Linear Parameter-varying Model Research Articles

A global approach to estimate continuous-time LPV models for wastewater nitrification

In wastewater treatment, understanding and modeling the nitrification process is crucial to implement control. However, the complexity of this process makes it challenging to create simplified models. This study introduces an innovative method for estimating linear parameter-varying (LPV) models in the context of biological nitrification processes. The research focuses on the development of a continuous-time LPV model for a submerged aerated biofiltration system, considering the conversion of ammonium to nitrate in wastewater treatment. The methodology adopts the reinitialized partial moment approach within a global identification framework. The resultant LPV model is structured to capture the dynamics of the biological nitrification process, considering various factors like flow rates, feed concentrations and environmental regulations. Application of this approach to measured data from a wastewater treatment plant, demonstrates its effectiveness in accurately estimating the LPV model parameters. The results not only offer valuable insights into the dynamics and the nonlinear behaviour of the nitrification process but also contribute to the design and optimization of wastewater treatment plants, particularly those employing submerged aerated nitrifying biofilters.

A Comparison of Two Methods of Adaptive Nonlinear Model Predictive Control

Conventional nonlinear model predictive control (NMPC) relies on an accurate process model. However, real-world systems’ models are often imperfect due to parametric variations, modelling errors and additive noise, leading to degraded control performance. This study investigates the effectiveness of two adaptive NMPC methods in solving this problem, namely, linear parameter varying (LPV) model predictive control (LPV-MPC) and model predictive control based on successive linearization (SL-MPC). In addition, an innovative approach for identifying LPV models is proposed and applied to three simulation examples. The identified LPV models gave a very strong fit. Also, simulation results demonstrate that both adaptive predictive NMPC (LPV-MPC and SL-MPC) exhibit performance similar to conventional NMPC and superior to Linear MPC. Notably, the two adaptive predictive controllers offer significantly reduced computational time compared to conventional NMPC.

An information theory approach for recursive LPV-ARX model identification via LS-SVM

When modeling a dynamical system, linear models are always the first choice due to their simplicity. However, many times the system is too complex so that employing linear models results in poor performances. In this context, Linear Parameter Varying (LPV) models allow to represent non-linear input/output relationships while preserving the simple structure of linear models. The method of Least Squares Support Vector Machines (LS-SVM) is one of the most common approaches to identify a LPV model in an ARX formulation (LPV-ARX). However, due to its computational cost, it is difficult to identify a LPV-ARX model using LS-SVM in online applications, where the model must be updated every time new data are collected. An efficient update algorithm has been presented for online identification of such models, where the idea is to update the model only upon certain data that are considered informative. However, this approach requires the tuning of some hyperparameters and in certain conditions can stop updating the model even when data are informative. This paper proposes an information-based algorithm to overcome these drawbacks. Evaluation on simulated and experimental data show the effectiveness of the proposed approach on both identification and computational sides.

Learning Reduced-Order Linear Parameter-Varying Models of Nonlinear Systems

In this paper, we consider the learning of a Reduced-Order Linear Parameter-Varying Model (ROLPVM) of a nonlinear dynamical system based on data. This is achieved by a two-step procedure. In the first step, we learn a projection to a lower dimensional state-space. In step two, an LPV model is learned on the reduced-order state-space using a novel, efficient parameterization in terms of neural networks. The improved modeling accuracy of the method compared to an existing method is demonstrated by simulation examples.

Linear Parameter-Varying Model Predictive Control for Intelligent Energy Management in Battery/Supercapacitor Electric Vehicles

Linear Parameter-Varying Model Predictive Control for Intelligent Energy Management in Battery/Supercapacitor Electric Vehicles

Application of gap metric for model selection in linear parameter varying model-based predictive control

Selecting the minimum number of linear models required to characterize a nonlinear dynamic system in the local approach to linear parameter varying (LPV) model identification remains a challenge. This study proposes the use of the gap metric for such selection. Subsequently, the identified LPV model is proposed for use in the design of LPV-based model predictive control (MPC). The results of the validation experiment reveal that the identified gap-based local and global LPV models provide a good fit, which is superior to that of any individual linear model, as measured by the computed mean squared error values. Furthermore, the closed-loop results show that the performance of the LPV-MPC closely matches that of the full-blown nonlinear MPC and multi-MPC, while LPV-MPC clearly outperforms Linear MPC. Additionally, the proposed LPV-MPC is found to be robust to process model mismatches and measurement noise, and is much less computationally intensive than nonlinear MPC.

