Influence of the Tensor Product Model Representation of qLPV Models on the Feasibility of Linear Matrix Inequality Based Stability Analysis

The paper investigates and proves the statement, that the convex hull of the polytopic tensor product (TP) model representation influences the feasibility of linear matrix inequality (LMI) based stability analysis methods. The proof is based on a complex stability analysis example of a given quasi linear parameter varying (qLPV) state‐space model. Specifically, the three degree of freedom (3‐DoF) aeroelastic wing section model including Stribeck friction is used as the tool for the example model. The proof is achieved by utilizing TP model transformation and LMI based tools. As a first step, numerous TP model type control solutions holding different convex hulls are systematically derived of the qLPV model via LMI based control design methods. As a second step, each control solution is further equivalently transformed for different TP model representations holding different convex hulls. Finally, the stability of all solutions over all TP model representations are checked via LMI based stability analysis methods. As a result of the two steps, a two dimensional (2D) convex hull space is attained for the 3‐DoF aeroelastic wing section model. The two dimensions are denoted by the LMI based control design and the LMI based stability analysis for different convex hulls. Based on the numerical results, a detailed, comprehensive analysis is provided. The paper as a novelty proves the statement, that the polytopic TP model representation of a given control solution strongly influences the feasibility of LMI based stability analysis methods.     

Minimal Volume Simplex (MVS) Polytopic Model Generation and Manipulation Methodology for TP Model Transformation

The vertices of a polytopic LPV/qLPV model determine the feasibility of any linear matrix inequality (LMI) based control design as well as the achievable control performance, since the polytope typically includes not only the actual system dynamics. This paper proposes a powerful method to determine suitable polytopic qLPV Tensor‐Product model forms that allows for fine tuning through constraints on the locations of the vertices. The paper shows that these tuning methods are capable of effectively influencing the closed‐loop control performance results from LMI‐based multi‐objective optimisation. The advantages of the proposed methods are their tractability, low computational complexity, as well as their determinism. The key idea behind the proposed methods is to determine vertices in each decoupled parameter‐dimensions of the TP model in such a way that they form a minimal volume enclosing simplex fulfilling further fine tuning conditions on its shape. This is achieved based on the extension of the Minimal Volume Simplex Analysis (MVSA) method, where the volume of the enclosing simplex polytope is minimized by applying Sequential Quadratic Programming with majorization minimization strategy extended with a set of additional constraints influencing the location of certain vertices according to physically established considerations. Besides the comprehensive presentation of the proposed design methodology an expressive numerical example is provided to demonstrate its effectiveness and usability.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

Computational relaxed TP model transformation: restricting the computation to subspaces of the dynamic model

The tensor‐product (TP) model transformation is a recently proposed numerical method capable of transforming linear parameter varying state‐space models to the higher order singular value decomposition (HOSVD) based canonical form of polytopic models. It is also capable of generating various types of convex TP models, a type of polytop models, for linear matrix inequality based controller design. The crucial point of the TP model transformation is that its computational load exponentially explodes with the dimensionality of the parameter vector of the parameter‐varying state‐space model. In this paper we propose a modified TP model transformation that leads to considerable reduction of the computation. The key idea of the method is that instead of transforming the whole system matrix at once in the whole parameter space, we decompose the problem and perform the transformation element wise and restrict the computation to the subspace where the given element of the model varies. The modified TP model transformation can readily be executed in higher dimensional cases when the original TP model transformation fails. The effectiveness of the new method is illustrated with numerical examples. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

GLOBAL ASYMPTOTIC STABILIZATION OF THE PROTOTYPICAL AEROELASTIC WING SECTION VIA TP MODEL TRANSFORMATION

A comprehensive analysis of aeroelastic systems has shown that these systems exhibit a broad class of pathological response regimes when certain types of non‐linearities are included. In this paper, we propose a design method of a state‐dependent non‐linear controller for aeroelastic systems that includes polynomial structural non‐linearities. The proposed method is based on recent numerical techniques such as the Tensor Product (TP) model transformation and the Linear Matrix Inequality (LMI) control design methods within the Parallel Distributed Compensation (PDC) frameworks. In order to link the TP model transformation and the LMI's in the proposed design method, we extend the TP model transformation with a further transformation. As an example, a controller is derived that ensures the global asymptotic stability of the prototypical aeroelastic wing section via one control surface, in contrast with previous approaches which have achieved local stability or applied additional control actuator on the purpose of achieving global stability. Numerical simulations are used to provide empirical validation of the control results. The effectiveness of the controller design is compared with a former approach.     

