Nonlinear State Space Equations Research Articles

Dynamic safety control of offshore wind turbine based on model predictive control

Offshore wind power is a clean energy source with excellent development prospects that are conducive to realising China's goal of carbon neutrality. However, offshore wind turbines (WTs) are prone to failure and damage owing to their long-term operation in marine environments with high humidity, high wind speeds, and frequent extreme weather. To ensure the safe operation of offshore WTs, a modified dynamic model predictive control (MPC) method was proposed in this study. First, a dynamic model of an offshore WT was established and its nonlinear state-space equations were obtained. These state-space equations were subsequently linearised and discretised, then fast trajectory tracking of the state variables was performed according to the conventional MPC principle. Finally, a control strategy was designed to realise the safe operational control of offshore WT using the safety index as a constraint condition in the MPC. Simulation results confirmed that the proposed MPC strategy using the safety index-based constraint exhibited superior tracking of the set trajectory and achieved the minimum energy loss under varying wind speeds.

Adaptive hybrid Kalman filter for attitude motion parameters estimation of space non-cooperative tumbling target

Attitude motion parameters estimation of the space non-cooperative tumbling target is the premise of the on-orbit target capture task. However, existing methods cannot balance the estimation accuracy and efficiency. In this paper, an adaptive hybrid Kalman filter is presented to estimate the attitude, angular velocity and inertia motion parameters of the space non-cooperative tumbling target. First, the dynamic model of the non-cooperative tumbling target is derived, and the nonlinear discrete state-space equation is obtained. Then, a hybrid Kalman filter is designed by combining the extended Kalman filter and the unscented Kalman filter. The adaptive technique is introduced to track time-varying uncertain measurement noise caused by complex space disturbances. After that, based on the first-order linearization technique, the sufficient condition for the local convergence of the proposed method is proved. Finally, the numerical simulation experiment was conducted to compare the proposed method with the adaptive extended Kalman filter and the adaptive unscented Kalman filter. The experimental results show that the estimation accuracy of the proposed method is much higher than that of the adaptive extended Kalman filter and slightly lower than that of the adaptive unscented Kalman filter, and the iteration-average Central-Processing-Unit time is 64.2% smaller than that of the adaptive unscented Kalman filter and slightly longer than that of the adaptive extended Kalman filter. The proposed method achieves a much higher accuracy with a slight loss of iteration-average Central-Processing-Unit time.

Turbo-shaft engine adaptive neural network control based on nonlinear state space equation

Intelligent Adaptive Control (AC) has remarkable advantages in the control system design of aero-engine which has strong nonlinearity and uncertainty. Inspired by the Nonlinear Autoregressive Moving Average (NARMA)-L2 adaptive control, a novel Nonlinear State Space Equation (NSSE) based Adaptive neural network Control (NSSE-AC) method is proposed for the turbo-shaft engine control system design. The proposed NSSE model is derived from a special neural network with an extra layer, and the rotor speed of the gas turbine is taken as the main state variable which makes the NSSE model be able to capture the system dynamic better than the NARMA-L2 model. A hybrid Recursive Least-Square and Levenberg-Marquardt (RLS-LM) algorithm is advanced to perform the online learning of the neural network, which further enhances both the accuracy of the NSSE model and the performance of the adaptive controller. The feedback correction is also utilized in the NSSE-AC system to eliminate the steady-state tracking error. Simulation results show that, compared with the NARMA-L2 model, the NSSE model of the turbo-shaft engine is more accurate. The maximum modeling error is decreased from 5.92% to 0.97% when the LM algorithm is introduced to optimize the neural network parameters. The NSSE-AC method can not only achieve a better main control loop performance than the traditional controller but also limit all the constraint parameters efficiently with quick and accurate switching responses even if component degradation exists. Thus, the effectiveness of the NSSE-AC method is validated.

