Nonlinear Subsystems Research Articles

Nonlinear energy sink to enhance the landing gear shimmy performance

The shimmy phenomenon is a significant concern in aircraft landing gear dynamics. The prediction of the shimmy instability is an essential issue in landing gear design to develop a passive or active suppression method. This work investigates the application of the nonlinear energy sink (NES) concept to mitigate the effects of shimmy in landing gears. The NES concept has been used in recent research on mechanical vibrations. It comprises a passive target energy transfer method that refers to a one-way energy transfer from a primary to a nonlinear subsystem. The landing gear model is based on torsional displacement coupled with the tyre classical elastic string analogy model. The NES device connects to the wheel shaft, and it comprises a mass, a linear damper, and a pure cubic spring. The numerical integration in time was used to assess the shimmy onset speed and the post-shimmy limit cycle oscillations. A parametric analysis of the landing gear nonlinear dynamics without the NES is presented. The design space of possible NES parameters is given, obeying design constraints, and inspected to assess adequate NES designs. The best NES samples are included in the landing gear dynamics to study their influence. Results have shown that the NES can adequately expand the operational speed range for no shimmy and lead to lower LCO amplitudes in the post-shimmy for a reasonable range of speeds. The NES concept’s successful employment for the landing gear dynamics suggests an enormous potential form of passive shimmy control.

Integration of vibration control and energy harvesting for whole-spacecraft: Experiments and theory

Giant magnetostrictive material (GMM) is integrated into a nonlinear energy sink (NES) to construct a NES-GMM device for vibration control and energy harvesting. The NES-GMM device is experimentally embedded in a scaled model of a whole-spacecraft system. LMS Test.Lab software and an oscilloscope are used to collect vibration and energy signals of the whole-spacecraft system with and without the NES-GMM device under sine sweeping frequency excitations. The transmissibility amplitudes, resonant frequencies, and voltage values demonstrate the effeteness of vibration control and energy harvesting of the NES-GMM device under different various conditions. The experimental system is modeled as a two-degrees-of-freedom (2DOF) linear primary system with a nonlinear NES-GMM additional subsystem. The complexification-averaging (CX-A) method verified with numerical verifications is developed to investigate analytically the vibration reduction and energy harvesting of the NES-GMM device for the whole-spacecraft system. The experimental and theoretical results show that the NES-GMM device reduces the vibration and produces the electricity and that the device hardly changes the resonance frequencies of the original whole-spacecraft system. The saddle-node (SN) bifurcation is detected in the steady-state periodic response of the system. The performance of the NES-GMM device is examined for its varying parameters.

Rao-Blackwellized Particle Filter for Asynchronously Dependent Noises

This paper develops Rao-Blackwellized particle filter with asynchronous dependence between system noise and measurement noise. It is pointed out that this dependence affects both the particle filter update step for the nonlinear sub-system and the Kalman filter update step for the conditionally linear sub-system in Rao-Blackwellized particle filter. A de-correlation method is suggested to deal with such influence. The optimal importance density function for sampling the nonlinear sub-state is found out, and a suboptimal one for approximating the optimal importance density function is proposed. The proposed methods are applied to target tracking to testify their effectiveness and superiority.

Adaptive cross backstepping control for a class of nonstrict feedback nonlinear systems with partial state constraints

AbstractThe tracking control problem for a class of partial state constrained nonlinear system is studied in this article. The system is divided into two semistrict feedback nonlinear subsystems, one is state constrained and the other is state free. By means of state transformation, the state constraint problem is transformed into the bounded problem of the transformed function. Compared with the barrier Lyapunov function (BLF) method, it not only solves the state constraint problem but also circumvents the feasibility check on virtual controllers. Based on the cross backstepping control, the constrained controller and unconstrained controller are designed simultaneously. It solves the coupling problem effectively in the design of cross processing control. On the other hand, dynamic surface control is used which effectively avoids “computation explosion” caused by backstepping control. The designed controllers can ensure the error signals converge to a small neighbourhood of zero and keep the asymmetric time‐varying constraints on system partial states are satisfied for all the time. Finally, simulation experiments are carried out on a hyperchaotic Rössler system to verify the efficacy of the control scheme.

