Stability Of Sampled-data Systems Research Articles

Stability of Sampled-Data Systems With Packet Losses: A Nonuniform Sampling Interval Approach.

In this article, inspired by the Halanay inequality, we study stability of sampled-data systems with packet losses by proposing a nonuniform sampling interval approach. First, a sampled-data controller with an exponential gain is put forward to reduce conservatism. We obtain the sufficient condition for linear sampled-data systems to be exponentially stable by extending the famous Halanay inequality to sampled-data systems. The obtained sufficient conditions indicate that the maximal-allowable bound of sampling intervals is determined by the constant terms in the Halanay inequality, and the decay rate is presented in the form of a Lambert function. Compared with some existing results on the stability of sampled-data systems by using the Gronwall-Bellman Lemma, the conservatism induced by the exponential term via the Gronwall-Bellman Lemma can be reduced to some extent. Considering the phenomenon of packet losses, a new lemma is further proposed to generalize the proposed Halanay-like inequality. The results derived by the new lemma permit that there exist some sampling intervals with the upper bound violating the desired condition of the Halanay-like inequality. This permits us to establish exponential stability in significant cases that do not satisfy the Halanay-like inequality needed in the previous results. Finally, the sampled-data local exponential stability is investigated for nonlinear systems with strong nonlinearity.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

Further Results on Stability of Sampled-Data Systems Considering Time-Delay and Its Application to Electric Power Markets

Further Results on Stability of Sampled-Data Systems Considering Time-Delay and Its Application to Electric Power Markets

Stabilization of Sampled-Data Systems With Noisy Sampling Intervals and Packet Dropouts via a Discrete-Time Approach

In real-world applications, sampled-data systems are often subject to undesirable physical constraints, which results in noisy sampling intervals that fluctuate around an ideal sampling period based on certain probability distribution. In the presence of noisy sampling intervals, this article is concerned with the stabilization problem for a class of sampled-data systems with successive packet dropouts. First, the relationship between two adjacent update times of zero-order hold is established. Then, by discrete-time approach, an equivalent discrete-time stochastic system is introduced. By the law of total expectation, together with confluent Vandermonde matrix, convolution formula, and Kronecker product operation, the mathematical expectation of a product of three matrices, including the system matrix and its transpose, is calculated. Based on this, a stabilization controller is designed such that the closed-loop system is stochastically stable. Finally, an illustrative example is provided to verify the effectiveness of the proposed design approach.

A sampling-period-partitioning approach for stability analysis of sampled-data systems with constant delay

This paper is concerned with the stability of sampled-data systems with constant delay. Firstly, by dividing the interval of sampling periods in two subintervals, two separate looped functionals are employed in each of these subintervals. Then, a new Lyapunov functional that combines classical Lyapunov functionals and looped-functionals is constructed. Furthermore, several zero equalities which consider the intrinsic relationships of state vectors in the system are introduced into the derivative of the constructed functional, and some stability criteria with less conservatism are obtained in forms of linear matrix inequalities (LMIs). Finally, two numerical examples are carried out as to verify the effectiveness and advantages of our method.

Input-to-State Stability and Stabilization of Sampled-Data Systems Under Aperiodic Sampling and Random Sampling

In this article, input-to-state stability (ISS) and stabilization are examined for sampled-data systems under deterministic aperiodic sampling and random sampling, respectively. Using the direct design method, the sampled-data systems are transformed into switched systems with switched time-varying delays. First, the ISS definition and criterion appropriate for these systems are provided. Based on this, the ISS criterion on sampled-data systems under deterministic aperiodic sampling is given. Second, after the stochastic ISS (SISS) definition and criterion are provided for switched nonlinear systems with randomly switching delays, the SISS criterion for the sampled-data systems under random sampling is provided. All of the ISS and SISS definitions are given in the form of $\mathcal {KL}$ function that is quite elegant and easy to work with. Then, sufficient conditions for input-to-state stabilization are obtained for sampled-data linear systems under deterministic aperiodic sampling and random sampling, respectively, via the Lyapunov-Krasovskii method. Finally, based on the criteria, a piecewise controller is designed by the matrix inequality approach for a sampled-data linear time invariant system, and simulation results are provided to illustrate our design method. The main conclusion of this article is that sampling intervals will affect the controller design of the systems, and the ISS properties are maintained using a piecewise controller.

