Discrete-time Systems Research Articles

Disturbance Compensation in Discrete-Time Sliding Mode Control – A Comparative Study

This work is intended to compare three methods for disturbances estimation and compensation in discrete-time variable structure systems with sliding modes. All methods detect disturbance, which appear in control channel, with time delay of one sampling period. The first method is based on nominal discrete-time plant model and can be used in any type of discrete-time control systems. The second and the third methods detect disturbance by measurement of sliding function, and it is applicable in discrete-time sliding mode control systems only. The main task of this work is to check efficiency of the given methods in the presence of unmodeled inertial dynamics in actuators and position and velocity sensors of a positional servo system. It is shown that all three methods give near identical results in the nominal case, while the second and the third methods are superior in the presence of unmodelled dynamics. The third method introduces zig-zag motion in the nominal case. Results of research are illustrated by computer simulation of an example of positional servo system.

Robust Observer-Based Proportional Derivative Fuzzy Control Approach for Discrete-Time Nonlinear Descriptor Systems with Transient Response Requirements

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

Max-consensus of multi-agent systems in random networks

This paper considers max-consensus of a discrete-time multi-agent system (MAS) in directed random networks. Interactions among agents in the MAS are probabilistic and independent with each other. By using max-plus algebra and random theory, a sufficient and necessary condition is given for achieving max-consensus of the MAS. Moreover, we demonstrate that the max-consensus in four probabilistic senses (almost surely, in probability, expectation and mean square) is equivalent when expected graph is strongly connected. This ensures that max-consensus can be achieved in multi-agent systems even if random failures occur in the communication network, which is of practical importance in the fields of wireless sensor networks and distributed computing. A simulation example is presented to illustrate the effectiveness of theoretical results.

A model reduction approach for discrete-time linear time-variant systems with delayed inputs

A model reduction approach for discrete-time linear time-variant systems with delayed inputs

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

Fractional-Order Model-Free Adaptive Control with High Order Estimation

This paper concerns an improved model-free adaptive fractional-order control with a high-order pseudo-partial derivative for uncertain discrete-time nonlinear systems. Firstly, a new equivalent model is obtained by employing the Grünwald–Letnikov (G-L) fractional-order difference of the input in a compact-form dynamic linearization. Then, the pseudo-partial derivative (PPD) is derived using a high-order estimation algorithm, which provides more PPD information than the previous time. A discrete-time model-free adaptive fractional-order controller is proposed, which utilizes more past input–output data information. The ultimate uniform boundedness of the tracking errors are demonstrated through formal analysis. Finally, the simulation results demonstrate the effectiveness of the proposed method.

Novel passivity and dissipativity criteria for discrete-time fractional generalized delayed Cohen–Grossberg neural networks

This paper pays attention to the passivity and dissipativity for discrete-time fractional generalized delayed Cohen-Grossberg neural networks. A new fractional passive lemma is firstly proposed for discrete-time system by means of the Lyapunov functional. This facilitates the discussion of system stabilization in terms of input and output energy. Some passive and dissipative conditions are established under comparison principle, reduction to absurdity, inequality techniques and the proposed passive lemma. Finally, two examples are performed to demonstrate the validity of proposed methods.

Stochastic LQ optimal control for Markov jumping systems with multiplicative noise using reinforcement learning

The stochastic linear quadratic (LQ) problem of discrete-time linear Markov jumping systems with multiplicative noise is investigated in this paper. Two reinforcement learning algorithms, one model-based and one model-free, are designed. The algorithms can be readily adapted to handle nonlinear cases by employing an appropriate function approximator. In the linear case, the network structure is designed to enable the problem to be solved without succumbing to modeling errors. For the model-free algorithm, an approach based on moving averages is proposed to reduce estimation variance. A proof of convergence is also provided. Finally, the effectiveness of the proposed algorithms is demonstrated through a numerical example.

A new synchronization result for fractional-order discrete-time chaotic systems via bandlimited channels

Recently, digital chaotic communication with limited bandwidth has seized the attention of several researchers. This interest stems from the fact that all physical communication channels are limited in bandwidth for data transmission. In furtherance of this, researchers and investigators in this area have put forward various new communication strategies based on synchronizing and controlling coupled chaotic systems evolving in discrete time. In this article, we present a new robust communication scheme based on the synchronization of two fractional-order discrete-time chaotic systems connected by bandlimited channels. The major contribution is to give a sufficient condition in term of the existing and updated parameters of the incorporated fractional-order system on one hand and the inserted filter coefficients, on the other hand. Once applied for data communications, the currently drawn up synchronization scheme pledges to enhance the transmission robustness. Indeed, the new considered parameters act like novel secret keys which widen the cipher key space. Primary and pertinent simulation curves and results performed on Matlab software portray and illustrate the effectiveness of our approach.

Dirac Structures for a Class of Port-Hamiltonian Systems in Discrete Time

Dirac Structures for a Class of Port-Hamiltonian Systems in Discrete Time

Generalization of Filtering Sensitivity Tradeoffs in Continuous- and Discrete-Time Systems: A Simplified Approach

Generalization of Filtering Sensitivity Tradeoffs in Continuous- and Discrete-Time Systems: A Simplified Approach

Constrained-Cost Adaptive Dynamic Programming for Optimal Control of Discrete-Time Nonlinear Systems.

