Stability Conditions In Terms Research Articles

Necessary Stability Conditions for Integral Delay Systems

In this article, necessary stability conditions for integral delay systems with distributed single delay are presented. Based on new properties that connect the Lyapunov delay matrix and the fundamental matrix of this class of equations, a complete type Lyapunov–Krasovskii functional is introduced. It allows us to present necessary stability conditions in terms of the delay Lyapunov matrix. The result is illustrated by some examples.

Robust design of iterative learning control for a batch process described by 2D Roesser system with packet dropouts and time‐varying delays

SummaryBatch process, working as a best choice for low‐volume and high‐value products in manufacturing, has been widely used in chemical industries. The actuator faults and time delays often occur in practical production. This paper develops an iterative learning control (ILC) design for a batch process described by two‐dimensional (2D) Roesser system with packet dropouts and time‐varying delays. The phenomenon of actuator faults is regarded as an arbitrary stochastic sequence satisfying the Bernoulli random binary distribution. Firstly, the ILC design for a batch process is transformed into stability analysis for a 2D stochastic system with time‐varying delays. Secondly, for analyzing the stability of 2D stochastic systems, we derive the stability condition in terms of linear matrix inequality. Then, we give a procedure to get the control gain for the ILC design. An injection modeling process as an example with simulations in different cases of data dropout is given to demonstrate the validity of the proposed method. Furthermore, the proposed method has a better result by comparing the existing methods.

Necessary and sufficient conditions to stability of discrete-time delay systems

This paper is concerned with the stability analysis of discrete-time linear systems with time-varying delays. The novelty of this paper lies in that a novel Lyapunov–Krasovskii functional that updates periodically along with the time is proposed to reduce the conservatism and eventually be able to achieve the non-conservativeness in stability analysis. It can be proved that the stability of a discrete-time linear delay system is equivalent to the existence of a periodic Lyapunov–Krasovskii functional. Two necessary and sufficient stability conditions in terms of linear matrix inequalities are proposed in this paper. Furthermore, the novel periodic Lyapunov–Krasovskii functional is employed to solve the ℓ2-gain performance analysis problem when exogenous disturbance is considered. The effectiveness of the proposed results is illustrated by several numerical examples.

On stability of neutral-type linear stochastic time-delay systems with three different delays

We study in this paper the mean square exponential stability of neutral-type linear stochastic time-delay systems with three different delays by using the Lyapunov–Krasovskii functionals (LKFs) based approach. First, for a specified augmented state vector, a complete augmented LKF is constructed to contain all possible integrals so that its time-derivatives (understood in the deterministic sense) can be written as a linear function of the augmented vector. The redundant variables in the LKF are eliminated so as to reduce the computational burden. Moreover, the absolute value requirement of certain terms in the constructed LKFs are removed by analyzing carefully the complex relationship among these three different time delays. Then, for ten different cases of time delays, sufficient stability conditions in terms of linear matrix inequalities are obtained with the help of the Ito formula and properties of stochastic integrals. The degenerated cases that the delays satisfy some equalities are also discussed. A numerical example is employed to illustrate the effectiveness of the proposed approach.

Delay range‐and‐rate dependent stability criteria for systems with interval time‐varying delay via a quasi‐quadratic convex framework

SummaryThis paper concerns the stability analysis of systems with interval time‐varying delay. A Lyapunov‐Krasovskii functional containing an augmented quadratic term and certain triple integral terms is constructed to integrate features of the truncated Bessel‐Legendre inequality less conservative than Wirtinger inequality that encompasses Jensen inequality, respectively, and to exploit merits of the newly developed double integral inequalities tighter than auxiliary function‐based, Wirtinger, and Jensen double integral inequalities. A new quadratic convex lemma is proposed to derive delay and its derivative dependent sufficient stability conditions in terms of linear matrix inequalities synthetically with reciprocal convex approach and affine convex combination. The efficiency of the presented method is illustrated on some classical numerical examples.

New stability criteria for asymptotic stability of time-delay systems via integral inequalities and Jensen inequalities

This paper investigates the new stability criteria for the asymptotic stability of time-delay systems via integral inequalities and Jensen inequalities. Firstly, not only the known constant time delay, but also the unknown time-varying delay is considered for the linear system. Secondly, the new delay-dependent Lyapunov–Krasovskii functional based on the double integral inequalities and Jensen inequalities is introduced, such that the linear system with time-delay is asymptotically stable. Thirdly, two classes of delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived, such that the control design conditions are relaxed and computation complexity is reduced. Compared with previous works, the larger feasible solution region and less conservative results are obtained. Finally, some numerical examples are performed to show the effectiveness and advantage of the proposed method.