Observer-Based Approximate Affine Nonlinear Model Predictive Controller for Hydraulic Robotic Excavators with Constraints

Given the highly nonlinear and strongly constrained nature of the electro-hydraulic system, we proposed an observer-based approximate nonlinear model predictive controller (ANMPC) for the trajectory tracking control of robotic excavators. A nonlinear non-affine state space equation with identified parameters is employed to describe the dynamics of the electro-hydraulic system. Then, to mitigate the plant-model mismatch caused by the first-order linearization, an approximate affine nonlinear state space model is utilized to represent the explicit relationship between the output and input and an ANMPC is designed based on the approximate nonlinear model. Meanwhile, the Extended Kalman Filter was introduced for state observation to deal with the unmeasurable velocity information and heavy measurement noises. Comparative experiments are conducted on a 1.7-ton hydraulic robotic excavator, where ANMPC and linear model predictive control are used to track a typical excavation trajectory. The experimental results provide evidence of convincing trajectory tracking performance.

Finite-time fault detection and reconstruction of permanent magnet synchronous generation wind turbine via sliding mode observer

In this study, a method for fault detection of a permanent magnet synchronous generator (PMSG) wind turbine is proposed. The dynamic model of the wind energy conversion system is simulated and the system’s fault is diagnosed using a proposed terminal sliding mode observer (TSMO). The nonlinear state-space equations of the closed-loop wind turbine are derived, which include a robust adaptive controller that is implemented for maximum power capture and a PI controller that is designed for pitch angle control. Then, utilising a new approach, a TSMO is designed for estimating the states and faults in a finite time. The stability and convergence of the states and fault estimation error are investigated via the Lyapunov stability theory. A PMSG wind turbine is simulated and using TSMO the states are estimated and a blade imbalance fault (BIF) fault is detected. The state and fault estimation errors converge to zero in a finite time.

Initialization Approach for Nonlinear State-Space Identification via the Subspace Encoder Approach

The SUBNET neural network architecture has been developed to identify nonlinear state-space models from input-output data. To achieve this, it combines the rolled-out nonlinear state-space equations and a state encoder function, both parameterised as neural networks The encoder function is introduced to reconstruct the current state from past input-output data. Hence, it enables the forward simulation of the rolled-out state-space model. While this approach has shown to provide high-accuracy and consistent model estimation, its convergence can be significantly improved by efficient initialization of the training process. This paper focuses on such an initialisation of the subspace encoder approach using the Best Linear Approximation (BLA). Using the BLA provided state-space matrices and its associated reconstructability map, both the state-transition part of the network and the encoder are initialized. The performance of the improved initialisation scheme is evaluated on a Wiener-Hammerstein simulation example and a benchmark dataset. The results show that for a weakly nonlinear system, the proposed initialisation based on the linear reconstructability map results in a faster convergence and a better model quality.

쿼드로터 UAV 의 PPO 기반 모델 참조 추종 제어

This paper presents a proximal policy optimization (PPO)-based model reference tracking controller design for quadrotor unmanned aerial vehicles (UAVs). First, the quadrotor UAV is divided into the attitude and position systems, and each system is expressed as a nonlinear state-space equation. Thereafter, we design a linear reference model, which is used to derive the tracking error dynamics. The proposed neural network model implements a PPO-based reinforcement learning algorithm consisting of an actor approximating a policy and a critic corresponding to a state-value function. The actor receives the state variables of the tracking error dynamics as input and outputs the thrust values to be applied to each axis. Further, we decentralize the PPO-based controller into the attitude and position controllers, which are then trained separately. For learning purposes, we implement an environment code that expresses the tracking error dynamics of the quadrotor by extending the OpenAI Gym environment. Finally, a simulation example is provided to show the position tracking performances of up to 0.0166 m and 0.0254 m for the horizontal and vertical axes, respectively.

Maximum likelihood-based extended Kalman filter for soft tissue modelling

Realistic modelling of human soft tissue is very important in medical applications. This paper proposes a novel method by dynamically incorporating soft tissue characterisation in the process of soft tissue modelling to increase the modelling fidelity. This method defines nonlinear tissue deformation with unknown mechanical properties as a problem of nonlinear filtering identification to dynamically identify mechanical properties and further estimate nonlinear deformation behaviour of soft tissue. It combines maximum likelihood theory, nonlinear filtering and nonlinear finite element method (NFEM) for modelling of nonlinear tissue deformation behaviour based on dynamic identification of homogeneous tissue properties. On the basis of hyperelasticity, a nonlinear state-space equation is established by discretizing tissue deformation through NFEM for dynamic filtering. A maximum likelihood algorithm is also established to dynamically identify tissue mechanical properties during the deformation process. Upon above, a maximum likelihood-based extended Kalman filter is further developed for dynamically estimating tissue nonlinear deformation based on dynamic identification of tissue mechanical properties. Simulation and experimental analyses reveal that the proposed method not only overcomes the NFEM limitation of expensive computations, but also absorbs the NFEM merit of high accuracy for modelling of homogeneous tissue deformation. Further, the proposed method also effectively identifies tissue mechanical properties during the deformation modelling process.

