Low-complexity linear parameter-varying modeling and control of a robotic manipulator

In this paper, a practical procedure for linear parameter-varying (LPV) modeling and identification of a robotic manipulator is presented, which leads to a successful experimental implementation of an LPV gain-scheduled controller. A nonlinear dynamic model of a two-degrees-of-freedom manipulator containing all important terms is obtained and unknown parameters which are required to construct an LPV model are identified. An important tool for obtaining a model of complexity low enough to be suitable for controller synthesis is the principle-component-analysis-based technique of parameter set mapping. Since the resulting quasi-LPV model has a large number of affine scheduling parameters and a large overbounding, parameter set mapping is used to reduce conservatism and complexity in controller design by finding tighter parameter regions with fewer scheduling parameters. A sufficient a posteriori condition is derived to assess the stability of the resulting closed-loop system. To evaluate the applicability and efficiency of the approximated model, a polytopic LPV gain-scheduled controller is synthesized and implemented experimentally on an industrial robot for a trajectory tracking task. The experimental results illustrate that the designed LPV controller outperforms an independent joint PD controller in terms of tracking performance and achieves a slightly better accuracy than a model-based inverse dynamics controller, while having a simpler structure. Moreover, it is shown that the LPV controller is more robust against dynamic parameter uncertainty.     

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.     

Fuzzy multi-objective design for a lateral missile autopilot

Gain scheduled control is one very useful control technique for linear parameter-varying (LPV) and nonlinear systems. A disadvantage of gain-scheduled control is that it is not easy to design a controller that guarantees the global stability of the closed-loop system over the entire operating range from the theoretical point of view. Another disadvantage is that the interpolation increases in complexity as number of scheduling parameters increases. As an improvement, this paper presents a gain-scheduling control technique, in which fuzzy logic is used to construct a model representing a quasi-LPV or a nonlinear missile and to perform a control law. The fuzzy inference system is generated using a multi-objective evolutionary algorithm to optimize the performance characteristics of the plant.     

Nonlinear internal model controller design for wastegate control of a turbocharged gasoline engine

Internal Model Control (IMC) has a great appeal for automotive powertrain control in reducing the control design and calibration effort. Motivated by its success in several automotive applications, this work investigates the design of nonlinear IMC for wastegate control of a turbocharged gasoline engine. The IMC design for linear time-invariant (LTI) systems is extended to nonlinear systems. To leverage the available tools for LTI IMC design, the quasi-linear parameter-varying (quasi-LPV) models are explored. IMC design through transfer function inverse of the quasi-LPV model is ruled out due to parameter variability. A new approach for nonlinear inversion, referred to as the structured quasi-LPV model inverse, is developed and validated. A fourth-order nonlinear model which sufficiently describes the dynamic behavior of the turbocharged engine is used as the design model in the IMC structure. The controller based on structured quasi-LPV model inverse is designed to achieve boost-pressure tracking. Finally, simulations on a validated high-fidelity model are carried out to show the feasibility of the proposed IMC. Its closed-loop performances are compared with a well-tuned PI controller with extensive feedforward and anti-windup built in. Robustness of the nonlinear IMC design is analyzed using simulations.     

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.     

LPV gain-scheduled control of SCR aftertreatment systems

Hydrocarbons, carbon monoxide and some of other polluting emissions produced by diesel engines are usually lower than those produced by gasoline engines. While great strides have been made in the exhaust aftertreatment of vehicular pollutants, the elimination of nitrogen oxide (NO _x ) from diesel vehicles is still a challenge. The primary reason is that diesel combustion is a fuel-lean process, and hence there is significant unreacted oxygen in the exhaust. Selective catalytic reduction (SCR) is a well-developed technology for power plants and has been recently employed for reducing NO _x emissions from automotive sources and in particular, heavy-duty diesel engines. In this article, we develop a linear parameter-varying (LPV) feedforward/feedback control design method for the SCR aftertreatment system to decrease NO _x emissions while keeping ammonia slippage to a desired low level downstream the catalyst. The performance of the closed-loop system obtained from the interconnection of the SCR system and the output feedback LPV control strategy is then compared with other control design methods including sliding mode, and observer-based static state-feedback parameter-varying control. To reduce the computational complexity involved in the control design process, the number of LPV parameters in the developed quasi-LPV (qLPV) model is reduced by applying the principal component analysis technique. An LPV feedback/feedforward controller is then designed for the qLPV model with reduced number of scheduling parameters. The designed full-order controller is further simplified to a first-order transfer function with a parameter-varying gain and pole. Finally, simulation results using both a low-order model and a high-fidelity and high-order model of SCR reactions in GT-POWER interfaced with MATLAB/SIMULINK illustrate the high NO _x conversion efficiency of the closed-loop SCR system using the proposed parameter-varying control law.     

