M. Dahleh

Coherent Control of Quantum Dynamics: The Dream Is Alive

Current experimental and theoretical progress toward the goal of controlling quantum dynamics is summarized. Two key developments have now revitalized the field. First, appropriate ultrafast laser pulse shaping capabilities have only recently become practical. Second, the introduction of engineering control concepts has put the required theoretical framework on a rigorous foundation. Extrapolations to determine what is realistically possible are presented.

Iterative learning control - A survey and new results

Learning control is an iterative approach to the problem of improving transient behavior for processes that are repetitive in nature. Some results on iterative learning control are presented. A complete review of the literature is given first. Then, a general formulation of the problem is given. Next, a complete analysis of the learning control problem for the case of linear, time-invariant plants and controllers is presented. This analysis offers: insight into the nature of the solution of the learning control problem by deriving sufficient convergence conditions; an approach to learning control for linear systems based on parameter estimation; and an analysis that shows that for finite-horizon problems it is possible to design a learning control algorithm that converges, with memory, in one step. Finally, a time-varying learning controller is given for controlling the trajectory of a nonlinear robot manipulator. A brief simulation example is presented to illustrate the effectiveness of this scheme. 56 refs.

Optimal control of two-level quantum systems

We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing an energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices of dimension two, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. This includes, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.

Brief Paper: Dynamical analysis and control of microcantilevers

In this paper, we study the dynamical behavior of a microcantilever-sample system that forms the basis for the operation of atomic force microscopes (AFM). We model the microcantilever by a single mode approximation and the interaction between the sample and cantilever by a van der Waals (vdW) potential. The cantilever is vibrated by a sinusoidal input, and its deflection is detected optically. We analyze the forced dynamics using Melnikov method, which reveals the region in the space of physical parameters where chaotic motion is possible. In addition, using a proportional and derivative controller we compute the Melnikov function in terms of the parameters of the controller. Using this relation it is possible to design controllers that will remove the possibility of chaos.

Energy amplification in channel flows with stochastic excitation

We investigate energy amplification in parallel channel flows, where background noise is modeled as stochastic excitation of the linearized Navier–Stokes equations. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves O(R3) amplification. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity fields. The dependence of these quantities on both the Reynolds number and the spanwise wave number are explicitly computed. We show how the amplification mechanism is due to a coupling between wall-normal velocity and vorticity disturbances, which in turn is due to nonzero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the non-normality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H/sub infinity /) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H/sub infinity /-norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered. >

Melnikov-Based Dynamical Analysis of Microcantilevers in Scanning Probe Microscopy

We study the dynamical behavior of a microcantilever-sample system that forms the basis for the operation of atomic force microscopes (AFM). We model the microcantilever by a single mode approximation. The interaction between the sample and the cantilever is modeled by a Lennard--Jones potential which consists of a short-range repulsive potential and a long-range van der Waals (vdW) attractive potential. We analyze the dynamics of the cantilever sample system when the cantilever is subjected to a sinusoidal forcing. Using the Melnikov method, the region in the space of physical parameters where chaotic motion is present is determined. In addition, using a proportional and derivative controller, we compute the Melnikov function in terms of the parameters of the controller. Using this relation, controllers can be designed to selectively change the regime of dynamical interaction.

Control of mixing in fluid flow: a maximum entropy approach

In many technological processes a fundamental stage involves the mixing of two or more fluids. As a result, the design of optimal mixing protocols is a problem of both fundamental and practical importance. In this paper, the authors formulate a prototypical mixing problem in a control framework, where the objective is to determine the sequence of fluid flows that will maximize entropy. By developing the appropriate ergodic-theoretic tools for the determination of entropy of periodic sequences, they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other. The authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols.

Capabilities and limitations of multirate control schemes

Abstract A fundamental obstacle in control system design is the presence of plant zeros. Recently it has been suggested that in digital control systems such problems may be overcome by using multirate sampling. In this paper we describe the capabilities of such multirate control schemes with regard to pole and zero assignment. We also point out a limitation of the approach due to degradation in the intersample behavior. In Mita and Chida (1988, Proc. 27th Conf. on Dec. and Control, pp. 1883–1888) it is shown that with a two-delay input control scheme it is possible to assign all the poles and zeros of the closed-loop system using state feedback. In this paper we generalize this result to the case of N -delay input control using dynamic compensators. In an N -delay input control scheme the input to the continuous system is changed N times more often than the output is sampled. Using such a scheme we show how to design output feedback controllers for SISO systems that ensure arbitrary placement of all the poles and zeros of the closed-loop system. Corresponding results for the MIMO regulator and the SISO servomechanism are given. We also show that a hidden cost associated with the use of N -delay input control is a degradation in the intersample behavior. We demonstrate this via simulation and present an analysis which explains why this degradation arises.

Piezoelectric scanners for atomic force microscopes: design of lateral sensors, identification and control

This paper presents the identification and control of piezoelectric positioners used in atomic force microscopes (AFM) with the goal of improving probe positioning on the sample surface. A novel sensor was developed for this task and employed to infer a sixth order linear two input two output model of the piezos lateral dynamics. The piezo model was used to design a controller for tracking reference signals common in AFM operation. The controller and sensor were shown to significantly improve the microscopes ability to position the probe on the samples surface, enabling the AFM user to precisely scan areas on a surface based on images from previous scans.