Feedback linearization vs. adaptive sliding mode control for a quadrotor helicopter

This paper presents two types of nonlinear controllers for an autonomous quadrotor helicopter. One type, a feedback linearization controller involves high-order derivative terms and turns out to be quite sensitive to sensor noise as well as modeling uncertainty. The second type involves a new approach to an adaptive sliding mode controller using input augmentation in order to account for the underactuated property of the helicopter, sensor noise, and uncertainty without using control inputs of large magnitude. The sliding mode controller performs very well under noisy conditions, and adaptation can effectively estimate uncertainty such as ground effects. Recommended by Editorial Board member Hyo-Choong Bang under the direction of Editor Hyun Seok Yang. This work was supported by the Korea Research Foundation Grant (MOEHRD) KRF-2005-204-D00002, the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government(MOST) R0A-2007-000-10017-0 and Engineering Research Institute at Seoul National University. Daewon Lee received the B.S. degree in Mechanical and Aerospace Engineering from Seoul National University (SNU), Seoul, Korea, in 2005, where he is currently working toward a Ph.D. degree in Mechanical and Aerospace Engineering. He has been a member of the UAV research team at SNU since 2005. His research interests include applications of nonlinear control and vision-based control of UAV. H. Jin Kim received the B.S. degree from Korea Advanced Institute of Technology (KAIST) in 1995, and the M.S. and Ph.D. degrees in Mechanical Engineering from University of California, Berkeley in 1999 and 2001, respectively. From 2002–2004, she was a Postdoctoral Researcher and Lecturer in Electrical Engineering and Computer Science (EECS), University of California, Berkeley (UC Berkeley). From 2004–2009, she was an Assistant Professor in the School of in Mechanical and Aerospace Engineering at Seoul National University (SNU), Seoul, Korea, where she is currently an Associate Professor. Her research interests include applications of nonlinear control theory and artificial intelligence for robotics, motion planning algorithms. Shankar Sastry received the B.Tech. degree from the Indian Institute of Technology, Bombay, in 1977, and the M.S. degree in EECS, the M.A. degree in mathematics, and the Ph.D. degree in EECS from UC Berkeley, in 1979, 1980, and 1981, respectively. He is currently Dean of the College of Engineering at UC Berkeley. He was formerly the Director of the Center for Information Technology Research in the Interest of Society (CITRIS). He served as Chair of the EECS Department from January, 2001 through June 2004. In 2000, he served as Director of the Information Technology Office at DARPA. From 1996 to 1999, he was the Director of the Electronics Research Laboratory at Berkeley (an organized research unit on the Berkeley campus conducting research in computer sciences and all aspects of electrical engineering). He is the NEC Distinguished Professor of Electrical Engineering and Computer Sciences and holds faculty appointments in the Departments of Bioengineering, EECS and Mechanical Engineering. Prior to joining the EECS faculty in 1983 he was a Professor with the Massachusetts Institute of Technology (MIT), Cambridge. He is a member of the National Academy of Engineering and Fellow of the IEEE.     

Feedback linearization based control of a rotational hydraulic drive

The technique of feedback linearization is used to design controllers for displacement, velocity and differential pressure control of a rotational hydraulic drive. The controllers, which take into account the square-root nonlinearity in the system's dynamics, are implemented on an experimental test bench and results of performance evaluation tests are presented. The objective of this research is twofold: firstly, to present a unified method for tracking control of displacement, velocity and differential pressure; and secondly, to experimentally address the issue of whether the system can be modeled with sufficient accuracy to effectively cancel out the nonlinearities in a real-world system.     

Feedback linearization and driftless systems

The problem of dynamic feedback linearization is recast using the notion of dynamic immersion. We investigate here a generic property which holds at every point of a dense open subset, but may fail at some points of interest, such as equilibrium points. Linearizable systems are then systems that can be immersed into linear controllable ones. This setting is used to study the linearization of driftless systems: a geometric sufficient condition in terms of Lie brackets is given; this condition is also shown to be necessary when the number of inputs equals two. Though noninvertible feedbacks are nota priori excluded, it turns out that linearizable driftless systems with two inputs can be linearized using only invertible feedbacks, and can also be put into a chained form by (invertible) static feedback. Most of the developments are done within the framework of differential forms and Pfaffian systems.This work was partially supported by INRIA, NSF Grant ECS-9203491, GR Automatique (CNRS), and DRED (Ministère de l'Éducation Nationale). Part of it was done while the first author was visiting the Center for Control Engineering and Computation, University of California at Santa Barbara.     