Improved control performance of the 3‐DoF aeroelastic wing section: a TP model based 2D parametric control performance optimization

Based on the most recent Tensor Product model transformation solutions, the paper presents an improved control performance for the most recent version of the three Degree of Freedom aeroelastic wing section model including Stribeck friction, according to signals pitch, plunge, trailing edge and control value, based on practical engineering criteria such as overshoot, undershoot, signal end values and settling time. This is achieved through proposing a novel two dimensional parametric convex hull manipulation based method for Tensor Product model transformation based Control Design Frameworks. The approach provides two TP model representations for the different requirements of the controller and observer of a given model, opening the possibility to utilize the TP model transformation's convex hull manipulation potential in control performance optimization for a separate optimization of the two TP model representations. Numerical simulation results are provided to illustrate the control performance improvements of the aeroelastic wing section model through the proposed 2D parametric convex hull manipulation based design method.     

基于状态观测器的约束鲁棒模型预测控制

模型不确定情况下的鲁棒问题是模型预测控制的一个根本问题。本文采用线性矩阵不等式(LMI)，研究多模型不确定性描述情况下的鲁棒模型预测控制问题。在输入输出约束条件下，最小化最坏情况下的无穷时域目标函数，获得保证系统稳定的基于状态观测器的状态反馈增益并且给出观测器增益的设计方法。实例说明算法可行且保证闭环系统渐近稳定。     

基于二维混合模型和状态观测器的重复控制设计

针对一类正则线性系统, 提出一种基于状态观测器和二维混合模型的重复控制系统设计方法. 首先, 通过构造一个状态观测器来重构系统的状态, 建立基于重构状态的线性控制律. 然后, 通过独立地考虑重复控制系统的连续控制过程与离散学习行为, 给出基于状态观测器和重构状态反馈的连续/离散二维混合模型. 针对这个混合模型, 运用二维Lyapunov泛函方法, 以线性矩阵不等式(Linear matrix inequality, LMI)的形式给出重复控制系统存在重复控制器和状态观测器的充分条件, 所给条件可用Matlab工具箱方便地求解. 数值仿真验证了本文所提方法的有效性.     

基于LPV观测器的永磁同步电机反推控制

针对永磁同步电机(Permanent Magnet Synchronous Motor,PMSM)无速度传感器转速跟踪控制精度问题,提出了一种基于线性变参数(Lineeo Parameteo Varying,LPV)转速观测器的永磁同步电机反推控制方法。该方法首先根据PMSM的LPV模型,推导出电机的凸胞形顶点方程;然后利用Lyapunov稳定性理论,获得了基于线性矩阵不等式(Lineeo Matrix Inequality,LMI)的观测器设计方法,构造了PMSM的LPV观测器,实现对电机转速及定子交轴电流的重构;最后运用反推控制策略,设计电机闭环系统控制器,实现对电机转速的高精度跟踪控制。仿真结果表明,该方法相较与传统PI矢量控制,跟踪精度高、响应快、抗负载扰动强,对实现PMSM的无速度传感器高精度转速跟踪控制具有重要意义。     

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.     

Parameter dependent H∞ control by finite dimensional LMI optimization: application to trade‐off dependent control

In this paper, we consider the design of an H_∞ trade‐off dependent controller, that is, a controller such that, for a given Linear Time‐Invariant plant, a set of performance trade‐offs parameterized by a scalar θ is satisfied. The controller state space matrices are explicit functions of θ. This new problem is a special case of the design of a parameter dependent controller for a parameter dependent plant, which has many application in Automatic Control. This last design problem can be naturally formulated as a convex but infinite dimensional optimization problem involving parameter dependent Linear Matrix Inequality (LMI) constraints. In this paper, we propose finite dimensional (parameter independent) LMI constraints which are equivalent to the parameter dependent LMI constraints. The parameter dependent controller design is then formulated as a convex finite dimensional LMI optimization problem. The obtained result is then applied to the trade‐off dependent controller design. A numerical example emphasizes the strong interest of our finite dimensional optimization problem with respect to the trade‐off dependent control application. Copyright © 2005 John Wiley & Sons, Ltd.     

基于状态观测器的多面体不确定随机广义系统鲁棒预测控制

针对It?o型多面体不确定随机广义系统,提出一种离线观测器型鲁棒预测控制器的综合方法.通过构造带有误差项的增广随机Lyapunov函数,运用多维It?o公式和LMI方法,将“min-max”随机规划问题等价转化为一组线性矩阵不等式的求解问题;给出了控制器存在的充分条件和参数表达式,证明了初始时刻的可行解可以保证闭环广义系统的随机容许性.仿真算例验证了该方法的有效性.     

