Nonlinear Parameter Varying Research Articles

Fault Estimation Observers for the Vehicle Suspension with a Varying Chassis Mass

This paper proposes a descriptor Nonlinear Parameter Varying (NLPV) observer for damper fault estimation in vehicle semi-active suspension systems, here considering the highly practical case of a varying sprung mass, which is an open problem. This mass and the control input are treated as known scheduling parameters for the design and implementation of the observers, thus making the structure adaptive. This novel modeling leads to parameter-dependent dynamics and output matrices in the nonlinear parameter-varying formulation. To solve this problem, we extend an existing reduced-order observer design method (following descriptor system modeling) to this new case. Both polytopic and grid-based methods are then considered to compute and implement the problem solutions. The designs are then assessed using frequency-domain analysis and realistic time-domain simulations.

Prescribed performance control for nonlinear parameter-varying systems with an application to turbofan engine

In this paper, we investigated the problem of the prescribed performance control for the turbofan engine described by nonlinear parameter-varying (NPV) systems formulation. NPV systems for describing the turbofan engine are set up. Compared with the existing dynamic linear models and nonlinear models, the turbofan engine NPV model can show its dynamics time-varying features and nonlinearity. Meanwhile, based on the turbofan engine NPV system, a prescribed performance tracking controller is designed by error mapping function, and a class of state-and-parameter-dependent (SAPD) control synthesis conditions are formulated. These SAPD conditions can be effectively solved by sum-of-squares technique, and steady-state and transient performance of tracking error can be ensured. Finally, simulation results on the turbofan engine have been given to verify the feasibility and effectiveness of the prescribed performance tracking control scheme.

Disturbance observer-based controller design for uncertain nonlinear parameter-varying systems

In this paper, the disturbance observer-based controller (DOBC) design problem is investigated for a class of uncertain nonlinear parameter-varying (NPV) systems subject to unknown uncertainty and unmeasurable state variables. First, aiming to deal with the unknown uncertainty, an efficient scheme for treating the uncertainty as an unknown disturbance is given. Second, based on the transformed model with unknown disturbance, a novel observer is presented to estimate the unmeasurable state variables and unknown disturbance, which is further used to design the state-and-parameter-dependent (SAPD) controller. By using the correspond closed-loop systems and Lyapunov stability theory, some SAPD conditions on designing the observer and controller are established by linear matrix inequalities (LMIs) forms. Based on sum-of-squares (SOS) techniques, these LMIs can be effective solved Remarkably, the SAPD controller and the SAPD observer are designed independently, which can significantly reduce the complexity of the disturbance-based control algorithm. Finally, some simulations and comparative results in two examples are provided to prove the feasibility of the proposed approach.

Non Linear Parameter Varying Observer based on Descriptor Modeling for Damper Fault Estimation

This paper proposes an H∞ Non Linear Parameter Varying (NLPV) observer for fault estimation in semi-active Electro-Rheological (ER) suspensions. The damper fault (a loss-of-efficiency factor) is modeled as a lost force of unknown/free dynamics to be estimated. Thanks to the parameter-dependent descriptor-form system modeling, there is no assumption made on the fault dynamics, thus making this method applicable to all considered types of damper faults. The nonlinearity in the damper model is bounded by its Lipschitz property, while the road disturbance and the measurement noise are handled using the H∞ condition. The observer is parameterized and then designed by solving Linear Matrix Inequalities (LMIs) and is implemented in a polytopic gain scheduling approach. Synthesis results including Bode plots and simulations illustrate the method in both the frequency and the time domains.

Multi-objective Grid-based Lipschitz NLPV PI Observer for Damper Fault Estimation

Dampers in semi-active suspension systems may be subject to various types of damper faults, e.g., oil leakage and electrical issues, that need to be estimated for diagnosis and isolation purposes. A new method to design the so-called multi-objective grid-based Lipschitz Nonlinear Parameter Varying (NLPV) Proportional Integral (PI) observer is here developed for damper fault estimation, where the fault is the loss of efficiency of the damper modeled as a slow-varying input. While the damper nonlinearity is bounded by the Lipschitz condition, the H∞ and generalized H2 conditions are used to minimize the effects of the input disturbance and the measurement noise, respectively, on the estimation error. Moreover, the observer is designed with Linear Matrix Inequalities (LMIs) formed and solved in a grid-based manner (considering a parameter-dependent Lyapunov function) to reduce the level of conservatism. Analyses in the frequency domain using Bode plots as well as in the time domain using realistic simulations illustrate the effectiveness of the proposed observer.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

