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.     

Singular PDEs and the single-step formulation of feedback linearization with pole placement

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.     

Normal forms and approximated feedback linearization in discrete time

The paper discusses approximated feedback linearization of nonlinear discrete-time dynamics which are controllable in first approximation and introduces two types of normal forms. The study is set in the context of differential/difference representations of discrete-time dynamics proposed in [Monaco, Normand-Cyrot, in: Normand-Cyrot (Ed.), Perspectives in Control, a Tribute to Ioan Doré Landau, Springer, Londres, 1998, pp. 191–205].     

Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277–283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127–132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del’Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517–522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Lévine J., Martin, P., Rouchon, P. (1999). A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922–937].     

Cascaded feedback linearization and its application to stabilization of nonholonomic systems

In this paper, a new cascaded feedback linearization problem is formulated and a set of conditions on the cascaded feedback linearizability are established for a class of two-input affine nonlinear systems. The proposed cascaded feedback linearization method enlarges the classes of nonlinear systems which can be dealt with using the feedback linearization technique. In particular, the proposed design can be applied to address the feedback stabilization problem for a few classes of nonlinear systems which have uncontrollable linearization and do not satisfy the standard feedback linearization conditions. As an illustrative application, the proposed cascade feedback linearization concept is used to solve the feedback stabilization problem of nonholonomic systems within the framework of continuously differentiable state feedback control. Simulation results are provided to illustrate the proposed method.     

On the largest feedback linearizable subsystem

A feedback invariant set of integers is associated with any nonlinear multivariable system which is linear with respect to the inputs: it is shown to be the set of controllability indices of the largest feedback linearizable subsystem, i.e. the largest subsystem which can be made locally linear and controllable by means of nonsingular feedback transformations.     

The effect of sampling on linear equivalence and feedback linearization

We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. For n = 2, we show, under the assumption of completeness of ad_FG, that if the discretized system is lineariable by state coordinate change and feedback, then the continuous time affine complete analytic system is linearizable by state coordinate change only. Also, we suggest a method of proof when n ≥ 3.     

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.     

Adaptive feedback linearization control of chaotic systems via recurrent high-order neural networks

In the realm of nonlinear control, feedback linearization via differential geometric techniques has been a concept of paramount importance. However, the applicability of this approach is quite limited, in the sense that a detailed knowledge of the system nonlinearities is required. In practice, most physical chaotic systems have inherent unknown nonlinearities, making real-time control of such chaotic systems still a very challenging area of research. In this paper, we propose using the recurrent high-order neural network for both identifying and controlling unknown chaotic systems, in which the feedback linearization technique is used in an adaptive manner. The global uniform boundedness of parameter estimation errors and the asymptotic stability of tracking errors are proved by the Lyapunov stability theory and the LaSalle-Yoshizawa theorem. In a systematic way, this method enables stabilization of chaotic motion to either a steady state or a desired trajectory. The effectiveness of the proposed adaptive control method is illustrated with computer simulations of a complex chaotic system.     

Approximate feedback linearization of discrete-time non-linear systems using virtual input direct design

The design of feedback-linearization and poles-placement controllers for discrete-time non-linear plants, using Input/Output/State measurements only, is typically addressed via indirect design. In this paper we propose the use of a new technique, based on a Virtual Input Direct Design (VID²) approach. The main feature of such a technique is to reduce the control design problem into a standard non-linear mapping approximation problem, without calling for the preliminary construction of an appropriate model of the plant. As compared with the existing methods, the new one requires less computational effort, while taking full advantage of the non-linear approximation software tools already available. In this paper, the new method is described, a simple theoretical analysis is given, and some numerical examples are presented.     

Global state and feedback equivalence of nonlinear systems

The problem of finding global state space transformations and global feedback of the form u(t)= α(x) + ν(t) to transform a given nonlinear system to a controllable linear system on Rⁿ or on an open subset of Rⁿ, is considered here. We give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem.     

Robust control with uncertainty estimation for feedback linearizable systems: application to control of distillation columns

The goal of this paper is to describe a linearizing feedback adaptive control structure which leads to a high quality regulation of the output error in the presence of uncertainties and external disturbances. The controller consists of three elements: a nominal input–output linearizing compensator, a state observer and an uncertainty estimator, which provides the adaptive part of the control structure. In this way, the feedback controller, based on the disturbance observer, compensates for external disturbances and plant uncertainties. The effectiveness of the controller is demonstrated on a distillation column via numerical simulations. ©     

Intelligent feedback linearization control of nonlinear electrohydraulic suspension systems using particle swarm optimization

The core factors governing the performance of active vehicle suspension systems (AVSS) are the inherent trade-offs involving suspension travel, ride comfort, road holding and power consumption. In addition to this, robustness to parameter variations is an essential issue that affects the effectiveness of highly nonlinear electrohydraulic AVSS. Therefore, this paper proposes a nonlinear control approach using dynamic neural network (DNN)-based input–output feedback linearization (FBL) for a quarter-car AVSS. The gains of the proposed controllers and the weights of the DNNs are selected using particle swarm optimization (PSO) algorithm while addressing simultaneously the AVSS trade-offs. Robustness and effectiveness of the proposed controller were demonstrated through simulations.     

