Existing Stability Conditions Research Articles

Policy Synthesis for Switched Linear Systems With Markov Decision Process Switching

In this article, we study the synthesis of mode switching protocols for a class of discrete-time switched linear systems in which the mode jumps are governed by Markov decision processes (MDPs). We call such systems MDP-JLS for brevity. Each state of the MDP corresponds to a mode in the switched system. The probabilistic state transitions in the MDP represent the mode transitions. We focus on finding a policy that selects the switching actions at each mode such that the switched system is guaranteed to be stable. Given a policy in the MDP, the considered MDP-JLS reduces to a Markov jump linear system (MJLS). We consider both mean-square stability and stability with probability one. For mean-square stability, we leverage existing stability conditions for MJLSs and propose efficient semidefinite programming formulations to find a stabilizing policy in the MDP. For stability with probability one, we derive new sufficient conditions and compute a stabilizing policy using linear programming. We also extend the policy synthesis results to MDP-JLS with uncertain mode transition probabilities.

Finite-Time Admissibility and Controller Design for T-S Fuzzy Stochastic Singular Systems with Distinct Differential Term Matrices

The finite-time admissibility analysis and controller design issues for extended T-S fuzzy stochastic singular systems (FSSSs) with distinct differential term matrices and Brownian parameter perturbations are discussed. When differential term matrices are allowed to be distinct in fuzzy rules, such fuzzy models can describe a wide class of nonlinear stochastic systems. Using fuzzy Lyapunov function (FLF), a new and relaxed sufficient condition is proposed via strict linear matrix inequalities (LMIs). Different from the existing stability conditions by FLF, the derivative bounds of fuzzy membership functions are not required in this condition. Based on admissibility analysis results, a design method for parallel distribution compensation (PDC) controller of FSSSs is given to guarantee the finite-time admissibility of the closed-loop system. Finally, the feasibility and effectiveness of the proposed methods in this article are illustrated with three examples.

Interval stability and interval stabilization of linear stochastic systems with time‐varying delay

AbstractThis article first investigates the criterion of interval stability for linear stochastic systems with time‐varying delay via equivalent systems and Lyapunov–Krasovskii (L‐K) functionals. Different from existing stability conditions, the criterion of interval stability can be used to make a more accurate judgment of the stability for linear time‐delay systems. In other words, the new criterion can judge not only the stability of the system but also the speed of its convergence. Meanwhile, based on the criterion of the interval stability, the condition of interval stabilization is derived, which cannot only ensure the stability of the linear time‐delay systems, but also adjust the convergence rate of the states to the ideal level. A numerical instance is presented to illustrate the advantages of the resulting interval stabilization criterion.

Delay‐dependent stability of a class of stochastic delay systems driven by G‐Brownian motion

This study discusses delay-dependent stability of a class of stochastic delay systems driven by G-Brownian motion in the sublinear expectation space. With the help of the degenerate Lyapunov functional, the mean square exponential stability and quasi-sure exponential stability criteria for stochastic delay systems driven by G-Brownian motion are established and an explicit upper bound of time delay is derived. In particular, for the delay-free case, the corresponding sufficient conditions are also obtained. Here, the stability conditions are directly related to the coefficients of the stochastic delay systems and are different from the existing stability conditions which are presented in terms of the G-Lyapunov function. Some examples are introduced to illustrate the delay-dependent stability criteria.

Global Asymptotic Stability of 2-D Digital Filters With a Saturation Operator on the State-Space

This brief investigates the global asymptotic stability (GAS) problem of two-dimensional (2- ${D}$ ) digital filters represented by Roesser model subject to saturation nonlinearities. A new sufficient GAS condition for the underlying 2- ${D}$ system is derived. The condition is compared with several existing stability conditions.

Unified stability criteria for slowly time-varying and switched linear systems

Abstract This paper presents a unified approach to formulating stability conditions for slowly time-varying linear systems and switched linear systems. The concept of total variation is generalized to the case of matrix-valued functions. Using this generalized concept, a result extending existing stability conditions for slowly time-varying linear systems is derived. As special cases of this result, two sets of stability conditions are derived for switched linear systems, which match known results in the literature. A numerical example is included to further illustrate the application of the main result.

