Nonlinear Time-varying Dynamics Research Articles

On Averaging Technique for Nonlinear Time-Varying Systems without Local Lipschitz Continuity

The majority of literature for averaging methods deal with a class of nonlinear time-varying (NLTV) dynamics and their time-invariant averaged systems with locally Lipschitz continuous (LLC) condition. In this work, a class of uniformly continuous NLTV systems and their uniformly continuous averaged systems are considered. The first result shows the closeness of solutions between the original NLTV system and its averaged system with a sufficiently small time-scale separation parameter ε on a subset of a time interval, in which both systems have well-defined solutions. If the averaged system is finite-time stable (FTS), the second result shows that the original NLTV system will uniformly converge to an arbitrarily small neighborhood of the origin on a finite-time interval with sufficiently small ε. Simulation results support the theoretical finding.

Design and Analysis of Data-Driven Learning Control: An Optimization-Based Approach.

Learning to perform perfect tracking tasks based on measurement data is desirable in the controller design of systems operating repetitively. This motivates this article to seek an optimization-based design and analysis approach for data-driven learning control systems by focusing on iterative learning control (ILC) of repetitive systems with unknown nonlinear time-varying dynamics. It is shown that perfect output tracking can be realized with updating inputs, where no explicit model knowledge but only measured input-output data are leveraged. In particular, adaptive updating strategies are proposed to obtain parameter estimations of nonlinearities. A double-dynamics analysis approach is applied to establish ILC convergence, together with boundedness of input, output, and estimated parameters, which benefits from employing properties of nonnegative matrices. Simulations are implemented to verify the validity of our optimization-based adaptive ILC.

Bioprocess Monitoring: A Moving Horizon Estimation Experimental Application

Microbial fermentation processes are most often described by nonlinear time-varying dynamics, which require the implementation of nonlinear state estimators to infer unmeasured metabolites in the cultivation broth. Among the various nonlinear available estimator strategies, the Moving Horizon Estimator (MHE) is an on-line optimization approach that easily allows to enforce hard constraints, an important feature that helps to avoid unfeasible concentrations. In this work we implemented an MHE by using experimental data from a fed-batch cultivation process of Corynebacterium glutamicum. Available real-time measurements of biomass and CO2 formation were used to infer sugar concentrations by combining the available measurements with a simple Monod model. We found that the MHE was able to estimate all the three variables of interest, including the unmeasured sugar concentrations, during the entire fed-batch cultivation process. Moreover, we show that the estimates are accurate in comparison to the reference offline samples. This work demonstrates the benefits of MHE as a soft sensor that can monitor bioprocesses in real-time.

Comparative study of the synchronous control for rigidly coupled electro-motors

Multiple electro-motors rigidly coupled by multi-points gear meshing is of a typical driving structure, and the synchronous control between parallel motors is the key issue about system stability and performance. Firstly, the drooping control is introduced to modify traditional speed parallel control (SPC) method; classical torque master-slave control (TMSC) using soft control parameter is also proposed as contrast. Subsequently, the basic principle of vector control with rotor flux orientation is described; the non-linear time-varying gear dynamics at the unbalanced pinion-gear meshing stage are also studied. Based on the mathematic analysis, simulation models under three different control methods are established in Simulink, and are comparatively studied against backlash, contact ratio and motor parameter deviations between multiple chains. Finally, SPC with drooping control (SPCD) and soft TMSC (STMSC) are applied on a test rig. The simulation and experimental results show that, SPC, SPCD and STMSC separately has advantages in speed tracking, system stability and torque synchronization, and should be used according to specific preference.

