Disturbances in the form of longitudinal vortices appear in boundary layers, where they can cause the transition towards turbulence.  In the case of concave wall boundary layer, these perturbations are called Gortler's vortices and in the case of flat plate boundary layer they are called algebraic instabilities.

We carried out a study of the amplification of various types of longitudinal vortices in a flat plate boundary layer.  We thus showed the decisive role played by the longitudinal component of the velocity of perturbation in the growth of the instability, as well as the general information of the profile of this same velocity produces the downstream, and sometimes called Klebanoff's mode.

We then sought a means of controlling this type of disturbance (in the case of flat or concave wall boundary layer), in the framework of optimal control.

We consider the disturbance which has the strongest amplification: the optimal perturbation.  We used a control by blowing/suction at the wall, and we found various profiles of the optimal velocity of control following the definition  of the cost functional to be minimized.  We thus showed that it is theoretically possible to control optimal disturbances of the boundary layer by action at the wall.

The control of physical systems to meet standard engineering performance specifications can become a complicated engineering design process.  Typically, the dynamics of physical systems must be characterized using nonlinear dynamic models.  The utility of these nonlinear models in the controller design process is contingent upon the type of nonlinear model developed and the controller design process.  Further complicating the controller design process is the need to meet standard engineering output performance specifications given actuator limitations such as rate and output saturation.

Presented in this seminar is a holistic controller design process for a limited class of nonlinear regulating systems subjected to external disturbances.  The design process begins with the nonlinear modeling of the physical system and ends with the design of a robust feedback controller.  The nonlinear modeling process is built on a truncated Volterra series model of the physical system.  The Volterra series representation is identified using a frequency domain approach.  Next, a set of conditions will be presented that guarantees closed loop stability.  Based on the controller, the maximum allowable external disturbance size (in the sense that its L_infinity norm) is determined.  Closed loop performance specifications will be enforced on the linearized model and has the form                               where y(t) denotes the output deviations of the linear model about the desired operating condition.  This performance specification will be formulated into a performance weight for the mixed sensitivity problem.  Additional frequency domain weights will be introduced on the complementary sensitivity transfer function to balance trade-offs between stability and performance.

The seminar will conclude with an application of the controller design process to the speed control of an internal combustion engine.  This application is particularly attractive, owing to the nonlinear dynamic behavior of engines and the induction-to-power delays.

Many systems consist of similar units which directly interact with their nearest neighbors.  Even when these units have tractable models and interact with their neighbors in a simple and predictable fashion, when viewed as a whole the resulting system often displays rich and complex behavior.  In this talk, we discuss new techniques for synthesizing control systems for these classes of systems, and show how the resulting control strategies naturally lend themselves to practical implementation.  We will also discuss current research in the control of autonomous systems, including the Cornell robot soccer team (www.mae.cornell.edu/robocup) that won the 1999 RoboCup F180 competition in Stockholm, Sweden.

First convergency properties are looked at for several iterative design methods.  In view of these, a general framework is presented for iterative controller design by model unfalsification.  Schemes relying on H_infinity and I_infinity formulations become special cases.  The most important feature of the scheme is its convergence to the best possible controller given a set of model and controller structures.  Computational problems are discussed and simulations are used to illustrate its application to flow control in aerodynamics and to PID controller tuning in general.

I will present a unifying theory of fast eye and arm movements based on the single physiological assumption that the neural control signals are corrupted by noise whose variance increases with the size of the control signal.  We propose that motor commands are selected to minimize the stochastic uncertainty in final position due to this signal-dependent noise.  This theory accurately predicts the trajectories of both saccades and arm movements.  These profiles are robust to changes in the dynamics of the eye or arm, as found empirically.  For the arm, we show that the theory captures the speed-accuracy trade-off described by Fitt's law as well as the relationship between path curvature and hand velocity during drawing movements reproducing the empirical "two-thirds power law".  For saccades we show the theory can reproduce the  main sequence, the relationship between saccadic amplitude and duration.  This theory provides a simple and powerful unifying perspective for both eye and arm movement control.  We suggest that controlling the statistics of movement in the presence of noise is the key consideration in biological motor control.

