Lower Triangular Structure Research Articles

Two-loop planar master integrals for NNLO QCD corrections to W-pair production in quark-antiquark annihilation

The planar two-loop scalar Feynman integrals contributing to the massive NNLO QCD corrections for W-boson pair production via quark-antiquark annihilation can be classified into three family branches, each of which is reduced to a distinct set of master integrals (MIs), totaling 27, 45 and 15, respectively. These MIs are analytically calculated using the method of differential equations, with solutions expanded as Taylor series in the dimensional regulator ϵ. For the first two family branches, the differential systems can be successfully transformed into canonical form by adopting appropriate bases of MIs. This enables the MIs of these family branches to be expressed either as Goncharov polylogarithms (GPLs) or as one-fold integrals over GPLs, up to O(ϵ4). In contrast, the differential system for the third family branch can only be cast into a form linear in ϵ due to the presence of elliptic integrals. The solution to this linear-form differential system is expressed in an iterated form owing to the strictly lower-triangular structure of the coefficient matrices at ϵ = 0. Our analytic expressions for these MIs are verified with high accuracy against the numerical results from the AMFlow package.

Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate

Article Adaptive Output Feedback Tracking Control for Nonlinear Systems with Unknown Growth Rate Manman Yuan 1,2,*, and Wei Qian 1,2,* 1 School of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454003, China 2 Henan Key Laboratory of Intelligent Detection and Control of Coal Mine Equipment, Jiaozuo 454003, China * Correspondence: ymmamhappy@163.com; qwei@hpu.edu.cn Received: 4 September 2023 Accepted: 7 November 2023 Published: 26 March 2024 Abstract: In this paper, the problem of adaptive output feedback tracking is considered for a class of nonlinear systems with lower-triangular structures. A novel dynamic gain is introduced to deal with the unknown growth rate. By coupling the dynamic gain with the observer and the controller, an adaptive output tracking controller is developed, which can guarantee that all signals of the closed-loop system are globally bounded. Finally, the effectiveness of the presented control scheme is illustrated by a numerical example.

Implementation of High Speed and Area Efficient Polar Encoder

Abstract: Polar codes were introduced by E.Arıkan in 2008. They are the first family of error-correcting codes that attain the capacity of binary memoryless and symmetric channels with efficient encoding, decoding, and construction algorithms. In this paper, implementation of high speed and area efficient polar encoder for systematic polar codes is presented. According to an iterative property of the generator matrix and particular lower triangular structure of the matrix, the number of XOR computations are reduced. In this implementation, total area of the encoder is decreased by 33.69%, delay is decreased by 54.6% and power consumption is decreased by 39.44%.

Back-stepping projective synchronization of fractional-order unified systems based on the lower triangular structure

The synchronization of fractional-order chaotic systems (FOCSs) plays an important role in modern control theory, the projective synchronization (PS) as a class of synchronization problems, also has huge applications and has attracted much attention. It is, however, shown in the obtained literature that the results on the PS of FOCSs either loss the rigorous theoretical demonstration or verify from the viewpoint of numerical simulations. How to derive a necessary and sufficient condition to guarantee the PS of complex FOCSs by a simple controller is still open. To this end, this article is concerned with the PS of fractional-order unified systems (FO-USs) that are important in FOCSs covering fractional-order Lorenz, Chen and Lü systems, where the controller is presented based on the lower triangular structure by use of the back-stepping technique. The necessary and sufficient criterion for the PS of FO-USs is proposed by solving an algebraic equation, and the controller for the PS of FO-USs is derived based on the lower triangular structure combined with back-stepping approach. Finally, the simulation results are reported to verify the correctness and efficiency of the obtained results.

Bayesian Exploratory Factor Analysis via Gibbs Sampling

Bayesian methods to infer model dimensionality in factor analysis generally assume a lower triangular structure for the factor loadings matrix. Consequently, the ordering of the outcomes influences the results. Therefore, we propose a method to infer model dimensionality without imposing any prior restriction on the loadings matrix. Our approach considers a relatively large number of factors and includes auxiliary multiplicative parameters, which may render null the unnecessary columns in the loadings matrix. The underlying dimensionality is then inferred based on the number of nonnull columns in the factor loadings matrix, and the model parameters are estimated with a postprocessing scheme. The advantages of the method in selecting the correct dimensionality are illustrated via simulations and using real data sets.

