Delayed State Vector Research Articles

Type-Dependent Average Dwell Time Method and Its Application to Delayed Neural Networks With Large Delays.

This article investigates the stability of delayed neural networks with large delays. Unlike previous studies, the original large delay is separated into several parts. Then, the delayed neural network is viewed as the switched system with one stable and multiple unstable subsystems. To effectively guarantee the stability of the considered system, the type-dependent average dwell time (ADT) is proposed to handle switches between any two sequences. Besides, multiple Lyapunov functions (MLFs) are employed to establish stability conditions. Adding more delayed state vectors increases the allowable maximum delay bound (AMDB), reducing the conservatism of stability criteria. A general form of the global exponential stability condition is put forward. Finally, a numerical example illustrates the effectiveness, and superiority of our method over the existing one.

An enhanced control parameterization technique with variable switching times for constrained optimal control problems with control-dependent time-delayed arguments and discrete time-delayed arguments

In this paper, we consider an optimal control problem whose dynamic system consists of control-dependent time-delayed arguments in the control and discrete time-delayed arguments in the state. Recently, control parameterization methods using hybrid time-scale transformation techniques have been developed to optimize both the control parameters and their switching times when the dynamic system consists of discrete time-delayed arguments only. This transformation maps the variable switching times in the original time horizon to fixed switching times in the new time horizon. Thus, the purpose of this paper is to extend the above technique to solve problems with both control-dependent time-delayed arguments and discrete time-delayed arguments. Using the above technique, we convert both the delayed state vector and the delayed control vector into non-delayed vectors in the new time horizon; hence both the control parameters and their switching times can be optimized. The gradients of the objective functions and the constraint functions with respect to the switching times and the controls are obtained. Two numerical examples, one of which is a practical problem involving the optimal transmission of messages in an internet, are solved to demonstrate the efficiency of our method.

New Delay-Partitioning LK-Functional for Stability Analysis with Neutral Type Systems

This paper investigates the stability issues associated with neutral-type delay systems. Firstly, the delay-partitioning method is employed to construct a brand-new LK-functional candidate. The discrete delay and a neutral delay are divided into several piecewise points through a relaxable sequence of constant numbers, are increasing at a steady rate and are not larger than 1. Secondly, to fully use the interconnection information among the delayed state vectors, a new LK-functional is constructed. Thirdly, the recently published single/multiple integral inequalities are employed to bound the derivative of the newly developed LK function. Finally, a novel stability criterion for neutral systems is developed based on the above treatment. Furthermore, a new corollary is also proposed for the condition of τ=h. The benefits and productivities of our method are demonstrated by numerical examples.

Stability analysis for systems with multiple/single time delays via a Cascade augmented L-K functional

This paper investigates stability of linear systems with multiple/single time-delays. Firstly, a three-level cascade augmented Lyapunov-Krasovskii (L-K) functional is introduced, in which interconnect information among delayed state vectors is fully taken into account. Based on a newly integral inequality and the cascade L-K functional, a novel stability criterion is derived for linear systems with multiple time-delays. Secondly, it is found that the proposed L-K functional is also suitable for linear systems with single time-delay if the delay-partitioning method is employed. This motivates us to obtain a less conservative stability condition for linear systems with single time-delay. Finally, Numerical examples are given to confirm the advantages of the proposed method.

A new robust discrete-time sliding mode control design for systems with time-varying delays on state and input and unknown unmatched parameter uncertainties

This paper proposes a new robust Discrete-Time Sliding Mode Control (dt-smc) scheme for linear discrete-time delay system. The delays, affecting the state and the input, are assumed to be known, time-varying and bounded. The considered system is also subject to time-varying Unknown But norm-Bounded (ubb) unmatched uncertainties, that act on the state and the delayed state vectors. The designed controller is based on a novel sliding surface. Using a new Lyapunov functional candidate and by introducing slack variables, a less conservative delay-dependent sufficient condition ensuring the asymptotic stability of the closed loop system is derived in terms of Linear Matrix Inequalities (lmis). With respect to that condition a new dt-smc law is synthesized. The proposed control algorithm is applied to an Autonomous Underwater Vehicle (auv). The simulation results demonstrate the effectiveness of the proposed controller.

Stability Analysis and Robust H∞ Controller Synthesis with Derivatives of Membership Functions for T-S Fuzzy Systems with Time-varying Delay: Input-output Stability Approach

This paper investigates the problem of delay dependent stability and H∞ control design with derivatives of membership functions of uncertain Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. A model transformation is employed by considering a three-term approximation of delayed state vector. Using Scaled Small Gain (SSG) theorem and fuzzy weighting-dependent Lyapunov functions with some useful slack variables, less conservative robust stability and stabilization criteria are formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by using standard numerical packages. Finally, numerical experiments are presented to illustrate the effectiveness of the proposed method.

Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators

Although most of the control design methods assume unbounded control signals, real systems do have saturating actuators, which may degenerate closed-loop performance or even lead to unstable behavior. Additionally, the delay is generally almost ubiquitous in processes, that is also imposing performance and stability constraints. Our main contribution is to provide a controller design methodology for the stabilization of delayed systems under saturating actuators. Specifically, we address the design of non-Parallel Distributed Compensation (non-PDC) state feedback fuzzy control laws that locally stabilize a class of nonlinear discrete-time systems with state time-varying delay and saturating actuators. The proposed non-PDC control law depends on the current state xk and the state delayed by d¯ samples. Based on the Lyapunov–Krasovskii approach, we characterize the safe region of initial conditions through two sets: an ellipsoidal one for the current state vector, and another set for the delayed state vectors. Through two convex optimization procedures, we can maximize the estimate of the region of attraction of the closed-loop control system. Additionally, a relaxation method inspired by the Frank-Wolfe algorithm is introduced, yielding better estimates of the region of attraction. The achievements are compared with other finds in the literature, illustrating the efficiency of this proposal.

Improved Delay-Dependent Stability of Nonlinear Quadratic TS Fuzzy Systems

This paper focuses on the problem of delay-dependent stability for nonlinear quadratic Takagi–Sugeno (TS) fuzzy systems with time-varying delay using the input–output approach. The results are based on the model transformation by employing a three-terms approximation of delayed state vector. By applying the scaled small-gain theorem and Lyapunov–Krasovskii functional, the stability criteria is obtained in terms of linear matrix inequalities. Furthermore, the Wirtinger-based integral inequality approach has been employed to derive less conservative results. Finally, the numerical examples are provided to demonstrate the effectiveness of the obtained results and for comparison with previous work.

Quiet standing: The Single Inverted Pendulum model is not so bad after all.

In the study of balance and postural control the (Single) Inverted Pendulum model (SIP) has been taken for a long time as an acceptable paradigm, with the implicit assumption that only ankle rotations are relevant for describing and explaining sway movements. However, more recent kinematic analysis of quiet standing revealed that hip motion cannot be neglected at all and that ankle-hip oscillatory patterns are characterized by complex in-phase and anti-phase interactions, suggesting that the SIP model should be substituted by a DIP (Double Inverted Pendulum) model. It was also suggested that DIP control could be characterized as a kind of optimal bi-axial active controller whose goal is minimizing the acceleration of the global CoM (Center of Mass). We propose here an alternative where active feedback control is applied in an intermittent manner only to the ankle joint, whereas the hip joint is stabilized by a passive stiffness mechanism. The active control impulses are delivered to the ankle joint as a function of the delayed state vector (tilt rotation angle + tilt rotational speed) of a Virtual Inverted Pendulum (VIP), namely a pendulum that links the ankle to the CoM, embedded in the real DIP. Simulations of such DIP/VIP model, with the hybrid control mechanism, show that it can reproduce the in-phase/anti-phase interaction patterns of the two joints described by several experimental studies. Moreover, the simulations demonstrate that the DIP/VIP model can also reproduce the measured minimization of the CoM acceleration, as an indirect biomechanical consequence of the dynamic interaction between the active control of the ankle joint and the passive control of the hip joint. We suggest that although the SIP model is literally false, because it ignores the ankle-hip coordination, it is functionally correct and practically acceptable for experimental studies that focus on the postural oscillations of the CoM.

LMI based stability analysis of non-linearly perturbed DC motor speed control system with constant and time-varying delays

This research work describes the time-delays effect on speed control of DC motor system with non-linear load disturbance in the states. With this task, the paper presents a linearised model of speed control system based on delay-dependent state equation. Since the non-linearly load disturbance in the DC motor system (with time delay) ultimately affects the evolution of the system states, the non-linear load disturbance is modelled (mathematical) as a non-bounded non-linear function of current state vector and delayed state vector, subsequently included into the system stability analyses. In the analysis, two cases are employed: constant time delay and time-varying delay with exogenous load disturbance. In the proposed results, for broad range of PI controller gains, the maximum value of upper bound delay of DC motor speed control system are determined and new stability criterion was identified.

