Showing papers on "Linearization published in 1992"

Output feedback stabilization of fully linearizable systems

Abstract: An observer-based controller is designed to stabilize a fully linearizable nonlinear system. The system is assumed to be left-invertible and minimum-phase. The controller is robust to uncertainties in modelling the nonlinearities of the system. The design of the controller and the stability analysis employs the techniques of singular perturbations. A new ‘Tikhonov-like’ theorem is presented and used to analyse the system when the control is globally bounded.

Nonlinear control via approximate input-output linearization: the ball and beam example

Abstract: Approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree is studied. For such systems, a method for constructing approximate systems that are input-output linearizable is provided. The analysis presented is motivated through its application to a common undergraduate control laboratory experiment-the ball and beam-where it is shown to be more effective for trajectory tracking than the standard Jacobian linearization. >

Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation

Abstract: This paper presents a complete formulation of a model of coupled associative thermoplasticity at finite strains, addresses in detail the numerical analysis aspects involved in its finite element implementation, and assesses the performance of the proposed mechanical and finite element models in a comprehensive set of numerical simulations. On the thermomechanical side, novel aspects of the proposed model of thermoplasticity are (1) the explicit characterization of the plastic (configurational) entropy as an independent internal variable, (2) a thermomechanical extension of the principle of maximum dissipation consistent with the multiplicative decomposition of the deformation gradient, and (3) the exploitation of this extended principle in the formulation of an associative flow which characterizes the evolution of the plastic entropy in terms of the change of the flow criterion with respect to temperature. On the numerical analysis side, salient features of the proposed approach are (4) a new global product formula algorithm constructed via an operator split of the nonlinear initial value problem, which leads to a two-step solution procedure, (5) a unified class of local return mapping algorithms which preserves exactly the incompressibility constraint on the plastic flow and reduces to the classical radial return method for isothermal J 2 - flow theory, and (6) the formulation of a mixed finite element method in terms of the elastic entropy and the temperature field which circumvents well-known difficulties associated with the incompressibility constraint on the plastic flow. The exact linearization of both the product formula algorithm and an alternative simulataneous solution scheme for the coupled thermomechanical problem is given in two appendices.

A new reformulation-linearization technique for bilinear programming problems

Abstract: This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This “RLT” process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.

Adaptive control of nonlinear systems using neural networks

Abstract: Layered networks are used in a nonlinear adaptive control problem. The plant is an unknown feedback-linearizable discrete-time system, represented by an input-output model. A state space model of the plant is obtained to define the zero dynamics, which are assumed to be stable. A linearizing feedback control is derived in terms of some unknown nonlinear functions. To identify these functions, it is assumed that they can be modelled by layered neural networks. The weights of the networks are updated and used to generate the control. A local convergence result is given. Computer simulations verify the theoretical result.

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

Abstract: This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.

Simulation of dispersion in heterogeneous porous formations: Statistics, first-order theories, convergence of computations

Abstract: This paper discusses the results of numerical analysis of dispersion of passive solutes in two-dimensional heterogeneous porous formations. Statistics of flow and transport variables, the accuracy and the role of approximations implicit in existing first-order theories, and the convergence of computational results are investigated. The results suggest that quite different rates of convergence with Monte Carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity (sigma(Y)2 < 1.6). This has implications for the extent of the spatial domain for single-realization numerical studies of the same type. A comparison of the variance of plumes with the results of linear theories (0.05 < sigma(Y)2 < 1.6) shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport. Reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory. The interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs, and their effects on the moments of dispersing plumes are compensating, thereby yielding consistent formulations. Unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed.

