Eigenvalue Constraints Research Articles

Reactor network synthesis with guaranteed robust stability

This paper proposes a systematic approach to design reactor networks with guaranteed robust stability. The approach is based on the superstructure approach for reactor network synthesis. A structured dynamic model for reactor network modeling is formulated and embedded in a MINLP with robust eigenvalue constraints. Design parameters, structural alternatives and parametric uncertainty are considered simultaneously as design degrees of freedom. Structural alternatives result from decisions on the existence of reactors and flow connections in the superstructure. Parametric uncertainty may either result from model uncertainties such as reaction kinetic constants or heat transfer coefficients, or from process uncertainties including slow disturbances in load or quality of raw materials. A tailored two-step solution strategy is proposed to tackle the robust mixed-integer optimization problem. A case study with five continuous stirred-tank reactors (CSTR) and five plug flow reactors (PFR) is presented for illustration.

Analytical sensitivity analysis of eigenvalues and lightweight design of composite laminated beams

The paper describes an optimisation technique for the lightweight design of composite laminated beams with a big ratio of span to thickness. The optimisation model for the lightweight design is to find the fibre volume fractions to minimise the mass of the composite laminated beams under the eigenvalue (or frequency) constraints. The analytical sensitivity of the eigenvalues, with respect to the fibre volume fractions, is formulated using the Euler–Bernoulli beam theory. The analytical formulae for sensitivity analysis are suitable for the different boundary conditions (pinned–pinned, fixed–fixed, fixed–free and fixed–pinned). The eigenvalues are linearised using Taylor’s series so that optimisation model is converted to a linear programming problem. The iterative computational procedure is proposed to solve the optimisation model and eliminate the Taylor’s series approximation since the current design point maybe not near the optimum design. Finally, the lightweight designs of the composite laminated beams with different boundary conditions are performed. The merits of the proposed method and lightweight designs obtained by proposed method are discussed.

Non-parametric shape optimization method for natural vibration design of stiffened shells

In this paper, we newly present an effective shape optimization method for natural vibration design of stiffened thin-walled or shell structures. Both the stiffeners and their basic structures are optimized by solving two kinds of optimization problems. The first is a specified eigenvalue maximization problem subject to a volume constraint, and the second is its reciprocal volume minimization problem subject to a specified eigenvalue constraint. The boundary shapes of a thin-walled structure are determined under the condition where the stiffeners and the basic structure are movable in the in-plane direction to their surface. Both problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are determined by applying the derived shape gradient function to the H1 gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical usages.

Modeling and Interpretation of Scattering Mechanisms in Polarimetric Synthetic Aperture Radar: Advances and perspectives

Recent advances in scattering modeling and model-based decomposition theorem were reviewed. The notable achievements include orientation compensation processing, nonnegative eigenvalue constraint, generalized scattering models, complete information utilization, full-parameter inversion strategy, and the polarimetric-interferometric decomposition scheme. These advances contribute to make scattering models more adaptive, better fit observations and guarantee physically meaningful decomposition solutions. The key features of these advances have been summarized. Performance evaluation and further development perspectives were also discussed. One promising way is to fuse multiple data to better model scattering mechanisms, such as the polarimetric-interferometric modeling attempts. Besides, with the progress in PolSAR sensors, imaging modes (e.g., bistatic, hybrid-polarization and multi-incident-angle modes) and application requirements, the development of specific scattering mechanism interpretation techniques, multiangular decomposition, and compact/hybrid decomposition techniques are also highly preferred.

Sparse Covariance Matrix Estimation With Eigenvalue Constraints

We propose a new approach for estimating high-dimensional, positive-definite covariance matrices. Our method extends the generalized thresholding operator by adding an explicit eigenvalue constraint. The estimated covariance matrix simultaneously achieves sparsity and positive definiteness. The estimator is rate optimal in the minimax sense and we develop an efficient iterative soft-thresholding and projection algorithm based on the alternating direction method of multipliers. Empirically, we conduct thorough numerical experiments on simulated datasets as well as real data examples to illustrate the usefulness of our method. Supplementary materials for the article are available online.

Four-Component Scattering Power Decomposition of Remainder Coherency Matrices Constrained for Nonnegative Eigenvalues

The motivation of this letter is to resolve the nonnegative eigenvalue constraint (NNEC) problem of four-component decomposition (FCD). It is analyzed that the NNEC is an essential requirement for remainder coherency matrices in the FCD, however the measured polarimetric synthetic aperture radar (POLSAR) data experiment shows there exits the NNEC problem that some remainder coherency matrices of the FCD do not satisfy the NNEC, which means these matrices are not positive semi-definite. In addition, it is analyzed that the scheme using the nonnegative eigenvalue decomposition (NNED) for three-component decomposition (TCD) cannot be directly extended to the FCD to overcome the NNEC problem, so a scheme using the NNED for the FCD is proposed as follow. From matrix theory, we draw a conclusion that if the last remainder coherency matrix satisfies the NNEC, then all remainder coherency matrices also satisfy the NNEC; we successively analyze that the NNEC problem of the last remainder coherency matrices results from the overestimation of scattering powers. Then a shrinkage coefficient is used to depress all possible overestimations of scattering powers, and the overestimation case with the minimum remainder power is chosen to resolve the NNEC problem. Moreover, we have simplified the solution to NNED, which is used to calculate the shrinkage coefficient. The measured POLSAR data experiment shows that the proposed FCD can further enhance double-bounce scattering and depress volume scattering for urban areas.

