Sequential Linearization Research Articles

A sequential LQG approach to nonlinear tracking problem

A new approach, named the sequential LQG solution for nonlinear and nonstationary systems tracking problem, has been proposed in the article. Proposed method for tracking of a desired reference trajectory uses a state space model of the multivariable nonlinear/nonstationary system. The method involves three steps. The first step is the design of nominal trajectory using the predictive control technique. The second step is the sequential linearization of the nonlinear system around the fixed operating points, chosen in accordance to the plant dynamics changes. The third step involves the choice of the weighting matrices in the classical LQG controller design for the sequence of linearized models. The feasibility of the proposed method has been demonstrated through its application to the control of aircraft, represented by the six-degree-of-freedom model around the prespecified nominal trajectory.

On the Estimation of Nonlinearly Aggregated Mixed Models

The article proposes an iterative algorithm for the estimation of fixed and random effects of a nonlinearly aggregated mixed model. The latter arises when an additive Gaussian model is formulated at the disaggregate level on a nonlinear transformation of the responses, but information is available in aggregate form. The nonlinear transformation breaks the linearity of the aggregate model, yielding a nonlinear tight observational constraint. The algorithm rests upon the sequential linearization of the nonlinear aggregation constraint around proposals that are iteratively updated until convergence. Two alternative pseudo maximum likelihood methods are discussed and compared. As a byproduct we provide a solution to the problem of distributing the aggregate responses over the units of analysis, enforcing the nonlinear observational constraints. Illustrations referring to the temporal disaggregation problem are provided.

Logic and Numerical Methods of Spacecraft Attitude Control and Reconfiguration

The problem of programmed control of the spacecraft attitude, including in worstcase situations of on-board equipment failures, is considered.During a programmed reorientation (traditionally, by solving an inverse problem of dynamics) there arise singular states of the gyrosystem that interrupt the process and require additional resources to reconfigure the gyroscope frames. A technique is proposed, which is not subject to these drawbacks, does not need an a priori specification of the phase trajectory and is based on a sequential linearization and controllability of the system.To ensure fault-tolerance of the control, a new method of knowledge-based control is suggested, which relies on an original, high-performance logical formalism and an advanced architecture of logic-dynamic control.Example of the application of methods of fault-tolerance control is considered.

SEQUENTIAL LINEARIZATION ADAPTIVE CONTROL OF SOLUTION POLYMERIZATION OF METHYL METHACRYLATE IN A BATCH REACTOR

A sequential linearization adaptive PI controller was designed to control the temperature of a batch MMA polymerization reactor. Using the mechanistic model of the polymerization, a transfer function was derived to relate the reactor temperature to the power of the heaters. The coefficients of the transfer function were calculated along the selected temperature trajectory by sequential linearization. In addition, a PI adaptive controller was designed, based on an integral type tuning criteria, to track the temperature trajectory. The controller showed an excellent performance and the maximum deviation was about 0.9°C around 90°C. Measurement of the viscosity average molecular weight shows the possibility of molecular weight control.

2556 同一表面で力/変位拘束を伴う構造最適化

The contact surface of deformable continuum body has importance on surface deformation and traction simultaneously, since the surface deformation due to the contacting object sometimes yields such surface traction that accompanies the severe stress conditions for the deformable body as well as for the contacting object. This study discusses a mechanical problem of deformable continuum body of hyper elasticity with surface traction and deformation prescribed simultaneously. Such a problem is considered as a design optimization problem in the context of homology design concept. The homology index for the prescribed deformation is defined on the body surface with the prescribed traction force, and the structural compliance minimization problem is formulated by epsilon-constraint technique for homology index in the context of the SIMP method for the topology optimization and solved by sequential linearization technique. Numerical case demonstrated the effectiveness of the proposed idea.

An LMI-Based Algorithm for Designing Suboptimal Static $\cal H_2/\cal H_\infty$ Output Feedback Controllers

We consider the problem of designing a suboptimal ${\cal H}_2/{\cal H}_{\infty}$ feedback control law for a linear time-invariant control system when a complete set of state variables is not available. This problem can be necessarily restated as a nonconvex optimization problem with a bilinear, multiobjective functional under suitably chosen linear matrix inequality (LMI) constraints. To solve such a problem, we propose an LMI-based procedure which is a sequential linearization programming approach. The properties and the convergence of the algorithm are discussed in detail. Finally, several numerical examples for static ${\cal H}_2/{\cal H}_{\infty}$ output feedback problems demonstrate the applicability of the considered algorithm and also verify the theoretical results numerically.

