Local Linear Time-invariant Research Articles

Experimental linear parameter-varying model identification of an elastic kinetic roof structure

Elastic kinetic structures are a recent approach to design transformable lightweight structures. Their transformation is based on elastic bending, exploiting the compliant material behavior of the structural members. This enables transformable structures with a stable transformation process. However, due to their lightweight and flexible design, elastic kinetic structures are highly sensitive to static and dynamic disturbances. Moreover, such transformable structures exhibit a transformation state dependent dynamic behavior, due to geometric nonlinearities. Nevertheless, most of current research focuses on the principles of elastic kinetic transformation than on effective disturbance mitigation. The latter is usually designed based on low-order control-oriented models. This paper describes a system identification methodology, suitable to identify low-order models for the transformation state dependent dynamics of elastic kinetic structures. The system identification is carried out using a local approach in the linear parameter-varying (LPV) framework. For the identification of local linear time-invariant (LTI) models, the system realization using the information matrix (SRIM) method is applied. The collection of local LTI models is internally balanced using a balancing transformation, in order to represent the locally estimated models with respect to a common state–space basis. Via linear interpolation, an LPV system in grid-based representation is derived. This methodology is experimentally validated with data from an active hybrid roof structure prototype.

Linear Parameter Varying Models-Based Gain-Scheduling Control for Lane Keeping System With Parameter Reduction

In this paper, a robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> controller based on Linear Parameter Varying (LPV) model with scheduling variable reduction is applied for a lane-keeping system. On a curved road section, varying longitudinal speed and roll motion lead to multiple parameter variations in the lateral vehicle dynamics. To trade-off between the complexity of the multiple scheduling variables and the accuracy of the LPV model, an order reducing method by Autoencoder (AE) is performed to obtain a reduced model and diminishes the conservatism of LPV-based controller design. Further, to ensure the continuous convex set membership of the reduced scheduling variables in online test scenarios, the Lipschitz constant of the offline trained neural network (NN) is tightly estimated by solving a Semidefinite Program (SDP). The convex set of local Linear Time-Invariant (LTI) controllers is designed according to the estimated Lipschitz bound of the trained NN. The LPV-based robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> feedback controller is then designed by solving a set of Linear Matrix Inequalities (LMIs). Numerical simulations with full vehicle dynamics from CarSim are given to demonstrate the performance of the proposed system on various test roads. The continuous finding of the reduced scheduling variable membership is guaranteed in the online tests, and the effectiveness of the proposed method is confirmed with the mean value of lateral offset error reduced by about 50% compared with LTI-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> controllers.

Interpolation-Based Modeling Methodology for Efficient Aeroelastic Control of a Folding Wing

The aeroelastic model of a folding wing varies with different configurations, so it actually represents a parameter-varying system. Firstly, a new approach based on interpolation of local models is proposed to generate the linear parameter-varying model of a folding wing. This model is capable of predicting the aeroelastic responses during the slow morphing process and is suitable for subsequent control synthesis. The underlying inconsistencies among local linear time-invariant (LTI) models are solved through the modal matching of structural modes and the special treatment of the rational functions in aerodynamic models. Once the local LTI models are represented in a coherent state-space form, the aeroservoelastic (ASE) model at any operating point can be immediately generated by the matrix interpolation technique. Next, based on the present ASE model, the design of a parameterized controller for suppressing the gust-induced vibration is studied. The receptance method is applied to derive fixed point controllers, and the effective independence method is adopted and modified for optimal sensor placement in variable configurations, which can avoid solving ill-conditioned feedback gains. Numerical simulation demonstrates the effectiveness of the proposed interpolation-based modeling approach, and the parameterized controller exhibits a good gust mitigation effect within a wide parameter-varying range. This paper provides an effective and practical solution for modeling and control of the parameterized aeroelastic system.

Estimating an LPV Model of Driver Neuromuscular Admittance Using Grip Force as Scheduling Variable

Humans can rapidly change their low-frequency arm dynamics to resist forces or give way to them. Quantifying driver time-varying arm dynamics is important to develop steer-by-wire and haptic support systems. Conventional linear time-invariant (LTI) identification, and even time-varying techniques such as wavelets, fail to capture fast changing dynamics. Moreover, such techniques require perturbation signals on the steering wheel (SW), which may affect steering feel and control behavior. We propose a novel two-step method to estimate time-varying driver admittance, using unobtrusive grip-force measurements of the hands on the wheel to schedule a linear parameter-varying (LPV) model that captures the full admittance range. A total of 18 subjects participated in two experiments in a simulator with an actuated SW. In a sensorimotor control experiment, we first establish the grip force and admittance relationship, requiring subjects to perform a boundary tracking task where perturbations on the wheel enabled local LTI identification. Six boundary widths is used to evoke admittance changes, after which a global LPV model is obtained through interpolation between the local models. Results show an inverse relationship between grip force and admittance and that the LPV model accurately captures the admittance settings (fit percentage > 90%). Second, a driving experiment is followed that aims to evoke differences in grip force and admittance in response to varying road widths, offering more realistic data to evaluate the LPV model predictions. Results show that the LPV model accurately describes adaptations in admittance to road width. Our method allows for online estimation of time-varying admittance during driving, without applying force perturbations.

