideration. Although CPRBT is a powerful algorithm for obtaining accurate PR reduced-order models, it cannot guarantee that the phase diagram of the reduced model remains inside the same interval as that of the original full-order system. We aim to address such a problem by modifying CPRBT in the way that the phase angle of the reduced transfer function always remains inside the conic and homolographic phase interval of the original system. This is proven through some matrix manipulations, which has added mathematical value to the paper. Finally, in order to assess the efficacy of the proposed method, two numerical examples are simulated.