Linear Fractional Transformation Form Research Articles

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

This paper proposes a new Linear Fractional Transformation (LFT) modeling approach for uncertain Linear Parameter Varying (LPV) multibody systems with parameter-dependent equilibrium. Traditional multibody approaches, which consist in building the nonlinear model of the whole structure and linearizing it around equilibrium after a numerical trimming, do not allow to isolate parametric variations with the LFT form. Although additional techniques, such as polynomial fitting or symbolic linearization, can provide an LFT model, they may be time-consuming or miss worst-case configurations. The proposed approach relies on the trimming and linearization of the equations at the substructure level, before assembly of the multibody structure, which allows to only perform operations that preserve the LFT form throughout the linearization process. Since the physical origin of the parameters is retained, the linearized LFT-LPV model of the structure exactly covers all plants, in a single parametric model, without introducing conservatism or fitting errors. An application to the LFT-LPV modeling of a robotic arm is proposed; in its nominal configuration, the model obtained with the proposed approach matches the model provided by the software Simscape Multibody, but it is enhanced with parametric variations with the LFT form; a robust LPV synthesis is performed using Matlab robust control toolbox to illustrate the capacity of the proposed approach for control design.

Sensor fault diagnosis for uncertain dissimilar redundant actuation system of more electric aircraft via bond graph and improved principal component analysis

This paper develops a sensor condition monitoring method integrating the model-based bond graph (BG) technique and data-driven principal component analysis (PCA) for the dissimilar redundant actuation system of more electric aircraft with uncertain parameters. The uncertain dissimilar redundant actuation system is modeled by BG in linear fractional transformation form. After that, the analytical redundancy relations containing the nominal part and the uncertain part can be derived, based on which the adaptive thresholds and the fault signature matrix (FSM) can be obtained for robust fault detection and fault isolation. To improve the fault isolation performance under the multiple faults condition, a new fault isolation method integrating FSM and improved PCA (IPCA) is developed, where the possible fault set generated from the FSM is further refined by the IPCA module with an improved reconstruction algorithm and cyclic PCA monitoring model to achieve a more efficient fault isolation result. The effectiveness of the proposed approach is validated by simulation investigations.

A novel fault-tolerant control strategy based on inverse bicausal bond graph model in linear fractional transformation

This article presents the development of a novel fault-tolerant control strategy. For this task, a bicausal bond graph model-based scheme is designed to generate online information to the inverse controller about the faults estimation. Secondly, a new approach is proposed for the fault-tolerant control based on the inverse bicausal bond graph in linear fractional transformation form. However, because of the time delay for fault estimation, the PI controller is used to reduce the error before the fault is estimated. Hence, the required input that compensates the fault is the sum of the control signal delivered by the PI controller and the control signal resulting from the inverse bicausal bond graph for fast fault compensation and for maintaining the control objectives. The novelties of the proposed approach are: (1) to exploit the power concept of the bond graph by feeding the power generated by the fault in the inverse model (2) to suitably combining the inverse bicausal bond graph with the PI feedback controller so that the proposed strategy can compensate for the fault with a very short time delay and stabilize the desired output. Finally, the experimental results illustrate the efficiency of the proposed strategy.

Optimized fault detection using bond graph in linear fractional transformation form

This article deals with the optimal robust fault detection problem using the bond graph in its linear fractional transformation form. Generally, this form of the bond graph allows the generation of two perfectly separate analytical redundancy relations, that are used as residual and threshold. However, the uncertainty calculation method gives overestimated thresholds. This may, for instance, lead to undetectable faults. Therefore, enhancing the robustness of fault detection and isolation algorithms is of utmost importance in designing a bond graph–based fault detection system. The main idea of this article is to develop optimized thresholds to ensure an optimal detection, otherwise this article proposes a method to detect tiny magnitude faults concerning parameter’s uncertainties. This work considers the issue of optimal fault detection as an optimization problem of the gap between the residuals and its threshold. New uncertainty values will be calculated in a way that these estimated parameters ensure the desired optimized gap between residuals and thresholds. These estimated uncertainty values will be used to generate optimized adaptive thresholds. Through these thresholds, we increase the sensitivity of the residuals to tiny magnitude faults, and we ensure an optimal and early detection.