Fast just-in-time-learning recursive multi-output LSSVR for quality prediction and control of multivariable dynamic systems

Aiming at quality prediction and control of blast furnace (BF) ironmaking process characterized by complicated nonlinear time-varying dynamics, this paper proposes a just-in-time-learning (JITL) recursive multi-output least squares support vector regression (JITL-R-M-LSSVR) algorithm with fast nonlinear local learning capability for multivariable dynamic systems. The proposed fast JITL-R-M-LSSVR effectively combines the online local learning of JITL with the multi-output LSSVR (M-LSSVR) based on multi-task transfer learning, and focuses on how to ensure the rapid verification of the local model during online learning of M-LSSVR, and how to perform model pruning while recursively updating the model parameters to improve the calculation efficiency. To this end, the proposed algorithm uses a derived multi-output incremental learning algorithm to recursively update model parameters online in a gentle way, which has better modeling stability and smoothness than the traditional way that discards old models. At the same time, when the model is pruned, a novel multi-output reverse decremental learning algorithm is proposed to adaptively delete the modeling data, so as to effectively control the sample size and reduces the calculation cost. In particular, the model verification of the proposed algorithm only needs to construct the M-LSSVR modeling matrix and the matrix inverse operation once, and the matrix after deleting each modeling sample can be easily obtained by reverse decremental learning of the original modeling matrix, which can achieve fast and efficient model verification. Finally, the effectiveness and practicability of the proposed method are verified by applying it to prediction modeling and predictive control of the molten iron quality in BF ironmaking process.

Fast adaptive fuzzy terminal sliding mode control of synergistic movement of the hip and knee joints (air-stepping) using functional electrical stimulation: A simulation study

Functional electrical stimulation (FES) is a practical method for rehabilitation and increasing the level of movement in paraplegic subjects. So far, various control methods such as the combination of fuzzy adaptive or neural network with classic sliding mode control have been used for generating FES control signals. Such hybrid methods have led to the chattering phenomenon, the singularity of the control signal, and low speed of the controller outside and on the sliding surface. In the current study, a fast adaptive fuzzy terminal sliding mode controller (FAFTSMC) for a musculoskeletal model of the hindlimb, including six muscles joined to a planar model of two-link rigid robotic manipulator is presented for controlling the air stepping through FES. To reduce the control complexity considerably, a decentralized FAFTSMC consisted of a set of independent multi-input multi-output (MIMO) controllers where each subsystem is controlled by a stand-alone controller, is used to control the tracking problem. Outside the sliding surface, a novel fast terminal reachability law is used to force the dynamics of the system to move toward the sliding surface. In the sliding phase, a continuous fast terminal sliding surface is designed for the convergence of the tracking error and its first derivative to a neighborhood of zero in a finite time without any occurrence of the chattering or singularity in the control effort. A fuzzy adaptive estimator is employed to identify nonlinear time-varying dynamics of the process model. The stability of the closed-loop system is proved using the Lyapunov method. Simulation results of the control algorithm show that the hip and knee joint reference trajectories can be tracked with the fast adaptive fuzzy terminal sliding mode controller with an excellent precision.

A self-organizing multi-model ensemble for identification of nonlinear time-varying dynamics of aerial vehicles

This article presents a novel identification approach which can deal with nonlinear and time-varying characteristics of complex dynamic systems, especially an aerial vehicle in the entire flight envelope. A set of local sub-models are first developed at different operating points of the system, and subsequently a self-organizing multi-model ensemble is introduced to aggregate the outputs of the local models as a single model. The number of employed local models in the proposed multi-model ensemble is optimized using a novel self-organizing approach. Also, wavelet neural networks, which combine both the universal approximation property of neural networks and the wavelet decomposition capability, are used as the local models of the proposed method. In addition, a generalized online sequential extreme learning machine is adopted in the introduced approach to determine the optimal validity function of the local models at each time step. Finally, the introduced self-organizing multi-model ensemble is applied to the NASA Generic Transport Model as a complex nonlinear system to demonstrate the effectiveness of the proposed identification approach. Furthermore, the results obtained from the conventional artificial neural networks are carefully compared with those from the wavelet neural networks, which are employed as the local models of the introduced multi-model ensemble. The simulation results suggest that the introduced wavelet neural network–based self-organizing multi-model ensemble can be used satisfactorily as the prediction model of model-based control systems for long prediction horizons.