An improved linear parameter-varying modeling, model order reduction, and control design process for flexible aircraft

This paper presents an improved process for the modeling, model order reduction (MOR), and control design of a flexible flying wing unmanned aerial vehicle based on the linear parameter-varying (LPV) technology. By applying the Lagrangian formulation and doublet lattice method, the aeroservoelastic (ASE) model of the flexible aircraft is first derived and then rewritten as an airspeed-dependent LPV system, which consists of rigid-body states, structural modes, aerodynamic states, etc. This leads to a high-order LPV model, and the stability of the system varies from stable to unstable when the speed of the aircraft exceeds the critical flutter speed. To facilitate the control design, an oblique projection-based MOR method is then adopted to reduce the order of the plant. Different from previous studies, the MOR method is improved to systematically generate a reduced-order LPV model with consistent states for both the stable and unstable dynamics parts. Finally, an output-feedback LPV controller is designed to simultaneously achieve the body freedom flutter suppression and longitudinal attitude control. Numerical simulation results show the effectiveness of the improved LPV-based ASE modeling, order reduction, and control design process for the flexible aircraft.

Open-Loop Control Co-Design of Semisubmersible Floating Offshore Wind Turbines Using Linear Parameter-Varying Models

Abstract This paper discusses a framework to design elements of the plant and control systems for floating offshore wind turbines in an integrated manner using linear parameter-varying models. Multiple linearized models derived from aero-elastic simulation software in different operating regions characterized by the incoming wind speed are combined to construct an approximate low-fidelity model of the system. The combined model is then used to generate open-loop, optimal control trajectories as part of a nested control co-design strategy that explores the system’s power production and stability using the platform pitch tilt as a proxy in the context of crucial plant and control design decisions. The radial distance between the central and outer columns and the diameter of the outer columns of the semisubmersible platform are the plant design variables. The platform stability and power production are studied for different plant design decisions. The effect of plant decisions on subsequent power production and stability response of the floating wind turbine is quantified in terms of the levelized cost of energy. The results show that the inner-loop constraints and the plant design decisions affect the turbine’s power and, subsequently, the cost of the system.

Linear parameter-varying model predictive control for nonlinear systems using general polytopic tubes

This paper deals with model predictive control (MPC) for nonlinear systems using linear-parameter varying (LPV) embedding of the nonlinear dynamics (LPVMPC). The proposed LPVMPC can incorporate information of the future evolution of the scheduling parameter over the MPC prediction horizon with uncertainty bounds, which are used to construct anticipated scheduling tubes for robustification. Therefore, this approach is less conservative than the methods that only consider knowledge of the bounds on the scheduling parameter’s rate of variation. The scheduling tubes are employed to synthesize online general polytopic invariant state tubes. The optimization problem of the proposed LPVMPC is a single quadratic program. Recursive feasibility is proven. A numerical example is presented for demonstrating the effectiveness of the proposed LPVMPC algorithm compared to nonlinear MPC and other standard approaches.

Robust control of gust-induced vibration of highly flexible aircraft

This paper mainly addresses the gust suppression and alleviation of highly flexible aircraft using the model predictive controller (MPC) based on linear parameter-varying (LPV) models. The dynamic behavior of highly flexible aircraft is modeled by a coupled nonlinear aeroelastic and flight dynamic formulation. Conventional trailing-edge flaps along the main wings and tails are deployed to provide the required loads for the wing vibration control and/or longitudinal flight attitude control of the slender vehicle. The coupled dynamic equations are performed with linearization and model reduction about a series of nonlinear equilibria to derive reduced-order LPV models. This work considers two quantities as the scheduling parameters in building the LPV models: the gust-induced angle of attack and the modal magnitude of wing deformation. With the reduced-order LPV model, MPC is designed to minimize the wing vibration and rigid-body motions excited by the time-domain Dryden gust perturbation. The proposed LPV modeling is capable of describing large wing deformations, and the LPV-MPC innovatively previews the scheduling parameter and gust disturbance in the prediction horizon. The closed-loop flight responses of a highly flexible aircraft with gust disturbance are presented in the numerical studies, which demonstrate the effectiveness of controlling coupled vibrations and rigid-body dynamics of such slender vehicles.