Robust predictive extended state observer for a class of nonlinear systems with time-varying input delay

ABSTRACTThis paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy.

Optimal Control of Nonlinear Systems With Time Delays: An Online ADP Perspective

Drawing upon Lyapunov stability theories and online adaptive dynamic programming (ADP) technique, we propose a novel optimal control scheme for the nonlinear time delay system. Our contribution is twofold. First, we investigate the asymptotical stability problem and obtain a generalized stability condition in terms of linear matrix inequalities (LMIs). An explicit, easy-computing delay bound is presented by virtue of Gronwall's inequality. Second, we propose the neural network (NN)-based optimal control strategy by utilizing two approximate NNs. The NN-based optimal control law converges to the real optimal control law since that the estimation errors of NNs weights converge to zero. Numerical examples are presented to illustrate our results.

Exponential stability of positive switched non‐linear systems under minimum dwell time

The positive and exponential stability for a class of switched non-linear systems under minimum dwell time switching is studied, whose non-linear functions for each subsystem are constrained in a sector field by two odd symmetric piecewise linear functions and whose system matrices for each subsystem are Metzler. A class of multiple time-varying Lyapunov functions is applied to obtain the computable sufficient stability conditions in terms of linear programming rather than non-linear programming. An example is provided to demonstrate the effectiveness of the proposed result.

Static Output Feedback Stabilization of Positive Polynomial Fuzzy Systems

This paper deals with the static output feedback stabilization of positive polynomial fuzzy-model-based control systems. The positive polynomial fuzzy model does not need to share the same premise membership functions with the static output feedback polynomial fuzzy controller. Unlike the state feedback control case, the static output feedback control usually leads to nonconvex stability conditions. To circumvent the problem, an approach is employed to transform the nonconvex stability conditions into convex ones by introducing a nonsingular transformation matrix. Initially, the conditions guaranteeing the resultant closed-loop systems to be positive and asymptotically stable are obtained. Moreover, the divisional approximated membership functions that carry the local information of the membership functions are employed to facilitate the stability analysis and controller synthesis. The relaxed stability conditions in terms of sum of squares are obtained based on Lyapunov stability theory. Finally, a simulation example is given to testify the validity of the analysis result.

Heterogeneity and chaos in congestion games

We analyze a class of congestion games where agents use resources to send a finite amount of goods from an initial location to a terminal one. The resources are costly and costs are load dependent. In this context we concentrate on the heterogeneity because we not only assume that agents have limited computational capability but also they differ in the quantities they must send and the reactivity. We introduce an appropriate dynamical system, which has the steady state exactly at the unique Nash equilibrium of the static congestion game, and we investigate the dynamical behavior of the game. We provide a closed form characterization on the unique Nash equilibrium in the underlying static congestion game and we prove that the equilibrium crucially depends on the aggregate congestion. Not only do we provide a local stability condition in terms of the agents’ reactivity and the nonlinearity of the cost functions but also we study the role of the heterogeneity in this context. We show analytically that heterogeneity can be destabilizing with a cascade of flip-bifurcation leading to periodic cycles and finally to chaos but also that mastering the degree of heterogeneity can be used to tame and control complex dynamics; moreover in relation to the reactivity levels their product must roughly small. However if both reactivity levels are near but outside the stability region then it is sufficient to act on just one to restore stability. Finally, if both levels are far form the stability region acting on one will not restore stability.

Design of Sliding Mode Controller With Proportional Integral Sliding Surface for Robust Regulation and Tracking of Process Control Systems

This paper deals with the design of sliding mode controller (SMC) with proportional plus integral sliding surface for regulation and tracking of uncertain process control systems. However, design method requires linear state model of the system. Tuning parameter of SMC has been determined using linear quadratic regulator (LQR) approach. This results in optimum sliding surface for selected performance index. Matched uncertainty is considered to obtain the stability condition in terms of its upper bound. A conventional state observer has been used to estimate the states. The estimated states are then fed to controller for determining control signal. The simulation study and experimentation on real-life level system have been carried out to validate performance and applicability of the proposed controller.

