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The internal model-based [H.sub.[infinity]] full information steady-state bumpless transfer synthesis procedure is developed and demonstrated to provide near optimal robust stability and robust performance for a much larger class of controller uncertainty than the earlier developed LQ state/output transfer. It is also pointed out that the steady-state bumpless transfer under controller uncertainty represents, in general, a hybrid mode of system behavior--a controlled discrete transition. Index Terms--Bumpless transfer, computational speedup, controller uncertainty, robust control.

A new class of systems characterized by admitting impulsive control action within their singular motion phases, such as dimension changes, state discontinuities, and other irregularities, is introduced. This class, termed dynamical systems with active, or controlled, singularities, encompasses various applications with sensing and/or actuation ultra-fast in comparison with the natural system time scale, including mechanisms with impact-induced motion, power and sensor networks under faults, fast positioning devices, smart skins, and switching electronic circuits. The present work focuses on systems with impact-type singularities induced through system interaction with controlled, or active, state constraints. The latter are assumed to be characterized by parameter-dependent elastic-type constraint violation that vanishes in the limit as the parameter value tends to infinity. A physically well justified representation of this class of systems is proposed, but found to be not well suited for controller synthesis in the singular phase. To address this problem, two equations are derived--one describing controlled rigid impacts in auxiliary stretched time and compatible with the regular control design techniques, termed controlled infinitesimal dynamics equation, and another incorporating controlled impact dynamics in the infinitesimal form in terms of the shift operator along the trajectories of the first equation, termed the limit model. The latter is demonstrated to generate a unique isolated discontinuous system motion, i.e., to provide a tight and well-behaved description of the collision with rigid constraint. It is then shown that the corresponding paths generated by the original physically-based and the limit representations can be made arbitrarily close to each other uniformly except, possibly, in the vicinities of the jump points, by the appropriate choice of the value of the constraint violation parameter. This is shown to permit enforcing, for sufficiently large values of the parameter, the desired limit system behavior onto the original system by simply taking the control signals found through the limit representation, time-rescaling them, and inserting the resulting signals directly into the original system. These features show that the procedure developed, in fact, provides a well-posed controller synthesis framework for the class of systems considered. Using this framework, a ball/racket rotationally controlled impact representation is developed, and on its basis, a singular phase control signal in the form of racket rotation velocity is designed, demonstrating that the soft racket provides the specified bounce-off angle increment under a noticeably lower racket rotation velocity and a longer rotation phase than the hard racket. Index Terms--Control of mechanical systems, controlled discrete transitions, generalized solutions, hybrid systems, impacts, impulsive control, nonsmooth analysis and synthesis, penalty method, singularities, systems with unilateral constraints.

The present work proposes a new class of algorithms for identification of fast linear time-varying systems on short time intervals, based on the biorthogonal function decomposition. When certain features of the system dynamics are known a priori, the algorithms admit their embedding into the identification procedure through the choice of the matching bases, yielding the rapidly convergent identification laws. The speed-up is attained via utilizing both time and frequency localized bases, permitting identification of fewer coefficients without noticeable loss of accuracy. Simulation shows that the resulting high speed identification algorithms can reject small persistent random disturbances as well as capture the fast changes in system dynamics. The algorithm development is based on the results of Part I where it is shown that the sets of all bounded-input-bounded-output (BIBO) stable or [l.sup.2]-stable linear discrete-time-varying (LTV) systems are Banach spaces, and modeling and identification of these systems are reducible to linear approximation problems in a Banach space setting.

An analytical framework is developed that permits the input-output representations of discrete-time linear time-varying (LTV) systems in terms of biorthogonal bases on compact time intervals. Using these representations, the companion paper, Part II develops computational procedures for rapid identification of fast nonsmooth LTV systems based on short data records. One of the representations proposed is also used in H. Zhao and J. Bentsman, 'Block Diagram Reduction of the Interconnected Linear Time-Varying Systems in the Time Frequency Domain,' accepted for publication by Multidimensional Systems and Signal Processing to form system interconnections, or wavelet networks, and develop subsystem connectibility conditions and reduction rules. Under the assumption that the inputs and the outputs of the plants considered in the present work belong to [l.sup.p] spaces, where p=2 or p=[infinity], their impulse responses are shown to belong to Banach spaces. Further on, by demonstrating that the set of all bounded-input bounded-output (BIBO) stable discrete-time LTV systems is a Banach space, the system representation problem is shown to be reducible to the linear approximation problem in the Banach space setting, with the approximation errors converging to zero as the number of terms in the representation increases. Three types of LTV system representation, based on the input-side, the output-side, and the input-output transformations, are developed and the suitability of each representation for matching a particular type of the LTV system behavior is indicated.

This paper presents a model reference adaptive control of a class of distributed parameter systems described by linear, n-dimensional, parabolic partial differential equations. Unknown parameters appearing in the system equation are either constant or spatially-varying. Distributed sensing and actuation are assumed to be available. Adaptation laws are obtained by the Lyapunov redesign method. It is shown that the concept of persistency of excitation, which guarantees the parameter error convergence to zero in finite-dimensional adaptive systems, in infinite-dimensional adaptive systems should be investigated in relation to time variable, spatial variable, and also boundary conditions. Unlike the finite-dimensional case, in infinite-dimensional adaptive systems even a constant input is shown to be persistently exciting in the sense that it guarantees the convergence of parameter errors to zero. Averaging theorems for two-time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed. The exponential stability of the adaptive system, which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics, is shown by applying averaging. A numerical example which demonstrates an averaged system and computer simulations are provided.

This paper develops the theory of vibrational control of nonlinear time lag systems with arbitrarily large but bounded delay. Averaging theory for fast oscillating, differential delay equations is presented and then applied to vibrational control. Conditions are given which ensure the existence of parametric vibrations that stabilize nonlinear time lag systems. Transient behavior is also discussed. Illustrative examples are given which show 1) the feasibility of the theory to important applications and 2) the differences in the theory presented and the existing known theory for vibrational control of ordinary differential equations.

In the first part (1) of this work the conditions for the existence of the stabilizing vibrations for a class of distributed parameter systems governed by parabolic partial differential equations with Neumann boundary conditions were derived, and the guidelines for the choice of the vibration parameters that ensure stabilization were given. The present note addresses the transient behavior analysis of vibrationally controlled systems of the same class.

This note shows that vibrational stabilization can be effective for linear systems with large bounded delays. Theorems are given that define the procedures for the search of the stabilizing vibrations. Robust oscillatory stabilization insensitive to the delay size is also shown to take place for some classes of systems.

This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.