Subspace identification of Bilinear and LPV systems for open- and closed-loop data

In this paper we present a novel algorithm to identify LPV systems with affine parameter dependence operating under open- and closed-loop conditions. A factorization is introduced which makes it possible to form a predictor that predicts the output, which is based on past inputs, outputs, and scheduling data. The predictor contains the LPV equivalent of the Markov parameters. Using this predictor, ideas from closed-loop LTI identification are developed to estimate the state sequence from which the LPV system matrices can be constructed. A numerically efficient implementation is presented using the kernel method. It turns out that if structure is present in the scheduling sequence the computational complexity reduces even more.     

Subspace identification of MIMO LPV systems using a periodic scheduling sequence

A novel subspace identification method is presented which is able to reconstruct the deterministic part of a multivariable state-space LPV system with affine parameter dependence, in the presence of process and output noise. It is assumed that the identification data is generated with the scheduling variable varying periodically during the course of the identification experiment. This allows to use methods from LTI subspace identification to determine the column space of the time-varying observability matrices. It is shown that the crucial step in determining the original LPV system is to ensure the obtained observability matrices are defined with respect to the same state basis. Once the LPV model has been identified, it is valid for other nonperiodic scheduling sequences as well.     

LPV state-feedback control of a control moment gyroscope

This paper presents the design and successful experimental validation of a linear parameter-varying (LPV) control strategy for a four-degrees-of-freedom control moment gyroscope (CMG). The MIMO plant is highly coupled and nonlinear. First, a linearized model with moving operating point is used to construct an LPV model. Then, a gridding-based LPV state-feedback control is designed that clearly outperforms linear time-invariant (LTI) controllers. Moreover, a way is proposed to select pre-filter gains for reference inputs that can be generalized to a large class of mechanical systems. Overall, the strategy allows a simple implementation in real-time and may be of interest for applications such as attitude control of a satellite. The method is applied to a laboratory scale CMG, and experimental results illustrate that the proposed LPV controller achieves indeed a better performance in a much wider range of operation than linear controllers reported in the literature.     

A virtual actuator and sensor approach for fault tolerant control of LPV systems

In this paper, a fault tolerant control (FTC) strategy using virtual actuators and sensors for linear parameter varying (LPV) systems is proposed. The main idea of this FTC method, initially developed for LTI systems, is to reconfigure the control loop such that the nominal controller could still be used without need of retuning it. The plant with the faulty actuator/sensor is modified adding the virtual actuator/sensor block that masks the actuator/sensor fault. The suggested technique is an active FTC strategy that reconfigures the virtual actuator/sensor on-line taking into account faults and operating point changes. The stability of the reconfigured control loop is guaranteed if the faulty plant is stabilizable/detectable. The LPV virtual actuator/sensor is designed using polytopic LPV techniques and linear matrix inequalities (LMIs). A two-tank system simulator is used to assess the performance of the proposed method. In particular, it is shown that the application of the proposed technique results in an improvement, in terms of performance, with respect to the LTI counterpart.     

A review on data-driven linear parameter-varying modeling approaches: A high-purity distillation column case study

Model-based control strategies are widely used for optimal operation of chemical processes to respond to the increasing performance demands in the chemical industry. Yet, obtaining accurate models to describe the inherently nonlinear, time-varying dynamics of chemical processes remains a challenge in most model-based control applications. This paper reviews data-driven, Linear Parameter-Varying (LPV) modeling approaches for process systems by exploring and comparing various identification methods on a high-purity distillation column case study. Several LPV identification methods that utilize input–output and series expansion model structures are explored. Two LPV identification perspectives are adopted: (i) the local approach, which corresponds to the interpolation of Linear Time-Invariant (LTI) models identified at different steady-state operating points of the system and (ii) the global approach, where a parametrized LPV model structure is identified directly using a global data set with varying operating points. For the local approach, various model interpolation schemes are studied under an Output Error (OE) noise setting, whereas in the global case, a polynomial parametrization based OE prediction error minimization approach, an Orthonormal Basis Functions (OBFs) based model estimator and a Least-Square Support Vector Machine (LS-SVM) based non-parametric approach are investigated. Through extensive simulation studies, the aforementioned LPV identification approaches are analyzed in terms of the attainable model accuracy and local frequency response behavior of the obtained models. Recommendations are provided to achieve adequate choice between the methods for a particular process system at hand.     

Robust multi-parametric model predictive control for LPV systems with application to anaesthesia

We present a multi-parametric model predictive controller (mpMPC) for discrete-time linear parameter-varying (LPV) systems based on the solution of the mpMPC problem for discrete-time linear time-invariant (LTI) systems. The control method yields a controller that adapts to parameter changes of the LPV system. This is accomplished by an add-on unit to the implementation of the mpMPC for LTI systems. No modification of the optimal mpMPC solution for LTI systems is needed. The mpMPC for LPV systems is entirely based on simple computational steps performed on-line. This control design method could improve the performance and robustness of a mpMPC for LPV systems with slowly varying parameters. We apply this method to process systems which suffer from slow variation of system parameters due, for example, to aging or degradation. As an illustrative example the reference tracking control problem of the hypnotic depth during intravenous anaesthesia is presented: the time varying system matrix mimics an external disturbance on the hypnotic depth. In this example the presented mpMPC for LPV systems shows a reduction of approximately 60% of the reference tracking error compared to the mpMPC for LTI systems.     

