Stability Analysis Of Nonlinear Systems Research Articles

Global inverse optimal tracking control of underactuated omni-directional intelligent navigators (ODINs)

This paper presents a design of optimal controllers with respect to a meaningful cost function to force an underactuated omni-directional intelligent navigator (ODIN) under unknown constant environmental loads to track a reference trajectory in two-dimensional space. Motivated by the vehicle’s steering practice, the yaw angle regarded as a virtual control plus the surge thrust force are used to force the position of the vehicle to globally track its reference trajectory. The control design is based on several recent results developed for inverse optimal control and stability analysis of nonlinear systems, a new design of bounded disturbance observers, and backstepping and Lyapunov’s direct methods. Both state- and output-feedback control designs are addressed. Simulations are included to illustrate the effectiveness of the proposed results.

Trajectory Based Approach for the Stability Analysis of Nonlinear Systems with Time Delays

We propose a new technique for stability analysis for nonlinear dynamical systems with delays and possible discontinuities. In contrast with Lyapunov based approaches, our trajectory based approach involves verifying certain inequalities along solutions of auxiliary systems. It applies to a wide range of systems, notably time-varying systems with time-varying delay, ODE coupled with difference equations, and networked control systems with delay. It relies on the input-to-state stability notion, and yields input-to-state stability with respect to uncertainty.

Finite-time and fixed-time stabilization: Implicit Lyapunov function approach

Theorems on Implicit Lyapunov Functions (ILF) for finite-time and fixed-time stability analysis of nonlinear systems are presented. Based on these results, new nonlinear control laws are designed for robust stabilization of a chain of integrators. High order sliding mode (HOSM) algorithms are obtained as particular cases. Some aspects of digital implementations of the presented algorithms are studied, it is shown that they possess a chattering reduction ability. Theoretical results are supported by numerical simulations.

Modified Lyapunov exponent, new measure of dynamics

Lyapunov exponents, defined as exponential divergent or convergent rate of initially infinitely close solution trajectories, have been widely used for diagnosing chaotic systems, as well for stability analysis of nonlinear systems. Although calculated from the evolution of disturbance vectors associated with the flow, Lyapunov exponents are not associated with any specific directions, and such evolutions are driven by the dynamics in all directions in the state space. It is desirable to explore the asymptotic behaviors of the dynamic systems along certain specific directions and the specific dynamics driving such behaviors. In this paper, the Lyapunov exponents are modified. The modified Lyapunov exponents can indicate the exponential divergent or convergent rates in certain directions, which are driven by the dynamics in the same directions. The existence and the invariance to the initial conditions of the proposed modified exponents are proven mathematically. The algorithm for calculating the modified Lyapunov exponents from mathematical models is also developed. A wide range of case studies, from classical nonlinear dynamic systems to engineering systems, are presented to demonstrate the proposed modified Lyapunov exponents, and the indications of the modified exponents are also discussed. The proposed modified Lyapunov exponents can reveal additional insights into the system dynamics to the conventional Lyapunov exponents. Such information can be instrumental for stability control design.

Stability Analysis of Nonlinear Systems with Slope Restricted Nonlinearities

The problem of absolute stability of Lur'e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.

Solving Large-Scale Robust Stability Problems by Exploiting the Parallel Structure of Polya's Theorem

In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured Linear Matrix Inequalities (LMIs). We then use a slight modification of a common interior-point primal-dual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

An Effective Computational Algorithm for a Class of Linear Multiplicative Programming

In this paper, an effective computational algorithm is proposed for a class of linear multiplicative problem (P), which have broad applications in financial optimization, economic plan, engineering designs and stability analysis of nonlinear systems, and so on. By utilizing piecewise linearization technique underestimates the objective function, linear relaxation programming of the original linear multiplicative programming problem (P) is established, and the proposed global optimization algorithm is convergent to the global optimal solution of the original problem (P). And finally the numerical experiments are given to illustrate that the feasibility of proposed algorithm and can be used to globally solve the class of linear multiplicative programming problem (P).