Maximal feedback linearization and its internal dynamics with applications to mechanical systems on

For systems that are not feedback linearizable, a natural question is: how to find the largest feedback linearizable subsystem and, if the partial linearization is not unique, what are the control‐theoretic properties of various partial linearizations. In this paper, we will consider the problem of how to choose a partially linearizing output that renders the zero dynamics asymptotically stable and when such an output exists. We will state general results solving completely the problem for systems whose linearizability defect is one by identifying and describing two classes of systems. For the first class, all maximal partial linearizations lead to the same zero dynamics. For the second class, any asymptotic behavior of the zero dynamics can be achieved by a suitable choice of a partially linearizing output. In the second part, we apply our results to mechanical systems with two‐degrees‐of‐freedom and provide a detailed study of their partial linearizations. We illustrate the obtained results by examples of Acrobot (which belongs to the second class) and Pendubot (which belongs to the first class).     

A state space embedding approach to approximate feedback linearization of single input nonlinear control systems

This contribution presents a numerical approach to approximate feedback linearization which transforms the Taylor expansion of a single input nonlinear system into an approximately linear system by considering the terms of the Taylor expansion step by step. In the linearization procedure, higher degree terms are taken into account by using a state space embedding such that the corresponding system representation has not to be computed in every linearization step. Linear matrix equations are explicitly derived for determining the nonlinear change of coordinates and the nonlinear feedback that approximately linearize the nonlinear system. If these linear matrix equations are not solvable, a least square solution by applying the Moore–Penrose inverse is proposed. The results of the paper are illustrated by the approximate feedback linearization of an inverted pendulum on a cart. Copyright © 2006 John Wiley & Sons, Ltd.     

非完整移动机器人目标环绕动态反馈线性化控制

本文针对多个非完整移动机器人对静止或运动目标的环绕追踪问题进行研究.每个机器人仅通过自身和其相邻的机器人的位置与方向信息以及所追踪的目标的位置信息来协调其运动.首先,提出了一种基于动态反馈线性化方法的分布式控制策略,并引入一个控制机器人之间相对角间距的非线性函数,控制机器人间的相对角间距.使多个机器人能够以期望的与目标之间的相对距离、环绕速度和机器人之间的相对角间距对目标进行追踪.然后,利用Lyapunov工具对控制算法进行了渐近稳定性和收敛性分析.最后构建了多移动机器人实验平台,进行了数值仿真和实验验证,仿真和实验的运行结果表明了所提出算法的有效性.     

A Faà di Bruno Hopf algebra for a group of Fliess operators with applications to feedback

A Faà di Bruno type Hopf algebra is developed for a group of integral operators known as Fliess operators, where operator composition is the group product. Such operators are normally written in terms of generating series over a noncommutative alphabet. Using a general series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. The result is applied to analytic nonlinear feedback systems to produce an explicit formula for the feedback product, that is, the generating series for the Fliess operator representation of the closed-loop system written in terms of the generating series of the Fliess operator component systems. This formula is employed to provide a proof that local convergence is preserved under feedback.     

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input-output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.     

Uncertainty modeling and robust minimax LQR control of multivariable nonlinear systems with application to hypersonic flight

For a class of multi‐input and multi‐output nonlinear uncertainty systems, a novel approach to design a nonlinear controller using minimax linear quadratic regulator (LQR) control is proposed. The proposed method combines a feedback linearization method with the robust minimax LQR approach in the presence of time‐varying uncertain parameters. The uncertainties, which are assumed to satisfy a certain integral quadratic constraint condition, do not necessarily satisfy a generalized matching condition. The procedure consists of feedback linearization of the nominal model and linearization of the remaining nonlinear uncertain terms with respect to each individual uncertainty at a local operating point. This two‐stage linearization process, followed by a robust minimax LQR control design, provides a robustly stable closed loop system. To demonstrate the effectiveness of the proposed approach, an application study is provided for a flight control problem of an air‐breathing hypersonic flight vehicle (AHFV), where the outputs to be controlled are the longitudinal velocity and altitude, and the control variables are the throttle setting and elevator deflection. The proposed method is used to derive a linearized uncertainty model for the longitudinal motion dynamics of the AHFV first, and then a robust minimax LQR controller is designed, which is based on this uncertainty model. The controller is synthesized considering seven uncertain aerodynamic and inertial parameters. The stability and performance of the synthesized controller is evaluated numerically via single scenario simulations for particular cruise conditions as well as a Monte‐Carlo type simulation based on numerous cases. It is observed that the control scheme proposed in this paper performs better, especially from the aspect of robustness to large ranges of uncertainties, than some controller design schemes previously published in the literature. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society