Makhin Thitsa

Left inversion of analytic nonlinear SISO systems via formal power series methods

Given a single-input, single-output (SISO) system, F , and a function y in the range of F , the left inversion problem is to determine an input u such that y = F u . The goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where F is an analytic mapping in the sense that it has a convergent Chen-Fliess functional expansion, and y is a real analytic function. In particular, it will be shown that given a certain condition on the generating series c of F , a corresponding unique analytic u can always be determined via operations on formal power series. The condition on c turns out to be equivalent to having a well-defined relative degree when F has an input-affine analytic state space realization with finite dimension. But the method is applicable even when F does not have such a realization. The technique is demonstrated on four examples, including a continuous stirred chemical reactor.

ON THE RADIUS OF CONVERGENCE OF INTERCONNECTED ANALYTIC NONLINEAR INPUT-OUTPUT SYSTEMS ∗

A complete analysis is presented of the radii of convergence of the parallel, product, cascade and feedback interconnections of analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions for the coefficients of the generating series for the subsystems, the radius of convergence of each interconnected system is computed assuming the subsystems are either all locally convergent or all globally convergent. In the process of deriving the radius of convergence for the feedback connection, it is shown definitively that local convergence is preserved under feedback. This had been an open problem in the literature until recently.

Laser power density dependent energy transfer between Tm 3+ and Tb 3+ : tunable upconversion emissions in NaYF 4 :Tm 3+ ,Tb 3+ ,Yb 3+ microcrystals

Energy transfer between Tm³⁺ and Tb³⁺ dependent on the power density of pump laser was investigated in NaYF₄: Tb³⁺,Tm³⁺,Yb³⁺ microcrystals. Under the excitation of a 976-nm near-infrared laser at various power densities, Tb³⁺-Tm³⁺-Yb³⁺ doped samples exhibited intense visible emissions with tunable color between green and blue. The ratio of blue and green emission were determined by energy transfer between Tm³⁺ and Tb³⁺. When the power density of pump laser was low, the energy transfer process from Tm³⁺ (³F₄) to Tb³⁺ (⁷F₀) occurred efficiently. Upconversion processes in Tm³⁺ were inhibited, only visible emissions from Tb³⁺ with green color were observed. When the power density increased, energy transfer from the ³F₄ (Tm³⁺) to ⁷F₀ level (Tb³⁺) was restrained and population on high energy levels of Tm³⁺ was increased. Contribution of upconversion emissions from Tm³⁺ gradually became dominant. The emission color was tuned from green to blue with increasing the power density. Energy transfer processes between low-lying levels of activators, such as Tm³⁺ will greatly reduce the population on certain levels for further high-order upconversion processes. The Tb³⁺-Tm³⁺-Yb³⁺ doped phosphors are promising materials for detecting the condition of power density of the invisible near-infrared laser.

On the radius of convergence of self-excited feedback connected analytic nonlinear systems

This paper describes the radius of convergence of a self-excited feedback interconnection of two convergent Fliess operators. For either locally convergent or globally convergent operators, a fundamental and achievable positive lower bound on the radius of convergence for the composite system is computed.

Band Gap Tuning of Macro-Porous Si Photonic Crystals by Thermally Grown SiO2 Interfacial Layer

By plane wave expansion method (PWM) we modeled the effects of thermal oxidation and oxide etching on Si photonic crystals (PCs) of triangular lattice and proposed that these two processes can be extremely advantageous methods for tuning the photonic band gap and defect frequency. Two dimensional photonic crystals consisting of air cylinders in silicon background are studied before and after oxidation and partial oxide etching. Before oxidation the absolute band gap can be tuned only in a small range of radii (0.404a-0.48a); a is the lattice constant. Within this range, the band gap changes rapidly with the air cylinder radius due to the high dielectric constant of silicon. When thermal oxide interfacial layers are incorporated, the band gap can be tuned by varying the oxide thickness in the range of (0.26a-0.48a). By supercell method, formation of a defect level inside the band gap by means of the interfacial oxide layer and tuning of the defect frequency by varying the oxide thickness are also studied. A monopole defect mode is observed by oxidizing the center cylinder in a 5x5 suppercell of silicon cylinders with radius 0.28a in air background. The defect frequency is tuned by varying the oxide thickness from 0.15a to 0.3a.

Effect of Oxidation, Etching, and Thin-Film Deposition on Silicon Photonic Crystals

The effects of oxidation, partial oxide etching, and thin-film deposition on the photonic bandgap of two-dimensional triangular photonic crystals are studied. Two structures, air cylinders in silicon background and silicon cylinders in air background, are analyzed by the plane wave expansion method. Fine tuning of the bandgap is demonstrated by varying the thickness of the interfacial layers of SiO 2 grown by thermal oxidation and Al 2 O 3 produced by atomic layer deposition (ALD). In the air cylinder structure the absolute bandgap can be tuned from 0 to 0.0365 a/λ during oxidation by varying the oxide thickness from 0 to 0.2125a, where a is the lattice constant. During etching it can be tuned from 0 to 0.0759 a/λ. In a silicon cylinder structure the transverse magnetic bandgap can be tuned from 0 to 0.15 a/λ during oxidation and from 0 to 0.16 a/λ during etching. Tuning of the defect frequency of a single defect formed by selective oxidation is demonstrated, and the dependence of the defect frequency on the thickness of the oxide is investigated. Results indicate that the effect of complementary metal oxide semiconductor compatible processes such as thermal oxidation and ALD on the photonic bandgap is significant.

On the radius of convergence of cascaded analytic nonlinear systems

A complete analysis is presented of the radius of convergence of the cascade connection of two analytic nonlinear input-output systems represented as Fliess operators. Such operators are described by convergent functional series, which are indexed by words over a noncommutative alphabet. Their generating series are therefore specified in terms of noncommutative formal power series. Given growth conditions on the coefficients of the generating series for the component systems, the radius of convergence of the cascaded system is computed.

Pre-Lie algebra characterization of SISO feedback invariants

Transformation groups have been used extensively in system theory since its inception. Recently, a feedback transformation group for a nonlinear input-output system characterized by a Chen-Fliess functional expansion was described by the authors. In particular, an algorithm was given to identify a class of feedback invariant series. Their relationship to series having well-defined relative degree was also developed. This paper is a continuation of that work, but with three innovations. First, newly developed algebraic tools from the field of pre-Lie algebras are applied to give more insight into the invariance theory. The role of relative degree in this paper is diminished in favor of systems with arbitrary generating series. Finally, dynamic output feedback, as described by the feedback product, is considered explicitly.

On the radius of convergence of mixed cascades of analytic nonlinear input-output systems

The primary goal is to compute the radius of convergence of any cascade connection involving analytic functions and analytic integral operators, provided that the composite system is well defined and resides in one of these two classes.

On the radius of convergence of self-excited feedback connected analytic nonlinear systems: The local case

This paper describes the radius of convergence of a self-excited feedback interconnection of two locally convergent Fliess operators. A fundamental and achievable positive lower bound on the radius of convergence for the composite system is computed.