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This paper considers the problem of inherent robustness analysis for nonlinear discrete–time systems using the concept of a finite–time Lyapunov function. The main contribution is to prove that, for sufficiently continuous dynamics and finite–time Lyapunov functions, inherent global input–to–state stability to general disturbances can be established for nominally stable nonlinear systems. Moreover, under mere continuity of the finite–time Lyapunov function and of the dynamics, inherent input–to–state stability on a compact set is obtained.

Energy scheduling plays a critical role in the safe and economical operation of the Microgrid. Several methods exist that can be leveraged to improve the cost efficiency of Microgrid operation in grid-connected mode. However, existing works overemphasize the economic benefit of Microgrid operation in grid-connected mode such that in case of sudden transition to islanded mode, the operation reliability, an important performance index in practical operation can't be guaranteed. In this paper, a two-stage scheduling scheme is proposed that takes the reliability issue into consideration by leveraging the upstream grid failure probability. Through real-world data sets based simulation evaluation, we evaluate our proposed two-stage scheduling scheme and validate its effectiveness.

We show that for any positive integer d , there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d , or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of a sum of squares Lyapunov function implies the finiteness property of the optimal product.

In this paper for the class of singular dynamical systems we present optimal control results. We develop unified framework for feedback optimal and inverse optimal control involving a nonlinear-nonquadratic performance functional. It is shown that the cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the nonlinear closed-loop singular system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, Hamilton-Jacobi-Bellman equation.

The aim of this work is to present an entropy-like Lyapunov function based dynamic feedback design technique for quasi-polynomial and Lotka-Volterra systems. It is shown, that the dynamic feedback design problem is equivalent to the feasibility of a bilinear matrix inequality. The problem is also formulated as a control Lyapunov function based feedback design when the Lyapunov function parameters are given, the solution of this problem can be obtained by solving a linear matrix inequality. The developed method is illustrated on a simple numerical example.

In this paper we consider the problem of global asymptotic stabilization with prescribed local behavior. We show that this problem can formulated in terms of control Lyapunov functions. Moreover, we show that if the local control law has been synthesized employing a LQ approach, then the associated Lyapunov function can be seen has the value function of an optimal problem with some specific properties. We illustrate these result on two specific classes of systems: backstepping and feedforward systems. Finally, we show how this framework can be employed when considering an orbital transfer problem.

For the class of singular time-delay dynamical systems we present optimal control results. We developed unified framework for feedback optimal and inverse optimal control involving a nonlinear-nonquadratic performance functional. It is shown that the cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the nonlinear closed-loop singular time-delay system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, Hamilton-Jacobi-Bellman equation.

Necessary conditions for the exponential stability of one delay linear systems expressed in terms of the Lyapunov matrix of the system are proved. The effectiveness of the proposed conditions is shown in illustrative examples.

Problem of the p -th mean exponential stabilizability of a class of stochastic linear control hybrid systems with unstable subsystems is considered. Sufficient conditions for the p -th mean exponential stabilizability under a feedback control and a stabilizing switching signal, that is constructed based on the Lyapunov techniques, the knowledge of regions of decreasing of Lyapunov functions for subsystems and the average dwell time approach, are derived. Three cases, including the existence of a single Lyapunov function, a multiple Lyapunov function and a single Lyapunov-like function, are discussed. Obtained results are illustrated by an example.

In this paper, a simple Lyapunov sampling is proposed. Contrary to a periodic fashion which samples the system uniformly in time, an event-based scheme updates the control signal only when the system trajectory sufficiently changes. Furthermore, the present triggering mechanism is based on a Lyapunov function in order to enforce the events only when required from a stability point of view. Nevertheless, whereas the Lyapunov sampling mechanism initially introduced in Velasco et al. (2009) requires to execute a computationally heavy off-line algorithm, a fully on-line version is developed in this paper. The different approaches are tested (in simulation and practice) to show the efficiency of such an event-based control and, eventually, the performance remains ensured even if the constraint on the stability is relaxed.

