This paper presents the methodology for the design and control of a high-stiffness fast tool servo (FTS) system for practical application in diamond machining of freeform surfaces. The electromechanical model of the FTS system is established by combining the piezoelectric actuator and the flexure hinge mechanism for the formulation of the control methodology. In order to accommodate the system nonlinearity, cutting force disturbances as well as to realize fast and accurate tracking of designated tool path required for diamond machining of freeform surfaces, a novel adaptive terminal sliding mode (NATS) control strategy capable of online self-tuning of the system gain and reach time related parameters are developed. Stability analysis of the proposed NATS controller is performed. The convergence behavior of tracking error in the presence of model parameter variations and cutting force disturbance is analyzed. Experimental verification was conducted to validate the effectiveness and robustness of the NATS control. By comparing with other related control methods, it shows significantly improved performance. The percentage of tracking error was only 0.1% for the varying trajectories on machining water-drop surface. Taking advantage of the excellent tracking performances of the FTS prototype on a diamond turning machine, various freeform surfaces were successfully produced with high quality.

Design and Adaptive Terminal Sliding Mode Control of a Fast Tool Servo System for Diamond Machining of Freeform Surfaces

On controllability of discrete-time bilinear systems by near-controllability

Integral terminal sliding mode control for nonlinear systems

Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linear and some forms of bilinear and nonlinear ensemble systems. In this paper, we study controllability of a broad class of bilinear ensemble systems defined on semisimple Lie groups, for which we define the notion of ensemble controllability through a Riemannian structure of the state space Lie group. Leveraging the Cartan decomposition of semisimple Lie algebras in representation theory, we develop a \emph{covering method} that decomposes the state space Lie group into a collection of Lie subgroups generating the Lie group, which enables the determination of ensemble controllability by controllability of the subsystems evolving on these Lie subgroups. Using the covering method, we show the equivalence between ensemble and classical controllability, i.e., controllability of each individual system in the ensemble implies ensemble controllability, for bilinear ensemble systems evolving on semisimple Lie groups. This equivalence makes the examination of controllability for infinite-dimensional ensemble systems as tractable as for a finite-dimensional single system.

Ensemble Control on Lie Groups

In this chapter we provide an overview of recent progress on the problem of controllability of parameter dependent systems. We explore a different control notions successfully developed through the last decade. The aim of the control function is to steer the system to a state satisfying some properties prescribed either at some time instant T > 0 or during a given time interval. These properties may be separated with respect to parameter values and can refer just to a single system itself (e.g. greedy control), or may consider solutions corresponding to the whole parameter range (e.g. ensemble control, averaged control). In the latter case control functions are designed as parameter invariant, implying a same control is to be applied to the system independently of a particular realisation of the parameter, while in the first case controls vary along with the parameter.
Beside the positive theoretical results, for each notion we provide a precise computational algorithm accompanied by a numerical example.

https://hal.archives-ouvertes.fr/hal-03035494/document

Control of parameter dependent systems

Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-cha...

On Separating Points for Ensemble Controllability

This paper presents a general form for the fast terminal sliding mode (FTSM). It is shown that the proposed form not only includes the conventional FTSM, but also provides new types of FTSM which can have a faster convergence rate than the conventional one. More importantly, the convergence time of the new FTSM models is bounded for any initial value, which means an upper bound of the convergence time that is determined only by parameters can be obtained. Therefore, when using the new FTSM models in the control of higher order systems, the systems can be stabilized in a certain time which can be designed and estimated out-line. Finally, the improved FTSM (IFTSM) models are applied in design of the sliding mode control for single-input single-output (SISO) nonlinear systems with uncertainty and external disturbance.

A general form and improvement of fast terminal sliding mode

Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-changing technologies in life and physical sciences and engineering, from neuroscience, biology, and quantum physics to robotics, where many control-enabled applications involve manipulating a large ensemble of structurally identical dynamic units, or agents. Analyzing fundamental properties of ensemble control systems in turn plays a foundational and critical role in enabling and, further, advancing these applications, and the analysis is largely beyond the capability of classical control techniques. In this paper, we consider an ensemble of time-invariant linear systems evolving on an infinite-dimensional space of continuous functions. We exploit the notion of separating points and techniques of polynomial approximation to develop necessary and sufficient ensemble controllability conditions. In particular, we introduce an extended notion of controllability matrix, called Ensemble Controllability Gramian. This means enables the characterization of ensemble controllability through evaluating controllability of each individual system in the ensemble. As a result, the work provides a unified framework with a systematic procedure for analyzing control systems defined on an infinite-dimensional space by a finite-dimensional approach.

Near-controllability is a system property which is defined for those systems that are uncontrollable but own a very large controllable region. In this paper, near-controllability of a class of three-dimensional discrete-time bilinear systems where the system matrix has complex eigenvalues is studied. Sufficient conditions which are almost necessary for the systems to be nearly controllable are presented.

On near-controllability of a class of three-dimensional discrete-time bilinear systems with system matrix having complex eigenvalues

Control of population systems is an indispensable task in many cutting-edge applications from nuclear magnetic resonance spectroscopy and imaging to deep brain stimulation for treatment of neurological disorders. Characterizing the fundamental limit of the ability to control such large-scale systems is critical towards the understanding of their functionality and dynamical structures. In this paper, we study controllability of time-invariant linear ensemble systems whose natural and control dynamics form a linear variety of the system parameter. We derive explicit controllability conditions in terms of the rank of the system and control matrices, and show that ensemble controllability is highly dependent on the spectrum structure of the parameter-dependent system matrix.

Controllability of linear ensemble systems with constant drift and linear parameter variation

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On Separating Points for Ensemble Controllability

A general form and improvement of fast terminal sliding mode

On Separating Points for Ensemble Controllability

On near-controllability of a class of three-dimensional discrete-time bilinear systems with system matrix having complex eigenvalues

Controllability of linear ensemble systems with constant drift and linear parameter variation