Discrete-time System Model Research Articles

Switching funnel transformation function-based discrete-time sliding-mode control for servo systems with time-varying external disturbances

This paper proposes a switching funnel transformation function-based discrete-time sliding-mode control with lower cost to prevent the issue of control failure caused by time-varying external disturbances leading to tracking errors exceeding the performance boundary. This scheme employs an offline spectral regularization-based neural network with good generalization capability to approximate the unknown nonlinear dynamics and modeling errors in the system, resulting in a more accurate discrete-time system model. Based on the discrete-time system model with time-varying external disturbances, a novel discrete-time switching funnel transformation function-based sliding surface is proposed to solve the problem when the tracking error exceeds performance boundaries. The discrete-time funnel boundaries are switched through a predefined event-triggered mechanism, ensuring that the tracking error remains within the performance boundaries at all times, thereby avoiding control failure. Furthermore, a time-varying sliding mode variable reaching rate is proposed to reduce control cost. Finally, theoretical analysis demonstrates that the tracking error remains within a smaller funnel region compared to the predefined one, and the sliding-mode variable ultimately stays within a bounded sliding-mode boundary. Experimental results on SCARA verify the effectiveness of the proposed method.

Stochastic modeling and optimization of discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism

This study develops novel reliability models for discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism. In view of the situation that the state of some engineering systems cannot be continuously monitored, discrete distribution is used to describe the distribution of random variables involved in this paper. For the case that multiple events can occur simultaneously in the discrete-time system model, the priority order of multiple events occurring simultaneously is defined. Two different models are proposed based on two types of priority rules. The retrial mechanism of the components is considered in the case that the failure information of the system components cannot be successfully transmitted to the repair facility. The preventive maintenance (PM) and repair after the breakdown of the repair facility are considered. The key reliability indices are derived by using the iterative algorithm of the difference equation and the generative function method. The algorithms for calculating system state probability, key reliability indices and cost–benefit ratio (CBR) are designed. To demonstrate the impact of each parameter on the system reliability indices, numerical results are provided. The optimal repair rate configuration scheme is chosen using the Particle Swarm Optimization (PSO) algorithm to achieve the lowest possible CBR. The impact of the system model with or without PM on stationary availability and CBR is shown. In addition, the impact of the number of system components on a series of system performance indices is analyzed, and the optimal number of components of the system corresponding to the minimum value of CBR is obtained.

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.

H∞ Controller Design for Networked Systems With Two-Channel Packet Dropouts and FDI Attacks.

In this article, the stochastic analysis and H∞ controller design problems of networked systems with packet dropouts and false data injection attacks are investigated. Different from the existing literature, we focus on the linear networked systems with external disturbances and both sensor-controller channel and controller-actuator channel are studied. First, we present a discrete-time modeling framework that leads to a stochastic closed-loop system with randomly varying parameters. To facilitate the analysis and H∞ control of resulting discrete-time stochastic closed-loop system, an equivalent yet analyzable stochastic augmented model is further constructed by matrix exponential computation. Based on this model, a stability condition is derived in the form of linear matrix inequality (LMI) with the aid of a reduced-order confluent Vandermonde matrix, Kronecker product operation, and law of total expectation. Specifically, the dimension of the LMI obtained in this article does not increase as the upper bound of consecutive packet dropouts does, which is also different from the existing literature. Subsequently, a desired H∞ controller is obtained such that the original discrete-time stochastic closed-loop system is exponentially mean-square stable with a prescribed H∞ performance. Finally, a numerical example and a direct current motor system are exploited to substantiate the effectiveness and practicability of the designed strategy.

High-order fully actuated system models for discrete-time strict-feedback systems with increasing dimensions

This work aims to propose a high-order fully actuated system (FAS) model for discrete-time strict-feedback systems with nonuniform dimensions. In most of existing works, the authors only consider strict-feedback systems with uniform dimension, i.e. all subsystems have the same dimension. In contrast, we assume that the subsystems may have different or nonidentical dimensions. In particular, the dimensions are assumed to be increasing. Both step-forward and step-backward FAS models for discrete-time strict-feedback systems with nonuniform dimensions are given, based on which the exponentially stabilised controllers can be directly obtained. We consider not only first-order strict-feedback systems but also second-order case.

