Linear Fractional Transformation Modeling of Multibody Dynamics Around Parameter-Dependent Equilibrium

International audience

In this paper, we have proposed a new method for identifying specified parameters in linear systems. The method is based on Youla-parameterization, which is derived by pulling out the deviations of specified parameters to the references. The resultant parameterization takes a form of linear fractional transformation (LFT) and has an arbitrariness on the stable factors of parameterizing. We can utilize the arbitrariness to enhance the accuracy of estimating specified parameters because the stable factorizing in the parameterization means introducing a filter for suppressing the effect of observation noises. Two examples are given for illustrating the effectiveness and for showing the possibility of applying the method to fault diagnosis.

In this paper, the development of a software tool for linear parameter varying (LPV)/linear fractional transformation (LFT) modeling and data-based validation techniques is described. This tool represents a step towards industrialization of these techniques, which although still suffering from a certain theoretical complexity are increasingly shown to help deal effectively with important real world applications. The tool is a key component for the development of an integrated LPV/LFT framework for control modeling, analysis, and design which is also a necessity for successful transfer of LPV/LFT techniques to industry. The focus of this paper is on describing the development of the integrated modeling/validation software and examining the applicability problems associated with these techniques. The technical readiness of the tool is exemplified through the development of validated LFT models used for LPV control design and analysis of a nonlinear longitudinal model of the NASA HL-20 re-entry vehicle.

Presents an alternative approach to solve robust performance problems for linear parameter-varying (LPV) control systems. In contrast to previous methods, which are focused on deterministic algorithms, this paper is based on a probabilistic setting. The proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a finite number of steps. The main advantages of this method over the previous literature are as follows: (i) the randomized algorithm gives a direct method for general LPV plants with state-space matrices depending on scheduling parameters in a nonlinear manner, i.e. the probabilistic setting does not require any approximation, such as a linear fractional transformation (LFT), of the state space matrices or a gridding on the set of scheduling parameters; (ii) the proposed algorithm is sequential and, at each iteration, it does not require heavy computational effort, such as simultaneously solving a large number of linear matrix inequalities.

In this work, control system design and implementation for active vibration control (AVC) of flexible structures with piezoelectric actuators is studied. The goal is to reduce the effect of harmonic disturbances with known time-varying frequencies acting on a system. As a test bed, a thin flexible aluminum cantilevered beam with two symmetrically bonded piezoelectric actuators is used. The harmonic excitation is generated by two DC motors each of them with an unbalanced mass. A discrete-time model is obtained through black-box system identification methods. The control algorithm design is based on a plant description with a disturbance model as a linear parameter varying (LPV) system in linear fractional transformation (LFT) form. This results in a gain-scheduled controller where the harmonic disturbance frequencies are the scheduling variables. The experimental real-time results show the effectiveness of the controller and its capability to suppress time-varying harmonic disturbances, whose frequencies are measured directly from the DC motors. The design method leads to a controller that stabilizes the closed-loop system even for arbitrarily fast changes in the disturbance frequencies. In the real-time experiment, the controller suppresses a disturbance consisting of two independent harmonics with frequencies that vary over a range of 15 Hz.

We characterize a probabilistic measure named Model Set Unfalsified Probability (MSUP) for model set validation, where the model set is described by an LFT (Linear Fractional Transformation) form. We derive upper and lower bounds of MSUP and show that the lower bound computation can be reduced to an LMI-based convex optimization. A necessary and sufficient condition for which MSUP=0.5 (50%) is also provided. A numerical example confirms that the probabilistic approach more appropriately evaluates the suitability of a model set in robust controller design than deterministic approaches.

An LFT Approach to Parameter Estimation

Coprime factor based anti-windup synthesis for parameter-dependent systems

This paper presents advances in the damping of container crane load swing via hoisting modulation based on linear parameter-varying (LPV) control techniques. We propose controllers based on constant, as well as parameter-dependent Lyapunov functions and formulate our problem in the linear fractional transformation (LFT) framework. The dynamics of a nonlinear observer are included into the generalized plant. Simulation and experimental results are compared to previous work using a polytopic LPV approach, as well as to earlier work based on the concept of resonant coupling control realized by a reduced normal form approach. The comparison indicates, that including the observer dynamics in the synthesis comes at the price of reduced performance, which is alleviated by the use of parameter-dependent Lyapunov functions. Furthermore, an a posteriori analysis verifies that the controller is robust against erroneously estimated scheduling signals and thus provides closed-loop guarantees for stability and performance for the new controller.