For discrete-time nonlinear systems, this research is concerned with optimal control problems (OCPs) with constrained cost, and a novel value iteration with constrained cost (VICC) method is developed to solve the optimal control law with the constrained cost functions. The VICC method is initialized through a value function constructed by a feasible control law. It is proven that the iterative value function is nonincreasing and converges to the solution of the Bellman equation with constrained cost. The feasibility of the iterative control law is proven. The method to find the initial feasible control law is given. Implementation using neural networks (NNs) is introduced, and the convergence is proven by considering the approximation error. Finally, the property of the present VICC method is shown by two simulation examples.

Supplementary Control for Quantized Discrete-Time Nonlinear Systems Under Goal Representation Heuristic Dynamic Programming.

This article is concerned with supplementary control of discrete-time nonlinear systems with multiple controllers in the framework of goal representation heuristic dynamic programming (GrHDP), where a logarithmic quantizer is used to govern the network communication. For the addressed problem, a neural network (NN)-based observer is first proposed to estimate the unknown system state in the simultaneous presence of quantized influence. In light of the estimated states and the ideal control inputs via a zero-sum game, a GrHDP algorithm with a reinforced term is developed to implement the supplementary control task, where some novel weight updating rules are constructed by virtue of an additional tunable parameter to improve the system performance. Furthermore, a set of conditions about the stability of estimated error dynamics of both observer states and updated NNs' weights are derived by resorting to the Lyapunov stability theory. Finally, the effectiveness of the developed method is verified by a power system and a numerical experiment.

Safe Reinforcement Learning via a Model-Free Safety Certifier.

This article presents a data-driven safe reinforcement learning (RL) algorithm for discrete-time nonlinear systems. A data-driven safety certifier is designed to intervene with the actions of the RL agent to ensure both safety and stability of its actions. This is in sharp contrast to existing model-based safety certifiers that can result in convergence to an undesired equilibrium point or conservative interventions that jeopardize the performance of the RL agent. To this end, the proposed method directly learns a robust safety certifier while completely bypassing the identification of the system model. The nonlinear system is modeled using linear parameter varying (LPV) systems with polytopic disturbances. To prevent the requirement for learning an explicit model of the LPV system, data-based λ -contractivity conditions are first provided for the closed-loop system to enforce robust invariance of a prespecified polyhedral safe set and the system's asymptotic stability. These conditions are then leveraged to directly learn a robust data-based gain-scheduling controller by solving a convex program. A significant advantage of the proposed direct safe learning over model-based certifiers is that it completely resolves conflicts between safety and stability requirements while assuring convergence to the desired equilibrium point. Data-based safety certification conditions are then provided using Minkowski functions. They are then used to seemingly integrate the learned backup safe gain-scheduling controller with the RL controller. Finally, we provide a simulation example to verify the effectiveness of the proposed approach.

Modified λ-Policy Iteration Based Adaptive Dynamic Programming for Unknown Discrete-Time Linear Systems.

In this article, the λ -policy iteration ( λ -PI) method for the optimal control problem of discrete-time linear systems is reconsidered and restated from a novel aspect. First, the traditional λ -PI method is recalled, and some new properties of the traditional λ -PI are proposed. Based on these new properties, a modified λ -PI algorithm is introduced with its convergence proven. Compared with the existing results, the initial condition is further relaxed. The data-driven implementation is then constructed with a new matrix rank condition for verifying the feasibility of the proposed data-driven implementation. A simulation example verifies the effectiveness of the proposed method.

Learning-Based Model-Free Adaptive Control for Nonlinear Discrete-Time Networked Control Systems Under Hybrid Cyber Attacks.

A novel learning-based model-free adaptive control (LMFAC) approach is presented in this article for a class of unknown nonaffine nonlinear discrete-time networked control systems (NCSs) subject to hybrid cyber attacks. The aperiodic denial-of-service (DoS) attacks and persistent deception attacks are assumed to arise in feedback channels, which could result in the absence or authenticity lackness of system signals sent to the controller. With the aid of dynamic linearizaton technology, the equivalent dynamic linearized data models of considered NCSs are first established only based on I/O information instead of the knowledge of mathematical models that are commonly used under the model-based control framework. Then, an LMFAC scheme is designed on the basis of occurred maximum DoS attacks interval to adaptively tune the attenuation coefficient of the input signal for improving system performance during the next DoS attacks interval. Finally, the boundedness of tracking error is rigorously proved through the contraction mapping principle and the effectiveness of the proposed pure data-driven LMFAC method is demonstrated via simulations.

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.

Matrix-Weighted Consensus of Second-Order Discrete-Time Multiagent Systems.

In this article, we study the matrix-weighted consensus issues for second-order discrete-time multiagent systems on directed network topology. Under the designed matrix-weighted consensus algorithm, based on the eigenvalues of the Laplacian matrix, coupling gains, and discrete interval, we build some consensus conditions for reaching discrete-time consensus and deduce some simplified and straightforward consensus conditions for undirected network topology. Besides, for a given network topology, we theoretically analyze the influence of the coupling gains and discrete intervals on the consensus conditions of the network dynamics. Finally, we offer several simulation examples to validate the obtained results.

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

Stability Radii of Infinite-Dimensional Discrete-Time Systems Discomfited by Stochastic Perturbations

Proving chaos for a system of coupled logistic maps: A topological approach.

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161-177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map.