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.

New Criterion for Stability of Linear Systems with Interval Time-Varying Delay

This paper is concerned with the delay-dependent stability for linear systems. Through constructing a new augmented LKF and using a new integral inequality, the improved delay-dependent stability criteria are derived in terms of linear matrix inequalities, and it is established that the results have less conservativ`e than some existing stability conditions. Finally, numerical examples are given to illustrate the effectiveness of the proposed result.

New mixed‐delay‐dependent robust stability conditions for uncertain linear neutral systems

In this study, mixed-delay-dependent robust stability problem is investigated for uncertain linear neutral systems with mixed delays. The existing stability conditions are obtained by employing the information of neutral delay and discrete delay independently, and are conservative to some extent. Different from most existing methods, this study attempts to introduce the interconnected information between neutral delay and discrete delay. Based on such an idea, the simple stability and robust stability conditions are firstly proposed by integral inequality method, then improved stability and robust stability conditions are obtained by incorporating delay-decomposition idea and augmented Lyapunov-Krasovskii functional. Theory analysis and examples show the benefits of the proposed techniques and conditions.

Assessment of Rockfall Hazard along the Road Cut Slopes of State Highway-72, Maharashtra, India

Rockfall is a major problem in high hill slopes and rocky mountainous regions and construction of highways at these rockfall prone areas often require stable slopes. The causes of rockfall are presence of discontinuities, high angle cut slopes, heavy rainfall, and unplanned slope geometry etc. Slope geometry is one of the most triggering parameters for rockfall, when there are variations in slope angle along the profile of slope. The Present study involves rockfall hazard assessment of road cut slopes for 15 km distance starting from Mahabaleshwar town along State Highway-72 (SH-72). The vertical to subvertical cut slopes are prone to instability due to unfavorable orientation of discontinuities in slope face of weathered and altered basaltic rockmass. The predominant type of instability has been found as wedge type failure involving medium to large size blocks. In order to investigate the existing stability conditions, analyses were carried out at two locations under different slope conditions. The kinematic analysis was performed using stereographic projection method. RockFall 4.0 numerical simulator software was used to calculate the maximum bounce heights, total kinetic energies and translational velocities of the falling rockmass blocks, and a comparative analysis is presented with increasing the mass of blocks and height of the slope. The result of numerical analysis shows that varying slope angle geometry creates more problems as compared to the mass of blocks in the scenario of rockfall.

New augmented Lyapunov functional method for stability of uncertain neutral systems with equivalent delays

This paper is concerned with the delay-dependent stability for neutral systems with equivalent delays and norm-bounded uncertainties. Through constructing a new augmented LKF and proving its positive definiteness, introducing some slack matrices and using integral inequality, the improved delay-dependent stability criteria are derived in terms of linear matrix inequalities, and it is theoretically established that obtained results are less conservative than some existing stability conditions. Finally, numerical examples are given to illustrate the significant improvement on the conservatism of the delay bound over some existing results.

Rockfall hazard and risk assessment: an example from a high promontory at the historical site of Monemvasia, Greece

Abstract. The paper presents the kinematics of rock instability of a high limestone promontory, where the Monemvasia historical site is situated, in Peloponnese in Southern Greece. The instability phenomena poses a significant threat to the town located at the base of the slope. Rockfall episodes occurred in the past due to the relaxation of the high cliff, whereas significant undermining of the castle frontiers has been observed at the slope crest. The predominant types of instability are of planar, wedge and toppling failure of medium to large blocks. In order to investigate the existing stability conditions and decide upon the protection measures, stability and rockfall analyses were carried out for numerous slope sections under different loading conditions and protection measures were suggested. A rock-fall risk rating system is proposed, which is based on morphological and structural criteria of the rock mass and on vulnerability and consequences. The rating system is applied for individual sections along the slope and a risk map was produced, which depicted areas having different degree of risk against rockfall occurrences.

A novel approach on stabilization for linear systems with time-varying input delay

This paper presents new results on delay-dependent stability and stabilization for linear systems with interval time-varying delays. Some less conservative delay-dependent criteria for determining the stability of the time-delay systems are obtained in this paper. Based on the stability conditions, we propose a new state transformation technology to facilitate controller designing efficiently and computationally. The method is also applicable to the existing stability conditions reported by now, while the existing technologies may fail to derive computational control procedures from the stability conditions. Finally, some numerical examples well illustrate the effectiveness of the proposed method.