Recursive Learning-Based Bilinear Subspace Identification for Online Modeling and Predictive Control of a Complicated Industrial Process

In this paper, a recursive learning based bilinear subspace identification (R-B-SI) algorithm is proposed for online modeling and data-driven predictive control of blast furnace (BF) ironmaking process with strong nonlinear time-varying dynamics. Different from the existing linear SI algorithms, the R-B-SI algorithm can make full use of the process data information by adding the Kronecker product term of input data and the Kronecker product term between input and output data into the data block Hankel matrices. Moreover, the parameters of the R-B-SI model can be updated online with the latest data using recursive learning. In order to adjust the sensitivity of the algorithm to the data at different sampling times, the forgetting factor is introduced into the involved recursive learning as well. Therefore, the nonlinear time-varying dynamics of process can be well captured and described by the R-B-SI model. As a result, the data-driven predictive controller designed by the R-B-SI prediction model with parameter adaption is always capable of achieving satisfactory control performances regardless of the variation in various process operating conditions. Data experiments using actual industrial data have verified the practicability and superiority of the proposed methods.

Passivity-Based Iterative Learning Control for Cycling Induced by Functional Electrical Stimulation With Electric Motor Assistance

This brief examines the use of a learning control method in a passivity-based framework to control a motorized cycle-rider system with functional electrical stimulation (FES) of the quadriceps muscle groups. FES cycling with motorized assistance has been used in the rehabilitation of people with neurological conditions. The concepts of adaptation and passivity are explored to compensate for the uncertain nonlinear time-varying dynamics of the motorized FES cycle-rider system. The system is modeled as a closed-loop feedback, state-dependent switched system such that in each cycle, the quadriceps muscle groups produce the functional torque and the electric motor provides assistance as needed. The output strictly passive feature of the closed-loop system is proven by considering a learning control input. Then, an adaptive update law, based on iterative learning control, is developed to guarantee the convergence of the cadence tracking error. Experimental results from seven able-bodied participants are presented and discussed to demonstrate the effectiveness of this approach. The average cadence tracking error is 0.00 ± 2.47 rpm for the desired trajectory of 50 rpm.

Robust Online Sequential RVFLNs for Data Modeling of Dynamic Time-Varying Systems With Application of an Ironmaking Blast Furnace.

By dealing with robust modeling and online learning together in a unified random vector functional-link networks (RVFLNs) framework, this paper presents a novel robust online sequential RVFLNs for data modeling of dynamic time-varying systems together with its application for a blast furnace (BF) ironmaking process. First, to overcome the difficulties caused by the nonlinear time-varying dynamics of process and to enable the RVFLNs to learn online and to avoid data saturation, an improved online sequential version of RVFLNs (OS-RVFLNs) is presented by sequential learning with forgetting factor. It has been shown that the improved OS-RVFLNs with forgetting factor is not only suitable for the large-scale and real-time data transfer situation but also can adjust the sensitivity of the algorithm to different samples. Second, in order to solve the issue of modeling robustness when the dataset is contaminated with various outliers, a Cauchy distribution function weighted M-estimator is introduced to strengthen the robustness of the improved OS-RVFLNs. The non-Gaussian Cauchy distribution function is used to estimate the weights of different data and thus the corresponding contribution on modeling can be properly distinguished. Experiments using actual industrial data of a large BF ironmaking process have demonstrated that the proposed algorithm produces a much stronger robustness and better estimation accuracy than other algorithms.