The stability of a linear system given in state-space form can be checked by looking at the location of the eigenvalues of its dynamic matrix.  Using Lyapunov stability theory, a symmetric instrumental matrix variable can be introduced and stability can be equivalently checked by looking at a set of LMI (Linear Matrix Inequalities).

This talk will show how an extended stability test can be obtained by adding to the Lyapunov LMI inequalities an extra instrumental variable.  The obtained condition can still be translated as an LMI and might be interpreted as a converse application of Finsler's Lemma.  This technique is also able to provide extended characterizations for  and  norm calculations.

While the introduction of an extra variable provides little advantage in the analysis of precisely known systems, it will be demonstrated how it can be used to build a parameter dependent Lyapunov function that is able to prove the stability of uncertain system described by convex polytopes.  It also provides an advantage for the synthesis of linear controllers.  Since a suitable controller parametrization involving only the extra variable is available, constrained controllers can be synthesized without imposing unnecessary and conservative constraints on the Lyapunov matrix.

Recent results on control of nonlinear oscillatory systems obtained in the "Control of Complex Systems" Lab. of IPME, RAS will be surveyed.  Starting with the problem of swinging the simple pendulum, step by step the general problem of feedback stabilization of the invariant manifold will be formulated.  The design of the stabilizing feedback based on speed-gradient method (Fradkov, 1979) and conditions ensuring the achievement of the control goal will be described.  The links between the proposed method and passivity-based approach (Byrnes-Isidori-Willems, 1991) as well as the partial stabilization methods (Rumyantsev, 1970; Vorotnikov, 1997) will be discussed.  Applications of the obtained results in physics will be addressed, e.g. studying escape from a potential well and feedback resonance phenomena.  The main ideas behind the talk are exposed in the recently published books:

A.L. Fradkov, A. Yu Pogromsky, "Introduction to Control of Oscillations and Chaos," World Scientific, 1998.
A.L. Fradkov, I.V. Miroshnik, V.O. Nikiforov, "Nonlinear and Adaptive Control of Complex Systems," Kluwer Acad. Publishers, 1999.

A tridiagonal canonical form.  Stability, controllability, and observability properties of tridiagonal systems.  Transfer functions, continued fractions, realization diagrams and state equations.  Specializations and generalizations.  Applications to model reduction and computation of frequency responses.

The project contributes to the improvement of control systems of glass producing facilities.  Due to increasing demands for product quality, increasing use of recycled glass and more flexible use of the facilities, the process control is crucial, especially during the conditioning in the forehearth.  Conventional controllers are frequently overtaxed because of their limited working range, neglect of coupling between zones and time variant behavior of the process.  Further, new measuring techniques are available to obtain important process data.  These variables could contribute to improved control but can not be used in traditional systems.

A new approach of forehearth control is desired to cover a wide range of operating conditions but requiring no time-consuming adjustment on site.

Therefore, a modular zone model is developed, which describes the dynamic behavior of the forehearth based on first principles.  Further, based on process data, a detailed model of a forehearth has been set up to verify the calculations of the simplified model.  Thus, a wide range of process conditions could be covered by simulations.

System identification techniques will be used to describe important relations between process variables which can hardly be modeled physically.  Dynamic models will be developed on the basis of experimental data.  It is crucial for practical use to fit the model to a given forehearth online.  Algorithms for identification of model parameters will be developed and verified on the basis of experimental data.

Once set up, the model can be used to provide solutions for a number of control problems which include:
            -    Tuning temperature setpoints to provide uniform conditions of the glass melt at the outflow.
            -    Utilization of information provided by new measuring devices.  The additional information is used to
                 detect product changes and to tune the control accordingly.
            -    Adaptation of existing controllers to varying conditions

Approaches to solve the problems are discussed.
 
