Discrete-time Nonlinear Control Systems Research Articles

Linear openness and feedback stabilization of nonlinear control systems

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and continuously differentiable stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.

Finite-horizon inverse optimal control for discrete-time nonlinear systems

In this note, we consider the problem of computing the parameters (or weights) of an optimal control objective function given optimal closed-loop state and control trajectories. We establish a method of inverse optimal control that exploits the discrete-time minimum principle. Under a testable matrix rank condition, our proposed method is guaranteed to recover the unknown objective-function parameters of finite-horizon discrete-time nonlinear optimal control problems.

Event-Triggered Control for Nonlinear Systems Under Unreliable Communication Links

This paper is concerned with the event-triggered $H_{\infty }$ control problem for discrete-time nonlinear networked control systems with unreliable communication links. First, an event-triggered scheme is proposed to determine whether the sampled data should be released into the network or not. Second, when the released data is transmitted in the network, a Bernoulli process is employed to model the phenomenon of data losses. Third, considering the instants at which the sampled data is not released or data losses occur, a new random process is first developed to model the input data sequence of the controller under the effect of the buffer. Consequently, a novel method is presented to address the stability analysis and control synthesis problems based on the polynomial fuzzy model approach. Finally, some simulation results are given to illustrate the effectiveness of the proposed method.

Learning control for discrete-time nonlinear systems with sensor saturation and measurement noises

ABSTRACTThe iterative learning control (ILC) is investigated for a class of nonlinear systems with measurement noises where the output is subject to sensor saturation. An ILC algorithm is introduced based on the measured output information rather than the actual output signal. A decreasing sequence is also incorporated into the learning algorithm to ensure a stable convergence under stochastic noises. It is strictly proved with the help of the stochastic approximation technique that the input sequence converges to the desired input almost surely along the iteration axis. Illustrative simulations are exploited to verify the effectiveness of the proposed algorithm.

Robust Adaptive Iterative Learning Control for Discrete-Time Nonlinear Systems With Time-Iteration-Varying Parameters

In this paper, the robustness problem in adaptive iterative learning control for discrete-time nonlinear systems with time-iteration-varying parameters is investigated. The time-iteration-varying parameter can be expressed as the sum of an iteration-invariant time-varying part and a bounded iteration-drift term. Then a novel dead-zone method is applied in estimations of the iteration-invariant part and the drift bound simultaneously. The boundedness of tracking error is ensured by rigorous analysis. The efficacy of the proposed method is illustrated through simulation results.

Event-triggered fault detection observer design for affine fuzzy systems

This paper is concerned with the event-based fault detection observer design problem for a class of discrete-time nonlinear networked control systems with output saturation. The system is represented by the T–S affine fuzzy models, and it is assumed that the premise variables of these models are not necessarily measurable. Different from the existing results, an adaptive event-triggered mechanism is introduced to reduce the burden of communication by judging whether the measured data should be transmitted. Based on the piecewise Lyapunov function and free-weighting matrices technique, the design conditions are given in terms of linear matrix inequalities, which can ensure the residual is sensitive to faults. Finally, a numerical example verifies the validity of the proposed method.

Fuzzy-model-based admissibility analysis and output feedback control for nonlinear discrete-time systems with time-varying delay

This paper is concerned with the admissibility analysis and stabilization problems for singular fuzzy discrete-time systems with time-varying delay. The novelty of this paper comes from the consideration of a new summation inequality which is less conservative than the usual Jensen inequality, the Abel-Lemma based inequality and the Seuret inequality. Based on the inequality, sufficient conditions are established to ensure the systems to be admissible. Moreover, the corresponding conditions for the existence of desired static output feedback controller gains are derived to guarantee that the closed-loop system is admissible. The conditions can be solved by a modified cone complementarity linearization (CCL) algorithm. Examples are given to show the effectiveness of the proposed method.

