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In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at particular points in the caster, each corresponding to a single discrete-in-time boundary measurement in the Stefan problem partial differential equation (PDE) mathematical model. The results for two previous estimators are summarized. The first estimator is based on the Stefan problem, using continuous instead of discrete-in-time boundary measurements. The second estimator employs a process model that is more detailed than the Stefan Problem, but with no output injection to reduce estimation error, other than model calibration. Both of these estimation frameworks are extended in the current paper to a more realistic sensing setting. First, an estimator is considered that uses the Stefan Problem under some simplifying but practically justified assumptions on the unknowns in the process. The maximum principle for parabolic PDEs is employed to prove that online calibration using a single discrete-in-time temperature measurement can provide removal of the estimation error arising due to mismatch of a single unknown parameter in the model. Although unproven, this result is then shown in simulation to apply to the more detailed process model.

Adaptive observers using projection-operator-based parameter update laws are considered for a class of linear infinite-dimensional systems with bounded input operator and full state measurement and subject to time-varying matched uncertainties and disturbances. The L1 adaptive control architecture, introduced recently for finite-dimensional plants to provide guaranteed transient performance via fast adaptation, is then extended to this class using the proposed observers. Existence and uniqueness of solutions for the resulting closed loop system and uniform boundedness of the observation error are established first. Then, provided certain assumptions on the plant transfer function and the solution of a Lyapunov inequality hold, uniform guaranteed transient performance bounds on the plant state and control signal under the L1 architecture are derived. Two examples satisfying the assumptions-control of a heat equation and a wave equation-are presented. Reference input tracking simulation results for the heat equation under the L1 adaptive control subject to time-varying matched uncertainties and disturbances are presented in support of the theory.

A controller synthesis setting for systems with controlled singularities under incomplete information is extended to the cases characterized by a constraint on the total physical impulse as well as on the time interval between the adjacent engagement phases. The use of this setting for the optimal control law calculation is illustrated on the ball/racket system representative of a large class of controlled impact problems, where there are two separate bodies interacting through controlled viscoelastic force.

The internal model-based H infin 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.

Presents the synthesis of adaptive identifiers for distributed parameter systems (DPS) with spatially varying parameters described by partial differential equations (PDEs) of parabolic, elliptic, and hyperbolic type. The features of the PDE setting are utilized to obtain the not directly intuitive parameter estimation algorithms that use spatial derivatives of the output data with the order reduced from that of the highest spatial plant derivative. The tunable identifier parameters are passed through the integrator block, which forms their orthogonal expansions. The latter are shown to be pointwise plant parameter estimates. In this regard, the approach of the paper is in the spirit of finite-dimensional observer realization in integrating rather than differentiating the output data, only applied to the spatial rather than temporal domain. The constructively enforceable identifiability conditions, formulated in terms of the sufficiently rich input signals referred to as generators of persistent excitation, are shown to guarantee the existence of a unique zero steady state for the parameter errors. Under such inputs, the tunable parameters in the adaptive identifiers proposed are shown to converge to plant parameters in L/sub 2/ and the orthogonal expansions of these tunable parameters-pointwise.

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. >

The transient behavior of vibrationally controlled distributed parameter systems governed by parabolic partial differential equations (PDEs) with Neumann boundary conditions is analyzed. The analysis uses a certain mapping and a time-invariant PDE whose trajectories under this mapping yield the approximate moving averages along the trajectories of the vibrationally controlled system. This approach considerably enhances the understanding of the behavior of vibrationally controlled distributed-parameter systems and facilitates a selection of the parameters of stabilizing vibrations that ensure transient behavior with more desirable properties. >

It is shown 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. >

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 are derived. The guidelines for the choice of the vibration parameters that ensure stabilization are given. Examples of vibrational stabilization of unstable systems by linear multiplicative and vector additive vibrations are given to support the theory. >

In the first part of this work [1], the criteria for the existence of stabilizing parametric oscillations have been derived. In the present paper, the problem of choosing the stabilizing vibrations is addressed and the properties of transient behavior of vibrationally controlled nonlinear systems are analyzed.