A New Predictive Current Control With Reduced Current Tracking Error and Switching Frequency for Multilevel Inverters

The forward Euler's integration method is widely used to compute the trajectories of the control variables in conventional predictive current control (PCC) techniques. However, the computational error caused by the forward Euler's method proportionally increases with the sampling time. Hence, the current tracking ability of conventional PCC techniques deteriorates when operating with a large sampling time. Furthermore, the conventional PCC methods produce more switching transitions to reduce the current tracking error, thereby leading to a high switching frequency operation. To overcome these problems, Heun's integration method is proposed for the PCC in this article. The proposed method has predictor and corrector stages to minimize the computational error caused by the large sampling time during the prediction process of the control variables. Thereby, the proposed Heun's method-based PCC techniques produce fewer switching transitions to minimize the current tracking error. Consequently, this results in a decrease in switching frequency. The proposed PCC method is applied to a four-level multilevel inverter (4L-MLI). The discrete-time system models are developed using Heun's integration method to implement the proposed PCC. The working philosophy of the proposed method is demonstrated through the dSPACE/DS1103 controlled 4L-MLI laboratory prototype. The experimental performance of the proposed PCC is compared with that of the conventional PCC in terms of average switching frequency, current tracking error, current harmonic distortion, and transient response time.

Nonsmooth funnel transformation function-based discrete-time sliding-mode control with deterministic performance margin for servo systems

This paper proposes a nonsmooth funnel transformation function-based discrete-time sliding-mode control strategy for the discrete-time servo systems with unknown frictions and disturbances. For obtaining a more accurate discrete-time system model, a filter-based adaptive identification algorithm (FAIA) is introduced, where the unknown measurement noises are considered. Based on the identified system model and a novel discrete-time nonsmooth funnel transformation function (improving the tracking performances), a discrete-time sliding-mode surface is designed to confine the tracking error to a smaller funnel region compared with the predefined one, which has a deterministic performance margin related to the upper bound of the lumped disturbance. Furthermore, selections of steady-state funnel boundary and sliding-mode surface parameter can be guided theoretically according to the upper bound of the lumped disturbance. Experimental results show that the tracking error is guaranteed in the predefined funnel with a deterministic performance margin by using the proposed control strategy.

A Virtual Impedance-Based Flying Start Considering Transient Characteristics for Permanent Magnet Synchronous Machine Drive Systems

A virtual impedance-based flying start considering transient characteristics for permanent magnet synchronous machine drive systems is proposed. The conventional flying start based on virtual resistance (VR) assumes that the load of the system is resistive. However, the maximum value of VR, which is determined by the machine parameter and sampling frequency, is sometimes small. In this case, the load of the system is non-resistive. This assumption error causes an estimated position error and degrades transient characteristics. In the proposed method, algebraic-type virtual inductance (VI) is added to the estimation current regulator of the flying start based on VR. This change improves the accuracy of the estimated rotor position and the transient characteristics. In addition, the discrete-time system model of the proposed flying start method is given, the stability was analyzed considering the change in VR caused by the proposed method, and the improvements were verified by PSIM simulations and experimental results.

Data-Driven Synthesis of Symbolic Abstractions With Guaranteed Confidence

In this letter, we propose a data-driven approach for the construction of finite abstractions ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a.k.a.,</i> symbolic models) for discrete-time deterministic control systems with unknown dynamics. We leverage notions of so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alternating bisimulation functions</i> (ABF), as a relation between each unknown system and its symbolic model, to quantify the mismatch between state behaviors of two systems. Accordingly, one can employ our proposed results to perform formal verification and synthesis over symbolic models and then carry the results back over unknown original systems. In our data-driven setting, we first cast the required conditions for constructing ABF as a robust optimization program (ROP). Solving the provided ROP is not tractable due to the existence of unknown models in the constraints of ROP. To tackle this difficulty, we collect finite numbers of data from trajectories of unknown systems and propose a scenario optimization program (SOP) corresponding to the original ROP. By establishing a probabilistic relation between optimal values of SOP and ROP, we formally construct ABF between unknown systems and their symbolic models based on the number of data and a required confidence level. We verify the effectiveness of our data-driven results over two physical case studies with unknown models including (i) a DC motor and (ii) a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor. We construct symbolic models from data as appropriate substitutes of original systems and synthesize policies maintaining states of unknown systems in a safe set within infinite time horizons with some guaranteed confidence levels.

Matched Pole–Zero Model Independent of the Properness of Systems

This paper extends a multi-input and multi-output (MIMO) system to the state-space form in the matched pole–zero (MPZ) model, one of the discrete-time models of continuous-time systems. The proposed method is applicable regardless of the properness of the system as it derives a model based on the Kronecker canonical form, which is the canonical form of matrix pencils. Three numerical examples are presented herein to illustrate the effectiveness of the proposed method for a combination of the number of inputs and outputs, and the system's properness. The MPZ model has applications in the discretization of continuous-time controllers and the stability guarantee discretization of closed-loop systems. Therefore, the proposed method can contribute to their implementation in MIMO systems and state-space form.

