Discrete-time Nonlinear Systems Research Articles

Finite-time cooperative tracking for high-maneuvering aerial target with sensor network

The finite-time distributed state estimation (DSE) problems based on sensor network for discrete-time nonlinear systems are considered in this paper. Considering the uncertainty of the target maneuvers, the interactive multiple models (IMM) based distributed estimation framework is established by local filter and consensus fusion. In the proposed distributed algorithm, partial of the output of the system can be collected by some sensors and the sensor network is supposed to be collectively observable. According to the consensus protocol, the final estimate results are obtained by fusing local estimate results of each sensor after the consensus fusion stage. The distributed estimate results can be obtained with finite-time when the consensus fusion step is larger than the diameter of the network, and the convergence rate of estimate error of the proposed distributed estimate algorithm can be guaranteed. Besides, the existence of the lower and upper bounds of the error covariance matrix are proved with some necessary assumptions. Finally, numerical simulation example is given for tracking the target with different maneuvers.

Decentralized Fuzzy Fault Estimation Observer Design for Discrete-Time Nonlinear Interconnected Systems

In this paper, a fault estimation technique is proposed for discrete-time nonlinear interconnected systems with uncertain interconnections. To achieve the fault estimation, the decentralized fuzzy observer is adopted based on the Takagi–Sugeno fuzzy model. Based on the estimation error model with the subsystems of the interconnected system and its decentralized fuzzy observer, the fault estimation condition with H∞ performance is presented. The main idea of this paper is that a novel inequality condition for H∞ performance is used, and the sufficient condition is presented to guarantee the good fault estimation performance. Also, the decentralized fuzzy observer design condition for the fault estimation is converted into linear matrix inequality formats. Finally, a simulation example is provided, and the effectiveness of the proposed fault estimation technique is verified by comparison of the fault estimation performance.

Set-valued state estimators for a class of uncertain discrete-time nonlinear systems

Set-membership state estimation problem for a certain class of discrete-time nonlinear systems is tackled in this work. First, this problem is formulated as a reachability analysis problem. Then, to avoid the wrapping effect, explicit expression of the general solution of this class of systems is used to design new set-valued state estimation schemes. Moreover, for practical considerations, two periodically re-initialized algorithms based on both interval analysis and zonotope computation are introduced. The merit of the proposed approach is illustrated on a numerical example and its performance is compared to that of other methods borrowed from the literature.

Stabilizing value iteration Q-learning for online evolving control of discrete-time nonlinear systems

Stabilizing value iteration Q-learning for online evolving control of discrete-time nonlinear systems

Near-Optimal Tracking Control of Partially Unknown Discrete-Time Nonlinear Systems Based on Radial Basis Function Neural Network

This paper proposes an optimal tracking control scheme through adaptive dynamic programming (ADP) for a class of partially unknown discrete-time (DT) nonlinear systems based on a radial basis function neural network (RBF-NN). In order to acquire the unknown system dynamics, we use two RBF-NNs; the first one is used to construct the identifier, and the other one is used to directly approximate the steady-state control input, where a novel adaptive law is proposed to update neural network weights. The optimal feedback control and the cost function are derived via feedforward neural network approximation, and a means of regulating the tracking error is proposed. The critic network and the actor network were trained online to obtain the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation within the ADP framework. Simulations were carried out to verify the effectiveness of the optimal tracking control technique using the neural networks.

Finite-time filtering design with past output measurements for interval type-2 fuzzy systems: A descriptor approach

This paper is interested in the design of full-order H∞ memory filters for discrete-time nonlinear systems within finite-time domain. First, the investigated nonlinear systems are modeled by a Takagi–Sugeno fuzzy technique for the interval type-2 case. And to reduce the size of the model, only the past measurements of systems are involved during the memory filters construction. Then, novel design processes with less conservatism are proposed through executing a decoupling transformation for the filtering error dynamics. Based on Lyapunov methods together with several strategies that apply convex optimization techniques and slack variables, some sufficient conditions are presented and certified to solve the memory filter parameters, such that the augmented error dynamics are singular finite-time bounded with prescribed H∞ performance. Finally, an example is introduced to illustrate the effectiveness of the proposed strategy in contrast to existent works.

