Ellipsoidal Bound Research Articles

Dynamical analysis and reachable set estimation of T-S fuzzy system with permanent magnet synchronous motor

In this study, we will present the dynamical analysis for permanent magnet synchronous motors (PMSM). Our goal is to identify an ellipsoid that contains the state trajectory of the system in as small a form as possible in the presence of control. First, we designed the non-fragile sampled-data fuzzy controller (NFSDFC) for the PMSM model. Second, the Lyapunov-Krasovskii functional (LKF) strategy, novel integral inequality mechanisms, and certain sufficient conditions are determined, which ensure an ellipsoidal bound of reachable sets for the closed-loop of a system with input constraints derived in terms of linear matrix inequalities (LMIs). Meanwhile, under the larger sampling interval, the corresponding sampled-data controller gains are designed. Finally, numerical examples are given to validate the derived theoretical results.

Fuzzy-based asynchronous controller design on reachable set synthesis of nonlinear hidden Markovian jump systems

This article investigates the reachable set synthesis problem for delay nonlinear hidden Markovian jump systems (NHMJSs) subject to asynchronous control strategy. The purpose is to find a suitable fuzzy controller to ensure that all the states of the generated closed-loop NHMJSs are bounded in the predetermined ellipsoidal region. First of all, Takagi–Sugeno (T–S) fuzzy model is used to approximate the nonlinear dynamics systems (NDS). On this basis, the hidden Markov model (HMM) is introduced to represent the asynchronous phenomenon among the controller modes and subsystem modes. Under taking time-varying delays into account, two descriptions of the reachable set ellipsoidal bound are provided. In addition, the relevant parameters of the fuzzy asynchronous controller are obtained by selecting the Lyapunov functionals with two different mode-dependent matrices, respectively. Finally, some numerical examples are given to verify the feasibility of the method proposed in this paper.

Safety monitoring under stealthy sensor injection attacks using reachable sets

Stealthy sensor injection attacks are serious threats for industrial plants as they can compromise the plant's integrity without being detected by traditional fault detectors. In this manuscript, we study the possibility of revealing the presence of such attacks by monitoring only the control input. This approach consists in computing an ellipsoidal bound of the input reachable set. When the control input does not belong to this set, this means that a stealthy sensor injection attack is driving the plant to critical states. The problem of finding this ellipsoidal bound is posed as a convex optimization problem (convex cost with Linear Matrix Inequalities constraints). Our monitoring approach is tested in simulation.

A non-uniform event-triggered distributed filtering scheme for discrete-time nonlinear systems over wireless sensor networks

This paper investigates the distributed set-membership filtering problem for discrete-time nonlinear systems subject to one-sided Lipschitz (OSL) and quadratic inner-boundedness (QIB) constraints over wireless sensor networks (WSNs). The investigation takes into account both system and sensor nonlinearities under unknown but bounded (UBB) perturbations. Moreover, a novel non-uniform event-triggering condition is introduced to achieve resource conservation while maintaining filter accuracy. The proposed triggering condition is more general and can be customized as per design requirements. Furthermore, a recursive condition for set-membership filter design is derived that guarantees the existence of optimal bounding ellipsoids for true system states. In contrast to the existing methods, the proposed approach considers set-membership filtering for discrete-time systems subject to OSL system and sensor nonlinearities over a WSN and deals with non-uniform amplitude-based triggering. Finally, a numerical example is presented to validate filter’s efficacy.

Reachable set estimation for Markov jump LPV systems with time delays

This paper investigates the reachable set estimation problem for discrete-time Markov jump LPV systems with time-varying delays and bounded disturbance. The main consideration is that under zero initial conditions, how to obtain a bound set as small as possible that includes system states. Based on this, by applying a novel summation inequality integrated with the reciprocally convex approach under the framework of the Lyapunov functional method, the delay-dependent conditions are provided to guarantee the presence of an ellipsoid that bounds the system state in the existence of bounded disturbances. Some derived conditions expressed by linear matrix inequalities can be solved via utilizing different computational tools, and the minimum possible ellipsoidal bound can also be determined. Additionally, the obtained results are extensively used in the case with partially known transition probability, and some conditions with more universality are derived. Finally, a numerical example is given to illustrate the effectiveness of the results.

