Linear Discrete Systems Research Articles

Cluster control of complex cyber-physical systems

To our minds, the real world appears as a composition of different interacting entitites, which demonstrate complex behavior. In the current paper, we primarly aim to study such networked systems by developing corresponding approaches to modeling them, given a class of tasks. We derive it from the primary concept of information and a system, with corresponding dynamics emerging from interactions between system components. As we progress through the study, we discover three possible levels of certain synchronous pattern composition in complex systems: microscopic (the level of elementary components), mesoscopic (the level of clusters), and macroscopic (the level of the whole system). Above all, we focus on the clusterization phenomenon, which allows to reduce system complexity by regarding only a small number of stable manifolds, corresponding to cluster synchronization of system component states—as opposed to regarding the system as a whole or each elementary component separately. Eventually, we demonstrate how an optimization problem for cluster control synthesis can be formulated for a simple discrete linear system with clusterization.

A Krein space-based approach to event-triggered [formula omitted] filtering for linear discrete time-varying systems

This paper is concerned with the problem of event-triggered H∞ filtering for linear discrete time-varying (LDTV) systems. Using the lifting technique, we firstly establish an equivalent relationship with a certain equivalent minimum problem of indefinite quadratic form subject to LDTV systems with non-uniform sampling periods. Then, based on Krein space projection and innovation analysis, sufficient and necessary conditions for the existence of desired filter are derived and a feasible solution is obtained in terms of Riccati recursions. Thus, an algorithm based on the time-update and event-update recursions is given for the implementation of event-triggered H∞ filtering. Different from some existing results, a new event-triggered H∞ filtering scheme is provided so that the estimation error can be completely decoupled from the event-triggered transmission error. Moreover, the new proposed Krein space approach is less conservative and more computational attractive than the existing methods based on recursive linear inequality matrix. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Reference Trajectory Based Quasi-Sliding Mode with Event-Triggered Control

The study presents a novel event-triggered quasi-sliding mode control algorithm for linear discrete time systems. The problem is divided into two main parts. Firstly, the sliding mode control of perturbed discrete time systems is considered. In order to limit the impact of external disturbances to one sampling step only, a reference trajectory-based control law is introduced. The proposed control method drives the system’s representative point to an a priori designed reference position in each control step, thus minimizing the influence of disturbance and improving the robustness. Moreover, the reference trajectory is generated according to a novel reaching law, which ensures the nonswitching movement within the quasi-sliding mode band. In the latter part of the study, the proposed control strategy is supplemented with an event-triggering algorithm. In the modified strategy the control signal is only updated when a certain triggering condition occurs. Therefore, the need for communication between system elements is reduced. As follows, the delays in the digital control process may be reduced as well, without compromising the system’s robustness.

Controllability Analysis of Propellant-Free Satellite Formation Flight

This paper presents a comprehensive analysis of the controllability for the relative dynamics of satellites using space environment forces. First, the constraints of the space environment forces are illustrated for circular orbits, and the integration of space environment forces for achieving full controllability is considered. Two approaches are used to estimate the controllability of various configurations of space environment forces. The first approach relies on controllability theory for linear continuous and discrete systems, including a derivation of a new linear formulation for solar radiation pressure with lower constraints. The second approach exploits extensive numerical simulations for controllability analyses of nonlinear dynamics models. The linear quadratic regulator is designed for different dynamics models of the space environment forces and implemented numerically for nonlinear dynamics using an anti-windup scheme with bounded control inputs. The numerical simulations demonstrate that the Lorentz forces and the differential atmospheric drag are more effective control forces in the low Earth orbits than the solar radiation pressure.

Guaranteed cost controller for discrete time-delayed systems with actuator saturation

The problem of guaranteed cost (GC) control using static-state feedback controllers for uncertain linear discrete time-delayed systems subjected to actuator saturation is studied in this paper. The stability analysis of closed-loop systems is carried out using a Lyapunov-Krasovskii functional. Conditions for the existence of state-feedback GC controllers are developed using a linear matrix inequality (LMI)-based criterion. The approach ensures a sufficient performance bound over all the acceptable parameter uncertainties. The scheme of the optimal GC controller problem is framed as a convex optimization problem with LMI constraints. The design of GC controllers for discrete-time systems subjected to actuator saturation without considering the effect of state-delay is also discussed. The effectiveness of the proposed approach is illustrated using suitable examples.