Active motor damping strategy for driveline vibrations of a hybrid electric vehicle during ABS operation

Torsional drivetrain vibrations in hybrid and electric drivetrains, which occur during traction and braking control, are commonly encountered. This problem occurs mainly due to the weak damping characteristic of the hybrid/electric drivetrain and the fast time response characteristics of the electric motor. An active motor damping (AMD) control algorithm for a hybrid electric vehicle, considering ABS braking maneuvers on different road conditions, is developed in this work. The control problem is structured such that the control objective is disturbance rejection against the estimated brake torque signal received from the ABS module. Communication delay between the ABS and motor control module is handled within the control strategy. State feedback control strategy is used with the linearized version of the non-linear state-space equations together with an Extended Kalman filter. Gear backlash is taken into account without the need to estimate the mode of backlash. In the development of the controller, state feedback gains are set such that the ABS functionality on tire slip regulation is not altered. Existing studies in the literature feedback only the angle of twist and axle wrap angular speed for damping the vibrations. The structure of the controller in this study is different in that it estimates and feeds back tire slip, in addition to these two. Simulation results show the effectiveness of the controller in reducing the angle of twist without deteriorating ABS tire slip control and braking performance of the vehicle.

Reduced-Order Extended Kalman Filter for Deformable Tissue Simulation

Modelling of soft tissue deformation is a key issue in surgical simulation. Despite extensive research studies on this issue, accurate modelling of soft tissue deformation in run time still remains challenging. This paper proposes a new reduced-order nonlinear Kalman filter to emulate nonlinear behaviors of biological deformable tissues. This approach defines the deformable modelling problem as a reduced-order filtering problem to accurately calculate soft tissue deformation in real time. Soft tissue deformation is discretized in space using nonlinear finite element method based on hyperelasticity and further formulated as a nonlinear state-space equation for filtering estimation. Subsequently, the order of this nonlinear state-space equation is reduced using proper orthogonal decomposition to reduce the computational cost. Upon this reduced-order state-space equation, an extended Kalman filter is constructed to online calculate nonlinear behaviors of tissue physical deformation. Simulation results and comparison analysis prove the effectiveness of the suggested method for accurate simulation of tissue physical deformation in real time.

Maximum likelihood-based extended Kalman filter for COVID-19 prediction.

Prediction of COVID-19 spread plays a significant role in the epidemiology study and government battles against the epidemic. However, the existing studies on COVID-19 prediction are dominated by constant model parameters, unable to reflect the actual situation of COVID-19 spread. This paper presents a new method for dynamic prediction of COVID-19 spread by considering time-dependent model parameters. This method discretises the susceptible-exposed-infected-recovered-dead (SEIRD) epidemiological model in time domain to construct the nonlinear state-space equation for dynamic estimation of COVID-19 spread. A maximum likelihood estimation theory is established to online estimate time-dependent model parameters. Subsequently, an extended Kalman filter is developed to estimate dynamic COVID-19 spread based on the online estimated model parameters. The proposed method is applied to simulate and analyse the COVID-19 pandemics in China and the United States based on daily reported cases, demonstrating its efficacy in modelling and prediction of COVID-19 spread.

Parallel distributed identification for block structure nonlinear system

The problem of identifying one block structure nonlinear system, consisting of four linear blocks and one static nonlinear block, is considered here. Under the assumptions that static nonlinear is a polynomial form, the block structure nonlinear system can be transformed into one state space form. Then this nonlinear system identification is formulated into identifying the state space form for these four linear blocks. After transforming the formal state space form, we obtain a polynomial nonlinear state space equation. Then the parallel distributed algorithm is used to identify each system matrix, while using the relationship between the formal state space equation and its transformed state space equation. Finally the simulation example is used to prove the efficiency of our algorithm.

