Linear Time-periodic Research Articles

Small-Signal Modeling and Analysis of MMC Under Unbalanced Grid Conditions Based on Linear Time-Periodic (LTP) Method

As in grid-tied Modular Multilevel Converters (MMC), the steady-state operation under asymmetric grid conditions is rather general, especially for the overhead line of the AC network. Therefore, the small-signal dynamics of MMC under unbalanced grid conditions have become a great important research topic. In this paper, the inherent periodic time-varying characteristic of MMC's original relationships is given priority consideration. Based on the linear time-periodic (LTP) method, a general small-signal LTP model under unbalanced grid conditions is built considering the internal dynamic of submodules (SMs). Then based on the eigenvalue locus analysis method from Floquet- Lyapunov theory, the effects of controller parameters and interactions between different controllers on small-signal dynamics are studied, respectively. Finally, the results from the detailed nonlinear MMC model are provided to verify the effectiveness of the proposed analysis.

Linear Time-Periodic System Identification with Grouped Atomic Norm Regularization

This paper proposes a new methodology in linear time-periodic (LTP) system identification. In contrast to previous methods that totally separate dynamics at different tag times for identification, the method focuses on imposing appropriate structural constraints on the linear time-invariant (LTI) reformulation of LTP systems. This method adopts a periodically-switched truncated infinite impulse response model for LTP systems, where the structural constraints are interpreted as the requirement to place the poles of the non-truncated models at the same locations for all sub-models. This constraint is imposed by combining the atomic norm regularization framework for LTI systems with the group lasso technique in regression. As a result, the estimated system is both uniform and low-order, which is hard to achieve with other existing estimators. Monte Carlo simulation shows that the grouped atomic norm method does not only show better results compared to other regularized methods, but also outperforms the subspace identification method under high noise levels in terms of model fitting.

High-Current Switched Capacitor Converter for On-Package VR With PDN Impedance Modeling

Digital ASIC devices are widely used in networking and computing applications. This kind of devices is implemented with a short-channel technology requiring high peak currents for high complexity systems and a low supply voltage. Digital ASICs are powered by an external voltage regulator with specifications similar to modern microprocessors’ power supply [voltage regulation modules (VRMs)]. In order to reduce the number of power pins and to reduce the power distribution network (PDN) issue, Intel’s Fourth-Generation Core integrates the voltage regulators. Moreover, many on-package conversion systems are present in the literature. In this article, a conversion solution based on a switched resonant tank is presented, yielding currents up to 300 A at 0.8 V, in an area of 10 cm2, with a resonant-driving technique. The novel converter is used to validate a new linear time-periodic (LTP) system modeling approach that can be applied to generic switched topologies; this contribution yields a mathematical description of the conversion chain, in particular enabling the precise calculation of the output impedance when a switched topology is used as the last stage.

Experimental Validation of Harmonic Impedance Measurement and LTP Nyquist Criterion for Stability Analysis in Power Converter Networks

This paper presents the first experimental validation of the stability analysis based on the online measurement of harmonic impedances exploiting the linear time-periodic (LTP) approach applied to ac networks of power converters. Previous studies have provided the theoretical framework for the method, enabling the stability assessment of an unknown system adopting a black-box approach, relying only on injected perturbations and local measurements. The experimental case study considered in this paper comprises two single-phase converters, one acting as source subsystem and the other as load subsystem . A third converter, the stability measurement unit , is controlled to inject small current perturbations at the point of common coupling (PCC). From the measured small-signal perturbations of PCC voltage, source current, and load current, the harmonic impedances of source and load subsystems are calculated. The LTP Nyquist criterion is then applied to the ratio of the two harmonic impedances in order to assess the stability of the whole system. Theoretical and experimental results from a 5-kW laboratory prototype are provided and confirm the effectiveness of the method. In addition, the measurements do not require sophisticated equipment or control boards and can be easily performed from data sampled by commercial micro-controllers.

