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This paper considers the locally and globally optimal solutions of the global optimisation problem whose objective function is a max‐plus vector‐valued function and constraint function is a real affine function. The formulas about global optimisation in the existing work are further explained and simplified. The necessary and sufficient conditions for the existence and uniqueness of locally optimal solutions of a single objective are established, which are then used to deduce these of globally optimal solutions. Furthermore, the general formulas of locally and globally optimal solutions are presented, respectively. The local optimisation is then applied to solve the load distribution problem of distributed systems with no globally optimal solution, and the optimal allocation scheme is proposed to complete the overall task at the earliest time. The proposed method is constructive, and the obtained results are illustrated by the numerical examples.

A quasi‐differential type event‐triggered model predictive control (ET‐MPC) framework for continuous‐time linear systems with additional disturbances is constructed. Different from the existing ET‐MPC, the triggering condition of the proposed method is focused on the differences of the errors between the actual states and the best prediction sequence at two consecutive sampling moments. Its advantage is that the dynamic characteristics of state changes can be better considered, which will achieve a more effective balance between communication load and system control performance. Then, to deal with persistent disturbances, a time‐varying tightening robustness state constraint is utilized in this scheme, instead of conventional state constraints. Finally, the new ET‐MPC algorithm is designed based on the dual‐mode control framework, and its feasibility, the system closed‐loop stability, as well as avoidance of Zeno behaviour are rigorously verified. Simulations and comparisons are provided to demonstrate the algorithm effectiveness.

The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semi-discrete version of the derivative NLS system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to the solve the inverse problem. One of these methods involves a discrete Marchenko system using as input the scattering data set consisting of the scattering coefficients and the bound-state information, and this method is presented in a way generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system is not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in the reflectionless case certain explicit solution formulas are presented in closed form expressed in terms of the two matrix triplets., 71 pages

This article presents a design strategy and stability analysis of modified repetitive sliding mode controller for uncertain linear systems. A modified repetitive controller is adopted to simultaneously track and reject periodic signals. A discrete-time sliding mode controller is combined to compensate the slow response of repetitive control and to provide robustness against plant parameters uncertainties. Stability analysis is provided to prove boundedness of the proposed control law and the convergence of sliding function and the tracking error. Comparative simulation results demonstrate that the proposed method is able to accurately track reference signal and to reject disturbance with fast transient response. The results also indicate that the closed-loop system remains stable in the presence of plant parameter changes.

Summary This article considers a distributed Kalman filtering problem for linear system contaminated by complex multi‐channel random uncertain parameter in which a number of nodes cooperative witho...

The problem of making the output of a linear system with polytopic uncertainties and discontinuities in the state evolution totally insensitive to an unknown disturbance input by state feedback is investigated. Suitable geometric notions are introduced and used to provide a structural, constructive solvability condition. The requirement of achieving global robust asymptotic stability of the compensated dynamics is then added and further solvability conditions are provided by requiring that the time instants at which discontinuities in the state evolution, or jumps, occur are sufficiently far from each other.

This paper investigates interval observer‐based controller design for switched linear systems involving additional disturbance and measurement noise, whose modes need not to be cooperative. First, by giving the upper bound and the lower bound of the disturbance and the measurement noise, we construct hybrid interval observer for the considered switched linear system by means of a switched coordinate transformation, which can transform the observer error systems into cooperative ones. The interval observer can supply certain state information at any instant. Compared with the interval observer design for switched systems with arbitrary switching sequences or dwell time switching based on common Lyapunov function, the difficulty consists in characterizing the jump of the multiple Lyapunov functions. Then, by using the multiple Lyapunov functions method and average dwell time scheme, some sufficient conditions are derived and applied to build the interval observer‐based state feedback controller. Finally, we provide an example to illustrate the validity of the derived results.

