Stability Of Discrete-time Nonlinear Systems Research Articles

Feedback Stabilization of Quasi-One-Sided Lipschitz Nonlinear Discrete-Time Systems with Reduced-Order Observer

The feedback stabilization problem for nonlinear discrete-time systems with a reduced-order observer is investigated, in which the nonlinear terms of the systems satisfy the quasi-one-sided Lipschitz condition. First, a discrete-time reduced-order observer for nonlinear systems is designed. Then, a feedback controller with a reduced-order observer is designed for realizing the stabilization of nonlinear discrete-time systems. We prove that the design of a feedback controller and reduced-order observer of systems can be carried out independently in the case of discrete-time with nonlinear terms, which largely reduces the computational complexity of the observer and controller. The introduction of the quasi-one-sided Lipschitz condition simultaneously enhances the robustness and stability of nonlinear control systems. Finally, the feasibility and effectiveness of the proposed design approach is verified by a numerical simulation.

On the Partial Stability of Nonlinear Discrete-Time Systems with Delay

A system of nonlinear discrete (finite-difference) of a general form with a bounded delay is considered. Interest in the tasks of qualitative analysis of such systems has increased significantly in recent years. At the same time, the problem of stability with respect to all variables of the zero equilibrium position, which has a great generality, is mainly analyzed in domestic and foreign literature. The main research method is a discrete-functional analogue of the direct Lyapunov method. In this article, it is assumed that the system under consideration admits a “partial” (in some part of the state variables) zero equilibrium position. The problem of stability of a given equilibrium position is posed, and stability is considered not in all, but only in relation to a part of the variables that determine this equilibrium position. Such a problem belongs to the class of problems of partial stability, which are actively studied for systems of various forms of mathematical description. The proposed statement of the problem complements the scope of the indicated studies in relation to the system under consideration. To solve this problem, a discrete version of the Lyapunov– Krasovskii functionals method is used in the space of discrete functions with appropriate specification of the functional requirements. To expand the capabilities of this method, it is proposed to use two types of additional auxiliary (vector, generally speaking) discrete functions in order to: 1) adjustments of the phase space region of the system in which the Lyapunov–Krasovskii functional is constructed; 2) finding the necessary estimates of the functionals and their differences (increment) due to the system under consideration, on the basis of which conclusions about partial stability are made. The expediency of this approach lies in the fact that as a result, the Lyapunov-Krasovskii functional, as well as its difference due to the system under consideration, can be alternating in the domain that is usually considered when analyzing partial stability. Sufficient conditions of partial stability, partial uniform stability, and partial uniform asymptotic stability of the specified type are obtained. The features of the proposed approach are shown on the example of two classes of nonlinear systems of a given structure, for which partial stability is analyzed in parameter space. Attention is drawn to the expediency of using a one-parameter family of functionals.

Global stability of discrete-time non-linear systems with descriptor standard and fractional positive linear parts and scalar feedbacks

Global stability of discrete-time non-linear systems with descriptor standard and fractional positive linear parts and scalar feedbacks

Discrete-time energy-balance passivity-based control

In this paper, new results for passivation and stabilization of discrete-time nonlinear systems via energy balancing are established. When specified on sampled-data systems, the approach is constructive for computing stabilizing digital controllers that assign, at all sampling instants, a target energy profile while stabilizing a target equilibrium. The class of mechanical systems is discussed as an example. Simulations are reported highlighting, for position regulation of a 2R robot, the effect of approximate solutions with respect to standard emulation.

Local Stabilization of Discrete-Time Nonlinear Systems

This article considers local stability analysis and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno fuzzy models. Conditions are established using Lyapunov functions and controller gains that depend also on past samples. Together with the stability and design conditions, an estimate of the region of attraction is also determined. The developed conditions are discussed and illustrated on several examples.

Evolved auxiliary controller with applications to aerospace

PurposeTo guarantee the asymptotic stability of discrete-time nonlinear systems, this paper aims to propose an evolved bat algorithm fuzzy neural network (NN) controller algorithm.Design/methodology/approachIn evolved fuzzy NN modeling, the NN model and linear differential inclusion representation are established for the arbitrary nonlinear dynamics. The control problems of the Fisher equation and a temperature cooling fin for high-speed aerospace vehicles will be described and demonstrated. The signal auxiliary controlled system is represented for the nonlinear parabolic partial differential equation (PDE) systems and the criterion of stability is derived via the Lyapunov function in terms of linear matrix inequalities.FindingsThis representation is constructed by sector nonlinearity, which converts the nonlinear model to a multiple rule base for the linear model and a new sufficient condition to guarantee the asymptotic stability.Originality/valueThis study also injects high frequency as an auxiliary and the control performance to stabilize the nonlinear high-speed aerospace vehicle system.

Interval centred form for proving stability of non-linear discrete-time systems

In this paper, we propose a new approach to prove stability of non-linear discrete-time systems. After introducing the new concept of stability contractor, we show that the interval centred form plays a fundamental role in this context and makes it possible to easily prove asymptotic stability of a discrete system. Then, we illustrate the principle of our approach through theoretical examples. Finally, we provide two practical examples using our method : proving stability of a localisation system and that of the trajectory of a robot.