Compound Fault Diagnosis and Sequential Prognosis for Electric Scooter with Uncertainties

This paper addresses diagnosis and prognosis problems for an electric scooter subjected to parameter uncertainties and compound faults (i.e., permanent fault and intermittent fault with non-monotonic degradation). First, the diagnostic bond graph in linear fractional transformation form is used to model the uncertain electric scooter and derive the analytical redundancy relations incorporating the nominal part and uncertain part, based on which the adaptive thresholds for robust fault detection and the fault signature matrix for fault isolation can be obtained. Second, an adaptive enhanced unscented Kalman filter is proposed to identify the fault magnitudes and distinguish the fault types where an auxiliary detector is introduced to capture the appearing and disappearing moments of intermittent fault. Third, a dynamic model with usage dependent degradation coefficient is developed to describe the degradation process of intermittent fault under various usage conditions. Due to the variation of degradation coefficient and the presence of non-monotonic degradation characteristic under some usage conditions, a sequential prognosis method is proposed where the reactivation of the prognoser is governed by the reactivation events. Finally, the proposed methods are validated by experiment results.

Robust model predictive control for switched linear systems

Developed is a robust model predictive control scheme for a class of discrete-time switched linear systems. The system is described in linear fractional transformation form in the presence of model uncertainty and induced norm bounded disturbances. The objective is to minimize the upper bound of an infinite horizon cost function subject to a terminal inequality. A Lyapunov function analysis for the switched system shows guaranteed closed-loop stability. Taking into account the switching structure of the system, the predictive control design problem along with sufficient conditions for the existence of a solution is expressed in terms of Riccati–Metzler inequalities. Then, these inequalities are turned into a linear matrix inequality feasibility problem. Three cases are analyzed to demonstrate the performance and effectiveness of the proposed robust model predictive controller for switched discrete-time linear systems.

Balanced truncation for temporal- and spatial-LPV interconnected systems based on the full block S-procedure

ABSTRACTThis paper presents a technique for model order reduction of exponentially stable temporal- and spatial-LPV (Linear Parameter Varying) interconnected systems represented in linear fractional transformation form, based on the application of full block S-procedure. The first stage of the proposed technique is to balance the system. The balancing transformation is applied to the nominal system; the reduced system is connected with the same scheduling block as the original system. The reduced models preserve exponential stability and the spatial structure of the system. The proposed approach simplifies the problem of model order reduction for LPV systems in the sense of avoiding gridding over the scheduling parameter range. The results are illustrated with the application to an experimentally identified spatially interconnected model of an actuated beam. In addition, extended reachability and observability matrices are presented.

Robust output regulation of uncertain linear fractional transformation systems with application to non‐linear Chua's circuit

This study deals with the robust H ∞ output regulation problem for a class of uncertain systems in a linear fractional transformation form. The problem is addressed under a measurement output-feedback framework, i.e. no full information of both plant and exosystem states is assumed for feedback control use. A new robust control law with a novel output regulator structure is proposed, based on which sufficient conditions for robust output regulability and L 2 stability are established in terms of linear matrix equations plus a set of linear matrix inequalities. As a result, the optimal H ∞ output regulation control solution can be synthesised effectively via convex optimisation. Finally, the proposed robust output regulation design scheme will be applied to solve the chaos tracking control problem for the non-linear Chua's circuit.

Almost output regulation of LFT systems via gain-scheduling control

ABSTRACTOutput regulation of general uncertain systems is a meaningful yet challenging problem. In spite of the rich literature in the field, the problem has not yet been addressed adequately due to the lack of an effective design mechanism. In this paper, we propose a new design framework for almost output regulation of uncertain systems described in the general form of linear fractional transformation (LFT) with time-varying parametric uncertainties and unknown external perturbations. A novel semi-LFT gain-scheduling output regulator structure is proposed, such that the associated control synthesis conditions guaranteeing both output regulation and disturbance attenuation performance are formulated as a set of linear matrix inequalities (LMIs) plus parameter-dependent linear matrix equations, which can be solved separately. A numerical example has been used to demonstrate the effectiveness of the proposed approach.

