Second Method Of Lyapunov Research Articles

Identification of nonlinear flight dynamics: theory and practice

This article presents an innovative time-domain nonlinear mapping-based identification method. The method reported is applied to identify the unknown parameters of multivariable dynamic systems which are mapped by nonlinear differential equations. A systematic identification method is introduced, and a novel algorithm is developed using nonlinear error maps. An analysis of parameter convergence is provided and the regions of convergence can be found using the second method of Lyapunov. Innovative nonquadratic Lyapunov functions are designed and used. Analytical and numerical studies are performed to illustrate and validate the identification concept. The unsteady flight of a high-alpha aircraft in the longitudinal axis is chosen as a nonlinear case study. The unknown parameters are identified. Simulation results show that the model dynamics match the experimental data. The reported example demonstrates that the time-domain nonlinear mapping-based identification method ensures robustness and reduces major shortcomings in stability, convergence, and computational efficiency compared with other algorithms available.

A findpath problem in the presence of moving obstacles

A solution of the findpath problem in which a moving object is required to avoid moving obstacles and move to the designated target in the plane is provided via the second method of Lyapunov. This paper presents a new control designed by a family of piecewise Lyapunov functions to solve a findpath problem and gives some simulation results of that.

Stability of a general class of difference systems

In this paper, we shall investigate several notions of stability of a general ordinary difference system using the prolific Second Method of Lyapunov.

An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability

The topic of this article is a nonlinear observer based on a slight modification of the extended Kalman filter. The purpose of this modification is twofold: first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities. Using the second method of Lyapunov it is proved that the proposed observer is an exponential observer. To examine the practical usefulness the proposed observer is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines.

Continuous-Time Adaptive Controller Design with Adjustments Working in Sliding Mode

This paper deals with a design of a adaptive controller with adjustments working in sliding mode. The mathematical description of the plant is linear continuous single input model. The structure of the adaptive law is chosen linear. Stability problem of the closed loop equation is solved using second method of Lyapunov with the LaSalle‘s extension. It is shown that the stability problem leads to the ‘Normal equations’ in closed loop, but here in differential form. They are solved as optimisation problem by sliding mode technique. The stochastic case in the presence of Gauss white noise is investigated using again the second method, but with stochastic Lyapunov function. Numerical simulations are also carried out.

Lyapunov stability of two-dimensional digital filters with overflow nonlinearities

In this paper, the second method of Lyapunov is utilized to establish sufficient conditions for the global asymptotic stability of the trivial solution of zero-input two-dimensional (2-D) Fornasini-Marchesini state-space digital filters which are endowed with a general class of overflow nonlinearities. Results for the global asymptotic stability of the null solution of the 2-D Fornasini-Marchesini second model with overflow nonlinearities are established. Several classes of Lyapunov functions are used in establishing the present results, including vector norms and the quadratic form. When the quadratic form Lyapunov functions are considered, the present results involve necessary and sufficient conditions under which positive definite matrices can be used to generate Lyapunov functions for 2-D digital filters with overflow nonlinearities.

On global uniform asymptotic stability of nonlinear time-varying systems in cascade

In this short paper we deal with the stability analysis problem of nonautonomous nonlinear systems in cascade. In particular, we give sufficient conditions to guarantee that (i) a globally uniformly stable (GUS) nonlinear time-varying (NLTV) system remains GUS when it is perturbed by the output of a globally uniformly asymptotically stable (GUAS) NLTV system, under the assumption that the perturbing signal is absolutely integrable; (ii) if in addition the perturbed system is GUAS, it remains GUAS under the cascaded interconnection; (iii) two GUAS systems yield a GUAS cascaded system, under some growth restrictions over the Lyapunov function. Our proofs rely on the second method of Lyapunov, roughly speaking on a “δ-ε stability analysis”.

Modelling and Control of Interaction Forces in Nonlinear-Stiffness Types of Contact

Abstract This paper analyses a force control scheme used for interactions between a manipulator and a nonlinearly stiff environment. Initially, a nonlinear model for the contact stiffness is developed, and the idea of using an accommodation law in controlling force is introduced. Accommodation is the inverse of damping, and corresponds to a force input/velocity output mapping. The stability of the scheme is analysed using the second method of Lyapunov. The analysis is new since it takes into account the variation in stiffness caused by changes in the deformation of the load (or environment). The results are applicable to many types of contact that can be described by nonlinear stiffness models. Such models are more general than the linear ones which have been the base for the results available in the literature.