A nonlinear parameter varying observer for real‐time damper force estimation of an automotive electro‐rheological suspension system

AbstractThis paper proposes a nonlinear parameter varying (NLPV) observer to estimate, in real‐time, the damper force of an electrorheological (ER) damper in a road vehicle suspension system. The objectives are to minimize the effects of bounded unknown road profile disturbances and measurement noises on the estimation errors in the framework. Furthermore, the nonlinearity coming from the damper model (and considered in the observer formulation) is handled through a Lipschitz condition. The observer inputs are data given by two low‐cost sensors (two accelerometers data from the sprung mass and the unsprung mass). For performance assessment, the observer is implemented on a 1/5‐scaled real vehicle testbench of GIPSA‐lab, namely INOVE (see www.gipsa‐lab.fr/projet/inove). Both simulation and experimental results demonstrate the effectiveness of the proposed observer in terms of the ability to estimate the damper force in real‐time and to minimize the effects of measurement noises and road disturbances.

Robust controllers design for constrained nonlinear parameter varying descriptor systems

AbstractThis article proposes a method for designing robust controller laws for a class of uncertain nonlinear parameter varying (NLPV) descriptor systems under input saturation and external disturbances. Both static and dynamic output feedback controllers are proposed. To synthesize the fuzzy controllers, the stability conditions are derived using polytopic parameter‐dependent (PD) nonquadratic Lyapunov functions with respect to the given saturation constraint on the control input. First, the designed conditions are established in terms of linear matrix inequalities (LMIs) and gain performance is used to attenuate the effect of the external disturbance signals. Then, the estimation of the largest domain of attraction (DoA) for the system is formulated and solved as an optimization problem. Two examples are used to illustrate the effectiveness of the proposed design methods.

A robust identification method for stochastic nonlinear parameter varying systems

<abstract><p>Successful identification procedures are undoubtedly important for accurate model description and the consequent implementation of control strategies. Linear Parameter Varying (LPV) models are nowadays standard for control design purposes and powerful identification techniques accordingly available. Anyhow, recent advances have brought to focus the class of Nonlinear Parameter Varying (NLPV) models, which keep some nonlinearities embedded to the formulation. Identification tools for this latter class are still not available. Therefore, this paper proposes a novel method for the robust identification of stochastic NLPV systems, considering that the nonlinear parameter part is <italic>a priori</italic> known and obeys a Lipschitz condition. The method is based on a modified extended Masreliez-Martin filter and yields the joint estimation of both NLPV systems states and model parameters. The method manages the stochasticity of the system by considering the presence of measurement outliers with non-Gaussian distributions. Results considering real data from a vehicle suspension system are presented in order to demonstrate the consistency of the proposed method.</p></abstract>

Gain‐scheduled control for discrete‐time non‐linear parameter‐varying systems with time‐varying delays

This study addresses the gain-scheduled control problem for discrete-time delayed non-linear parameter-varying (NLPV) and linear parameter-varying (LPV) systems. First, by constructing the parameter-dependent Lyapunov–Krasovskii functional and employing multiple auxiliary functions, delay-dependent reciprocally convex inequality, and selecting a suitable augmented vector, novel delay-dependent linear matrix inequality conditions for the static output-feedback control design and state-feedback control design for delayed NLPV are provided. Second, the results obtained for discrete-time delayed NLPV systems are modified in a simple way to deal with discrete-time delayed LPV systems. Finally, the effectiveness of the proposed methods is illustrated by numerical examples.

Ensuring performance requirements for semiactive suspension with nonconventional control systems via robust linear parameter varying framework

SummaryIn the article a method which is able to provide the required performance level of a system is proposed. Its principle is to combine the results of conventional control methods with those of methods based on nonconventional, for example, machine‐learning‐based ones. In more detail, it designs a robust linear parameter varying (LPV) control in a predefined form, whose output is equivalent to the output of a machine‐learning‐based control inside a predefined operational range. Outside of the operation range the output of the machine‐learning‐based control is overridden, while the intervention with the performance level is guaranteed. The efficiency of the proposed method is illustrated through an example on the semiactive suspension control design. The nonlinearities in the dynamics of the magneto‐rheological damper are considered through a nonlinear parameter varying (NLPV) model. It designs an NLPV model‐based LPV control, which is combined with a neural network to achieve preview capability.

