Showing papers on "Lyapunov function published in 1995"

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Abstract: Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Control of a nonholonomic mobile robot: backstepping kinematics into dynamics

Abstract: A dynamical extension that makes possible the integration of a kinematic controller and a torque controller for nonholonomic mobile robots is presented. A combined kinematic/torque control law is developed using backstepping and asymptotic stability is guaranteed by Lyapunov theory. Moreover, this control algorithm can be applied to the three basic nonholonomic navigation problems: tracking a reference trajectory, path following and stabilization about a desired posture. A general structure for controlling a mobile robot results that can accommodate different control techniques ranging from a conventional computed-torque controller, when all dynamics are known, to adaptive controllers.

New conditions for global stability of neural networks with application to linear and quadratic programming problems

Abstract: In this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced for solving linear and quadratic programming problems. In this application, the principal result here obtained is that these networks are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation. >

Tools for Semiglobal Stabilization by Partial State and Output Feedback

Abstract: We develop tools for uniform semiglobal stabilization by partial state and output feedback. We show, by means of examples, that these tools are useful for solving a variety of problems. One application is a general result on semiglobal output feedback stabilizability when global state feedback stabilizability is achievable by a control function that is uniformly completely observable. We provide more general results on semiglobal output feedback stabilization as well. Globally minimum phase input--output linearizable systems are considered as a special case. Throughout our discussion we demonstrate the usefulness of considering local convergence separate from boundedness of solutions. For the former we employ a sufficient small gain condition guaranteeing convergence. For the latter we rely on Lyapunov techniques.

Nonlinear adaptive control of active suspensions

Abstract: In this paper, a previously developed nonlinear "sliding" control law is applied to an electro-hydraulic suspension system. The controller relies on an accurate model of the suspension system. To reduce the error in the model, a standard parameter adaptation scheme, based on Lyapunov analysis, is introduced. A modified adaptation scheme, which enables the identification of parameters whose values change with regions of the state space, is then presented. These parameters are not restricted to being slowly time-varying as in the standard adaptation scheme; however, they are restricted to being constant or slowly time varying within regions of the state space. The adaptation algorithms are coupled with the control algorithm and the resulting system performance is analyzed experimentally. The performance is determined by the ability of the actuator output to track a specified force. The performance of the active system, with and without the adaptation, is analyzed. Simulation and experimental results show that the active system is better than a passive system in terms of improving the ride quality of the vehicle. Furthermore, both of the adaptive schemes improve performance, with the modified scheme giving the greater improvement in performance. >

Closed loop steering of unicycle like vehicles via Lyapunov techniques

Abstract: With a special choice for the system state equations, the use of the simplest quadratic form as candidate Lyapunov function directly leads to the definition of very simple, smooth and effective closed loop control laws for unicycle-like vehicles, suitable to be used for steering, path following, and navigation. The authors provide simulation examples to show the effectiveness and, in a sense, the "natural behavior" of the obtained closed loop motions (when compared with everyday driving experience). >

Global stability for a class of predator-prey systems

Abstract: This paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey systems. The Dulac’s criterion is applied and Liapunov functions are constructed to establish the global stability.

Adaptive nonlinear design with controller-identifier separation and swapping

Abstract: We present a new adaptive nonlinear control design which achieves a complete controller-identifier separation. This modularity is made possible by a strong input-to-state stability property of the new controller with respect to the parameter estimation error and its derivative as inputs. These inputs are independently guaranteed to be bounded by the identifier. The new design is more flexible than the Lyapunov-based design because the identifier can employ any standard update law gradient and least-squares, normalized and unnormalized. A key ingredient in the identifier design and convergence analysis is a nonlinear extension of the well-known linear swapping lemma. >

Automated fault detection and accommodation: a learning systems approach

Abstract: The detection, diagnosis, and accommodation of system failures or degradations are becoming increasingly more important in modern engineering problems. A system failure often causes changes in critical system parameters, or even, changes in the nonlinear dynamics of the system. This paper presents a general framework for constructing automated fault diagnosis and accommodation architectures using on-line approximators and adaptation/learning schemes. In this framework, neural network models constitute an important class of on-line approximators. Changes in the system dynamics are monitored by an on-line approximation model, which is used not only for detecting but also for accommodating failures. A systematic procedure for constructing nonlinear estimation algorithms is developed, and a stable learning scheme is derived using Lyapunov theory. Simulation studies are used to illustrate the results and to gain intuition into the selection of design parameters. >

Stability of queueing networks and scheduling policies

Abstract: We develop a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady-state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. >

Smooth Sliding-Mode Control for Spacecraft Attitude Tracking Maneuvers

Abstract: A sliding-mode control (SMC) algorithm is derived and applied to quaternion-based spacecraft attitude tracking maneuvers. Based on some interesting properties related to the spacecraft model, a class of linear sliding manifolds is selected. Significantly, a Lyapunov function is introduced in the SMC design, which can avoid the inverse of the inertia matrix and thus simplify the controller design. To improve the transient response before reaching the sliding manifold, the smoothing model-reference sliding-mode control (SMRSMC) is further developed, which requires well-estimated initial conditions. Simulation results are included to demonstrate the usefulness of the SMRSMC method.

Lyapunov-Like Techniques for Stochastic Stability

Abstract: The purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a Wiener process. Sufficient conditions for the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium point of the system, are provided by means of stochastic Lyapunov-like techniques. The notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied.