Output feedback robust MPC using general polyhedral and ellipsoidal true state bounds for LPV model with bounded disturbance

ABSTRACTThis paper considers the dynamic output feedback robust model predictive control (MPC) for a system with both polytopic model parametric uncertainty and bounded disturbance. For this topic, the techniques for handling the unknown true state are crucial, and the strict guarantee of the input/output/state constraints requires replacing the true state by its bounds in the optimisation problems. Previously, in the separate works, we (i) gave the general polyhedral bound; (ii) proposed the general ellipsoidal bound; (iii) applied some special polyhedral bounds to tighten the ellipsoidal bound since the latter is crucial for guaranteeing recursive feasibility. In this paper, (i)–(iii) are unified, and the up-to-date least conservative treatment of the true state bound is given, so the control performance can be greatly improved. The contribution mainly lies in overcoming the difficulties in developing technical details for the unification. A numerical example is given to illustrate the effectiveness of the new method.

Improved Reachable Set Bounding for Linear Neutral Systems with Disturbances via Triple Integral Functionals

This paper focuses on bound of reachable sets for linear neutral systems with bounded peak disturbances. To best of our knowledge, there were few results related to the ellipsoidal bound of reachable sets for linear neutral systems. By utilizing the Lyapunov-Krasovskii functional theory, delay decomposition technique, reciprocally convex method and free-weighting matrix approach, some new results expressed in terms of linear matrix inequalities are derived. Furthermore, triple integral functionals are introduced first time to derive bound of reachable sets for linear neutral systems. A tighter bound of the reachable set is obtained.

Observer-Based Output Feedback MPC for T-S Fuzzy System With Data Loss and Bounded Disturbance.

This paper investigates the output feedback model predictive control (OFMPC) for Takagi-Sugeno fuzzy networked control systems with bounded disturbance, where data quantization and data loss occur simultaneously. The quantization error is treated as sector bound uncertainties by using the sector bound approach and the data loss process is modeled as a time-homogeneous Markov chain. Invoking S -procedure and the notion of quadratic boundedness which can specify closed-loop stability for system with disturbance, the state observer is offline designed and the networked output feedback model predictive controller is provided which explicitly considers the satisfaction of input constraints. Two online synthesis algorithms of OFMPC are presented, one parameterizing the infinite horizon control moves into a single feedback law, the other into one free control move followed by the single feedback law based on the state observer. A new formula is introduced to refresh the ellipsoidal bound of estimation error which can guarantee the recursive feasibility of optimization problem. An example is given to demonstrate the effectiveness of the proposed new design techniques.

Reachable set estimation for discrete‐time switched system with application to time‐delay system

SummaryThis paper addresses the problem of reachable set estimation and synthesis for a class of discrete‐time switched linear systems with time delay and bounded peak disturbance. Combined with the feature of mode‐dependent average dwell time switching, a new algorithm is developed to estimate the reachable set of switched system, which is both quasi‐time‐dependent and mode‐dependent. Then, the proposed method is applied to time‐delay system and a sufficient condition is presented to guarantee the asymptotic stability and estimate the bounding ellipsoid. Furthermore, the quasi‐time‐dependent controller is designed to stabilize the system and restrict the closed‐loop system states to an ellipsoidal bound. Examples are presented to illustrate the effectiveness and advantages of the obtained theorems.