Predictive LPV control of a liquid–gas separation process

The problem of controlling a liquid–gas separation process is approached by using LPV control techniques. An LPV model is derived from a nonlinear model of the process using differential inclusion techniques. Once an LPV model is available, an LPV controller can be synthesized. The authors present a predictive LPV controller based on the GPC controller [Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part I. Automatica 1987;23(2):137–48; Clarke D, Mohtadi C, Tuffs P. Generalized predictive control – Part II. Extensions and interpretations. Automatica 1987;23(2):149–60]. The resulting controller is denoted as GPC–LPV. This one shows the same structure as a general LPV controller [El Gahoui L, Scorletti G. Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 1996;32(9):1273–84; Scorletti G, El Ghaoui L. Improved LMI conditions for gain scheduling and related control problems. International Journal of Robust Nonlinear Control 1998;8:845–77; Apkarian P, Tuan HD. Parametrized LMIs in control theory. In: Proceedings of the 37th IEEE conference on decision and control; 1998. p. 152–7; Scherer CW. LPV control and full block multipliers. Automatica 2001;37:361–75], which presents a linear fractional dependence on the process signal measurements. Therefore, this controller has the ability of modifying its dynamics depending on measurements leading to a possibly nonlinear controller. That controller is designed in two steps. First, for a given steady state point is obtained a linear GPC using a linear local model of the nonlinear system around that operating point. And second, using bilinear and linear matrix inequalities (BMIs/LMIs) the remaining matrices of GPC–LPV are selected in order to achieve some closed loop properties: stability in some operation zone, norm bounding of some input/output channels, maximum settling time, maximum overshoot, etc., given some LPV model for the nonlinear system. As an application, a GPC–LPV is designed for the derived LPV model of the liquid–gas separation process. This methodology can be applied to any nonlinear system which can be embedded in an LPV system using differential inclusion techniques.     

Fault detection using interval LPV models in an open-flow canal

This paper proposes a new method of fault detection using Linear Parameter Varying (LPV) interval models and its application to an open-flow canal. The use of such models is motivated because the parameters and transport delay in the canal transfer function model vary with the operating point. LPV models allow to consider these variations by characterizing the parameters/delay variation law with the operating point while intervals are used to bound the parameter/delay uncertainty. Additionally, a LPV parameter estimation algorithm that allows to estimate parameter/delay uncertainty intervals is also proposed. As an application case study, an open-flow canal system based on the Lunax dam-gallery system located in France is used to show the effectiveness of the proposed method to detect faults. The satisfactory results obtained allow to assess the effectiveness of the proposed approach.     

Bridging the gap between theory and practice in LPV fault detection for flight control actuators

Two different approaches for fault detection, the geometric and the detection filter based methods, are compared in the paper from practical aspects, using the linear parameter-varying (LPV) framework. Presenting two designs allows a comparison of global, system level, and local component level fault detection methods with special emphasis on their relevance to aircraft industry. Practical engineering design decisions are highlighted via applying them to a high-fidelity commercial aircraft problem. The successive steps of the design, including fault modeling, LPV model generation, and LPV FDI filter synthesis, including implementation aspects, are discussed. Results are presented according to the industrial assessment perspectives phrased within the EU ADDSAFE project.     

Linear parameter varying PID controller design for charge control of a spark-ignited engine

This paper presents the design and experimental test of a fixed-structure LPV controller for the charge control of a spark-ignition engine. A nonlinear model of the plant is transformed into an affine LPV model in the form of an LFT representation. Using a hybrid evolutionary-algebraic synthesis approach that combines LMI techniques based on K-S iteration with evolutionary search, a scheduled PID controller is designed. To reduce conservatism, the technique of quadratic separators is used in the analysis step. To improve tracking behavior, the gain scheduled feedback controller is supported by an LTI feedforward controller. The controller has been implemented on a standard electronic control unit, and experimental results on a test car illustrate that it meets the performance requirements in a wide range of operation.     

Detection filter design for LPV systems—a geometric approach

This paper investigates fault detection and isolation of linear parameter-varying (LPV) systems by using parameter-varying (C,A)-invariant subspace and parameter-varying unobservability subspaces. The so called “detection filter” approach, formulated as the fundamental problem of residual generation (FPRG) for linear time-invariant (LTI) systems, is extended for a class of LPV systems. The question of stability is addressed in the terms of Lyapunov quadratic stability by using linear matrix inequalities. The results are applied to the model of a generic small commercial aircraft.     

Estimation and model predictive control of non-linear batch processes using linear parameter varying models

In this paper, feedback control is implemented for batch processes using linear models which describe the batch dynamics locally along its optimal trajectory. A Linear Parameter Varying (LPV) model obtained by interpolation between these multiple models is used to emulate the behaviour of the non-linear batch. The interpolation functions and state estimates are computed using a recursive Bayesian technique. The control technique is based on model predictive control (MPC) which is used for regulation and targeting the product specifications at the end of the batch.     