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

In 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.     

风力机的线性变参数主动容错控制

针对风力机具有非线性和参数的不确定性的特征,提出了基于线性变参数(linear parameter varying,LPV)增益调度的风力机主动容错控制方法,降低故障对机组动态特性的影响.基于LPV凸分解方法,将风力机的非线性模型转化为具有凸多面体结构LPV模型,利用线性矩阵不等式(linear matrix inequalities,LMIs)技术对凸多面体各个顶点分别设计满足性能要求的控制器,再利用各顶点设计的反馈控制器得到具有凸多面体结构LPV容错控制器.仿真结果表明,LPV增益调度技术可以成功地应用于风力机系统的容错控制.     

A Multi-step Output Feedback Robust MPC Approach for LPV Systems with Bounded Parameter Changes and Disturbance

This paper considers a multi-step output feedback robust model predictive control (OFRMPC) approach for the linear parameter varying (LPV) systems with bounded changes of scheduling parameters and bounded disturbance. Less conservative bounds of future estimation error sets and system parametric uncertain sets are predicted by considering bounded changes of scheduling parameters in LPV systems. In the multi-step OFRMPC approach, an optimization problem is solved to obtain a sequence of controller gains, which considers predictions of future bounds of estimation error sets and system parametric uncertain sets. The optimized sequence of controller gains corresponding to a sequence of Lyaponov matrices have less constraint conditions and also introduce more degree of freedom for the optimization. The proposed multi-step OFRMPC guarantees robust uniform ultimately bounded of the estimation error and robust stability of the observer system. A numerical example is given to demonstrate the effectiveness of the approach.     

Reduced Order Robust Gain‐Scheduling Control Of The Diesel Apu For Series Hybrid Vehicles

This paper presents a reduced order robust gain‐scheduling approach for the control of the diesel auxiliary power unit (APU) for series hybrid vehicles. The nonlinear plant dynamics are converted into a linear parameter‐varying (LPV) form with parametric uncertainties, in which only partial information of the plant states is available. For this type of LPV system, a new reduced order robust gain‐scheduling synthesis method is proposed based on partial state feedback. The parametric uncertainties are considered using multipliers to reduce the conservatism. The reduced order synthesis problem is solved offline by means of linear matrix inequalities (LMIs), and the synthesis result requires much simpler online computation than the explicit controller formulas do. The synthesis method is applied to the diesel APU controller design, and simulation results are given to demonstrate the controller performance.     

Synthesis of a linearly interpolated gain scheduling controller for large flexible spacecraft ETS-VIII

This paper illustrates a design procedure for a linearly interpolated gain scheduling controller for Engineering Test Satellite VIII (ETS-VIII) using its linear parameter-varying (LPV) model. The LPV model here consists of piecewise-linear functions of the paddle rotation angle and a norm-bounded perturbation. The main purpose of this research is to derive a simple structured scheduling law that can be easily implemented in a satellite onboard computer. The derived gain has only two grid points and is scheduled simply by linear interpolation, which is desirable from the standpoint of implementability. Moreover, since the synthesis condition is based on parameter-dependent Lyapunov functions, it gives less conservative results than existing methods. Simulation results are presented to show the effectiveness of the proposed synthesis.     

LPV model development and control of a solution copolymerization reactor

In this paper, linear parameter-varying (LPV) control is considered for a solution copolymerization reactor, which takes into account the time-varying nature of the parameters of the process. The nonlinear model of the process is first converted to an exact LPV model representation in the state-space form that has a large number of scheduling variables and hence is not appropriate for control design purposes due to the complexity of the LPV control synthesis problem. To reduce such complexity, two approaches are proposed in this paper. First, an approximate LPV representation with only one scheduling variable is obtained by means of a parameter set mapping (PSM). The second approach is based on reformulating the nonlinear model so that it provides an LPV model with a fewer number of scheduling parameters but preserves the same input–output behavior. Moreover, in the implementation of the LPV controllers synthesized with the derived models, the unmeasurable scheduling variables are estimated by an extended Kalman filter. Simulation results using the nonlinear model of the copolymerization reactor are provided in order to illustrate the performance of the proposed controllers in reducing the convergence time and the control effort.     

