Time-varying stabilizing feedback for a class of nonlinear discrete-time control systems

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

The problem of dynamic output feedback equivalence of nonlinear observable discrete-time control systems with outputs has been studied by Pawluszewics-Bartosiewics. In this paper we use those results to achieve equivalence of a nonlinear system to a linear one. In this way we obtain an algebraic criterion of dynamic output linearization for discrete-time systems. We recall the concept of anticipating output equivalence, where transformations of outputs and controls depend on future values, and the concept of dynamic output equivalence, depending on past and present controls and outputs values. In order to give the criterion of linearisation of a discrete-time control system we have to use the difference output algebra and the difference output universe associated with the given system.

This chapter investigates the problem of stabilization of nonlinear discrete-time networked control systems (NCS) with event-triggering communication scheme in the presence of signal transmission delay. A Takagi–Sugeno (T–S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy event-triggered controller for the stabilization of nonlinear discrete-time NCS. The idea of the event-triggering communication scheme (a soft computation algorithm) under consideration is that the current sensor data is transmitted only when the current sensor data and the previously transmitted one satisfy a certain state-dependent trigger condition. By taking the signal transmission delay into consideration and using delay system approach, a T–S fuzzy delay system model is established to describe the nonlinear discrete-time NCSs with event-triggering communication scheme. Attention is focused on the design of fuzzy event-triggered controller which ensures asymptotic stability of the closed-loop fuzzy systems. Linear matrix inequality- (LMI-) based conditions are formulated for the existence of admissible fuzzy event-triggered controller. If these conditions are feasible, a desired fuzzy event-triggered controller can be readily constructed. A nonlinear mass-spring-damper mechanical system is presented to demonstrate the effectiveness of the proposed method.

In this work we study the design and analysis for nonlinear discrete time active control system with boring charter. It is shown that most analysis result for continuous time nonlinear system can be extended to the discrete time case. In previous studies, a method of nonlinear Model Following Control System (MFCS) was proposed by Okubo (1985). In this study, the method of nonlinear MFCS will be extended to nonlinear discrete time system with boring charter. Nonlinear systems which are dealt in this study have the property of norm constraints ║ƒ (v (k))║≤α + β║v (k)║^γ , where α≥0, β≥0, 0≤γ≤1. When 0≤γ<1. It is easy to extend the method to discrete time systems. But in the case γ = 1 discrete time systems, the proof becomes difficult. In this case, a new criterion is proposed to ensure that internal states are stable. We expect that this method will provide a useful tool in areas related to stability analysis and design for nonlinear discrete time systems as well.

Design of output feedback semiglobally practically input-to-state (SPIS) stabilizing controllers for nonlinear strict-feedback networked systems is considered. We first design continuous-time state feedback globally asymptotically (GA) stabilizing controllers and discrete-time reduced-order observers and combine the emulation of state feedback controllers and the designed observers to construct discrete-time output feedback controllers. Then we show that the designed controllers SPIS stabilize nonlinear strict-feedback networked control systems when transmission intervals are sufficiently small and communication networks satisfy suitable assumptions. A numerical example is also given to illustrate the efficiency of proposed design of controllers.

Robust Failure Detection and Isolation in Non-Linear Control Systems

Event-triggered controller design of nonlinear discrete-time networked control systems in T-S fuzzy model

We consider the problem of formally verifying almost-sure (a.s.) asymptotic stability in discrete-time nonlinear stochastic control systems. While verifying stability in deterministic control systems is extensively studied in the literature, verifying stability in stochastic control systems is an open problem. The few existing works on this topic either consider only specialized forms of stochasticity or make restrictive assumptions on the system, rendering them inapplicable to learning algorithms with neural network policies. In this work, we present an approach for general nonlinear stochastic control problems with two novel aspects: (a) instead of classical stochastic extensions of Lyapunov functions, we use ranking supermartingales (RSMs) to certify a.s. asymptotic stability, and (b) we present a method for learning neural network RSMs. We prove that our approach guarantees a.s. asymptotic stability of the system and provides the first method to obtain bounds on the stabilization time, which stochastic Lyapunov functions do not. Finally, we validate our approach experimentally on a set of nonlinear stochastic reinforcement learning environments with neural network policies.

This paper is concerned with fuzzy-model based stabilization of nonlinear networked control systems (NCSs) with time-varying transmission delays and transmission intervals based on a random-delay approach. The real-time distribution of input delays resulting from transmission delays and intervals is modeled as a dependent and nonidentically distributed process. Then a randomly switched Takagi-Sugeno fuzzy system with multiple input-delay subsystems is proposed to model the nonlinear NCSs. Based on an improved Lyapunov-Krasovskii method, which takes into account the real-time distribution of input delays in estimating cross-product integral terms, new sufficient conditions are deriveed for the exponential stability of the overall systems. The resulting controller design method is equivalent to a nonlinear convex optimization problem with LMI constraints. Numerical examples are presented to substantiate the effectiveness of our results.

Fundamentals of the theory of non-linear pluse control systems

This paper describes discretized PID control and robust stabilization for continuous plants on a grid pattern that is composed of integers. Although all control systems are currently realized using discretized signals, the analysis and design of discrete-time and discrete-value (digital) control systems has not been elucidated. In this paper, a designing problem of such nonlinear discrete-time control systems based on the PID algorithm is presented in a frequency domain. In the design procedure, the concept of a modified Nyquist and Hall diagram (off-axis M-circles) for nonlinear control systems is applied. Numerical examples for continuous plants with time delay are provided to verify the design method.

Generalized sampling solutions for discontinuous stabilization of nonlinear 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.

The package NLControl, developed in the Institute of Cybernetics at Tallinn University of Technology within the Mathematica environment, has been made partially available over the internet using webMathematica tools. The package consists of functions that assist the solution of different modelling, analysis, and synthesis problems for nonlinear control systems, described either by state or by input-output equations. This paper focuses on describing the webMathematica-based tools for discrete-time nonlinear control systems.

This paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input single-output (SISO) discrete-time non-linear systems. The condition is presented in terms of common factors of a non-commutative polynomial expression. This extends the result on controllability of linear systems and the recent result on controllability of continuous-time non-linear systems to expose controllability of discrete-time non-linear systems in the input–output framework. The controllability condition yields a finite step computing procedure for examining controllability of discrete-time input–output non-linear systems.