Minimum Phase Condition Research Articles

Bias control for optical single-sideband transmitter based on single IQ modulator

The optical in-phase and quadrature modulator (IQM) is typically utilized as an optical single-sideband (SSB) transmitter for its simple structure and low cost. It requires the modulator biases to be slightly offset for generating the optical carrier. However, this poses significant challenges in the bias control circuits, as most off-the-shelf automatic bias control (ABC) modules are designed to stabilize the IQM at its carrier suppression point for coherent optical communication. In this paper, a novel arbitrary ABC scheme is proposed to stabilize the carrier-to-signal power ratio (CSPR) of an optical SSB signal over long-term operation. This scheme uses SSB rather than double-sideband (DSB) sinusoidal signals as the dither signals. It does not violate the minimum phase condition, thus avoiding performance penalties compared to the conventional ABC scheme in which the DSB dither signal is used. We also develop a closed-form expression to reveal the relationship between the control parameter in the ABC algorithm and the CSPR of the optical SSB signal. By this means, the optical SSB transmitter can automatically adjust its bias voltages and stabilize them at a given CSPR value. The experimental results with ∼80 Gb/s optical SSB signal show that our proposed ABC scheme is effective in stabilizing the bias of the IQM at any point and maintains the CSPR value of the optical SSB signal over hours. It is also shown that the proposed method leads to 1 dB improvement in the receiver sensitivity compared with the traditional ABC technique.

Output Feedback Active Fault Tolerant Control for a 3-DOF Laboratory Helicopter With Sensor Fault

In this paper, an output feedback-based active fault tolerant control (FTC) scheme is proposed for an unstable three degree-of-freedom (3-DOF) helicopter equipped only with angular position sensors, of which a single sensor can be faulty. Limited by the available outputs, existing fault estimation (FE) and FTC schemes cannot be applied to this helicopter because when the sensor measuring the elevation or travel angle is faulty, it does not satisfy the minimum-phase and matching conditions required by standard observers. To circumvent this problem, an adaptive interval observer is firstly designed as a fault detection and isolation (FDI) unit to indicate the occurrence and location of a fault, in contrast to the existing FDI methods that require a bank of observers and incur a high computational cost. Then a high-gain observer is combined with a sliding mode observer to form an FE unit that does not require the matching and minimum-phase conditions. Based on the estimate of the fault, an FTC scheme is constructed to ensure an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_{\infty}$</tex-math> </inline-formula> performance of the faulty system. Finally, physical experiments on the 3-DOF helicopter verify the effectiveness of the proposed scheme. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The 3-DOF lab helicopter serves as an ideal experimental platform for control strategies. In this paper, an active fault tolerant control (FTC) scheme is developed for a 3-DOF lab helicopter when any single sensor can be faulty. The helicopter is nonlinear and subject to external disturbances. There are only encoders measuring the attitude angles and no sensors for angular velocities. In this paper, we firstly design a fault detection and isolation (FDI) unit based on an adaptive interval observer to detect the fault and identify its location; it incurs a lower computational cost compared to existing methods that use a bank of observers. Then a high-gain observer is combined with a sliding mode observer to estimate the fault. Based on the estimate of the fault, an FTC scheme is constructed and designed using Linear Matrix Inequalities to ensure an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_{\infty}$</tex-math> </inline-formula> performance of the faulty system.

Sliding mode observers for fault identification in linear systems not satisfying matching and minimum phase conditions

Sliding mode observers for fault identification in linear systems not satisfying matching and minimum phase conditions

A sliding mode technique for reconstruction of unknown control input in linear systems

The unknown input estimation problem for linear dynamic systems exposed to the disturbances is studied in the paper. A novel approach to designing sliding mode observer is suggested for systems not satisfying matching, minimum phase, and detectability conditions. To apply this approach, the reduced-order model of the original system is used. Such a model allows to reduce the dimension of the sliding mode observer, besides the restrictions placed on the original system are relaxed.

Leader-Follower Asymptotic Consensus Control of Multiagent Systems: An Observer-Based Disturbance Reconstruction Approach.

