Krasovskii Method Research Articles

A new delay-dependent exponential stability criterion for Itô stochastic systems with Markovian switching and time-varying delay

On the basis of the Lyapunov–Krasovskii method, the exponential stability in the mean square sense is investigated for Itô stochastic systems with Markovian switching and time-varying delay. The statistic properties of the Markov process and Brownian motion are employed to compute the constructed Lyapunov–Krasovskii functional of a rather general form. This enables us to make sense of the challenging problems in the stochastic framework, and then find a way to extend the techniques developed in the deterministic framework. Therefore, the stability conditions are established with the aid of some slack matrices and the boundary conditions on time-varying delay. Numerical examples are given to demonstrate the reduced conservatism.

Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

AbstractThis paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Absolute stability of Lurie networked control systems

AbstractThis paper is concerned with the absolute stability problem of networked control systems (NCSs) with the controlled plant being Lurie systems (Lurie NCSs), in which the network‐induced delays are assumed to be time‐varying and bounded. First, in consideration of both the time‐varying network‐induced delays and data packet dropouts, the Lurie NCSs can be modeled as a multiple‐delays Lurie system. Then, a delay‐dependent absolute stability condition is established by using the Lyapunov–Krasovskii method. Next, two approaches to controller design are proposed in the terms of simple algebra criteria, which are easily solved via the toolbox in Matlab. Furthermore, the main results can be extended to robust absolute stability of Lurie NCSs with the structured uncertainties, where robust absolute stability conditions and approaches to robust controller design are presented. Finally, two numerical examples are worked out to illustrate the feasibility and the effectiveness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.

Control under quantization, saturation and delay: An LMI approach

This paper studies quantized and delayed state-feedback control of linear systems with given constant bounds on the quantization error and on the time-varying delay. The quantizer is supposed to be saturated. We consider two types of quantizations: quantized control input and quantized state. The controller is designed with the following property: all the states of the closed-loop system starting from a neighborhood of the origin exponentially converge to some bounded region (both, in R n and in some infinite-dimensional state space). Under suitable conditions the attractive region is inside the initial one. We propose decomposition of the quantization into a sum of a saturation and of a uniformly bounded (by the quantization error bound) disturbance. A Linear Matrix Inequalities (LMIs) approach via Lyapunov–Krasovskii method originating in the earlier work [Fridman, E., Dambrine, M., & Yeganefar, N. (2008). On input-to-state stability of systems with time-delay: A matrix inequalities approach. Automatica, 44, 2364–2369] is extended to the case of saturated quantizer and of quantized state and is based on the simplified and improved Lyapunov–Krasovskii technique.

N.N. Krasovskii’s extremal shift method and problems of boundary control

For the boundary-controlled dynamic system obeying a parabolic differential equation with the Neumann boundary condition, the problems of following the reference motion, following the reference control, and guaranteed control (at domination of the controller resource) were solved on the basis of the N.N. Krasovskii method of extremal shift from the theory of positional differential games.

Delay‐dependent H∞ control of linear discrete‐time systems with time‐varying delay via switched system approach

AbstractThis paper deals with the problem of H∞ control of linear discrete‐time systems with time‐varying delay. In order to highlight the relations between all combinations of delayed states, a general Lyapunov–Krasovskii functional is introduced, from which the relation between the switched system transformation and the classical Lyapunov–Krasovskii method is established. The basic idea is to formulate the original time‐delay systems as switched systems with arbitrary switching. Therefore, the stability and the H∞ performance problems can be reduced to the corresponding problems of such switched systems. This fact bridges two distinct research fields. Finally, numerical examples are given to illustrate the effectiveness of the method and its advantage over existing results. Copyright © 2009 John Wiley & Sons, Ltd.

Exponential stability of linear distributed parameter systems with time-varying delays

Exponential stability analysis via the Lyapunov–Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems.

Fuzzy output regulator design of discrete affine systems with multiple time-varying delays

This paper presents a Takagi–Sugeno (T–S) fuzzy model-based regulator for a class of discrete affine nonlinear systems with multiple time-varying delays. First, combining the T–S fuzzy representation and Lyapunov–Krasovskii method, we derive the sufficient conditions for the unforced affine system with multiple time-varying delays, disturbances, and non-vanishing bias at the origin. These stability conditions are not derived in current literature. Next, the fuzzy output regulator is proposed by introducing a memoryless fuzzy observer and a fuzzy integral compensator. Then, asymptotic output regulation and state estimation is achieved by solving linear matrix inequalities (LMIs) in a two-step manner. Moreover, an H ∞ performance criterion is guaranteed for disturbance attenuation. Compared to traditional approaches, this paper does not require exact information of partial states, each time-varying delay, and the desired operational point. The proposed controller yields a multi-task regulation without needing controller redesign once the output command is changed. Finally, numerical simulations are carried out on two intuitive time-delay systems to illustrate the expected performance.

