Stability Criterion For Nonlinear Systems Research Articles

The Over-Damped String Stability Condition for a Platooning System

The over-damped string stability criterion is a very strong stability condition that not only addresses the stability in a stricter sense but also adequately captures the safety performance of a platoon. However, the mathematical representation of this criterion is incomplete in the literature. Here, this representation is completely described. Moreover, this article presents the mathematical test method to evaluate this stability condition for linear or linearized systems from the transfer function. The classical sting stability condition does not address the transient undesired convergent dynamics of a platoon, such as over-shooting, under-shooting or damped oscillating dynamics. This paper demonstrates that the over-damped string stability characteristic significantly attenuates these undesired convergent dynamics in the upstream direction. Thus, the advantage of this condition over the classical criterion for linear system is clarified theoretically and by simulation. Later, the numerical method to analyze the over-damped string stability criterion for nonlinear systems is discussed. Additionally, numerical simulations of an over-damped string stable adaptive cruise control (ACC) vehicle model are compared with that of some experimental test results on platoons of commercially implemented ACC equipped vehicles.

Novel fixed-time stability criteria of nonlinear systems and applications in fuzzy competitive neural network and Chua’s oscillator

Novel fixed-time stability criteria of nonlinear systems and applications in fuzzy competitive neural network and Chua’s oscillator

A Novel Approach for Adaptive Partial Sliding Mode Controller Design and Tuning in Non-Minimum Phase Switch-Mode Power Supplies

In this article, a novel systematic approach is proposed for a partial sliding mode controller (SMC) design and tuning in non-minimum phase switch-mode power supplies (SMPS). To achieve a more simplified controller in comparison with the conventional SMCs, the partial SMC (PSMC) is introduced in this article, which just requires a part of the sliding surface for controller formulation. The accuracy of the developed PSMC is proved mathematically within the entire range of operation. Since the control parameters of the PSMC are not selected by trial and error, it can maintain the stability and robustness of the closed-loop system in a broad operational range. In this regard, and to develop a systematic approach for robust control of SMPS, a constant frequency equivalent SMC is designed using the converter nominal parameters. Then, the extracted controller is combined with an adaptive component to ensure asymptotical stability against load and line changes. Considering the Lyapunov stability criteria for nonlinear systems, it is proved that the presented SPMC can be used for output voltage regulation in both discontinuous and continuous operating modes with zero steady state error. To avoid the trial and error method during the controller tuning and parameters selection, the system characteristic equation is extracted using the Jacobian approach. Considering the roots of the characteristic equation and the stable range of the closed-loop system, the controller parameters are tuned. Furthermore, in addition to simulation, the developed approach is evaluated practically using the TMS3220F2810 digital signal processor. It is shown that the dynamic response of the proposed approach is faster than the standard double-loop SMC during load and line changes. Additionally, it is seen that the developed controller is robust against model changes in both continuous and discontinuous operations.

Explicit criteria for exponential stability of time-varying systems with infinite delay

Time-varying differential systems with infinite delay are considered. Explicit criteria for global exponential stability of linear (nonlinear) systems are presented. Furthermore, an explicit robust stability bound for linear systems subject to time-varying perturbations is given. The exponential stability criteria for nonlinear systems are used to investigate exponential stability of equilibria of neural networks. Three examples are given to illustrate obtained results. To the best of our knowledge, the results of this paper are new.

Indefinite derivative Lyapunov–Krasovskii functional method for input to state stability of nonlinear systems with time-delay

In this paper, the problem of the input to state stability (ISS) for nonlinear systems with time-delay is investigated. A continuously differentiable Lyapunov–Krasovskii functional with indefinite derivative is introduced to derive the ISS of the systems, which generalizes the classic Lyapunov–Krasovskii functional with positive definite derivative. As a result, the uniform stability and uniform asymptotic stability criteria of nonlinear systems with time-delay are also established by employing the proposed Lyapunov–Krasovskii functional.

Stability criterion for nonlinear systems

In the example of the Rossler model problem, a method for the practical analysis of the stability of nonlinear systems is considered. It is shown that besides the stability-instability conditions, the proposed criterion makes it possible in the analysis of the eigenvalue spectrum of the evolution matrix to determine the possibility of chaotic behavior of systems upon loss of stability.

Two-sided bounds for the largest Lyapunov exponent and exponential stability criteria for nonlinear systems with arbitrary delays

We consider a system of nonlinear differential equations with a given linear part, a nonlinear term bounded in the norm, and variable concentrated and distributed delays. We find two-sided bounds on the maximal Lyapunov exponent expressed via the norm of the nonlinear term and maxima of the delay functions. For some systems, we find the exact value of this exponent. These results give sufficient (and, in some cases, necessary) conditions for a system's exponential stability which are invariant with respect to the delay. We give examples that illustrate our method.

