Lipschitz Nonlinear Term Research Articles

Interval estimation of sensor fault in rotary steerable drilling tools based on set-membership approach

The rotary steerable drilling tools (RSDTs) are invented for high-precision wellpath control. Fault diagnosis is essential for the RSDT as it can improve the reliability of the drilling processes. To estimate the sensor fault of the RSDT, this paper investigates the interval estimation problem for Lipschitz nonlinear systems with sensor fault via the set-membership approach. Firstly, the RSDT is modeled by a discrete-time Lipschitz nonlinear system with unknown inputs, noises, and parameter uncertainties. Next, based on the ellipsoid bundles, a set-membership fault estimator is presented, where the unknown inputs are decoupled. Additionally, a sufficient condition, which is derived based on the P-radius of the ellipsoid bundles, is put forward to design the parameters of the estimator with Lipschitz nonlinear terms. Then, the interval estimations of the states and fault are obtained via ellipsoid bundles. Finally, the efficiency of the proposed approach is evaluated through numerical simulations and experiments.

Weak mean attractors of stochastic p-Laplacian delay lattice systems driven by nonlinear noise

This paper deals with the existence and uniqueness of weak pullback mean random attractors for a class non-autonomous stochastic p-Laplacian delay lattice systems with local Lipschitz nonlinear terms driven by nonlinear noise defined on the entire integer set Z. We first establish the global well-posedness to non-autonomous stochastic p-Laplacian delay lattice system in C([τ,∞),L2(Ω,ℓ2)) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions, based on the uniform estimates of approximate solutions in a finite time-interval as well as an appropriate stopping time. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in product Hilbert space L2(Ω,F;ℓ2)×L2(Ω,F;L2((−ρ,0),ℓ2)) under the condition that the constant λ in the system is relatively large.

Load frequency predictive control for power systems concerning wind turbine and communication delay

AbstractThis article addresses a model predictive control (MPC) technique for load frequency control (LFC) system in the presence of wind power, communication delay, and denial‐of‐service (DoS) attack. In this article, communication delay is incorporated into a single area control error transmission for simplicity, wind power and load disturbance are regarded as Lipschitz nonlinear terms, as for the randomly occurring DoS attack, it is modeled as Bernoulli processes with known conditional probability. Thinking all these adverse factors to stability and the limitation of input constraint synthetically, the stability of LFC system can be guaranteed by delay‐dependent Lyapunov function lemma and a state feedback MPC controller is designed to solve the LFC problems by minimizing the infinite‐horizon objective function. Although some scholars have studied the performance degradation and instability of LFC system caused by cyber attack and/or communication delay and some very nice results have been addressed, limited works have considered the MPC approach to deal with both the problems of cyber attack and communication delay which explicitly considers the physical constraints. In addition, the delay‐dependent Lyapunov function is adopted to deal with the problem of communication delay, which results in less conservatism of the presented method. Finally, the optimization problem with input constraint is solved and proven to be recursive feasibility, and the closed‐loop system turns out to be stable. The reasonability and validity of the provided strategy is verified through several groups of simulation experiments. It illustrates that the proposed control method can keep the system frequency steady in the standard range in spite of various attack conditions.

Adaptive adjustable dimension observer based fault estimation for switched fuzzy systems with unmeasurable premise variables

This paper focuses on the sensor fault estimation problem for a kind of switched nonlinear systems. A novel fault estimation observer design method is proposed, which can reconstruct the sensor fault and system state, simultaneously. T-S fuzzy model is considered to describe the nonlinear dynamics in switched systems, and the premise variables of the fuzzy models are assumed to be dependent on the unmeasurable system state rather than the measurable variables, which leads to that there is a Lipschitz nonlinear term in the error dynamics. The Lipschitz nonlinear term is divided into two parts, and their influence on estimation result is decoupled and canceled by selecting appropriate observer parameter matrices and adding adaptive compensator, respectively. Compared with the existing results, the proposed method does not require the Lipschitz constant information. Thus, it avoids calculating the Lipschitz constant and reduces the conservatism caused by introducing the Lipschitz constant into LMIs. Besides, the order of the designed observer is not fixed, and can be selected in a certain range, which implies that the proposed method is with more design freedom for different conditions. At last, two examples are listed to verify the proposed fault estimation approach.

Stability and applications of multi-order fractional systems

<p style='text-indent:20px;'>This paper establishes conditions for global/local robust asymptotic stability for a class of multi-order nonlinear fractional systems consisting of a linear part plus a global/local Lipschitz nonlinear term. The derivation order can be different in each coordinate and take values in <inline-formula><tex-math id="M1">\begin{document}$ (0, 2) $\end{document}</tex-math></inline-formula>. As a consequence, a linearized stability theorem for multi-order systems is also obtained. The stability conditions are order-dependent, reducing the conservatism of order-independent ones. Detailed examples in robust control and population dynamics show the applicability of our results. Simulations are attached, showing the distinctive features that justify multi-order modelling.</p>

Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity

We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficient linear numerical scheme with the help of an artificial regularization term of the form $A\tau^k\frac{\partial}{\partial t}\mathcal{L}^{p(k)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $\tau$ is the uniform time step-size. The exponent $p(k)$ is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $\mathcal{L}$ together with the desired temporal order of accuracy $k$. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results.