 

Hybrid estimation refers to estimation of a hybrid random process with jump as well as diffusion components, such as state estimation for a hybrid system, which includes both continuous- and discrete-valued state variables.  It unifies conventional estimation and decision in some sense, and provides a powerful framework for many inference problems (e.g. detection, estimation, recognition and identification) involving both structural and parametric uncertainties or changes.  It is also a good tool for decomposing a complex problem (system) into simpler, sub-problems (subsystems).  An introduction to hybrid estimation is given.  The multiple-model method, a major approach to hybrid estimation, is emphasized.  Its three generations (static, interacting and variable structure) are described.  Their application in target tracking and recognition, and in fault detection and identification, are addressed.  Model-set design is covered.  An innovative approach to performance analysis and prediction of hybrid estimation algorithms is also presented.

The idea of decomposing dynamical systems, and their related signals, in terms of orthogonal components other than the standard Fourier series, dates at least back to the work of Lee and Wiener in the thirties. The issue is that a dynamical system being represented in terms of a series expansion of a pre-chosen sequence of orthogonal functions, can be modeled approximately, by considering only a finite length of the expansion. The accuracy of this model will be very much dependent on the specific sequence of basis functions chosen. Laguerre functions have been used extensively in an attempt to increase the flexibility of the basis functions, providing a scalar design variable to the modeler, to be tuned according to the expected dynamics of the system. Over the last years, a general theory has been developed, in line with the ideas of Kautz in the fifties, for the construction of generalized basis functions for the class of stable systems H₂. Being based on orthogonal exponentials, the considered functions correspond to rational transfer functions. These basis functions show an extensive flexibility, i.e. there are numerous degrees of freedom that can be tuned by the model builder. Construction schemes encompass the classical Gram-Schmidt orthogonalization scheme as well as system theoretic approaches based on balanced realizations of square inner (stable all-pass) functions. The considered functions generalize the classical pulse, Laguerre and Kautz basis functions.

Besides other application areas, the use of these basis functions in identification problems has shown attractive results. The parameterization of linear models in terms of finite expansion coefficients is attractive because of the linear-in-the-parameters model structure, while a high rate of convergence of the series expansion (being the result of an appropriately chosen basis) leads to estimated models with small bias and variance errors. The model structure is closely related to a so-called "fixed-pole" or "fixed-denominator" model structure. However in addition to these closely related forms, the orthogonal structure has particular advantages.

In this seminar, the construction of the basis functions will be discussed, and it will be shown how they can be effectively used for solving problems in e.g. system identification based on either time-domain or frequency domain data, system approximation, model uncertainty estimation and (approximate) realization.

Mathematical models for simulating physical, biological and economic systems are now often more accurate than the data that is available to drive them. In particular, complete information describing the initial state of an evolutionary system is seldom known. In this case it is desirable to use the measured output data that is available from the system over an interval of time, in combination with the model equations, to derive accurate estimates of the expected system behavior. The problem of constructing a state-estimator, or observer, is the dual of the feedback control design problem. For very large nonlinear systems arising in numerical weather prediction and in ocean circulation modeling, traditional control system design techniques are not practicable, and 'data assimilation' schemes are used instead to generate accurate state-estimates. The aim of these schemes is to incorporate observed data into computational simulations in order to improve the accuracy of the numerical forecasts.

Currently, variational data assimilation schemes are under development. These schemes are attractive because they deliver the best statistically linear unbiased estimate of the model solution given the available observations and their error covariances. The problem is formulated as an optimal control problem where the cost functional measures the mismatch between the model predictions and the observed system states, weighted by the inverse of the covariance matrices. The model equations are treated as strong constraints and the controls to be determined are the initial states of the system. The constrained minimization problem is typically solved by a gradient iterative procedure for finding the optimal controls. The gradient directions needed in the iteration are obtained by solving the linear adjoint equations associated with the problem.

In practice the model equations do not represent the system behavior exactly and model errors arise due to lack of resolution, to inaccurate physical parameters, or to errors in boundary conditions, in topography or in other forcing terms. To account for model error, the system equations can be treated as weak constraints in the optimization problem. The residual errors in the model equations at every time point are then treated as control parameters. Statistically the model error is assumed to be unbiased white noise that is uncorrelated in time. This approach is not practicable, however, due to the excessive size of the optimization problem. Moreover, the statistical assumptions are not generally satisfied, since the model errors are expected to be time-correlated.