Frequency-Domain Subspace Identification of Linear Time-Periodic (LTP) Systems

This paper proposes a new methodology for subs-pace-based state-space identification for linear time-periodic (LTP) systems. Since LTP systems can be lifted to equivalent linear time-invariant (LTI) systems, we first lift input–output data from an unknown LTP system as if they were collected from an equivalent LTI system. Then, we use frequency-domain subspace identification methods to find the LTI system estimate. Subsequently, we propose a novel method to obtain a time-periodic realization for the estimated lifted LTI system by exploiting the specific parametric structure of Fourier series coefficients of the frequency-domain lifting method. Our method can be used to obtain state-space estimates for unknown LTP systems as well as to obtain Floquet transforms for known LTP systems.

Modeling and Stability Assessment of Single-Phase Grid Synchronization Techniques: Linear Time-Periodic Versus Linear Time-Invariant Frameworks

The grid synchronization unit, which is often based on a phase-locked loop (PLL) or a frequency-locked loop (FLL), highly affects the power converter performance and stability, particularly under weak grid conditions. It implies that a careful stability assessment of grid synchronization techniques (GSTs) is of vital importance. This task is most often based on obtaining a linear time-invariant (LTI) model for the GST and applying standard stability tests to it. Another option is modeling and dynamics/stability assessment of GSTs in the linear time-periodic (LTP) framework, which has received a little attention. In this letter, the procedure of deriving the LTP model for single-phase GSTs is first demonstrated. The accuracy of the LTP model in predicting the GST dynamic behavior and stability is then evaluated and compared with that of the LTI one. Two well-known single-phase GSTs, i.e., the second-order generalized integrator-based FLL (SOGI-FLL) and enhanced PLL (EPLL), are considered as the case studies.

Stability Assessment of High-Bandwidth DC Voltage Controllers in Single-Phase Active Front Ends: LTI Versus LTP Models

In recent years, a considerable effort has been made to minimize the size of dc-link capacitors in single-phase active front ends (SP-AFEs), to reduce cost, and to increase power density. As a result of the lower energy storage, a high-bandwidth outer dc voltage control loop is required to respond to fast load changes. Linearized modeling is usually performed according to the power-balance method and the control is designed using LTI techniques. This is done assuming negligible voltage ripple at twice the grid frequency, and the model is considered valid up to the grid frequency. However, its precise validity limits are usually unknown and the control design becomes empirical when approaching these boundaries. To overcome this drawback, linear time-periodic (LTP) theory can be exploited, defining the range of validity of the LTI model and providing precise stability boundaries for the dc-link voltage loop. The main result is that LTP models more accurately describe the system behavior and provide superior results compared to the LTI ones. Theoretical analysis, simulations, and extensive experimental tests on a 10-kW converter are presented to validate the claims.

Frequency domain analysis of regenerative chatter in machine tools with Linear Time Periodic dynamics

Abstract Unstable vibrations (Chatter) during machining operations is caused by the flexibility of various components of the machine tool or the workpiece, which can be stationary and symmetrical or rotating and non-symmetrical. Therefore, the structural dynamics of machining systems are best described by Linear Time Periodic (LTP) systems, but the existing solutions of chatter use Linear Time Invariant (LTI) descriptions of the machine tool dynamics. This paper presents an update to the widely used Multi-Frequency Solution of regenerative chatter by modelling the machine tool dynamics as a LTP system. The stability of the LTP system is then determined by utilizing the concept of Harmonic Transfer Functions (HTF) and Nyquist stability criterion. The presented method provides a unified solution to determine the stability of vibrations in both stationary and rotating coordinate frames, regardless of the stationary or rotating nature of the vibration modes. The accuracy of the presented approach is verified by comparing the predicted stability diagrams with the results of Multi-Frequency Solution, Semi-Discretization Method, and with experimental results.