The control synthesis problem involving implicit model following (IMF) is considered in the context of linear parameter-varying (LPV) and H/sub 2//H/spl infin/ theories. The well-known quadratic or nominal H/sub 2/ IMF problem is first extended to encompass LPV system models with a linear fractional transformation (LFT) structure. This problem is then embedded in the framework of LPV theory. Conditions for dealing with additional mixed H/sub 2//H/spl infin/ criteria are discussed. The solvability conditions are provided with little conservatism by a previous multi-channel LFT/LPV result in discrete time. Finally, an illustrative example is used to validate this new formulation. Also. we demonstrate through this example that the IMF formulation is an effective technique to achieve a desired transient behavior for LPV systems.

A technique to synthesize distributed linear parameter-varying (LPV) controllers for the control of heterogeneous LPV systems interconnected through switching directed interaction topologies is presented. Groups of subsystems are defined with undirected interaction within, but directed interconnections between each other. This allows to construct a virtual symmetric interconnection matrix representation of the graph topology. The symmetry guarantees the existence of a diagonalizing transformation, which renders both analysis and synthesis problems particularly simple. Structural constraints on multipliers reduce the complexity of the resulting coupled matrix inequalities to be of the order of a single subsystem times the number of groups. The problem can be solved by modified linear fractional transformation (LFT)-based LPV gain-scheduling synthesis methods.

The paper presents the results of application of linear parameter varying (LPV) decoupling control and a prefilter to improve the tracking performance in the air path of Diesel engines modeled as a quasi-LPV system. The proposed decoupling method benefits the multi-variable control of multiinput multi-output (MIMO) systems with variable operating conditions, variable parameters and nonlinear behavior. The results of this paper illustrate the reduced variability and performance enhancement of the two inputs (EGR valve effective area and VGT effective area) and two outputs (boost pressure and mass air flow) dynamic system of the air path of Diesel engines, where there is a significant coupling in the system dynamics. The proposed design method combines a prefilter (used to shape the reference input) with the LPV feedback control (based on an LPV decoupling method) proposed here to achieve the reference tracking with desired transient performance specifications. The prefilter is designed based on the closed-loop dynamics resulting from the LPV design, and a systematic input shaping prefilter design process is developed. The designed prefilter successfully extends the closed-loop bandwidth. Simulation results demonstrate the effectiveness of the input shaping prefilter. Moreover, the designed prefilter is structurally simple and computationally efficient.

Due to the universal approximation capability of Takagi–Sugeno (T–S) fuzzy models for nonlinear dynamics, many control issues have been investigated based on fuzzy control theory. In this paper, a transformation procedure is proposed to convert fuzzy models into linear fractional transformation (LFT) models. Then, T–S fuzzy systems can be regarded as a special case of linear parameter-varying (LPV) systems which proved useful for nonlinear control problems. The newly established connection between T–S fuzzy models and LPV models provides a new perspective of the control problems for T–S fuzzy systems and facilitates the fuzzy control designs. Specifically, an output feedback gain-scheduling control design approach for T–S fuzzy systems is presented to ensure globally asymptotical stability and optimize H∞ performance of the closed-loop systems. The control synthesis problem is cast as a convex optimization problem in terms of linear matrix inequalities (LMIs). Two examples have been used to illustrate the efficiency of the proposed method.

SOME EXPERIMENTAL RESULTS OF AN ℋ∞ -LPV CONTROLLER APPLIED TO A DIESEL ENGINE COMMON RAIL INJECTION SYSTEM

Linear parameter varying (LPV) systems are defined as linear systems whose dynamics depend on exogenous time-varying parameters. Typically, the parameter values are assumed to be confined to a prescribed set. Further assumptions may include restrictions on the parameter rates of variation. In this paper, the authors examine the stability and stabilization of LPV systems under various restrictions on the parameter variations. In particular, the authors employ computational methods to explicitly construct invariant and controlled invariant sets for LPV systems. These sets will allow one to either infer stability or construct stabilizing feedback. Such methods have been used extensively throughout the controls literature from a variety of contexts. The authors present methods for stability analysis and stabilization. These methods are applicable to a variety of constraints on parameter variations.