Fast geometry scaling of miniaturized microwave couplers with power split correction

Redesigning a microwave circuit for various operating conditions is a practically important yet challenging problem. The purpose of this article is development and presentation of a technique for fast geometry scaling of miniaturized microwave couplers with respect to operating frequency. Our approach exploits an inverse surrogate model constructed using several reference designs that are optimized for a set of operating frequencies within a range of interest. For the sake of computational efficiency, the reference designs are obtained for an equivalent network model of the coupler. The surrogate directly predicts the optimum values of geometry parameters of the structure at hand corresponding to a requested operating frequency. By introducing appropriate correction, the model allows for coupler scaling at the EM simulation model level. Because the surrogate does not carry information about the power split ratio of the coupler, an additional analytical corrective procedure is developed to ensure an equal power split of scaled structure. The computational cost of the scaling procedure corresponds to only two EM analyses of the circuit at hand (including both correction steps). The operation and performance of our technique is demonstrated using a compact microstrip rat‐race coupler scaled for the operating frequency range of 0.5‐2.5 GHz. Experimental validation is also provided.     

Missile autopilot design via a multi‐channel LFT/LPV control method

In this paper, the missile pitch‐axis autopilot design is revisited using a new and recently available linear parameter‐varying (LPV) control technique. The missile plant model is characterized by a linear fractional transformation (LFT) representation. The synthesis task is conducted by exploiting new capabilities of the LPV method: firstly, a set of H₂/H_∞ criteria defined channel‐wise is considered; secondly, different Lyapunov and scaling variables are used for each channel/specification which is known to reduce conserva tism; and finally, the controller gain‐scheduling function is constructed as affine matrix‐valued function in the polytopic co‐ordinates of the scheduled parameter. All these features are examined and evaluated in turn for the missile control problem. The method is shown to provide additional flexibility to tradeoff conflicting and demanding performance and robustness specifications for the missile while preserving the practical advantage of previous single‐objective LPV methods. Finally, the method is shown to perform very satisfactorily for the missile autopilot design over a wide range of operating conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

应用于传感器系统噪声模拟的随机数发生器

传统用于电压噪声模拟的随机数发生器受到工作电压的限制,且产生的随机序列复杂度一般。针对这2个问题,研究了不同工作电压下的电路单元特性,设计相应的随机数电路结构,在保证输出的同时,解决了工作电压的变化带来的时序问题。此外,通过采用非线性的反馈逻辑,提高了随机序列的复杂度。     

一种基于LPV系统的$H_infty$切换控制器设计方法

针对线性参数变化(LPV)系统提出一种切换控制器参数化设计方法.基于Youla参数化思想,将控制器设计过程分解为两个步骤.首先,设计一个中心控制器保证闭环系统的全局$H_\infty$性能;其次,将参数变化区域划分为若干个子区域,在每个子区域中将中心控制器进行线性分式变换,得到切换控制器自由参数的状态空间实现,将切换控制器转换为自由参数之间的切换.基于所提出的切换LPV控制器线性分式变换实现方法,不仅可以保证在任意切换的情况下子系统各自局部的$H_\infty$性能,而且可以保证整个闭环系统满足某一整体的$H_\infty$性能,并通过仿真结果验证了所提出方法的有效性.     

Generation of a reduced-order LPV/LFT model from a set of large-scale MIMO LTI flexible aircraft models

In the civilian aviation industry, the aeroelastic behavior of an aircraft is often modeled at frozen flight and mass configurations using high fidelity numerical tools. Unfortunately, the resulting large-scale models cannot be handled in such form by modern analysis and control techniques, which generally require the considered models to be written as low-order Linear Fractional Representations (LFR). In this context, a methodology is described to derive a reduced-order Linear Parameter Varying (LPV) model from a reference set of large-scale Multiple Input Multiple Output (MIMO) Linear Time Invariant (LTI) models describing a given system at frozen configurations. The proposed approach is in two steps. The reference models are first reduced using recent advances in Krylov methods, leading to a set of low-order state-space representations with consistent state vectors. An LPV model is then obtained by polynomial approximation and converted into an LFR of reasonable size. A special effort is made to avoid data overfitting by using as simple as possible approximation formulas. The method is applied to a long-range commercial aircraft model developed in an industrial context: a set of large-scale flexible models linearized at different mass configurations is converted into a single low-order LPV model. More generally, any kind of purely numerical models for which the analytical structure is unknown can be considered.     

Quasi-LPV modeling,identification and control of a twin rotor MIMO system

This paper describes the quasi-linear parameter varying (quasi-LPV) modeling, identification and control of a Twin Rotor MIMO System (TRMS). The non-linear model of the TRMS is transformed into a quasi-LPV system and approximated in a polytopic way. The unknown model parameters have been calibrated by means of the non-linear least squares identification approach and validated against real data. Finally, an LPV state observer and state-feedback controller have been designed using an LPV pole placement method based on LMI regions. The effectiveness and performance of the proposed control approach have been proved both in simulation and on the real set-up.