Lyapunov Stability Analysis of an Orbit Determination Problem

Statistical orbit determination is an important part of tracking satellites and predicting their future location. This is accomplished by processing the raw observation data from ground stations tracking satellites in orbit and estimating the location of these satellites. Kalman filter theory is a popular statistical process but it is not commonly used for satellite orbit determination because it exhibits poor stability characteristics in this application. In particular, an Extended Kalman Filter (EKF) is needed for this application because the orbit determination problem is a nonlinear one. The EKF does not lend itself well to the traditional eigenvalue stability analysis, which is applicable for linear, time-invariant systems. If the stability characteristics of the EKF used for the orbit determination problem were better understood, it could become useful for these applications. A stability analysis concept which has received much attention is Lyapunov’s stability theory. The focus of this paper is to apply the second method of Lyapunov (or the direct method) to evaluate the stability characteristics of an EKF estimating a satellite’s orbit using simulated radar data from ground tracking stations at Mahe Island and Thule. Specifically, a Lyapunov function is used to analyze the stability characteristics of that EKF while selecting the weighting matrices of the filter for the satellite orbit determination problem.

Model Nonlinearities and Stability Analysis of Implicit Differential Systems

This note aims to contribute to the field of implicit and singular systems with new results on general methodology for stability and qualitative analysis of these systems via the second method of Lyapunov. The model nonlinearities are used to enhance such analysis. Algebraic conditions are derived based on algebraic constraints in the model to ensure information on time-evolution for all system variables. General type stability and boundedness conditions are given in terms of Lyapunov's direct method.

Stability of a Spinning Axisymmetric Rocket with Dissipative Internal Mass Motion

Onehypothesisforexplainingtheconinginstabilitiesobservedincertainspinningaxisymmetricsolidfuelrockets attributes the source of the instability to the accumulation of slag near the exit of the rocket casing. Previously this hypothesis was explored using a particle model with elastic restoring forces to represent the slag motion. A recent paper further examines how the results are modie ed when dissipative forces are added to the model. The work is based on the second method of Lyapunov. An alternative derivation of the dissipative results is now provided. The new derivation is based on perturbation analysis of the undamped characteristic values and yields additional insight into the stability results. A numerical example simulating the coning instability is also given. It shows that a coning growth similar to that observed in space is possible assuming certain time histories of spacecraft parameters. The example requires the natural frequency of moving mass to beclose to thenominal spacecraft spin rate to produce the coning instability. In addition to their relevance for the specie c problem of spinning rocket stability, the results presented provide insight on how the maximum axis spin rule for torque free spinning bodies is modie ed in the presence of thrust.

The Properties of an Output Feedback Scheme and Control System Design. Approach from Low-Order Model.

The optimal regulator is a key device in modern linear control theory. This paper deals with a problem of computation of the output feedback gain which minimizes the quadratic performance criterion. Numerous methods of computation of the output feedback have been presented in other reports, but they all require iterative calculation. However, use of such iterative algorithms requires solution of nonlinear matrix equations. Furthermore, the conditions for the existence of the output feedback gain which satisfies an algebraic equation are unclear. In this paper, a low-order model for computation of the feedback gain is proposed. This method is based on the second method of Lyapunov. The computational method for the output feedback gain and comparison of the Lyapunov functions of the original system and the low-order model are presented. This computational procedure does not involve use of any iterative algorithms. We have identified the conditions for the existence of the output feedback gain and the sufficient conditions for attainment of the optimal solution.

A new robust control for a class of uncertain discrete-time systems

A new linear robust state-feedback controller that guarantees uniform boundedness for a class of discrete-time uncertain systems with multi-inputs is presented, based on the second method of Lyapunov. Furthermore, an upper bound of the uncertainties has been derived in order to make the closed-loop system stable.

Practical stabilization of nonlinear/uncertain dynamic systems with bounded controllers

We address the problem of practically stabilizing nonlinear and/or uncertain continuous-time dynamic systems when the norm of the control input is subject to a given fixed bound. We consider a class of systems whose nominal part is linear and whose nonlinear/uncertain part does not satisfy the matching condition. In this treatment no statistical information regarding the uncertainties is needed. Only a norm-bound on each nonlinear/uncertain quantity is assumed. We propose a bounded state-feedback controller whose design incorporates elements from both variable structure control theory and the deterministic control methodology. Using the second method of Lyapunov we obtain sliding domains, regions of uniform ultimate boundedness, and estimates of the domain of attraction for systems employing this controller. The closed-loop stability analysis as well as the design of the controller is facilitated by introducing a special coordinate transformation. The approach is illustrated by a numerical example.