This presentations deals with designing of Lyapunov functions for the analysis of a finite-time stabilization of the class of nonlinear systems containing the, so-called, second order type sliding-mode controllers. The design of the corresponding Lyapunov functions is based on the solution of a specic Partial Differential Equation (PDE) which can be resolved by the application of “Method of Characteristics” transforming the considered PDE into a system of Ordinary Didderential Equations (ODE). The collection of the first integrals of this ODE-system permits to find an analytical expression for the desired Lyapunov Function, in particular, for “twisting”, “super-twisting” and “nested” (imbedded) controllers. This paper presents the survey of the works fulfilled in CINVESTAV-IPN during three last years.

This paper addresses the problem of stabilization of discrete-time systems including a cone-bounded nonlinearity and a saturating actuator. In the sense of Lyapunov stability, we introduce a new candidate Lyapunov function which takes nonlinearity behavior into account. The local stability criterion is formulated as a set of Bilinear Matrix Inequalities (BMI) conditions. We present an optimization problem in order to guarantee the closed-loop stability aiming the largest basin of attraction, which may be nonconvex, and/or, nonconnected. Furthermore, a simple iterative algorithm is proposed in order to solve our BMI problem. Some numerical examples are presented to highlight the relevance of the new Lyapunov function in regard to the classical quadratic function.

We present an algorithm to calculate control inputs when available processing resources are time-varying. The basic idea is to calculate the control input to decrease a Lyapunov function value as compared to as many time steps in the past as allowed by the system to be controlled and the available processing resources. We analyze the stability of the resulting closed loop system using stochastic Lyapunov functions and indicate, through numerical simulations, that the performance gains obtained can be significant.

This paper focuses on the design of control Lyapunov function for control affine systems to guarantee the stability for the states of interest in a specified region. Without restrictive assumptions found in previous approaches, a min-max optimization problem is formulated to solve for a quadratic Lyapunov function. A derivative-free coordinate search method is employed to optimize a non-differentiable objective function. Approximation of the objective function as a piecewise linear function gradually reduces search space, leading to an effective Lyapunov function. A CSTR example with actuator saturation is illustrated to demonstrate the efficacy of the proposed approach.

Systems of Lyapunov inequalities of arbitrary order are considered. A way to analytically find solutions of dual linear matrix inequality systems is presented. Some particular cases are considered in detail, providing sufficient conditions of original Lyapunov inequality systems’ infeasibility.

A numerical procedure for the construction of Lyapunov matrices is proposed. It is shown that the matrices satisfy an auxiliary two-point boundary value problem for a special delay free system of matrix equations. Some applications of the functionals are also given. In this paper we study the problem of computation of Lyapunov functionals with a prescribed time derivative for the case of delay systems with periodic coefficients. Similar to the case of time invariant systems the functionals are defined by special Lyapunov matrices.

This paper is concerned with the design of Model Predictive Control (MPC) for Linear Parameter Varying (LPV) discrete-time systems. Sufficient linear matrix inequality (LMI) conditions are provided for the existence of a path-dependent Lyapunov function which generalizes previous results based on affine parameter-dependent Lyapunov functions. At each sampling time the control law is obtained from a convex optimization problem under LMI constraints. As illustrated by examples, the proposed approach yields less conservative results than other available methods for MPC.

The paper considers the observer design problem for a class of scalar distributed transport-reaction systems with boundary measurements, mixed Dirichlet-Neumann boundary conditions, linear (diffusive-convective) transport and nonlinear conically bounded source. A direct Lyapunov method is developed for the exponential stability analysis of the observation error. The Lyapunov functional is chosen with a constant kernel for the small values of the so-called Peclet number (i.e. the quotient of diffusion to convection time) and with a spatial-dependent kernel for high Peclet numbers. The derived Lyapunov-based stability conditions correspond to the ones based on the frequency-domain analysis. A numerical example illustrates the efficiency of the method.

The problem of designing feedback controllers for dynamical systems with time-delay is addressed in this paper. Previous work has imposed significant restrictions on the structure of the candidate Control Lyapunov Functions in order to develop appropriate LMI conditions for the design. This paper addresses this issue and provides two new results. The first result is a step towards controller synthesis using the “complete quadratic” Lyapunov functional. Specifically given such a “complete quadratic” functional, defined by polynomials, we give an algorithm for constructing the inverse of the linear operator which defines that functional. Following this, we derive semidefinite programming conditions, expressed as a Sum-of-Squares program, for state feedback synthesis of these systems using a restricted structure of the Lyapunov functional.

