Solving the nonlinear least square problem: Application of a general method

1976 ◽  
Vol 18 (4) ◽  
pp. 469-483 ◽  
Author(s):  
J. T. Betts
Keyword(s):  
Least Square ◽  
Least Square Problem ◽  
General Method

scholarly journals Numerical method to a class of boundary value problems

2018 ◽  
Vol 22 (4) ◽  
pp. 1877-1883 ◽  
Author(s):  
Yu-Yang Qiu
Keyword(s):  
Boundary Value Problems ◽  
Numerical Method ◽  
Conjugate Gradient ◽  
Iteration Method ◽  
Boundary Value ◽  
Solution Process ◽  
Least Square ◽  
Least Square Problem ◽  
Matrix Iteration

A class of boundary value problems can be transformed uniformly to a least square problem with Toeplitz constraint. Conjugate gradient least square, a matrix iteration method, is adopted to solve this problem, and the solution process is elucidated step by step so that the example can be used as a paradigm for other applications.

Suboptimal control synthesis for state and input delayed quadratic systems

2022 ◽  
pp. 095965182110674
Author(s):  
Mohamed Karim Bouafoura ◽  
Naceur Benhadj Braiek
Keyword(s):  
State Feedback ◽  
Closed Loop ◽  
Chemical Reactor ◽  
Open Loop ◽  
Quadratic System ◽  
Least Square ◽  
Suboptimal Control ◽  
Control Synthesis ◽  
Least Square Problem ◽  
Linear State

In this article a suboptimal linear-state feedback controller for multi-delay quadratic system is investigated. Optimal state and input coefficients resulting from the expansion over a hybrid basis of block pulse and Legendre polynomials are first obtained by formulating a nonlinear programming problem. Afterwards, suboptimal control gains are found by solving a least square problem constructed with optimal coefficients of the open loop study. A sufficient condition for the exponential stability of the closed loop is obtained from generalized Grönwall–Bellman lemma. The Van de Vusse chemical reactor case is handled allowing to validate the proposed technique.

scholarly journals A Novel Method for Asynchronous Time-of-Arrival-Based Source Localization: Algorithms, Performance and Complexity

Sensors ◽  
2020 ◽  
Vol 20 (12) ◽  
pp. 3466
Author(s):  
Yuanpeng Chen ◽  
Zhiqiang Yao ◽  
Zheng Peng
Keyword(s):  
Source Localization ◽  
Signal Propagation ◽  
Least Square ◽  
Time Of Arrival ◽  
Propagation Time ◽  
Performance Loss ◽  
Localization Algorithms ◽  
Least Square Problem ◽  
Localization Performance ◽  
Asynchronous State

In time-of-arrival (TOA)-based source localization, accurate positioning can be achieved only when the correct signal propagation time between the source and the sensors is obtained. In practice, a clock error usually exists between the nodes causing the source and sensors to often be in an asynchronous state. This leads to the asynchronous source localization problem which is then formulated to a least square problem with nonconvex and nonsmooth objective function. The state-of-the-art algorithms need to relax the original problem to convex programming, such as semidefinite programming (SDP), which results in performance loss. In this paper, unlike the existing approaches, we propose a proximal alternating minimization positioning (PAMP) method, which minimizes the original function without relaxation. Utilizing the biconvex property of original asynchronous problem, the method divides it into two subproblems: the clock offset subproblem and the synchronous source localization subproblem. For the former we derive a global solution, whereas the later is solved by a proposed efficient subgradient algorithm extended from the simulated annealing-based Barzilai–Borwein algorithm. The proposed method obtains preferable localization performance with lower computational complexity. The convergence of our method in Lyapunov framework is also established. Simulation results demonstrate that the performance of PAMP method can be close to the optimality benchmark of Cramér–Rao Lower Bound.

Virtual Locator Trimming in Pre-Production: Rigid and Non-Rigid Analysis

2005 ◽  
Author(s):  
Lars Lindkvist ◽  
Johan S. Carlson ◽  
Rikard So¨derberg
Keyword(s):  
Least Square ◽  
Computer Tool ◽  
Number Of Components ◽  
Form Errors ◽  
Influence Coefficients ◽  
Least Square Problem ◽  
Inspection Data ◽  
Linear Least Square

In pre-production, during assembly of newly produced components geometrical deviations, caused by form errors of the parts, are discovered that can cause either functional or esthetical problems. One commonly used way of solving this is to reposition the components by adjusting their locators, also known as trimming. Traditionally this is done by assembling a number of components, measuring the deviations to surrounding parts, adjusting the locator points, reassembling the components and measuring the result. This is repeated until the result is satisfactory, and is a quite time and effort consuming activity. This paper presents a method and a tool that simplifies this process. Based on inspection data from the initial components all trimming activities are performed in the computer tool presented. After the locators are adjusted, the result is presented directly, which eliminates the need for physical inspection in order to verify the result of the trimming. The presented tool also includes optimization of the trimming. By formulating the problem of minimizing the geometrical deviations as a linear least square problem a general optimization package can be used. The optimization handles both boundaries on the allowed trimming and weighting of the different inspection features. By using the method of influence coefficients, also compliant (non-rigid) components can be handled.

Synthesis of Room Responses Using the Virtual Source Representation and the Comb-Nested Allpass Filters

2005 ◽  
Vol 128 (6) ◽  
pp. 690-698 ◽  
Author(s):  
Mingsian R. Bai ◽  
Kwuen-Yieng Ou
Keyword(s):  
Sound Field ◽  
Sampling Procedure ◽  
Least Square ◽  
Synthesis Process ◽  
Virtual Source ◽  
Allpass Filters ◽  
Least Square Problem ◽  
Human Hearing ◽  
Allpass Filter ◽  
Rectangular Room

An artificial reverberator is proposed in this paper to synthesize room responses. The method employs the virtual source representation and the comb-nested allpass filters to generate the early reflection and late reverberation, respectively, of room responses. The virtual source method is based on a simple representation of sound field with a distribution of discrete simple sources on the boundary. The complex strengths of the virtual sources are then calculated by solving a frequency domain least-square problem. Parameters such as room geometry, size, and wall absorption are naturally incorporated into the synthesis process. The filtering property of human hearing is also exploited in a nonuniform sampling procedure to further simplify the computation. Apart from the early reflection, a comb-allpass filter network is adopted to simulate late reverberations. Optimal parameters of the comb-allpass filter network are obtained using the genetic algorithm (GA). The energy decay curve (EDC) is chosen as the objective function in the GA procedure. Numerical simulations are carried out for a rectangular room and a concert hall model to verify the proposed technique. Subjective listening experiments demonstrate that the present technique is capable of conferring remarkable realism of room responses.

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