A primal-dual potential reduction method for problems involving matrix inequalities

Abstract: We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration. We describe in detail how the algorithm works for optimization problems withL Lyapunov inequalities, each of sizem. We prove an overallworst-case operation count of O(m 5.5L1.5). Theaverage-case complexity appears to be closer to O(m 4L1.5). This estimate is justified by extensive numerical experimentation, and is consistent with other researchers' experience with the practical performance of interior-point algorithms for linear programming. This result means that the computational cost of extending current control theory based on the solution of Lyapunov or Riccatiequations to a theory that is based on the solution of (multiple, coupled) Lyapunov or Riccatiinequalities is modest.

Stability of conditionally invariant sets and controlleduncertain dynamic systems on time scales

Abstract: A basic feedback control problem is that of obtaining some desired stability property from a system which contains uncertainties due to unknown inputs into the system. Despite such imperfect knowledge in the selected mathematical model, we often seek to devise controllers that will steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3–9], [11] models described by difference and differential equations, respectively. Recently, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle differences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invariant sets which are then applied to discuss controlled systems with uncertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to the problem in question.

Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions

Abstract: From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.

Semi-global stabilizability of linear null controllable systems with input nonlinearities

Abstract: An H/sub /spl infin//-based Lyapunov proof is provided for a result established by Lin and Saberi (1993): if a linear system is asymptotically null controllable with bounded controls then, when subject to input saturation, it is semi-globally stabilizable by linear state feedback. A new result is that if the system is also detectable then it is semi-global stabilizable by completely linear output feedback. Further, an extension which relaxes the requirements on the input characteristic is obtained. >

Dynamics of large constrained nonlinear systems-a taxonomy theory [power system stability]

Abstract: This paper provides an overview of the taxonomy theory which has been proposed as a fundamental platform for solving practical stability related problems in large constrained nonlinear systems such as the electric power system. The theory reveals a two-level intertwined cellular nature of the constrained system dynamics which serves as a unifying structure, a taxonomy, for analyzing nonlinear phenomena in large system models. These broadly divide into the state space aspects (related to dynamic stability issues among others) and the parameter space aspects (connected with bifurcation phenomena among others). In the state-space formulation, the boundary of the region of attraction for the operating point is shown (under certain Morse-Smale like assumptions) to be composed of stable manifolds of certain anchors and portions of the singularity surface. Such boundary characterization provides the foundation for rigorous Lyapunov theoretic transient stability methods. In the parameter space analysis, the feasibility region which is bounded by the feasibility boundary provides a safe opening region for guaranteeing local stability at the equilibrium under slow parametric variations. The feasibility boundary where the operating point undergoes loss of local stability is characterized in the form of three principal bifurcations including a new bifurcation called the singularity induced bifurcation.

Stability and stabilizability of fuzzy-neural-linear control systems

Abstract: This paper discusses stability analysis of fuzzy-neural-linear (FNL) control systems which consist of combinations of fuzzy models, neural network (NN) models, and linear models. The authors consider a relation among the dynamics of NN models, those of fuzzy models and those of linear models. It is pointed out that the dynamics of linear models and NN models can be perfectly represented by Takagi-Sugeno (T-S) fuzzy models whose consequent parts are described by linear equations. In particular, the authors present a procedure for representing the dynamics of NN models via T-S fuzzy models. Next, the authors recall stability conditions for ensuring stability of fuzzy control systems in the sense of Lyapunov. The stability criteria is reduced to the problem of finding a common Lyapunov function for a set of Lyapunov inequalities. The stability conditions are employed to analyze stability of FNL control systems. Finally, stability analysis for four types of FNL control systems is demonstrated.

A tuning procedure for stable PID control of robot manipulators

Abstract: In this paper we propose some simple rules for PID tuning of robot manipulators. The procedure suggested requires the knowledge of the structure of the inertia matrix and the gravitational torque vector of the robot dynamics, but only upper bounds on the dynamics parameters are needed. This tuning procedure is extracted from the stability analysis by using a suitable Lyapunov function together with the LaSalle invariance principle. We show that with this guideline, the overall closed-loop system is asymptotically stable. This procedure is illustrated for a two degrees-of-freedom robot

Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis

Abstract: Many practical applications of robust feedback control involve constant real parameter uncertainty, whereas small gain or norm-bounding techniques guarantee robust stability against complex, frequency-dependent uncertainty, thus entailing undue conservatism. Since conventional Lyapunov bounding techniques guarantee stability with respect to time-varying perturbations, they possess a similar drawback. In this paper we develop a framework for parameter-dependent Lyapunov functions, a less conservative refinement of "fixed" Lyapunov functions. An immediate application of this framework is a reinterpretation of the classical Popov criterion as a parameter-dependent Lyapunov function. This result is then used for robust controller synthesis with full-order and reduced-order controllers. >

Autonomous rendezvous using artificial potential function guidance

Abstract: 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.

Nonsmooth control-Lyapunov functions

Abstract: It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative that has been studied in set-valued analysis and the theory of differential inclusions with various names such as "upper contingent derivative". This result generalizes to the nonsmooth case the theorem of Artstein (1983) relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "nonstrict" version of the results, analogous to the LaSalle invariance principle, is also provided.

Direct neural model reference adaptive control

Abstract: The paper investigates in detail the possible application of neural networks to direct model reference adaptive control. The difficulties involved in training the neural controller embedded within the closed loop are discussed in detail. A training structure is suggested that removes the need for a generalised learning phase. Techniques are discussed for the backpropagation of errors through the plant to the controller. In particular, dynamic plant Jacobian modelling is proposed that uses a parallel neural forward model of the plant. The benefits of neural control are then demonstrated by comparison with Lyapunov adaptive control for a number of example plants. A continuously stirred tank reactor and a nonlinear guidance system are chosen as two realistic nonlinear case studies for the demonstration of the techniques discussed. In both cases nonlinear neural control was found to provide greatly improved performance over conventional approaches.