Harmonic transfer functions based controllers for linear time-periodic systems

The analysis, identification and control of periodic systems has gained increasing interest during the last few decades due to the increased use of dynamical systems that exhibit periodic motion. The vast majority of these studies focus on the analysis and control problem for a known state-space formulation of the linear time-periodic (LTP) system. On the other hand, there are also some studies that focus on data-driven identification of LTP systems with unknown state-space formulations. However, most of these methods provide numerical estimates for the harmonic transfer functions (HTFs) of an LTP system that are extremely difficult to work with during controller design. The goal of this paper is to provide a simple controller design methodology for unknown LTP systems by utilizing so-called HTFs estimates. To this end, we first build a mathematical basis of LTP controller design for known LTP systems using the Nyquist diagrams and analytically derived HTFs. We propose a new methodology to design P-, PD- and PID-type controllers for LTP systems using Nyquist diagrams and the eigenlocus of the HTFs. Having established the HTF-based controller design procedure, we extend our methodology to unknown LTP systems by presenting a new sum-of-cosine signal-based data-driven system identification method. We show that the proposed data-driven controller design method allows estimation of the HTFs and it provides simple tools for optimizing certain time-domain performance metrics. We provide numerical examples for both known and unknown LTP system cases to illustrate the performance of the proposed controller design methodology.

Stability Analysis of AC–DC Full-Bridge Converters With Reduced DC-Link Capacitance

Voltage ripple in single-phase ac–dc converters is usually disregarded to design the dc-link voltage control system, since large dc-link electrolytic capacitors are typically employed. However, replacement of electrolytic capacitors by film capacitors has been widely considered for increasing reliability. Consequently, due to cost constraints, such replacement usually employ low capacitance and may be combined with fast dc-bus voltage controllers. This paper shows that the conventional linear time-invariant (LTI) model may not represent the real converter behavior for reduced capacitance and/or faster controllers, leading to poor system performance or even instability. In this way, for any of these cases, the linear time-periodic (LTP) model is highly encouraged for stability and transient analysis as a tool for the controller design. Experimental results confirm that stability margins are precisely obtained from the LTP model but may not from the LTI model. Finally, this paper also compares both LTI and LTP models while considering the control gain and the dc-bus capacitance size. It is graphically revealed that the phase margin of the LTI model diverges from the LTP model for low dc-bus capacitances and high control gains.

Stability Assessment of Power-Converter-Based AC systems by LTP Theory: Eigenvalue Analysis and Harmonic Impedance Estimation

Stability analysis of power-converter-based ac systems poses serious challenges not only because of the nonlinear nature of power converters, but also because linearization is not generally applied around a steady-state operating point, as in the dc case, but around a time-periodic operating trajectory. Typical examples are single-phase and unbalanced three-phase systems. In this paper, two general methods to assess stability of the aforementioned systems are presented. Both are based on the linear time periodic (LTP) systems theory. The first is model-based and relies on the evaluation of the eigenvalues of the linearized model, assuming a complete knowledge of the parameters. By contrast, the second proposes a set of small-signal current injections to measure the harmonic impedances and applies the LTP Nyquist criterion, so that the stability of the system can be assessed with a black-box approach, without relying on the knowledge of the system parameters. The basic LTP systems theory is reviewed in order to provide a mathematical justification for the second method. As case study, a simple network, consisting of a source full-bridge converter in ac voltage-control mode and a load full-bridge converter in ac current-control mode including phase locked loop, is considered. Analytical results based on average modeling and simulations based on both average and switching models are presented, showing good accuracy in the identification of the stability thresholds for both the proposed methods.

Linear Time Invariant Approximations of Linear Time Periodic Systems

Several methods for analysis of linear time periodic (LTP) systems have successfully been demonstrated using harmonic decompositions. One method recently examined is to create a linear time invariant (LTI) model approximation by expansion of the LTP system states into various harmonic state representations, and formulating corresponding linear time invariant models. Although this method has shown success, it relies on a second order formulation of the original LTP system. This second order formulation can prove problematic for degrees of freedom not explicitly represented in second order form. Specifically, difficulties arise when performing the harmonic decomposition of body and inflow states as well as interpretation of LTI velocities. Instead this paper will present a more generalized LTI formulation using a first order formulation for harmonic decomposition. The new first order approach is evaluated for a UH-60 rotorcraft model, and is used to show the significance of particular harmonic terms; specifically that the coupling of harmonic components of body and inflow states with the rotor states has a significant contribution to the LTI model fidelity in the prediction of vibratory hub loads.