Vibration control of structural systems via robust non-fragile sampled-data control scheme

This paper investigates the dissipative non-fragile output feedback sampled-data control problem for uncertain structural systems. In the proposed system, uncertainties are assumed to be time-varying and bounded, which are considered in the linear fractional transformation form. The main objective of this paper is to design a non-fragile sampled-data controller such that the resulting closed-loop system is strictly (Q,S,R)-α-dissipative. On the basis of a suitable Lyapunov–Krasovskii functional and linear matrix inequality technique, a new set of sufficient conditions is established to achieve the required result for both nominal and uncertain systems. In particular, an output feedback dissipative sampled-data controller is designed by solving a set of matrix inequalities. More precisely, Schur complement lemma, free weighting matrix approach and Wirtinger-based double integral inequality are utilized to substantially simplify the derivation of the main results. Finally, simulations based on structural systems are conducted to illustrate the effectiveness of the proposed control scheme.

Hybrid bond graph model based for robust fault detection and isolation

The article deals with the robust fault detection and isolation based on linear fractional transformation hybrid bond graph. The main objective is to improve the robustness of fault detection step in the presence of parametric uncertainties in hybrid models in order to minimize the false alarms. The scientific interest of the proposed method is the use of only one tool – the bond graph not only for dynamic modelling of uncertain hybrid system but also for generation of adaptive thresholds needed in residual evaluation step. For this task, hybrid bond graph uncertain model approach with controlled junctions in linear fractional transformation form (allowing to represent graphically the parametric uncertainties) is first proposed to represent all the modes of the hybrid system. Second, based on causal and structural properties of the hybrid bond graph, analytical redundancy relations (with nominal and uncertain part) valid for all the modes are then derived systematically from the diagnostic hybrid bond graph. An application to a hydraulic system is used to illustrate this method.

Output feedback control of linear fractional transformation systems subject to actuator saturation

ABSTRACTIn this paper, the control problem for a class of linear parameter varying (LPV) plant subject to actuator saturation is investigated. For the saturated LPV plant depending on the scheduling parameters in linear fractional transformation (LFT) fashion, a gain-scheduled output feedback controller in the LFT form is designed to guarantee the stability of the closed-loop LPV system and provide optimised disturbance/error attenuation performance. By using the congruent transformation, the synthesis condition is formulated as a convex optimisation problem in terms of a finite number of LMIs for which efficient optimisation techniques are available. The nonlinear inverted pendulum problem is employed to demonstrate the effectiveness of the proposed approach. Moreover, the comparison between our LPV saturated approach with an existing linear saturated method reveals the advantage of the LPV controller when handling nonlinear plants.

Robust reliable H∞ control for fuzzy systems with random delays and linear fractional uncertainties

This article studies the reliable robust stabilization problem for a class of uncertain Takagi–Sugeno (TS) fuzzy systems with time-varying delays. The delay factor is assumed to be random delay which belongs to a given interval and parameter uncertainties are considered with linear fractional transformation form. By implementing a proper novel Lyapunov functional together with linear matrix inequality (LMI) approach, a new set of delay-dependent sufficient conditions is derived to guarantee the asymptotic stability of TS fuzzy system with a prescribed H∞ performance index. Further, a reliable robust H∞ control design with an appropriate gain matrix has been derived to achieve the robust asymptotic stability for uncertain TS fuzzy system. Further, Schur complement and Jensen's integral inequality are used to simplify the derivation in the main results. The set of sufficient conditions is established using the relationship among the random time-varying delay and its lower and upper bounds, which can be easily solved by MATLAB LMI toolbox. Finally, an illustrative example based on the truck-trailer model is provided to show the effectiveness of the proposed new design technique.

Robust ℋ2 and ℋ∞ switched feedforward control of uncertain LFT systems

This paper presents a new robust switched feedforward control scheme for a class of uncertain systems described in a standard linear fractional transformation form. First, the analysis conditions for switching stability are derived by using a piecewise Lyapunov function incorporated with the min-switching control technique. Based on the analysis results, the synthesis conditions are then formulated as a special type of bilinear matrix inequalities, which can be solved by means of linear matrix inequalities and line search. Both robust and feedforward control problems are considered. The proposed robust switched control scheme outperforms existing robust feedforward control approaches for uncertain systems based on single quadratic Lyapunov function, and leads to less conservative control design. Numerical examples will be used to illustrate the effectiveness and advantages of the proposed results. Copyright © 2015 John Wiley & Sons, Ltd.

