Construction of Lyapunov–Krasovskii functionals for switched nonlinear systems with input delay

This paper studies the stability issue for switched nonlinear systems with input delay and disturbance. It is assumed that for the nominal system an exponential stabilizing controller is predesigned such that the switched system is stable under a certain switching signal, and a piecewise Lyapunov function for the corresponding closed-loop system is known. However, in the presence of input delay and disturbance, the system may be unstable under the same switching signal. For this case, a new Lyapunov–Krasovskii functional is firstly constructed based on the known Lyapunov function. Then, by employing this new functional, a new switching signal satisfying the new average dwell time conditions is constructed to guarantee the input-to-state stability of the system under a certain delay bound. The bound on the average dwell time is closely related to the bound on the input delay. Finally, numerical examples are given to illustrate the effectiveness of the proposed theory.     

Stabilization of retarded systems of neutral type by control Lyapunov–Krasovskii functionals

This paper deals with the stabilization and the practical stabilization of nonlinear systems described by neutral functional differential equations in Hale’s form, affine in the control input. Artstein’s methodology and Sontag’s universal formula are investigated for this class of systems, by means of invariantly differentiable control Lyapunov–Krasovskii functionals.     

Diagonal Lyapunov–Krasovskii functionals for discrete-time positive systems with delay

We consider the existence of diagonal Lyapunov–Krasovskii (L–K) functionals for positive discrete-time systems subject to time-delay. In particular, we show that the existence of a diagonal functional is necessary and sufficient for delay-independent stability of a positive linear time-delay system. We extend this result and provide conditions for the existence of diagonal L–K functionals for classes of nonlinear positive time-delay systems, which are not necessarily order preserving. We also describe sufficient conditions for the existence of common diagonal L–K functionals for switched positive systems subject to time-delay.     

Lyapunov–Krasovskii functionals and application to input delay compensation for linear time-invariant systems

For linear systems with pointwise or distributed delay in the inputs which are stabilized through the reduction approach, we propose a new technique of construction of Lyapunov–Krasovskii functionals. These functionals allow us to establish the ISS property of the closed-loop systems relative to additive disturbances.     

A novel control design for delayed teleoperation based on delay-scheduled Lyapunov–Krasovskii functionals

This paper addresses the controller design problem for bilateral teleoperation over unreliable networks. The stability and tracking performance analyses are presented for a novel force-reflecting emulator control scheme. The performance (stability, synchronisation, transparency) is guaranteed by H_∞ control theory and delay-scheduled Lyapunov–Krasovskii functionals (LKF), which could improve the existing stability criterion. The design is achieved by using linear matrix inequality optimisation. For the simulation, first, numerical examples are given to demonstrate the effectiveness and benefits of the delay-scheduled LKF-based stability results; second, the proposed controller design solution is illustrated by various simulations and compared with other recent approaches under different working conditions, e.g. abrupt tracking motion and wall contact.     

Improved stability of a class of switched neutral systems via Lyapunov–Krasovskii functionals and an average dwell-time scheme

This article is concerned with the stability analysis for a class of switched neutral systems with mixed time-varying delays. The upper bound of derivative of the discrete time-varying delay is not restricted to one. A delay-dependent criterion of exponential stability under switching signals with an average dwell time is developed by employing free weighting matrices. The criterion is given in the form of linear matrix inequality. A state decay estimation for the switched system is explicitly given by considering the individual rate of decay of state for each subsystem. Two simulation examples are given to illustrate the effectiveness of the proposed method.     

Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided.     

Estimating stable delay intervals with a discretized Lyapunov–Krasovskii functional formulation

In general, a system with time delay may have multiple stable delay intervals. Especially, a stable delay interval does not always contain zero. Asymptotically accurate stability conditions such as discretized Lyapunov–Krasovskii functional (DLF) method and sum-of-square (SOS) method are especially effective for such systems. In this article, a DLF-based method is proposed to estimate the maximal stable delay interval accurately without using bisection when one point in this interval is given. The method is formulated as a generalized eigenvalue problem (GEVP) of linear matrix inequalities (LMIs), and an accurate estimate may be reached by iteration either in a finite number of steps or asymptotically. The coupled differential–difference equation formulation is used to illustrate the method. However, the idea can be easily adapted to the traditional differential–difference equation setting.     

The input–output version of the Aizerman–Myshkis problem for retarded systems

Non-linear retarded systems are considered. Conditions for the input-output stability are derived. They are formulated in terms of the zeros of the corresponding quasi-polynomials. In addition, it is proved that retarded systems with positive impulse (Green) functions satisfy the Aizerman-Myshkis conjecture in the input-output version.     

Efficient formulation of the Gibbs–Appell equations for constrained multibody systems

Multibody System Dynamics - In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell...     

A Lyapunov–Razumikhin approach for stability analysis of logistics networks with time-delays

Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. Complexity of the system, time-delays and perturbations in a customer demand may cause unstable behaviour of the network. This leads to the loss of the customers and high inventory costs. Thus the investigation of the network on stability is desired during its design. In this article we consider local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. We are looking for verifiable conditions that guarantee stability of the network under consideration. Lyapunov–Razumikhin functions and the local small gain condition are utilised to obtain such conditions. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate the proposed approach.     