Stability analysis of nonlinear systems using higher order derivatives of Lyapunov function candidates

The Lyapunov method for stability analysis of an equilibrium state of a nonlinear dynamic system requires a Lyapunov function v(t,x) having the following properties: (1) v is a positive definite function, and (2) v̇ is at least a negative semi-definite function. Finding such a function is a challenging task. The first theorem presented in this paper simplifies the second property for a Lyapunov function candidate, i.e. this property is replaced by negative definiteness of some weighted average of the higher order time derivatives of v. This generalizes the well-known Lyapunov theorem. The second theorem uses such weighted average of the higher order time derivatives of a Lyapunov function candidate to obtain a suitable Lyapunov function for nonlinear systems’ stability analysis. Even if we have a suitable Lyapunov function then this theorem can be used to prove a bigger region of attraction. The approach is illustrated by some examples.

A Converse Sum of Squares Lyapunov Result With a Degree Bound

Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

A radial-basis-function network-based method of estimating Lyapunov exponents from a scalar time series for analyzing nonlinear systems stability

Lyapunov exponents indicate the asymptotic behaviors of nonlinear systems, the concept of which is a powerful tool of the stability analysis for nonlinear systems, especially when the dynamic models of the systems are available. For real world systems, however, such models are often unknown, and estimating the exponents reliably from experimental data is notoriously difficult. In this paper, a novel method of estimating Lyapunov exponents from a time series is presented. The method combines the ideas of reconstructing the attractor of the system under study and approximating the embedded attractor through tuning a Radial-Basis-Function (RBF) network, based on which the Jacobian matrices can be easily derived, making the model-based algorithm applicable. Three case studies are presented to demonstrate the efficacy of the proposed method. The Henon map and the Lorenz system feature spectra including not only the positive exponent, but also the negative one, while the standing biped balance system is characterized by four negative exponents. Compared with the existing methods, the numerical accuracy of the Lyapunov exponents derived through the newly proposed method is much higher regardless of their signs even in the presence of measurement noise. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either unknown or difficult to model due to complexities.

Stochastic Barbalat's Lemma and Its Applications

In the deterministic case, a significant improvement on stability analysis of nonlinear systems is caused by introducing Barbalat's lemma into control area after Lyapunov's second method and LaSalle's theorem were established. This note considers the extension of Barbalat's lemma to the stochastic case. To this end, the uniform continuity and the absolute integrability are firstly described in stochastic forms. It is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily Ito diffusions. When it is applied to Ito diffusion processes, many classical results on stochastic stability are covered as special cases.

On stability analysis via Lyapunov exponents calculated based on radial basis function networks

The concept of Lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. For real world systems, such models are often unknown. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. However the time-series-based methods are believed to be reliable only for estimating positive exponents. Furthermore, when nonlinear mapping is applied for deriving the neighbourhood-to-neighbourhood matrices, the loads of mathematical deduction and programming increase significantly, which makes it unfeasible to nonlinear systems with high dimensions. In contrary, the model-based methods are constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantage of the model-based methods. In this article, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the radial basis function network. The proposed method inherits the advantage of the model-based methods, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.

On expansion of estimated stability region: Theory, methodology, and application to power systems

The expansion of the estimated stability region plays an important role in the stability analysis of nonlinear systems. However, current literatures have not provided a complete mathematical description for this problem. This paper reveals that essentially the enlargement or the compression of the estimated stability region results directly from the diffeomorphism map, which is induced by the flow contained in the stability region. By proving that any integration algorithm with an order higher than one can approximately trace the flow of the system, a generalized methodology is proposed to construct various algorithms to realize the enlargement or the compression of the estimated stability region. With this methodology, two new algorithms based on symbolic calculation are suggested to reduce the computational burden. Furthermore, this methodology is applied to construct a scalable numerical algorithm to calculate the critical clearing time (CCT) of the power system for given faults. Tests on the IEEE 10-machine 39-bus system show that the computational results coincide well with the step-by-step simulation with high accuracy.

Stability analysis and robustness design of nonlinear systems: An NN-based approach

In this study a neural-network (NN) based approach is developed which combines H ∞ control performance with Tagagi–Sugeno (T–S) fuzzy control for the purpose of stabilization and stability analysis of nonlinear systems. A Takagi–Sugeno (T–S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy controller for the stabilization of nonlinear systems. The neural-network model is adopted to overcome the modeling error problems found with nonlinear systems. A novel stability condition based on an NN-based controller design is derived to ensure the stability of the nonlinear system. The control problem can now be reformulated as a linear matrix inequality (LMI) problem. A simulation is provided in order to explore the feasibility of the proposed fuzzy controller design method.