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。     

On linear‐parameter‐varying (LPV) slip‐controller design for two‐wheeled vehicles

This paper describes the application of linear‐parameter‐varying (LPV) control design techniques to the problem of slip control for two‐wheeled vehicles. A nonlinear multi‐body motorcycle simulator is employed to derive a control‐oriented dynamic model. It is shown that, in order to devise a robust controller with good performance, it is necessary to take into account the dependence of the model on the velocity and on the wheel slip. This dependence is modeled via an LPV system constructed from Jacobian linearizations at different velocities and slip values. The control problem is formulated as a model‐matching control problem within the LPV framework; a specific modification of the LPV control synthesis algorithm is proposed to alleviate controller interpolation problems. Linear and nonlinear simulations indicate that the synthesized controller achieves the required robustness and performance. Copyright © 2008 John Wiley & Sons, Ltd.     

Discretization and event triggered digital output feedback control of LPV systems

This paper investigates the problem of discretization and digital output feedback control design for continuous-time linear parameter-varying (LPV) systems subject to a time-varying networked-induced delay. The proposed discretization procedure converts a continuous-time LPV system into an equivalent discrete-time LPV system based on an extension of the Taylor series expansion and using an event-based sampling. The scheduling parameters are continuously measured and modeled as piecewise constant. A new transmission of the measured output to the controller is triggered by significant changes in the parameters, yielding time-varying transmission intervals. The obtained discretized model has matrices with polynomial dependence on the time-varying parameters and an additive norm-bounded term representing the discretization residual error. A two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the LPV model. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through numerical examples.     

Control of nonlinear systems represented by Galerkin models using adaptation-based linear parameter-varying models

This paper studies the control of nonlinear Galerkin systems, which are an important class of nonlinear systems that arise in reduced-order modeling of infinite-dimensional systems. A novel approach is proposed in which a linear parameter-varying (LPV) model representing the Galerkin model is built, where the parameter variation is dictated by a specially designed adaptation scheme. The controller design is then carried out on the simpler LPV model, instead of dealing directly with the complicated nonlinear Galerkin system. An automatically scheduled H-infinity controller is designed using the LPV model, and it is proven that this controller will indeed achieve the desired stabilization when applied to the nonlinear Galerkin model. The approach is illustrated with an example on cavity flow control, where the design is seen to produce satisfactory results in suppressing unwanted oscillations.     

Gain-scheduled PID controller design

Gain scheduling (GS) is one of the most popular approaches to nonlinear control design and it is known that GS controllers have a better performance than robust ones. Following the terminology of control engineering, linear parameter-varying (LPV) systems are time-varying plants whose state space matrices are fixed functions of some vector of varying parameters. Our approach is based on considering that the LPV system, scheduling parameters and their derivatives with respect to time lie in a priori given hyper rectangles. To guarantee the performance we use the notion of guaranteed costs. The class of control structure includes centralized, decentralized fixed order output feedbacks like PID controller. Numerical examples illustrate the effectiveness of the proposed approach.     

Observer‐based synthesis of linear parameter‐varying mixed sensitivity controllers

This article presents a computationally efficient way of synthesizing linear parameter‐varying (LPV) controllers. It reviews the possibility of a separate observer and state feedback synthesis with guaranteed performance and shows that a standard mixed sensitivity problem can be solved in this way. The resultant output feedback controller consists of an LPV observer, augmented with dynamic filters to incorporate integral control and roll‐off properties, and an LPV state feedback gain. It is thus highly structured, which is beneficial for implementation. Moreover, it does not depend on scheduling parameter rates regardless of whether parameter‐dependent Lyapunov matrices are used during synthesis. A representative control design for active flutter suppression on an aeroelastic unmanned aircraft demonstrates the benefits of the proposed method in comparison with state‐of‐the‐art LPV output feedback synthesis.     

Robust attitude controller design for miniature quadrotors

This paper addresses the robust attitude control problem of miniature quadrotors. A simplified linear dynamical model is obtained for each attitude angle, whereas nonlinear dynamics, interaxis coupling, parameter perturbations, and external disturbances are considered as uncertainties. For each channel, a linear time‐invariant and decoupled robust controller are proposed based on a linear reduced‐order observer and a robust compensator. The observer is applied to estimate the angular velocities, and the robust compensator is introduced for reducing the effects of uncertainties. It is proven that the estimation errors of angular velocities and angular tracking errors can converge to the given neighborhood of the origin in a finite time. Experimental results on the miniature quadrotor are presented to verify the effectiveness of the proposed control approach. Copyright © 2015 John Wiley & Sons, Ltd.