Feedback linearization control for a distributed solar collector field

This article describes the application of a feedback linearization technique for control of a distributed solar collector field using the energy from solar radiation to heat a fluid. The control target is to track an outlet temperature reference by manipulating the fluid flow rate through the solar field, while attenuating the effect of disturbances (mainly radiation and inlet temperature). The proposed control scheme is very easy to implement, as it uses a numerical approximation of the transport delay and a modification of the classical control scheme to improve startup in such a way that results compared with other control structures under similar conditions are improved while preserving short commissioning times. Experiments in the real plant are also described, demonstrating how operation can be started up efficiently.     

Automatic differentiation and nonlinear controller design by exact linearization

Various modern algorithms for controller design are based on differential-geometric concepts. A method of particular importance is called exact linearization via feedback. In this case, the implementation of the controller requires the computation of Lie derivatives, which have been computed symbolically. This can be very time consuming. We present a new computation method relying on automatic differentiation.     

Nonlinear system identification of a small-scale unmanned helicopter

This paper presents a comprehensive method for identifying the nonlinear model of a small-scale unmanned helicopter. The model structure is obtained by first principles derivation, and the model parameters are determined by direct measurement and system identification. A new adaptive genetic algorithm is proposed to identify the parameters that cannot be directly measured. To simplify the identification process, the overall system is divided into two subsystems for identification: the heave–yaw dynamics and the lateral–longitudinal dynamics. On the basis of the input–output data collected from actual flight experiments, these two subsystems are identified using the proposed algorithm. The effectiveness of the identified model is verified by comparing the response of the simulation model with the actual response during the flight experiments. Results show that the identified model can accurately predict the response of the small-scale helicopter. Furthermore, the identified model is used for the design of an attitude controller. The experiment results show that the identified model is suitable for controller design.     

Feedback linearization of a flexible manipulator near its rigid body manifold

It is shown that a ‘slow’ 2n-dimensional manifold exists in the 4n-dimensional state space of an n-link manipulator with n flexible joints. While the flexible manifold is not linearizable by feedback, its restriction to'the slow manifold is. A method is given for computing a feedback control which achieves an approximate linearization.     

Feedback linearization of discrete-time nonlinear uncertain plants via first-principles-based serial neuro-gray-box models

In this paper, discrete-time control schemes based on feedback linearization of serial gray-box models are considered for partially known nonlinear processes. These techniques combine the benefits of feedback linearization, neural networks, and serial gray-box modeling, which result in larger dynamic operating ranges, better extrapolation properties, and fewer data acquisition efforts in comparison with the corresponding black-box-based schemes. First-principles-based serial gray-box models are classified into invertible and non-invertible structures for training purposes, and an improved approximate feedback linearization scheme based on Taylor series terms of a non-affine gray-box model is proposed. Moreover, an affine gray-box model is developed for applying the exact feedback linearization scheme. Simulation results on a fermentation process show that the proposed methods yield significant improvement in modeling and control performance in comparison with that of the black-box feedback linearization schemes.     

On the problem of feedback linearization

The paper studies and solves in a geometric framework the problem of partial feedback linearization for discrete-time dynamics. An algorithm for computing the largest linearizable subsystem is proposed. This approach can be considered as dual to the one already proposed in literature in an algebraic context.     

On approximate linearization of nonlinear control systems

A method for the approximate linearization of nonlinear control systems based on the ‘state-space exact linearization’ method is presented. An explicit procedure, both for the single-input and for the multiple-input case, is given, which is straightforward to implement.     

纵列式无人直升机建模及其精确线性化方法研究

利用牛顿-欧拉法对纵列式无人直升机的近似悬停模态进行数学建模，得出标准仿射非线性状态方程，然后应用状态反馈精确线性化方法进行设计．仿真结果表明，系统输出可稳定跟踪给定信号．     

离散时间非线性系统线性化的泛模型方法

考虑离散时间非线性系统线性化问题，提出一种“泛模型”方法，泛模型是具有时变参数，形成为线性的模型族，证明了工程上能实现的非线性系统，在输入－输出等价的意义下可用泛模型来描述。     

Quotient submanifolds for static feedback linearization

A procedure for finding locally the linearizing output of a single input nonlinear affine system is proposed. It relies on successive integrations of one-dimensional distributions and projections along these submanifolds. The algorithm proceeds recursively reducing the dimension one by one of both the number of coordinates and the number of vector fields, until the solution is obtained. A variant of the algorithm is also proposed, which does not require the computation of the full initial distribution. The proof of convergence of this second algorithm shows the importance of a new anti-symmetrical product. Besides providing a new insight into the involutivity condition, the algorithm can lead to a simple way of integrating the system of partial differential equations defining the linearizing output.     

