Relaxed Stability Conditions Research Articles

Elastic full-waveform inversion based on GPU accelerated temporal fourth-order finite-difference approximation

Elastic full-waveform inversion (FWI) estimates the subsurface velocity model by inverting multi-component seismic data. Most of the current elastic FWI solves the elastic wave equation by a temporal second-order finite-difference time-domain (FDTD) method. However, the traditional temporal second-order FDTD introduces large temporal dispersion in the case of a large time-stepping size. To suppress the temporal dispersion, a relatively small time step should be used, which increases the computational cost of FWI. We introduce a temporal fourth-order and spatial arbitrary even-order FDTD method for elastic wave modeling and inversion. The temporal fourth-order scheme can simulate a highly accurate wavefield, even when a large time step is used for temporal extrapolation. Compared with the traditional temporal second-order FDTD scheme, the temporal fourth-order FDTD enables a larger time step for its more relaxed stability condition and smaller dispersion. As a result, FWI with temporal fourth-order FDTD scheme can achieve high accuracy inversion without reducing much computational efficiency. We run our elastic FWI codes on a Graphic Processing Unit (GPU) computing platform, and to accelerate the GPU computing, we propose to use the GPU shared memory to reduce the data access delay. By transferring the data frequently accessed for the FDTD approximations, from the global memory to the shared memory, we realize the FDTD calculation in a more efficient way. Forward modeling and FWI examples are carried out to confirm the effectiveness of our elastic FWI.

Relaxed Sum-of-Squares Based Stabilization Conditions for Polynomial Fuzzy-Model-Based Control Systems

This paper presents a relaxed stabilization condition for polynomial-fuzzy-model-based control system by using sum-of-squares (SOS) approach. Full-block S-procedure is utilized to equivalently decouple the fuzzy weighting constraint into two constraints, which can be computed directly with the SOS approach. Besides, a free full-block multiplier imposed by this procedure also contributes to reducing the conservativeness of stability analysis. The copositivity verification relaxation performed on the membership-functions-related constraint can both obtain more relaxed results and reduce the computation complexity. The concept of polynomial Lyapunov function (PLF) with square matrical representation, which simplifies the stabilization condition by avoiding partial derivatives of a polynomial matrix, is also introduced to generalize the existing expression of PLFs. Then, the derived stabilization condition, which is represented in terms of SOS can be numerically solved. Three simulation examples are conducted to illustrate the effectiveness and applicability of the proposed method. These examples also demonstrate the advantages of the proposed method over the existing convex SOS approach.

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

SummaryIn adaptive control of uncertain dynamical systems, it is well known that the presence of actuator and/or unmodeled dynamics in feedback loops can yield to unstable closed‐loop system trajectories. Motivated by this standpoint, this paper focuses on the analysis and synthesis of multiple adaptive architectures for control of uncertain dynamical systems with both actuator and unmodeled dynamics. Specifically, we first analyze model reference adaptive control architectures with standard, hedging‐based, and expanded reference models for this class of uncertain dynamical systems and develop sufficient stability conditions. We then synthesize a robustifying term for the latter architecture and analytically show that this term can allow for a relaxed sufficient stability condition. The proposed theoretical treatments involve Lyapunov stability theory, linear matrix inequalities, and matrix mathematics. Finally, we compare the resulting sufficient stability conditions of the considered adaptive control architectures on a benchmark mechanical system subject to actuator and unmodeled dynamics.

New relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems

AbstractThis paper develops relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems based on the extended nonquadratic Lyapunov function, the nonparallel distributed compensation law, and the convexity of the fuzzy blending coefficients. Three new main results are proposed to further reduce the conservatism by fully exploring the slack matrix technique and introducing new slack matrices and extra collection matrices. The new stabilization conditions are gradually less and less conservative, and more advantageous than the existing results by overall consideration of the conservatism and the computational efforts. A well‐known numerical case and a practical case are carried out to demonstrate the effectiveness of the proposed stabilization conditions.