Reachable Set Estimation and Decentralized Controller Design for Large-Scale Nonlinear Systems With Time-Varying Delay and Input Constraint

This paper investigates the problem of reachable set estimation and decentralized controller design for large-scale nonlinear systems, where time-varying delays and nonlinearities appear in interconnections connected to other subsystems. The Takagi–Sugeno model is used to describe each nonlinear subsystem. The aim of this paper is to design a decentralized state-feedback fuzzy controller such that the reachable set of the resulting closed-loop system with input constraint is bounded by an intersection of ellipsoids. First, a model transformation is proposed to reformulate the closed-loop system as several feedback interconnections with extra inputs and outputs. Then, based on a combined application of the Lyapunov–Krasovskii functional and the scaled small-gain theorem, the input–output approach is developed for the reachable set estimation and synthesis. Several conditions for the existence of a decentralized state-feedback fuzzy controller that ensures an ellipsoidal bound of reachable sets for the closed-loop system with input constraint are derived in terms of linear matrix inequalities. Finally, two numerical examples are given to validate the effectiveness of the proposed strategy.

Almost sure estimation and synthesis of ultimate bounds for discrete-time Markov jump linear systems

ABSTRACTThis paper is concerned with the problems of almost sure ultimate bound estimation and controller design for Markov jump linear systems with bounded stochastic disturbances. By utilising a Lyapunov-based scheme proposed in this paper, an almost sure estimation of ellipsoidal ultimate bound (EUB) of the system is obtained through tractable matrix inequalities. On the basis of the estimation results, the problem of designing mode-dependent state feedback controllers that make the closed-loop system admit a prescribed ellipsoid as an EUB is considered. The obtained results on estimation and synthesis are then extended to the case of systems with deficient mode information. Finally, a practical example in DC motor devices is presented to demonstrate the applicability of the obtained results.

Stochastic Light Culling for VPLs on GGX Microsurfaces

AbstractThis paper introduces a real‐time rendering method for single‐bounce glossy caustics created by GGX microsurfaces. Our method is based on stochastic light culling of virtual point lights (VPLs), which is an unbiased culling method that randomly determines the range of influence of light for each VPL. While the original stochastic light culling method uses a bounding sphere defined by that light range for coarse culling (e.g., tiled culling), we have further extended the method by calculating a tighter bounding ellipsoid for glossy VPLs. Such bounding ellipsoids can be calculated analytically under the classic Phong reflection model which cannot be applied to physically plausible materials used in modern computer graphics productions. In order to use stochastic light culling for such modern materials, this paper derives a simple analytical solution to generate a tighter bounding ellipsoid for VPLs on GGX microsurfaces. This paper also presents an efficient implementation for culling bounding ellipsoids in the context of tiled culling. When stochastic light culling is combined with interleaved sampling for a scene with tens of thousands of VPLs, this tiled culling is faster than conservative rasterization‐based clustered shading which is a state‐of‐the‐art culling technique that supports bounding ellipsoids. Using these techniques, VPLs are culled efficiently for completely dynamic single‐bounce glossy caustics reflected from GGX microsurfaces.

On reachable set estimation of two‐dimensional systems described by the Roesser model with time‐varying delays

SummaryIn this paper, the problem of reachable set estimation of two‐dimensional (2‐D) discrete‐time systems described by the Roesser model with interval time‐varying delays is considered for the first time. New 2‐D weighted summation inequalities, which provide a tighter lower bound than the commonly used Jensen summation inequality, are proposed. Based on the Lyapunov‐Krasovskii functional approach, and by using the 2‐D weighted summation inequalities presented in this paper, new delay‐dependent conditions are derived to ensure the existence of an ellipsoid that bounds the system states in the presence of bounded disturbances. The derived conditions are expressed in terms of linear matrix inequalities, which can be solved by various computational tools to determine a smallest possible ellipsoidal bound. Applications to exponential stability analysis of 2‐D systems with delays are also presented. The effectiveness of the obtained results are illustrated by numerical examples.