Stability Analysis and Stabilization of Nonlinear Systems via Locally Defined Density Functions

This paper considers local stability analysis of nonlinear systems with deriving a positively invariant set based on the Rantzer's stability theory by using density functions. We define a noti...

A robust method on estimation of Lyapunov exponents from a noisy time series

Lyapunov exponents can indicate the asymptotic behaviors of nonlinear systems, and thus can be used for stability analysis. However, it is notoriously difficult to estimate these exponents reliably from experimental data due to the measurement error (noise). In this paper, a novel method for estimating Lyapunov exponents from a time series in the presence of additive noise corruption is presented. The method combines the ideas of averaging the noisy data to form new neighbors and of nonlinear mapping to determine neighborhood mapping matrices. Two case studies of balancing control of a bipedal robot and the Lorenz systems are presented to demonstrate the efficacy of the proposed method. The bipedal robot system has two negative Lyapunov exponents while the Lorenz system has one positive, zero, and negative exponents, respectively. It is shown that, as compared with the existing methods, our proposed one is more robust to the ratio of signal to noise, and is particularly effective in estimating negative Lyapunov exponents. We believe that the work can contribute significantly to the stability analysis of nonlinear systems using a noisy time series.

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.

A SYMPLECTIC ALGORITHM FOR THE STABILITY ANALYSIS OF NONLINEAR PARAMETRIC EXCITED SYSTEMS

A symplectic numerical approach to the stability analysis of nonlinear parametric excited systems with multi-degree-of-freedom is developed and the stability of a nonlinear vibration system depends on the dynamic behavior of its perturbation. The nonlinear parameter-varying differential equations for the perturbation motion are expressed in the form of Hamiltonian equations with time-varying Hamiltonian, and are converted further into the classical Hamiltonian equations with extended time-invariant Hamiltonian by augmenting the state variables. The solution to the augmented Hamiltonian equations has the symplectic structure in terms of the symplectic transformation. Then the difference equations of the symplectic Runge-Kutta algorithm with a sufficient condition are constructed, which is proved to preserve the intrinsic symplectic structure of the original solution. In particular, the symplectic Gauss-Runge-Kutta algorithm with stage 2 and order 4 is proposed and applied to the stability analysis of a nonlinear system. Unstable regions based on the nonlinear periodic-parameter perturbation equation are obtained by using the symplectic Gauss-Runge-Kutta algorithm, analytical solution method and non-symplectic conventional Runge-Kutta algorithm to verify the higher accuracy of the proposed algorithm. Unstable regions based on the nonlinear perturbation are given to illustrate the improvement over those based on the linear perturbation. The developed symplectic approach to the stability analysis can preserve the symplectic structure of the original system and is applicable to nonlinear parametric excited systems with multi-degree-of-freedom.

A new theorem on higher order derivatives of Lyapunov functions

The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive first derivative of the Lyapunov functions along the system’s trajectories. In this paper, a new method is developed to relax this requirement. A new sufficient condition is developed for the stability analysis of nonlinear systems, introducing some inequalities for higher order derivatives of the Lyapunov function. The differential inequalities can be considered as a new controllable canonical form linear time invariant system with negative inputs. The stability analysis of a given nonlinear system is then reduced to check if the characteristic equation for the new linear time invariant system is Hurwitz. Some examples are presented to establish the approach.

Energy absorption capacity; a new concept for stability analysis of nonlinear dynamic systems

AbstractStability is the main concern considered for every system. Generally the well‐known Lyapunov and input‐output stability methods are utilized for the stability analysis of nonlinear systems. These methods face serious difficulties as the size and complexity of the systems increases. In this paper a new approach is presented to overcome this problem by introducing a new concept “Energy Absorption Capacity” (EAC) for every component. The EAC of the system can be derived from its component EACs considering their interaction. It is shown that the stability of every individual component is assured if its EAC has a positive value. The proposed approach is less conservative compared to a Lyapunov‐based approach. This is due to its reliance on EAC as the extreme value of energy function rather than the function itself. Some examples are given to support the proposed approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society