Lyapunov Function For Stability Analysis Research Articles

A New Class of Lyapunov Functions for Stability Analysis of Singular Dynamical Systems. Elements of p-Regularity Theory

A New Class of Lyapunov Functions for Stability Analysis of Singular Dynamical Systems. Elements of p-Regularity Theory

Stability and stabilisation of heterogeneous switched systems with arbitrary switching constraints

Heterogeneous switched systems are a class of special hybrid systems containing different dimensional subsystems. To research the stability and stabilisation of heterogeneous switched systems, a novel approach is presented to design static output feedback controllers for asymptotically stabilising heterogeneous continuous-time and discrete-time switched systems under arbitrary switching rules. This approach, which has less computation complexity, includes a main diagonal square block common Lyapunov function for stability analysis of heterogeneous switched systems and a main diagonal square block common homogeneous polynomial Lyapunov function for stabilisation of heterogeneous switched systems. In the study of stabilisation of heterogeneous switched systems, a two-step iterative method including two convex optimisation functions is established by sums of squares to express existing properties condition and solvable properties condition for determining block controller. At last, a practical example and two numerical examples are presented to show the feasibility of the proposed results.

Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases

In this paper, we develop a method to generate the Lyapunov function for stability analysis for chemical reaction networks. Based on the Chemical Master Equation, we derive the Lyapunov Function partial differential equations (PDEs), whose solution approximates the scaling non-equilibrium potential and serves as the candidate Lyapunov function for the given network. We further prove that for any chemical reaction network the solution (if exists) of the PDEs is dissipative. Moreover, the proposed method of Lyapunov Function PDEs is qualified for analyzing the asymptotic stability of complex balanced networks, all networks with $1$-dimensional stoichiometric subspace and some special networks with more than $2$-dimensional stoichiometric subspace if some moderate conditions are added. Several examples are presented to illustrate the efficiency of the method.

Energy-based Hamiltonian approach in H∞ controller design for n-degree of freedom mechanical systems

This paper studies the energy-based approach for controller design of -degree of freedom mechanical systems. In this approach, the Hamiltonian function, which is the sum of kinetic and potential energies of the system, is considered as the Lyapunov function for stability analysis. The stability analysis is done based on the port-controlled Hamiltonian (PCH) model. In this regard, two theorems are given and proved that the proposed controllers lead to disturbance attenuation for both absolutely known system model and unknown ones with parametric uncertainties. In the case of parametric uncertainties, the energy-based controller has an adaptive approach. Performance of proposed controllers is illustrated through simulations taken on a 2-link robot manipulator system, which validate the theoretical achievements of this paper.

Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems

Based on proximal subdifferentials and subgradients, and instrumented with an extended Caputo differintegral operator, the stability analysis of a general class of fractional-order nonlinear systems is considered by means of non-smooth but convex Lyapunov functions. This facilitates concluding the Mittag–Leffler stability for fractional-order systems whose solutions are not necessarily differentiable in any integer-order sense. As a solution to the problem of robust command of fractional-order systems subject to unknown but Lebesgue-measurable and bounded disturbances, a unit-vector-like integral sliding mode controller is proposed. Numerical simulations are conducted to highlight the reliability of the proposed method in the analysis and design of fractional-order systems closed by non-smooth robust controllers.

Quadratic Lyapunov functions for stability analysis in fractional-order systems with not necessarily differentiable solutions

Solutions of fractional-order differintegral equations are generally not necessarily integer-order differentiable, neither in the strong nor in the weak sense, thus limiting the stability analysis in systems based on the most conventional fractional-order operators. In this paper, a consistent and well-posed definition for fractional-order systems is performed based on the study of alternative fractional-order operators that preserve the most interesting and useful properties of differintegrals, even in the case of not necessarily integer-order (weakly) differentiable functions. In addition, it is shown that these operators comply to a recently verified well-known inequality, which allows us to demonstrate Mittag-Leffler stability in a more general class of fractional-order systems, considering quadratic Lyapunov functions, by demonstrating a generalization of the Lyapunov direct method for a class of fractional-order nonlinear systems. Illustrative examples are given to highlight the feasibility of the proposed method, and a multivariable fractional integral sliding mode control application is presented.

Convex Lyapunov functions for stability analysis of fractional order systems

This study presents an inequality which can be used to analyse the stability of fractional order systems by constructing Lyapunov functions. By using the presented inequality, it is shown that the fractional order system is Mittag-Leffler stable if there is a convex and positive definite function such that its fractional order derivative is negative definite. This result generalises the existing works and gives a useful method to construct the Lyapunov function for the stability analysis of the fractional order systems. Finally, the authors illustrate the advantages of the proposed method by two examples and their numerical simulation.