H∞Tracking Control of Non-lower Triangular Nonlinear Systems: Coronary Artery System

This work presents a novel control approach that integrates backstepping and dynamic surface control techniques to design anH∞tracking control system. Our focus is on a particular class of nonlinear systems that do not possess lower triangular structure, with a specific application to the coronary system. By employing dynamic surface control, the issue of differential explosion encountered during the backstepping process is effectively addressed, while also guaranteeing robustness. Finally, simulation of coronary artery system proves the effectiveness of the proposed control strategy.

Homogeneity, Forward Completeness, and Global Stabilization of a Family of Time-Delay Nonlinear Systems by Memoryless Non-Lipschitz Continuous Feedback

For a family of genuinely nonlinear systems with delays in the input and state, global strong stabilization (GSS) in the sense of Kurzweil is shown to be possible by memoryless state and/or output feedback under two conditions: 1) The input delay is within an appropriate range although the state delays can be sufficiently large; 2) the time-delay bounding system is homogeneous of degree zero and has a lower triangular structure. The proof is based on the Lyapunov–Krasovskii functional theorem combined with the homogeneous domination philosophy. Using the emulation approach, we design memoryless homogeneous state and output feedback controllers, respectively, achieving GSS of the time-delay closed-loop systems with severe nonlinearity. Extensions to a wider class of time-delay nonlinear systems in the Hessenberg form are also given. Examples and counter-examples presented in this article illustrate not only the necessity of the homogeneous growth conditions but also the significance of the memoryless non-Lipschitz continuous feedback control strategies.

Cooperative Control of Heterogeneous Connected Vehicles with Directed Acyclic Interactions

Cooperation of multiple connected vehicles has the potential to benefit the road traffic greatly. In this paper, we consider analysis and synthesis problems of the cooperative control of a platoon of heterogeneous connected vehicles with directed acyclic interactions (characterized by directed acyclic graphs). In contrast to previous works that view heterogeneity as a type of uncertainty, this paper directly takes heterogeneity into account in the problem formulation, allowing us to develop a deeper understanding of the influence of heterogeneity on the collective behavior of a platoon of connected vehicles. Our major strategies include an application of the celebrated internal model principle and an exploitation of lower-triangular structures for platoons with directed acyclic interactions. The major findings include: 1) we explicitly highlight the tracking ability of heterogeneous platoons, showing that the followers can only track the leader's spacing and velocity; 2) we analytically derive a stability region of feedback gains for platoons with directed acyclic interactions; 3) and consequently we propose a synthesis method based on the solution to an algebraic Riccati equation that shares the dimension of single vehicle dynamics. Numerical experiments are carried out to validate the effectiveness of our results.

One-Point Spectrum Nordsieck Almost Collocation Methods

A family of multivalue collocation methods for the numerical solution of differen- tial problems is proposed. These methods are developed in order to be suitable for the solu- tion of stiff problems, since they are highly stable and do not suffer from order reduction, as they have uniform order of convergence in the whole integration interval. In addition, they permits to have an effcient implementation, due to the fact that the coeffcient matrix of the nonlinear sys- tem for the computation of the internal stages has a lower triangular structure with one-point spectrum. The uniform order of convergence is numerically computed in order to experimentally verify theoretical results.

Back‐stepping stabilization of fractional‐order triangular system with applications to chaotic systems

AbstractThis paper addresses the stabilization of fractional‐order nonlinear lower triangular systems and applies the obtained results to the synchronization of fractional‐order chaotic systems (FOCSs). Three cases are considered as follows: (i) FOCSs have a lower triangular structure themselves. (ii) FOCSs can be converted into the form in full by appropriate nonsingular coordinate transformations. (iii) FOCSs can be transformed into the structure in part by the conversions. It is shown that the above three cases combining with back‐stepping approach are useful for the control of FOCSs. Different from the previous literature, the system structure studied is more concise with more applications. A numerical example is presented to show the validity of the proposed results.