Computational model for 3‐D contact problems with friction based on the penalty method

Abstract: The friction forces are assumed to follow the Coulomb law, with a slip criterion treated in the context of a standard return mapping algorithm Consistent linearization of the field equations is performed which leads to a fully implicit scheme with non-symmetric tangent stiffness which preserves asymptotic quadratic convergence of the Newton-Raphson method

Input/Output Linearization Using Time Delay Control

Abstract: A control procedure that uses Time Delay Control to achieve input/output linearization of a class of nonlinear systems is presented. The control system is characterized by a simple algorithm and enhanced robustness properties in comparison with current control algorithms. The paper first reviews the fundamentals of input/output linearization. The use of Time Delay Control is then shown to result in an exact linear system for sufficiently small delay time. Modified controllers for systems with a low-pass filter are also investigated. Simulation results show that the algorithm works well with measurement noise. The controller is also tested on a single-link flexible arm to show the effectiveness of the simple algorithm in the control of complicated systems.

Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems

Abstract: In this paper, a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given, which is the generalization of the Brunovsky form of linear systems. A set of invariants under the quadratic changes of coordinates and feedbacks is found. It is then proved that any linearly controllable nonlinear system is linearizable to second degree by a dynamic state feedback.

On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity: formulation and integration algorithms

Abstract: The continuum basis and numerical implementation of a finite deformation plasticity model formulated within the framework of the geometrically exact shell model presented in Parts I and III of this work, is discussed in detail. The model is formulated entirely in stress resultants, and hence the expensive integration through the thickness associated with the traditional degenerated solid approach is entirely by-passed. In particular, the classical Ilyushin-Shapiro plasticity model for shells is extended to accommodate kinematic and isotropic hardening, and consistently formulated to accommodate finite deformation. The corresponding closest-point-projection return mapping algorithm is shown to reduce to the solution of a system of two nonlinear scalar equations, and proved to be amenable to exact linearization leading to a closed form expression of the consistent elastoplastic tangent moduli. Numerical simulations are presented and comparisons with exact and approximate solutions are made which demonstrate the excellent performance of the proposed methodology.

Stable control of nonlinear systems using neural networks

Abstract: A neural-network-based direct control architecture is presented that achieves output tracking for a class of continuous-time nonlinear plants, for which the nonlinearities are unknown. The controller employs neural networks to perform approximate input/output plant linearization. The network parameters are adapted according to a stability principle. The architecture is based on a modification of a method previously proposed by the authors, where the modification comprises adding a sliding control term to the controller. This modification serves two purposes: first, as suggested by Sanner and Slotine,1 sliding control compensates for plant uncertainties outside the state region where the networks are used, thus providing global stability; second, the sliding control compensates for inherent network approximation errors, hence improving tracking performance. A complete stability and tracking error convergence proof is given and the setting of the controller parameters is discussed. It is demonstrated that as a result of using sliding control, better use of the network's approximation ability can be achieved, and the asymptotic tracking error can be made dependent only on inherent network approximation errors and the frequency range of unmodelled dynamical modes. Two simulations are provided to demonstrate the features of the control method.

A mass conservative numerical solution for two‐phase flow in porous media with application to unsaturated flow

Abstract: A numerical algorithm for simulation of two-phase flow in porous media is presented. The algorithm is based on a modified Picard linearization of the governing equations of flow, coupled with a lumped finite element approximation in space and dynamic time step control. Numerical results indicate that the algorithm produces solutions that are essentially mass conservative and oscillation free, even in the presence of steep infiltrating fronts. When the algorithm is applied to the case of air and water flow in unsaturated soils, numerical results confirm the conditions under which Richards's equation is valid. Numerical results also demonstrate the potential importance of air phase advection when considering contaminant transport in unsaturated soils. Comparison to several other numerical algorithms shows that the modified Picard approach offers robust, mass conservative solutions to the general equations that describe two-phase flow in porous media.

