Dynamic output feedback linearization and global stabilization

We present a class of single-input single-output nonlinear systems which are globally transformable by a dynamic output feedback control and a time-varying state space transformation into a linear, observable and minimum phase system. We then show how those systems can be globally stabilized by a dynamic output feedback nonlinear control and how global output tracking can be achieved as well.     

On semi-global stabilizability of MIMO nonlinear systems by output feedback

For multi-input multi-output (MIMO) nonlinear systems, we prove that global stabilizability via smooth state feedback and uniform observability imply semi-global stabilizability by dynamic output feedback. This result incorporates and generalizes Teel-Praly's theorem established previously for single-input single-output (SISO) nonlinear systems. The generalization is made possible by employing, in contrast to the complicated proof in Teel and Praly [(1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22, 313-325], a simple and intuitive argument that involves no intricate Lyapunov functions.     

Global stabilization of linear discrete-time systems with bounded feedback

This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less than or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (“single hidden layer neural networks”) of simple saturation functions.     

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.     

Linearization of discrete-time systems by exogenous dynamic feedback

It is shown that a discrete-time system may be linearizable by exogenous dynamic feedback, even if it cannot be linearized by endogenous feedback. This property is completely unexpected and constitutes a fundamental difference with respect to the continuous-time case. The notion of exogenous linearizing output is introduced. It is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability. Necessary and sufficient conditions for the existence of an exogenous linearizing output are provided. The results of the paper are obtained using transformal operator matrices. The properties of such operators are studied. The theory is applied to the exact discrete-time model of a mobile robot, showing that the above-mentioned property concerns not only academic examples, but also physical systems.     

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.     

Parametric quadratic stabilizability of uncertain nonlinear systems

The objective of this paper is to introduce the notion of parametric quadratic stabilizability (PQ-stabilizability) of nonlinear control systems

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. When we consider nonlinear systems, reference inputs and disturbances may alter dynamics in an essential way by moving the equilibrium to a new location, or destroying it altogether. For this reason, when we consider quadratic stabilizability (Q-stabilizability) of nonlinear systems, we need to combine the two concepts of parametric stability and Q-stabilizability, and consider PQ-stabilizability. When nonlinear function φ belongs to a subclass of passive functions, it is shown that the necessary and sufficient condition for quadratic stabilizability (Q-stabilizability) of linear systems via dynamic state feedback, which was derived by Zhou and Khargonekar, is also a necessary and sufficient condition for that of nonlinear systems considered in this paper. Moreover, when nonlinear function φ belongs to a subclass of incresing functions and when ΔB = 0, it shown that the necessary and sufficient condition for Q-stabilizability of linear systems via static state feedback, which was derrived by Petersen, is also a necessary and sufficient condition for that of nonlinear systems considered in this paper.     

Stabilization of nonholonomic chained systems via nonregular feedback linearization

This paper addresses the problem of feedback stabilization of nonholonomic chained systems within the framework of nonregular feedback linearization. Firstly, the nonsmooth version of nonregular feedback linearization is formulated, and a criterion for nonregular feedback linearization is provided. Then, it is proved that the chained form is linearizable via nonregular feedback control, thus enable us to design feedback control laws using standard techniques for linear systems. The obtained discontinuous control laws guarantee convergence of the closed-loop system with exponential rates. Finally, simulation results are presented to show the effectiveness of the approach.     

一类非线性离散时间系统的模糊辨识

对一类非线性离散时间系统提出了模糊辨识方法,此方法用与未知参数向量成线性关系的模糊逻辑系统作为辨识模型,并通过自适应学习律对此模糊逻辑系统中的未知参数进行自适应调节,文中证明了此方法可使辨识误差收敛到原点的一个邻域内。仿真结果验证了此方法的有效性。     

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].     

Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization

Without imposing any growth condition, we prove that every chain of odd power integrators perturbed by a C¹ triangular vector field is globally stabilizable via non-Lipschitz continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback because the Jacobian linearization may have uncontrollable modes whose eigenvalues are on the right half-plane. The proof is constructive and accomplished by developing a machinery – adding a power integrator – that enables one to explicitly design a C⁰ globally stabilizing feedback law as well as a C¹ control Lyapunov function which is positive definite and proper.     

Remarks on linearization of discrete-time autonomous systems and nonlinear observer design

This paper presents necessary and sufficient conditions under which a discrete-time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form (Krener and Isidori, 1983). In particular, an open problem raised in Lee and Nam (1991), namely the observer linearization problem, is solved for a nonlinear system which may not be invertible (i.e., the mapping f may not be a local diffeomorphism). As a consequence, the nonlinear observer design problem is solved by means of exact linearization techniques.     

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.     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

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.     

Passivity and feedback passification of switched discrete-time linear systems

This paper focuses on the passivity analysis and feedback passification for switched discrete-time linear systems using multiple storage functions under a certain switching law. Controllers and switching laws are designed based on complete and partial state measurements, respectively. Conditions for strict passivity of a switched discrete-time system are obtained without the requirement of strict passivity of subsystems. In the case of partial state measurements, dynamic output feedback controllers for subsystems are designed, and a switching law depending only on the state of the output feedback controllers is constructed to guarantee strict passivity of the closed-loop switched system. Finally, a numerical example is provided to demonstrate the feasibility of the theoretical results.     

Multiple model-based event-triggered adaptive control of a class of discrete-time nonlinear systems

In this study, the problem of event-triggered-based adaptive control (ETAC) for a class of discrete-time nonlinear systems with unknown parameters and nonlinear uncertainties is considered. Both neural network (NN) based and linear identifiers are used to approximate the unknown system dynamics. The feedback output signals are transmitted, and the parameters and the NN weights of the identifiers are tuned in an aperiodic manner at the event sample instants. A switching mechanism is provided to evaluate the approximate performance of each identifier and decide which estimated output is utilised for the event-triggered controller design, during any two events. The linear identifier with an auxiliary output and an improved adaptive law is introduced so that the nonlinear uncertainties are no longer assumed to be Lipschitz. The number of transmission times are significantly reduced by incorporating multiple model schemes into ETAC. The boundedness of both the parameters of identifiers and the system outputs is demonstrated though the Lyapunov approach. Simulation results demonstrate the effectiveness of the proposed method.     

一种新的非线性离散时间系统的模糊辨识方法

对一类非线性离散时间系统提出一种新的模糊的辨识方法。该方法在假设逼近误差界已知的情况下,基于死区函数对模糊逻辑系统中的未知参数设计自适应学习律;在逼近误差界未知的情况下,基于时变死区函数对模糊逻辑系统中的未知参数设计自适应学习律,并对时变死区进行自适应调节。证明了所设计的自适应学习律均可使辨识误差收敛到原点的一个小邻域内。仿真结果表明了该算法的有效性。     

On H∞-control of discrete-time nonlinear systems

Following recent works on continuous-time nonlinear H_∞-control, where connections with game theory and passivity conditions have been set, the present paper studies the corresponding problem for discrete-time systems. The paper describes sufficient conditions for the existence and the construction of a feedback law which imposes a prescribed level of disturbance attenuation with internal stability. Both cases of state feedback and measurement feedback are considered.