Robustness bound for receding horizon finite memory control: Lyapunov–Krasovskii approach

This article presents a robustness bound (RB) of receding horizon finite memory control (RHFMC) (Kwon, W.H., and Han, S. (2004), ‘Receding Horizon Finite Memory Controls of Output Feedback Controls for State Space Systems’, IEEE Transactions on Automatic Control, 49, 1905–1915) for continuous-time state-space systems with norm-bounded uncertainties. The proposed RB is easily obtained by solving a convex problem in terms of a linear matrix inequality. We show through a numerical example that the RHFMC can guarantee the robust stabilisation for a larger class of uncertain systems than linear quadratic Gaussian controls when some poles of closed-loop systems are close to an imaginary axis in the complex plane.     

Robust sampled-data H∞ control of uncertain singularly perturbed systems using time-dependent Lyapunov functionals

In this paper, the problem of robust sampled-data H_∞ control of linear uncertain singularly perturbed systems is investigated. The parametric uncertainties are assumed to be time-varying and norm-bounded. Two types of controller design are considered: (1) with a fast sampling in the fast state and a slow one in the slow state, and (2) with a fast sampling in both states. For each type, a time-dependent Lyapunov functional associated with the sampling pattern is introduced to analyse the exponential stability and L₂-gain performance of the closed-loop system. Linear matrix inequalities based solutions of the robust sampled-data H_∞ control problem are derived. The new results are proved theoretically to be less conservative than the existing results. An illustrative example is given which substantiates the usefulness of the proposed method.     

Construction of regions of stability for non-linear systems containing time delays†

A method has been developed for constructing regions of stability for non-linear systems containing time delays. This method, based on modified Liapunov stability theorems, is applicable to higher-dimensional systems containing several time delays or time-varying delays. A reactor example is presented to illustrate the method.     

The control of linear time-periodic systems using Floquet–Lyapunov theory

In this paper we use Floquet–Lyapunov theory to derive the Floquet factors of the state-transition matrix of a given linear time-periodic system. We show how the periodicity of one of the factors can be determined a priori using a constant matrix, which we call the Yakubovich matrix, based upon the signs of the eigenvalues of the monodromy matrix. We then describe a method for the numerical computation of the Floquet factors, relying upon a boundary-value problem formulation and the Yakubovich matrix. Further, we show how the invertibility of the controllability Gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed-loop system. The controller can be full-state or observer-based. It also allows the engineer to assign all the invariants of the system; i.e. the full monodromy matrix. Deriving the feedback matrix requires solving a matrix integral equation for the periodic Floquet factor of the new state-transition matrix of the closed-loop system. This is achieved via a spectral method, with further refinement possible through a boundary-value problem formulation. The computational efficiency of the scheme may be further improved by performing the controller synthesis on the transformed system obtained from the Lyapunov reducibility theorem. The effectiveness of the method is illustrated with an application to a quick-return mechanism using a software toolbox developed for MATLAB?.     

Distributed δ-consensus in directed delayed networks of multi-agent systems

In this article, we propose a distributed δ-consensus protocol in directed networks of dynamic agents having communication delays. The δ-consensus protocol is an average consensus protocol where agents exchange the information with their neighbours at some discontinuous moments. We provide convergence analysis for such consensus algorithm under stochastic switching communication graphs, and then present some generic criteria for solving the average consensus problem. We also show that directed delayed networks of dynamic agents can achieve average consensus even when each agent in the networks intermittently exchanges the information with its neighbours only at some discrete moments. Subsequently, a typical numerical example illustrates and visualises the effectiveness and feasibility of the theoretical results.     

When retarded nonlinear time-delay systems admit an input–output representation of neutral type

The paper shows that nonlinear retarded time-delay systems can admit an input–output representation of neutral type. This behaviour represents a strictly nonlinear phenomenon, for it cannot happen in the linear time-delay case where retarded systems always admit an input–output representation of retarded type. A necessary and sufficient condition for a nonlinear system to exhibit this behaviour is given, and a strategy for finding an input–output representation of retarded type is outlined. Some open problems that arise consequently are discussed as well. All the systems considered in this work are time-invariant and have commensurable delays.     

Analytical modeling of interconnection networks in heterogeneous multi-cluster systems

The study of interconnection networks is important because the overall performance of a distributed system is often critically hinged on the effectiveness of its interconnection network. This paper addresses the problem of interconnection networks performance modeling of large-scale distributed systems with emphases on heterogeneous multi-cluster computing systems. We present an analytical model to predict message latency in multi-cluster systems in the presence of node, network and system organization heterogeneity. The model is validated through comprehensive simulation, which demonstrates that the proposed model exhibits a good degree of accuracy for various system organizations and under different working conditions.

On Sontag’s formula for the input-to-state practical stabilization of retarded control-affine systems

In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on Artstein’s theory of control Lyapunov functions and related Sontag’s formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distributed time-delays, of any size, as long as they are constant and, in general, known.