A polynomial approximation of the Lyapunov matrix appearing in the complete type Lyapunov Krasovskii functionals associated to a class of retarded time delay systems is proposed. The results are concordant with the available semi-analytic solution. The comparison of the approximation error with that of the piecewise linear approximation previously introduced in the literature shows a significant improvement.

We explore the connections of Margreta Kuijper's parametrization of stabilizing controllers, with some issues arising in inverse optimal control and in Lyapunov stability theory for higher-order linear time-invariant differential systems.

A smooth patchy control Lyapunov function for a nonlinear control system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a certain arrangement property. In contrast to (Goebel et al., IEEE CDC, 2006), on each intersection of the domains, we do not impose a decrease condition of the local control Lyapunov functions when the index increases. We show a construction, based on such a patchy control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise. Copyrighte 2007 IFAC

This paper focuses on the problem of stabilization of homogeneous systems with dilations using implicit Lyapunov functions. In this paper, we define implicit homogeneous Lyapunov functions and implicit homogeneous control Lyapunov functions. Moreover, we proposed a design method of implicit homogeneous control Lyapunov functions and a controller design method based on the functions. Finally, we confirmed the effectiveness of the proposed method by computer simulation.

We consider strongly connected networks of input-to-state stable (ISS) systems. Provided a small gain condition holds it is shown how to construct an ISS Lyapunov function using ISS Lyapunov functions of the subsystems. The construction relies on two steps: The construction of a strictly increasing path in a region defined on the positive orthant in ℝ n by the gain matrix and the combination of the given ISS Lyapunov functions of the subsystems to a ISS Lyapunov function for the composite system. Novelties are the explicit path construction and that all the involved Lyapunov functions are nonsmooth, i.e., they are only required to be locally Lipschitz continuous. The existence of a nonsmooth ISS Lyapunov function is qualitatively equivalent to ISS.

In this paper a procedure is proposed for construction of the Lyapunov matrices for time delay systems. This algorithm provides continuous piece-wise linear approximations. An example illustrates the procedure.

This paper addresses the robust stability of linear continuous-time systems affected by uncertain time-varying parameters which belong to a polytope and have bounded rates of variation. The proposed conditions rely on homogeneous polynomially parameter-dependent Lyapunov functions of arbitrary degree, being written as a set of linear matrix inequalities at the vertices of the polytope of uncertainties, taking into account the bounds on the time-derivatives of the uncertain parameters and the degree of the Lyapunov function. Progressively less conservative results can be obtained when the degree of the Lyapunov function is increased, as shown by numerical examples which also illustrate that the proposed conditions yield, with lower computational effort, better results than those from similar tests in the literature.

In this paper, two procedures are proposed for construction of the Lyapunov matrices for time delay systems with commensurable delays. The first one is semianalytical, while the second one provides continuous piece-wise linear approximations.

Lyapunov functions for various single machine power systems models are presented. The sharpness of the analysis with these functions is illustrated in a benchmark. Their use as control Lyapunov functions in the presence of voltage regulators is discussed. The need for sharper Lyapunov functions is discussed and illustrated by examples.

Given a finite dimensional nonlinear control system, we study the problem of designing dynamic output feedbacks rendering the origin a globally asymptotically stable equilibrium. First we give necessary structure of an Output Control Lyapunov function. Then we exhibit necessary conditions on a given Control Lyapunov function to allow the derivation of an Output Control Lyapunov function. With the help of a minimax framework, we give also a sufficient condition for the existence of an Output Control Lyapunov function. We check that our necessary condition and sufficient condition coincide for linear systems and are equivalent to the standard necessary and sufficient conditions of stabilizability and detectability. Finally we consider specific structures, in particular “Euler-Lagrange systems” and “integral output” systems.

A problem of designing a bounded feedback control for a linear dynamical system is considered. A feedback control law is proposed which is bounded and steers any state in a neighborhood of the origin to the origin in a finite time. The construction is based on the notion of a common Lyapunov function. The existence of a common quadratic Lyapunov function for matrices of specific forms is proven. Some remarkable properties of this function are established.

