Linear Time Periodic Systems: Transfer Function, Poles, Transmission Zeroes and Directional Properties

This paper focuses on the parameter estimation of multi-input multi-output (MIMO), linear time-periodic (LTP) systems in the time domain. Discrete-time state-space models are selected for the description of LTP systems and the Covariance Based Realization Algorithm (CoBRA), one branch of subspace methods, is used for the identification of the generalized time-lifted state-space model for an LTP system. The use of the CoBRA method for estimation is motivated by the robustness against output noise with (high-order) noise dynamics and a focus on the estimation of low rank deterministic model for the LTP system. In addition to the use of CoBRA, a novel method is proposed to compute a topologically equivalent realization for the original LTP system. The method includes two different but theoretically equivalent approaches of calculating the state matrices, which can be also used as a measure for the estimation quality. Finally, a simulation is given to illustrate the efficiency of the proposed method.

CRONE control of continuous linear time periodic systems: Application to a testing bench

This paper proposes a new methodology for subspace identification of linear time-periodic (LTP) systems with periodic inputs. This method overcomes the issues related to the computation of frequency response of LTP systems by utilizing the frequency response of the time-lifted system with linear time-invariant structure instead. The response is estimated with an ensemble of input-output data with periodic inputs. This allows the frequency-domain subspace identification technique to be extended to LTP systems. The time-aliased periodic impulse response can then be estimated and the order-revealing decomposition of the block-Hankel matrix is formulated. The consistency of the proposed method is proved under mild noise assumptions. Numerical simulation shows that the proposed method performs better than multiple widely-used time-domain subspace identification methods when an ensemble of periodic data is available.

As per Floquet theory, a transformation matrix (Lyapunov Floquet transformation matrix) converts a linear time periodic system to a linear time-invariant one. Though a closed form expression for such a matrix was missing in the literature, this method has been widely used for studying the dynamical stability of a time periodic system. In this paper, the authors have derived a closed form expression for the Lyapunov Floquet (L-F) transformation matrix analytically using intuitive state augmentation, Modal Transformation and Normal Forms techniques. The results are tested and validated with the numerical methods on a Mathieu equation with and without damping. This approach could be applied to any linear time periodic systems.

We present the experimental demonstration of the occurrence of exceptional points of degeneracy (EPDs) in a single resonator by introducing a linear time-periodic variation of one of its components. This is in contrast with the requirement of two coupled resonators with parity time-symmetric systems with precise values of gain and loss. In the proposed scheme, only the tuning of the modulation frequency is required, which is easily achieved in electronic systems. The EPD is a point in a system parameters’ space at which two or more eigenstates coalesce, and this leads to unique properties not occurring at other non-degenerate operating points. We show theoretically and experimentally the existence of a second-order EPD in a time-varying single resonator. Furthermore, we measure the sensitivity of the proposed system to a small structural perturbation and show that the two shifted system’s eigenfrequencies are well detected even for relative perturbations of 0.3%, with distinguished peaks well above the noise floor. We show that the regime of operation of the system at an EPD leads to a unique square-root-like sensitivity, which can devise new exceptionally sensitive sensors based on a single resonator by simply applying time modulation.

Statistical Analysis and H2 - Norm of Finite Dimensional Linear Time-Periodic Systems