An optimized encoding algorithm for systematic polar codes

Many different encoding algorithms for systematic polar codes (SPC) have been introduced since SPC was proposed in 2011. However, the number of the computing units of exclusive OR (XOR) has not been optimized yet. According to an iterative property of the generator matrix and particular lower triangular structure of the matrix, we propose an optimized encoding algorithm (OEA) of SPC that can reduce the number of XOR computing units compared with existing non-recursive algorithms. We also prove that this property of the generator matrix could extend to different code lengths and rates of the polar codes. Through the matrix segmentation and transformation, we obtain a submatrix with all zero elements to save computation resources. The proportion of zero elements in the matrix can reach up to 58.5% from the OEA for SPC when the code length and code rate are 2048 and 0.5, respectively. Furthermore, the proposed OEA is beneficial to hardware implementation compared with the existing recursive algorithms in which signals are transmitted bidirectionally.

Globally exponential stabilization for a class of nonlinear systems with time delays both in nonlinearities and input

This paper stabilizes a class of nonlinear systems with lower-triangular structure and different time delays at exponential convergence rate via output feedback domination method. An explicit construction for the output feedback stabilizer guaranteeing the globally exponential stabilization of the system is proposed, in which a scaling gain is introduced and can be designed to dominate the nonlinearities. We then analyze the exponential stability of the system via constructing a Lyapunov functional, and result in some constraints subject to the scaling gain and time-delays. Note that it is the first time to analyze the effects from time delays on the stability of lower-triangular systems. Two simulation results are provided to support the theoretical developments.

Semiglobal Asymptotic Stabilization of Lower Triangular Systems by Digital Output Feedback

Digital control of nonlinear systems is studied by output feedback. It is proved that for nonlinear systems with a lower triangular structure, semiglobal asymptotic stabilization is still possible via discretized output feedback as long as a sampling time is small. The establishment of semiglobal asymptotic stabilizability needs no restrictive condition on the nonlinearities and/or unmeasurable states of the system, such as a linear growth condition or an output-dependent growth condition as commonly required in the case of global output feedback stabilization. It involves, however, tedious but subtle stability analysis due to the hybrid nature of the closed-loop system. A design method through “sample and hold” is developed to construct a semiglobally asymptotically stabilizing, digital output feedback compensator that consists of a high-gain nonlinear observer and controller, both with saturated states.

Reducing time headway for platooning of connected vehicles via V2V communication

In a platoon of connected vehicles, time headway plays an important role in both traffic capacity and road safety. It is desirable to maintain a lower time headway while satisfying string stability in a platoon, since this leads to a higher traffic capacity and guarantees the disturbance attenuation ability. In this paper, we study a multiple-predecessor following strategy to reduce time headway via vehicle-to-vehicle (V2V) communication. We first introduce a new definition of desired inter-vehicle distances based on the constant time headway (CTH) policy, which is suitable for general communication topologies. By exploiting lower-triangular structures in a time headway matrix and an information topology matrix, we derive a set of necessary and sufficient conditions on feedback gains for internal asymptotic stability. Further, by analyzing the stable region of feedback gains, a necessary and sufficient condition on time headway is also obtained for the string stability specification. It is proved that a platoon can be asymptotically stable and string stable when the time headway is lower bounded. Moreover, this bound can be reduced by increasing the number of predecessors. These results explicitly highlight the benefits of V2V communication on reducing time headway for platooning of connected vehicles.

A Novel Neural-Network-Based Adaptive Control Scheme for Output-Constrained Stochastic Switched Nonlinear Systems

In this paper, a novel neural-network (NN)-based adaptive tracking controller design method is presented for the single-input/single-output nonlinear stochastic switched systems in lower triangular structures with an output constraint. First, a well-designed nonlinear mapping is introduced to transform the switched stochastic system to a new system without constraints, which implies the controller design of the transformed system is equivalent to that of the stochastic switched system. Then radial basis function NNs are applied to model the unknown nonlinearities and the adaptive backstepping technique is employed to construct two classes of adaptive neural controllers under different adaptive laws. It is proved that both controllers can assure all the signals in the closed-loop remain bounded in probability, and the tracking error finally converges to a neighborhood of the origin without violating the constraint. Furthermore, the use of the nonlinear mapping to deal with the asymmetric output constraint is also studied as a generalization result. Two illustrative examples with numerical data and simulation results are given to show the validity and performance of the proposed control schemes.

NUMERICAL APPROXIMATION OF ELLIPTIC PROBLEMS WITH LOG-NORMAL RANDOM COEFFICIENTS

In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.