Stabilization of Subba Rao-Liporace models

The stability of time-varying autoregressive (TVAR) models is an important issue in many applications such as time-varying spectral estimation, EEG simulation and analysis, and time-varying linear prediction coding (TVLPC). For stationary AR models there are methods that guarantee stability, but the for nonadaptive time-varying approaches there are no such methods. On the other hand, in some situations, such as in EEG analysis, the models that temporarily exhibit roots with almost unit moduli are difficult to use. Thus we may need a tighter stability condition such as stability with margin 1??. In this paper we propose a method for the estimation of TVAR models that guarantees stability with margin 1??, that is, the moduli of the roots of the time-varying characteristic polynomial are less than or equal to some arbitrary positive number ? for every time instant. The model class is the Subba Rao-Liporace class, in which the time-varying coefficients are constrained to a subspace of the coefficient time evolutions. The method is based on sequential linearization of the associated nonlinear constraints and the subsequent use of a Gauss-Newton-type algorithm. The method is also applied to a simulated autoregressive process.

Automated optimization of transverse frame layouts for ships by elastic-plastic finite element analysis

This paper presents a model for the optimum design of ship transverse frames. An elastic-plastic finite element analysis algorithm for plane frames has been incorporated in the model to evaluate the ultimate strength of the overall frame, and different effects of design loads. Using these strengths and load effects, appropriate design constraints are then formulated to prevent different failure categories; the overall collapse, ultimate limit state failures and serviceability failures. Possible instabilities and effects of combined loads are accounted for in formulating these constraints. Scantlings of the frame structure have been modelled as free design variables. The weight function and different constraint functions are then derived relating design variables in such a way that once parameters for finite element analysis are input, the scheme automatically forms the objective function and all constraints, and then interacts with the simplex algorithm through sequential linearization to find the optimum solution. Thus the scheme is almost automatic. Different layouts of the frame structure have been designed by executing this scheme, which demonstrates the capability of the model and the possibility of weight savings by choosing the appropriate layout. Finally, it is suggested how this model would interact with the design of longitudinal materials to ensure the overall optimality in ship hull module design, to prevent grillage buckling and to validate underlying assumptions in analysis.

Sequential Linearization of Empirical Likelihood Constraints with Application to U-Statistics

Abstract Empirical likelihood for a mean is straightforward to compute, but for nonlinear statistics significant computational difficulties arise because of the presence of nonlinear constraints in the underlying optimization problem. It is certainly the case that these difficulties can be overcome with sufficient time, care, and programming effort. However, they do make it difficult to write general software for implementing empirical likelihood, and therefore these difficulties are likely to hinder the widespread use of empirical likelihood in applied work. The purpose of this article is to suggest an approximate approach that sidesteps the difficult computational issues. The basic idea, which may be described as “sequential linearization of constraints,” is a very simple one, but we believe it could have significant ramifications for the implementation and practical use of empirical likelihood methodology. One application of the linearization approach, which we consider in this article, is to the probl...

Sequential Linearization of Empirical Likelihood Constraints with Application to U-Statistics

Sequential Linearization of Empirical Likelihood Constraints with Application to U-Statistics

Structure identification of non-linear models for QMB polymer-coated sensors

This article deals with the representation of quartz-microbalance (QMB) polymer-coated sensors by block-structured models, their structural identification and their functional identification. The block-structured modelling approach is one of the techniques available for obtaining a complete description of the dynamic behaviour of a non-linear sensing device without having detailed information about its inner structure. A QMB sensor, employed for the detection of mixtures of n-octane and toluene, is studied at the high concentration range (up to 100000 ppm), where a non-linear behaviour is expected. The sensor is exposed to time-varying concentrations of toluene and n-octane having quasi-white power spectral density. It is recognized that the sensor is very likely to admit a special model structure, called the ′Wiener model′, which is a simple non-linear cascade consisting of sequential dynamic linear, static non-linear and dynamic linear processes. The model is completely identified and gives a good approximation to the response of the sensor to any time-varying concentration. Our approach has not yet been widely applied because of the complication of the structure testing procedure, but it is demonstrated that the method is well suited for the study of most chemical transducers provided certain considerations and preliminary analyses are made, as described in this paper.

Methods for optimization of nonlinear problems with discrete variables: A review

The methods for discrete-integer-continuous variable nonlinear optimization are reviewed. They are classified into the following six categories: branch and bound, simulated annealing, sequential linearization, penalty functions, Lagrangian relaxation, and other methods. Basic ideas of each method are described and details of some of the algorithms are given. They are transcribed into a step-by-step format for easy implementation into a computer. Under “other methods”, rounding-off, heuristic, cutting-plane, pure discrete, and genetic algorithms are described. For nonlinear problems, none of the methods are guaranteed to produce the global minimizer; however, “good practical” solutions can be obtained.

