Mir Shahrouz Takyar

Metrics for Power Spectra: An Axiomatic Approach

We present an axiomatic framework for seeking distances between power spectral density functions. The axioms require that the sought metric respects the effects of additive and multiplicative noise in reducing our ability to discriminate spectra, as well as they require continuity of statistical quantities with respect to perturbations measured in the metric. We then present a particular metric which abides by these requirements. The metric is based on the Monge-Kantorovich transportation problem and is contrasted with an earlier Riemannian metric based on the minimum-variance prediction geometry of the underlying time-series. It is also being compared with the more traditional Itakura-Saito distance measure, as well as the aforementioned prediction metric, on two representative examples.

The fractional integrator as a control design element

It is a well-known fact that set-point following, in linear control systems, requires an integrator in the feedback loop. However, such an integrator introduces phase lag which may often have a destabilizing effect. A variety of options exist for adding lead to mediate this effect. In this paper we consider yet another option, a fractional integrator. We study possible implementations of the fractional integrator and the effect of such an element in achieving set-point following specifications.

Weight Selection in Feedback Design With Degree Constraints

We present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint. Performance is cast as an interpolation problem with bounded analytic functions. Minimizers of a certain weighted-entropy functional provide interpolants having degree less than the number of constraints. The choice of weight parameterizes all such bounded degree solutions. However, the relationship between the weights and the shape of corresponding transfer functions is not direct. Thus, in this paper we develop a formalism that guides weight selection.

Weight Selection in Interpolation with a Dimensionality Constraint

The topic of the paper relates to a recent parametrization of analytic interpolants with a bound on their dimension, as solutions to certain weighted entropy minimization problems. The analytic interpolation problem arises in the context of shaping closed-loop transfer functions via a suitable choice of controller. Our goal is to shed light on how the choice of weights affects the shape of the corresponding closed-loop transfer functions. Further, given a desirable shape, we indicate how a suitable weight can be obtained as a solution of a certain quasi-convex problem

Sensitivity shaping with degree constraint via convex optimization

We present an approach for shaping closed-loop operators while keeping their Mcmillan degree bounded by the sum of unstable plant-poles and non-minimum phase plant-zeros. We make use of recent developments in analytic interpolation with degree constraint and we focus on the paradigm of sensitivity minimization. The sensitivity function can be obtained as the minimizer of a convex weighted-entropy functional. It is the choice of this weight that we formulate as a convex optimization problem in this paper

Analytic Interpolation With a Degree Constraint for Matrix-Valued Functions

We consider a Nehari problem for matrix-valued, positive-real functions, and characterize the class of (generically) minimal-degree solutions. Analytic interpolation problems (such as the one studied herein) for positive-real functions arise in time-series modeling and system identification. The degree of positive-real interpolants relates to the dimension of models and to the degree of matricial power-spectra of vector-valued time-series. The main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint, where the class of (generically) minimal-degree solutions is characterized by an arbitrary choice of ¿spectral-zeros¿. Naturally, in the current matricial setting, there is freedom in assigning the Jordan structure of the spectral-zeros of the power spectrum, i.e., the spectral-zeros as well as their respective invariant subspaces. The characterization utilizes Rosenbrocks theorem on assignability of dynamics via linear state feedback.

Fractional-order Systems and the Internal Model Principle

We consider the fractional-integrator as a feedback design element. It is shown, in a simple setting, that the fractional integrator ensures zero steady-state tracking. This observation should be contrasted with the typical formulation of the internal model principle which requires a full integrator in the loop for such a purpose. The use of a fractional integrator allows increased stability margin, trading-off phase margin against the rate of convergence to steady-state. A similar rationale can be applied to tracking sinusoidal signals. Likewise, in this case, fractional poles on the imaginary axis suffice to achieve zero steady-state following and disturbance rejection. We establish the above observations for cases with simple dynamics and conjecture that they hold in general. We also explain and discuss basic implementations of a fractional integrating element.

Transport metrics for power spectra

We present a family of metrics for power spectra based on the Monge-Kantorivic transportation distances. These metrics are constructed so that distances reduce with additive and multiplicative noise, reflecting the intuition that noise typically reduces our ability to discriminate spectra. In addition, perturbations measured in these metrics are continuous with respect to the statistics of the underlying time series. A general framework for constructing such metrics is put forth and these are contrasted with an earlier Riemannian metric which is based on prediction theory and the relevant geometry of the underlying time-series.

On the Jordan structure of the spectral-zero dynamics in multivariable analytic interpolation

The parametrization of solutions to scalar interpolation problems with a degree constraint relies on the concept of spectral-zeros - these are the poles of the inverse of a corresponding spectral factor. In fact, under a certain degree constraint, the spectral-zeros are free (modulo a stability requirement) and parameterize all solutions. The subject of this paper is the multivariable analog of a Nehari-like analytic interpolation with a degree constraint. Our main result is based on Rosenbrock¿s pole assignability theorem and addresses the freedom in assigning the Jordan structure of the spectral-zero dynamics.