Stabilization of Discrete-Time Switched Nonlinear Systems

PurposeNonlinear systems identification from experimental data without any prior knowledge of the system parameters is a challenge in control and process diagnostic. It determines mathematical model parameters that are able to reproduce the dynamic behavior of a system. This paper aims to combine two fundamental research areas: MIMO state space system identification and nonlinear control system. This combination produces a technique that leads to robust stabilization of a nonlinear Takagi–Sugeno fuzzy system (T-S).Design/methodology/approachThe first part of this paper describes the identification based on the Numerical algorithm for Subspace State Space System IDentification (N4SID). The second part, from the identified models of first part, explains how we use the interpolation of linear time invariants models to build a nonlinear multiple model system, T-S model. For demonstration purposes, conditions on stability and stabilization of discrete time, T-S model were discussed.FindingsStability analysis based on the quadratic Lyapunov function to simplify implementation was explained in this paper. The linear matrix inequalities technique obtained from the linearization of the bilinear matrix inequalities was computed. The suggested N4SID2 algorithm had the smallest error value compared to other algorithms for all estimated system matrices.Originality/valueThe stabilization of the closed-loop discrete time T-S system, using the improved parallel distributed compensation control law, was discussed to reconstruct the state from nonlinear Luenberger observers.

This paper has proposed a novel class of the stable discrete-time multilayer perceptron (MLP) neural observer for a class of discrete-time multiple-input multiple-output uncertain nonlinear systems. It is trained online and is used to estimate the states of the system especially when there is no prior knowledge about the dynamics of the system and can be applied as a basis for designing an intelligent control. The weight updating for the MLP is the developed backpropagation algorithm. Online training, convergence of the observer error to neighborhood of origin, robustness against uncertainties and disturbance, low computational expense and fast convergence rate are the main qualities of the suggested method. Lyapunov’s direct method is used to guarantee the stability of the proposed system. To demonstrate the performance of the discrete-time MLP observer, two discrete-time nonlinear dynamic systems are simulated in MATLAB/Simulink. Simulation results confirm the proficiency of the system even at different operating conditions and the presence of disturbance and measurement noise.

The positivity and asymptotic stability of the discrete-time and continuous-time nonlinear systems are addressed. Sufficient conditions for the positivity and asymptotic stability of the nonlinear systems are established. The proposed stability tests are based on an extension of the Lyapunov method to the positive nonlinear systems.

The positivity and asymptotic stability of the discrete-time and continuous-time nonlinear systems are addressed. Sufficient conditions for the positivity and asymptotic stability of the nonlinear systems are established. The proposed stability tests are based on an extension of the Lyapunov method to the positive nonlinear systems.

This paper analyzes the nonlinear phenomena and stability of discrete-time and discrete-value (discretized/quantized) control systems in a frequency domain. By partitioning the discretized nonlinear characteristic into two sections and by defining a sectorial area over a specified threshold, the concept of the robust stability condition for nonlinear discrete-time systems, which was developed in the previous studies, is applied to the discretized nonlinear control system in question. As a result, the nonlinear phenomena of a discretized control system are clarified, and the robust stability of discretized nonlinear feedback systems is elucidated. Numerical examples are shown to verify the result.

The control problem of nonlinear systems affected by external perturbations and parametric uncertainties has attracted the attention for many researches. Artificial Neural Networks (ANN) constitutes an option for systems whose mathematical description is uncertain or partially unknown. In this paper, a Recurrent Neural Network (RNN) is designed to address the problems of identification and control of discrete-time nonlinear systems given by a gray box. The learning laws for the RNN are designed in terms of discrete-time Lyapunov stability. The control input is developed fulfilling the existence condition to establish a Quasi Sliding Regime. In means of Lyapunov stability, the identification and tracking errors are ultimately bounded in a neighborhood around zero. Numerical examples are presented to show the behavior of the RNN in the identification and control processes of a highly nonlinear discrete-time system, a Lorentz chaotic oscillator.