The trade-off between optimality and complexity has been one of the most important challenges in the field of robust Model Predictive Control (MPC). To address the challenge, we propose a flexible robust MPC scheme by synergizing the multi-stage and tube-based MPC approaches. The key idea is to exploit the non-conservatism of the multi-stage MPC and the simplicity of the tube-based MPC. The proposed scheme provides two options for the user to determine the trade-off depending on the application: the choice of the robust horizon and the classification of the uncertainties. Beyond the robust horizon, the branching of the scenario-tree employed in multi-stage MPC is avoided with the help of tubes. The growth of the problem size with respect to the number of uncertainties is reduced by handling \emph{small} uncertainties via an invariant tube that can be computed offline. This results in linear growth of the problem size beyond the robust horizon and no growth of the problem size concerning small magnitude uncertainties. The proposed approach helps to achieve a desired trade-off between optimality and complexity compared to existing robust MPC approaches. We show that the proposed approach is robustly asymptotically stable. Its advantages are demonstrated for a CSTR example.

Summary The paper considers a block‐oriented nonlinear Hammerstein system that consists of a linear block with real‐valued time‐varying unknown coefficients and a pure time delay, preceded by the s...

This work introduces the design of finite-gain L1 interval observers based on an event-triggered mechanism for linear continuous-time systems in the presence of unknown-but-bounded uncertainties with a priori known bounds (state disturbances and measurement noises). In addition, the times when the measurements are required in the estimation procedure are driven by an event-triggered mechanism. A dynamic event-triggered condition, which depends on the width of the feasible domain of the system's uncertainties and the width of the estimated state enclosures, is established. The proposed event-triggered strategy guarantees the existence of a positive lower bound on the inter-event times, which avoids the Zeno phenomenon. Moreover, the L1-stability of the estimation error is ensured while using as less as possible sensors data. Finally, simulation results are given to illustrate the effectiveness of the proposed method.

Due to increasing water demands globally, freshwater ecosystems are under constant pressure. Groundwater resources, as the main source of accessible freshwater, are crucially important for irrigation worldwide. Over-abstraction of groundwater leads to declines in groundwater levels; consequently, the groundwater inflow to streams decreases. The reduction in baseflow and alteration of the streamflow regime can potentially have an adverse effect on groundwater-dependent ecosystems. A spatially distributed, coupled groundwater-surface water model can simulate the impacts of groundwater abstraction on aquatic ecosystems. A constrained optimization algorithm and a simulation model in combination can provide an objective tool for the water practitioner to evaluate the interplay between economic benefits of groundwater abstractions and requirements to environmental flow. In this study, a holistic catchment-scale groundwater abstraction optimization framework has been developed that allows for a spatially explicit optimization of groundwater abstraction, while fulfilling a predefined maximum allowed reduction of streamflow (baseflow [Q95] or median flow [Q50]) as constraint criteria for 1484 stream locations across the catchment. A balanced K-Means clustering method was implemented to reduce the computational burden of the optimization. The model parameters and observation uncertainties calculated based on Bayesian linear theory allow for a risk assessment on the optimized groundwater abstraction values. The results from different optimization scenarios indicated that using the linear programming optimization algorithm in conjunction with integrated models provides valuable information for guiding the water practitioners in designing an effective groundwater abstraction plan with the consideration of environmental flow criteria important for the ecological status of the entire system.

This paper presents an algorithm that computes polytopic robust control-invariant (RCI) sets for rationally parameter-dependent systems with additive disturbances. By means of novel linear matrix inequalities (LMI) feasibility conditions for invariance along with a newly developed method for volume maximization, an iterative algorithm is proposed for the computation of RCI sets with maximized volumes. The obtained RCI sets are symmetric around the origin by construction and have a user-defined level of complexity. Unlike many similar approaches, the proposed algorithm directly computes the RCI sets without requiring control inputs to be in a specific feedback form. In fact, a specific control input is obtained from the LMI problem for each extreme point of the RCI set. The outcomes of the proposed algorithm can be used to construct a piecewise-affine controller based on offline computations.

In this work, an energy stable numerical scheme is proposed to solve the PFC model with a nonlocal Lagrange multiplier. The construction of the numerical scheme is based on invariant energy quadratization (IEQ) technique to transform a nonlinear system into a linear system, while the time variables is discretized by second order scheme. The stability term in the new scheme can balance the influence of nonlinear term. Moreover, we obtain the results of unconditional energy stability, uniqueness and uniform boundedness of numerical solution, and the numerical scheme is convergent with order of $\mathcal{O}(\delta t^2)$. Several numerical tests are conducted to confirm the theoretical results.

