Sum Of Squares Method Research Articles

Application of Sum of Squares Method in Nonlinear H∞ Control for Satellite Attitude Maneuvers

The Hamilton–Jacobi–Issacs (HJI) inequality is the most basic relation in nonlinear H∞ design, to which no effective analytical solution is currently available. The sum of squares (SOS) method can numerically solve nonlinear problems that are not easy to solve analytically, but it still cannot solve HJI inequalities directly. In this paper, an HJI inequality suitable for SOS is firstly derived to solve the problem of nonconvex optimization. Then, the problems of SOS in nonlinear H∞ design are analyzed in detail. Finally, a two‐step iterative design method for solving nonlinear H∞ control is presented. The first step is to design an adjustable nonlinear state feedback of the gain array of the system using SOS. The second step is to solve the L2 gain of the system; the optimization problem is solved by a graphical analytical method. In the iterative design, a diagonally dominant design idea is proposed to reduce the numerical error of SOS. The nonlinear H∞ control design of a polynomial system for large satellite attitude maneuvers is taken as our example. Simulation results show that the SOS method is comparable to the LMI method used for linear systems, and it is expected to find a broad range of applications in the analysis and design of nonlinear systems.

Sum of Squares Approach for Nonlinear H∞ Control

A proper Hamilton-Jacobi-Isaacs (HJI) inequality must be solved in a nonlinear H∞ control problem. The sum of squares (SOS) method can now be used to solve an analytically unsolvable nonlinear problem. A HJI inequality suitable for SOS approach is derived in the paper. The SOS algorithm for solving the HJI inequality is also provided. Conservativeness of the SOS method is then discussed in the paper. The conservativeness of the SOS approach is caused by the method itself, because it is really a synthesis method over the entire state space. To reduce the conservativeness, a local H∞ design on a restricted state-space region is proposed. But the SOS approach for the local H∞ design also suffers from the conservativeness problem, because the S-procedure for solving the set-containment constraint provides only a sufficient condition. The above-mentioned sources of conservativeness are peculiar for the SOS approaches. So a proper approach must be carefully selected in the design process to get a reasonable result. A design example is also given in the paper.

Disponibilidad de residuos forestales y su potencial para la generación de energía en los bosques templados de El Salto, Durango

Los residuos de la cosecha forestal son una fuente de energía renovable que no ha sido considerada todavía a gran escala en México y para su estimación se requiere de evaluaciones cada vez más precisas. En este estudio se desarrollaron ecuaciones para estimar el volumen de residuos forestales y se cuantificó la disponibilidad de biomasa y su potencial para generar energía eléctrica y reducir emisiones de CO2 en la región de El Salto, Durango. La metodología se dividió en tres fases: i) se obtuvo el volumen total autorizado de los programas de manejo forestal maderable de 20 ejidos de la región; ii) se derribaron y cubicaron 2156 árboles en las áreas de corta de los 20 ejidos para generar ecuaciones de volumen de residuos a escala de árbol individual; y iii) el volumen de residuos fue convertido a toneladas de biomasa empleando los valores de densidad de la madera para posteriormente transformar este valor a unidades de energía eléctrica. Dado que las especies estudiadas constituyen bosques mezclados, manejados con el mismo tratamiento silvícola y edad de rotación, se evaluaron las diferencias entre especies en las ecuaciones de volumen de residuos, utilizando el método de la suma adicional de cuadrados no lineales. Los parámetros de las ecuaciones de volumen de residuos fueron significativamente diferentes entre la mayoría de las especies. La cantidad de residuos a escala de árbol fue mayor en las especies del género Quercus, mientras que la cantidad de residuos forestales disponibles por ciclo de corta fue mayor en las especies del género Pinus. Alrededor de 31 000 Mg año-1 de residuos forestales pueden utilizarse en la región para generar aproximadamente 65.6 GW h año-1, considerando el uso de una tecnología con una eficiencia mínima de 40%.

Event-triggered containment control for multi-agent systems with constant time delays

The paper investigates containment control for the first-order and second-order multi-agent systems with constant time delays under event-triggered conditions, respectively. By applying existing sum of square method to stability analysis of containment control for multi-agent systems, sufficient containment conditions for multi-agent systems are obtained. First, we discussed the case of containment control for multi-agent system with single time delay, and event-triggered conditions are proposed. Then, we extend the results of containment control for multi-agent systems with single time delay to the case with multiple time delays. Finally, simulation examples are given to illustrate the effectiveness of our theoretical results.

