Dual algebraic framework for discrete-time nonlinear control systems

In this paper we propose definitions of equivalence via stochastic bisimulation and of equivalence of stochastic external behavior for the class of discrete-time stochastic nonlinear control systems with possibly degenerate distributed disturbances. The notions are inspired by the analogue notions that exist for probabilistic chains and for behavioral systems, respectively. Connections between the two notions and also with stochastic reachability are discussed. For the subclass of discrete-time stochastic linear control systems with multivariate possibly degenerate normal distributed disturbances, geometric necessary and sufficient conditions for checking stochastic bisimulation equivalence and stochastic external behavior equivalence are derived. The proposed results nicely extend the theory of bisimulation equivalence and equivalence of external behavior known for non-deterministic control systems to a stochastic setting.

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 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 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.

Exponential stabilizability and robustness analysis for discrete-time nonlinear systems

This paper analyzes the amplitude dependent behavior of discrete-time and discrete-value (discretized/quantized) nonlinear control systems in a frequency domain. First, the robust stability condition which was derived in our previous papers is applied to a discrete-time control system with a stepwise nonlinear element. Then, an instability condition for that type of nonlinear feedback system is given. By considering restricted areas (two sectors) in the stepwise nonlinear characteristic, the amplitude of sustained oscillations for the feedback system will be estimated.

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.

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

The paper focuses on unification of the study of continuous- and discrete-time nonlinear control systems using the mathematical tools of pseudo-linear algebra. The realization problem is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of the state space realization of the nonlinear input-output equation defined in terms of the pseudo-linear operator. The sufficiency part of the proof gives a constructive procedure (up to integration of the one-forms) for obtaining the observable state equations. The special cases for continuous- and discrete-time systems, described either in terms of the shift or difference operator, follow from the general result as the special cases. Moreover, the result accommodates also the systems described via q-shift or q-difference operators.

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.

Surprisingly, computer algebra systems have not been used for the analysis of discrete-time nonlinear control systems. The only exceptions seem to be papers based on Reduce and on Maple. This paper presents certain modelling problems of discrete-time nonlinear system as an example of what can be done with symbolic computation in nonlinear discrete-time control with Mathematica.

In the paper we give more general then previous ([l],[7],[8]) definition of dynamic feedback output (state) equivalence of two nonlinear discrete-time input-output control system. Feedback transformations that are presented depend both on past and future values of outputs and controls (or states and controls).

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.

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.

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.