In this article, a leader-follower asymptotic consensus control strategy is developed for a class of linear multiagent systems (MASs) with unknown external disturbances and measurement noises. First, the preconditions, the minimum phase condition (MPC) and observer matching condition (OMC), are discussed in detail, and an equivalent result under these two preconditions is given. In this way, the corresponding results from Corless and Tu (1998) are improved. Meanwhile, a reduced-order observer is designed for a constructed augmented system to estimate the system states and noises of each agent. Next, with the help of a traditional interval observer, a novel unknown disturbance reconstruction method is developed, and the reconstruction can converge to the unknown disturbance asymptotically and decouple from the control input. The subsequent asymptotic consensus is accomplished by utilizing an observer-based control scheme, with its design satisfying the so-called separation principle. Finally, two simulation examples are given to verify the effectiveness and show the advantages of the proposed methods.

A limit Kalman filter and smoother for systems with unknown inputs

This paper derives the limit of the Kalman filter and smoother as the inverse of the process noise covariance tends to zero (the zero informational limit) in the case that there is direct feedthrough (of full column rank) of the process noise input to the measurements. Two forms of the filter in the limit are derived with the second being a standard Kalman filter without unknown inputs. The latter form is used to derive necessary and sufficient conditions for convergence and stability of the filter. These consist of a controllability condition and a minimum phase condition. The filter and smoother are applied to an automotive example to estimate an unknown road profile. The example illustrates the usefulness of the stability and convergence conditions to inform the choice of a suitable set of sensors.

Low Cost and Sustainable Test Methods to Study Vulnerabilities of Large-Scale Systems against EMP

Intense electromagnetic pulses are electromagnetic waves with sharp rise time, high field strength and short duration. They have attracted more and more attention in recent years because they can cause destructions or malfunctions of some key national core infrastructures, such as power grids, communication and financial networks, etc. Hence, it is important to harden these facilities to ensure that they can survive in the face of electromagnetic pulse attacks. A direct way to investigate the vulnerabilities of these facilities is placing them in the electromagnetic environments generated by EMP simulators. However, the scale of facilities under test are limited by the working volumes of simulators. With the increase in working volumes and field strength, the price and technical difficulty of simulators are increased. Therefore, low cost and sustainable test methods to investigate vulnerabilities of large-scale systems against EMP are proposed in this study. The study takes advantage of continuous wave immersion (CWI) test and pulse current injection (PCI) test methods, which are low cost and sustainable to predict the pulse responses and assess the nonlinear effect of large-scale facilities under EMP attacks. In a CWI test, the magnitude of the transfer function of large-scale systems or facilities can be measured, and the corresponding phase information of the transfer function can be reconstructed by minimum phase algorithm (MPA) if the systems meet the minimum phase condition. After acquiring the entire information of the transfer function, we can predict the responses of a system under threat-level EMP attacks. However, these responses are obtained under the assumption that the system does not have nonlinear effects. Because the CWI is a low-level test, it cannot simulate the threat-level EMP attacks. In the PCI test proposed here, a bulky pulse current is coupled into the system to stimulate enough current intensity, just as the system was attacked by threat-level EMPs. In this situation, the system would be destroyed, or any other nonlinear effect would occur in the system. After that, the problem is to determine the quantities of the injected current, and a few kinds of norms are introduced in this paper to define the quantities. The method proposed here innovatively takes the experimental results of CWI as reference inputs of PCI tests. In this paper, the accuracy of the response prediction is validated by means of simulations and experiments. Results show that as a low-level test method in the frequency domain, the CWI test method can not only analyze couplings of external electromagnetic energy from frequency domain but also predict responses of the facilities under high amplitude electromagnetic pulses. The nonlinear effect of large-scale facilities can be assessed by applying the PCI test method with the results from CWI prediction. Therefore, if infrastructures or facilities are too large to be tested under EMP simulators, an alternative approach is to carry out the CWI and PCI experiments.

Symbol-interleaved precoding technique for DDO-OFDM systems with the KK receiver.