[EN] A Kalman reduced form is obtained for linear systems over Hermite rings. This reduced form gives information of the set of assignable polynomials to a given linear system. SI

This paper studies the global leader-following consensus problem for a multi-agent system with intermittent directed communication in the presence of actuator saturation. Both the follower agents and the leader agent are described by a chain of integrators of an arbitrary length. A bounded consensus algorithm is constructed for each follower agent which utilizes the information obtained from the communication network intermittently. Global leader-following consensus is achieved by the consensus algorithms when the communication topology among all agents, when active, contains a spanning tree rooted at the leader agent. Simulation results illustrate the theoretical conclusions.

In this article, the dynamics and stability of a linear system with stochastic delay and additive noise are investigated. It is assumed that the delay value is sampled periodically from a stationary distribution. A semi-discretization technique is used to time-discretize the system and derive the mean and second-moment dynamics. These dynamics are used to obtain the stationary moments and the corresponding necessary and sufficient stability conditions. The application of the proposed method is illustrated through the analysis of the Hayes equation with stochastic delay and additive noise. The method is also applied to the control design of a connected automated vehicle. These examples illuminate the effects of stochastic delays on the robustness of dynamical systems.

Summary This article considers the design of interval functional observers to estimate a linear function of the state vector of time‐delay systems subject to both input and output additive disturba...

This paper explores the influence of linear hysteretic damping on the performance of passive tunedinerter devices. An inerter is a device that produces a force proportional to the relative acceleration across its two terminals; devices incorporating inerters have received widespread attention in the earthquake engineering community, because they offer the ability to improve the seismic response of structures. However, the majority of this research has assumed that the damping components within the tuned-inerter device exhibit viscous, rather than hysteretic, damping. This restriction imposes an essential question on how the hysteretic damping model will change the performance of the device compared to the viscous damping model. It is shown that the response of viscous and hysteretic inerter systems have significant differences in displacement amplitude due to the frequency-dependency of the damping. Therefore, a new formulation for obtaining the optimum loss factor of the hysteretic damping in the inerter system is proposed. Next, the challenges associated with accurately predicting the time-response of a hysteretically damped system are discussed. A numerical time-integration method is extended to address these challenges, using a new formulation that has the benefit of being broadly applicable to multi-degree-of-freedom hysteretic linear systems and nonstationary random signals. The results show that the earthquake responses from the hysteretic damping model can differ significantly from the ones obtained via the viscous model.

In this chapter, we present a model that characterizes the propagation of stepping‐stone attacks in the Internet of Things (IoT) using a vulnerability graph with fixed and switching topology. The cost of a stepping‐stone stack path is modeled as a first‐order dynamical system where the control input is the biased min‐consensus protocol, which converges to the minimum cost, giving us the tool for the calculation of the shortest stepping‐stone attack path from every source node to every target node. The model is discretized, and for the convergence analysis of the stepping‐stone dynamics, the min‐plus algebra is introduced. In this framework, the stepping‐stone dynamic is formulated as a linear system in a min‐plus algebra. The model is expanded to a more realistic scenario where the vulnerability graph is not fixed, that is, when the vulnerability graph changes due to the attack have been detected, triggering the intrusion detection system, or because the attacker that has exploited a vulnerability in any node will exploit the same vulnerability more efficiently in any other node. In this scenario, the stepping‐stone dynamics are modeled as a switched linear system in a min‐plus algebra. Necessary and sufficient conditions for the convergence of the stepping‐stone dynamics to the minimum cost in the fixed topology case and a sufficient condition for the switching topology case are provided. For the validation of the model, simulations for the fixed and switching topology case are presented. A preliminary version of this work was published at the 2018 IEEE ICC conference (M. Gamarra, S. Shetty, O. Gonzalez, D. Nicol, C. A. Kamhoua, and L. Njilla. “Analysis of stepping stone attacks in dynamic vulnerability graphs.” In: IEEE International Conference on Communications (ICC) , Kansas City, MO, 20–24 May 2018.). In this chapter, we have extended the preliminary work to frame the model within IoT context with more detailed mathematical results.

Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow linear solution and a pure advective transport solution with different right-hand sides. The new solver was compared with other preconditioned iterative methods, the MODFLOW's GMG solver, and direct sparse solvers. Test problems include two- and three-dimensional benchmarks spanning homogeneous and highly heterogeneous and anisotropic formations. For the groundwater flow problems, using the algebraic multigrid preconditioning speeds up the numerical solution by one to two orders of magnitude. The algebraic multigrid preconditioner efficiency was preserved for the three dimensional heterogeneous and anisotropic problem unlike for the MODFLOW's GMG solver. Contrarily, a sparse direct solver was the most efficient for the pure advective transport processes such as the forward travel time simulations. Hence, the best sparse solver for the more general advection-dispersion transport equation is likely to be Péclet number dependent. When equipped with the best solvers, processing multimillion grid blocks by the dual delineation approach is a matter of seconds. This paves the way for its routine application to large geological models. The paper gives practical hints on the strategies and conditions under which algebraic multigrid preconditioning would remain competitive for the class of nonlinear and/or transient problems.

Finite‐difference modeling with a cross‐rhombus stencil with high‐order accuracy in both spatial and temporal derivatives is a potential method for efficient seismic simulation. The finite‐difference coefficients determined by Taylor‐series expansion usually preserve the dispersion property in a limited wavenumber range and fixed angles of propagation. To construct the dispersion‐relationship‐preserving scheme for satisfying high‐wavenumber components and multiple angles, we expand the dispersion relation of the cross‐rhombus stencil to an over‐determined system and apply a regularization method to obtain the stable least‐squares solution of the finite‐difference coefficients. The new dispersion‐relationship‐preserving based scheme not only satisfies several designated wavenumbers but also has high‐order accuracy in temporal discretization. The numerical analysis demonstrates that the new scheme possesses a better dispersion characteristic and more relaxed stability conditions compared with the Taylor‐series expansion based methods. Seismic wave simulations for the homogeneous model and the Sigsbee model demonstrate that the new scheme yields small dispersion error and improves the accuracy of the forward modelling.

Many problems in computer graphics and vision can be formulated as a nonlinear least squares optimization problem, for which numerous off-the-shelf solvers are readily available. Depending on the structure of the problem, however, existing solvers may be more or less suitable, and in some cases the solution comes at the cost of lengthy convergence times. One such case is semi-sparse optimization problems, emerging for example in localized facial performance reconstruction, where the nonlinear least squares problem can be composed of hundreds of thousands of cost functions, each one involving many of the optimization parameters. While such problems can be solved with existing solvers, the computation time can severely hinder the applicability of these methods. We introduce a novel iterative solver for nonlinear least squares optimization of large-scale semi-sparse problems. We use the nonlinear Levenberg-Marquardt method to locally linearize the problem in parallel, based on its firstorder approximation. Then, we decompose the linear problem in small blocks, using the local Schur complement, leading to a more compact linear system without loss of information. The resulting system is dense but its size is small enough to be solved using a parallel direct method in a short amount of time. The main benefit we get by using such an approach is that the overall optimization process is entirely parallel and scalable, making it suitable to be mapped onto graphics hardware (GPU). By using our minimizer, results are obtained up to one order of magnitude faster than other existing solvers, without sacrificing the generality and the accuracy of the model. We provide a detailed analysis of our approach and validate our results with the application of performance-based facial capture using a recently-proposed anatomical local face deformation model.

For separation of enantiomers in presence of a chiral selector, data obtained with the 1D dynamic simulators SIMUL5complex and GENTRANS are compared to data predicted by PeakMaster 6, a recently released generalized model of the linear theory of electromigration. Four electrophoretic systems with stereoisomers of weak bases were investigated. They deal with the estimation of input data for complexation together with the elucidation of the origin of observed system peaks, the interference of analyte and system peak migration, the change of enantiomer migration order as function of the selector concentration and the inversion of analyte migration direction in presence of a multiply negatively charged selector. For all systems, data predicted with PeakMaster 6 are in agreement with those of the dynamic simulators and simulation data compare well with experimental data that were monitored with setups featuring conductivity and/or UV absorbance detection along the capillary. SIMUL5complex and GENTRANS provide the full dynamics of any buffer and sample arrangement and require very long execution time intervals. PeakMaster 6 is restricted to conventional CZE, is based on an approximate solution of the transport equations, provides data for realistic experimental conditions within seconds and represents a practical tool for an experimentalist.