Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces

We investigate general differential relations connecting the respective behavior s of the phase and modulo of probability amplitudes of the form \amp{\psi_f}{\psi}, where $\ket{\psi_f}$ is a fixed state in Hilbert space and $\ket{\psi}$ is a section of a holomorphic line bundle over some complex parameter space. Amplitude functions on such bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions involving the U(1) Berry-Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulo, therefore allowing for the reconstruction of the phase from the modulo (or vice-versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space ${\cal R} = {\mathbb C}P^n$, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini-Study metric. In conjunction with the generalized Cauchy-Riemann conditions, this geodesic interpretation leads to additional relations, in particular a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.Comment: 11 pages, 1 figure, revtex

Adiabatic driving and parallel transport for parameter-dependent Hamiltonians

We use the Van Vleck-Primas perturbation theory to study the problem of parallel transport of the eigenvectors of a parameter-dependent Hamiltonian. The perturbative approach allows us to define a non-Abelian connection $\mathcal{A}$ that generates parallel translation via unitary transformation of the eigenvectors. It is shown that the connection obtained via the perturbative approach is an average of the Maurer-Cartan 1-form of the one-parameter subgroup generated by the Hamiltonian. We use the Yang-Mills curvature and the non-Abelian Stokes' theorem to show that the holonomy of the connection $\mathcal{A}$ is related to the Berry phase

HOSM State Estimation and Robust PID Control of a Chemical Process

This paper presents some results of the authors’ studies on robust estimation and control for chemical processes. Here, an observer-based controller is designed for a chemical process. A High Order Sliding Mode Observer (HOSMO) for state and parameter estimation is synthesized and, a Multi-Variable PID (MV-PID) controller is calculated using the estimated variables. The HOSMO presents insensitivity and robustness against a class of uncertainties in the system, and the MV-PID allows the tracking of slow-varying and piecewise constant references, which are often proposed to drive chemical processes. Numerical simulations show that the observer-based controller presents a good performance in presence of parametric variations, which often are presented in chemical processes; the proposed structure is compared with a First Order Sliding Mode Observer (FOSMO) coupled with a MV-PID.CinvestavUniversidad Nacional de Colombi

State and Parameter Estimation of a CSTR

In the continuation of authors’ studies on estimation and control for Continuous Stirred-Tank Reactors (CSTR), a new structure to estimate the concentration of reactive state, the global heat transfer coefficient, and the heat of reaction parameters is proposed here. This scheme consist of an Observer Based Estimator (OBE) connected in cascade with a High Order Sliding Mode Observer (HOSMO). The OBE estimates the global heat transfer coefficient, and the HOSMO estimates the heat of reaction, and the concentration of reactive. Numerical simulations show that the whole structure presents a good performance in presence of parametric variations, which often are presented in chemical processes.Universidad Nacional de ColombiaCINVESTA

Estrategias de control y estimación de estado por modos deslizantes para un horno eléctrico

Este artículo presenta el diseño de un observador por modos deslizantes de segundo orden para una planta con modelo lineal, asumiendo ruido en la variable medida. Adicionalmente se propone un controlador de alto orden por modos deslizantes para una de las variables de estado asociadas al proceso. Para finalizar, se realiza un ejemplo con las respectivas simulaciones. This paper presents the design of a sliding mode observer for a second order linear plant model, assuming noise in the measured variable. Additionally, high order controller for sliding for state variables associated with the process modes is proposed. Finally, an example is made with the respective simulations.Universidad Nacional de ColombiaCINVESTAVColciencia

Diseño un controlador robusto basado en observador para el modelo lineal de un helicóptero de un grado de libertad (VTOL)

Este artículo propone un controlador basado en observador para un modelo de helicóptero tipo VTOL de un grado de libertad. El objetivo de control es mantener la posición del VTOL en un punto de operación, a pesar de las incertidumbres ocasionadas por inexactitud en el modelo y ruido en las mediciones. El controlador y el observador son diseñados con base en algoritmos que inducen modos deslizantes en el sistema en lazo cerrado, ofreciendo características de robustez y convergencia en tiempo finito. Los resultados de simulación muestran un buen desempeño del esquema propuesto tanto en condiciones de ausencia de ruido como cuando se consideran mediciones ruidosas.Universidad Nacional de ColombiaCinvestavColcienciasBanco Mundia