We present results on changing supply rates for input-output to state stable (IOSS) discrete-time nonlinear systems. Our results can be used to combine two Lyapunov functions, none of which can be used to verify that the system has a certain property, into a new Lyapunov function from which the property of interest can be concluded. We present two applications of our results to discrete-time systems, i.e. a LaSalle criterion for input to state stability (ISS) and input to state stability with positive semidefinite Lyapunov functions.

For systems which are linear up to nonlinear output injection or which have an open loop monotonicity property, there exists observers with the same dimension as the system and such that the error system admits a Lyapunov function depending only on the error. Here we investigate for general systems under which necessary or sufficient conditions such a property holds. This leads us to introduce the notions of Observer Lyapunov Function and State Independent Error Lyapunov Function.

The problem of robust mixed H ∞ / H 2 control is addressed in this paper. The considered control law is static state-feedback. The studied uncertain systems are Linear Time Invariant (LTI) and are subject to parametric uncertainty. They are modelled such that the state-space representation of the system belongs to a polytope of matrices. The main contribution is to revisit the Lyapunov Shaping paradigm deriving sufficient LMI -based conditions for various mixed H 2 / H ∞ synthesis problems via the use of parameter-dependent Lyapunov functions. This allows to drastically reduce the conservatism of existing methods.

This paper deals with the problems of H 2 analysis and control synthesis for linear continuous-time parameter-dependent systems. The admissible values of the parameters and their rates of variation are supposed to belong to a given polytope. Based on a parameter-dependent Lyapunov function, which is quadratic on the parameters, we derive LMI conditions for both robust H 2 performance analysis and state-feedback control. Both constant and parameter-dependent control laws are developed.

This paper deals with the estimation of a robust domain of attraction for nonlinear systems with unstructured uncertainties in the system matrix. For this goal a piecewise constant parameter-dependent Lyapunov function is used. This type of a Lyapunov function is based on dividing the uncertainty bounding set into a finite number of partitions, yielding a common Lyapunov function for each partition.

A parameter dependent Lyapunov function is introduced for stability analysis of linear systems with slowly varying parameters. The Lyapunov function is a quadratic form using a linear fractional transformation (LFT) of the parameters. The parameters are assumed to be bounded and with bounded derivatives. Other types of uncertainties can be included in the problem. The new criterion yields less conservative bounds than previously proposed criteria.

In this paper some properties and applications of the Lyapunov mapping are considered. First, we show the spectrums of the restrictions of the Lyapunov mapping on subspaces of symmetric and skew symmetric matrices. Secondly, the norm of the Lyapunov mapping is investigated. In third part, Lyapunov mapping is used for the problem of common quadratic Lyapunov functions. Some generalizations of results in Ooba et al. 1997 a and b are obtained. Then norm estimation is used for common Lyapunov functions of a set of matrices.

This paper gives a new generalization of Lyapunov theorems. Tt shows how to explore the stability properties of a given system by using a Lyapunov function which is nonnegative semidefinite, rather than positive definite. Several examples and an application to the algebraic Riccati Equation are given to illustrate the new approach.

In this paper, we further develop the generalized Lyapunov equations for discrete-time descriptor systems given by Bender. We associate a stable discrete-time descriptor system with a Lyapunov equation which has unique solution. Furthermore, under the assumptions of reachability and observability, the solutions are guaranteed to be positive definite. All results are valid for causal and noncausal descriptor systems. This provides a unification of Lyapunov equations and theories established for both normal and descriptor systems. Based on the developed Lyapunov equation, a Riccati equation is also obtained for solving the state-feedback stabilization problem.

Lyapunov's stability theory has been extended to the stability analysis of non-smooth systems in which non-smooth Lyapunov functions are often constructed. It is believed that the main difficulty in Lyapunov stability analysis of non-smooth systems is the determination of the generalized derivative on a discontinuity surface, which involves the estimation of the intersection of a number of convex sets. Although in this work, it still requires above determination, the form of the sets is simplified, and the number of the sets involved in the generalized derivative is significantly reduced as compared with previous work by Shevitz and Paden (1994), The current extension does not require much advanced mathematical machinery and therefore, makes the stability analysis of non-smooth systems practically easier. Two examples, including a system with stick-slip friction and a robotic system having interaction with the environment, are used to demonstrate the applicability of the method proposed in this paper.