WITHDRAWN: Deep learning for Koopman Operator Optimal Control

Nonlinear dynamics are ubiquitous in complex systems. Their applications range from robotics to computational neuroscience. In this work, the Koopman framework for globally linearizing nonlinear dynamics is introduced. Under this framework, the nonlinear observable states are lifted into a higher dimensional, linear regime. The challenge is to identify functions that facilitate the coordinate transformation to this raised linear space. This point is tackled using deep learning, where nonlinear dynamics are learned in a model-free manner, i.e., the underlying dynamics are uncovered using data rather than the nonlinear state-space equations. The main contributions include an implementation of the Linearly Recurrent Encoder Network (LREN) that is faster than the existing implementation and is significantly faster than the state-of-the-art deep learning-based approach. Also, a novel architecture termed Deep Encoder with Initial State Parameterization (DENIS) is proposed. By deriving an energy-budget control performance evaluation method, we demonstrate that DENIS also outperforms LREN in control performance. It is also on-par with and sometimes better than the iterative linear quadratic regulator (iLQR), which requires access to the state-space equations. Extensive experiments are done on DENIS to validate its performance. Also, another novel architecture termed Double Encoder for Input Nonaffine systems (DEINA) is described. Additionally, DEINA's potential ability to outperform existing Koopman frameworks for controlling nonaffine input systems is shown. We attribute this to using an auxiliary network to nonlinearly transform the inputs, thereby lifting the strong linear constraints imposed by the traditional Koopman approximation approach. Koopman model predictive control (KMPC) is implemented to verify that our models can also be successfully controlled under this popular approach. Overall, we demonstrate the deep learning-based Koopman framework shows promise for optimally controlling nonlinear dynamics.

A New Dynamic Stall Approach for Investigating Bifurcation and Chaos in Aeroelastic Response of a Blade Section with Flap Free-Play Section

This paper investigates the effects of the unsteady nonlinear aerodynamic, plunge/pitch cubic nonlinearities, flap free-play nonlinearity, and coupled nonlinear aeroelasticity on the dynamics of the three-dimensional blade section. The dynamic stall model is developed based on the unsteady Wagner aerodynamics. Coupling the developed nonlinear aerodynamic model and nonlinear elasticity model results in the nonlinear aeroelastic model. The nonlinear aeroelastic equation of motion is converted into a state-space form. The resulting nonlinear state-space equation of motion is simulated by a standard Runge–Kutta algorithm in MATLAB. The proposed model is validated against test data of distinct two- and three-degrees-of-freedom studies and is compared to the ONERA model. Bifurcation diagrams show that there is distinct airspeed, in which the system experiences limit cycle oscillations (LCOs) or chaos. Both hysteresis air loads and structural nonlinearity make the system unstable at airspeed less than linear flutter speed. The nonlinearity of the structure causes supercritical pitchfork bifurcation. Elastic-aerodynamic nonlinearity interaction causes sub-supercritical bifurcation at the lower airspeed and chaotic motion at a higher airspeed. Furthermore, the effects of the initial condition on the response of the nonlinear aero-servo-elastic system are investigated by the Lyapunov exponent method.

Nonlinear Control System Design for Active Lubrication of Hydrostatic Thrust Bearing

The active controlled hydrostatic bearing is becoming more and more popular because of its accuracy, safety, as well as low vibration and noise. In this paper, we present a design approach for a hydrostatic thrust bearing system, where the analytical nonlinear state space equation of the system is established first, and then three kinds of control inputs are investigated and compared to each other. It is found that, by selecting the supply pressure as the control input, we could obtain an affine nonlinear system, which could be linearized by the feedback linearization method, and its robustness could be enhanced by the sliding mode control method. The tracking control law could be easily obtained with the linearized system. The simulation verifies the effectiveness of the nonlinear control law. The proposed nonlinear control model might have a positive effect on the improvement of the machining accuracy, safety, and vibration absorption.

On-line calibration of spectroscopic sensors based on state observers

Spectroscopic sensors provide on-line information about process variables and they have been widely used for monitoring and control. These sensors measure the spectral responses at a large number of wavelengths correlated with the process variables of interest. However the spectral measurement can also be affected by external factors such as changes in temperature. In order to estimate the process variables from the acquired spectrum it is necessary the use multivariate calibration methods. Additive effects of external factors can be easily compensated by standard calibration methods, but multiplicative effects require complex off-line calibration procedures. This work, shows that this problem can be modeled by a non-linear state space equation. In addition, it also proposes an on-line calibration method based on a state observer for compensating multiplicative effects and at the same time estimating the desired process variable from the spectrum. The convergence of the observer requires a uniform observability condition to be satisfied. Simulation results obtained by using a spectral sensor for monitoring a mixing process under time-varying temperature show the main features and potential of the proposed approach. More complex spectral models for modeling the effect of temperature and other variables can be considered and included in the proposed framework.

