Robust analysis and control for a class of uncertain nonlinear discrete-time systems

This paper deals with the problem of robust analysis and control of a class of nonlinear discrete-time systems with (constant) uncertain parameters. For the analysis problem we use a polynomial Lyapunov function and we generalize, for nonlinear systems, the “extended stability” notion proposed by Oliveira et al. (1999) in the context of linear discrete-time uncertain systems. As a result, we propose an LMI optimization problem to maximize an estimate of the domain of attraction, and also extend this approach to the synthesis problem by considering parameter-dependent Lyapunov functions and nonlinear multipliers. Numerical examples illustrate the approach and show its potential for solving analysis and control problems of nonlinear discrete-time systems.     

A delay-partitioning approach to the stability analysis of discrete-time systems

This paper revisits the problem of stability analysis for linear discrete-time systems with time-varying delay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms of linear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. We have also established that the conservatism of the conditions is a non-increasing function of the number of partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.     

Robust stabilization of linear uncertain discrete-time systems via a limited capacity communication channel

The paper considers the problem of robust stabilization of linear uncertain discrete-time systems via limited capacity communication channels. We consider the case when the control input is to be transmitted via communication channel with a bit-rate constraint. A constructive method to design a robustly stabilizing controller is proposed.     

Performance impacts of autocorrelated flows in multi-tiered systems

This paper presents an analysis of the performance effects of burstiness in multi-tiered systems. We introduce a compact characterization of burstiness based on autocorrelation that can be used in capacity planning, performance prediction, and admission control. We show that if autocorrelation exists either in the arrival or the service process of any of the tiers in a multi-tiered system, then autocorrelation propagates to all tiers of the system. We also observe the surprising result that in spite of the fact that the bottleneck resource in the system is far from saturation and that the measured throughput and utilizations of other resources are also modest, user response times are very high. When autocorrelation is not considered, this underutilization of resources falsely indicates that the system can sustain higher capacities.

We examine the behavior of a small queuing system that helps us understand this counter-intuitive behavior and quantify the performance degradation that originates from autocorrelated flows. We present a case study in an experimental multi-tiered Internet server and devise a model to capture the observed behavior. Our evaluation indicates that the model is in excellent agreement with experimental results and captures the propagation of autocorrelation in the multi-tiered system and resulting performance trends. Finally, we analyze an admission control algorithm that takes autocorrelation into account and improves performance by reducing the long tail of the response time distribution.     

Stability conditions for a class of time-varying discrete-time systems

This paper deals with the stability conditions for a class of dynamical discrete-time systems, called ‘P-invariants’ and ‘not P-invariants’. Stability conditions are given in terms of polyhedral convex cones and are obtained by using some extensions on M-matrices.     

On near-controllability and stabilizability of a class of discrete-time bilinear systems

In this paper, the near-controllability and stabilizability of a class of discrete-time bilinear systems are studied. A necessary and sufficient condition for the near-controllability of the systems is first derived. Then, on the basis of the corresponding near-controllability criterion, the stabilizability problem of the systems is solved in turn.     

Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

In this paper, a new methodology is developed for the closed-form solution of a generalized version of the finite-horizon linear-quadratic regulator problem for LTI discrete-time systems. The problem considered herein encompasses the classical version of the LQ problem with assigned initial state and weighted terminal state, as well as the so-called fixed-end point version, in which both the initial and the terminal states are sharply assigned. The present approach is based on a parametrization of all the solutions of the extended symplectic system. In this way, closed-form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions. By taking advantage of standard software routines for the solution of the algebraic Riccati and Stein equations, our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the Riccati difference equation.     

Analysis of a class of discrete-time systems with power rule

In this communique, the properties of a class of discrete-time dynamic systems with power rule are studied and comparisons with their continuous-time counterparts as well as the linear discrete-time systems are made. It is shown that using power rule can be beneficial for improving dynamic behaviors of discrete-time systems. For example, a power control law may stabilize a discrete-time system which is not stabilizable by the linear control law.     

New characterization of positive realness and control of a class of uncertain polytopic discrete-time systems

This paper deals with the problems of positive real analysis and control synthesis for a class of discrete-time polytopic systems with uncertainties. The systems under consideration are modelled in a polytopic form with linear fractional uncertainties in its vertices. A new linear matrix inequality (LMI) characterization of positive realness for this class of systems is given. It enables one to check the positive realness by using parameter-dependent Lyapunov function. This new characterization exhibits a kind of decoupling between the Lyapunov matrix and the system matrices, which is subsequently exploited for control design. Based on the new result, sufficient conditions with reduced conservatism are obtained. A numerical example is also included to demonstrate the applicability of the proposed results.     

Comments on “Adaptive ILC for a class of discrete-time systems with iteration-varying trajectory and random initial condition”

This note points out some errors in the proof of the asymptotic learning convergence of the tracking error in Chi, Hou and Xu [Chi, R.H., Hou, Z.S., & Xu, J.X. (2008) Adaptive ILC for a class of discrete-time systems with iteration-varying trajectory and random initial condition. Automatica, 44(8), 2207-2213].     