Structural sensitivity analysis and optimization in the RIPAK system. Part 1

An approach to structural optimization within the framework of the RIPAK FEM package is presented. The first part of the paper is devoted to the description of sensitivity analysis and optimization methods in the RIPAK package. Structural mass is the goal function. A broad spectrum of constraints is considered. Both mass distribution and shape design variables are varied. The original algorithm of sequential linearization in the optimality criteria method is described. A numerical test is presented.

Numerical construction of an optimal control in the mayer problem with constraints for a nonlinear mechanical system

A methodology is proposed for the numerical construction of an optimal control for a nonlinear system in the presence of nonclassical constraints. The sequential linearization and reduction of all the constraints of the problem to a single integral form allow one on the different stages of solution of the problem to use their discretization step and thereby to preserve the dimension of the resulting nonlinear programming problem.

Stacking sequence optimization of simply supported laminates with stability and strain constraints

An integer programming formulation for the design of symmetric and balanced rectangular composite laminates with simply supported boundary conditions subject to buckling and strain constraints is presented. The design variables that define the stacking sequence of the laminate are ply-identity zero-one integers. The buckling constraint is linear in terms of the ply-identity design variables, but strains are nonlinear functions of these variables. A linear approximation is developed for the strain constraints so that the problem can be solved by sequential linearization using the branch and bound algorithm. Examples of graphite-epox y plates under biaxial compression are presented. Optimum stacking sequences obtained using the linear approximation are compared with global optimum designs obtained using a genetic search procedure.

Optimal control of a nonlinear mechanical system by instantaneous impulses

A method is proposed for the numerical construction of an optimal control for a nonlinear system moving under the action of instantaneous impulses. The magnitudes of impulses and the moments of their action are variable. The algorithm is based on the sequential linearization of the system of equations of motion and the projection of the antigradient of the objective function on the set of constraints.

Computing Cournot-Nash Equilibria

This paper examines convergence criteria of an algorithm for the computation of Cournot-Nash economic equilibria. The method is based on formulating the equilibrium problem as that of finding a solution to a nonlinear complementarity problem, solved by sequential linearization and Lemke's algorithm. Conditions for local and global convergence are developed and the technique is applied to homogeneous, segmented and differentiated product markets.

Computational Implementation and Tests of a Sequential Linearization Algorithm for Mixed-Discrete Nonlinear Design Optimization

A previous article gave the theoretical background and motivation for a new sequential linearization approach to the solution of mixed-discrete nonlinear optimal design problems. The present sequel article gives the implementation details of program MINSLIP based on this approach. Illustrative examples, modeling issues, and program parameter selection are discussed. A report on extensive computational results with test problems, as well as comparisons with other methods, shows advantages in both robustness and efficiency. Sample design applications are included.

A Sequential Linearization Approach for Solving Mixed-Discrete Nonlinear Design Optimization Problems

Design optimization models of often contain variables that must take only discrete values, such as standard sizes. Nonlinear optimization problems with a mixture of discrete and continuous variables are very difficult, and existing algorithms are either computationally intensive or applicable to models with special structure. A new approach for solving nonlinear mixed-discrete problems with no particular structure is presented here, motivated by its efficiency for models with extensive monotonicities of the problem’s objective and constraint functions with respect to the design variables. It involves solving a sequence of mixed-discrete linear approximations of the original nonlinear model. In this article, a review of previous approaches is followed by description of the resulting algorithm, its convergence properties and limitations. Several illustrative examples are given. A sequel article presents a detailed algorithmic implementation and extensive computational results.

Dissection of a nonlinear cascade model for sensory encoding

Action potential encoding in the cockroach tactile spine neuron may be treated as a single-input, single-output dynamic nonlinear process, where the input is the electric current flowing across the neuronal membrane and the output is the resultant train of action potentials. The nonlinear behavior of the system may be characterized by a functional expansion method which efficiently and accurately yields similar kernels to the Wiener method. A simple nonlinear cascade consisting of sequential dynamic linear, static nonlinear, and dynamic linear components was identified and gives a good approximation to the response of the neuron to random stimulation. Next, we attempted to study the components of the cascade by the use of a drug, phentolamine, which selectively modifies the dynamic behavior of the encoder. Application of phentolamine to the neuron caused a significant change in the first dynamic linear component of the cascade without affecting the other components. The change was much larger than the variability between results obtained from individual animals. This finding has implications for the biophysical processes which are involved in the components of the cascade.