Delay Dependent Local Stabilization Conditions for Time-delay Nonlinear Discrete-time Systems Using Takagi-Sugeno Models

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

Robust Stability of Networked Load Frequency Control Systems with Time-Varying Delays

In deregulated power system scenario coupled with smart grid technology, for networked load frequency control (LFC), an open communication structure, owing to low cost and flexibility, is preferred over dedicated networks in the feedback control loop for transmitting/receiving the data between the geographically displaced power system and the control center. In such a control scheme, closing the feedback loop through an open communication channel, in turn, introduces two additive time-varying delays of dissimilar characteristics in the feedback path. These delays degrade the performance of the closed-loop system, and exert a destabilizing effect on the overall system. In this paper, using Lyapunov-Krasovskii functional approach, a less conservative stability criterion is presented to ascertain delay-dependent stability of such network-controlled LFC systems with two additive time-varying delays in the feedback path in the presence of uncertain load disturbance conditions. Unlike the existing results, which are derived by combining these two delays into one, the proposed result considers the two delays as separate entities thereby imparting more generalization into the stability analysis. The effect of unknown exogenous load disturbance is incorporated by mathematically modeling them as a bounded non-linear time-varying function of current and delayed state vectors.

Finite-time stability for a linear discrete-time delay systems by using discrete convolution: An LMI approach

The problem of finite-time stability for a class of linear discrete-time systems with state delay is studied. The Lyapunov-Krasovskii-like functional which is based on discrete convolutions of delayed state vector and time-dependent discrete vector functions is used. In order to obtain much less conservative results, new sufficient condition is derived in form of linear matrix inequalities. Numerical example is given to demonstrate the effectiveness of the proposed stability criterion. It was shown that the obtained results are less conservative than some existing ones in the literature. A computer simulation was performed for the analysis of the dynamical behaviour of this system.

Stability Criteria for Nonlinearly Perturbed Load Frequency Systems With Time-Delay

In this paper, new criteria are presented for ascertaining delay-dependent stability of networked multi-area load frequency control (LFC) systems with feedback loop delay in presence of unknown and time-varying exogenous load disturbance. The proposed stability criteria are based on Lyapunov-Krasovskii (LK) functional approach, and they are expressed in linear matrix inequality (LMI) framework. Since the unknown exogenous load disturbance in the power system (cause) ultimately affects the evolution of the state variables of the system (effect), in the reported stability criteria, the load disturbance is mathematically modelled as a norm-bounded nonlinear time-varying function of current and delayed state vectors, and subsequently incorporated into the delay-dependent stability analysis. Two cases of time-delays are considered in the stability analysis: time-varying single feedback-loop delay element and time-invariant multiple feedback-loop delay elements. In addition, to realize less conservative stability criteria with minimal number of decision variables, in the stability analysis, tighter integral inequalities, viz., reciprocal convex combination lemma (for time-varying delay case) and Jenson integral inequality (for time-invariant delay case) are employed. The proposed results are illustrated on standard benchmark LFC systems, and supported suitably by simulation results.

Input delay compensation of linear systems with both state and input delays by adding integrators

This paper studies stabilization of linear systems with both state and input delays. A dynamic input-delay compensator obtained by adding integrators is established to compensate the input delays that can be arbitrarily large. With the input delay compensator, the original stabilization problem reduces to the problem of stabilizing an augmented linear time-delay system without input delay. Three methods are also proposed to design stabilizing controllers for the augmented linear time-delay system. The first method is based on linear matrix inequalities (LMIs) and the second method is based on model reduction. The third method is based on pole placement and is built for the particular case that the original time-delay system has only a pure delayed state vector on its right hand side. For this method, the optimal gain such that the decay rate of the closed-loop system is maximized is also proposed. The effectiveness of the proposed approaches is illustrated by three linear time-delay systems that are open-loop unstable.

Decentralized output-feedback stabilization for interconnected discrete systems with unknown delays

The decentralized feedback stabilization problem of a class of nonlinear interconnected discrete-time systems is considered. This class of systems has unknown-but-bounded state-delay and uncertain nonlinear perturbations satisfying quadratic constraints that are functions of the overall state and delayed state vectors. A decentralized output feedback scheme is proposed and analyzed such that the overall closed-loop system guarantees global delay-dependent stability condition, derived in terms of local subsystem variables. Incorporating feedback gain perturbations, new resilient decentralized feedback scheme is subsequently developed. The proposed approach is formulated within the framework of convex optimization over linear matrix inequalities. Simulation results illustrate the effectiveness of the proposed decentralized output-feedback controllers. Copyright © 2009 John Wiley & Sons, Ltd.

Singular extremal control problem with time delay

In this article, the singular extremal control problem with time delay is studied from the viewpoint of the calculus of variation and matrix theory. Some sufficient conditions for the non-negativity of the second variation are obtained for an optimal control problem with a delayed state vector. The hypothesis of Hass is first assumed, and the total singular situation of Jacobson and Bell is also considered. A second-order necessary condition for optimality is also presented.

A conjugate point condition for control problems with delayed state vectors

The new result presented is a conjugate point condition for an optimal control problem in which there is a constant delay in the state vector. The proof is outlined, and the solution method and a differential game application are discussed.