Control Lyapunov functions for adaptive nonlinear stabilization

For the problem of stabilization of nonlinear systems linear in unknown constant parameters, we introduce the concept of an adaptive control Lyapunov function (aclf) and use Sontag's constructive proof of Artstein's theorem to design an adaptive controller. In this framework the problem of adaptive stabilization of a nonlinear system is reduced to the problem of nonadaptive stabilization of a modified system. To illustrate the construction of aclf's we give an adaptive backstepping lemma which recovers our earlier design.     

Global adaptive stabilization of infinite-dimensional systems

The purpose of this paper is to show that a first-order adaptive regulator globally stabilizes a class of infinite-dimensional systems having no right half plane zeros. Knowledge of the spectrum of the model's high-frequency gain is not required.     

On the adaptive stabilization of linear systems

The problem of adaptively stabilizing a linear system to zero is addressed. A new result is presented, showing that a finite family of linear systems is adaptively stabilizable if and only if the family is simultaneously stabilizable in the wellknown algebraic sense. This result is all the more surprising as the observations available consist of the signal corrupted by additive white Gaussian noise.     

A passification approach to adaptive nonlinear stabilization

In this paper, we unify and extend previously obtained results on adaptive stabilization of nonlinear systems from a passive systems viewpoint. It is shown that, for an interconnected multi-input nonlinear system composed of a chain of feedback strictly passive subsystems, an adaptive controller can be designed to render the closed loop system passive. Further, adaptive regulation is guaranteed if the zero-dynamics equation of each subsystem satisfies an appropriate input/state property. A systematic use of feedback passification in our proposed adaptive control scheme allows a simpler reinterpretation of the backstepping design with tuning functions in Krsti? et al. (1992).     

Sufficient information for the adaptive stabilization of delay systems

A sufficient condition for the adaptive stabilization of a large class of delay systems modeled by functional differential equations is given. Knowledge of the order of a finite dimensional controller, such that for every system in the class there exists a stabilizing controller of that order, is shown to constitute sufficient a priori information. The results of this paper generalize the recent result for finite dimensional systems.     

Simple adaptive stabilization of high-gain stabilizable systems

It is shown that the simple adaptive feedback strategy

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is a universal adaptive stabilizer for the class of single-input, single-output, finite-dimensional, linear systems which are stabilizable by either negative or positive high-gain feedback.     

A self-stabilizing autonomic recoverer for eventual Byzantine software

We suggest modeling software package flaws (bugs) by assuming eventual Byzantine behavior of the package. We assume that if a program is started in a predefined initial state, it will exhibit legal behavior for a period of time but will eventually become Byzantine. We assume that this behavior pattern can be attributed to the fact that the manufacturer had performed sufficient package tests for limited time scenarios. Restarts are useful for recovering such systems. We suggest a general, yet practical, framework and paradigm for the monitoring and restarting of systems where the framework and paradigm are based on a theoretical foundation. An autonomic recoverer that monitors and initiates system recovery is proposed. It is designed to handle a task, given specific task requirements in the form of predicates and actions. A directed acyclic graph subsystem hierarchical structure is used by a consistency monitoring procedure for achieving a gracious recovery. The existence and correct functionality of the autonomic recovery is guaranteed by the use of a self-stabilizing kernel resident (anchor) process. The autonomic recoverer uses a new scheme for liveness assurance via on-line monitoring that complements known schemes for on-line safety assurance.     

High-pain adaptive stabilization of multivariable linear systems — revisited

We consider the class of multi-input multi-output, finite dimensional, state space systems of the form , where the state dimension is unknown, the system is minumum phase, and it is known that det(CB) ≠ 0. For this class a universal adaptive high gain controller — not based on identification or estimation algorithms — is presented which ensures exponential decay of the motion of the closed-loop system. It is shown that the controller is robust with respect to certain nonlinear perturbations.     