Finite-Time Stabilization Procedures for Discrete-Time Nonlinear Systems in Feedback Form

New control procedures for finite-time stabilization of discrete-time nonlinear systems in the strict-feedback form are proposed. The proposed designs enjoy the properties of being dependent on one design parameter and the ability to bring the state vector to the origin in finite time. Moreover, the settling-time functions are estimated in terms of the initial state vector and the unique design parameter that regulates the rate of the finite-time convergence. Finally, a trajectory-tracking control algorithm is presented and approved by numerical simulations. The simulation results have shown the excellent performance of the closed-loop system for different values of the free design parameter.

Stabilization of Nonlinear Discrete-Time Systems to Target Measures Using Stochastic Feedback Laws

In this article, we address the problem of stabilizing a discrete-time deterministic nonlinear control system to a target invariant measure using time-invariant stochastic feedback laws. This problem can be viewed as an extension of the problem of designing the transition probabilities of a Markov chain so that the process is exponentially stabilized to a target stationary distribution. Alternatively, it can be seen as an extension of the classical control problem of asymptotically stabilizing a discrete-time system to a single point, which corresponds to the Dirac measure in the measure stabilization framework. We assume that the target measure is supported on the entire state space of the system and is absolutely continuous with respect to the Lebesgue measure. Under the condition that the system is locally controllable at every point in the state space within one time step, we show that the associated measure stabilization problem is well-posed. Given this well-posedness result, we then frame an infinite-dimensional convex optimization problem to construct feedback control laws that stabilize the system to a target invariant measure at a maximized rate of convergence. We validate our optimization approach with numerical simulations of two-dimensional linear and nonlinear discrete-time control systems.

Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators

Although most of the control design methods assume unbounded control signals, real systems do have saturating actuators, which may degenerate closed-loop performance or even lead to unstable behavior. Additionally, the delay is generally almost ubiquitous in processes, that is also imposing performance and stability constraints. Our main contribution is to provide a controller design methodology for the stabilization of delayed systems under saturating actuators. Specifically, we address the design of non-Parallel Distributed Compensation (non-PDC) state feedback fuzzy control laws that locally stabilize a class of nonlinear discrete-time systems with state time-varying delay and saturating actuators. The proposed non-PDC control law depends on the current state xk and the state delayed by d¯ samples. Based on the Lyapunov–Krasovskii approach, we characterize the safe region of initial conditions through two sets: an ellipsoidal one for the current state vector, and another set for the delayed state vectors. Through two convex optimization procedures, we can maximize the estimate of the region of attraction of the closed-loop control system. Additionally, a relaxation method inspired by the Frank-Wolfe algorithm is introduced, yielding better estimates of the region of attraction. The achievements are compared with other finds in the literature, illustrating the efficiency of this proposal.

Reduction-based stabilization of nonlinear discrete-time systems through delayed state measurements

In this paper, the problem of stabilizing nonlinear discrete-time systems affected by delayed state measures is addressed under average passivity-based control. The contribution stands in the introduction of a new delay-free dynamics which is used for the design of the stabilizing feedback. Simulations over an academic example illustrate the performances in a comparative sense.

Evolved Fuzzy NN Control for Discrete-Time Nonlinear Systems

To guarantee the asymptotic stability of discrete-time nonlinear systems, this paper proposes an Evolved Bat Algorithm (EBA) fuzzy neural network (NN) controller. In the evolved fuzzy NN modeling, an NN model and linear differential inclusion (LDI) representation are established for arbitrary nonlinear dynamics. This representation is constructed by the use of sector nonlinearity to convert a nonlinear model to the multiple rule base of the linear model, and a new sufficiency condition to guarantee asymptotic stability in the Lyapunov function is implemented in terms of linear matrix inequalities. The proposed method is an enhancement of existing methods which produces good results.

Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

To Problem of Partial Stability of Nonlinear Discrete-Time Systems

It is can identify three main classes of problems broadly characterizing stability of a dynamical systems, viz., (1) stability with respect to a part of the variables of the zero equilibrium position (Lyapunov-Rumyantsev stability problem), (2) stability of the partialzero equilibrium position, and (3) stability with respect to a part of the variables of the partialzero equilibrium position. In the problem of stability with respect to a part of the variables of the zero equilibrium position of systems of ordinary differential equations with continuous right-side assumes the domain of initial perturbations to be a sufficiently small neighborhood of the zero equilibrium position. Along with this statement, the case then initial perturbations can be large with respect to one part of non-controlled variables and arbitrary with respect to their other part is also considered. On the other hand, for stability problem of partial zero equilibrium positions of systems of ordinary differential equations also naturally assume that initial perturbations of variables that do not define the given equilibrium position can be large with respect to one part of the variables and arbitrary with respect to their other part. Contrary the assumptions that initial perturbations of this variables are either only arbitrary or only large the combined assumption made it possible an admissible trade-off between the meaning sense for notion of stability and the respective requirements on the Lyapunov functions. The article studies the problem of stability for nonlinear discrete-time systems: stability with respect to a part of the variables of partial equilibrium position. Initial perturbations of variables that do not define the given equilibrium position can be large (belonging to an arbitrary compact set) with respect to one part of the variables and arbitrary with respect to their other part. A conditions of stability of this type are obtained in the context of a discrete analog of the Lyapunov functions method, which generalize a number of existing results. Example is given. The problem of unification of process of studying stability problems of stationary and non-stationary nonlinear discrete-time systems is also discussed