A novel state-of-health notion and its use for battery aging monitoring of zinc-air batteries

Accurate monitoring of battery aging is pivotal for optimizing the performance of zinc-air batteries (ZABs). To obtain precise information about battery aging, an appropriate definition of state-of-health (SOH) is essential. However, the definition of SOH for ZABs has not been well-defined. This work introduces a novel SOH concept based on energy efficiency and integrates it with an underlying linear parameter varying (LPV) model. Crucially, the LPV model is chosen for its capability to represent nonlinear behavior, making it apt for capturing the intricacies of battery aging dynamics. With SOH defined as the scheduling parameter, electrochemical impedance spectroscopy (EIS) is employed to assess the influence of SOH. The results validate the model's robustness, as its predictions align closely with the empirical data. This newly proposed SOH definition exhibits a strong association with the model parameters. Furthermore, the efficacy of the LPV technique, especially when paired with the proposed SOH, suggests significant potential for refining battery aging monitoring in ZABs.

Multi-Input Multi-Output Robust Control of Turbofan Engines Based on Off-Equilibrium Linearization Linear Parameter Varying Model

Abstract The conventional linear parameter varying (LPV) modeling for turbofan engines is performed at a set of equilibrium points, which may result in inaccurate description of dynamics of engines when the engine accelerates/decelerates in a broad range and operates away from the equilibria. Meanwhile, to ensure stable operation of turbofan engines, the engine pressure ratio (EPR) needs to be regulated when turning on afterburner. According to this complex multi-input multi-output (MIMO) problem mixed with disturbances, this paper proposes a hybrid H∞/H2 robust control framework based on off-equilibrium linearization. We first establish an off-equilibrium linearization based polynomial LPV model for the MIMO turbofan engine, and then design an H∞/H2 robust controller by solving sum-of-squares (SOS) programing. The simulation results show that the controller can simultaneously achieve fast dual command tracking and disturbance suppression when turning on afterburner. Furthermore, this control framework is superior to the equilibrium linearization-based controller in many aspects, which further proves the accuracy of the off-equilibrium based LPV model.

Longitudinal model identification of multi-gear vehicles using an LPV approach

This paper aims to provide a data-driven approach for modeling the longitudinal dynamics of a typical ground vehicle with a gasoline engine and automatic transmission. In the identification process, a Linear Parameter Varying (LPV) model is considered, whose inputs are throttle level and road grade and whose parameters vary as a function of throttle level and gear number to capture the nonlinear dynamics. Three parametric structures based on time-series modeling (ARX, ARMAX, BJ) are investigated, whose performances are discussed comparatively. In addition to selecting the best structure, an optimization problem is proposed to acquire the optimal model order of these structures. It is shown, using semi-experimental tests on the CarSim® software package, that the dynamics of different vehicles can best be represented by different model orders. To evaluate the versatility and utility of the proposed LPV model, a PI controller is tuned using the model, and the closed-loop performance of the system is compared against the model. It is shown that the LPV model is very accurate in closed-loop settings.

ROS Implementation of Planning and Robust Control Strategies for Autonomous Vehicles

Autonomous vehicles are rapidly emerging as a crucial sector within the automotive industry. Several companies are investing in the development and enhancement of technologies, which presents challenging problems in the context of robotics and control. In particular, this work primarily focuses on the creation of an autonomous vehicle architecture utilizing the robotic operating system (ROS2) framework, accompanied by advanced control algorithms. In order to facilitate the development and implementation of lateral vehicle dynamics controllers, a reduced-size automated car available in GIPSA-Lab is used as an experimental platform. The objective is to design robust controllers capable of achieving optimal tracking and stability. Accordingly, this problem is tackled under different robust control syntheses, considering the H∞ approach: using linear time-invariant (LTI) and linear parameter-varying (LPV) model representations. Several simulation and experimental results are included to demonstrate the efficiency of the controllers.