An input‐to‐state stable model predictive control framework for Lipschitz nonlinear parameter varying systems

SummaryThis article develops a state‐feedback model predictive control (MPC) framework for nonlinear parameter varying (NLPV) systems with explicit Lipschitz nonlinearities along the input trajectory. Considering a guess for the evolution of the scheduling parameters along the prediction horizon, the proposed optimization procedure for the MPC design includes a terminal stage cost, a contracting terminal region constraint and nominal predictions scheduled with respect to this guess. The terminal region is taken so that it is monotonically decreasing to a pre‐determined set, which guarantees recursive feasibility of the algorithm with respect to a bound on the admissible uncertainties (introduced from the scheduling parameter guess). The terminal set is a scheduled robust control invariant set for Lipschitz nonlinear systems, computed through some proposed LMIs. The inclusion of the terminal ingredient also serves to demonstrate input‐to‐state quadratic stability. This article ends with a successful numerical simulation example of the technique applied to the control of a semiactive automotive suspension system equipped with electro‐rheological dampers. Comparisons are given with respect to open‐loop behavior and to a robust LQR controller.

Robust model predictive control for nonlinear parameter varying systems without computational delay

SummaryThis article proposes a one‐step ahead robust model predictive control (MPC) for discrete‐time Lipschitz nonlinear parameter varying (NLPV) systems subject to disturbances. Within the proposed design framework, the optimization that generates the MPC policy to be implemented at next time instant is executed in advance during the current sampling period based on future state prediction. This new feature allows avoidance of the online computational delay existing in the traditional MPC settings and improves the control performance. The proposed MPC is proved to be recursively feasible with the guarantee for robust closed‐loop system stability and satisfaction of input and output constraints. A tractable linear matrix inequality (LMI) optimization problem is formulated to compute the control gains at each time instant. The computational complexity of the obtained LMI problem is also analyzed. The one‐step ahead robust MPC is further developed to cover discrete‐time Lipschitz NLPV systems with disturbance compensation. Efficacy and performance improvement of the design are demonstrated through a numerical example and an application to adaptive cooperative cruise control for automated vehicles under variable road geometry.

On the domain of attraction and local stabilization of nonlinear parameter‐varying systems

SummaryIn this paper, the domain of attraction (DA) and the local stabilization problem are investigated under the nonlinear parameter‐varying (NPV) framework. Specifically, to take into account the time‐varying parameter, we generalize the definition of robust DA (RDA) and propose the concept of parameter‐dependent DA (PDA) for NPV systems, together with the sum‐of‐squares (SOS) conditions for their estimations. Differently from the existing DA‐related works, the theoretical results in this paper can be applied to a large class of nonlinear polynomial systems including the time‐invariant, the parameter‐dependent, and the uncertain ones. Moreover, the commonly used iterative/coordinate‐wise search is avoided because we eliminate the bilinear product terms between the Lyapunov functional and the SOS multipliers. We also consider the local stabilization problem of NPV systems, in which the RDA can be specified as a control performance index. Finally, several examples are given for illustration purposes.

Real-time Damper Force Estimation of Vehicle Electrorheological Suspension: A NonLinear Parameter Varying Approach

This paper proposes a nonlinear parameter varying (NLPV) observer to estimate in real-time the damper force of an electrorheological (ER) damper in road vehicle suspension system. First, a nonlinear quarter-car model equipped with the dynamic nonlinear model of ER damper is presented, which captures the main behavior of the suspension system. The estimation method of the damper force is developed using a NLPV observer whose objectives are to minimize the effects of bounded unknown road profile disturbances and measurement noises on the estimation errors in the H∞ framework. Furthermore, the nonlinearity coming from damper model (and considered in the observer formulation) is handled through a Lipschitz condition. The observer inputs are given by two low-cost sensors data (two accelerometers data from the sprung mass and the unsprung mass). For performance assessment, the observer is implemented on the INOVE testbench of GIPSA-lab (1/5-scaled real vehicle). Both simulation and experimental results demonstrate the effectiveness of the proposed observer in terms of the ability of estimating the damper force in real-time and of minimizing the effects of measurement noises and road disturbances.