Relaxed Robust Stabilization Conditions for Nonhomogeneous Markovian Jump Systems with Actuator Saturation and General Switching Policies

This paper investigates the robust control synthesis problem of nonhomogeneous Markovian jump systems with actuator saturation and uncertainties. First of all, to make a comprehensive mode-transition description, three separate mode sets are established according to the property of transition rates. And then, to improve the numerical solvability of inequality conditions dependent on time-varying transition rates, two useful relaxation methods are developed on the basis of the mode sets. Especially, by taking advantage of a new zero-sum constraint, the relaxation process is further evolved such that the conservatism arising from incomplete knowledge of transition rates can be reduced. Lastly, this paper provides the LMI-based conditions capable of estimating the attraction domain as well as of designing a mode-dependent robust control.

An efficient numerical model for liquid water uptake in porous material and its parameter estimation

The goal of this study is to propose an efficient numerical model for the predictions of capillary adsorption phenomena in a porous material. The Scharfetter–Gummel numerical scheme is proposed to solve an advection–diffusion equation with gravity flux. Its advantages such as accuracy, relaxed stability condition, and reduced computational cost are discussed along with the study of linear and nonlinear cases. The reliability of the numerical model is evaluated by comparing the numerical predictions with experimental observations of liquid uptake in bricks. A parameter estimation problem is solved to adjust the uncertain coefficients of moisture diffusivity and hydraulic conductivity.

Semi‐implicit FDTD TW scheme for modelling effective surface plasmons

A semi-implicit finite-difference time-domain thin-wire (TW) scheme with a relaxed stability condition is proposed for modelling effective surface plasmons in waveguide metamaterials. Its formulation is based on introducing the Newmark-Beta method into the TW formulation based on the contour-path integral of Maxwell's equations around a wire. The resultant scheme is magnetically implicit. Its stability, accuracy and efficiency are assessed in the application to a wire-added parallel-plate waveguide perturbed with a small gap.

A Relaxed Stability Condition of Weakly Renegotiation-Proof International Environmental Agreement Between Asymmetric Countries

This study shows that asymmetry among countries can develop a stability of international environmental agreements, using a repeated game model. Our model reveals relaxed conditions for a weakly renegotiation-proof (WRP) equilibrium by selecting the punishment levels according to the types of deviating country. Furthermore, we identified the minimum number of participating countries necessary for relaxation of a WRP condition.

Hybrid Newmark-conformal FDTD modeling of thin spoof plasmonic metamaterials

The aim of this paper is to present a new efficient and accurate semi-implicit finite-difference time-domain (FDTD) method for modeling thin plasmonic metamaterials supporting spoof surface plasmons. The presented method is formulated by combining Yu–Mittra's conformal FDTD technique for dielectrics, the simplified conformal finite integration technique for perfectly electric conductors (PECs) and the Newmark-Beta method. The resultant scheme is magnetically implicit, has a more relaxed stability condition than that of Yee's FDTD scheme, and allows for accurate modeling of both PEC and dielectric curved surfaces. Energy conservation and stability of the presented scheme is theoretically proved with the energy inequality method. Its formulation can be generalized to a class of magnetically implicit FDTD schemes with weakly conditional and unconditional stability in a unified manner. The presented scheme is used to simulate thin structures supporting spoof surface plasmon polaritons and spoof localized surface plasmons, and also applied to recent plasmonic devices such as planar frequency splitter, ultrathin rainbow trapping device and compact planar metadisk with split-ring resonators. The accuracy and efficiency of the presented scheme are assessed in comparison with the conventional explicit and implicit FDTD methods, and their conformal variants.