Fault estimation and fault-tolerant control for linear discrete time-varying stochastic systems

Fault estimation and fault-tolerant control for linear discrete time-varying stochastic systems

Observer-Based Stabilization of Linear Discrete Time-VaryingDelay Systems

Abstract This paper presents the design of an observer-based stabilizing controller for linear discrete-time systems subject to interval time-varying state-delay. In this work, the problem has been formulated in convex optimization framework by constructing a new Lyapunov–Krasovskii (LK) functional to derive delay-dependent stabilization criteria. The summation inequality and the extended reciprocally convex inequality are exploited to obtain a less conservative delay upper bound in linear matrix inequality (LMI) framework. The derived stability conditions are delay-dependent and thus ensure global asymptotic stability in presence of any time-delay less than the obtained delay upper bound. Numerical examples are included to demonstrate the usefulness of the developed results.

Periodic Event-Triggered Estimation for Networked Control Systems

This paper considers the problem of remote state estimation in a linear discrete invariant system, where a smart sensor is utilized to measure the system state and generate a local estimate. The communication depends on an event scheduler in the smart sensor. When the channel between the remote estimator and the smart sensor is activated, the remote estimator simply adopts the estimate transmitted by the smart sensor. Otherwise, it calculates an estimate based on the available information. The closed-form of the minimum mean-square error (MMSE) estimator is introduced, and we use Gaussian preserving event-based sensor scheduling to obtain an ideal compromise between the communication cost and estimation quality. Furthermore, we calculate a variation range of communication probability, which helps to design the policy of event-triggered estimation. Finally, the simulation results are given to illustrate the effectiveness of the proposed event-triggered estimator.

Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

AbstractThe direct and inverse scattering problems are analyzed for a first‐order discrete system associated with the semi‐discrete version of the derivative nonlinear Schrödinger (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 solve the inverse problem to recover the potential values in the first‐order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first‐order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first‐order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are 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.

Distributed fault detection and isolation for uncertain linear discrete time-varying heterogeneous multi-agent systems

The distributed fault detection and isolation of linear discrete time-varying multi-agent systems subject to heterogeneous dynamics and norm bounded model uncertainties is investigated. By combining the model uncertainties and external disturbances into a new generalized disturbance, a distributed closed-loop residual generator is constructed based on the estimate of the generalized disturbance. Then, the concerned fault detection is transformed into an indefinite quadratic minimum problem by using the finite-horizon robust H∞ filtering method, for which necessary and sufficient minimum conditions are provided by the Krein-space theory. Afterwards, a computationally efficient recursive algorithm is further developed to determine the residual for individual agent. Based on the resulting residuals, the fault occurrence can be alerted by devising a novel distributed fault detection scheme, which is then applied for the fault isolation by some appropriate transformation of the output measurements. Finally, both numerical and practical simulations are proposed to verify the effectiveness of the proposed algorithm.

A method for constructing an optimal control strategy in a linear terminal problem

This paper deals with an optimal control problem for a linear discrete system subject to unknown bounded disturbances, where the control goal is to steer the system with guarantees into a given terminal set while minimising the terminal cost function. We define an optimal control strategy which takes into account the state of the system at one future time instant and propose an efficient numerical method for its construction. The results of numerical experiments show an improvement in performance under the optimal control strategy in comparison to the optimal open-loop worst-case control while maintaining comparable computation times.

Singular Perturbation Method Applied to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Initial Value Problem (IVP) and Boundary Value Problem (BVP). It is applied to two state switching power converters to provide rigorous justification of the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for IVP and BVP. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

An adaptive sparse grid rational Arnoldi method for uncertainty quantification of dynamical systems in the frequency domain

AbstractIn this paper, we address discrete linear systems in the frequency domain, where both frequency and random parameters are considered. Sampling such a system many times is computationally challenging if the size is too large, which is often the case for a finite element discretization of partial differential equations. We propose an adaptive method, steered by dual error indicators, which combines rational Arnoldi model order reduction and sparse grid interpolation with hierarchical Leja nodes. At each Leja node, a reduced order model (ROM) is constructed such that the ROM‐error in the frequency domain is balanced with the sparse grid error in the random parameter domain. Both the ROM basis and the sparse grid set are enlarged in a hierarchical manner to achieve a prescribed accuracy in statistical moments. Moreover, parameter sensitivities over the frequency range can be easily extracted from the combined reduced order‐surrogate model. In the numerical tests considered in the paper, the method employs sampling sets, which are reduced by at least an order of magnitude, compared to Monte Carlo simulation. Additionally, for an example from vibroacoustics, building a ROM reduces the system size by roughly a factor of 100, while still providing an acceptable accuracy.