Reliable stabilization of discrete-time nonlinear singular systems with time-varying delays

The problem of delay-dependent reliable control for uncertain discrete-time nonlinear singular system with time-varying delays is investigated. A new set of delay-dependent sufficient conditions are established for the existence of a reliable controller which ensures the singular system to be regular, causal (absence of impulses) and stable for all admissible parameter uncertainties. First, the controller for singular system in the presence of known actuator fault matrix is designed. Further, the result is extended to study the robust stabilization of singular systems with unknown actuator fault matrix. The sensor failures of actuator are considered by a variable, which is varying in a given interval. The conditions for the existence of proposed controller design are established in the form of linear matrix inequality (LMI), which can be easily solved via standard numerical software. Finally, numerical examples with simulation results are provided to illustrate the usefulness of control design and results are compared with existing results to show the less conservatism of the proposed theory.

Safety enhancement for nonlinear systems via learning-based model predictive control with Gaussian process regression

This paper proposes a novel safe Gaussian process model predictive control scheme for discrete-time nonlinear systems subject to additive uncertainties. The scheme is implemented using an online learning framework that provides safety guarantees based on a nominal model, and employs Gaussian process regression (GPR) to learn the uncertainties to improve the control performance. The advantage of this framework lies in the ability to decouple safety and performance in the robust controller design. Furthermore, the inaccuracy measure given by GPR can be used to adaptively tighten the constraints, leading to the less conservative behaviors. A rigorous analysis of the constraint satisfaction, recursive feasibility and closed-loop stability is also presented. Two simulation examples, including a regulation problem of a nominally linear system and a tracking problem of a car whose model is nonlinear, are performed to verify the effectiveness of the proposed scheme.

A new approach for dynamic output feedback control design of time-delayed nonlinear systems

This paper introduces a full-order dynamic output-feedback (DOF) controller for discrete-time nonlinear systems with time-varying delays represented by Linear Parameter-Varying (LPV) systems. The controller design is carried out by developing delay-dependent Linear Matrix Inequality based conditions using Lyapunov–Krasovskii stability arguments. A notable characteristic of the proposed approach is that the dynamic controller gains are directly obtained, without the need for variable transformations, equality constraints, or iterative algorithms. This feature sets it apart from many existing approaches in the literature and simplifies the design process. Furthermore, the proposed approach guarantees the local asymptotic stability of the origin of the closed-loop system and provides an estimated admissible initial condition region. The correct operation of the system is ensured once the state trajectories initiated within the admissible initial condition region remain enclosed in the validity domain of the LPV model. Numerical examples are provided to illustrate the potential and effectiveness of the proposed conditions.

Unified Solutions to Optimal Fuzzy Observer-Based Fault Detection for Discrete-Time Nonlinear Systems

Unified Solutions to Optimal Fuzzy Observer-Based Fault Detection for Discrete-Time Nonlinear Systems

Model-free anti-disturbance tracking control for high-order discrete-time nonlinear system based on concurrent learning extended state observer

The research objective of this paper is to address the tracking control problem of high-order discrete-time nonlinear systems in the presence of completely unknown internal uncertainties, unknown external disturbances, and unknown control input gain. To tackle this challenge, the paper proposes a high-order discrete-time model-free anti-disturbance tracking control method without relying on prior knowledge of system parameters. The core of the approach is the development of a data-driven concurrent learning extended state observer. The uniqueness of this observer lies in its ability to estimate unknown control input gains without the persistent excitation and ensure the convergence of the estimates. Based on the concurrent learning extended state observer, a high-order discrete-time tracking control law is designed to precisely track the reference signal, ensuring the stability of the nonlinear closed-loop system. The effectiveness of this method was verified by applying simulation verification to the heading tracking control of surface automatic vehicles.