Active distributed vibration control of anisotropic piezoelectric laminated plates

Abstract A design strategy is proposed for the active vibration control of fully anisotropic plates in which the active elements are laminated, spatially distributed, piezoelectric layers. The control methodology results from stability criteria established through the second method of Lyapunov, and is based on the consideration of the total system energy. The results show that for a fully anisotropic plate it is sufficient to ensure asymptotic stability provided that three criteria are met: (1) for each piezoelectric actuator laminate above the composite structure mid-plane there exists a corresponding identically polarized sensor laminate, also located above the mid-plane; (2) a linear control law governing each conjugate sensor/actuator pair is enforced such that the input to a given actuator is always proportional and opposite in sign to the current induced by the corresponding sensor; (3) for each conjugate pair above the mid-plane there exists an identical pair below the mid-plane. The analysis shows that these design prerequisites may be relaxed significantly in the absence of bending-stretching coupling. When the design criteria are satisfied, the measure of active vibration suppression becomes directly dependent on the choice of transducer spatial distribution functions. A weaker sufficient criterion is also identified, which could potentially be utilized to relax the design constraints further for fully anisotropic media. Previously employed design strategies for bending vibration control of beams and isotropic plates are shown to be a subclass of general anisotropic plate control theory, and often destabilizing in the presence of anisotropy.

Guaranteed cost stabilization for a class of uncertain discrete-time systems

A technique to design stabilizing state feedback controllers is presented for a class of linear discrete-time systems subject to possibly time-varying uncertainties that affect the system and input matrices. The controller design procedure is based on the solution of one of three modified algebraic Riccati matrix equations. Stability considerations rely on the second method of Lyapunov. To evaluate the system's performance degradation due to uncertainly, an upper bound (guaranteed cost) is derived for the cost of the controlled uncertain system with respect to a designer-chosen quadratic cost function. An example adapted from the literature is used to illustrate the method.

Autonomous rendezvous using artificial potential function guidance

A novel methodology has been developed for the guidance and control of a maneuvering chase vehicle undergoing terminal rendezvous in the presence of path constraints and multiple obstructions. The method hinges on defining a suitable scalar function which represents an artificial potential field describing the locality of the target vehicle. Using a set of bounded impulses the chase vehicle is guided by the local topology of this potential function. Obstructions and path constraints are introduced by superimposing regions of high potential around these regions. Exact, analytical expressions are then obtained for the required control impulse magnitude, direction and switching times using the second method of Lyapunov. These control impulses ensure that the potential function monotonically decreases so that convergence of the chaser to the target is ensured analytically, without violating the path constraints. Since the components of the potential and control impulses may be represented analytically, the method appears suitable for autonomous, real-time control of complex maneuvers with a minimum of onboard computational power.

A DIRECT NONLINEAR ADAPTIVE CONTROL OF STATE FEEDBACK LINEARIZABLE SYSTEMS

Abstract A direct nonlinear adaptive control of state feedback linearizable single-input single-output systems is proposed in the case when parametric uncertainties are represented linearly in the unknown parameters. The main feature of the proposed nonlinear adaptive control system is that the linearizing coordinate transformation and the state feedback are updated by parametric adaptive law, derived using the second method of Lyapunov. The proposed adaptive control scheme is relatively straightforward and simple in the sense that it does not use the concept of augmented error. This adaptive control scheme is numerically applied to an exothermic chemical reactor system and is compared with the nonadaptive stale feedback linearization which has an integral action. The simulation shows that the proposed adaptive control scheme can be applied effectively to highly nonlinear, uncertain chemical systems.

Effect of battery energy storage system on load frequency control considering governor deadband and generation rate constraint

Since a battery energy storage system (BES) can provide fast active power compensation, it also can be used to improve the performance of load-frequency control. In this paper a new incremental model of a BES is presented and merged into the load-frequency control of a power system. A comprehensive digital computer model of a two-area interconnected power system including governor deadband and generation rate constraint is employed for a realistic response. Computer simulations show that the BES is very effective in damping the oscillations caused by load disturbances. The BES model is suitable for charging mode and discharging mode operations. Optimization of controller gains is obtained by the second method of Lyapunov.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

AUTOMATIC GENERATION CONTROL WITH SUPERCONDUCTING MAGNETIC ENERGY STORAGE IN POWER SYSTEM

ABSTRACT This paper describes the development of a digital computer model for a two area interconnected power system which includes the governor deadband nonlinearity, steam reheat constraints and boiler dynamics. The improvement in automatic generation control (AGO with the addition of a small capacity Superconducting Magnetic Energy Storage (SMES) unit is studied. Time domain simulations are used to study the performance of the power system and the controller. Parameter optimization of the controller are carried out by the second method of Lyapunov, which ensures the stability of the system. Suitable methods for the control of SMES unit are also described.