Trajectory Modeling by Distributed Gaussian Processes in Multiagent Systems.

This paper considers trajectory a modeling problem for a multi-agent system by using the Gaussian processes. The Gaussian process, as the typical data-driven method, is well suited to characterize the model uncertainties and perturbations in a complex environment. To address model uncertainties and noises disturbances, a distributed Gaussian process is proposed to characterize the system model by using local information exchange among neighboring agents, in which a number of agents cooperate without central coordination to estimate a common Gaussian process function based on local measurements and datum received from neighbors. In addition, both the continuous-time system model and the discrete-time system model are considered, in which we design a control Lyapunov function to learn the continuous-time model, and a distributed model predictive control-based approach is used to learn the discrete-time model. Furthermore, we apply a Kullback–Leibler average consensus fusion algorithm to fuse the local prediction results (mean and variance) of the desired Gaussian process. The performance of the proposed distributed Gaussian process is analyzed and is verified by two trajectory tracking examples.

UAV formation control based on distributed Kalman model predictive control algorithm

To address the perturbation of formation of multiple unmanned aerial vehicles (UAVs) subject to external disturbances, an algorithm of distributed Kalman model predictive control is proposed in this paper to improve the accuracy of maintaining a formation in flight. A UAV two-order discrete-time system model was built before devising a Kalman prediction model based on the standard prediction model. The desired formation configuration and neighbor Kalman optimal state estimation were conducted to determine the reference state of UAVs. While taking into account the formation tracking error and input stability, a logarithmic barrier function was introduced in the design of the overall cost function to ensure flight safety. Meanwhile, information was exchanged with neighbors with the directed and time-invariant communication topological structure. With the Lyapunov stability theorem, sufficient conditions were defined for the asymptotic stability of the formation system. Simulation results revealed that the algorithm could effectively suppress the perturbation in the formation of UAVs arising from external disturbances, allowing the formation to cope with the conflicts between individual UAVs.

On the consistency and asymptotic normality of discrete-time LTI models identified from concatenated data sets

Even-though data concatenation is a well-known technique for identifying Linear Time-Invariant models from multiple records, the study of the asymptotic properties of the estimator continues to be limited. Therefore, we investigated consistency and asymptotic normality as the number or records tend to infinity, with focus on the identification of discrete-time parametric models for single-input single-output systems operating in open loop. This paper presents the results of a consistency and asymptotic normality study based on the analysis of the prediction error cost function and Monte Carlo simulations. We show that for persistently exciting input signals (filtered white noise), model structures such as Output-Error, AR and ARX are consistently estimated, and the estimated parameters are asymptotically normally distributed. On the other hand, ARMA, ARMAX and Box–Jenkins present a bias on the estimated parameters. However, this bias asymptotically disappears for longer records

Algorithmic Differentiation of the MWGS-Based Arrays for Computing the Information Matrix Sensitivity Equations within the Problem of Parameter Identification

The paper considers the problem of algorithmic differentiation of information matrix difference equations for calculating the information matrix derivatives in the information Kalman filter. The equations are presented in the form of a matrix MWGS (modified weighted Gram–Schmidt) transformation. The solution is based on the usage of special methods for the algorithmic differentiation of matrix MWGS transformation of two types: forward (MWGS-LD) and backward (MWGS-UD). The main result of the work is a new MWGS-based array algorithm for computing the information matrix sensitivity equations. The algorithm is robust to machine round-off errors due to the application of the MWGS orthogonalization procedure at each step. The obtained results are applied to solve the problem of parameter identification for state-space models of discrete-time linear stochastic systems. Numerical experiments confirm the efficiency of the proposed solution.

Learning Discrete-Time Uncertain Nonlinear Systems With Probabilistic Safety and Stability Constraints

This paper presents a discrete-time dynamical system model learning method from demonstration while providing probabilistic guarantees on the safety and stability of the learned model. The controlled dynamic model of a discrete-time system with a zero-mean Gaussian process noise is approximated using an Extreme Learning Machine (ELM) whose parameters are learned subject to chance constraints derived using a discrete-time control barrier function and discrete-time control Lyapunov function in the presence of the ELM reconstruction error. To estimate the ELM parameters a quadratically constrained quadratic program (QCQP) is developed subject to the constraints that are only required to be evaluated at sampled points. Simulations validate that the system model learned using the proposed method can reproduce the demonstrations inside a prescribed safe set while converging to the desired goal location starting from various different initial conditions inside the safe set. Furthermore, it is shown that the learned model can adapt to changes in goal location during reproductions without violating the stability and safety constraints.