Stability of Nonautonomous Difference Equations with Bounded Parameters

For a nonautonomous difference equation with bounded parameters, stability conditions in terms of estimates of the Cauchy function are obtained.

Network-Aware Design of State-Feedback Controllers for Linear Wireless Networked Control Systems

We study the problem of stabilizing the origin of a plant, modeled as a discrete-time linear system, for which the communication with the controller is ensured by a wireless network. The transmissions over the wireless channel are characterized by the so-called stochastic allowable transmission intervals (SATI), that is a stochastic version of the maximum allowable transmission interval (MATI). Instead of deterministic transmissions, SATI gives stability conditions in terms of the cumulative probability of successful transmission over N steps. We argue that SATI is well-suited for wireless networked control systems to cope both with the stochastic nature of the communications and the design of energy-efficient communication strategies. Our objective is to synthesize a stabilizing state-feedback controller and SATI parameters simultaneously. We model the overall closed-loop system as a Markov jump linear system and we first provide linear conditions for the stability of the wireless networked control systems in a mean-square sense. We then provide linear matrix inequalities conditions for the design of state-feedback controllers to ensure stability of the closed-loop system. These conditions can be used to obtain both the controller and the SATI. A numerical example is presented to illustrate our results.

Stability conditions for time delay systems in terms of the Lyapunov matrix

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proof strategies are outlined in the retarded type case. The state of the art and potential extensions to other classes of delay systems are discussed.

Chaotic congestion games

We analyze a class of congestion games where two agents must send a finite amount of goods from an initial location to a terminal one. To do so the agents must use resources which are costly and costs are load dependent. In this context we assume that the agents have limited computational capability and they use a gradient rule as a decision mechanism. By introducing an appropriate dynamical system, which has the steady state exactly at the unique Nash equilibrium of the static congestion game, we investigate the dynamical behavior of the game. We provide a local stability condition in terms of the agents’ reactivity and the nonlinearity of the cost functions. In particular we show numerically that there is a route to complex dynamics: a cascade of flip-bifurcation leading to periodic cycles and finally to chaos.

Asymptotic Stability of LTV Systems With Applications to Distributed Dynamic Fusion

We investigate the behavior of Linear Time-Varying (LTV) systems with randomly appearing, sub-stochastic system matrices. Motivated by dynamic fusion over mobile networks, we develop conditions on the system matrices that lead to asymptotic stability of the underlying LTV system. By partitioning the sequence of system matrices into slices, we obtain the stability conditions in terms of slice lengths and introduce the notion of unbounded connectivity , i.e., the time-intervals, over which the multi-agent network is connected, do not have to be bounded as long as they do not grow faster than a certain exponential rate. We further apply the above analysis to derive the asymptotic behavior of a dynamic leader-follower algorithm.

Lur\u2019e-Postnikov Lyapunov function approach to global robust Mittag-Leffler stability of fractional-order neural networks

In this paper, the global robust Mittag-Leffler stability analysis is preformed for fractional-order neural networks (FNNs) with parameter uncertainties. A new inequality with respect to the Caputo derivative of integer-order integral function with the variable upper limit is developed. By means of the properties of Brouwer degree and the matrix inequality analysis technique, the proof of the existence and uniqueness of equilibrium point is given. By using integer-order integral with the variable upper limit, Lur’e-Postnikov type Lyapunov functional candidate is constructed to address the global robust Mittag-Leffler stability condition in terms of linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the validity of the theoretical results.

Stability in terms of two measures of solutions to stochastic partial differential delay equations with switching

In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed‐index sequence monotonicity condition and under the average dwell‐time switching are considered.

Regions of exponential stability in coefficient space for linear systems on nonuniform discrete domains

The paper addresses the problem of exponential stability of linear control systems defined on nonuniform discrete domains. The main goal of this research is to find necessary and sufficient stability conditions in terms of the coefficients of the characteristic polynomial, associated with a system, and to study the geometry of stability regions, determined by the derived conditions in the coefficient space. As the first step in this direction, this paper presents such conditions for a special case of second-order systems defined on a discrete time scale with two asymptotic graininesses. The geometry of the corresponding regions in the coefficient space is illustrated for different values of the asymptotic graininesses.