Decentralized adaptive stabilization of interconnected nonlinear systems with unknown non-symmetric dead-zone inputs

The decentralized adaptive stabilization method is proposed for uncertain interconnected nonlinear systems with unknown non-symmetric dead-zone inputs. The class of systems considered in this paper consists of strict-feedback nonlinear subsystems with unknown non-symmetric dead-zone inputs which interact through their outputs. The unknown nonlinear interaction terms are assumed to be bounded by nonlinear functions with unknown parameters. For the simple controller design, the local controller for each subsystem is systematically derived based on the dynamic surface design technique, without constructing the dead-zone inverse and requiring the bound information of dead-zone parameters (slopes and break-points). All unknown parameters of interconnected nonlinear systems are compensated by the adaptive technique. From Lyapunov stability theorem, it is proved that all signals in the interconnected closed-loop system with decentralized adaptive controllers are semi-globally bounded. Simulation results for tripled inverted pendulums demonstrate the effectiveness of the proposed approach.     

Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance

We address the class of stochastic output-feedback nonlinear systems driven by noise whose covariance is time varying and bounded but the bound is not known a priori. This problem is analogous to deterministic disturbance attenuation problems. We first design a controller which guarantees that the solutions converge (in probability) to a residual set proportional to the unknown bound on the covariance. Then, for the case of a vanishing noise vector field, we design an adaptive controller which, besides global stability in probability, guarantees regulation of the state of the plant to zero with probability one.     

Robustness of an adaptive backstepping controller without modification

This paper examines the robustness of the adaptive controller designed using the backstepping technique proposed in Kristic et al. (Nonlinear and Adaptive Control Design, Wiley, New York, 1995). It is shown that the adaptive controller without any modification still possesses some robustness against a class of unmodeled dynamics. Moreover, the system transient performance in the presence of unmodeled dynamics can be improved to an arbitrary level by adjusting the controller design parameters.     

Self-tuning adaptive frequency tracker

An automatic gain tuning algorithm is proposed for a recently introduced adaptive notch filter. Theoretical analysis and simulations show that, under Gaussian random-walk type assumptions, the proposed extension is capable of adjusting adaptation gains of the filter so as to minimize the mean-squared frequency tracking error without prior knowledge of the true frequency trajectory. A simplified one degree of freedom version of the filter, recommended for practical applications, is proposed as well.     

Unifying stabilization and termination in message-passing systems

The paper dispels the myth that it is impossible for a message-passing program to be both terminating and stabilizing. We consider a rather general notion of termination: a terminating program eventually stops its execution after the environment ceases to provide input. We identify termination-symmetry to be a necessary condition for a problem to admit a solution with such properties. Our results do confirm that a number of well-known problems (e.g., consensus, leader election) do not allow a terminating and stabilizing solution. On the flip side, they show that other problems such as mutual exclusion and reliable-transmission allow such solutions. We present a message-passing solution to the mutual exclusion problem that is both stabilizing and terminating. We also describe an approach of adding termination to a stabilizing program. To illustrate this approach, we add termination to a stabilizing solution for the reliable transmission problem.Published online: 15 November 2004Anish Arora: Supported in part by DARPA contract OSU-RF #F33615-01-C-1901,NSF grant NSF-CCR-9972368, Ohio State University Fellowship,and 2002-2003,2003-2004 grants from Microsoft Research.Mikhail Nesterenko: Supported in part by DARPA contract OSU-RF #F33615-01-C-1901 and byNSF CAREER Award 0347485Some of the results in this paper were presented at the 21st International Conference on Distributed Computing Systems, Mesa, Arizona, April 2001, pp 99-106. Correspondence to: Mikhail Nesterenko     

Robust decentralized adaptive stabilization for a class of interconnected systems

The robust decentralized adaptive output-feedback stabilization for a class of interconnected systems with static and dynamic interconnections by using the MT-filters and backstepping design method is studied. By introducing a new filtered tramfomnation, the adaptive laws were derived for measurement. Under the assurnption of the nonlinear growth conditions imposed on the nonlinear interconnections and by constructing the error system and using a new proof method, the global stability of the closed-loop system was effectively analyzed, and the exponential convergence of all the signals except for parameter estimates were guaranteed.     