Stability analysis of sampled-data control system and its application to electric power market

The stability of sampled-data systems is investigated using a new type of Lyapunov functional. The interval from the sampling point t k to t k+1 is assumed to be a sampling interval. By fully utilizing the characteristic information on the whole sampling interval, a new two-sided closed-loop Lyapunov functional is proposed, which utilizes the information on both the intervals from the sampling point t to t k and from t to t k+1 . Based on the two-sided closed-loop Lyapunov functional and modified free-matrix-based inequality, a less conservative stability criterion is derived for a sampled-data control system, and three numerical examples are provided to verify the effectiveness and reduced conservativeness of the proposed method. Furthermore, the proposed method is applied to solve the stability problem of electric power markets, and the practical significance of reducing the conservativeness is discussed.

Improved results on H∞ stability analysis of sampled-data systems via looped-functionals and zero equalities

This paper revisits the H∞ stability analysis for sampled-data system by employing Lyapunov functional and loop functionals. Through existing methods such as the looped-functional approach and zero equality in sampled-data systems, an improved result is derived in terms of linear matrix inequalities. Based on the proposed result, by utilizing the relation of state vectors belonging to augmented vectors, less conservative criteria are introduced. The main objective of this study is to increase a maximum sampling value in both periodic and aperiodic sampling cases according to minimum sampling value. Using the proposed results, H∞ stability for sampled-data systems with disturbance is studied. Lastly, some numerical examples are utilized to confirm the superiority and effectiveness of the results by comparing them with the existing ones.

Novel stability criteria for sampled-data systems with variable sampling periods

This paper is concerned with a novel Lyapunovlike functional approach to the stability of sampled-data systems with variable sampling periods. The Lyapunov-like functional has four striking characters compared to usual ones. First, it is time-dependent. Second, it may be discontinuous. Third, not every term of it is required to be positive definite. Fourth, the Lyapunov functional includes not only the state and the sampled state but also the integral of the state. By using a recently reported inequality to estimate the derivative of this Lyapunov functional, a sampled-interval-dependent stability criterion with reduced conservatism is obtained. The stability criterion is further extended to sampled-data systems with polytopic uncertainties. Finally, three examples are given to illustrate the reduced conservatism of the stability criteria.

Stability of Sampled-data Systems with Uncertain Time-varying Delays and Its Application to Consensus Control of Multi-agent Systems

In this paper, we propose a new stability condition for sampled-data systems with uncertain time-varying delays less than a sampling interval. The derivation is based on the robustness of the corresponding discrete-time systems subject to perturbation caused by an uncertain time-varying delay. Then, we apply the proposed stability condition to a consensus control problem of two-wheeled robots. Numerical simulations and experiments show the validity and usefulness of the derived condition.

Recent developments on the stability of systems with aperiodic sampling: An overview

This article presents basic concepts and recent research directions about the stability of sampled-data systems with aperiodic sampling. We focus mainly on the stability problem for systems with arbitrary time-varying sampling intervals which has been addressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as time-delay systems, hybrid systems, Input/Output interconnections, discrete-time systems with time-varying parameters, etc. The goal of the article is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive, which would be neither possible nor useful, we try to bring together results from diverse communities and present them in a unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems (when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunov and frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stability conditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems that remain open and the possible directions of improvement. At last, some emerging research directions, such as the design of stabilizing sampling sequences, are briefly discussed.

Robust Exponential Stabilization for Sampled-Data Systems with Variable Sampling and Packet Dropouts

This paper investigates the problem of robust exponential stabilization for sampled-data systems with variable sampling and packet dropouts. It is assumed that the system parameter uncertainties ar...

Stabilization for sampled-data systems under noisy sampling interval

In engineering practice, the sampling interval for a sampled-data system often fluctuates around a nominal/ideal value based on certain probability distributions that can be specified a priori through statistical tests. In this paper, a fundamental stabilization problem is investigated for a class of sampled-data systems under noisy sampling interval. The stochastic sampled-data control system under consideration is first converted into a discrete-time system whose system matrix is represented as an equivalent yet tractable form via the matrix exponential computation. Then, by introducing a Vandermonde matrix, the mathematical expectation of the quadratic form of the system matrix is computed. By recurring to the Kronecker product operation, the sampled-data stabilization controller is designed such that the closed-loop system is stochastically stable in the presence of noisy sampling interval. Subsequently, a special case is considered where the sampling interval obeys the continuous uniform distribution and the corresponding stabilization controller is designed. Finally, a numerical simulation example is provided to demonstrate the effectiveness of the proposed design approach.