Receiver sensitivity of direct-detection optical orthogonal frequency-division multiplexing (DDO-OFDM) systems can be significantly improved by using the Kramers-Kronig (KK) receiver. However, the KK algorithm is sensitive to the high signal-to-average power ratio (PAPR) of the OFDM signal, which may violate the minimum phase condition (MPC) and lead to severe signal distortions during the optical field reconstruction. Channel-independent discrete Fourier transform (DFT) precoding technique can efficiently reduce the PAPR and realize subcarrier signal-to-noise ratio (SNR) equalization, but the PAPR of the precoded signal will increase due to chromatic dispersion. Besides, the relevant researches exhibit that it is hard to accurately reconstruct the complex field from the sampled signals based on KK relation at the low carrier-to-signal power ratio (CSPR) even if the MPC is satisfied. To combat the signal distortions induced by the KK algorithm and improve the bit error rate (BER) performance, we propose and verify DFT/Walsh-Hadamard transform (WHT)-based precoding with a simple symbol-interleaving technique for DDO-OFDM transmission systems. In addition to PAPR reduction, the proposed technique can equalize both subcarrier and inter-symbol SNRs efficiently. The simulated results show that the BER performance can be improved by up to one order of magnitude at a CSPR of 6 dB after up to 1000km single-mode fiber transmission, compared to the conventional DFT precoding technique. It is expected that the proposed technique can be employed to improve the receiver sensitivity of medium-reach DDO-OFDM transmission systems with the KK receiver.

Output tracking control of non‐minimum phase systems using on‐line output redefinition approach based on barrier function method

This paper considers the output tracking problem of non-minimum phase non-linear systems using the inverse dynamics control which is impossible in conventional methods due to unstable zero dynamics. The proposed method is based on the output redefinition approach defined as a parametric function. The tracking error is decomposed into two terms. The first one named redefined output tracking error is the difference between the redefined output and the reference signal, and the second one is the distance between the redefined output and the system output. For the first term, due to minimum phase property of the system with the redefined output, an inverse dynamics control is used to guarantee the convergence of the redefined output tracking error to zero. To minimize the second term, the parameters of the redefined output are set by solving an optimization problem whose constraints include minimum phase conditions. To solve the optimization problem which satisfies all constraints, the barrier function method as an interior point method is adopted. The optimization parameters are set using steepest descent algorithm. Due to on-line updating the redefined output parameters, the related zero dynamics is time varying. Therefore, sufficient conditions for exponential stability of the resulting time-varying zero dynamics are obtained.

Impact of the carrier contribution factor in the self-coherent DC-value method

The carrier contribution factor (CCF) is used in the DC-value method to assure the minimum phase condition (MPC) at the transmitter. We analyze the impact of the CCF estimation accuracy at the receiver on the DC-value system performance. We found that the CCF estimation accuracy at the receiver should be within ±5% to exploit all advantages of the DC-value method. We propose an accurate method to estimate the CCF at the receiver that works effectively with both direct current (DC) and alternating current (AC) coupled photodetectors. We experimentally validate the proposed method employing a 24 GBaud 16QAM signal. The experimental results show that the minimum BER value can be approached within ±5% offset of the estimated CCF value almost independently of the carrier to signal power ratio (CSPR) value. The proposed method can also be used to reconstruct the DC component in other minimum phase signal self-coherent techniques, such as the Kramers-Kronig receiver.

Quantized-output feedback model reference control of discrete-time linear systems

This paper proposes a model reference control (MRC) scheme for general discrete-time linear time-invariant (LTI) systems subject to output quantization and saturation. In such a scheme, the quantized-output feedback MRC law is analytically specified only by using external reference input and saturated-and-quantized output. It is proven that, by appropriately designing the sensitivity of the quantizer, the MRC law can ensure that all closed-loop signals are bounded and the output tracking error converges to a certain small residual set in a certain finite time only under the minimum-phase condition. Particularly, the proposed MRC scheme does not rely on the coprimeness of zero and pole polynomials or initial conditions that are commonly used in the related literature. A representative example is given to demonstrate the design procedure and verify the validity of the proposed MRC scheme.

Fault identification in nonlinear dynamic systems not satisfying matching, minimum phase, and detectability conditions

The paper studies the fault identification problem for nonlinear control systems under unmatched disturbances. A novel approach to construct a sliding mode observer is proposed for systems that do not satisfy common conditions required for fault estimation, in particular matching condition, minimum phase condition, and detectability conditions. The suggested approach is based on the reduced order model of the original system and on the canonical form of matrices describing such a model. This allows to reduce the complexity of sliding mode observers and relax the limitations imposed on the original system. Three different cases for disturbances are considered and the appropriate solutions are suggested.

Investigation of the low-complexity Hilbert FIR filter enhanced 112-Gbit/s SSB 16-QAM transmission with parallelized Kramers-Kronig reception over 1440-km SSMF.