Volume Scattering Power Constraint Based on the Principal Minors of the Coherency Matrix

This letter proposes a constraint for a volume scattering power employing the principal minors, which can be used for polarimetric synthetic aperture radar (POLSAR) model-based decomposition. This constraint effectively allows for avoiding unreasonable results which yield negative eigenvalues. The proposed constraint is derived so that all the principal minors of the coherency matrix after volume scattering subtraction are nonnegative. Mathematically, the constraint is exactly the same as that based on the nonnegative eigenvalues. Thus, it is guaranteed that the result is physically reasonable and that the volume scattering power is not overestimated. A significant advantage of the proposed method compared to the constraint based on the nonnegative eigenvalues is the high computation efficiency, since the maximal volume scattering power can be derived analytically, while the nonnegative eigenvalue constraint requires a numerical calculation. In our experiment, the computation of the maximal power is six times faster using the approach based on the principal minors than that based on the nonnegative eigenvalues.

Free-Boundary Shape Optimization Method for Natural Vibration Problem of Thin-Walled Structure

In this paper, we present a shape optimization method for the natural vibration problem of thin-walled or shell structures with or without sitiffeners. The boundary shapes of a shell or stiffeners are determined under the condition where the boundary is movable in the in-plane direction to the surface. The design problems deal with eigenvalue maximization problem and volume minimization problem, which are subject to a volume constraint and an eigenvalue constraint respectively. The both optimization problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are obtained by applying the derived shape gradient functions to the H1 gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical utility.

2408 固有振動問題に対する補強リブ付きシェル構造の形状最適化手法

This paper presents a shape optimization method for the natural vibration problem of stiffened thin-walled or shell structures. The boundary shapes of stiffeners are determined under the condition where the boundary is movable in the inplane direction to the surface. The design problems deal with eigenvalue maximization problem and volume minimization problem, which are subject to a volume constraint and an eigenvalue constraint respectively. The both optimization problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are obtained by applying the derived shape gradient functions to the H^1 gradient method for shells, which is a parameterfree shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical utility.

Nonlinear Acceleration of Transport Criticality Problems

We present a nonlinear acceleration algorithm for a transport criticality problem. The algorithm combines the well-known nonlinear diffusion acceleration (NDA) algorithm with a recently developed, Newton-based nonlinear criticality acceleration (NCA) algorithm. The algorithm first employs NDA to reduce the system to scalar flux, then NCA is applied to the resulting drift-diffusion system. We apply a nonlinear elimination technique to eliminate the eigenvalue constraint equation from the Jacobian matrix. Numerical results show that the algorithm can reduce the CPU time by a factor of 30 to 400 compared to traditional power iterations (PIs) combined with standard source iterations and by a factor of 3 to 5 compared to application of NDA combined with inner PIs.

Linear Precoder Optimization for MIMO Systems with Joint Power Constraints

This paper considers linear precoder optimization problems for multiple-input multiple-output (MIMO) systems. In addition to the conventionally used sum-power constraint, maximum eigenvalue constraint on the precoding matrix is also considered so as to account for power limitations imposed on each antenna by the linearity of its own power amplifier in practical implementations. A framework employing directional derivative is developed to obtain optimal precoder designs for different criteria including maximizing the information rate and minimizing the sum of mean-square error (MSE). It turns out that power allocations in such situations are piecewise linear in sum-power space. The piecewise linear property allows us to generate the entire path of solution through finding out a finite number of breakpoints. A Homotopy-type algorithm is then proposed to obtain the solution for an arbitrary sum-power constraint. The number of breakpoints to be determined in our exact piecewise linear solution is in fact only about two times of the number of transmit antennas, so that our method is super fast and outperforms existing approximate solutions in the literature in both effectiveness and efficiency. Simulated experiments are performed to verify our theoretical analysis.