The accuracy of static pressure measurement with water-filled urodynamic systems.

To quantify the inaccuracy of pressure measurement by water-filled systems, with a view to recommending better practice. Papers examining quality of measurements were reviewed, and standards documents examined. Technical data from manufacturers were gathered and summarized. Measurements were made on two different urodynamic systems. Individual inaccuracies were combined using the root sum of squares method. The total inaccuracy in measurement may reach up to 9.6 cmH2 O on pdet , but with good practice this may be mitigated to 1.3 cmH2 O. Water-filled systems, while being the recommended method for urodynamic measurements, are subject to a potential inaccuracy during poor use of up to approximately 10 cmH2 O on pdet . Good practice can reduce this inaccuracy to a tolerable level, that is, approximately 1 cmH2 O.

Stem taper and volume models for natural cedar and Taurus fir mixed stands in Bucak District

Abstract: In this study, we assessed the performance of different types of taper equations for predicting tree diameters at specific heights and total stem volumes for mixed stands of Taurus cedar ( Cedrus libani A. Rich.) and Taurus fir ( Abies cilicica Carr.). We used data from mixed stands containing a total of 131 cedar and 124 Taurus fir trees. We evaluated six commonly used and well-known forestry taper functions developed by a variety of researchers (Biging (1984), Zakrzewski (1999), Muhairwe (1999), Fang et al. (2000), Kozak (2004), and Sharma and Zhang (2004)). To address problems related to autocorrelation and multicollinearity in the hierarchical data associated with the construction of taper models, we used appropriate statistical procedures for the model fitting. We compared model performances based on the analysis of three goodness-of-fit statistics and found the compatible segmented model of Fang et al. (2000) to be superior in describing the stem profile and stem volume of both tree species in mixed stands. The equation used by Zakrzewski (1999) exhibited the poorest fitting results of the three taper equations. In general, we found segmented taper equations to provide more accurate predictions than variable-form models for both tree species. Results from the non-linear extra sum of squares method indicate that stem tapers differ among tree species in mixed stands. Therefore, a different taper function should be used for each tree species in mixed stands in the Bucak district. Using individual-specific taper equations yields more robust estimations and, therefore, will enhance the prediction accuracy of diameters at different heights and volumes in mixed stands. Keywords: Segmented model, compatible equation, stem form, merchantable volume, autocorrelation, F- test Received (Gelis): 08.02.2017 - Revised (Duzeltme): 17.03.2017 - Accepted (Kabul): 28.04.2017 Cite (Atif): Ozcelik, R., Dirican, O., 2017. Individual taper models for natural cedar and Taurus fir mixed stands of Bucak Region, Turkey. Journal of the Faculty of Forestry Istanbul University 67(2): 336-354. DOI: 10.17099/jffiu.290845

The REPOP Toolbox: Tackling Polynomial Optimization Using Relative Entropy Relaxations

Polynomial optimization is an active field of research which can be used in a broad range of applications including the synthesis of control policies for non-linear systems, and solution methods such as approximate dynamic programming. Finding the optimal solution of a generic polynomial optimization problem remains a computationally intractable problem. Several studies in the literature resort to hierarchical schemes that converge to the optimal solution, by employing appropriate convex relaxations of the original problem. In this direction, sum of squares methods have shown to be effective in addressing problems of low degree and dimension, with numerous MATLAB toolboxes allowing for efficient implementation. An alternative solution method is to cast the problem as a signomial optimization and solve it using a hierarchy of relative entropy relaxations. In contrast to sum of squares, this method can tackle problems involving high degree and dimension polynomials. In this paper, we develop the publicly available REPOP toolbox to address polynomial optimization problems using relative entropy relaxations. The toolbox is equipped with appropriate pre-processing routines that considerably reduce the size of the resulting optimization problem. In addition, we propose a convergent hierarchy which combines aspects from sum of squares and relative entropy relaxations. The proposed method offers computational advantages over both methods.

Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination.

We revisit the problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program (SDP). However, often the NOE distance restraints are too imprecise and sparse for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy, a hierarchy of convex relaxations with increasing complexity and approximation power. Unlike classical global optimization approaches, SOS optimization returns a certificate of optimality if the global optimum is found. Based on the SOS method, we proposed two algorithms-RDC-SOS and RDC-NOE-SOS, that have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. To the best of our knowledge this is the first time SOS relaxation is introduced to solve non-convex optimization problems in structural biology. We further introduce a statistical tool, the Cramér-Rao bound (CRB), to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements using any unbiased estimator. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements obtained in two alignment media, achieving more accurate and faster reconstructions compared to the current state of the art.