Adaptive TS‐FNN control for a class of uncertain multi‐time‐delay systems: The exponentially stable sliding mode‐based approach

AbstractThis paper presents an adaptive Takagi–Sugeno fuzzy neural network (TS‐FNN) control for a class of multiple time‐delay uncertain nonlinear systems. First, we develop a sliding surface guaranteed to achieve exponential stability while considering mismatched uncertainty and unknown delays. This exponential stability result based on a novel Lyapunov–Krasovskii method is an improvement when compared with traditional schemes where only asymptotic stability is achieved. The stability analysis is transformed into a linear matrix inequalities problem independent of time delays. Then, a sliding mode control‐based TS‐FNN control scheme is proposed to achieve asymptotic stability for the controlled system. Since the TS‐FNN combines TS fuzzy rules and a neural network structure, fewer numbers of fuzzy rules and tuning parameters are used compared with the traditional pure TS fuzzy approach. Moreover, all the fuzzy membership functions are tuned on‐line even in the presence of input uncertainty. Finally, simulation results show the control performance of the proposed scheme. Copyright © 2008 John Wiley & Sons, Ltd.

Mixed [formula omitted] output-feedback control of second-order neutral systems with time-varying state and input delays

A mixed H 2 / H ∞ output-feedback control design methodology is presented in this paper for second-order neutral linear systems with time-varying state and input delays. Delay-dependent sufficient conditions for the design of a desired control are given in terms of linear matrix inequalities (LMIs). A controller, which guarantees asymptotic stability and a mixed H 2 / H ∞ performance for the closed-loop system of the second-order neutral linear system, is then developed directly instead of coupling the model to a first-order neutral system. A Lyapunov–Krasovskii method underlies the LMI-based mixed H 2 / H ∞ output-feedback control design using some free weighting matrices. The simulation results illustrate the effectiveness of the proposed methodology.

Minimum-energy control in the integro-differential linear systems

The problem of minimum-energy control in the integro-differential linear systems with Fredholm integral was solved using the N.N. Krasovskii method of the problem of moments. An explicit solution of the boundary problem for the integro-differential equation was established using the K.A. Kasymov method.

On input-to-state stability of systems with time-delay: A matrix inequalities approach

Nonlinear matrix inequalities (NLMIs) approach, which is known to be efficient for stability and L 2 -gain analysis, is extended to input-to-state stability (ISS). We first obtain sufficient conditions for ISS of systems with time-varying delays via Lyapunov–Krasovskii method. NLMIs are derived then for a class of systems with delayed state-feedback by using the S -procedure. If NLMIs are feasible for all x , then the results are global. When NLMIs are feasible in a compact set containing the origin, bounds on the initial state and on the disturbance are given, which lead to bounded solutions. The numerical examples of sampled-data quantized stabilization illustrate the efficiency of the method.

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.

New criteria on global robust stability of Cohen–Grossberg neural networks with time-varying delays

This paper considers the problem of robust stability analysis of Cohen–Grossberg neural networks with time-varying delays and norm-bounded parameter uncertainties. The activation functions are assumed to be bounded and globally Lipschitz continuous. Both the monotonic increasing and non-monotonic increasing activations are considered. In terms of linear matrix inequalities (LMIs), sufficient conditions are obtained by using the Lyapunov–Krasovskii method, which guarantee the existence, uniqueness and global robust asymptotic stability of the equilibrium point of the delayed Cohen–Grossberg neural network. It is theoretically established that the derived LMI conditions are less conservative than certain existing ones in the literature. Finally, numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed LMI conditions.