Design of quadrature mirror filters with linear phase in the frequency domain

Abstract: The design of quadrature mirror filter (QMF) banks whose analysis and synthesis filters have linear phase is considered. Because the design problem in the frequency domain is a highly nonlinear optimization problem, a linearization technique is proposed. An analytical solution formula is obtained, leading to a very efficient procedure. Computer simulations show that the design technique achieves better results in fewer iterations than conventional approaches when starting at the same preset initial guess. Moreover, the technique produces almost the same good results in six iterations if it starts at a better initial guess compared to the preset initial guess. By incorporating the technique with a weighted least squares, (WLS) algorithm, the design of QMF banks whose overall reconstruction error is minimized in the minimax sense over the entire frequency band is facilitated. Computer simulations for illustration and comparison are provided. >

The GS algorithm for exact linearization to Brunovsky normal form

Abstract: An algorithm utilizing the minimal number of integrations for the exact linearization of nonlinear systems to Brunovsky normal form under nonlinear feedback is presented. The tools which are involved are based on classical constructions appearing in the theory of exterior differential systems. >

Genuinely resultant shell finite elements accounting for geometric and material non-linearity

Abstract: The underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. This paper examines the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method, the treatment of finite rotations through an arbitrary parametrization of the rotation group, the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis.

Dynamical Systems: An Introduction with Applications in Economics and Biology

Abstract: This book gives a comprehensive account of dynamical systems in non-technical language Starting from the first steps of differential equations, on the assumption that readers only have a modest mathematical background, it quickly takes them to nonlinear dynamical systems, linearization theory, limit cycles, gradient, Lagrangean and Hamiltonian dynamical systems and more advanced material such as bifurcation, chaos, catastrophes and optimal dynamical systems A chapter reviewing linear algebra makes the book self-contained, and a chapter devoted to applications in economics and biology will improve reader motivation

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

Abstract: 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.

A system level study of the longitudinal control of a platoon of vehicles

Abstract: This paper presents a preliminary system study of a longitudinal control law for a platoon of nonidentical vehicles using a simplified nonlinear model for the vehicle dynamics This study advances the art of automatic longitudinal control for a platoon of vehicles in the sense that is considers longer platoons composed of nonidentical vehicles; furthermore, the longitudinal control laws presented in this study take advantage of communication possibilities not available in the recent past

A linearization procedure for quadratic and cubic mixed-integer problems

Abstract: Several techniques of linearization have appeared in the literature. The technique of F. Glover, which seems to be the most efficient, linearizes a binary quadratic integer problem of n variables by introducing n new continuous variables and 4n auxiliary linear constraints. The new technique proposed in this paper is not only useful in linearizing binary quadratic and cubic integer problems, but also applicable to the case of quadratic and to a certain class of cubic “mixed-integer” problems. It is shown that the new technique further reduces the number of auxiliary linear constraints from 4n to n, while keeping the number of new continuous variables at n for the binary quadratic integer problem of n variables. And, it requires, in the case of a certain class of cubic mixed-integer problems having 2n of 0–1 variables, only 3n auxiliary linear constraints and the same number of new continuous variables. The analytical superiority of the new linearization technique has also been observed, in terms of the nu...

The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: intrinsic definition and geometric interpretation

Abstract: The tangent stiffness matrix (i.e., the Hessian) for nonlinear structural models in computational solid mechanics is typically computed by linearization of the weak form of the equilibrium equations via the directional derivative formula. Depending on the specific mechanical model, away from equilibrium this procedure will in general yield a nonsymmetric tangent stiffness matrix. By contrast, if the directional (Gateaux) derivative is replaced by the covariant derivative (relative to a certain Riemannian metric) an intrinsic definition of the Hessian is obtained which is always symmetric away from equilibrium. It is shown that by endowing the rotation group with the standard bi-invariant metric, the resulting (symmetric) Hessian is simply the symmetrization of the usual expression obtained via the Gateaux derivative. In a finite element context, this property provides a rigorous justification for the common practice of symmetrizing the ‘seemingly nonsymmetric tangent’ away from equilibrium. This apparently ad hoc symmetrization procedure yields, in fact, the actual Hessian.