Although the general form of the model error is not known, some simple assumptions about the evolution of the error can be made. The control variables then reduce to the unknown initial values of the model error and the corresponding optimization problem can be solved efficiently. A major advantage of this approach is that the gradient directions with respect to the model errors can be obtained directly from the adjoint equations for the original problem at very little extra cost. We show here, using a nonlinear shallow water model, that a constant model error, or bias error, can be used as a control to correct for model errors in topography and/or in the Coriolis force. This technique gives significantly greater improvement in the subsequent forecast than assimilation using the initial data as the control. An extension of this approach to the treatment of time-correlated advection errors is also demonstrated.

The asymptotic behavior of receding horizon optimal control problems with terminal cost chosen as control Liapunov function is analyzed for regulator as well as disturbance attenuation problems. Both the continuous as well as the discrete-time cases are treated. Further, the approximation of the continuous time optimal control problem by the discrete-time receding horizon problems is studied. Applications to the Navier-Stokes equations are also presented.

The introduction of feedback control in a copier paperpath enables tomorrow’s copiers to better handle high throughputs, a wide range of media, disturbance rejection, varying operating conditions and automatic jam clearance. The paperpath dynamics will be analyzed from a control perspective. An overview of several control strategies will be presented. The strategies are based on optimal control theory and on a heuristic, hierarchical approach. The demands that the control systems put on the copier architecture in terms of actuators and sensors will be investigated together with an analysis of controllability and stability properties. Simulation results and comparisons between the different control algorithms will be shown as well as the similarities to other control structures of complex systems such as Automated Highway Systems. This research is an ongoing project between the University of California, Berkeley and Xerox Corp. An overview of the experimental work to date will be given.

Asymptotic regimes of geophysical dynamics are described for different Burger number limits. Rotating three-dimensional Euler-Boussinesq equations are analyzed in the asymptotic limit of strong stratification in the Burger number of order one situation as well as in the asymptotic regimes of strong stratification and weak rotation relevant to flows at the tropopause and the lower stratosphere. Reduced prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure/ thermal wind imbalance) are analyzed. In the asymptotic regime of strong stratification and weak rotation we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak rotation. This frontogenesis yields vertical "glueing" of anisotropic pancake dynamics, in contrast to the independent dynamics of horizontal layers in strongly stratified turbulence without rotation. DNS simulations of stably stratified turbulence will be presented. Implications of our rigorous asymptotic interaction theory between synoptic scales (potential vorticity) and inertio-gravity waves to the data assimilation problem for mesoscale flows in the middle atmosphere will be discussed.

* This seminar is jointly sponsored with Fluid Mechanics

Recently, the possibility of using flow control to reduce resonant oscillations in subsonic and transonic flows over open cavities has attracted much attention. Significant reductions in sound pressure levels (around 20 dB) have been achieved through upstream mass as well as piezoelectric flaps, and closed-loop feedback control promises similar reductions in noise, but with much lower power input. The need for low-order models of cavity resonance, in order to affect an efficient closed-loop control, is the motivation for the present study.

This talk will focus on the results of numerical simulations of the instabilities in the two dimensional flow past an open cavity, and our recent efforts to use the data to develop low-order models. The compressible Navier-Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field. Under certain conditions, the oscillations are well characterized by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare favorably with predictions based on linear stability analysis of the Kelvin-Helmholtz mode. The linear stability modes are also qualitatively similar to the modes of a Proper Orthogonal Decomposition (POD) of the data, and we discuss how these modes can be used to construct low-order models of the flow.