Modelos de volumen específicos y regionales para 12 especies forestales en el estado de Durango, México

Introduction: Estimating the volume of individual trees is an important aspect in dendrometry and in the conducting of forest inventories. Objective: Significant differences were evaluated in additive equation systems for estimating the total volume of individual trees in nine species of the genus Pinus (P. cooperi, P. durangensis, P. arizonica, P. leiophylla, P. teocote, P. engelmannii, P. lumholtzii, P. strobiformisand P. herrerae), and three of the genus Quercus (Q. sideroxyla, Q. durifoliaand Q. rugosa), and among regional forest management units (UMAFOR) when dealing with the same species. Materials and methods: To evaluate whether equation systems are different among tree species of the same genus and among the UMAFORs for the same species, two complementary statistical analyzes were used based on the fitting of a reduced and a full equation system: the F test of nonlinear extra sum of squares method, and the parameter significance analysis. Results and discussion: For most of the species studied, equation systems are significantly different. The need to use regional models in 10 of the 12 tree species, except for P. strobiformis and Q. rugosa, is also reported. Conclusion: The use of a state model for P. strobiformis and Q. rugosa is recommended.

A Sum-of-Squares and Semidefinite Programming Approach for Maximum Likelihood DOA Estimation.

Direction of arrival (DOA) estimation using a uniform linear array (ULA) is a classical problem in array signal processing. In this paper, we focus on DOA estimation based on the maximum likelihood (ML) criterion, transform the estimation problem into a novel formulation, named as sum-of-squares (SOS), and then solve it using semidefinite programming (SDP). We first derive the SOS and SDP method for DOA estimation in the scenario of a single source and then extend it under the framework of alternating projection for multiple DOA estimation. The simulations demonstrate that the SOS- and SDP-based algorithms can provide stable and accurate DOA estimation when the number of snapshots is small and the signal-to-noise ratio (SNR) is low. Moveover, it has a higher spatial resolution compared to existing methods based on the ML criterion.

SOS 제어기법을 이용한 입력제한이 있는 2관절 로봇팔의 조정제어

This paper proposes the controller design for regulation control of two-link robot arm using sum of squares (SOS) control method that takes into account the input constraint. The existing studies of two link robotic arm system used a linear model of all the non-linearity of the system is linearized. For a linear controller, since the model of the system is simplified, it is possible to design a controller in consideration of constraints on the disturbance. However, there is a limit to the performance using a linearized model for a system with a complex nonlinear properties. To compensate for this in the case of using a fuzzy LMI method, it is necessary to have a large number of linear models and thus there is a disadvantage that the system becomes complicated. To solve these problems, we represents a two-link robot arm system with a polynomial model using a Taylor series expansion and design the controller considering the case where the magnitude of the control input is limited using SOS method. We demonstrate by simulations the feasibility of the proposed algorithm.

Real time security framework for detecting abnormal events at ATM installations

Automated Teller Machines (ATM) transactions are quick and convenient, but the machines and the areas surrounding them make people and ATM vulnerable to felonious activities if not properly put under the protection. Responsibility for providing security needs to be fixed, however, most machines have very less or no security. It is imminent to develop security framework that would identify event as their happening. In this paper we propose a robust computer vision approach for identifying abnormal activity at ATM premises in real time. For effective identification of activity, we propose a novel method in which different Window size is used to record magnitude of pixel intensity using root of sum of square method. To describe this pattern, histogram of gradients is used. Further random forest is applied to infer the most likely class. The average accuracy of our security system is 93.1 %. For validation of our approach we have tested it on two standard datasets, HMDB and Caviar. Our approach achieved 52.12 % accuracy on HMDB dataset and 81.48 % on Caviar dataset.