D-stable Controller Design for Lipschitz NLPV System

This paper addresses the design of a state-feedback controller for a class of nonlinear parameter varying (NLPV) systems in which the nonlinearity can be expressed as a parameter-varying Lipschitz term. The controller is designed to satisfy a D-stability specification, which is akin to imposing constraints on the closed-loop pole location in the case of LTI and LPV systems. The design conditions, obtained using a quadratic Lyapunov function, are eventually expressed in terms of linear matrix inequalities (LMIs), which can be solved efficiently using available solvers. The effectiveness of the proposed method is demonstrated by means of a numerical example.

Exponential stabilisation of nonlinear parameter-varying systems with applications to conversion flight control of a tilt rotor aircraft

ABSTRACTThe exponential stabilisation problem of nonlinear parameter-varying (NPV) systems with input constraints is investigated. Unlike the existing NPV-related works, this paper directly addresses NPV models to conduct the stabilisation problem, without hiding nonlinearities or ignoring the time-varying nature. Existence conditions of a nonlinear time-varying (NTV) controller to render the uniformly exponential stability of an NPV system are given in terms of state-and-parameter-dependent linear matrix inequalities (SPLMIs). Specifically, a new controller structure and a novel Lyapunov functional are adopted such that the exact variation rates of the state and parameters can be incorporated into the SPLMIs. The generalised S-procedure is used to convexify the input constraints such that the resulting closed-loop system satisfies input constraints for any state starting from an admissible set. The derived SPLMIs can then be efficiently solved via sum-of-squares programming. The proposed approach is applied to the conversion flight control of a tilt rotor aircraft.

A continuous dynamic modeling approach for an omnidirectional mobile robot

MY wheel-II is one of switch omnidirectional wheel mechanisms. The omnidirectional mobile robot based on MY wheels-II is a switched non-linear system (i.e. discontinuous system). The aim of this paper is to propose a continuous modeling approach which can be employed to derive a continuous model from any given discontinuous robot dynamic model. This approach results in a continuous non-linear parameter varying (NLPV) model, and offers one solution for model-based control design. Firstly, our previously proposed average dynamic modeling approach is analyzed. We find that this modeling approach is effective only for a specific class of robot configurations. To overcome this problem, we first derive the switching conditions of MY wheel-II. Based on derived switching conditions, we then propose a simple continuous NLPV modeling approach. The new approach replaces the real discontinuous contact radius in the discontinuous dynamic model with an adaptive continuous curve. An illustrative example of the adaptive continuous curve design is provided. Both simulation and experimental results verify the effectiveness of the proposed NLPV modeling approach against the average modeling approach.

An Interval NLPV Parity Equations Approach for Fault Detection and Isolation of a Wind Farm

In this paper, the problem of fault diagnosis of a wind farm is addressed using interval nonlinear parameter-varying (NLPV) parity equations. Fault detection is based on the use of parity equations assuming unknown but bounded description of the noise and modeling errors. The fault detection test is based on checking the consistency between the measurements and the model, by finding if the formers are inside the interval prediction bounds. The fault isolation algorithm is based on analyzing the observed fault signatures online and matching them with the theoretical ones obtained using structural analysis. Finally, the proposed approach is tested using the wind farm benchmark proposed in the context of the wind farm fault-detection-and-isolation/fault-tolerant-control competition.

Nonlinear MPC Design for Identified Nonlinear Parameter Varying (NPV) Model

In this paper, a novel nonlinear model predictive controller (MPC) is proposed based on an identified nonlinear parameter varying (NPV) model. Firstly, an NPV model scheme is present for process identification, which is featured by its nonlinear hybrid Hammerstein model structure and varying model parameters. The hybrid Hammerstein model combines a normalized static artificial neural network with a linear transfer function to identify general nonlinear systems at each fixed working point. Meanwhile, a model interpolating philosophy is utilized to obtain the global model across the whole operation domain. The NPV model considers both the nonlinearity of transition dynamics due to the variation of the working-point and the nonlinear mapping from the input to the output at fixed working points. Moreover, under the new NPV framework, the control action is computed via a multistep linearization method aimed for nonlinear optimization problems. In the proposed scheme, only low cost tests are needed for system identification and the controller can achieve better output performance than MPC methods based on linear parameter varying (LPV) models. Numerical examples validate the effectiveness of the proposed approach. Keywords: Nonlinear parameter varying (NPV), Hammerstein model, nonlinear MPC.