Gain scheduling output feedback control of linear plants with actuator saturation

From a gain-scheduling perspective, we will study the output feedback control problem for linear systems with some of control channels subject to actuator saturation. This includes the scenario of all actuator saturation as a special case. A feedback controller, expressed in the form of linear fractional transformation, is proposed to guarantee regional stability of the closed-loop system and provide disturbance/error attenuation measured in L2 gain. The resulting synthesis condition is formulated as linear matrix inequalities (LMIs) and can be solved efficiently. Moreover, explicit formulas are derived to calculate controller gains, which reduces the computational cost compared to the method of directly solving the LMI-based condition. Numerical examples are provided to demonstrate the proposed saturation control approach.

Integrated Diagnosis and Prognosis of Uncertain Systems: A Bond Graph Approach

Bond Graph (BG) methodology is used to model the dynamic uncertain systems. Uncertainty is considered on the system parameters in form of intervals. The uncertain parameters are allowed to deviate within their prescribed interval limits. Single fault hypothesis is considered in this work such that the parameter undergoing degradation is known a priori. A new method for generation of interval valued thresholds is briefly described in the framework of BG models in Linear Fractional Transformation form. The diagnostic module is formed using such thresholds which detect the beginning of degradation of a parameter in the real system. The new concept of Interval Extension of Analytical Redundancy Relations (IE-ARRs) is introduced which consider the parametric uncertainties and the evolution of degrading parameter in real time. Then, the Centre and Range method for fitting linear regression models to interval symbolic data is adapted to fit piece wise linear models to the interval valued times series data of IE-ARRs. Further, the new concept of generation of failure thresholds from a nominal system model is introduced and developed. Finally, the fitted linear model is used to estimate the remaining useful life of the parameter under degradation. Simulations are carried out on an example DC motor model. Linear and non-linear parametric degradations are considered. Results are presented in form of simulations.

Gain Scheduling Output Feedback Control of Linear Plants with Partial Actuator Saturation

From a gain-scheduling control perspective, we will study the output feedback control problem for linear systems with some of control channels subject to actuator saturation. This includes the scenario of all actuator saturation as a special case. A feedback controller, expressed in the form of linear fractional transformation, is proposed to guarantee regional stability of the closed-loop system and minimizes disturbance/error effect measured in ***L2 gain. The resulting synthesis condition is formulated as linear matrix inequalities (LMIs) and can be solved efficiently. A modified inverted pendulum is utilized to demonstrate the proposed approach.

3자유도 차량모델을 이용한 차선추종 µ 제어기 설계

Many articles have been published about a 2-degree of freedom model that includes the lateral and yaw motions for controller synthesis in intelligent transport system applications. In this paper, a 3-degree of freedom linear model that includes the roll motion is developed to design a robust steering controller for lane following maneuvers using <TEX>${\mu}$</TEX>-synthesis. This linear perturbed system includes a set of parametric uncertainties in cornering stiffness and unmodelled dynamics in steering actuators. The state-space model with parametric uncertainties is represented in linear fractional transformation form. Design purpose can be obtained by properly choosing the frequency dependent weighting functions. The objective of this study is to keep the tracking error and steering input energy small in the presence of variations of the cornering stiffness coefficients. Furthermore, good ride quality has to be achieved against these uncertainties. Frequency-domain analyses and time-domain numerical simulations are carried out in order to evaluate these performance specifications of a given vehicle system. Finally, the simulation results indicate that the proposed robust controller achieves good performance over a wide range of uncertainty for the given maneuvers.

Stability Margin Clearance of Missile Autopilot Using μ Anylysis

A new method of stability margin clearance of missile autopilot is described. First, a model of linear fractional transformation form is established for missile has uncertain parameters; then, the system stability margin requirements are seen as a virtual uncertainty and introduced into the model; Finally, the model is analyzed, and the clearance results about the missile autopilot performance is obtained. The results show this is a more efficient and accurate clearance method than traditional ones.

3910 既約分解表現を利用したパラメータ同定と構造物の故障診断への応用(G10-2 計測,診断,推定法,G10 機械力学・計測制御)

It is called modeling or system identification to express a target system by the mathematical model, is important for model based technology and many researches have been presented. In this paper, we have proposed a new method for identifying only the specified physical 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 arbitrariness on the stable factors of parametrizing. 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. In this paper we describe an experimental study of applying the method to fault diagnosis.