In this paper, a necessary and sufficient condition for asymptotic stability is derived for continuous-time linear time-invariant descriptor systems using the Lyapunov approach. The result is presented in terms of the solutions of a generalized Lyapunov equation which is valid for both impulsive and non-impulsive descriptor systems. Conditions for asymptotic stability based on R-controllability or R-observability are also provided. Numerical examples are employed to illustrate the results.

The purpose of this paper is to present some stability results using piecewise linear Lyapunov functions, which axe extensions of previous results using polytope Lyapunov functions. To demonstrate the usefulness of the results, we show sonic examples.

In this paper we show how the Hamiltonian formulation of systems may be used in order to derive Lyapunov functions for forced systems with constant input. In a first part the structure of Hamiltonian systems with dissipation is recalled and it is shown that, under certain integrability assumptions on the input vector fields, a candidate Lyapunov function for the forced system may be derived. In a second part, an interpretation of this candidate Lyapunov function is given in terms of the total energy of the forced system, composed of the sum of its internal energy and the energy function associated with the source element. In the third part, such a candidate Lyapunov function is used to derive a stabilizing controller for the boost converter.

In this paper, we use a state-dependent Lyapunov function to reduce the conservatism in stability analysis of Takagi-Sugeno (T-S) fuzzy systems via Linear Matrix Inequalities (LMIs). Our approach uses T-S fuzzy rules to define Lyapunov functions, and therefore substantially enlarges the class of Lyapunov functions which can be used to prove stability of T-S fuzzy systems. This approach greatly reduces the conservatism and generalizes previous results in this area which relied on finding a. common positive definite Lyapunov function. Since the resulting Lyapunov function is state-dependent, this method requires a priori bounds on the variations of those states that are the argument of the membership functions. The effectiveness of this approach is finally presented via a simple numerical example.

A generic parameterization of Lyapunov function candidates is suggested for a quite general class of non-autonomous smooth nonlinear systems. It is shown that conditions for the existence of a Lyapunov function that guarantees uniform exponential stability can be formulated as a set of linear matrix inequalities. Hence, computation of the Lyapunov function involves the solution of a convex optimization problem for which reasonably efficient algorithms exist. The tradeoff between accuracy and computational complexity as well as possible conservativeness of the procedure is given particular attention.

The new method for the computation of Lyapunov quantities for the second-order dynamical system, permitting us to narrow the requirements on a smoothness of system, is obtained.

This paper deals with the stabilization problem of nonlinear systems using vector Lyapunov methods. The stabilization of general nonlinear systems is posed in terms of the existence of some vector Lyapunov function which together with comparison principles allows us to compute the stabilizing controller. The design procedure consists of giving a comparison system satisfying the desired stability requirements and proposing some vector Lyapunov function then, the stabilizing control law is computed from certain differential inequality. For nonlinear systems affine in the control input the computation of a closedform expression for the control law is indeed guaranteed by the existence of some control Lyapunov function.

The generalized Lyapunov equation, in its continuous-time and discretetime versions, plays an essential role in control theory, as in the stabilization and optimal control of linear systems. This article focuses on solving the generalized Lyapunov equation by the Bartels-Stewart method. Sequential and block-oriented implementations using standard software libraries for linear algebra computations BLAS and LAPACK are presented. Furthermore, parallel versions on distributed memory computers using the parallel library ScaLAPACK are described.

The decomposition of nonlinear output feedback control into an observer and a state feedback control This paper presents a method to design an explicit control Lyapunov function for affine and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”. For these systems a positive definite function is known that can only be made non increasing by feedback; a control Lyapunov function is obtained via a deformation of this function. As an example of its applications, this method allows one to construct an explicit control Lyapunov function for the stabilization of the angular velocity of an underactuated rigid body.

Chaotic phenomena can be controlled in nonlinear chaotic dynamical plants by controlling the maximum Lyapunov exponent. In the case of unknown or time varying chaotic plants, the Lyapunov exponents may vary during the plant operation. In this paper, an effective adaptive strategy is proposed for on-line identification and assignment of the maximum Lyapunov exponent. Simulation results are provided to show the effectiveness of the proposed methodology.