Comparison of stem taper equations for eight major tree species in the Spanish Plateau

Aim of study: A stem taper function and a compatible merchantable volume system are compared to evaluate which provides a better description of the stem profile for the main species in central Spain.Area of study: This research was carried out in the region of Castile-Leon, located in Central Spain.Material and Methods: A total of 6,357 trees were selected for destructive sampling. All models were fitted using a first-order continuous autoregressive error structure to address the problem of autocorrelation.Main results: In terms of accuracy, the root mean square error (RMSE) in both models ranged from 0.75 to 2.72 depending on the species analyzed, presenting values similar to those reported in other studies. Small differences in the goodness-of-fit for both procedures were also found, and the Stud model provided better accuracy for 6 of the 8 species studied, with RMSE reductions of 0.5% to 8.6%. The RMSE obtained in the cross-validation phase was on average 1.22 times higher than what was obtained in the fitting phase.Research highlights: The non-linear extra sum of squares method indicated that the stem taper differs among the five softwood species and three hardwood species. In hardwoods, the first inflection point is lower than in softwoods (at around 5%) and the second inflection point is higher (at around 85%) than those of softwoods.Keywords: taper function; volume system; Central Spain; softwoods; hardwoods.

Design procedure for dynamic response of concrete rectangular liquid storage tanks using generalized SDOF system

In this paper, a design procedure using the generalized single degree of freedom (SDOF) system is proposed for two-dimensional dynamic response of rectangular liquid containing structures (LCS). The proposed model considers the effect of flexibility of tank wall on hydrodynamic pressures and uses the consistent mass approach in dynamic analysis. The contribution of higher modes to the dynamic response of LCS is included in the proposed model. The square root of sum of squares method is proposed for the combination of the first two modes. One set of calculations for a tall tank is presented and compared with the results obtained using the current practice as well as the finite element method reported by the authors in previous investigations. The proposed method using the generalized SDOF system can be simply used in seismic design of LCS.

Development of ecoregion-based merchantable volume systems for Pinus brutia Ten. and Pinus nigra Arnold. in southern Turkey

Estimating individual tree volume is one of the essential building blocks in forest growth and yield models. Ecologically based taper equations provide accurate volume predictions and allow classification by merchantable sizes, assisting in sustainable forest management. In the present study, ecoregion-based compatible volume systems for brutian pine and black pine in the three ecoregions of southern Turkey were developed. Several well-known taper functions were evaluated. A second-order continuous-time autoregressive error structure was used to correct the inherent autocorrelation in the hierarchical data, allowing the model to be applied to irregularly spaced and unbalanced data. The compatible segmented model of Fang et al. (For Sci 46:1–12, 2000) best described the experimental data. It is therefore recommended for estimating diameter at a specific height, height to a specific diameter, merchantable volume, and total volume for the three ecoregions and two species analyzed. The nonlinear extra sum of squares method indicated differences in ecoregion and tree-specific taper functions. A different taper function should therefore be used for each pine species and ecoregion in southern Turkey. Using ecoregion-specific taper equations allows making more robust estimations and, therefore, will enhance the accuracy of diameter at different heights and volume predictions.

A computational study of pressure wave reflections in the pulmonary arteries.

Experiments using wave intensity analysis suggest that the pulmonary circulation in sheep and dogs is characterized by negative or open-end type wave reflections, that reduce the systolic pressure. Since the pulmonary physiology is similar in most mammals, including humans, we test and verify this hypothesis by using a subject specific one-dimensional model of the human pulmonary circulation and a conventional wave intensity analysis. Using the simulated pressure and velocity, we also analyse the performance of the P-U loop and sum of squares techniques for estimating the local pulse wave velocity in the pulmonary arteries, and then analyse the effects of these methods on linear wave separation in the main pulmonary artery. P-U loops are found to provide much better estimates than the sum of squares technique at proximal locations, but both techniques accumulate progressive error at distal locations away from heart, particularly near junctions. The pulse wave velocity estimated using the sum of squares method also gives rise to an artificial early systolic backward compression wave. Finally, we study the influence of three types of pulmonary hypertension viz. pulmonary arterial hypertension, chronic thromboembolic pulmonary hypertension and pulmonary hypertension associated with hypoxic lung disease. Simulating these conditions by changing the relevant parameters in the model and then applying the wave intensity analysis, we observe that for each group the early systolic backward decompression wave reflected from proximal junctions is maintained, whilst the initial forward compression and the late systolic backward compression waves amplify with increasing pathology and contribute significantly to increases in systolic pressure.

Using ROC curves to choose minimally important change thresholds when sensitivity and specificity are valued equally: the forgotten lesson of pythagoras. theoretical considerations and an example application of change in health status.