Adaptive fuzzy integral sliding mode control of blood glucose level in patients with type 1 diabetes: In silico studies

Currently, artificial pancreas is an alternative treatment instead of insulin therapy for patients with type 1 diabetes mellitus. Closed-loop control of blood glucose level (BGL) is one of the difficult tasks in biomedical engineering field due to nonlinear time-varying dynamics of insulin-glucose relation that is combined with time delays and model uncertainties. In this paper, we propose a novel adaptive fuzzy integral sliding mode control scheme for BGL regulation. System dynamics is identified online using fuzzy logic systems. The presented method is evaluated in silico studies by nine different virtual patients in three different groups for two continuous days. Simulation results demonstrate effective performance of the proposed control scheme of BGL regulation in presence of simultaneous meal and physical exercise disturbances. Comparison of the proposed control method with proportional–integral–derivative (PID) control and model predictive control (MPC) shows a superiority of the adaptive fuzzy integral sliding mode control with regard to two conventional methods of BGL regulation (PID and MPC) and sliding mode control.

On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems

This paper considers the filtering problem for a class of multi-input multi-output systems with nonlinear time-varying uncertain dynamics, random process andmeasurement noise. An extended state based Kalman filter, with the idea of timely estimating the unknown dynamics, is proposed for better robustness and higher estimation precision. The stability of the proposed filter is rigorously proved for nonlinear time-varying uncertain system with weaker stability condition than the extended Kalman filter, i.e., the initial estimation error, the uncertain dynamics and the noises are only required to be bounded rather than small enough. Moreover, quantitative precision of the proposed filter is theoretically evaluated. The proposed algorithm is proved to be the asymptotic unbiased minimum variance filter for constant uncertainty. The simulation results of some benchmark examples demonstrate the feasibility and effectiveness of the method.

Flight control system design for hypersonic reentry vehicle based on LFT–LPV method

An angle-of-attack tracking control system is designed for the hypersonic reentry vehicle, whose aerodynamic parameters vary dramatically during reentry phase. The linear parameter-varying (LPV) theory based on linear fractional transformation (LFT) model (named as LPV–LFT method) is applied to design the controller for hypersonic reentry vehicle. Longitudinal moment trim of the hypersonic reentry vehicle is made along the desired flight trajectory, and a damping feedback loop is firstly designed to improve the system’s damping and static stability. Then, the linear dynamics model with damping feedback loop is established in LFT structure and treated as the controlled plant, and a parameter-varying reference model is utilized to guarantee the transient performance. The effectiveness of the proposed angle-of-attack tracking control system is validated through the frequency domain analysis and step response simulations. Finally, the actual angle-of-attack command tracking simulations using the nonlinear time-varying mathematical dynamics model are carried out to verify the accuracy and robustness of the hypersonic reentry vehicle control system.

Nonlinear Elastodynamic Behaviour Analysis of High-Speed Spatial Parallel Coordinate Measuring Machines

In order to study the elastodynamic behaviour of 4- universal joints- prismatic pairs- spherical joints / universal joints- prismatic pairs- universal joints 4-UPS-UPU high-speed spatial PCMMs(parallel coordinate measuring machines), the nonlinear time-varying dynamics model, which comprehensively considers geometric nonlinearity and the rigid-flexible coupling effect, is derived by using Lagrange equations and finite element methods. Based on the Newmark method, the kinematics output response of 4-UPS-UPU PCMMs is illustrated through numerical simulation. The results of the simulation show that the flexibility of the links is demonstrated to have a significant impact on the system dynamics response. This research can provide the important theoretical base of the optimization design and vibration control for 4-UPS-UPU PCMMs.

Moving mass control system in conjunction with brain emotional learning-based intelligent control for rate regulation of suborbital reentry payloads

An adaptive intelligent control strategy based on a brain emotional learning model is investigated in the application of the rate regulation of suborbital reentry payloads. Because of nonlinear time-varying dynamics of these payloads, choosing an appropriate control mechanism and stability strategy can be an engineering challenge. Thus in a new approach, a moving mass control system in conjunction with brain emotional learning-based intelligent control is used to fulfill payload de-tumbling. The contribution of brain emotional learning-based intelligent control in handling the nonlinear time-varying dynamics is shown by comparison with results obtained from a linear proportional–integral controller. The results demonstrate excellent performance of brain emotional learning-based intelligent control in learning dynamic couplings and improvement of behavior without any considerable control system complexity.