Data-based adaptive predictive bipartite consensus for nonlinear multiagent systems against DoS attacks

This article investigates a Denial-of-Service (DoS) attack problem for nonlinear unknown discrete-time multiagent systems (MASs) to implement bipartite consensus tracking tasks with fixed and switching topologies. Firstly, an equivalent linearization data model of each agent is constructed using a pseudo partial derivative approach, where only one parameter needs to be estimated using input/output data of the controlled MASs. Meanwhile, the DoS attack behavior is described by a Bernoulli distribution process, and both cooperative and competitive relationships among agents are investigated. Moreover, an increment prediction compensator is designed to reduce the effect of DoS attacks. A data-based adaptive predictive bipartite consensus control algorithm is formulated. The corresponding theoretical analysis indicates that tracking errors of MASs with fixed and switching topologies converge to a small range around zero. Finally, several simulations and hardware tests further verify the proposed scheme’s effectiveness.

PAC-Bayes Generalisation Bounds for Dynamical Systems including Stable RNNs

In this paper, we derive a PAC-Bayes bound on the generalisation gap, in a supervised time-series setting for a special class of discrete-time non-linear dynamical systems. This class includes stable recurrent neural networks (RNN), and the motivation for this work was its application to RNNs. In order to achieve the results, we impose some stability constraints, on the allowed models. Here, stability is understood in the sense of dynamical systems. For RNNs, these stability conditions can be expressed in terms of conditions on the weights. We assume the processes involved are essentially bounded and the loss functions are Lipschitz. The proposed bound on the generalisation gap depends on the mixing coefficient of the data distribution, and the essential supremum of the data. Furthermore, the bound converges to zero as the dataset size increases. In this paper, we 1) formalize the learning problem, 2) derive a PAC-Bayesian error bound for such systems, 3) discuss various consequences of this error bound, and 4) show an illustrative example, with discussions on computing the proposed bound. Unlike other available bounds the derived bound holds for non i.i.d. data (time-series) and it does not grow with the number of steps of the RNN.

Reducing the wrapping effect of set computation via Delaunay triangulation for guaranteed state estimation of nonlinear discrete-time systems

AbstractSet computation methods have been widely used to compute reachable sets, design invariant sets and estimate system state for dynamic systems. The wrapping effect of such set computation methods plays an essential role in the accuracy of their solutions. This paper studies the wrapping effect of existing interval, zonotopic and polytopic set computation methods and proposes novel approaches to reduce the wrapping effect for these set computation methods based on the task of computing the dynamic evolution of a nonlinear uncertain discrete-time system with a set as the initial state. The proposed novel approaches include the partition of a polytopic set via Delaunay triangulation and also the representation of a polytopic set by the union of small zonotopes for the following set propagation. The proposed novel approaches with the reduced wrapping effect has been further applied to state estimation of a nonlinear uncertain discrete-time system with improved accuracy. Similar to bisection for interval and zonotopic sets, Delaunay triangulation has been introduced as a set partition tool for polytopic sets, which has opened new research directions in terms of novel set partition, set representation and set propagation for reducing the wrapping effect of set computation.

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.

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.

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.

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.

Adaptive Optimal Control of Networked Nonlinear Systems With Stochastic Sensor and Actuator Dropouts Based on Reinforcement Learning.

This article investigates the adaptive optimal control problem for networked discrete-time nonlinear systems with stochastic packet dropouts in both controller-to-actuator and sensor-to-controller channels. A Bernoulli model-based Hamilton-Jacobi-Bellman (BHJB) equation is first developed to deal with the corresponding nonadaptive optimal control problem with known system dynamics and probability models of packet dropouts. The solvability of the nonadaptive optimal control problem is analyzed, and the stability and optimality of the resulting closed-loop system are proven. Two reinforcement learning (RL)-based policy iteration (PI) and value iteration (VI) algorithms are further developed to obtain the solution to the BHJB equation, and their convergence analysis is also provided. Furthermore, in the absence of a priori knowledge of partial system dynamics and probabilities of packet dropouts, two more online RL-based PI and VI algorithms are developed by using critic-actor approximators and packet dropout probability estimator. It is shown that the concerned adaptive optimal control problem can be solved by the proposed online RL-based PI and VI algorithms. Finally, simulation studies of a single-link manipulator are provided to illustrate the effectiveness of the proposed approaches.