Planar microwave bandpass filters (BPFs) are an essential component in many communication systems. Traditional BPF design focuses on the application of frequency-invariant K-/J-inverters inserted between the two resonant modes or served as the external coupling to feed the resonators. This design method is most suitable for narrow bandwidth BPFs, but for wideband cases it is difficult for a direct application. To solve this problem, a synthesis method based on a modified general Chebyshev function and the Richard’s transformation has been proposed to characterize the frequency response over the whole frequency range for the wideband BPF designs. As for the corresponding circuit structure, short-circuited stepped-impedance multi-mode resonators (MMRs) serve as the foundation to the designs in this dissertation. By parallel-connecting or folding the MMRs, a class of dual-wideband BPFs has been developed. Based on the above discussion, this dissertation presents the wideband BPF designs in two aspects, the proposed filter synthesis procedure and a new class of circuit structures to realize the target filtering responses. As the foundation of this research work, short-circuited MMRs have been researched in the first place for a single-wideband BPF design. n sections of cascaded commensurate lines are used to group the resonances, and short-ended stubs are shunt on its two sides to inductively feed the MMR. For the filter design, a Chebyshev filtering function is used to characterize the frequency response of the circuit. By equating the filtering function to the transfer function of the circuit, the design parameters are obtained. The first set of dual-wideband BPFs is formed by parallel-connecting two dissimilar single-wideband BPFs, both formed by stepped-impedance MMRs. The two transmitting paths of 180o phase difference are used to generate up to two pairs of transmission zeroes (TZs) with symmetry to the dual-wide passbands. By appropriately choosing the characteristic impedances of the two paralleled MMRs, one pair of TZs is located in real frequencies contributing to the attenuation between the two passbands, whereas the second pair of TZs is located in imaginary frequencies adjusting the group delay within the passbands. The short-circuited stubs, in shunt connection on the two sides of the MMR, act as the inductive loading element to feed the MMR and to flatten the passband. In the meanwhile, they introduce TZs at multiple of π to suppress the DC-component. Following a similar design principle, the second set of dual-wideband BPF has been proposed as an improved design in terms of realizing a more compact size, the greater flexibility in terms of reflection zeros in the passband and an extra degree of freedom in choosing characteristic impedances. The generic structure of the proposed filters is formed by folding a MMR along its symmetrical plane and by coupling a pair of short-circuited stubs on the two ends of the MMR. Thus, two signal transmitting paths, one from the folded MMR and another from the loaded stubs, are used to generate additional TZs between the two passbands. By increasing sections of the loaded MMR, the reflection zeros in the passband is enlarged. Under fully-coupled structure, the merit of compact size and extra design freedom is obtained. For these two sets of dual-wideband BPFs mentioned above, the even-/odd-mode analysis method has been used to derive the transfer function to characterize their frequency responses. Given the properties of the proposed filters, the commonly used generalized Chebyshev filtering function as well as its transferred dual-band function is not appropriately used herein. Instead, a generalized Chebyshev filtering function, originally developed for a single-wideband BPF design, is readily applied. By directly formulating a filtering function with mirror to the origin, it has dual passbands respectively located at positive and negative frequency ranges. To accurately account for the wideband response, a TZ is assigned at the origin. Next, the Richard’s transformation is applied to transfer the normalized frequency response to its dual-band counterpart. With this proposed synthesis procedure, the targeted filtering function is characterized by a ratio of two polynomials. By equating these two aforementioned functions, which are the ideal filtering function from the specifications and the transfer function from the circuits, all the circuit design parameters are obtained. Finally, the proposed single-/dual-wideband bandpass filters have been synthesized and designed for practical implementation. The experimental results of the fabricated filter circuits have strongly verified these new filter structures and their corresponding synthesis method proposed.

An analytical approach for the synthesis of narrow-band lossy filters with arbitrary transmission zeros is proposed in this paper. The scattering parameters are deduced from filtering polynomials once the transmission zeros are assigned. The formulas defining the transformation from scattering matrix to admittance matrix can be used by reconstructing the non-paraconjugate transmission zeros as paraconjugate. A canonical $$N + 2$$ transversal or $$N \times N$$ coupling matrix is then identified using partial fraction expansion of the normalized admittance functions. An increased order of the final network (with respect to the order of the characteristic polynomials) is required for obtaining this result, provided there are non-paraconjugate transmission zeros. Because of its versatility, the proposed approach can also be employed for lossless networks that have a transfer function with symmetrical transmission zeros. Two typical examples are presented to validate the proposed technique.

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.

We design an open-loop active flow control for separated flows around a plunging circular cylinder based on resolvent analysis. The cylinder is plunging at a Strouhal number of 0.36 and a Reynolds number of 500. A linear time-periodic system for control is obtained by linearizing the non-inertial incompressible vorticity equation in the cylinder-fixed frame about a time-averaged base flow. Using the Lyapunouv–Floquet transformation, the linear time-periodic system is transformed into a similar linear time-invariant system, whose resolvent is analysed to obtain an optimal actuating Strouhal number of 0.1464 for the transformed linear system. Simulations show that the active control with tangential actuations is capable of reducing the lift fluctuation by up to 25.7 % when the flow is actuated near the predicted harmonic and subharmonic frequencies.