Stability Analysis of Vector-Controlled Modular Multilevel Converters in Linear Time-Periodic Framework

Stability analysis of average value models (AVMs) of vector-controlled modular multilevel converters (MMCs) is the subject matter of this paper. Stability analysis of fundamental frequency phasor-based AVMs of MMCs can be conducted in a traditional linear time-invariant framework through eigenvalue computation. This class of models does not consider circulating current control loop and hence fails to capture system instability that occurs in a certain range of gains of the circulating current controller. We propose stability analysis in a linear time-periodic (LTP) framework to solve this issue. To that end, a nonlinear AVM is presented that considers the submodule capacitor insertion dynamics and takes into account the output and the circulating current control schemes in the vector control approach. Upon linearization, an LTP model is derived from this averaged model. It is shown that the Poincar $\acute{e}$ multipliers are indicative of system instability corresponding to a certain range of gains of the circulating current controller.

Identification of a vertical hopping robot model via harmonic transfer functions

A common approach to understanding and controlling robotic legged locomotion is the construction and analysis of simplified mathematical models that capture essential features of locomotor behaviours. However, the representational power of such simple mathematical models is inevitably limited due to the non-linear and complex nature of biological locomotor systems. Attempting to identify and explicitly incorporate key non-linearities into the model is challenging, increases complexity, and decreases the analytic utility of the resulting models. In this paper, we adopt a data-driven approach, with the goal of furnishing an input–output representation of a locomotor system. Our method is based on approximating the hybrid dynamics of a legged locomotion model around its limit cycle as a Linear Time Periodic (LTP) system. Perturbing inputs to the locomotor system with small chirp signals yield the input–output data necessary for the application of LTP system identification techniques, allowing us to estimate harmonic transfer functions (HTFs) associated with the local LTP approximation to the system dynamics around the limit cycle. We compare actual system responses with responses predicted by the HTF, providing evidence that data-driven system identification methods can be used to construct models for locomotor behaviours.

Exact LTP Representation of the Generalized Periodic-Reference FxLMS Algorithm

The analysis of the Filtered-x Least Mean Square (FxLMS) algorithm can be based either on a stochastic or deterministic approach. One of the drawbacks of the stochastic analysis is that it relies on several assumptions which are not justified if the reference signal is time periodic. To overcome these limitations, several deterministic methods have been suggested. However, useful results have been reported only for setups with certain filter length, secondary path model, or reference signal. In this paper, an exact Linear Time-Periodic (LTP) representation is proposed assuming only that the reference is synchronously-sampled. The representation is derived for a general FxLMS system, which also covers the multichannel topology of several parallel filters and its common narrowband modification. The representation is found to have a Linear Time-Invariant (LTI) form if the filter length is suitably chosen. The usability of the method is demonstrated in a numerically computed example by comparing it to an approximate LTI model.

Frequency domain, parametric estimation of the evolution of the time-varying dynamics of periodically time-varying systems from noisy input–output observations

This paper presents a frequency domain, parametric identification method for continuous- and discrete-time, slow linear time-periodic (LTP) systems from input–output measurements. In this framework, the output as well as the input is allowed to be corrupted by stationary noise (i.e. an errors-in-variables approach is adopted). It is assumed that the system under consideration can be excited by a broad-band periodic signal with a user-defined amplitude spectrum (i.e. a multisine), and that the periodicity of the excitation signal Texc can be synchronized with the periodicity of the time-variation Tsys (i.e. Texc/Tsys∈Q), such that the system reaches a steady state (a periodic solution). Tsys is also known as the pumping period. Once the parametric estimation of the time-evolution of the system parameters has been performed, the system model is evaluated at the level of the instantaneous transfer function (also known as system function, or parametric transfer function), which rigorously characterizes LTP systems. If the dynamics of the LTP system are slowly varying or the system is linear parameter varying (LPV), a frozen transfer function approach is provided to easily visualize and assess the quality of the estimated model. To give the estimated quantities a quality label, uncertainty bounds on the model-related quantities (such as the time-periodic (TP) system parameters, the frozen transfer function, the frozen resonance frequency, etc.) are derived in this paper as well. Besides, a clear distinction between the instantaneous and the frozen transfer function concept is made, and both can be estimated with the proposed identification scheme. The user decides which transfer function definition suits best its purpose in practice. Finally, the identification algorithm is applied to a simulation example and to real measurements on an extendible robot arm.