Asynchronous Event-Triggered Finite-Time Filtering for Networked Switched T–S Fuzzy Systems

This paper investigates the problem of event-triggered finite-time robust filtering for a class of networked switched T–S fuzzy systems. By adopting the extended dissipative performance index, the $$H_{\infty }$$ and $$L_{2 }-L_{\infty }$$ filtering problems can be solved in a unified framework. We propose an event-triggered scheme to save the network resources. Under the event-triggered scheme, the filter is not synchronized with the corresponding subsystem. By constructing a reasonable Lyapunov–Krasovskii function, we give sufficient conditions for the finite-time extended dissipative of the filtering error system. We employ the T–S fuzzy models to represent the nonlinear subsystems. The filter gains can be obtained by solving some matrix inequalities. Finally, an example is presented to show the efficiency of the method.

A Multi-Index Feedback Linearization Control for a Buck-Boost Converter

Due to the nonlinear and nonminimum phase characteristics of the buck-boost converter, the design of its controller has always been a challenging problem. In this paper, a multi-index feedback linearization control strategy is proposed to design the controller of the buck-boost converter. Firstly, by constructing an appropriate output function, the original nonlinear system is mapped into a combination of a linear subsystem and a nonlinear subsystem. Then, according to the structural characteristics of these two subsystems, the linear optimal control theory is adopted for the control design of the linear subsystem to make it have a good output performance, while for the nonlinear subsystem, the coefficient of the output function is adjusted to ensure its stability. Finally, based on the Hartman–Grobman theorem, the internal mechanism and coefficient adjustment basis of the proposed method are revealed; that is, by adjusting the coefficient of the output function and the feedback coefficient of the linear control law, the poles of the system are configured to achieve the purpose of adjusting the static and dynamic performance of the system. The simulation results show the feasibility and superiority of using the multi-index feedback linearization control strategy to design the nonlinear control law of the buck-boost converter.

Reachable set bounding of output-feedback control for discrete-time large-scale IT-2 fuzzy descriptor systems using distributed event-based broadcasts

This paper studies the distributed event-based broadcasting of output-feedback control for discrete-time large-scale fuzzy descriptor systems subject to reachable set bounding. First, each nonlinear subsystem is represented by the interval type-2 fuzzy model, where fuzzy representation is assumed to be appearing not only in both the state and input matrices but also in the derivative matrix. By using a descriptor redundancy approach, the fuzzy representation in the derivative matrix is reformulated into the linear one. Then, we introduce an output-feedback fuzzy controller with distributed event-based broadcasting, where all subsystems communicate with each other via a distributed network, and each subsystem broadcasts its decision-making of transmission based on a prescribed event. Two event-based mechanisms (EBMs) are respectively proposed to examine when the system output and fuzzy premise variable should be transmitted to the controller. Moreover, by further employing a descriptor redundancy representation, combined with reachable set analysis method, it will be shown that the proposed fuzzy controller guarantees that the estimation error is bounded within the ellipsoidal boundary while communication data are transmitted to the controller as little as possible, and sufficient condition for designing the desired controller is derived in terms of linear matrix inequalities (LMIs). Finally, a simulation study is given to show the effectiveness of the proposed method.

The impact of static output nonlinearities on the control strategies that humans use in command-following tasks

The results of a human-in-the-loop experiment are used to investigate the control strategies that humans use to interact with nonlinear dynamic systems. Two groups of human subjects interact with a dynamic system and perform a command-following task. The first group interacts with a linear time-invariant (LTI) dynamic system. The second group interacts with a Wiener system, which consists of the same LTI dynamics cascaded with a static output nonlinearity. Both groups exhibit improved performance over the trials, but the average of the linear group’s performance is better on more than three-fourths of the trials. A new nonlinear subsystem identification algorithm is presented and used to identify the feedback and feedforward control strategies used by the subjects in both groups. The identification results for the linear group agree with prior studies suggesting that adaptive feedforward inversion is a primary control strategy used by humans for command-following tasks. The main results of this paper address an open question of whether a similar control strategy is used for nonlinear systems. The identification results for the nonlinear group suggest that those subjects also use adaptive feedforward inversion. However, the static output nonlinearity inhibits the human’s ability to approximate the inverse.