Multi-step heuristic dynamic programming for optimal control of nonlinear discrete-time systems

Policy iteration and value iteration are two main iterative adaptive dynamic programming frameworks for solving optimal control problems. Policy iteration converges fast while requiring an initial stabilizing control policy, which is a strict constraint in practice. Value iteration avoids the requirement of initial admissible control policy while converging much slowly. This paper tries to utilize the advantages of policy iteration and value iteration, and avoids their drawbacks at the same time. Therefore, a multi-step heuristic dynamic programming (MsHDP) method is developed for solving the optimal control problem of nonlinear discrete-time systems. MsHDP speeds up value iteration and avoids the requirement of initial admissible control policy in policy iteration at the same time. The convergence theory of MsHDP is established by proving that it converges to the solution of the Bellman equation. For implementation purpose, the actor-critic neural network (NN) structure is developed. The critic NN is employed to estimate the value function and its NN weight vector is computed with a least-square scheme. The actor NN is used to estimate the control policy and a gradient descent method is proposed for updating its NN weight vector. According to the comparative simulation studies on two examples, the effectiveness and advantages of MsHDP are verified.

Stability of finite horizon model predictive control with incremental input constraints

Model predictive control of discrete-time nonlinear systems with incremental input constraints is proposed in this paper. Firstly, the existence of the terminal set and terminal penalty is proven on the assumption that the considered system is twice continuously differentiable. Secondly, properties of the optimal cost function are exploited. It shows that the optimal cost function is positive semi-definite, continuous at the equilibrium and monotonically decreasing along the predicted trajectory. The systems under control converge to the equilibrium since the optimal cost function is monotonically decreasing. Thirdly, stability of nonlinear systems is proven in terms of the classical Lyapunov Theorem, where an upper bound of the optimal cost function in the terminal set is chosen as a candidate Lyapuonv function. Finally, the system is asymptotically stable since the system state converges to the equilibrium and the system is stable.

H2−H∞ control of discrete-time nonlinear systems using the state-dependent Riccati equation approach

ABSTRACTA novel H2−H∞ State-dependent Riccati equation control approach is presented for providing a generalized control framework to discrete-time nonlinear system. By solving a generalized Riccati equation at each time step, the nonlinear state feedback control solution is found to satisfy mixed performance criteria guaranteeing quadratic optimality with inherent stability property in combination with H∞ type of disturbance attenuation. Two numerical techniques to compute the solution of the resulting Riccati equation are presented: The first one is based on finding the steady-state solution of the difference equation at every step and the second one is based on finding the minimum solution of a linear matrix inequality. The effectiveness of the proposed techniques is demonstrated by simulations involving the control of an inverted pendulum on a cart, a benchmark mechanical system.

Output Stabilization and Weighted Suppression of Disturbances in Discrete-Time Control Systems

Output stabilizability conditions for some classes of discrete-time linear and nonlinear control systems are formulated by using dynamic controllers. Methods for constructing static and dynamic controllers that provide a specified evaluation of the weighted damping level of external and initial disturbances are proposed for systems with controllable and observable outputs. The implementation of these methods with the use of static state feedback or complete order dynamic controllers is based on the solution of systems of linear matrix inequalities. The example of a controller synthesis for a two-mass mechanical system is given.

Estimated plant’s sensitivity based on data-driving observer for a class of nonlinear discrete-time control systems

This paper presents an adaptive controller based on Fuzzy rule emulated network (FREN) for a class of nonlinear discrete-time systems. The learning algorithm of FREN is conducted via plant’s sensitivity estimated by the data-driven observer unit. The novel learning rate for data-driven scheme is proposed with convergence analysis established by Lyapunov direct method. The control direction can be omitted and boundaries of sensitivity can be assumed to be unknown. Only the relation between plant’s output and control effort is required to design this controller within the format of IF--THEN rules. The closed-loop performance can be guaranteed beside of the convergence of tracking error, adjustable parameters and observer’s output. Results from two practical systems, DC-motor current control and pressing force control, demonstrate that the proposed controller is capable of controlling unknown discrete-time systems with satisfactory performance.

Symbolic Models for Networks of Control Systems

In this note, we propose symbolic models for networks of discrete-time nonlinear control systems. If each subsystem composing the network admits an incremental input-to-state stable Lyapunov function and if some small gain theorem-type conditions are satisfied, a network of symbolic models, each one associated with each subsystem composing the network, is proposed and shown to be approximately bisimilar to the original network with any desired accuracy.