Discretisation method and stability criteria for non-linear systems under discrete-time time delay control

Time delay control (TDC) for non-linear systems has rapidly drawn attention as a result of its unusually robust performance and yet its extraordinarily compact form. In many real applications, TDC has been implemented digitally and the time delay, λ, was set to the sampling period of the control system, which is a constant during the control process. The existing stability analysis, however, has been made based on the assumption of the continuous-time TDC and infinitesimal time delay (λ→0). The assumption not only fails to reflect the reality that the closed-loop system (CLS) is a sampled-data system, but also leads to a stability criterion in which important parameters, such as λ, are absent. In this paper, therefore, sufficient stability criteria for a non-linear system based on the premise of discrete-time TDC and λ that is equal to the sampling period are presented. To this end, we have first proposed a discretization method to derive the approximate discrete-time model of CLS. Then by using the model and the concepts of consistency and Lyapunov stability, we have derived stability criteria for the exact discrete-time model of CLS. The suggested criteria consist of the sampling period and other parameters of TDC. These criteria have been verified by simulation results.

Nonlinear Control Synthesis by Convex Optimization

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used.

Typical Takagi–Sugeno PI and PD fuzzy controllers: analytical structures and stability analysis

In the first part of this paper, we investigate explicit structure of the typical Takagi–Sugeno (TS) fuzzy PI and PD controllers. By typical, we mean the use of at least three trapezoidal or triangular input fuzzy sets for each input variable, fuzzy rules with linear consequent, Zadeh fuzzy logic AND operator and the centroid defuzzifier. This configuration covers many TS fuzzy controllers in the literature. We mathematically prove these fuzzy controllers to be nonlinear PI (or PD) controllers with proportional-gain and integral-gain (or derivative-gain) changing with output of the controlled system. We reveal these fuzzy controllers to be inherently nonlinear gain scheduling PI (or PD) controllers with different variable gains in different regions of input space. The gains constantly change even in the same region, and switch continuously and smoothly between adjacent regions. The explicit structural expressions of the fuzzy controllers are derived and their characteristics, including the bounds and geometrical shape of the gain variation, are studied. In the second part of the paper, we apply the Small Gain Theorem to the derived structures of the fuzzy controllers and establish a sufficient global BIBO stability criterion for nonlinear systems controlled by the fuzzy controllers. Local stability is also examined.

Liapunov functions and stability criteria for nonlinear systems with multiple critical eigenvalues

Efficient criteria are derived via explicit construction of Liapunov functions for local asymptotic stability inference of nonlinear systems, whose linearizations possessc ≥ 2 critical modes at an equilibrium point. The stability criteria are obtained in the context of two novel notions, relaxed definiteness and relaxed stability. A real symmetricc×c matrixQ isrelaxed negative definite ifw TQw<0 for any 0≠weℜ + c , ℜ+=[0, ∞); a matrixR isrelaxed stable if there is aP>0 such thatPR+R TP is relaxed negative definite. The construction leads to some characterizations of the nonlinear system's local structure,in the sense of Liapunov, and the so-called stability characteristic matrices and tensors. It is shown that a nonlinear system with multiple critical modes is locally asymptotically stable generically if the stability characteristic matrix is relaxed stable and less generically if the stability characteristic tensor is trivial or degenerate in certain way and the perturbed stability characteristic matrix is relaxed stable.

Simple stability criterion for non-linear systems with multiplicative noise

A new sufficient condition for mean square stability of stochastic systems with sector-bounded non-linearity is presented. Verification of modern stability criteria for systems under consideration is a complicated numerical problem. As a rule, in the process of verification of most of these criteria, we repeatedly have to solve non-linear matrix equations. We have used estimates of the solution of this equation and non-traditional matrix calculus. As a result, we obtained a simple, easy-to-check mean-square stability criterion.

Stability criterion for non-linear systems—integral restrictions

Using simple transformations of the block-diagram a criterion is derived for Instability (Zames 1964, 1966 a, b) of conventional feedback systems with the non-linearity f(˙) satisfying an integral condition, namely, where β is a positive constant, It is shown that the criterion is more general than the Popov criterion (Popov 1962) in that a sector condition need not be satisfied and the non-linearity can lie in all the four quadrants. This is because the criterion fully exploits the characteristics of the open-loop transfer-function G(s). A more general case of the criterion with the non-linearity satisfying an integral plus a slope restriction is also presented.

A stability criterion for nonlinear systems

IN ANOTHER PAPER1 Wolf reports on the recurrence relations in the solution of a certain class of nonlinear systems. In Wolf's dissertation2 a mathematical theory for the analysis of a class of nonlinear systems has been developed. This paper reports on a stability criterion based on the aforementioned dissertation and its application to nonlinear as well as linear feedback control systems.