Quantized control of nonlinear quadratic discrete-time systems

International audience

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.

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

This paper deals with the controller design problem of nonlinear quadratic Markov jumping systems with input saturation. Both mode-dependent and mode-independent state feedback controllers are designed. By using the concept of domain of attraction in mean square sense, sufficient conditions for stochastic stabilisation for nonlinear quadratic systems are derived in terms of linear matrix inequalities. Certain existing results are improved. Simulation examples are presented to illustrate the effectiveness of the proposed technique.

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.

In this work, an enhanced data-driven optimal iterative learning control (eDDOILC) is proposed for nonlinear nonaffine systems where a new iterative sliding mode surface (ISMS) is designed to replace the traditional tracking error in the controller design and analysis. It is the first time to extend the sliding mode surface to the iteration domain for systems operate repetitively over a finite time interval. By virtual of the new designed ISMS, the control design becomes more flexible where both the time and the iteration dynamics can be taken into account. Before proceeding to the controller design, an iterative dynamic linear data model is built between two consecutive iterations to formulate the linear input-output data relationship of the repetitive nonlinear nonaffine discrete-time system. The linear data model is virtual and does not have any physical meanings, which is very different to the traditional mechanism mathematical model. In the sequel, the eDDOILC is proposed by designing an objective function with respect to the proposed two-dimensional ISMS. Rigorous proof is provided to show the convergence of the proposed eDDOILC method. Furthermore, the results have been extended to a multiple-input multiple-output (MIMO) nonaffine nonlinear discrete-time repetitive system. In general, the proposed eDDOILC is data-driven where no explicit model information is included. It is illustrated that the presented eDDOILC is effective when applied to the nonlinear nonaffine uncertain systems.

This paper deals with the problem of output feedback control for a class of discrete-time nonlinear quadratic systems with quantization effects and missing measurements. The measured output of system is quantized by a logarithmic quantizer. Moreover, the missing measurements are modeled by introducing a diagonal matrix composed of a series of mutually independent random variables satisfying certain probabilistic distributions on the interval [0 1]. In this paper, the output feedback controller design method, which guarantees the zero equilibrium point of the closed-loop system is stochastic stable, is provided. By using LMI technique, the sufficient condition is derived for computing the desired controller gain matrix and observer gain matrix of the system. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.

In this paper, the output feedback based finite-horizon near optimal regulation of nonlinear affine discrete-time systems with unknown system dynamics is considered by using neural networks (NNs) to approximate Hamilton-Jacobi-Bellman (HJB) equation solution. First, a NN-based Luenberger observer is proposed to reconstruct both the system states and the control coefficient matrix. Next, reinforcement learning methodology with actor-critic structure is utilized to approximate the time-varying solution, referred to as the value function, of the HJB equation by using a NN. To properly satisfy the terminal constraint, a new error term is defined and incorporated in the NN update law so that the terminal constraint error is also minimized over time. The NN with constant weights and time-dependent activation function is employed to approximate the time-varying value function which is subsequently utilized to generate the finite-horizon near optimal control policy due to NN reconstruction errors. The proposed scheme functions in a forward-in-time manner without offline training phase. Lyapunov analysis is used to investigate the stability of the overall closed-loop system. Simulation results are given to show the effectiveness and feasibility of the proposed method.

In this paper, the direct neural dynamic programming technique is utilized to solve the Hamilton Jacobi-Bellman (HJB) equation online and forward-in-time for the decentralized nearly optimal control of nonlinear interconnected discrete-time systems in affine form with unknown internal subsystem and interconnection dynamics. Only the state vector of the local subsystem is considered measurable. The decentralized optimal controller design for each subsystem consists of an action neural network (NN) that is aimed to provide a nearly optimal control signal, and a critic NN which approximates the cost function. The NN weights are tuned online for both the NNs. It is shown that all subsystems signals are uniformly ultimately bounded (UUB) and that the subsystem inputs approach their corresponding nearly optimal control inputs with bounded error.

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.

An algorithm for control of nonlinear discrete-time systems is presented in the paper. Controlled system is linearized by dynamic output feedback so that the linearized closed loop system is equivalent to a predefined discrete-time transfer function representing reference model of the control system. Choice of the reference model provides placement of zeroes and poles of the closed loop system. In the proposed approach at least one of the poles is not fixed and moves during the time of transient process. Evolution of the pole occurs according to certain rules formalized and implemented in the algorithm by means of fuzzy logic. Therefore, the parameters of the transfer function of the linearized closed loop system may be understood as nonlinear functions which depend on the current control error and its derivative. Thus the poles of the closed loop system are placed dynamically according to the predefined rules providing necessary behavior of the control system. Controlled system has to be represented by a nonlinear model with no couplings between different time instances what can be performed by training an Artificial Neural Network of the specific structure. The developed theory and control algorithm are illustrated by means of numerical example.

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

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.

Optimal control of uncertain nonlinear quadratic systems

Exponential admissibility and [formula omitted] control for nonlinear singular discrete systems with mixed time-varying delays