Active Vibration Control of a Convertible Structure based on a Polytopic LPV Model Representation

This paper deals with modeling and control of lightweight convertible structures for the application in civil and structural engineering. Such structures are prone to vibrations due to their lightweight design. In addition, they exhibit transformation state dependent dynamic behavior. In order to guarantee a reliable operation, the use of active vibration control (AVC) is an effective means. For the example of a simplified convertible structure, modeling is demonstrated using the linear parameter-varying (LPV) framework. Based on local linear time-invariant (LTI) models, derived from a finite element model of the convertible structure, a polytopic LPV model is established. This LPV model is then utilized to design a polytopic LPV output-feedback controller for AVC during the structure’s transformation. The effectiveness of the designed controller is validated in simulation.

Active Vibration Control of a Convertible Structure based on a Linear Parameter-Varying Model

This paper investigates modelling and active vibration control (AVC) of ultra-lightweight convertible structures based on the elastic kinetic motion mechanism with a focus on civil and structural engineering applications. Model building is carried out in the linear parameter-varying (LPV) framework for the example of a convertible Fin Ray structure. In a first step local linear time-invariant (LTI) models are derived from a finite element model of the structure for different transformation states. On the basis of this collection of local LTI models, a grid-based LPV model is established. An AVC of multiple modes is implemented by means of a modal velocity feedback control. Single-input single-output (SISO) control design is carried out with the root locus method for the first and second mode using the same control input under the assumption of well separated eigenfrequencies. The implemented AVC is validated in simulation.

Linear Parameter-Varying Models for Convertible Structures in Civil and Structural Engineering

This paper investigates the feasibility of the linear parameter-varying (LPV) framework for modelling the dynamic behaviour of ultra-lightweight convertible structures based on the elastic kinetic motion mechanism with a focus on civil and structural engineering applications. Model building is carried out for the example of a Fin Ray structure. In a first step local linear time-invariant (LTI) models are derived from a finite element model of the structure for different transformation states. On the basis of this collection of local LTI models, a grid-based LPV model is established. The derived LPV model is validated in simulation by comparison with data from a finite element model.

LPV System Local Model Interpolation Based on Combined Model Reduction

The local approach to linear parameter varying (LPV) system identification consists in interpolating a collection of linear time invariant (LTI) models, which have been estimated from data acquired at different working points of a nonlinear system. Interpolation is essential in this approach. When the local LTI models are in state-space form, as each local model can be estimated with an arbitrary state basis, it is widely acknowledged that the local models should be made coherent before their interpolation. In order to avoid the delicate task of making local state-space models coherent, a new interpolation method of local state-space models is proposed in this paper, which does not require coherent local models. This method is based on the reduction of the large state-space model built by combining the local models. Numerical examples are presented to illustrate the effectiveness of this method.

On experiment design for local approach identification of LPV systems

The local approach for the estimation of a Linear Parameter-Varying (LPV) model consists in an interpolation of a finite number of Linear Time-Invariant (LTI) systems called local LTI systems. Each local LTI system is obtained by performing an identification experiment at a fixed constant value of the parameter describing the dynamic variation of the LPV model. This parameter is referred in the literature as scheduling variable and the fixed values of the scheduling variable are simply called operating points or scheduling points. In order to improve the accuracy of the local method for the identification of LPV systems, the choice of the scheduling points and the inputs used at each scheduling point for the identification of the local LTI systems is addressed in this paper. To deal with this problem, an accuracy measure is first introduced. This measure is shown to be a linear combination of the classic A-optimality accuracy measure of the local LTI systems. Using this result, an algorithm is finally proposed to solve the experiment design problem.

From structurally independent local LTI models to LPV model

The local approach to linear parameter varying (LPV) system identification consists in interpolating individually estimated local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, individually estimated local state-space LTI models do not contain sufficient information for determining similarity transformations making them coherent. It is possible to estimate these similarity transformations from input–output data under appropriate excitation conditions.