BackgroundReceiver Operator Characteristic (ROC) curves are being used to identify Minimally Important Change (MIC) thresholds on scales that measure a change in health status. In quasi-continuous patient reported outcome measures, such as those that measure changes in chronic diseases with variable clinical trajectories, sensitivity and specificity are often valued equally. Notwithstanding methodologists agreeing that these should be valued equally, different approaches have been taken to estimating MIC thresholds using ROC curves.Aims and objectivesWe aimed to compare the different approaches used with a new approach, exploring the extent to which the methods choose different thresholds, and considering the effect of differences on conclusions in responder analyses.MethodsUsing graphical methods, hypothetical data, and data from a large randomised controlled trial of manual therapy for low back pain, we compared two existing approaches with a new approach that is based on the addition of the sums of squares of 1-sensitivity and 1-specificity.ResultsThere can be divergence in the thresholds chosen by different estimators. The cut-point selected by different estimators is dependent on the relationship between the cut-points in ROC space and the different contours described by the estimators. In particular, asymmetry and the number of possible cut-points affects threshold selection.ConclusionChoice of MIC estimator is important. Different methods for choosing cut-points can lead to materially different MIC thresholds and thus affect results of responder analyses and trial conclusions. An estimator based on the smallest sum of squares of 1-sensitivity and 1-specificity is preferable when sensitivity and specificity are valued equally. Unlike other methods currently in use, the cut-point chosen by the sum of squares method always and efficiently chooses the cut-point closest to the top-left corner of ROC space, regardless of the shape of the ROC curve.

Model Equivalence in General Linear Models: Set-to-Zero, Sum-to-Zero Restrictions, and Extra Sum of Squares Method

The paper is drawn from the authors' experience in teaching general and generalized linear fixed effects models at the university level. The steps followed include model specification, model estimation, and hypothesis testing in general linear model setting. Among these steps, estimation of model parameters such as the main effect least squares means and contrasts were among the most challenging for students. Since no unique solution exists, students are first exposed to the equivalence between two popular techniques that an over-parameterized model can be subjected to in order to obtain the parameter estimates. This is particularly important because existing software do not necessarily follow the same path to produce an Analysis of Variance (or Covariance) of the general, generalized linear fixed or mixed effects models. These steps are generally hidden from the users. It is therefore crucial for the students to understand the intermediary processes that ultimately produce the same results regardless of the software one uses. The equivalent techniques, the set-to-zero and sum-to-zero restrictions, used to obtain solution of the normal equations of the fixed effects model, are presented. The relationship between them is also presented and in the process, data analysis makes use of two important concepts: the generalized inverse and estimable function. The invariance property of estimable functions is also explained in details in addition to the extra sum of squares principle which is introduced to supplement the other concepts. To exemplify these ideas and put them in practice, a simple one-way treatment structure analysis of variance is performed.

Singularity-free workspace analysis of general 6-UPS parallel mechanisms via convex optimization

This paper investigates the singularity-free zones in the workspace of general 6-UPS parallel mechanisms. The emphasis is placed on obtaining the maximum volume ellipsoid or sphere in the singularity-free subregions, while taking into account the actuator strokes. The proposed algorithms are based on the convex optimization and have several advantages over the existing methods such as Lagrange multipliers approach, which makes it appropriate for other applicable optimization problems. For determining the maximal singularity-free zones, in order to find the maximum volume ellipsoid, a judicious iterative procedure, referred to as Improved Lower Bound Semidefinite Programming, is proposed. Additionally, different reformulations of the problem are proposed in order to solve other interesting problems in the robotics community such as obtaining the maximum volume sphere. Furthermore, an approach based on the sum of squares method is proposed to solve this problem for a general Gough–Stewart platform with any arbitrary geometric parameter which is conducive to a polynomial optimization problem. The computational time for the proposed algorithms are considerably low compared to other methods proposed in the literature for obtaining the singularity-free workspace which opens an avenue to use them as systematic algorithms for on-line purposes.

Fuzzy control of continuous-time recurrent fuzzy systems

This article presents a synthesis method for stabilization of equilibria in continuous-time recurrent fuzzy system. It is first shown how to obtain a hybrid system representation from the given linguistic differential equations defining the recurrent fuzzy system. Based on this, the controller synthesis algorithm presented is then a two step procedure. By means of the synthesis step, a fuzzy controller is obtained by solving a single optimization problem, which guarantees that no chattering effects will occur in the controlled recurrent fuzzy system. Stability of the equilibrium is then proven by a verification procedure utilizing sum of squares methods. A performance optimization by means of additional polynomial controllers is discussed. The main advantage of our method relies in the computational efficiency of the synthesis problem as well as in the linguistic interpretability of the resulting controller. In addition, chattering effects are proven not to occur.