Forward and Backward Shifts of Vector Fields: Towards the Dual Algebraic Framework

An algebraic framework for discrete-time nonlinear control systems is introduced, based on the forward and backward shifts of the vector fields. It appears to be the dual of the framework based on the differential 1-forms. As applications, the characterization of accessibility and the solution to the static state feedback linearization problem are stated in terms of the given vector fields.

Fuzzy Tracking Control for Nonlinear Networked Systems.

This paper studies the observer-based tracking control problem for discrete-time nonlinear networked control systems with parameter uncertainties and unmeasurable state variables. A network-induced constraint, i.e., the intermittent measurement loss, is considered in the controller design. The uncertain nonlinear system is described by an interval type-2 (IT2) fuzzy Takagi-Sugeno model, in which the lower and the upper membership functions with corresponding coefficients are used to capture and express uncertainties existing in the system. A premise-variables-independent IT2 fuzzy observer is constructed to estimate the unmeasurable state variables, and then a novel IT2 fuzzy tracking controller is designed. Furthermore, sufficient criteria are established to guarantee the resulting closed-loop system to be stochastically stable. Finally, two examples are provided to show the effectiveness of the proposed approach.

A NN-Based Hybrid Intelligent Algorithm for a Discrete Nonlinear Uncertain Optimal Control Problem

A discrete-time nonlinear optimal control system with uncertain perturbation is studied in this paper. A value-iteration-based hybrid intelligent algorithm (HIA), incorporating heuristic dynamic programming algorithm and uncertain simulation, is presented. And some characters of the HIA are provided. Two neural networks are used in the algorithm: a critic network is aimed at approximating the objective function, as well as an active network is used to obtain an uncertain optimal control law. As an application, a schistosomiasis compartment SI model is solved to illustrate the feasibility and effectiveness of the HIA.

Indirect Adaptive Control for Discrete-Time Nonlinear Systems based on T-S Fuzzy Model

The main goal of this paper is to present an indirect adaptive fuzzy control of discrete-time non affine nonlinear systems with parametric variations. The synthesis of the state feedback control law is based on the Takagi-Sugeno (T-S) fuzzy models developed by a local description of the considered system. In the first step, the model parameters locally estimated by the fuzzy model are adjusted using gradient method. In the second step, the local control gain based on pole placement is computed. After that, the global state feedback control law is applied to the nonlinear system. Based on the Lyapunov stability theory, the asymptotic stability of the proposed state feedback adaptive fuzzy control method is studied to ensure the global stability of the system. To illustrate the performance of the proposed controller, inverted pendulum and two links robot manipulator arm are presented.

Novel observer-based output feedback control synthesis of discrete-time nonlinear control systems via a fuzzy approach

The paper focuses on less conservative design of observer-based output feedback control synthesis of discrete-time Takagi–Sugeno fuzzy systems on account of an improved homogenous polynomial approach. Compared with those previous methods in the cited literature, a novel fuzzy observer-based output feedback law is constructed in virtue of utilizing one promising solution to homogenous polynomials that are parameter-dependent on both the current-time and those multi-steps past-time normalized fuzzy weighting functions with a pair of prescribed degrees. It is rather remarkable that more information contained in the normalized fuzzy weighting functions is integrated to build the proposed control law, and thus the conservatism left behind previous results can be released in this work. Moreover, a robust version of the underlying result is also proposed. Finally, the effectiveness of the given approach is demonstrated via some illustrative experiments.

Reliable H∞ control for nonlinear discrete-time systems with multiple intermittent faults in sensors or actuators

ABSTRACTThis study proposes a fault-tolerant control method for stochastic systems with multiple intermittent faults (IFs) and nonlinear disturbances, and both sensor and actuator faults are considered. The occurrence and disappearance of IFs are governed by Markov chain, and its transition probabilities are partly known. Hence, the faulty system can be described by a Markovian jump system (MJS). In order to ensure that the MJS is stochastically stable and satisfies H∞ performance index, mode-dependent output feedback controllers are modelled using linear matrix inequalities. Numerous sufficient conditions for stochastic stability are obtained on the basis of Lyapunov stability theory. Finally, the effectiveness of the developed method is evaluated on the three-tank system.

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.