Experimental Study on an Electric Variable Valve Timing Actuator: Linear Parameter Varying Modeling and Control

This paper presents experimental investigation results of an electric variable valve timing (EVVT) actuator using linear parameter varying (LPV) system identification and control. For the LPV system identification, a number of local system identification tests were carried out to obtain a family of linear time-invariant (LTI) models at fixed engine speed and battery voltage. Using engine speed and battery voltage as time-varying scheduling parameters, the family of local LTI models is translated into a single LPV model. Then, a robust gain-scheduling (RGS) dynamic output-feedback (DOF) controller with guaranteed H∞ performance was synthesized and validated experimentally. In contrast to the vast majority of gain-scheduling literature, scheduling parameters are assumed to be polluted by measurement noises and the engine speed and battery voltage are modeled as noisy scheduling parameters. Experimental and simulation results show the effectiveness of the developed approach.

A local identification method for linear parameter-varying systems based on interpolation of state-space matrices and least-squares approximation

This paper proposes a novel state-space matrix interpolation technique to generate linear parameter-varying (LPV) models starting from a set of local linear time-invariant (LTI) models estimated at fixed operating conditions. Since the state-space representation of LTI models is unique up to a similarity transformation, the state-space matrices need to be represented in a common state-space form. This is needed to avoid potentially large variations as a function of the scheduling parameters of the state-space matrices to be interpolated due to underlying similarity transformations, which might degrade the accuracy of the interpolation significantly. Underlying linear state coordinate transformations for a set of local LTI models are extracted by the computation of similarity transformation matrices by means of linear least-squares approximations. These matrices are then used to transform the local LTI state-space matrices into a form suitable to achieve accurate interpolation results. The proposed LPV modeling technique is validated by pertinent numerical results.

LPV System Common State Basis Estimation from Independent Local LTI Models

For the identification of a linear parameter varying (LPV) system steered by a scheduling variable evolving within a finite set, the local approach consists in separately estimating local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, the local state-space LTI models do not contain the necessary information about the similarity transformations making them coherent. Nevertheless, it is possible to estimate these similarity transformations from input-output data under appropriate input excitation conditions. These estimations result in a common state basis of the transformed local LTI models, so that they form a coherent global LPV model, suitable for numerical simulations in the case of fast scheduling variable evolutions.

Interpolated Modeling of LPV Systems Based on Observability and Controllability

This paper presents a State-space Model Interpolation of Local Estimates (SMILE) technique to compute linear parameter-varying (LPV) models for parameter-dependent systems through the interpolation of a set of linear time-invariant (LTI) state-space models obtained for fixed operating conditions. Since the state-space representation of LTI models is not unique, a suitable coherent representation needs to be computed for the local LTI models such that they can be interpolated. In this work, this coherent representation is computed based on observability and controllability properties. It is shown that compared with the state of the art in the literature, this new method has three strong appeals: it is general, fully automatic and results in numerically well-conditioned LPV models. An example demonstrates the potential of the new SMILE technique.

Optimal experimental design for LPV identification using a local approach

A common approach for dealing with non-linear systems is to describe the system by a model with parameters that vary as a function of the operating point. Consequently, the non-linear system is seen as a combination of local Linear Time-Invariant (LTI) systems, one for each value of the operating point. Such representations are called Linear Parameter-Varying (LPV) models. Due to the importance of this representation for the control of nonlinear systems, numerous algorithms have recently been developed to identify LPV models. However, the optimal design of such identification experiments remains completely unexplored. In this paper, we consider the so-called local approach for the LPV identification. In the local approach, the local linear models, corresponding to a series of fixed operating points, are identified by performing one identification experiment at each of these operating points. The LPV nature of the system is then retrieved by interpolating the value of the parameters at other operating points for example with a polynomial function which is fitted through the parameters identified at the operating points considered. We present an approach to choose optimally the value of the operating-points at which the local identification experiments will be performed. By optimal, we mean that the value of the operating points are optimized in such a way that the LPV model obtained after interpolation has a maximum accuracy.

Identification of Interpolating Affine LPV Models for Mechatronic Systems with one Varying Parameter

This paper presents a technique to identify an affine parameter-dependent model based on a set of Linear- Time-Invariant (LTI) models that are identified in local operating points of the varying system. The systems under investigation are mechatronic systems in which the dynamic behaviour is depending on a single varying parameter. By fitting specific pole and zero loci on the poles and zeros of the local LTI models, an affine state-space model can be constructed. Since the resulting affine model is well conditioned and has a low complexity, it is suitable to be used for Linear- Parameter-Varying (LPV) control. The applicability of the presented technique is shown on an industrial pick-and-place machine with position-dependent dynamics.