Input-to-state stabilization of nonlinear discrete-time systems with event-triggered controllers

This paper studies an event-triggered control problem of nonlinear discrete-time systems in the presence of external disturbances. In particular, we focus on the practically interesting situation where the controller update at step k is determined by the feedback information x(k−1) measured at step k−1. The presence of external disturbance makes it hard to predict the magnitude of x(k) based on x(k−1) and then to design a desired event-triggering data transmission mechanism. In this paper, refined tools of input-to-state stability (ISS) and the nonlinear small-gain theorem are developed to estimate the influence of external disturbances, before an ISS-induced design is proposed to solve the problem. The proposed solution leads to a new event-triggering data transmission mechanism, which drastically differs from the existing event-triggered control results.

Local stability and local stabilization of discrete-time T–S fuzzy systems with time-delay

The goal of this paper is to propose linear matrix inequality (LMI) formulations to analyze the local stability and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno (T–S) fuzzy systems with time-delay. In order to estimate the domain of attraction (DA) of the time-delay systems, the so-called fuzzy Lyapunov-Krasovskii functionals are used to characterize the subsets of the DA as sublevel sets of the Lyapunov- Krasovskii functionals. We use the information of the variation rates of the membership functions to obtain less conservative results. Finally, examples are given to illustrate the proposed methods.

Stabilization of time-delay nonlinear discrete-time systems with saturating actuators through T-S models

In this paper the problem of local stabilization of nonlinear discrete-time systems with time-varying delay and saturating actuators is studied. Firstly, through a fuzzy Lyapunov-Krasovskii (L-K) function, we develop convex conditions to synthesize fuzzy state feedback gain controllers that stabilize the nonlinear system subject to saturating actuators. Next, we introduce a new approach to compute an estimate of the region of attraction where the initial condition sequence is split into two subsequences. The first one is composed of the state vector at the actual instant of sampling, i.e. for k = 0. The second one is composed of the state vectors at the delayed samplings. Then, we propose a convex optimization problem to maximize the estimated region of attraction of the closed loop control system. Finally, we give a numerical example to illustrate the obtained results.

Data-driven asymptotic stabilization for discrete-time nonlinear systems

In this paper, we propose a data-driven feedback controller design method based on Lyapunov approach, which can guarantee the asymptotic stability of the closed-loop and enlarge the estimate of domain of attraction (DOA) for the closed-loop. First, sufficient conditions for a feedback controller asymptotically stabilizing the discrete-time nonlinear plant are proposed. That is, if a feedback controller belongs to an open set consisting of pairs of control input and state, whose elements can make the difference of a control Lyapunov function (CLF) to be negative-definite, then the controller asymptotically stabilizes the plant. Then, for a given CLF candidate, an algorithm, to estimate the open set only using data, is proposed. With the estimate, it is checked whether the candidate is or is not a CLF. If it is, a feedback controller is designed just using data, which satisfies sufficient conditions mentioned above. Finally, the estimate of DOA for closed-loop is enlarged by finding an appropriate CLF from a CLF candidate set based on data. Because the controller is designed directly from data, complexity in building the model and modeling error are avoided.

Robust inverse optimal control for discrete-time nonlinear system stabilization

This paper presents an inverse optimal control approach in order to achieve stabilization of discrete-time nonlinear systems, avoiding the need to solve the associated Hamilton–Jacobi–Bellman equation, and minimizing a cost functional. Then, the proposed approach is extended to discrete-time disturbed nonlinear systems. The synthesized stabilizing optimal controller is based on a discrete-time control Lyapunov function. The applicability of the proposed approach is illustrated via simulations.

Robust stability and stabilization of a class of nonlinear discrete time stochastic systems: An LMI approach

A problem of robust state feedback stability and stabilization of nonlinear discrete-time stochastic processes is considered. The linear rate vector of a discrete-time system is perturbed by a nonlinear function that satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of nonlinear discrete-time systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The state dependent diffusion is modeled by a normal sequence of identically independently distributed random variables. The new formulation provides a suitable setting for robust stabilization of nonlinear discrete-time systems where the underlying deterministic systems satisfy the generalized matching conditions. Our method which is based on linear matrix inequalities (LMIs) is distinctive from the existing robust control and absolute stability techniques. Examples are given to demonstrate the obtained results.