A hybrid metaheuristic algorithm for identification of continuous-time Hammerstein systems

This paper presents a new hybrid identification algorithm called the Average Multi-Verse Optimizer and Sine Cosine Algorithm for identifying the continuous-time Hammerstein system. In this paper, two modifications were employed on the conventional Multi-Verse Optimizer. Our first modification was an average design parameter updating mechanism to solve the local optima issue. The second modification was the hybridization of Multi-Verse Optimizer with Sine Cosine Algorithm that will balance the exploration and exploitation processes and thus improve the poor searching capability. The proposed hybrid method was used for identifying the parameters of linear and nonlinear subsystems in the Hammerstein model using the given input and output data. A continuous-time linear subsystem was considered in this study, while there were a few methods that utilize such models. Furthermore, various nonlinear subsystems such as the quadratic and hyperbolic functions had been used in those experiments. The efficiency of the novel technique is illustrated using a numerical example and two real-world applications, which are a twin rotor system and a flexible manipulator system. The numerical and experimental results analysis were observed with respect to the convergence curve of the fitness function, the parameter deviation index, time-domain and frequency-domain responses of the identified model, and the Wilcoxon's rank test. The results showed that the proposed method was efficient in identifying both the Hammerstein model subsystems in terms of the quadratic output estimation error and parameter deviation index. The proposed hybrid method also achieved better performance in modeling of the twin-rotor system as well as the flexible manipulator system and provided better solutions compared to other optimization methods including Particle Swarm Optimizer, Grey Wolf Optimizer, Multi-Verse Optimizer and Sine Cosine Algorithm.

Estimating multiple latencies in the auditory system from auditory steady-state responses on a single EEG channel

The latency of the auditory steady-state response (ASSR) may provide valuable information regarding the integrity of the auditory system, as it could potentially reveal the presence of multiple intracerebral sources. To estimate multiple latencies from high-order ASSRs, we propose a novel two-stage procedure that consists of a nonparametric estimation method, called apparent latency from phase coherence (ALPC), followed by a heuristic sequential forward selection algorithm (SFS). Compared with existing methods, ALPC-SFS requires few prior assumptions, and is straightforward to implement for higher-order nonlinear responses to multi-cosine sound complexes with their initial phases set to zero. It systematically evaluates the nonlinear components of the ASSRs by estimating multiple latencies, automatically identifies involved ASSR components, and reports a latency consistency index. To verify the proposed method, we performed simulations for several scenarios: two nonlinear subsystems with different or overlapping outputs. We compared the results from our method with predictions from existing, parametric methods. We also recorded the EEG from ten normal-hearing adults by bilaterally presenting superimposed tones with four frequencies that evoke a unique set of ASSRs. From these ASSRs, two major latencies were found to be stable across subjects on repeated measurement days. The two latencies are dominated by low-frequency (LF) (near 40 Hz, at around 41–52 ms) and high-frequency (HF) (> 80 Hz, at around 21–27 ms) ASSR components. The frontal-central brain region showed longer latencies on LF components, but shorter latencies on HF components, when compared with temporal-lobe regions. In conclusion, the proposed nonparametric ALPC-SFS method, applied to zero-phase, multi-cosine sound complexes is more suitable for evaluating embedded nonlinear systems underlying ASSRs than existing methods. It may therefore be a promising objective measure for hearing performance and auditory cortex (dys)function.

Analytical Stability Conditions on Interconnected Nonlinear Systems With Delays

This paper is concerned with stability of interconnected systems with time delays. We develop a self-contained approach to stability analysis for linear and nonlinear systems in a unified framework. New lemmas are established on matrix properties and used as the key to make negativity of the derivative of the Lyapunov function. The scalar and simple analytical stability conditions are given. Unlike the majority of the literature on stability of delay systems, no matrix equations/inequalities are involved in our conditions, which is true even for large-scale systems and nonlinear subsystems with delayed interconnections. They are applicable to the more general nonlinear, time-varying, and/or interconnected systems than the relevant results reported in the literature. The examples are presented for illustration of the new results.

