First and Second Order Systems

In this paper, the vibration behavior of atomic force microscopy (AFM) micro-cantilever (MC) with a piezoelectric layer in the non-contact mode considering the axial load in the air medium was studied. In the dynamic modes of AFM nano-robot, parameters such as amplitude, frequency and motion phase are used as feedback factors in the control system in order to extract sample surface information and the nanoparticle voltage. Consequently, to extract these parameters, the vibration analysis of the MCs of this nano-robot is essential. Additionally, because of the wide-spread applications of oblique MCs, applying the axial and transverse forces into the model is absolutely necessary. In order to simulate the AFM four-layered piezoelectric MC with the axial displacement, Timoshenko beam theory as well as modified couple stress theory has been used. Considering the axial displacement in the modeling not only leads to more degrees of freedom, but also has increased the amount of calculations. To extract the equations of the system, energy method has been used, and then, the FEM was applied to discrete the equations. To verify the accuracy of the results, the frequency and time responses of FEM have been compared with the results of the 3D mode simulation obtained by COMSOL software and experimental results. The comparison of results indicates that modifying the modeling by considering the axial displacement has increased the accuracy of the frequency and time responses. The error percentage of the first frequency response is 0.42% and the error percentage of the time response is 0.0053%. Furthermore, to examine the effect of MC geometry on its vibrational behavior, frequency and time responses of three common geometries of AFM MCs have been extracted. The results illustrate the improvement of modeling in all three geometries. Then, the sample surface topography of MCs at the first and second excitation frequencies in contact with rectangular roughnesses demonstrated the decline of time delay at the second excitation frequency. The results of the topography simulation in the air environment show that the roughness’s trend is better at the second mode than at the first mode. Finally, the effect of MC angle on the topography of the rectangular MC surfaces in contact with the rectangular roughness in 2D and 3D modes was examined in the presence and the absence of axial load. The results show that in the presence of the axial displacement in modeling, by increasing of MC angle, the topography quality increases drastically.

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

This paper develops generalized analytical first and second order transfer functions for the nonlinear second order system. A periodic input is also conducted to characterize the overall system response from the fundamental components. The proposed analytical solution provides more understanding of the influence of each linear and nonlinear component on the overall system behavior.

The notion of the transfer function of the discrete-time nonlinear control system is defined. The definition is based on a non-commutative twisted polynomial ring, which can be by the Ore condition extended into its quotient ring (field of fractions). Some properties of the transfer function, related to accessibility and observability of the system, are studied and the transfer functions of different composite systems (series, parallel, and feedback connections) are given. The resulting theory is, in principle, similar to that in the linear case, except that the polynomial description relates now the differentials of inputs and outputs, and the resulting polynomial ring is non-commutative.

The purpose of this paper is to explore the role of Battery Energy Storage System (BESS) in improving the frequency response of a low inertia power system. For the recent years, prolific amount of renewable energy resources that include wind and photovoltaic (PV) sources have been introduced in South Australian power system. These wind generators are mostly variable speed machines, hence they neither have inertia response nor can they perform governor like response following a contingency. The demand in South Australia can be significantly catered utilizing renewable sources and considerable power import from the adjacent state Victoria. Therefore, the number of synchronous generators in the South Australian network can be small. In such situations, the frequency response following a contingency can be concerning and the network may undergo substantial under frequency load shedding. To overcome this challenge, this paper proposes a frequency responsive BESS model to improve the frequency response of a low inertia power system that resembles the South Australian network. The network is investigated under various operating conditions to explore the performance of BESS regarding frequency response enhancement.

odinger operators. 2000 Mathematics Subject Classification Primary 35J10, 45P05; Secondary 35P05, 47B10. The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper “Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves”, published by F. Gesztesy, K. Zumbrun and the second author in J. Math. Pures Appl. 90, 160‐200 (2008), where the Evans and Jost functions for the Schr¨ odinger equations have been considered. In the current work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman-Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so-called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation. One of the main results of the current paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above-mentioned generalized Jost solutions. Copyright line will be provided by the publisher

This paper presents a novel closed-form solution for a low-order system frequency response (SFR) model that is accurate for all time periods and an accompanying approximation for representing primary frequency responses at two different speeds while still maintaining mathematical tractability. This allows for the inclusion of both fast frequency responses (e.g. from battery energy storage systems) and more conventional responses (e.g. from thermal generation) in a single SFR formulation. The closed-form expressions can be efficiently used in applications that use the SFR model such as frequency stability studies and security-constrained unit commitment (SCUC) studies.