Passivity Enforcement for Admittance Models of Distributed Networks Using an Inverse Eigenvalue Method

Most transient circuit simulators are based on admittance representations of the constituent circuit elements. It is therefore natural to use admittance parameter descriptions of linear networks, preferably in the form of rational transfer functions that can be directly implemented in the analysis. A problem arises when the measured or calculated frequency-domain response of a linear distributed network must be derived from data that has inherent error, is of limited bandwidth, or is not in the appropriate rational form. A reduced-order rational model that is causal, stable, and passive must be developed. Previous methods of deriving rational functions for the admittance parameters of a network do guarantee stability and causality, but passivity of the model must be assured through subsequent post-processing. Enforcing passivity requires modification of the state-space parameters of the model with consequent introduction of errors. This paper reports on a procedure to simultaneously achieve passivity, accuracy, causality, and stability in the development of an admittance macromodel described using a matrix of rational functions. An iterative inverse eigenvalue algorithm enforces passivity, and is applied by conjoining sets of eigenvalue and admittance constraints. These constraints form a monolithic projection matrix, which simultaneously addresses both passivity and accuracy of the model.

Cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields

We determine all Hermitian O Q ( d ) -matrices for which every eigenvalue is in the interval [ − 2 , 2 ] , for each d ∈ { − 2 , − 7 , − 11 , − 15 } . To do so, we generalise charged signed graphs to L -graphs for appropriate finite sets L , and classify all L -graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix/graph is contained in a maximal example with all eigenvalues ±2.

Hydrodynamic stability control in CFD by shape optimization

AbstractWe propose a numerical scheme which aims at solving shape optimization problems in the context of hydrodynamic stability. It is the goal to minimize the drag and ensure hydrodynamic stability by modification of parts of the underlying geometry. The stability issues are represented by eigenvalue constraints. Numerical results for a test case show that a transition from an unstable design into a stable one is attainable by considering an adequate change of the geometry. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Exploiting Group Symmetry in Truss Topology Optimization

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental fre-quency of vibration of the truss structure.These SDP formulations were introduced in: [M.Ohsaki, K.Fujisawa, N.Katoh and Y.Kanno, Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints, Comp.Meth.Appl.Mech.Engng., 180: 203-217, 1999].We show how one may automatically obtain symmetric designs, by eliminating the 'redundant' symmetry in the SDP problem formulation.This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.

Minimum Weight Shape Design For The NaturalVibration Problem Of Plate And Shell Structures

In this paper we present a solution to a shape optimization problem involving plate and shell structures subject to natural vibration. The volume is chosen as the response to be minimized under a specified eigenvalue constraint with mode tracking. The designed boundaries are assumed to be movable in the in-plane direction so as to maintain the initial curvatures. The surfaces are discretized by plane elements based on the Mindlin-Reissner plate theory. A non-parametric or a distributed shape optimization problem is formulated and the shape gradient function is theoretically derived using the material derivative method and the Lagrange multiplier method. The traction method, a shape optimization method developed by the authors, is applied to obtain the optimal shape in this problem. The validity of the numerical solution for minimizing the weight of the plate and shell structures is verified through application to fundamental design problems and an actual automotive suspension component.

Symmetricity of the solution of semidefinite programming

Symmetricity of an optimal solution of Semi-Definite Programming (SDP) is discussed based on the symmetry property of the central path that is traced by a primal-dual interior-point method. A symmetric SDP is defined by operators for rearranging elements of matrices and vectors, and the solution on the central path is proved to be symmetric. Therefore, it is theoretically guaranteed that a symmetric optimal solution is always obtained by using a primal-dual interior-point method even if there exist other asymmetric optimal solutions. The optimization problem of symmetric trusses under eigenvalue constraints is shown to be formulated as a symmetric SDP. Numerical experiments illustrate convergence to strictly symmetric optimal solutions.

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables.

Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints

Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as Semi-Definite Programming (SDP), and an algorithm is presented based on the Semi-Definite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed, the SDPA is especially useful for the case where the optimal design has multiple fundamental eigenvalues. Global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view. It is shown in the examples, that SDPA has advantage over existing methods in view of computational efficiency and accuracy of the solutions, and an optimal topology with five-fold fundamental eigenvalue is found without any difficulty.

Non-linear dynamic systems fault detection and isolation using fuzzy observers

This paper presents a novel fault detection and isolation scheme for non-linear dynamic systems. This scheme utilizes a fuzzy observer to generate the diagnostic residual signal for fault detection and isolation. This fuzzy observer, based upon the idea of Takagi-Sugeno fuzzy models, comprises a number of locally linear observers and the final state estimate is a fuzzy fusion of all local observer outputs. To ensure good estimation performance, the eigenvalues of the fuzzy observer are assigned in a pre-defined region in the complex plane. The stability as well as eigenvalue constraint conditions for the fuzzy observer design are presented and solved in the linear matrix inequality framework in this paper. Finally, the paper demonstrates the application of fuzzy observers in detecting and isolating intermittent faults in the induction motor of a railway traction system using a real-time test-rig implementation.