Darbeli DC Sinterleme Sistemi Konteynerinin Soğumasına Yönelik Termal Devre Modeli

Pulse DC Sintering System (PDCS) is a cheap and quick way of producing different types of materials. After sintering of the sample, the cooling of the PDCS container is needed before taking it out. The sample production time is the sum of the sintering and the cooling time. Therefore, estimation of the sample cooling time must be made accurately to model sample production. Its container dimensions and the material, it is made of, determines its temperature during cooling. In this paper, a thermal circuit which models a pulse DC sintering system container during cooling is given. Its thermal circuit model is made assuming that some heat leaks from the steel container to the cooper bars, the copper bars and the container all have natural convection and also radiate heat to cool down. The thermal model is described with a set of nonlinear state-space equations. The state-space equations are solved numerically using Runge-Kutta 4 method. The time required to make the sample cool down to ambient temperature is calculated using simulations. The temperatures of the container and the copper bars of an PDCS system are measured to find the experimental cooling time. The results are compared. The PDCS thermal model is able to verify the experimental results. It has also been experimentally shown that the cooling time is not dependent on the sample type produced for the examined PDCS system. Such a model can be easily implemented in an engineering software which aims to model the sample production process of the PDCS system and can also be used for its optimization considering its physical parameters such as dimensions, electrical and mechanical constants etc.

Controlling opinions in Deffuant model by reconfiguring the network topology

This paper proposes a topology reconfiguration approach for improving the rate of convergence of opinions to the opinion of a pre-specified leader for the class of convergent Deffuant models with a limited number of links. From a systems theory perspective, this problem can be viewed as a constrained stochastic nonlinear on–off control problem. Accordingly, we first propose a deterministic version of the Deffuant model and rewrite its dynamic equations to reach a set of nonlinear state-space equations where opinions are state variables and link connectivities are inputs. For that model, we then design an on–off controller based on a short-sighted predictive control strategy that dynamically changes the topology of the network by a low computational burden process. Results confirm that the proposed control strategy reaches faster convergence rates of opinions to the leader’s opinion in comparison with the well-known Erdős–Rényi structure with a similar number of links. The proposed control strategy also provides a higher rate of link connectivity for the links that are connected to the leader. Furthermore, it is observed that if the network has a fixed topology based on the obtained rate of link connectivity, it will still have a relatively rapid convergence rate which is comparable with that of a fully connected topology.

A thermal model for calculating axial temperature distribution of overhead conductor under laboratory conditions

Most traditional thermal models of overhead conductor neglect the axial temperature distribution along the length of conductor. In order to provide a necessary foundation for calculating axial conductor temperature distribution in real weather conditions, a thermal model is proposed for the axial conductor temperature calculation under laboratory environment. This proposed model is established with the equivalent thermal network, and expressed as a nonlinear state-space equation. An identification method is presented to determine the conductor thermal parameters, which are non-weather related and hardly to be calculated theoretically. The interior-point algorithm is employed to optimize the objective function of parameter identification. The effectiveness of identification method and proposed thermal model are validated by experiment results. This study will provide the basic modelling principle, parameter identification method and conductor thermal parameters to develop the conductor thermal model under the real weather conditions in the future.

Nonlinear system identification of fractional Wiener models

The Wiener model is a case of block-oriented model, which is suitable for modeling a large class of nonlinear systems. It consists of the cascade of two parts, where a linear dynamic part is followed by a static nonlinear part. On the other hand, fractional order models have gained increasing interest over the last decades due to their better representation of some physical phenomena. In this paper, the fractional Wiener system identification is aimed. The polynomial nonlinear fractional state space equations are used to describe the nonlinear model. Thus, the extension of an output error method (Levenberg Marquard algorithm) is developed to estimate the fractional Wiener system. The efficiency of the identification method is evaluated and confirmed by simulation examples.