Stabilization of a class of discrete-time switched systems via observer-based output feedback

In this paper, observer-based static output feedback control problem for discrete-time uncertain switched systems is investigated under an arbitrary switching rule. The main method used in this note is combining switched Lyapunov function (SLF) method with Finsler’s Lemma. Based on linear matrix inequality (LMI) a less conservative stability condition is established and this condition allows extra degree of freedom for stability analysis. Finally, a simulation example is given to illustrate the efficiency of the result.     

Analysis of a large number of Markov chains competing for transitions

We consider the behaviour of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyse the first time at which one of the Markov chains reaches its absorbing state. When the number of Markov chains goes to infinity, we analyse the asymptotic behaviour of the system for an arbitrary probability mass function governing the competition. We give conditions that ensure the existence of the asymptotic distribution and we show how these results apply to cluster-based distributed storage when the competition is handled using a geometric distribution.     

Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions

In this paper, output feedback adaptive control is investigated for a class of nonlinear systems in output-feedback form with unknown control gains. To construct output feedback control, the system is transformed into the form of the NARMA (nonlinear-auto-regressive-moving-average) model, based on which future output prediction is carried out. With employment of the predicted future output, a constructive output feedback adaptive control is given with the discrete Nussbaum gain exploited to overcome the difficulty due to unknown control directions. Under the global Lipschitz condition of the system functions, the boundedness of all the closed-loop signals and asymptotical output tracking are achieved by the proposed control. Simulation results are presented to show the effectiveness of the proposed approach.     

Time-varying discrete-time linear systems with bounded rates of variation: Stability analysis and control design

This paper investigates the problems of robust stability analysis and state feedback control design for discrete-time linear systems with time-varying parameters. It is assumed that the time-varying parameters lie inside a polytopic domain and have known bounds on their rate of variation. A convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed. A feasible solution provides a parameter-dependent Lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems. Extensions to deal with robust control design as well as gain-scheduling by state feedback are also provided in terms of linear matrix inequalities. Numerical examples illustrate the results.     

Existence of unmixed solutions of the discrete-time algebraic Riccati equation and a nonstrictly bounded real lemma

The existence of a solution of the discrete-time algebraic Riccati equation is established assuming modulus controllability and positive semidefiniteness on the unit circle of the Popov function. As an application a nonstrictly bounded real lemma is obtained.     

Stability of discrete-time systems joined with a saturation operator on the state–space: Generalized form of Liu-Michel’s criterion

A novel criterion for the global asymptotic stability of discrete-time systems in a state–space realization using saturation arithmetic was previously given by Liu and Michel. In their approach, as in most (or probably all) existing approaches, the matrix is assumed to be symmetric. In this communique, it is shown that their approach is such as to be extended to allow the use of an unsymmetric matrix . Taking note of this, a more general form of their criterion is presented. A modified form of the criterion is also presented. Examples show the effectiveness of the generalized approach. To the best of author’s knowledge, this communique is the first report of exploiting an unsymmetric to obtain a stability condition with reduced conservatism.     

Waiting time distribution in a class of discrete-time cyclic service multi-queue systems

A new approximate method is developed for finding the waiting and sojourn time distributions in a class of multi-queue systems served in cyclic order at discrete intervals. An immediate application for such a model is in communication networks where a number of different traffic sources compete to access a group of transmission channels operating under a time-slotted sharing policy. This system maps naturally onto a model in which the inter-visit time has a probability mass function of phase-type. We derive a set of matrix equations with easily tractable iterative procedures for their solution and controllable accuracy in their numerical evaluation. We then validate the analytical model against simulation and discuss the validity of the assumptions. This methodology can be extended to several other polling strategies.     

Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels

This paper deals with the problem of Fault Detection and Isolation (FDI) for discrete-time Markovian Jump Linear Systems (MJLSs). A geometric property related to the unobservable subspace of an MJLS is first presented and the concept of unobservability subspace is introduced. Sufficient conditions for designing an H_∞-based FDI algorithm for MJLSs subject to input disturbances and measurement noise are presented and developed. Our proposed approach is then applied to the problem of fault detection and isolation in a network of multi-agent systems when imperfect communication channels exist among the agents. A discrete-time communication link with a stochastic packet dropping effect is considered based on the Gilbert–Elliott model and the entire network is modeled as a discrete-time MJLS. Simulation results are presented for formation flight of satellites to demonstrate and verify the effectiveness and performance capabilities of our proposed FDI algorithm.     

Addressing a workload characterization study to the design of consistency protocols

Shared Virtual Memory (SVM) provides a low-cost and effective way to implement the shared-memory programming paradigm. SVMs utilize a number of concepts that include consistency models/protocols, sharing patterns, false sharing, and fragmentation issues. The range of issues encountered in an SVM introduces a level of complexity and presents a challenge to many SVM researchers. This paper presents a careful study of SVM systems focusing on how the workload characteristics can affect the performace of consistency protocols. This knowledge is used to propose a novel consistency protocol that improves the system performance. This paper pursues two main goals: (i) to illustrate how different SVM workload characteristics are interrelated, and (ii) to motivate the design of a new multiple-writer memory consistency protocol. To achieve the first goal, we provide a detailed workload characterization analysis and discussion on how consistency models and protocols work. To achieve the second goal, we describe a software-based SVM protocol that achieves better performance than a hardware protocol proposed in the literature. In some workloads, the speedup obtained over the baseline protocol is more than 20%.