Reduced-order model based dynamic tracking for soft manipulators: Data-driven LPV modeling, control design and experimental results

We propose a generic nonlinear reduced-order tracking control method for elastic soft robots. To this end, a new linear parameter varying (LPV) control framework is developed using the data collected from the soft robots. Specifically, for LPV modeling we first derive a nonlinear robot model, which is large-scale by nature, using finite element methods (FEM). Then, a proper orthogonal decomposition (POD) algorithm is used to generate a set of linearized reduced-order models, representing the local behaviors of the soft robot at different operating points within the workspace. Via a unified POD projector, not only the order of these linearized models can be significantly reduced but also their mechanical structure and stability properties can be preserved for LPV modeling and control design. Next, using radial basis function (RBF) networks, we propose an iterative training method to build the LPV robot model by interpolating a subset of selected linearized models with a specified interpolation error. For LPV control design, the equivalent-input-disturbance (EID) concept is exploited to develop a dynamic tracking control scheme, which is composed of three core components: feedforward control, disturbance-estimator control and feedback control. The feedforward control is designed to account for the effects of the trajectory reference and the time-varying affine term, issued from the FEM-based model linearization. The disturbance-estimator control is obtained from a generalized proportional integral LPV observer, which also provides the estimated reduced-order states for feedback control. The observer-based feedback control design is reformulated as a convex optimization problem under linear matrix inequality (LMI) constraints. The globally uniformly ℓ∞ stability of the closed-loop LPV robot model is guaranteed by means of Lyapunov stability theory. Experimental tests are conducted with a soft Trunk robot, inspired by the elephant trunk, under several scenarios with small and large deformations to show the effectiveness of the proposed LPV tracking control framework. A comparative study is also performed with a recent linear EID-based controller and an iterative learning controller to emphasize the interests this nonlinear control method for soft elastic robots. This paper is complemented with a series of demonstration videos: https://bit.ly/3D8C4Vd.

A Generalized and Efficient Control-Oriented Modeling Approach for Vibration-Prone Delta 3D Printers Using Receptance Coupling

Delta 3D printers can significantly increase throughput in additive manufacturing by enabling faster and more precise motion compared to conventional serial-axis 3D printers. Further improvements in motion speed and part quality can be realized through model-based feedforward vibration control, as demonstrated on serial-axis 3D printers. However, delta machines have not benefited from model-based controllers because of the difficulty in accurately modeling their position-dependent, coupled nonlinear dynamics. In this paper, we propose an efficient framework to obtain accurate linear parameter-varying models of delta 3D printers at any position within their workspace from a few frequency response measurements. We decompose the dynamics into two sub-models&#x2013;(1) an experimentally-identified sub-model containing decoupled vibration dynamics; and (2) an analytically-derived sub-model containing coupled dynamics&#x2013;which are combined into one using receptance coupling. We generalize the framework by extending the analytical model of (2) to account for differing mass profiles and dynamic models of the printer&#x2019;s end-effector. Experiments demonstrate reasonably accurate predictions of the position-dependent dynamics of a commercial delta printer, augmented with a direct drive extruder, at various positions in its workspace. <i>Note to Practitioners</i>&#x2014;This work aims to equip high-speed 3D printers, like delta machines, with model-based controllers to complement their speed with high-accuracy. Due to the coupled kinematic chains of the delta, complex control methodologies, some of which require real-time state measurements, are often used to achieve satisfactory control performance. Our modeling approach provides an efficient methodology for obtaining accurate linear models without the need for real-time measurements, thus enabling practitioners to design linear model-based feedforward controllers to achieve the high throughput and accuracy desired in additive manufacturing (AM). The models we develop in this paper are intended for use with feedforward vibration compensation methods, which can be beneficial for both industrial-scale AM machines that have high-powered servo motors and feedback controllers, as well as consumer-grade AM machines which use stepper motors in feedforward control.