Sampling-interval-dependent stability for sampled-data systems with state quantization

This paper is concerned with the stability of sampled-data systems with state quantization. A new piecewise differentiable Lyapunov functional is first constructed by fully utilizing information about sampling instants. This functional has two features: one is that it is of the second order in time t and of every term being dependent on time t explicitly and the other is that it is discontinuous and is only required to be definite positive at sampling instants. Then, on the basis of this piecewise differentiable Lyapunov functional, a sampling-interval-dependent exponential stability criterion is derived by applying the technique of a convex quadratic function with respect to the time t to check the negative definiteness for the derivative of the piecewise differentiable Lyapunov functional. In the case of no quantization, a new sampling-interval-dependent stability criterion is also obtained. It is shown that the new stability criterion is less conservative than some existing one in the literature. Finally, two examples are given to illustrate the effectiveness of the stability criterion. Copyright © 2013 John Wiley & Sons, Ltd.

Taking into account period variations and actuator saturation in sampled-data systems

This paper deals with the problem of stability and stabilization of sampled-data systems under asynchronous samplings and actuator saturation. The method is based, on the one hand, on the use of a novel class of Lyapunov functionals whose derivative is negative along the trajectories of the continuous-time model of the sampled data system. It is shown that this fact guarantees that a quadratic Lyapunov function is strictly decreasing for the discrete-time asynchronous system. On the other hand, the control saturation is taken into account from the use of a generalized sector condition. These ingredients lead to the formulation of improved LMI conditions that can be cast in optimization problems aiming at enlarging estimates of the region of attraction of the closed-loop system or maximizing the bounds on the sampling period jitter for which stability and stabilization are ensured.

An Improved Stabilization Method for Sampled-Data Control Systems With Control Packet Loss

This technical note presents a new method for stability analysis and stabilization of sampled-data systems with control packet loss. It is assumed that if the control packet from the controller to the actuator is lost, then the actuator input to the plant is set to zero. The new method is based on a novel construction of piecewise differentiable Lyapunov functionals by using an impulsive system representation of sampled-data systems. A significant feature of the new Lyapunov functionals is that they are continuous at impulse times but not necessarily positive definite inside the impulse intervals. Applying the new Lyapunov functionals to sampled-data systems with control packet loss, improved criteria for stability and stabilization are derived. The new criteria are proved theoretically to be less conservative than the existing results. Illustrative examples are given which substantiate the usefulness of the proposed method.

Stability Analysis for Sampled-Data Systems with Input Delay

This article proposes a novel approach to assess the stability of linear systems with delayed and sampled-data inputs. It proposes an extension of existing results on the stability of sampled-data systems to the case where a delay is introduced in the control loop. The method is based on a continuous-time modelling of the systems together with the discrete-time Lyapunov theorem, which provides easy tractable sufficient conditions for asymptotic stability. Those conditions cope the problem of stability under asynchronous samplings and time-varying delays. The period and delay-dependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency of the stability criteria.

Exponential stability and stabilization of sampled-data systems with time-varying period

This article proposes a novel approach to assess the exponential stability of linear systems with sampled-data inputs. The paper considers both uncertainties in the model parameters and in the sampling period. Inspired by the input-delay approach and the stability of impulsive systems, the proposed method provides easy tractable stability conditions. Sufficient stability and stabilization conditions are provided to deal with both cases of constant and time-varying sampling periods. The period-dependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency and the limitation of such stability criteria.

Analysis of Stability of Sampled-Data Systems with a Nonlinear Continuous Part by Means of Lyapunov Functions

Analysis of Stability of Sampled-Data Systems with a Nonlinear Continuous Part by Means of Lyapunov Functions

Computation-oriented expression of a non-conservative condition for robust stability of sampled-data systems

A new expression of a non-conservative robust stability condition is derived for sampled-data systems taking into consideration its convenience in numerical computation. Our expression is described using a global maximum of a certain differentiable function, while the conventional expression includes a global maximum of a structured singular value, which is expensive to compute. When the sampling period is long, the advantages of our expression become clear. A possible problem is that the maximized function may be multimodal. However, the conventional expression has the same problem and even a local maximum is shown to give useful information. Our expression is applicable to a fairly large class of systems including the multi-input multi-output additive and multiplicative perturbation models. Using an example, we show how our expression is computed and make a comparison with the conventional expression.