Sample and hold functions have been described for periodic linear system control applications where they are used to assign the eigenvalues of the system monodromy matrix. If the system is controllable in one period it was shown that the complete monodromy matrix can be assigned. In this paper we apply these methods to non periodic continuous time linear systems. When the system is controllable across each sampling interval, the state transition matrix associated with each interval can be completely assigned with full state feedback. Thus the resulting closed loop system resembles a closed loop periodic system whose monodromy matrix is completely specified.

This paper presents an original and simple coupling-matrix-based synthesis methodology for the design of a new class of bandpass filters (BPFs) that employ hybrid acoustic-wave-lumped-element resonator (AWLR) modules with improved out-of-band isolation (IS). The proposed BPFs feature quasi-elliptic-type frequency response—shaped by $N$ poles and $2N$ transmission zeros (TZs) for an $N$ th-order transfer function, compact physical size, and high effective quality factors $(Q_{\rm eff})$ of the order of 1000. Despite the use of acoustic wave (AW) resonators, passbands exhibiting fractional bandwidths (FBWs) that are no longer limited by the electromechanical coupling coefficient $(k_{t}^{2})$ of the constituent AW resonators are obtained. A coupling-matrix-based model of a multi-mode AW resonator is also reported. It facilitates the incorporation of high- and low-frequency spurious modes that are present in a realistic filter response so that they can be anticipated at the synthesis and simulation levels. For proof-of-concept validation purposes, two BPF prototypes at 418 MHz made up of commercially-available surface acoustic wave (SAW) resonators and surface mounted devices (SMD) were built and measured. They perform first- (one pole and two TZs) and second-order (two poles and four TZs) transfer functions with measured passband insertion losses (IL) between 2.4–5.4 dB, $Q_{\rm eff}$ between 1600–1900, 3-dB absolute bandwidths ranging from 0.52 to 1 MHz (i.e., 1.6–3.2 times $k_{t}^{2}$ ), and minimum IS levels between 25–46 dB.

A novel approach for the prediction of the milling stability based on the Simpson method

A class of multi‐band bandpass filters (BPFs) based on glass‐integrated multi‐resonant cells (MRCs) are reported. Each MRC comprises multiple in‐parallel connected series‐type resonators that create passbands in‐between transmission zeros (TZs)—located at the resonators’ resonance. The MRCs can be combined in multi‐stage configurations to create high‐order multi‐band transfer functions or used in the RF input and output ports of a conventional transversal multi‐band BPF to enhance its selectivity by adding poles and TZs without increasing its size. A compact integration concept using high‐Q 3D integrated‐passive devices (IPDs) is used for the realization of sub‐6‐GHz multi‐band BPFs. For practical validation purposes: (1) a tri‐band BPF with passbands centred at 1.58, 2.07, 2.83 Hz and fractional bandwidths (FBWs): 10.9, 9.06, 10% respectively and in‐band insertion loss (IL) between 2 and 2.8 dB and (2) a dual‐band BPF with passbands centred at 2 and 2.9 GHz with FBW 14.9% and 16.14% we manufactured and tested.

Most design techniques for power factor correction converters considers the cascade control technique assumes that output voltage ripple is removed from the feedback path using a compensator with low-pass characteristic. This strategy allows power factor correction and output voltage regulation but result in voltage loop with a poor dynamic performance. In order to mitigate this problem many designers have used notch filters in the control loop to increase the bandwidth of the controller without compromise the power factor characteristic. However, conventional linear time invariant approach is no longer valid to analyze the stability of these systems. This paper presents the linear time periodic systems approach to analyze the stability of these systems using harmonic transfer functions formulation. The paper details the analysis of a full-bridge power factor correction boost rectifier, including harmonic transfer function converter modeling, frequency response description, time-domain simulation and a stability analysis. Simulation results for two different controllers are carried out to compare linear time invariant and linear time periodic approaches.