Event-Based Control for Networked T–S Fuzzy Systems via Auxiliary Random Series Approach

This paper presents an auxiliary random series approach to model the effect of network induced problems, such as data losses and transmission delay subject to event-based communication scheme for nonlinear continuous time systems. T-S fuzzy model is employed to describe the nonlinear systems. In order to save the bandwidth and energy, we introduce the event-triggered mechanism to reduce the number of data for transmission and computation. Thus, it is necessary to consider the influence of data losses, data disorder, and transmission delay since the transmitted data packets become more important. Consequently, it is very complicated to analyze the performance of such networked system and one of the most difficult part, in the authors' opinion, is to construct the mathematical model of closed-loop systems. In this paper, we present an auxiliary random series approach to describe the data transmitted in the system, and therefore, the closed-loop systems can be obtained. Associated with a tailor-made Lyapunov-Krasovskii functional, the stability analysis is processed and a fuzzy controller is designed. Asynchronous membership functions are considered to obtain more relaxed stability conditions. To clarify the effectiveness of the proposed method, a cart-damper-spring system is employed for simulation.

New Relaxed Stability Conditions for Uncertain Two-Dimensional Discrete Systems

This paper is concerned with the problem of robust stability of uncertain two-dimensional (2-D) discrete systems described by the Roesser model with polytopic uncertain parameters. Based on a newly developed parameter-dependent Lyapunov–Krasovski functional combined with Finsler’s lemma, new sufficient conditions for robust stability analysis are derived in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness and less conservatism of the proposed results.

Optimally robust H∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm

ABSTRACTThis paper proposes an optimally robust H∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm (QEA) for continuous/discrete time polynomial fuzzy systems with model uncertainties and external disturbances. To improve control performance, QEA is adopted to evolve optimal control gains with a fitness function that is defined by performance requirements. The stability and robustness of the control system are then guaranteed by the proposed robust H∞ stability conditions, which are formed by the sum of squares (SOS) method. By using the principle of copositivity, novel relaxed SOS-based stability conditions are derived to reduce the conservativeness of solving SOS-based stability conditions, while the feasible solution space is broadened. Four numerical examples demonstrate the effectiveness of the proposed approaches.

Structural Health Monitoring of Damaged Beams Using an Improved Variational Vibration Model

This paper presents a novel modeling and analysis process that can be used in the structural health monitoring process. A previously developed variational vibration model for structural health monitoring purposes of damaged Euler–Bernoulli beams with geometric variations used the central difference method to model the beam. This formulation had a severe restrictive stability condition, which increased computational cost and could produce unstable results. This research proposes a new method to overcome this drawback by introducing the weighted-average and Du Fort–Frankel method into the damage modeling procedure. In the presented two examples, the stability analyses of the proposed two methods are discussed clearly. They show that the weighted-average method is unconditionally stable, whereas the Du Fort–Frankel method has a relaxed stability condition. Moreover, the uncertainties in machining tolerances are considered in two examples, and a stochastic simulation process (namely, the Monte Carlo simulation) is applied to quantify the corresponding uncertainty effects on the system performance. The results from the validation examples clearly show that the proposed method can induce a significant reduction in the computational cost and can be used in structural health monitoring applications.

Membership-Function-Dependent Stability and Stabilization Conditions for T–S Fuzzy Time-Delay Systems

ABSTRACTIn this paper, the stability analysis and stabilization of time-delay nonlinear systems represented by a Takagi–Sugeno fuzzy model are investigated. Combining with a recently developed integral inequality, relaxed delay-dependent stability conditions are derived by means of linear matrix inequalities. To obtain the further relaxed stability criteria, the membership-function-dependent method is applied via the piecewise membership functions (PMFs) and some slack variables. The PMFs are selected to approximate the membership functions (MFs), and then the characteristics of the MFs are taken into consideration adequately. Under the imperfect premise matching design scheme, this paper also proposes a flexible fuzzy controller design strategy that the fuzzy controllers do not employ the identical MFs and the number of rules as the fuzzy models. Finally, this proposed approach is tested successfully and its effectiveness is validated by the simulation examples.

Static Output Feedback Stabilization of Positive Polynomial Fuzzy Systems

This paper deals with the static output feedback stabilization of positive polynomial fuzzy-model-based control systems. The positive polynomial fuzzy model does not need to share the same premise membership functions with the static output feedback polynomial fuzzy controller. Unlike the state feedback control case, the static output feedback control usually leads to nonconvex stability conditions. To circumvent the problem, an approach is employed to transform the nonconvex stability conditions into convex ones by introducing a nonsingular transformation matrix. Initially, the conditions guaranteeing the resultant closed-loop systems to be positive and asymptotically stable are obtained. Moreover, the divisional approximated membership functions that carry the local information of the membership functions are employed to facilitate the stability analysis and controller synthesis. The relaxed stability conditions in terms of sum of squares are obtained based on Lyapunov stability theory. Finally, a simulation example is given to testify the validity of the analysis result.