Feedback stabilization of connected nonlinear oscillators with uncontrollable linearization

A nonlinear system with uncontrollable linearization is under investigation. The uncontrollable part of the linearization is assumed to have no eigenvalues in the right half-plane and have eigenvalues with zero real and nonzero imaginary part. A nonlinear feedback loop is assumed to carry out a connection between the controllable part of the system and the uncontrollable one. The control law of the controllable part is defined to be a linear function of the state variables. This linear feedback loop can maintain the stability of a whole control system provided its feedback law coefficients satisfy a system of algebraic inequalities. The consistence of this system of inequalities is proved.     

Approximate linearization of nonlinear control systems

The paper address the problem of approximate linearization of a nonlinear control system by state feedback or dynamic feedback. The main result proved is that if the fast k d-relative degrees are equal to each other, then the input-output response of a nonlinear control system can be linearized to degree k. Any system having linear d-relative degree can be linearized to any degree by dynamic feedbacks. One example of signal tracking using dynamic feedback linearization method to improve the performance is also given.     

一类非线性直升机模型的滑模降阶控制器设计

针对带有交叉耦合的多输入CE150直升机模型,研究了一类多输入仿射非线性系统的控制设计问题,基于滑模变结构控制理论,采用了一种新的控制器设计方法:滑模降阶方法,即反复运用变结构控制理论,对一类高阶的仿射非线性系统,构造了合适的微分同胚变换函数,把初始高阶系统降至低阶系统,并构造了变结构控制律,再利用当前级和上一级控制输入的映射关系反推出初始系统的控制输入.通过CE150直升机模型仿真结果表明,该方法有效可行.     

Nonlinear robust control of a small-scale helicopter on a test bench

A nonlinear robust controller design procedure is presented, which is designed to simultaneously satisfy multiple conflicting closed-loop performance specifications. Significantly, a robust performance specification for the experimental system, developed for studying the attitude control of a small-scale helicopter in our previous work, is discussed quantitatively. The robust performance specifications and nominal multiple closed-loop performance specifications are conflicting. Use of the Convex Integrated Design (CID) method can provide, where feasible, a single closed-loop controller which satisfies a set of multiple conflicting performance specifications. However, the resultant controller has a complex form. Here, the standard CID method is extended to a more general control system framework to solve the conflicting simultaneous performance design problem. When compared with the standard CID design, the extended CID design procedure generates a relatively simple closed-loop controller. Finally, the synthesised controller is tested in simulation and is validated with an experimental small-scale test helicopter, demonstrating the performance of the proposed controller.     

基于两时间尺度模型的直升机非线性控制器设计

提出一种基于两时间尺度模型的直升机非线性控制方法. 该方法利用直升机不同状态达到稳定的时间不同的特点, 将直升机模型分为快速和慢速两种模型. 反步控制方法和逆动力学控制方法分别被用于进行快慢两种模型控制器的设计, 并在控制过程中采用了不同的控制周期. 仿真结果表明, 利用上述方法设计的控制器, 对于阶跃变化和正弦变化的速度轨迹具有良好的跟踪效果.

     

遥控模型直升机分通道模型辨识方法研究

设计了嵌入式控制系统,研究了小型遥控直升飞机在悬停状态的模型辨识方法.按照遥控直升机的4个控制输入(纵向周期变距、横向周期变距、总距、以及尾距),直升机的运动可分解为俯仰、倾斜、升降和航向4个通道,分别采用分通道飞行实验数据,通过Matlab辨识工具,得到了遥控直升机在悬停状态下各通道的单榆入单输出控制模型.飞行实验数据的验证结果表明,分通道数学模型可以很好地反映相应通道的动态特性,对实现遥控直升机自动驾驶控制具有重要的应用价值.     

基于扩张状态观测器的反馈线性化无人飞行器姿态控制

随着集成电子等技术逐渐成熟,无人飞行器得到了广泛的关注和应用;其中,由于飞行器在飞行阶段具有高响应速度、耦合动力学和非线性等特点,使其在飞行阶段的姿态控制成为该领域重点研究的方向之一;针对该研究方向,文章提出了一种基于反步法技术的反馈线性化无人飞行器姿态跟踪控制方案来解决其在受到环境干扰的条件下可以精准跟踪姿态角度的问题;在该方法中,设计了反步法技术拆分简化模型和反馈线性化减少调节参数,并利用扩张状态观测器(ESO)来对扰动进行估计和补偿,同时给出了ESO的收敛性和闭环系统的稳定性来证明该方法的可实施性;最后给出了仿真结果,验证了该方法在干扰的环境下仍可以精准控制无人飞行器姿态。