Design and Analysis of Non-Binary LDPC-CPM System for Hybrid Check Matrix Construction Algorithm of WSN.

In order to enhance the reliability and anti-interference performance of wireless sensor network (WSN) data transmission, this paper designs the low power scheme of the WSN from the angle of error correction coding and proposes the hybrid check matrix construction (HC) algorithm based on iterative coding algorithms with linear coding complexity. The algorithm first improves the traditional iterative coding algorithm, making it suitable for non-binary low-density parity check (LDPC) codes. Then, the algorithm applies the backward iteration method to change the coding scheme and uses the check matrix construction method so that the progressive edge growth (PEG) algorithm has a lower triangular structure, which is used as a base matrix. An improved quasi-cyclic LDPC (QC-LDPC) algorithm, with a lower triangular structure, is used to generate a cyclic shift matrix and a finite domain coefficient matrix. Simultaneously, the short loop is eliminated and the optimal check matrix is selected for use in the channel coding process. The non-binary LDPC-CPM system is modeled and simulated. The simulation results show that the non-binary LDPC code constructed by the HC algorithm not only has linear coding and storage complexity but also has strong error correction capability. The design of non-binary LDPC-CPM system parameters can enhance the reliability, anti-jamming capability and reduce the complexity and reduce the complexity of the WSN.

An improved nonlinear robust control design for grid-side converter of VSC-HVDC connected to wind power generation system

This article proposes an improved nonlinear (IN) robust control strategy for the grid-side voltage-source converter (GSVSC) of a VSC-based high-voltage direct current (VSC-HVDC) transmission system connected to a large wind farm by the using Hamiltonian function method. With the help of variable transformation, the nonlinear model with parameter uncertainties and external disturbances of the GSVSC is changed into the port-controlled dissipative Hamiltonian (PCDH) system. Based on the PCDH system, an IN robust control law is established. In order to demonstrate the effectiveness and robustness of the IN robust control, the backstepping power control (BPC), which is developed by using the backstepping design procedure in the sense of Lyapunov stability theorem for the GSVSC to satisfy the control objectives of a stable HVDC bus and the grid connection with a unity power factor, is selected as a comparison object. Generally speaking, the system used by the backstepping method must have special strict feedback of the lower triangular structure, but the wind farm connected to the power grid with the VSC-HVDC is a multivariable structure and highly coupled nonlinear system. So, the BPC cannot effectively maintain and utilize the nonlinear characteristics of the VSC-HVDC system, while this nonlinear physical structure characteristics is very useful for the design of a nonlinear controller. The greatest advantage of the Hamilton function method is that it can effectively keep and utilize the nonlinear characteristics of the system. The simulation results indicate that the proposed IN control designed by using the Hamilton function method gains the advantage over the BPC under a variety of operating conditions.

Semi-Global Asymptotic Control by Sampled-Data Output Feedback

This paper shows that for a class of nonlinear systems with a lower-triangular structure, the problem of semi-global asymptotic stabilization is solvable by sampled-data output feedback, without requiring restrictive conditions on the nonlinearities and unmeasurable states of the system, such as linear growth, output-dependent growth or homogeneous growth conditions as commonly assumed in the case of global output feedback stabilization. The main contribution is to point out that semi-global asymptotic rather than practical stabilizability of certain classes of nonlinear systems is still possible by sampled-data output feedback if a sampling time is small enough. A design method is also given for the construction of semi-globally stabilizing, sampled-data output feedback controllers.

Iterative Changing Supply Rates, Dynamic State Feedback, and Adaptive Stabilization of Time-Delay Systems

Global adaptive stabilization by partial state feedback is studied for time-delay cascade systems with nonlinear parameterization. The inverse-dynamics of time-delay nonlinear systems under consideration is of a lower-triangular form and assumed to satisfy certain ISS-like conditions. By taking advantage of the lower-triangular structure, we present an iterative algorithm for changing supply rates so that the time-delay zero-dynamics can be handled effectively. With the aid of the iterative technique of changing supply rates, we develop a dynamic gain-based control strategy that, together with the feedback domination design, leads to a construction of partial-state, delay-free adaptive controllers. As a result, all the states of the time-delay cascade system are regulated to the origin and the boundedness of the solution of the closed-loop system is achieved.