Optimization of dynamically loaded nonlinear technical systems

Modeling and simulation technologies play an important role in the operation, diagnosis of the current state and prediction of changes in various mechanical systems. The most effective approach for the study of nonlinear friction systems is the use of the method of physical and mathematical modeling, which is based on the construction of a mathematical model of a quasi-linear subsystem of a machine or mechanism and the creation of a physical model of a substantially nonlinear friction subsystem. Fundamentally new approaches have been developed to solve the problems of optimization, increasing the efficiency and competitiveness of non-linear technical systems using the example of the wheel-rail tribosystem. Proposed technology of metal cladding of locomotive wheel band and implementation of anisotropy of friction coupling provide significant increase of efficiency of “wheel-rail” tribosystem.

Model-free predictive control for a class of switched nonlinear systems

A model-free predictive control (MFPC) method is devised towards a kind of high order discrete nonlinear switched systems whose models are undefined. Firstly, the estimation models are designed by using the compact dynamic linearisation technique and improved projection algorithm to approach the controlled nonlinear subsystems. Then, on the basis of the estimation models, the controller's explicit analytic solutions are obtained by solving the finite time domain rolling optimisation quadratic functions. Finally, an appropriate switching law is designed to make the subsystems switch reasonably in different subsystems, thus ensuring the stability of the whole system. The simulation results show that the control strategy is effective.

Compositional (In)Finite Abstractions for Large-Scale Interconnected Stochastic Systems

This article is concerned with a compositional approach for constructing both infinite (reduced-order models) and finite abstractions [a.k.a. finite Markov decision processes (MDPs)] of large-scale interconnected discrete-time stochastic systems. The proposed framework is based on the notion of stochastic simulation functions enabling us to employ an abstract system as a substitution of the original one in the controller design process with guaranteed error bounds. In the first part of this article, we derive sufficient small-gain-type conditions for the compositional quantification of the probabilistic distance between the interconnection of stochastic control subsystems and that of their infinite abstractions. We then construct infinite abstractions together with their corresponding stochastic simulation functions for a particular class of discrete-time nonlinear stochastic control systems. In the second part of this article, we leverage small-gain-type conditions for the compositional construction of finite abstractions. We propose an approach to construct finite MDPs as finite abstractions of concrete models or their reduced-order versions satisfying an incremental input-to-state stability property. We also show that for the particular class of nonlinear stochastic control systems, the aforementioned property can be readily checked by matrix inequalities. We demonstrate the effectiveness of the proposed results by applying our approaches to a fully interconnected network of 20 nonlinear subsystems (totally 100 dimensions). We construct finite MDPs from their reduced-order versions (together 20 dimensions) with guaranteed error bounds on their output trajectories. We also apply the proposed results to a temperature regulation in a circular building and construct compositionally a finite abstraction of a network containing 1000 rooms. We employ the constructed finite abstractions as substitutes to compositionally synthesize policies regulating the temperature in each room for a bounded time horizon.

A Novel Connectivity-Preserving Control Design for Rendezvous Problem of Networked Uncertain Nonlinear Systems.

This article proposes a novel and robust connectivity-preserving rendezvous control design for a group of uncertain nonlinear multiagent systems with communication constraint where each agent has a limited sensing range. The control design can work under the assumption that the communication network is initially connected and is characterized by two distinguishing features. First, a new potential function is provided not only to maintain the existing and newly added links by the hysteresis rule but also to overcome the difficulty imposed by the nonlinear terms from system dynamics. Second, by constructing a series of lemmas, a connectivity-preserving stabilizing control law is presented to solve the robust stabilization problem with connectivity preservation for a time-varying nonlinear system, which is a special case of the augmented system with both dynamic and static uncertainties obtained via internal model design. After further incorporating the adaptive control technique, regardless of uncertain parameters and external disturbances in the multiple nonlinear subsystems, the leader-following rendezvous with connectivity preservation problem is finally solved by a distributed connectivity-preserving controller with parameter update law.

Compositional abstraction-based synthesis of general MDPs via approximate probabilistic relations

We propose a compositional approach for constructing abstractions of general Markov decision processes (gMDPs) using approximate probabilistic relations. The abstraction framework is based on the notion of δ-lifted relations, using which one can quantify the distance in probability between the interconnected gMDPs and that of their abstractions. This new approximate relation unifies compositionality results in the literature by incorporating the dependencies between state transitions explicitly and by allowing abstract models to have either infinite or finite state spaces. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract models, and then carry the results over concrete ones. To this end, we first propose our compositionality results using the new approximate probabilistic relation which is based on lifting. We then focus on a class of stochastic nonlinear dynamical systems and construct their abstractions using both model order reduction and space discretization in a unified framework. We provide conditions for simultaneous existence of relations incorporating the structure of the network. Finally, we demonstrate the effectiveness of the proposed results by considering a network of four nonlinear dynamical subsystems (together 12 dimensions) and constructing finite abstractions from their reduced-order versions (together 4 dimensions) in a unified compositional framework. We benchmark our results against the compositional abstraction techniques that construct both infinite abstractions (reduced-order models) and finite MDPs in two consecutive steps. We show that our approach is less conservative than the ones available in the literature.

Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy singular perturbation method

Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2.

Observer Design for Nonlinear Invertible System from the View of Both Local and Global Levels

This paper emphasizes the importance of the influences of local dynamics on the global dynamics of a control system. By considering an actuator as an individual, nonlinear subsystem connected with a nonlinear process subsystem in cascade, a structure of interconnected nonlinear systems is proposed which allows for global and local supervision properties of the interconnected systems. To achieve this purpose, a kind of interconnected observer design method is investigated, and the convergence is studied. One major difficulty is that a state observation can only rely on the global system output at the terminal boundary. This is because the connection point between the two subsystems is considered unable to be measured, due to physical or economic reasons. Therefore, the aim of the interconnected observer is to estimate the state vector of each subsystem and the unmeasurable connection point. Specifically, the output used in the observer of the actuator subsystem is replaced by the estimation of the process subsystem observer, while the estimation of this interconnection is treated like an additional state in the observer design of the process subsystem. Expression for this new state is achieved by calculating the derivatives of the output equation of the actuator subsystem. Numerical simulations confirm the effectiveness and robustness of the proposed observer, which highlight the significance of the work compared with state-of-the-art methods.

Neural-network-based nonlinear model predictive tracking control of a pneumatic muscle actuator-driven exoskeleton

Pneumatic muscle actuators &#x0028 PMAs &#x0029 are compliant and suitable for robotic devices that have been shown to be effective in assisting patients with neurologic injuries, such as strokes, spinal cord injuries, etc., to accomplish rehabilitation tasks. However, because PMAs have nonlinearities, hysteresis, and uncertainties, etc., complex mechanisms are rarely involved in the study of PMA-driven robotic systems. In this paper, we use nonlinear model predictive control &#x0028 NMPC &#x0029 and an extension of the echo state network called an echo state Gaussian process &#x0028 ESGP &#x0029 to design a tracking controller for a PMA-driven lower limb exoskeleton. The dynamics of the system include the PMA actuation and mechanism of the leg orthoses; thus, the system is represented by two nonlinear uncertain subsystems. To facilitate the design of the controller, joint angles of leg orthoses are forecasted based on the universal approximation ability of the ESGP. A gradient descent algorithm is employed to solve the optimization problem and generate the control signal. The stability of the closed-loop system is guaranteed when the ESGP is capable of approximating system dynamics. Simulations and experiments are conducted to verify the approximation ability of the ESGP and achieve gait pattern training with four healthy subjects.

Dwell-time-based energy scheduling and distributed control for large-scale nonlinear systems under Round-Robin protocol

In this paper, a distributed control problem is investigated for large-scale nonlinear systems under Round-Robin (RR) protocol and the energy constraint. The large-scale system is composed of several interconnected nonlinear subsystems that are equipped with independent sensors. RR protocol is adapted to coordinate the data transmission of each sensor from the viewpoint of fairness. Moreover, an energy scheduling scheme is employed in response to the energy constraint by distributing high energy or low energy to sensors. The transmissions (between sensors and controllers) with high energy can be accurate. Correspondingly, the accessed data may be lost with a certain probability when sensors use low energy. This paper aims to design the scheduling scheme and distributed controllers jointly such that the stochastic finite-time boundedness (FTB) can be guaranteed for large-scale nonlinear systems with limited energy and communication protocols over a finite horizon. In light of the energy allocations and the induced packet dropout, novel distributed controllers are established for large-scale systems. Furthermore, the explicit expressions of the controller for each subsystem and the energy scheduling scheme with average dwell time (ADT) are derived by using recursive design methods combined with stochastic analysis. Finally, two simulation examples are given to demonstrate the effectiveness of the designed scheme and controllers.