The Hilbert transform links the log-magnitude and the phase of the field modulated signals as long as the minimum phase condition is satisfied in the Kramer-Kronig (KK) receiver. In discrete-time signal processing, the Hilbert transform is generally replaced by a finite impulse response (FIR) filter to reduce the computational complexity, that is the so-called Hilbert transform FIR (HT-FIR) filter. The performance of the HT-FIR filter is extremely important, as the in-band flatness, the ripple, the group delay, the Gibbs phenomenon, and the edge effect, which indeed impair the phase retrieval. Hence, we investigate four different HT-FIR filter schemes that are in the form of type III and type IV based on the frequency-domain (FD) sampling approach and the time-domain (TD) windowing function approach. Also, we analyze the performance for each filter under different digital upsampling scenarios and conclude that a trade-off between the reduced inter-symbol-interference (ISI) and the Gibbs phenomenon is essential to obtain an optimal sampling rate and an improved KK performance when the HT-FIR filter with a short length is adopted. The results show that the FD-based HT-FIR filter can relax the upsampling requirement while having a better in-band flatness and a lower edge effect. The experiment is conducted in the parallelized block-wise KK reception-based 112-Gbit/s SSB 16-QAM optical transmission system over a 1920-km cascaded Raman fiber amplifier (RFA) link to investigate the limit transmission performance of the practical KK receiver. The experimental results show that when the transmission distance is up to 1440-km, the BER of the FD-based HT-FIR filter can be lower than the soft decision-forward error correction (SD-FEC) threshold of 2 × 10-2 with only 3 samples per symbol (3-SPS) upsampling rate and 8 non-integer tap coefficients are used, while other TD-based HT-FIR filter schemes with a BER lower than the SD-FEC threshold require at least 4-SPS upsampling rate.

Enhanced CSPR for a multichannel Kramers-Kronig receiver by self-seeded stimulated Brillouin scattering.

We demonstrate carrier-to-signal power ratio (CSPR) enhancement by self-seeded stimulated Brillouin scattering to improve the performance of Kramers-Kronig (KK) detection for multichannel single-sideband (SSB) signals. By virtue of low-CSPR transmission and high-CSPR detection, our proposed scheme effectively advances system performance by reducing propagation-induced distortion while maintaining the minimum phase condition. We experimentally demonstrate the improvement in CSPR and bit error rate of 5×10-Gbaud 16-QAM SSB signals by applying the carrier recovery block after 80-km transmission. Under optimum pump power, the average Q factor improvement of all five channels is 3.0 dB. We also analyze the performances of different channels and the major limiting factor. The results verify that our scheme offers a promising solution to enhance SSB self-coherent KK detection in wavelength-division multiplexing systems.

Fault Identification in Nonlinear Systems not Satisfying Matching and Minimum Phase Conditions

Abstract The paper studies the fault identification problem for nonlinear control systems under the unmatched disturbances. A novel approach to construct sliding mode observer is proposed for systems that do not satisfy common conditions required for fault estimation, in particular, minimum phase, matching, and detectability conditions. The suggested approach is based on the reduced order model of the original system invariant with respect to the disturbances. This allows to reduce complexity of sliding mode observer and relax the limitations imposed on the original system.

Employing a Kramers–Kronig receiver with a directly-modulated laser

The Kramers–Kronig (KK) receiver has been used for dispersion compensation in direct detection systems employing external-modulator-based transmitters. In this paper, we investigate the use of the KK receiver for dispersion compensation in a direct detection system based on a directly-modulated laser. We show the validity of the KK receiver by considering a linear contribution to the optical phase arising from a numerical solution of the laser rate equations as a frequency shift of the optical spectrum. The modulated optical signal can then meet the minimum-phase condition required by the KK algorithm. A system model is presented based on the direct modulation of a 10 Gb/s distributed feedback laser with 16-ary quadrature amplitude modulation (16-QAM) subcarrier modulation. The modulated signal is optically filtered, and a 53.5 Gb/s vestigial sideband signal is transmitted over 65 km of standard single-mode fiber. Dispersion compensation is applied after direct detection with a KK receiver. In addition, the impact of the optical filter parameters and receiver sampling rate on system performance are investigated.