Consensus/synchronisation of networked nonlinear multiple agent systems with event-triggered communications

ABSTRACT This paper describes a solution to the problem of consensus/synchronisation for a general class of networked nonlinear multi-agent systems (MAS) using a distributed control strategy with an event-triggered communications (ETC) mechanism. We consider the case where the dynamics of each agent contain linear and Lipschitz nonlinear terms and the underlying communication graph is directed. The strategy proposed has two important properties: (i) it achieves practical consensus, i.e. the synchronisation error that measures the disagreement among the agents' states converges to a ball centred at the origin, with a radius that can be made arbitrarily small and (ii) the minimum of the inter-event times for each agent is lower bounded by a strictly positive number, hence Zeno behaviour is excluded. Furthermore, it affords system designers adequate tools to trade off the frequency of communications among agents against the level of performance achieved in MAS consensus/synchronisation. Numerical simulation results are also given.

Guaranteed‐Performance Consensualization for High‐Order Lipschitz Nonlinear Multi‐Agent Systems with Balanced Directed Switching Topologies

AbstractGuaranteed‐performance consensus design problems for both the leaderless and leader‐following high‐order multi‐agent systems with balanced directed switching topologies and Lipschitz nonlinear dynamics are investigated. Consensus protocols and quadratic performance functions based on state errors are proposed to obtain the guaranteed‐performance consensus. By a state decomposition method, the consensualization problems for leaderless cases are transformed into asymptotic stability ones of a reduced‐order subsystem and the guaranteed‐performance cost is determined by the Lyapunov stability analysis of the reduced‐order subsystem, and the state error method is utilized to deal with the guaranteed‐performance consensualization problems for leader‐following cases. Then, by using the structure property of the transformation matrix and the Lipschitz condition, the impacts of Lipschitz nonlinear terms can be determined. Moreover, guaranteed‐performance consensualization criteria are presented based on linear matrix inequalities for both leaderless and leader‐following cases with directed switching topologies. Furthermore, the guaranteed‐performance costs are determined, which can respectively reflect the structures of the interaction topologies of leaderless and leader‐following cases. Finally, two numerical simulations are presented to verify the validity of theoretical results.

Multiple solutions for a class of perturbed second-order differential equations with impulses

The present paper is an attempt to investigate the existence of weak solutions for perturbed impulsive problems containing a Lipschitz nonlinear term. The study bases itself on the most recent variational approaches to the smooth functionals which are defined on reflexive Banach spaces. The findings of the study, finally, revealed that, under appropriate conditions, such problems possess at least three weak solutions. According to the results, these solutions are generated by impulses when the Lipschitz nonlinear term is zero.

Improved results on generalised robust H ∞ filtering for Lipschitz descriptor non‐linear systems with uncertainties

Recently, a generalised framework for robust non-linear H ∞ filtering of uncertain Lipschitz descriptor systems has been developed by Abbaszadeh and Marquez. The main feature of the associated H ∞ filter is that it holds a general structure, which can capture both dynamic and static-gain filter structures. In this study, the authors aim to propose improved results on the general H ∞ filter synthesis for Lipschitz descriptor non-linear systems in the presence of disturbance and model uncertainties. Based on a technique to directly deal with the Lipschitz non-linear terms, new H ∞ filter synthesis conditions that guarantee asymptotic stability of the estimation error dynamics with prescribed disturbance attention level are established and formulated in terms of linear matrix inequalities. They also give some remarks on the results of Abbaszadeh and Marquez and compare them with the author's results. More precisely, it is shown that the proposed conditions are less conservative than those in the existing results. Simulation results on an example are given to numerically verify the proposed design.

On a Class of Stochastic Semilinear PDEs

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L 2(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.

Structure of the solution set for a vector valued elliptic boundary value problem with a Lipschitz nonlinear term

We consider the boundary value problem with the Dirichlet condition in a Banach space for a semilinear elliptic equation on a bounded domain in R n whose nonlinear term satisfies the Lipschitz condition. If the Lipschitz constant L is less than λ 1 , then this problem has a unique solution, where λ 1 is the least eigenvalue of the corresponding (real valued) eigenvalue problem. On the other hand, for any L > λ 1 we can construct a nonlinear term with the Lipschitz constant L such that the solution set is homeomorphic to any prescribed closed subset of the Banach space.

Mobile Point Controls Versus Locally Distributed Ones for the Controllability of the Semilinear Parabolic Equation

It is well known now that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly null-controllable in L2 (\Omega)$, when governed in a bounded domain by the locally distributed controls. In this paper we intend to show that, in fact, in one space dimension ($\Omega= (0, 1)$) the very same results can be achieved by employing {\em at most} two mobile point controls with support on the curves properly selected within an arbitrary subdomain of QT = (0,1) × (0,T). We will show that such curves can be described by a certain differential inequality and the explicit examples are provided. We also discuss some extensions of our main results to the superlinear terms and to the case of several dimensions.