Stabilization of switched systems with time-varying delay: a state transformation method

In this paper, the problems of exponential stability and control design for switched systems with time-varying delay are investigated. Under a class of switching signals counting with average dwell time, a new less conservative stability condition is proposed for the systems. To facilitate controller design for the switched systems with time-varying input delay, a new state transformation method is developed, with which the non-linear term in matrix inequality is removed. The method is also applicable to the existing stability conditions for controller design, while the existing methods may be not able to obtain computational control procedures from the stability conditions. In this paper, the computational algorithms are established for the stability analysis and control synthesis. Finally, some numerical examples are provided to illustrate the effectiveness of the proposed techniques.

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

New Delay-Dependent Approach on Stability for Systems with Interval Time-Varying Delay

In this paper, the stability of systems with interval time-varying delay is investigated. The time delay varies in an interval. By employing a new and tighter integral inequality and constructing an appropriate type of Lyapunov functional, the delay-dependent stability criteria are derived. Because neither any model transformation nor free weighting matrices are employed in the theoretical derivation, the developed stability criteria significantly improve and simplify the existing stability conditions.

Mean Square Exponential Stability for Uncertain Delayed Stochastic Neural Networks with Markovian Jump Parameters

This paper is concerned with the problem of delay-dependent mean square exponential stability for a class of delayed stochastic Hopfield neural networks with Markovian jump parameters. The delays here are time-varying delays. Based on a new Lyapunov–Krasovskii functional, delay-dependent stability conditions are derived by means of linear matrix inequalities (LMIs). It is shown that the proposed results can contain some existing stability conditions as a special case. Finally, three numerical examples are given to illustrate the effectiveness of the proposed method, and the simulations show that our results are less conservative than the existing ones.

New Approach on Robust Delay-Dependent ${H}_\infty$ Control for Uncertain T--S Fuzzy Systems With Interval Time-Varying Delay

This paper investigates the robust <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> control for Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. By employing a new and tighter integral inequality and constructing an appropriate type of Lyapunov functional, delay-dependent stability criteria are derived for the control problem. Because neither any model transformation nor free weighting matrices are employed in our theoretical derivation, the developed stability criteria significantly improve and simplify the existing stability conditions. Also, the maximum allowable upper delay bound and controller feedback gains can be obtained simultaneously from the developed approach by solving a constrained convex optimization problem. Numerical examples are given to demonstrate the effectiveness of the proposed methods.

On Delay-Independent Diagonal Stability of Max-Min Congestion Control

Network feedback in a congestion-control system is subject to delay, which can significantly affect stability and performance of the entire system. While most existing stability conditions explicitly depend on delay Di of individual flow i, a recent study shows that the combination of a symmetric Jacobian A and condition p(A) < 1 guarantees local stability of the system regardless of Di . However, the requirement of symmetry is very conservative and no further results have been obtained beyond this point. In this technical note, we proceed in this direction and gain a better understanding of conditions under which congestion-control systems can achieve delay-independent stability. Towards this end, we first prove that if Jacobian matrix A satisfies ||A|| < 1 for any monotonic induced matrix norm ||.||, the system is locally stable under arbitrary diagonal delay Di. We then derive a more generic result and prove that delay-independent stability is guaranteed as long as A is Schur diagonally stable , which is also observed to be a necessary condition in simulations. Utilizing these results, we identify several classes of well-known matrices that are stable under diagonal delays if rho(A) < 1 and prove stability of MKC with arbitrary parameters alphai and betai.

New results on robust exponential stability for discrete recurrent neural networks with time-varying delays

This paper is concerned with the problem of robust exponential stability analysis for uncertain discrete recurrent neural networks with time-varying delays. In terms of linear matrix inequality (LMI) approach, some novel stability conditions are proposed via a new Lyapunov function. Neither any model transformation nor free-weighting matrices are employed in our theoretical derivation. The established stability criteria significantly improve and simplify some existing stability conditions. Numerical examples are given to demonstrate the effectiveness of the proposed methods.