Dynamic Output Feedback-Predictive Control of a Takagi–Sugeno Model With Bounded Disturbance

This paper considers predictive control of a Takagi–Sugeno fuzzy model with bounded disturbance, strictly satisfying the input and state constraints. The membership-function-dependent dynamic output feedback law, as in a previous work, is utilized. The novelty lies in the following technical improvements. For the notion of quadratic boundedness, which specifies closed-loop stability and invariance properties, the full Lyapunov matrix is utilized. The real-time ellipsoidal bound of true state is recursively optimized by invoking the S-procedure. Some relaxation scalars, being optimized by the norm-bounding technique, are introduced for better handling the input and state constraints. The multistep approach, where a sequence of dynamic output feedback laws are optimized at each sampling instant, is given to improve the single-step approach. Numerical examples are given to illustrate the effectiveness of the proposed controllers.

Estimation and synthesis of reachable set for discrete-time periodic systems

Summary This paper is concerned with the problems of reachable set estimation and synthesis for discrete-time periodic systems under bounded peak disturbances. For the reachable set estimation problem, the lifting approach and the pseudo-periodic Lyapunov function approach are utilized to determine the bounding ellipsoids for the reachable set. By using the lifting approach, the periodic system is transformed into several time-invariant systems; then, the bounding ellipsoids are determined through the transformed time-invariant systems. By applying the pseudo-periodic Lyapunov function approach, the bounding ellipsoids are determined through the original periodic system directly. Genetic algorithm is adopted in the pseudo-periodic Lyapunov function approach to search for the optimal value of the decision variables. Moreover, based on the reachable set estimation results, state-feedback controllers are designed for manipulating the reachable set. Finally, numerical examples are presented to verify the effectiveness of the theoretical findings. Copyright © 2015 John Wiley & Sons, Ltd.

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

On dynamic output feedback robust MPC for constrained quasi-LPV systems

This paper considers the dynamic output feedback robust model predictive control (MPC) of a quasi-linear parameter varying (quasi-LPV) system with bounded noise. In our previous works, for the unknown true state, either its ellipsoidal bounds or its polyhedral bounds were solely applied in the main optimisation problem. The recursive feasibility of the main optimisation problem was guaranteed by a simple refreshment of the ellipsoidal bound, but might be lost by applying the polyhedral bounds. This paper shows how and to what extent the recursive feasibility can be restored when the polyhedral bounds are still utilised. First, we propose a new approach which, at each sampling time, utilises either the ellipsoidal bound or the polyhedral bound in the main optimisation problem, the latter being used if and only if it is contained in the former. Then, we show the sufficient conditions under which the approaches based on polyhedral bounds preserve the property of recursive feasibility. A numerical example is given to illustrate the effectiveness of the controller.

Variational Approach to Percolation Threshold of Nanocomposites Considering Clustering Effect

AbstractTo mathematically model the phenomenon of complete dispersion, a crucial step is taken in a recent paper on isotropic formulation of the homogenized Eshelby’s tensor, leading to derivation of the ellipsoidal bound. In this paper, the writers show that in the same variational framework, the effect of clustering can be mathematically modeled as anisotropy of the homogenized Eshelby’s tensor. The degree of clustering or dispersion is accordingly represented by the aspect ratio of the ellipsoidal shape associated with the anisotropic Eshelby’s tensor. The asymptotic results and calculation demonstrate that such a new anisotropic formulation consistently leads to an increase of the percolation threshold due to the clustering effect.

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

The framework of set membership filter (SMF) with unknown-but-bounded noise assumption provides an attractive alternative for probabilistic filters. However, the potential computational burden and conservation consideration may seriously limit the usage of this filter in practical applications. In this paper, based on guaranteed bounding ellipsoid approximation, a new enhanced set membership filter with better real-time property and reduced conservation is proposed for state estimation problem of nonlinear systems. The nonlinear model is firstly linearized and the DC programming method is used to outer-bound the linearization error, which is incorporated to the model noise with ellipsoidal approximations. A classical two-step prediction-correction procedure consisting vector sum computation between ellipsoids and an iterative outer-bounding ellipsoid algorithm to intersect ellipsoid with strip is presented to compute the ellipsoidal feasible set of the estimated states. Simulation results with comparisons to the nonlinear extended set membership filter are given to demonstrate the effectiveness and improved performances of our proposed algorithm.