Robust control of a nonlinear non-minimum phase supersonic flight vehicle based on stable system center

Design and synthesis of a nonlinear time-varying and non-minimum phase supersonic flight vehicle longitudinal dynamics control for g-commands output tracking are presented. Differential geometric approach utilized to outline system relative degree and unstable internal dynamics explored based on the vehicle dynamics transformation into a nonlinear normal form. Unstable internal dynamics are then invoked to identify ideal state reference profiles leading to a desired output-tracking problem. Compensated internal dynamics converged to a solution on the system center manifold for ideal internal dynamics asymptotically. Finally, sliding mode control employed based on an appropriate sliding manifold to track the reference input while being robust against uncertainties. Simulation results are performed and effectiveness of the designed approach demonstrated.

Rate regulation of a suborbital reentry payload by moving-mass actuators

In order to save cost and time, sounding rockets are effectively used to develop space technologies. In a biological payload that is under the study on this investigation, reentry rate regulation is one of the critical issues to be solved. Because of non-linear time-varying dynamics of these payloads and presence of high aerodynamic disturbances during reentry, choosing an appropriate stability and control mechanism can be an engineering challenge. Having a lot of benefits, nowadays, moving-mass actuation systems are used in a variety of aerospace applications. As an innovative approach, a moving-mass system is designed and analysed to regulate the payload body rates during reentry phase. Simulation results indicate significant performance of suggested application for moving-mass control systems.

Switched Predictive Control Design for Optimal Wet-clutch Engagement

Identification and control of wet-clutches has been recognized as a challenging control problem, due to their non-linear time-varying dynamics and their discontinuous switching between filling and slipping phases. In this paper, our approach is based on a full (non)linear system identification of the filling and slipping phases, followed by two closed loop model predictive controllers (MPC) for the two phases and switching between these controllers at suitable moments for optimal torque transfer. Further, it is shown that an intelligent formulation of the switching function is beneficial to the robustness against time-varying model parameters. Finally, our control technique is validated on a real-life wet-clutch test-bench to obtain optimal engagement.

Modeling and fuzzy control of the engine coolant conditioning system in an IC engine test bed

Mechanical and thermodynamical performance of internal combustion engines is significantly affected by the engine working temperature. In an engine test bed, the internal combustion engines are tested in different operating conditions using a dynamometer. It is required that the engine temperature be controlled precisely, particularly in transient states. This precise control can be achieved by an engine coolant conditioning system mainly consisting of a heat exchanger, a control valve, and a controller. In this study, constitutive equations of the system are derived first. These differential equations show the second-order nonlinear time-varying dynamics of the system. The model is validated with the experimental data providing satisfactory results. After presenting the dynamic equations of the system, a fuzzy controller is designed based on our prior knowledge of the system. The fuzzy rules and the membership functions are derived by a trial and error and heuristic method. Because of the nonlinear nature of the system the fuzzy rules are set to satisfy the requirements of the temperature control for different operating conditions of the engine. The performance of the fuzzy controller is compared with a PI one for different transient conditions. The results of the simulation show the better performance of the fuzzy controller. The main advantages of the fuzzy controller are the shorter settling time, smaller overshoot, and improved performance especially in the transient states of the system.

Observer-Based Robust Synchronization of Dynamical Systems

The paper deals with the problem of robust synchronization of dynamical systems. The design procedure is based on the concept of observers with absolutely stable error dynamics. In the general case of nonlinear time-varying error dynamics the procedure requires exact knowledge of a Lyapunov function while in case of the linearizable error dynamics frequency domain conditions which ensure existence of such a function can be employed. Two examples are considered: synchronization of two Lorenz systems and Rössler systems.