Multiscale finite-difference method for frequency-domain acoustic wave modeling

Conventional finite-difference frequency-domain (FDFD) methods can describe wave attenuation and velocity dispersion more easily than time-domain methods. However, there are significant challenges associated with the computational costs for solving the linear system when frequency-domain methods are applied in models with large dimensions or fine-scale property variations. Direct-iterative solvers and parallel strategies attempt a trade-off between memory and time costs. We have followed the general framework of a heterogeneous multiscale method and developed a multiscale FDFD approach to solve the Helmholtz equation with lower memory and time costs. To achieve this, the discrete linear system approximating the Helmholtz equation is constructed on a coarse mesh, making its dimension much smaller than that of conventional methods. The coefficient matrix in the linear system of dimension-reduction captures fine-scale heterogeneity in the media by coupling fine- and coarse-scale meshes. Several test models are used to verify the accuracy of our multiscale method and investigate potential sources of error. Numerical results demonstrate that our method accurately approximates the wavefields of fine-scale solutions at low frequencies of the source and could produce solutions with small errors by reducing the size of the coarse-mesh cells at high frequencies as well. Comparisons of computational costs with conventional FDFD methods show that our multiscale method significantly reduces computation time and memory consumption.

Augmented Perpetual Manifolds and Perpetual Mechanical Systems—Part I: Definitions, Theorem, and Corollary for Triggering Perpetual Manifolds, Application in Reduced-Order Modeling and Particle-Wave Motion of Flexible Mechanical Systems

Abstract Perpetual points in mechanical systems were defined recently. Herein, they are used to seek specific solutions of N-degrees-of-freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, the perpetual points proved that they form the perpetual manifolds, they are associated with rigid body motions, and herein these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem defining the conditions of the external forces applied in an N-degrees-of-freedom system led to a solution in the exact augmented perpetual manifold of rigid body motions is proven. In this case, the motion by only one differential equation is described; therefore, it forms reduced-order modeling (ROM) of the original equations of motion. Further on, a corollary is proven that for harmonic motion in the augmented perpetual manifolds, the system moves in dual mode as wave-particle. The developed theory is certified in three examples, and the analytical solutions are in excellent agreement with the numerical simulations. This research is significant in several sciences, mathematics, physics, and mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally, in mechanical engineering, the theory in all mechanical structures can be applied, e.g., cars, airplanes, spaceships, and boats, targeting only the rigid body motions.

Observer-Based PID Security Control for Discrete Time-Delay Systems Under Cyber-Attacks

This article deals with the observer-based proportional-integral-derivative (PID) security control problem for a kind of linear discrete time-delay systems subject to cyber-attacks. The cyber-attacks, which include both denial-of-service and deception attacks, are allowed to be randomly occurring as regulated by two sequences of Bernoulli distributed random variables with certain probabilities. A novel observer-based PID controller is proposed such that the closed-loop system achieves the desired security level and the quadratic cost criterion (QCC) has an upper bound. Sufficient conditions are derived under which the exponentially mean-square input-to-state stability is guaranteed and the desired security level is then achieved. Subsequently, an upper bound of the QCC is obtained and the explicit expression of the desired PID controller is also parameterized. Finally, the validity of the developed design approach is verified via an illustrative example.

Anisotropy-Based Filtering for Linear Discrete Time-Varying Systems with Multiplicative Noises on a Finite Horizon

Anisotropy-Based Filtering for Linear Discrete Time-Varying Systems with Multiplicative Noises on a Finite Horizon

Robust state estimation for uncertain linear discrete systems with d‐step state delay

This paper discusses the state estimation problems of an uncertain linear discrete time-varying state space model with d-step state delay. Based on the principle of minimising the expectation of estimation errors and the method of state augmentation, a robust state estimation algorithm is proposed. Specially, this estimator retains the form of Kalman-like filter and the characteristics of fast recursive calculation. Moreover, the conditions of bounded estimation error covariance and the proof of asymptotic unbiasedness of the filter are given. Finally, numerical examples are used to verify the effectiveness and the wide applicability of this estimator.

Multiparametric/explicit nonlinear model predictive control for quadratically constrained problems

Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a second-order Taylor approximation of Fiacco’s Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited.

Zero Controllability Criterion for Discrete Positive Systems with Multiple Delays on Both States and Inputs

Zero controllability criterion for positive linear discrete systems with multiple delays in both states and inputs is obtained and proved. An example is given to support our main result.