Optimal control of a class of pseudo Euler‐Lagrange systems

SummaryThis paper presents a solution of the optimal control problem for a class of pseudo Euler‐Lagrange systems and proposes a systematic approach to find a Lyapunov function for stability analysis and controller synthesis for such systems. There are three main contributions of the paper. First, a systematic procedure is proposed and proved to construct a Lyapunov function for pseudo Euler‐Lagrange system directly from the mathematical structure of the differential equations, without the need to determine any kinetic or potential energy of the system first. Second, control methodologies for pseudo Euler‐Lagrange systems are also developed. In particular, an optimal controller is found for the case of second order dynamics yielding the same structure for the closed‐loop Lyapunov function as the one derived from the systematic procedure outlined as the first contribution. Finally, the optimal control methodology is extended to systems with order higher than two for a class of triangular systems. The method proposed here works for any mathematical model in the class of pseudo Euler‐Lagrange systems and is therefore not restricted to models of physical systems. Several examples illustrate the application of the novel approach, including mass‐spring‐damper systems and Van der Pol oscillators. Copyright © 2016 John Wiley & Sons, Ltd.

Parameter-Dependent Lyapunov Functions in the Robust Control of Discrete Linear Repetitive Processes Using Previous Pass-Windowed Information

Abstract The variables in multidimensional systems are functions of more than one indeterminate, and such systems cannot be controlled by standard systems theory. This paper considers a subclass of these systems that operate over a subset of the upper-right quadrant of the two-dimensional (2D) plane in the discrete domain with a specified recursive structure known as repetitive processes. Physical examples of such processes are known and also their representations can be used in the analysis of other classes of systems, such as iterative learning control. This paper gives new results on the use of the parameter-dependent Lyapunov functions for stability analysis and controls law design of a subclass of repetitive processes that arise in application areas. These results aim to eliminate or reduce the effects of model parameter uncertainty.

A Class of Vector Lyapunov Functions for Stability Analysis of Nonlinear Impulsive Differential Systems

A novel and effective approach to stability of the solutions of nonlinear systems with impulsive effect is considered. The investigations are carried out by means of a class of vector Lyapunov functions and differential inequalities for piecewise continuous functions. Simulation examples are given to illustrate the presented results.

Stability Analysis of MEMS Gyroscope Dynamic Systems

In this paper, the existence of a common quadratic Lyapunov function for stability analysis of MEMS Gyroscope dynamic systems has been studied then a new method based on stochastic stability of MEMS Gyroscope system has been proposed.

Dynamics and control of chemical process networks: Integrating physics, communication and computation

This paper provides the theoretical foundation for the modeling, analysis and control of integrated chemical process networks, or, in short, “process networks.” The dynamics of process networks is represented using state-space descriptions derived from classical irreversible thermodynamics and constrained by the second law so that dissipation is always non-negative. The state descriptions (models) derived from this point of view provide exact process representations. A unique, quadratic Lyapunov function for stability analysis and control design is derived directly from the entropy. The resulting process models are complex and simplifications may be needed in practical applications. Time-scale decomposition and singular perturbation theory provide the basis for exploring the network-level dynamic behavior that emerges as a result of tight inventory integration, and developing appropriate reduced-order models and a hierarchy of control systems for managing inventories and inventory flows. Model-based networked control and Lyapunov theory are leveraged to develop an integrated control and communication strategy that manages the information flows between the network components and explicitly accounts for communication constraints.

Sum of squares decomposition: theory and applications in control

The sum of squares (SOS) decomposition technique allows numerical methods such as semidefinite programming to be used for proving the positivity of multivariable polynomial functions. It is well known that it is not an easy task to find Lyapunov functions for stability analysis of nonlinear systems. An algorithmic tool is used in this work for solving this problem. This approach is presented as SOS programming and solutions were obtained with a Matlab toolbox. Simple examples of SOS concepts, stability analysis for nonlinear polynomial and rational systems with uncertainties in parameters are presented to show the use of this tool. Besides these approaches, an alternative stability analysis for switched systems using a polynomial approach is also presented.