State and delay reconstruction for nonlinear systems with input delays

In this paper, the problem of simultaneously reconstructing the states and delays for nonlinear systems subject to input delays is studied. To relax some limitations of the classical Lipschitz condition, the one-sided Lipschitz and quadratically inner-bounded conditions are employed to handle nonlinearities. The control inputs are subject to bounded unknown time-varying delays. To deal with this open and interesting problem, we propose a sliding mode observer based method to simultaneously estimate the states and identify the unknown delays of the systems without making assumption on differentiability of the delays. Furthermore, our method does not require the minimum phase condition that is common in sliding mode observer based schemes. The effectiveness of the proposed method is illustrated by a numerical example.

Minimum Phase Conditions in Kramers-Kronig Optical Receivers

Theoretically, for continuous signals, Kramers-Kronig (KK) receivers should be able to precisely reconstruct a band-limited signal's phase from its envelope if the minimum phase (MP) condition is satisfied. However, simulations have shown that even if the MP condition is satisfied, for sampled signals the reconstructed signal can contain large errors. In this article, we explore the effectiveness of DSP-based KK receivers close to the MP condition using single-tone and multi-tones. For a single-tone signal, we use a precise mathematical analysis to demonstrate large distortions in the reconstructed signal even when the MP condition is clearly satisfied and illustrate this on the complex-plane. We then study multi-tone signals spectrally, and show that there are strong intermodulation distortion tones when MP is satisfied that either directly fall in-band, or could be aliased back in-band during down-sampling. Satisfying MP does, however, eliminate signal-signal beat interference, as expected with KK processing. High up-sampling ratios reduce the spectral density of these tones, which explains the improved performance seen in experimental systems. We then explore the error-vector magnitude close to the boundary of MP for various carrier-to-signal power ratios and up-sampling ratios, to illustrate that there is a soft transition around the MP condition. Overall, this work shows that the MP condition is not sufficient to guarantee error-free performance in noiseless conditions.

Plant Extracts Containing Saponins Affects the Stability and Biological Activity of Hempseed Oil Emulsion System.

In this study, two saponins-rich plant extracts, viz. Saponaria officinalis and Quillaja saponaria, were used as surfactants in an oil-in-water (O/W) emulsion based on hempseed oil (HSO). This study focused on a low oil phase content of 2% v/v HSO to investigate stable emulsion systems under minimum oil phase conditions. Emulsion stability was characterized by the emulsification index (EI), centrifugation tests, droplet size distribution as well as microscopic imaging. The smallest droplets recorded by dynamic light scattering (droplets size v. number), one day after the preparation of the emulsion, were around 50–120 nm depending the on use of Saponaria and Quillaja as a surfactant and corresponding to critical micelle concentration (CMC) in the range 0–2 g/L. The surface and interfacial tension of the emulsion components were studied as well. The effect of emulsions on environmental bacteria strains was also investigated. It was observed that emulsions with Saponaria officinalis extract exhibited slight toxic activity (the cell metabolic activity reduced to 80%), in contrast to Quillaja emulsion, which induced Pseudomonas fluorescens ATCC 17400 growth. The highest-stability samples were those with doubled CMC concentration. The presented results demonstrate a possible use of oil emulsions based on plant extract rich in saponins for the food industry, biomedical and cosmetics applications, and nanoemulsion preparations.

Enhanced Kramers-Kronig Single-Sideband Receivers

Theoretically, Kramer–Kronig (KK) receivers are able to fully cancel signal-signal beat interference (SSBI), which results from a field-modulated single-sideband (with carrier) transmitter being used with a direct-detection receiver. However, even for strong carriers, which should meet the minimum-phase (MP) condition, bits are frequently erroneous. In this article, we simulate two methods of reducing error rates for weak carriers. The first is to limit the minimum value of the received signal, so that strong negative-going peaks are not produced by the logarithm function of the KK's ‘correction path’. The second is to replace the square-root function in the main signal path with a logarithmic function. This is to allow analog processing using semiconductor diodes. Our numerical simulations show that grossly approximating the square-root function is actually beneficial to the system, as it reduces the power in the signal close to the clipping events. When both the first and second techniques are combined, low error rates can be obtained at 2-dB less carrier power than for conventional processing, and 0.25-dB better receiver sensitivity. When 100-km of standard fiber is included, the carrier can be reduced by 3 dB, and the system has a 0.18 dB better sensitivity, and potentially supports analog processing. This work suggests that there may be other improvements possible, in signal quality and computational load, by moving away from a theoretically perfect implementation of the KK algorithm.