Second-order Systems Research Articles

Modeling and Admittance Control of a Piezoactuated Needle Insertion Device for Safe Puncture of Spinal Membranes

Abstract The most challenging procedure for lumbar puncture (LP) is accurately puncturing the spinal membrane (dura mater) based on an automatic needle insertion device (NID). Piezoactuated NID has shown its advantages in robotic-assisted LP with high precision and compact structure. Soft control of the NID is important for insertion safety; however, for stick-slip piezo-actuated NID, there are few studies due to the complex mechanism of stick-slip motion. Here, a modeling and admittance control method for a proposed stick-slip piezoactuated NID is proposed for safe puncture of the spinal membrane. To analytically model the NID, the compliant mechanism (CM) in the NID is reduced to a second-order system. The stick-slip friction and the spinal membrane are modeled based on the LuGre model and the Hunt-Crossley model, respectively. Based on these models, an admittance controller (AC) for the proposed NID is established to realize the precise control of the position and the safety protection against puncture errors. Simulations and preliminary experiments based on a prototype of the NID and a phantom of the spinal membrane were carried out to test the proposed modeling and control method. Results show that the proposed NID with AC has a maximum insertion error of 0.62 mm and the insertion depth decays by 80% when an unexpected force is applied. Therefore, the proposed model and control method have the potential to be used in real LP procedures by further development.

Research on Periodic Solutions of Non-autonomous Second-order Hamiltonian Systems

In the field of nonlinear science, Hamiltonian systems are crucial models extensively applied in engineering, physics, and biology. The existence of periodic solutions in non-autonomous second-order Hamiltonian systems, as a relatively specialized form, has been a focal and challenging research area. This paper employs variational methods to analyze the existence of periodic solutions in such systems, aiming to provide insights for related research endeavors.

On the Homotopy-First Integral Method for Non-conservative Oscillators

This paper presents a ready-to-use formula for determining the number and approximate location of periodic orbits in second-order Lienard systems. As a result of the exact closed-form derived in [16], in which an ordinary differential equation (ODE) must be solved to determine the existence and location of periodic orbits for general non-conservative oscillators, a homotopy functional is defined for Lienard-type systems. This provides a closed-form and ready-to-use polynomial formula with roots as an approximation of the periodic orbit's amplitude. In addition, some examples are analyzed, along with conclusions and future plans.

A fixed-time nonsingular fast terminal sliding mode control scheme with time-varying sliding mode surface

This article proposes a fixed-time nonsingular fast terminal sliding mode control scheme with time-varying sliding mode surface for a class of second-order uncertain system. The control scheme makes the tracking error converge to the origin in fixed time without singularity problem and guarantees fast convergence rate given any initial states. In addition, the proposed control scheme ensures system states to converge to the origin instead of a neighbourhood of the origin which is so called practical fixed-time convergence. As a result, high precision of the trajectory tracking is guaranteed by the control scheme. A comparative simulation with a two-link manipulator is given and the results show that the proposed control scheme can achieve a good performance.

Event-triggered secure consensus for second-order nonlinear multiagent systems against asynchronous DoS attacks

This paper investigates secure consensus of the second-order nonlinear multiagent systems under malicious asynchronous DoS attacks by using event-triggered control. Due to the presence of diverse attackers, network protocols, and different defense strategies, asynchronous launched DoS attacks are always considered in communication and control channels. Moreover, to alleviate the pressure on communication and computing resources, a distributed event-triggered control strategy is presented, where the control strategy relies on the agent’s own and its neighbors’ information. Then, several secure consensus criteria for multiagent systems with directed and undirected topologies are established based on the designed distributed controller and different Lyapunov functions. Furthermore, it is demonstrated that the trigger sequences eliminate Zeno behavior by utilizing a hybrid event-triggered strategy. Finally, the usefulness of the achieved outcomes is substantiated via two simulation examples.

Adaptive output-feedback trajectory tracking for a class of uncertain second-order linear systems

Abstract In this work, we present a novel design of an explicit control strategy to solve the trajectory tracking problem for an uncertain second-order linear system with only partial availability of the state. The system parameters are unknown, and an additive constant disturbance is considered. Despite the apparent simplicity of this system, the solution requires an intricate combination of available methodologies. The proposed approach corresponds to a Lyapunov-based adaptive control strategy consisting of a reduced-order observer, a dynamic slave system and an adaptive controller with dynamic extension. Simulation results are provided to highlight the effectiveness of the proposed adaptive control strategy in accomplishing the regulation and trajectory tracking problems.

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

Automatic control of constant temperature and humidity in building air conditioning systems based on frequency domain analysis

How to solve their automatic control of constant temperature and humidity gradually becomes a research hotspot as the continuous upgrading of air conditioning systems. This study aims to optimize the traditional proportional-integral-differential controller for improvement to solve the time-delay instability phenomenon in temperature and humidity control. The objective of this study is to optimize existing proportional-integral-differential controllers to improve the time-delay instability problem that is common in temperature and humidity control. Firstly, it treats the controlled object as a first-order and second-order system with time-delay characteristics. Next, the Smith predictor controller is generalized equivalent to ensure that the equivalent system does not contain time-delay. Finally, an analysis of the first-order and second-order closed-loop control system is conducted by combining Smith predictive controller and proportional-integral-differential controller. The system achieves the goal of automatic control of constant temperature and humidity by adjusting the control parameters. The experiment showcased that the temperature control time of the proposed control scheme under first-order and second-order time-delays was 16 s and 3 s, respectively. Meanwhile, the humidity control time was 14 s and 13 s, respectively. In practical applications, the proposed control scheme achieved good control effects in all four seasons. This indicates that the controller designed in this study possesses good control performance. It also can achieve the goal of constant temperature and humidity control. This can provide technical support for the automation control of air conditioning systems.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.