STABILIZATION OF DYNAMICAL SYSTEMS BY DIRECT MINIMIZATION OF KINETIC ENERGY

It was established long ago that the stability of continuous time dynamical systems can be studied by using positive non-increasing energy functions, which came to be known as Lyapunov functions. While the process of establishing these functions for linear systems has been easy, the case has been different for nonlinear systems where trial and error methods are still used. This paper revives the application of the system kinetic energy as a Lyapunov function for stability analysis of that particular system. The kinetic energy was the foundation that gave rise to the notion of the generalized energy functions, which became prominent in linear optimal control with names such as cost functions or objective functions. Although generalized energy functions have been very successful with linear systems, no equivalent generalized energy function has ever been developed for nonlinear systems. This can be attributed to the fact that the generalized energy function expressions for linear systems have the same structure as the kinetic energy for all linear systems, while the corresponding expression for nonlinear systems depend on the system itself; therefore, it is difficult to develop one generalized energy function structure for all nonlinear systems. The paper recalls the use of kinetic energy in studying the stability of continuous time dynamical systems, and it develops the optimality equation for stabilization of such systems. The paper shows further that most existing stabilizing controllers for linear systems can be defined in terms of this optimality equation. The paper closes by providing some examples of using kinetic energy in analyzing and stabilizing some known nonlinear systems.

Global Directional Control of a Slender Autonomous Underwater Vehicle

C ONVENTIONAL streamlined autonomous underwater vehicles (AUVs) are underactuated by design. Typically, they include a propulsor and torque actuators, such as control planes. In this Note, it is assumed that the 6 degrees of freedom vehicle is actuated in 4 degrees of freedom: surge, roll, pitch, and yaw. The control objective is to globally asymptotically stabilize longitudinalaxis translation of the vehicle in a desired inertial direction. A provably convergent, global directional controller could enhance the operation of agile vehicles in dynamic environments, such as rivers, tidal basins, or other coastal areas. Energy-based tools are appealing for designing such controllers because they provide natural candidate Lyapunov functions for stability analysis. As an example, a passivity-based technique known as interconnection and damping assignment was used to stabilize the unstable dynamics of a slender, underactuatedAUV in [1]. The paper assumed an idealized (inviscid) dynamic model and did not consider the attitude kinematics. A nonlinear directional controller was developed in [2] using potential energy shaping, although that model also omitted viscous effects. This Note treats the directional stabilization problem for a vehicle model which requires minimal assumptions about the viscous effects. The result therefore allows a broad range of viscous force and moment models. Directional stabilization for AUVs may be considered a natural, intermediate step between waypoint navigation and general, threedimensional path following, as described in [3], for example. Several approaches have been proposed to address three-dimensional path following for underactuated, 6 degrees of freedom AUVs. These approaches include adaptive feedback [4], switching theory [5], and backstepping [3,6]. Here, we consider the more fundamental problem of directional stabilization for a slender, underactuated AUV and we approach the problem from an energy perspective. The stabilization result, which is adapted from [7], is based on themethod of feedback passivation described in [8]. In this approach, a given system is transformed into a feedback interconnection of two passive subsystems. It follows from passive system theory that the interconnection is passive and therefore that the system is stable. Asymptotic stability may be shown through further analysis, with suitably defined feedback dissipation included, if necessary. As is the case in this Note, exploiting intrinsic properties such as passivity often allows one to derive control algorithms which are globally effective and which work with the natural dynamics rather than dominate or supplant them.

Algorithms for qualitative analysis of complex systems

Algorithms are proposed that construct first integrals and Lyapunov functions for stability analysis of stationary solutions of differential equations describing mechanical systems of linked bodies.

Generalized decompositions of dynamic systems and vector Lyapunov functions

The notion of decomposition is generalized to provide more freedom in constructing vector Lyapunov functions for stability analysis of nonlinear dynamic systems. A generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state-space of a given system. An inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system, so that stabliity of the expansion implies stability of the original system. Stability of the expansion can then be established by standard disjoint decompositions and vector Lyapunov functions. The applicability, of the new approach is demonstrated using the Lotka-Voiterra equations.

On the inclusion of transfer conductances in Lyapunov functions for multimachine power systems

It is the usual practice to neglect transfer conductances while forming the Lyapunov functions for stability analysis of multimachine power systems. It is shown in this note that with certain approximations it is possible to construct valid Lyapunov functions for such systems which includes effects of transfer conductances.

The circle criterion and quadratic Lyapunov functions for stability analysis

In this note the general equivalence is pointed out between the stability results obtained by means of the circle criterion and those derived from an optimum quadratic Lyapunov function.

Kinetic Lyapunov Function for Stability Analysis of Nonlinear Control Systems

A kinetic Lyapunov function is a Lyapunov function of the first derivatives of the state variables. Its use leads to a sufficient condition for the asymptotic stability in the large of a general nonlinear system without hysteresis. The foregoing sufficient condition is similar to but more stringent than the local stability condition for linearized systems.