This chapter deals with sufficient conditions of asymptotic stability and stabilization for nonlinear discrete-time systems represented by a Takagi-Sugeno type fuzzy model whose state variables take only nonnegative values at all times for any nonnegative initial state. This class of systems is called positive T–S fuzzy systems. The conditions of stabilizability are obtained with state feedback control. This work is based on multiple Lyapunov functions. The results are presented in LMI form. A real plant model is studied to illustrate this technique.

This paper studies deterministic and stochastic fixed-time stability of autonomous nonlinear discrete-time (DT) systems. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness results from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed-time attractive systems, and a stochastic settling-time function fixed upper bound is derived for stochastic DT systems. Illustrative examples are given along with simulation results to verify the introduced results.

Discrete-time stability with perturbations: application to model predictive control

This paper discusses about the stabilization of unknown nonlinear discrete-time fixed state delay systems. The unknown system nonlinearity is approximated by Chebyshev neural network (CNN), and weight update law is presented for approximating the system nonlinearity. Using appropriate Lyapunov-Krasovskii functional the stability of the nonlinear system is ensured by the solution of linear matrix inequalities. Finally, a relevant example is given to illustrate the effectiveness of the proposed control scheme.

This article concerns the theoretical analysis and mechanical verification of input-to-state stability (ISS) for nonautonomous discrete-time switched systems. To start with, based on a bounded function and the average dwell time, we successively propose less conservative sufficient conditions for uniform input-to-state stability, global uniform asymptotic input-to-state stability, and global uniform exponential input-to-state stability of nonautonomous switched nonlinear systems. Then, for systems with zero inputs, we apply our bounded function and average dwell time based method to further relax the sufficient conditions for their uniform stability, global uniform asymptotic stability, and global uniform exponential stability. Particularly, we propose a linear semidefinite programming based computable approach for mechanical verification of our current theoretical results for the rational (and even certain nonrational) nonautonomous switched systems. Note that our theoretical results and mechanical approach are both illustrated by examples.

In this paper, stability of the projection-based constrained discrete-time extended Kalman filter (EKF) as applied to nonlinear systems in a stochastic framework has been studied. It has been shown that like the unconstrained EKF, the estimation error of the EKF with known constraints on the states remains bounded when the initial error and noise terms are small, and the solution of the Riccati difference equation remains positive definite and bounded. Stability results are verified and performance of the constrained EKF is demonstrated through simulations on a nonlinear engineering example.

The stability and stabilization problems for a class of switched discrete-time nonlinear systems are studied in this paper. Each nonlinear subsystem of the presented switched system is modeled as a piecewise affine (PWA) one by splitting the state space into polyhedron regions. With the aid of a simple searching strategy for active state transition pairs at a switching instant, i.e., the so-called $\mathbb {S}$ -arbitrary switching approach, the stability criteria are derived via the relaxed piecewise quadratic Lyapunov function technique. Then, using the descriptor system approach, a family of PWA stabilizing controllers are designed to guarantee exponential stability of the resulting closed-loop control system, and the corresponding PWA controller gains could be calculated using numerical software. The validity and potential of the developed techniques are verified through a numerical example.

Exponential stability of discrete-time systems, stability domains, and state bounds are obtained with the use of vector norm Lyapunov functions. For linear systems, necessary and sufficient conditions are given for asymptotic stability, and exponential bounds are derived. For nonlinear systems, sufficient conditions for exponential stability also yield a stability domain and exponential bounds. Applications are demonstrated in observer design for uncertain discrete-time nonlinear systems.

This paper describes robust stabilization and PID control for discrete-time and discrete-value (discretized/quantized) control systems. Although all control systems are currently realized using discretized signals, the analysis and design of such nonlinear discrete-time control systems has not been elucidated. In this paper, the robust stability analysis of discrete-time and discrete-value (digital) control systems with discretizing units at the input and output sides of a nonlinear continuous element (sensor/actuator) are examined in a frequency domain, and a method of designing PID control and robust stabilization for nonlinear discretized systems on a grid pattern in the time and control variables space is presented. A modified Nichols diagram and parameter specifications are used in this study. Numerical examples are provided to verify the validity of the designing method.