A Semi-Global Finite-Time Dynamic Control Strategy of Stochastic Nonlinear Systems

In the article, the semi-global finite-time control strategy for stochastic nonlinear systems is studied. Firstly, the general stochastic nonlinear system is considered and the required conditions are provided. An important theorem that helps to construct the controller directly is subsequently obtained by adopting a dynamic gain and homogeneous domination method. The equilibrium of the whole system is semi-global finite-time stable in probability (SGFSP) under the designed controller. Finally, the presented method is successfully applied to a second-order system. Simulation results indicate the effectiveness of the method.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

Scaled consensus of second-order nonlinear multi-agent systems with fully distributed adaptive aperiodically intermittent communication: A non-reduced order approach

Scaled consensus of second-order nonlinear multi-agent systems with fully distributed adaptive aperiodically intermittent communication: A non-reduced order approach

Cloud-based time-varying formation active fault-tolerant predictive control of networked multi-agent systems with actuator and sensor faults

This paper addresses the time-varying formation control problem of second-order networked multi-agent systems consisting of a virtual leader and multiple followers, where random communication constraints in the forward and feedback channels as well as actuator and sensor faults are considered. A state observer is designed to estimate the system state and the potential faults, based on which a time-varying formation active fault-tolerant predictive control scheme is proposed. Then, a necessary and sufficient condition is derived to guarantee the stability of the closed-loop system and to achieve the time-varying formation, which is independent of random communication constraints as well as actuator and sensor faults. Finally, simulation results of three contrastive cases are presented to verify the effectiveness of the proposed control scheme.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Distributed optimal coordination algorithm for nonlinear second-order multi-agent systems and its application to vehicle platoon

This paper addresses the problem of distributed optimal coordination over second-order multi-agent systems, where the dynamics are nonlinear and unknown. To solve this problem, a state-based observer is designed to handle the unknown matched and unmatched nonlinear terms. Using this observer and the proportional–integral control concept, a distributed optimal coordination algorithm with fixed control parameters is constructed through the consensus scheme and gradient flow method. With the help of the Routh–Hurwitz stability criterion, the exponential convergence proof is guaranteed under the condition that the nonlinear term and the convex local cost function are all locally Lipschitz. Finally, we present four simulation examples, including the vehicle platoon’s cooperative control, to demonstrate the efficiency of our proposed algorithms.

Engineering the passband shape of coupled-cavity filters for low loss and/or narrow bandwidth.

Emerging applications of photonic integrated circuits are calling for extremely narrowband and/or low-insertion-loss bandpass filters. Both properties are limited by cavity losses or intrinsic quality factors. However, the choice of inter-cavity and bus couplings establishes trade-offs between these two properties and the passband shape, which have been little explored. Using the widely used second-order resonant system as an example, we present new, to the best of our knowledge, classes of filter passband shapes that provide the lowest insertion loss and the narrowest bandwidth for a given loss Q. A normalized design and novel properties based on a temporal coupled-mode theory model are presented, including a design tool to apply these results. These results may benefit loss-sensitive filtering applications such as quantum-correlated photon pair sources and RF-photonic integrated circuits.

A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.

Adaptive neural event-triggered consensus control for unknown nonlinear second-order delayed multi-agent systems

In this paper, we discuss event-triggered consensus control for a heterogeneous second-order multi-agent systems with unknown nonlinear functions and state delays (SODMASs). Firstly, a broad adaptive dynamic event-triggered mechanism (ETM), which includes many existing ETM, is proposed. Based on this ETM and the approximation property of neural networks, an adaptive event-triggered protocol with weight estimation of neural network and time-varying integral gain compensation is designed to reduce communication frequency and to eliminate the uncertainty of nonlinear function and state delays. Then, the consensus problem for the heterogeneous SODMASs under the protocol is solved successfully by selecting a suitable Lyapunov–Krasovskii functional. In addition to these, it is also proved that all agents can exclude Zeno behavior under the proposed event-triggered protocol. Finally, a numerical example is given to verify the effectiveness of the proposed consensus protocol algorithm.

Finite-time consensus of second-order multi-agent connectivity preserving based on adaptive sliding mode control

This study addresses the problem of robust finite-time connectivity preserving consensus for second-order multi-agent systems (MASs) with a limited communication range. Considering that the communication ability of agents in practical applications is limited, this study introduces an innovative approach to the consensus protocol, which involves the incorporation of a potential function aimed at ensuring sustained communication connection within a MAS. In addition, a finite-time fault-tolerant consensus protocol for a second-order MAS with actuator additive faults and external disturbances is designed by combining sliding mode control (SMC) and adaptive control. Following the homogeneous theorem and the finite-time consensus theory, this study concludes that, under specific conditions, a MAS can achieve finite-time consensus. Moreover, an advanced control protocol named a super-twisting sliding mode control protocol is introduced to mitigate the undesirable chattering phenomenon in the SMC. Finally, the effectiveness and superiority of the proposed method are verified by numerical examples.

Periodic second-order systems and coupled forced Van der Pol oscillators

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.