A methodology to design stable nonlinear fuzzy control systems

The stability of nonlinear systems has to be investigated without making linear approaches. In order to do this, there are several techniques based on Lyapunov's second method. For example, Krasovskii's method allows to prove the sufficient condition for the asymptotic stability of nonlinear systems. This method requires the calculation of the Jacobian matrix. In this paper, an equivalent mathematical closed loop model of a multivariable nonlinear control system based on fuzzy logic theory is developed. Later, this model is used to compute the Jacobian matrix of a closed loop fuzzy system. Next, an algorithm to solve the Jacobian matrix is proposed. The algorithm uses a methodology based on the extension of the state vector. The developed algorithm is completely general: it is independent of the type of membership function that is chosen for building the fuzzy plant and controller models, and it allows the compound of different membership functions in a same model. We have developed a MATLAB's 1 1 MATLAB is a registered trademark of The Math Works, Inc. function that implements the improved algorithm, together with a series of additional applications for its use. The designed software provides complementary functions to facilitate the reading and writing of fuzzy systems, as well as an interface that makes possible the use of all the developed functions from the MATLAB's environment, which allows to complement and to extend the possibilities of the MATLAB's Fuzzy Logic Toolbox. An example with a fuzzy controller for a nonlinear system to illustrate the design procedure is presented. The work developed in this paper can be useful for the analysis and synthesis of fuzzy control systems.

Robust adaptive control for uncertain time‐delay systems

AbstractThe problem of robust stabilization for uncertain dynamic time‐delay systems is considered. Firstly a class of time‐delay systems with uncertainties bounded by high‐order polynomials and unknown coefficients are considered. The corresponding controller is designed by employing adaptive method. It is shown that the controller designed can render the closed‐loop system uniformly ultimately bounded stable based on Lyapunov–Krasovskii method and Lyapunov stability theory. Then the proposed adaptive idea is applied to stabilizing a class of large‐scale time‐delay systems with strong interconnections. A decentralized feedback adaptive controller is designed which guarantees the closed‐loop large‐scale systems uniformly ultimately bounded stable. Finally, numerical examples are given to show the potential of the proposed techniques. Copyright © 2005 John Wiley & Sons, Ltd.

Variable structure adaptive fuzzy control for a class of nonlinear time-delay systems

In this paper, the problem of variable structure adaptive fuzzy control for a class of nonlinear time-delay systems is investigated. Sliding surface dependent of the time delay is constructed based on the Lyapunov–Krasovskii method. Two adaptive fuzzy logic systems are proposed to approximate two unknown continuous nonlinear functions containing the current state and the delayed state, respectively. The corresponding reaching law is designed, which can drive the state trajectory of the closed system onto the sliding surface within limited time. Simulation investigation verifies the effect of the main results of the paper.

Robust stabilization of a class of nonlinear time-delay systems

The problem of robust stabilization for a class of uncertain dynamic systems with multiple delayed state perturbations is considered. It is assumed that perturbations of the time-delayed sections are not bounded by first-order linear functions, but bounded by high-order functions with unknown gains. And the time delay considered is time varying. Two classes of controllers are proposed. When the time derivative of each time-varying time delay is less than 1, a class of adaptive state feedback controllers are proposed based on Lyapunov–Krasovskii method, which can render the closed-loop systems uniformly ultimately bounded stable. Novel nonlinear feedback controllers are developed by employing Razumikhin lemma, and the controller also can render the closed-loop systems stable in the sense of uniform ultimate boundedness. Finally, several examples are given to show the potential of the proposed techniques.

P-1-119 - Surgical treatment for cervical internal carotid artery stenosis with ischemic heart disease

This paper is concerned with the problem of networked H∞ filtering under a weighted try-once-discard (TOD) protocol. Firstly, a weighted TOD protocol is investigated to orchestrate the sampled-data transmission from sensors to filter via a bandwidth-limited network. Taking into account the network-induced delay, the studied system is transformed into an impulsive system with an interval time-varying delay. Secondly, by using the Lyapunov–Krasovskii method, sufficient conditions for ensuring the exponential stability with a prescribed H∞ performance and the existence of suitable H∞ filter are obtained in the form of linear matrix inequalities (LMIs). Then, for minimizing H∞ performance, an algorithm is presented by using the binary search method. Finally, an example of quarter-car model with an active suspension is used to verify the validity of proposed method.

Asymptotic stability of grazing systems with unbounded delay

A system of nonlinear integro-differential equations with unbounded delay for grazing a grassland by a cattle population is considered. Sufficient conditions for asymptotic stability of a positive equilibrium are investigated under fairly general types of growth conditions on the nonlinear functions. The main tools in our analysis are the Liapunov-Razumikhin technique, construction of Liapunov functionals, and Krasovskii's method. Several examples are presented and a few remarks are made to illustrate the generality of the hypotheses.