Tensor Product Model-based Control Design with Relaxed Stability Conditions for Perching Maneuvers

Tensor Product Model-based Control Design with Relaxed Stability Conditions for Perching Maneuvers

Efficient Stochastic Asymptotic-Preserving Implicit-Explicit Methods for Transport Equations with Diffusive Scalings and Random Inputs

For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based asymptotic-preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales. Compared with previous methods for these problems, our new method uses the implicit-explicit time discretization to gain higher order accuracy, and by using a modified diffusion operator based penalty method, a more relaxed stability condition---a hyperbolic, rather than parabolic, CFL stability condition---is achieved in the case of a small mean free path in the diffusive regime. The stochastic asymptotic-preserving property of these methods will be shown asymptotically and demonstrated numerically, along with a computational cost comparison with previous methods.

Novel Lyapunov–Krasovskii functional with delay-dependent matrix for stability of time-varying delay systems

This paper investigates the stability criteria of time-varying delay systems with known bounds of the delay and its derivative. To obtain a tighter bound of integral term, quadratic generalized free-weighting matrix inequality (QGFMI) is proposed. Furthermore, a novel augmented Lyapunov–Krasovskii functional (LKF) are constructed with a delay-dependent matrix, which impose the information for a bound of delay derivative. Relaxed stability condition using QGFMI and LKF provides a larger delay bound with low computational burden. The superiority of the proposed stability condition is verified by comparing to recent results.

Robust Constrained Model Predictive Control for Discrete‐Time Uncertain System in Takagi‐Sugeno's Form

AbstractIn this paper, we investigate a robust constrained model predictive control synthesis approach for discrete‐time Takagi‐Sugeno's (T‐S) fuzzy system with structured uncertainty. The key idea is to determine, at each sampling time, a state feedback fuzzy predictive controller that minimizes the performance objective function in the infinite time horizon by solving a class of linear matrix inequalities (LMIs) optimization problem. To do this, the fuzzy predictive controller is designed on the basis of non‐parallel distributed compensation (non‐PDC) control law, relaxed stability conditions of the closed‐loop fuzzy system are developed by employing an extended nonquadratic Lyapunov function and introducing additional slack and collection matrices. In addition, the presented approach is capable of ensuring the robust asymptotic stability as well as the recursive feasibility of the closed‐loop fuzzy system. Simulations on a highly nonlinear continuous stirred tank reactor (CSTR) are eventually presented to demonstrate the effectiveness of the developed theoretical approach.

Design of Fuzzy Functional Observer-Controller via Higher Order Derivatives of Lyapunov Function for Nonlinear Systems.

In this paper, we investigate the stability of Takagi-Sugeno fuzzy-model-based (FMB) functional observer-control system. When system states are not measurable for state-feedback control, a fuzzy functional observer is designed to directly estimate the control input instead of the system states. Although the fuzzy functional observer can reduce the order of the observer, it leads to a number of observer gains to be determined. Therefore, a new form of fuzzy functional observer is proposed to facilitate the stability analysis such that the observer gains can be numerically obtained and the stability can be guaranteed simultaneously. The proposed form is also in favor of applying separation principle to separately design the fuzzy controller and the fuzzy functional observer. To design the fuzzy controller with the consideration of system stability, higher order derivatives of Lyapunov function (HODLF) are employed to reduce the conservativeness of stability conditions. The HODLF generalizes the commonly used first-order derivative. By exploiting the properties of membership functions and the dynamics of the FMB control system, convex and relaxed stability conditions can be derived. Simulation examples are provided to show the relaxation of the proposed stability conditions and the feasibility of designed fuzzy functional observer-controller.