Robust actuator fault detection for quadrotor UAV with guaranteed sensitivity

This paper proposes an actuator fault detection method for quadrotor UAV, in which sensitivity to fault and robustness to disturbance are simultaneously guaranteed. First, a quadrotor UAV system is converted to a linear-parameter-varying (LPV) form related to the altitude velocity and attitude angular velocity. Secondly, based on this LPV model, a fault detection method is designed, in which H− performance is applied to enhance fault sensitivity and L∞ performance is applied for guarantee robustness of the residual against the disturbance. And then the L∞ performance provides a detection threshold combined with initial flight condition for residual evaluation. Finally, based on the QDrone testbed, both numerical simulations and plentiful online experiments are implemented to validate the performance of the proposed fault detection scheme.

Robust H∞ Output Feedback Trajectory Tracking Control for Steer-by-Wire Four-Wheel Independent Actuated Electric Vehicles

This paper investigates the trajectory tracking control issue of four-wheel independently actuated electric vehicles (FWIA EVs) with steer-by-wire devices concerning parameter uncertainties, external disturbances, input saturation, and vehicle sideslip angle not easily obtained. A robust H∞ dynamic output feedback control strategy is proposed for the integrated control of the steering motor current and direct yaw moment without using sideslip angle information to ensure the reference trajectory tracking and the improvement of handling performance and yaw stability. In the proposed integration control framework, the steer-by-wire device dynamic is involved in the polyhedral linear parameter-varying (LPV) trajectory tracking error model considering the time-varying longitudinal velocity, and the norm-bounded parameter uncertainties such as road adhesion coefficient, tire cornering stiffness, vehicle mass, and vehicle moment of inertia. With the help of the LPV model of all the states of the steer-by-wire FWIA EV, a dynamic output feedback trajectory tracking controller is designed using the robust H∞ technique. The controller gain matrices are obtained by solving the linear matrix inequalities. Finally, the high-fidelity full-vehicle model based on the CarSim-MATLAB/Simulink joint simulation platform verifies the robustness and advantages of the designed control strategy in the accelerated lane-change scenario.

Event-Triggered Control for LPV Modeling of DC–DC Boost Converter

This study presents the event-triggered control (ETC) for linear parameter varying (LPV) model of boost converters. We examine the nonlinear dynamics of boost converters in the LPV framework. The proposed controller is duty-ratio-dependent and provides better performance while requiring less computation. Using the parameter-dependent Lyapunov function (PDLF), we demonstrate the stablity analysis of the proposed approach. Furthermore, we demonstrate that the inter-event time is lower bound by a positive constant, which indicates Zeno behavior free performance. In comparison to earlier time-invariant synthesis techniques, the LPV formulation offers for increased robustness and performance properties. Simulation and experimental results validate the effectiveness of the proposed method.

Passive Fuzzy Controller Design for the Parameter-Dependent Polynomial Fuzzy Model

This paper discusses a passive control issue for Nonlinear Time-Varying (NTV) systems subject to stability and attenuation performance. Based on the modeling approaches of Takagi-Sugeno (T-S) fuzzy model and Linear Parameter-Varying (LPV) model, a Parameter-Dependent Polynomial Fuzzy (PDPF) model is constructed to represent NTV systems. According to the Parallel Distributed Compensation (PDC) concept, a parameter-dependent polynomial fuzzy controller is built to achieve robust stability and passivity of the PDPF model. Furthermore, the passive theory is applied to achieve performance, constraining the disturbance effect on the PDPF systems. To develop the stability criteria, by introducing a parameter-dependent polynomial Lyapunov function, one can derive some stability conditions, which belong to the term of Sum-Of-Squares (SOS) form. Based on the Lyapunov function, two stability criteria are proposed to design the corresponding PDPF controller, such that the NTV system is robustly stable and passive. Finally, two examples are applied to demonstrate the effectiveness of the proposed stability criterion.