具有状态观测器的鲁棒重复控制系统设计

针对一类时变不确定系统,研究基于状态观测器的鲁棒重复控制系统设计问题.通过状态观测器重构系统状态,利用重构状态进行状态反馈.基丁2维系统稳定性理论,应川线性矩阵不等式(LMI)方法,推导出了重复控制系统存在状态观测器和重复控制器的一个LMI条件,并用LMI的町行解给出了状态观测器和重复控制器参数的具体形式.最后用数值仿真验证了本文所提方法的有效性.     

An LMI approach to design observer for unknown inputs Takagi‐Sugeno fuzzy models

This paper considers the design of an observer for a Takagi‐Sugeno (T‐S) fuzzy model subject to unknown inputs affecting states and outputs of the system simultaneously. Uncertainties affecting state matrices are also considered. Based on the Lyapunov method, sufficient conditions in Linear Matrix Inequalities (LMI) terms are proposed to design the given unknown input T‐S observer. In order to improve the performances of the proposed T‐S observer, the pole placement in an LMI region is also considered. An numerical example is given to illustrate the validity of the derived results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Fuzzy observer-based control for maximum power-point tracking of a photovoltaic system

This paper presents a novel fuzzy control design method for maximum power-point tracking (MPPT) via a Takagi and Sugeno (TS) fuzzy model-based approach. A knowledge-dynamic model of the PV system is first developed leading to a TS representation by a simple convex polytopic transformation. Then, based on this exact fuzzy representation, a H_∞ observer-based fuzzy controller is proposed to achieve MPPT even when we consider varying climatic conditions. A specified TS reference model is designed to generate the optimum trajectory which must be tracked to ensure maximum power operation. The controller and observer gains are obtained in a one-step procedure by solving a set of linear matrix inequalities (LMIs). The proposed method has been compared with some classical MPPT techniques taking into account convergence speed and tracking accuracy. Finally, various simulation and experimental tests have been carried out to illustrate the effectiveness of the proposed TS fuzzy MPPT strategy.     

IS A MIXED DESIGN OF OBSERVER‐CONTROLLERS FOR TIME‐DELAY SYSTEMS INTERESTING?

This paper deals with H∞ observer‐based feedback control for linear time‐delay systems, in the framework of delay independent stability. We will propose a new LMI solution to observer‐controller design that ensures a disturbance attenuation level for the controlled output as well as for the state estimation error, which is an open problem. This will be compared with a well‐known solution and with a usual strategy in control which consists in designing the observer and the controller separately. Our aim is to try to bring a positive answer to the following question: is there an interest to solve the problem in a single (unique) formulation or should we design separately the observer and the controller? An application to a wind tunnel model is provided to emphasize the interest of the given results, particularly in comparison with existing results on H∞ observer‐based control.     

Robust output‐control for uncertain linear systems: Homogeneous differentiator‐based observer approach

This paper deals with the design of a robust control for linear systems with external disturbances using a homogeneous differentiator‐based observer based on a implicit Lyapunov function approach. Sufficient conditions for stability of the closed‐loop system in the presence of external disturbances are obtained and represented by linear matrix inequalities. The parameter tuning for both controller and observer is formulated as a semi‐definite programming problem with linear matrix inequalities constraints. Simulation results illustrate the feasibility of the proposed approach and some improvements with respect to the classic linear observer approach. Copyright © 2016 John Wiley & Sons, Ltd.     

中立型时滞系统基于状态观测器的控制设计-LMI方法

研究一类中立型时滞系统基于状态观测器的反馈控制问题。目的是利用线性矩阵不等式(LMI)方法设计状态观测器和动态输出反馈控制器,使得相应的闭环系统渐近稳定。最后给出一个数值例子验证了本文所给结果的有效性。     

一类不确定离散非线性时滞系统保性能控制

针对状态不完全可测的一类不确定连续离散时间非线性系统,基于状态观测器研究其保性能控制问题。采用T-S模型对非线性系统进行建模,根据李雅普诺夫稳定性理论,基于线性矩阵不等式(LMI),利用分散化并行分布补偿(PDC)的方法设计了基于观测器的控制器,给出了实现该非线性系统保性能控制的充分条件。在此基础上通过求解相应的线性矩阵不等式组给出了控制器和观测器的设计方法,并通过数值仿真验证了提出的方法,仿真结果表明该方法是正确的。