A novel model reduction method for viscously damped structures with statically determinate interfaces, such as spacecraft flexible appendages, is proposed. The paper presents a derivation of the complete complex modal expansion of the interface dynamic stiffness of these structures. Based on the identity relation for all complex modes, which is obtained during the derivation, it is found that the interface acceleration impedance can be expressed as a rational fraction with high accuracy using only low-order complex modes. Using this rational fraction as an approximation model, numerical results of the frequency response can be fitted. The fitted interface acceleration impedance can be applied to real-time control as a reduced model in the form of a transfer function. Furthermore, it can be transformed into the form of system matrices by introducing auxiliary variables, which then participate in the dynamic analysis of the assembly. The reduction process circumvents complex modal analysis and necessitates only the results of frequency responses. Thanks to the powerful ability of conventional finite element software to perform frequency response analysis, this reduction method can be used for large-scale complex models in actual engineering applications.

Frequency response is a key concern for power system security. Usually, it is estimated in a deterministic approach. But in most cases, power systems behave stochastically. Moreover, presence of wind power imposes further variability on generation profile of a power system. Hence, it appears more pragmatic to address these uncertainties during an evaluation of frequency response. This paper develops an index to assess frequency response of a power system with high wind power penetration. A probabilistic method is used for this purpose. K-means algorithm, a well-known data mining technique is employed. The developed index is called a ‘probabilistic frequency response index’. A realistic interconnected power system is investigated for a case study. Impact of increased wind power penetration on probabilistic frequency response index is analysed.

In this paper, the vector method design schemes of fractional order proportional differential controllers have been proposed. This method is to map the controller to get a set of vectors in the complex plane. According to the method of vector and geometric in the complex plane, the complicated derivation and calculation can be simplified in the process of the controller design. In addition, the established geometric relationship equations not only can still be used when the plant is different, but also can be programmed in MATLAB. Therefore, the repetitive process of formula derivation can be omitted. From the simulation results presented, the obtained parameters based on MATLAB program is unique and the designed fractional order proportional differential controllers work effective.

Pressure measurements in the microcirculation with active and passive transducers

This paper proposes a dynamical model and the governing equations of motion of the micro-cantilever beams based MEMS with piezoelectric actuator (PZT). The Rayleigh–Ritz method is used to reduce the order of the system and the state equations are presented in modal space. The first ten mode frequencies and mode shapes of the micro-cantilever beam with and without PZT are studied. The effects of PZT on the modal frequencies and shapes of the beam system can be ignored for the reason that the beam holds larger nature frequencies and Q values in micro-scale. A rational linearizing feedback controller with a high gain observer is designed to eliminate the unwanted deflection of the micro-cantilever beam system. The open-loop step response and the effects of situated places of PZT on the frequency responses of the system are discussed. Various frequency responses of the beam system, subject to different applied control voltages and feedback gains, are illustrated. The four resonances are well controlled, while the anti-resonance has little change. Computer simulations are provided to demonstrate the performance of the designed control scheme.

Chapter 4 - Frequency Analysis: The Fourier Series

This paper presents a systematic way of using linear analysis techniques to find and rank candidate locations for controlled shunt and series compensations for power system damping purposes. A two-stage approach based on frequency response computations is introduced using a 4780-bus system model of the Nordic power system for assessment of controlled shunt compensations. In the first stage, their controllability is determined by means of long-distance frequency responses. In the second stage, the frequency responses of the transfer functions from the bus shunt susceptance to the bus voltage magnitude are computed for buses that have shown high controllability. This paper shows that the interpretation of the frequency responses of these local transfer functions, for buses that have exhibited high controllability, is in most cases sufficient to predict the behavior of controlled shunt compensation at different locations in the system. The interpretation is based on the observation of amplitudes, phase angles, and the rate of change of the phase angles around the interarea resonance frequencies. An important finding was that controlled shunt compensation located in an area with high-power import over ac interconnections lowered the damping of at least one of the interarea oscillation modes if operating in normal voltage control. This finding was also verified with nonlinear simulations. The first stage of the proposed frequency response-based approach has also been introduced for assessment of controlled series compensations. The strength of the new methods is that they can be used with both linear and nonlinear system models.

FIR Adaptive Identification L∞ Bounds on Frequency Response Error