关于自适应非线性镇定的某些结果

不确定非线性系统的反馈控制一直是控制科学的中心问题之一,迄今已经取得很大进展。然而,目前现有大部分工作所研究的反馈控制规律,或是连续时间形式的,或是采样反馈形式但需要采样频率充分快,或是离散时间反馈形式,但需要被控离散时间系统的非线性函数增长速度不超过线性。要消除或减弱这些约束条件,一般来讲是相当困难的。这就促使我们探究反馈机制的最大能力和根本局限。尽管近年来在这个方向有许多重要进展,但仍有许多非平凡的重要问题有待研究。例如,在反馈通道中有时滞情形,或者系统状态是高维的情形。在本文中,我们将探索两类比较特殊的离散时间不确定非线性动力系统的控制问题,给出关于全局自适应反馈镇定的某些初步结果。     

Two results for adaptive output feedback stabilization of nonlinear systems

The problem of (adaptive) stabilization by means of output feedback of a class of nonlinear systems is addressed and solved. The proposed method relies on the asymptotic reconstruction of a stabilizing state feedback control law, does not require stable zero dynamics nor the construction of a Lyapunov function for the closed loop system, and treats in a unified way unknown parameters and unmeasured states. The applicability of the proposed method is discussed via theoretical examples. Finally, it is shown that the proposed method yields a solution to the problem of output feedback regulation for a DC-to-DC power converter and the efficacy of the resulting controller is verified via experiments.     

Adaptive stabilization of not necessarily minimum phase plants

We show that a first-order adaptive regulator can stabilize any linear time-invariant plant (LTI) whose transfer function has arbitrary relative degree and order, and is not necessarily minimum phase, provided the dominant slow part of the plant is minimum phase and of relative degree one and the parasitic fast part is stable. The results are extended to r × r multivariable systems whose dominant parts are minimum phase and the spectrum of their high-frequency gains is either in Re[s] < 0 or in Re[s] > 0, and their parasitic fast parts are stable.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems

In this paper, the problem of decentralized adaptive output-feedback stabilization is investigated for large-scale stochastic nonlinear systems with three types of uncertainties, including parametric uncertainties, nonlinear uncertain interactions and stochastic inverse dynamics. Under the assumption that the inverse dynamics of the subsystems are stochastic input-to-state stable, an adaptive output-feedback controller is constructively designed by the backstepping method. It is shown that under some general conditions, the closed-loop system trajectories are bounded in probability and the outputs can be regulated into a small neighborhood of the origin in probability. In addition, the equilibrium of interest is globally stable in probability and the outputs can be regulated to the origin almost surely when the drift and diffusion vector fields vanish at the origin. The contributions of the work are characterized by the following novel features: (1) even for centralized single-input single-output systems, this paper presents a first result in stochastic, nonlinear, adaptive, output-feedback asymptotic stabilization; (2) the methodology previously developed for deterministic large-scale systems is generalized to stochastic ones. At the same time, novel small-gain conditions for small signals are identified in the setting of stochastic systems design; (3) both drift and diffusion vector fields are allowed to be dependent not only on the measurable outputs but some unmeasurable states; (4) parameter update laws are used to counteract the parametric uncertainty existing in both drift and diffusion vector fields, which may appear nonlinearly; (5) the concept of stochastic input-to-state stability and the method of changing supply functions are adapted, for the first time, to deal with stochastic and nonlinear inverse dynamics in the context of decentralized control.     

Differential Riccati equation‐based adaptive stabilization for switched linear systems

In this paper, a novel adaptive control design is proposed to stabilize a class of switched linear systems with parametric uncertainties and disturbances. In particular, a family of differential Riccati equations that holds for the finite switching intervals are established, of which the closed‐form solutions are exploited to develop the Lyapunov function for the closed‐loop system. This Lyapunov function is imposed to be decreasing at and between two arbitrary switching instants by solving a group of matrix inequalities, based on which the adaptive controllers for switched linear system with and without disturbances are proposed, respectively. In the nondisturbed setting, asymptotic stability of the adaptive switched linear system is achieved, while the system is bounded in a mean square sense in the disturbed setting. A numerical example of flight control for F4E fighter aircraft is used to illustrate the proposed adaptive control methodology.