The results show a transition from a shear layer mode, for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The wake mode is characterized by large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow over an axisymmetric gap with a laminar upstream boundary layer. Under these conditions, it is shown that the Kelvin-Helmholtz instability grows to sufficient strength such that a strong recirculating flow is induced in the cavity. It appears in this case that the shear layer becomes absolutely unstable, leading to vortex shedding that is unconnected to any acoustic feedback from the trailing edge. The results suggest that periodic laminar separation upstream of the leading edge is also connected with the transition to wake mode, and a reasonably accurate predictive criteria for the onset of wake mode is developed based on a separation criterion. The wake mode has been observed in previous RANS calculations, but not in experiments at higher Reynolds numbers, for reasons that are discussed.

* This seminar is jointly sponsored with Fluid Mechanics

A main problem in modern technology is the difficulty to master the more and more complex systems, which are conceived to achieve always higher quality performances. With their strong nonlinearity and extreme heterogeneity, new problems arise in their control, related to the description of system degrees of freedom and their interactions. Aside difficulty to get accurate dynamical equations when complexity increases due to uncertainties and disturbances, more fundamental is the existence of physical uncertainty related to the birth of chaotic behavior leading to trajectory indistinguishability in phase space and produced by system complexification. The reaction of this irreducible measurement inaccuracy onto dynamics of system components affects control efficiency which looses asymptotic stability property and may even become unstable.

As uncertainties are also on functional dependence in terms of system state variables, classical PID-type controllers are insufficient. Improvements have been obtained before by concepts of adaptiveness and robustness. In the later approach the system is approximated by a simpler one for which the control is determined. If the actual system belongs to robustness domain of the approximate one, the two systems belong to the same equivalence class with close performances. Aside amplitude limitation of robustness bound depending on controller quality, best results usually give simple system stability, i.e., the error ball produced by unknown part cannot be eliminated and propagates to the output.

Improvement by use of intelligent controls, based on system trajectory geometric properties in phase space converted into a space of logical rules, have also shown limitations for complex enough systems because of the level of required details needed to output a significant change in control quality and of the corresponding intricacy of combinatorics when applying the logical rules, as trajectory indistinguishability reduces trajectory related information, and single trajectory descriptive variables become inadequate as too much detailed without any relation to control action which can only, by internal system structure, act at the more global level of available invariants for system description. Elementary example of thermodynamics shows that following single molecule trajectory is irrelevant to improve temperature control as the only invariant here is energy surface on which the motion of all molecules takes place.

Corresponding state space description should be replaced by more global function space approach, adapted to deal with equivalence classes of trajectories appearing from dynamical behavior. Classical tracking of single unobservable trajectory transforms into imposing the solution of system dynamics to belong to a prescribed function space, a problem solvable by Fixed Point method giving asymptotic stability toward desired trajectory. To combine both advantages of functional and robust controls, i.e., asymptotic stability and simplicity, exponentially decaying function space used in application of Lyapunov theorem is enlarged without requiring extra system information. Extension to the case of poorly or completely unknown systems is obtained by using projective method.

The resulting asymptotic decay is not necessarily exponential, but its rate is of importance for overall system performance, because of passivation requirement for the controlled system while interacting with outer elements. So the problem is to retain exponential asymptotic decay of adaptive approach without usually required detailed knowledge of system dynamics, here not precisely accessible, in agreement with robust and functional methods which only need a norm bound on the unknown part. New performance domain is isolated from original PID-control applicable to classical systems, and from usual adaptive and robust approaches. Intelligent controls are there downgraded from their well-tailored qualitative concept for decisional type reasoning to quantitative action because of absence of global variables suited to natural system invariants. Only the functional control used here is providing as it uses natural system invatiants. The question raised here is to reach the new domain from adaptive side in order to keep exponential decay property. The method is a combination of majorant type approach used for fixed point application and of projection onto a base set with adjustable coefficients determined by adaptive type conditions. This guarantees convergence rate as the decay is a consequence of dynamical behavior of constructed side system describing parameter evolution. The difference with usual adaptive method is that only representation of global norm bounding expression is used, with corresponding simplification of controller structure. The method is developed after casting the system in canonical form, and explicit control expression and parameter actualization equations are given with application as an example to the class of Lagrangian systems as a whole.