A Decomposition-Based Approach to Linear Time-Periodic Distributed Control of Satellite Formations

In this paper, we consider the problem of designing a distributed controller for a formation of spacecraft following a periodic orbit. Each satellite is controlled locally on the basis of information from only a subset of the others (the nearest ones). We describe the dynamics of each spacecraft by means of a linear time-periodic (LTP) approximation, and we cast the satellite formation into a state-space formulation that facilitates control synthesis. Our technique exploits a novel modal decomposition of the state-space model and uses linear matrix inequalities (LMIs) for suboptimal control design of distributed controllers with guaranteed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance for formations of any size. The application of the method is shown in two case studies. The first example is inspired by a mission in a low, sun-synchronous Earth orbit, namely the new Dutch-Chinese Formation for Atmospheric Science and Technology demonstration mission (FAST), which is now in the preliminary design phase. The second example deals with a formation of spacecraft in a halo orbit.

Cramer–Rao Bound Development for Linear Time Periodic Systems

System identification techniques are often used to determine the parameters required to define a model of a linear time invariant (LTI) system. The Cramer–Rao bound can be used to validate those parameters in order to ensure that the system model is an accurate representation of the system. Unfortunately, the Cramer–Rao bound is only valid for LTI systems and is not valid for linear time periodic (LTP) systems such as a helicopter rotor in forward flight. This paper describes an extension of the Cramer–Rao bound to LTP systems and demonstrates the methodology for a simple LTP system.

Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or linear time-periodic (LTP). Examples include anisotropic rotor-bearing systems, wind turbines, and nonlinear systems linearized about a periodic trajectory. Many of these have been treated analytically in the literature, yet few methods exist for experimentally characterizing LTP systems. This paper presents a set of tools that can be used to identify a parametric model of a LTP system, using a frequency-domain approach and employing existing algorithms to perform parameter identification. One of the approaches is based on lifting the response to obtain an equivalent linear time-invariant (LTI) form and the other based is on Fourier series expansion. The development focuses on the preprocessing steps needed to apply LTI identification to the measurements, the postprocessing needed to reconstruct the LTP model from the identification results, and the interpretation of the measurements. This elucidates the similarities between LTP and LTI identification, allowing the experimentalist to transfer insight between the two. The approach determines the model order of the system and the postprocessing reveals the shapes of the time-periodic functions comprising the LTP model. Further postprocessing is also presented, which allows one to generate the state transition and time-varying state matrices of the system from the output of the LTI identification routine, so long as the measurement set is adequate. The experimental techniques are demonstrated on simulated measurements from a Jeffcott rotor mounted on an anisotropic flexible shaft supported by anisotropic bearings.

Dynamic Phasor Analysis of Periodic Systems

The paper considers stability analysis of linear time-periodic (LTP) systems based on the dynamic phasor model (DPM). The DPM exploits the periodicity of the system by expanding the system state in a Fourier series over a moving time window. This results in an L 2-equivalent representation in terms of an infinite-dimensional LTI system which describes the evolution of time varying Fourier coefficients. To prove stability, we consider quadratic time-periodic Lyapunov candidates. Using the DPM, the corresponding time-periodic Lyapunov inequality can be stated as a finite dimensional inequality and the Lyapunov function can be found by solving a linear matrix inequality.