Stability Preserving Model Reduction Technique for Weighted and Limited Interval Discrete-Time Systems With Error Bound

The pioneer frequency weighted and limited Gramians based model order reduction techniques presented by Enns and Wang &#x0026; Zilouchian produce unstable reduced-order models for discrete-time systems. To overcome these main drawbacks, many researchers provided a solution to preserve the reduced-order model&#x2019;s stability. However, these existing techniques also produce an unstable reduced-order model in some conditions and produce a large variation to the original system, producing a large approximation error. In this brief, the frequency weighted and limited Gramians based model order reduction technique is presented for the discrete-time systems, which ensure the stability of the reduced-order models and provide low-frequency response approximation error. Furthermore, the proposed technique also provides an easily calculable <i>a priori error bound</i> formula. The proposed work produces steady and precise outcomes compared to conventional reduction methods that show the efficacy of the proposed algorithm.

Cramer-Rao Low Bound of Channel Estimation for Orthogonal Time Frequency Space Modulation System

Orthogonal time-frequency space (OTFS) modulation is supposed to outperform the classical orthogonal frequency division multiplexing (OFDM) in the high-mobility communication scenarios. Channel estimation is an important issue for OTFS, and Cramer-Rao low bound (CRLB) provides a benchmark for different channel estimation algorithms. In this paper, we give an exact and detailed formulation of the discrete-time system model for non-ideal pulse-shaping OTFS in both single cyclic prefix (SCP) and multiple cyclic prefix (MCP) cases, and derive the CRLB of channel estimation for SCP-OTFS, MCP-OTFS, and multiple cyclic prefix approximation (MCPA) OTFS cases.

Global exponential stability of discrete-time higher-order Cohen–Grossberg neural networks with time-varying delays, connection weights and impulses

In this paper, an auxiliary model-based nonsingular M-matrix approach is used to establish the global exponential stability of the zero equilibrium, for a class of discrete-time high-order Cohen–Grossberg neural networks (HOCGNNs) with time-varying delays, connection weights and impulses. A new impulse-free discrete-time HOCGNN with time-varying delays and connection weights is firstly constructed, and the relationship between the solutions of original systems and new HOCGNNs is indicated by a technical lemma. From which, the global exponential stability criteria for the zero equilibrium are derived by using an inductive idea and the properties of nonsingular M-matrices. The effectiveness of the obtained stability criteria is illustrated by numerical examples. Compared with the previous results, this paper has three advantages: (i) The Lyapunov–Krasovskii functional is not required; (ii) The obtained global exponential stability criteria are applied to check whether a matrix is a nonsingular M-matrix, which can be conveniently tested; (iii) The proposed approach applies to most of discrete-time system models with impulses and delays.

Defining velocities for accurate kinetic statistics in the Grønbech-Jensen Farago thermostat.

We expand on two previous developments in the modeling of discrete-time Langevin systems. One is the well-documented Grønbech-Jensen Farago (GJF) thermostat, which has been demonstrated to give robust and accurate configurational sampling of the phase space. Another is the recent discovery that also kinetics can be accurately sampled for the GJF method. Through a complete investigation of all possible finite-difference approximations to the velocity, we arrive at two main conclusions: (1) It is not possible to define a so-called on-site velocity such that kinetic temperature will be correct and independent of the time step, and (2) there exists a set of infinitely many possibilities for defining a two-point (leap-frog) velocity that measures kinetic energy correctly for linear systems in addition to the correct configurational statistics obtained from the GJF algorithm. We give explicit expressions for the possible definitions, and we incorporate these into convenient and practical algorithmic forms of the normal Verlet-type algorithms along with a set of suggested criteria for selecting a useful definition of velocity.

Decentralized Dynamic Event-Triggered $\mathcal {H}_{\infty }$ Control for Nonlinear Systems With Unreliable Communication Channel and Limited Bandwidth

This article investigates the dynamic event-triggered $\mathcal {H}_{\infty }$ control problem for nonlinear networked control systems with unreliable communication channel, variable communication delays, and limited bandwidth. The nonlinear plant is represented by discrete-time polynomial fuzzy model. First, a decentralized dynamic event-triggered mechanism is proposed to determine whether the measured data are transmitted or not, and in order to exclude data collision caused by limited bandwidth, novel try-once-discard and flexible round-robin scheduling protocols are proposed to assign communication channel to certain sensor node. Then, Bernoulli distribution is employed to model the unreliable communication channel, and a new random sequence is developed to model the received data sequence under the effect of data losses and scheduling protocols. Furthermore, a discrete-time stochastic system model with both state and error delays is constructed, and sufficient conditions in the form of sum-of-squares are developed for the design of $\mathcal {H}_{\infty }$ controllers such that the closed-loop system is stochastically stable and preserves guaranteed $\mathcal {H}_{\infty }$ performance. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed results.