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Linearized system

It should be a symmetrical form in the impulse response of a linear system. [Pg.370]

Minimizing the square of the gradient vector under the condition c/ = I yields the following linear system of equations... [Pg.2338]

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. [Pg.289]

The second consideration is the geometry of the molecule. The multipole estimation methods are only valid for describing interactions between distant regions of the molecule. The same is true of integral accuracy cutoffs. Because of this, it is common to find that the calculated CPU time can vary between different conformers. Linear systems can be modeled most efficiently and... [Pg.44]

The sequence of amino acids in a peptide can be written using the three-letter code shown in Figure 45.3 or a one-letter code, both in common use. For example, the tripeptide, ala.ala.phe, could be abbreviated further to AAF Although peptides and proteins have chain-like structures, they seldom produce a simple linear system rather, the chains fold and wrap around each other to give complex shapes. The chemical nature of the various amino acid side groups dictates the way in which the chains fold to arrive at a thermodynamically most-favored state. [Pg.331]

Zuazua E. (1995) Controllability of the linear system of thermoelasticity. J. Math. Pures Appl. 74 (4), 291-315. [Pg.386]

Eor a linear system f (c) = if, so the wave velocity becomes independent of concentration and, in the absence of dispersive effects such as mass transfer resistance or axial mixing, a concentration perturbation propagates without changing its shape. The propagation velocity is inversely dependent on the adsorption equiUbrium constant. [Pg.261]

Young, D. M. Iterative Solution for- Large Linear Systems. Academic, New York (1971). [Pg.424]

Example The differential equation My" + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2 VkM. the roots of the characteristic equation... [Pg.454]

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. [Pg.466]

Pivoting in Gauss Elimination It might seem that the Gauss elimination completely disposes of the problem of finding solutions of linear systems, and theoretically it does. In practice, however, things are not so simple. [Pg.467]

Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use or transfer functions and block diagrams. A transfer func tion can be obtained by starting with a physical model as... [Pg.720]

Bioprocess Control An industrial fermenter is a fairly sophisticated device with control of temperature, aeration rate, and perhaps pH, concentration of dissolved oxygen, or some nutrient concentration. There has been a strong trend to automated data collection and analysis. Analog control is stiU very common, but when a computer is available for on-line data collec tion, it makes sense to use it for control as well. More elaborate measurements are performed with research bioreactors, but each new electrode or assay adds more work, additional costs, and potential headaches. Most of the functional relationships in biotechnology are nonlinear, but this may not hinder control when bioprocess operate over a narrow range of conditions. Furthermore, process control is far advanced beyond the days when the main tools for designing control systems were intended for linear systems. [Pg.2148]

As an illustration, let us consider the collinear reaction AB -I- C -> A -I- BC. It is known that the collinear motion of the linear system ABC relative to its center-of-mass reduces to the motion of the... [Pg.32]

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). [Pg.2]

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. [Pg.15]

A dynamic system is linear if the Principle of Superposition can be applied. This states that The response y t) of a linear system due to several inputs x t),... [Pg.69]

The steady-state response of a linear system will be... [Pg.145]

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... [Pg.274]

The Linear Quadratic Regulator (LQR) provides an optimal control law for a linear system with a quadratic performance index. [Pg.274]

Luenberger, D.G. (1964) Observing the State of a Linear System, IEEE Trans. Military Electronics, MIL-8, pp. 74-80. [Pg.430]

When eompiling the material for the book, deeisions had to be made as to what should be ineluded, and what should not. It was deeided to plaee the emphasis on the eontrol of eontinuous and diserete-time linear systems. Treatment of nonlinear systems (other than linearization) has therefore not been ineluded and it is suggested that other works (sueh as Feedbaek Control Systems, Phillips and Harbor (2000)) be eonsulted as neeessary. [Pg.455]

In an alternative approaeh, Jezequel (2001) proposes the eoneept of the sealeable reaetor based on the observation that in highly non-linear systems sueh as preeipitation, seale-up is better assured when all the eritieal parts of the proeess are exaetly at the same seales. Thus, the eritieal parts are sealed by replieation, whilst the main holding vessel in whieh slow proeesses take plaee is sealed in volume (Figure 8.9). [Pg.229]

Given the partition functions, the enthalpy and entropy terms may be calculated by carrying out the required differentiations in eq. (12.8). For one mole of molecules, the results for a non-linear system are (R being the gas constant)... [Pg.303]

Calculation of the SMB system As the enantioselectivity is 1.5, we need between 200 and 400 theoretical plates in the system with a working pressure of 10 bar. The system constraints are similar to those of the linear system, one has ... [Pg.274]

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... [Pg.171]

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... [Pg.270]

Practically, the solution is carried out by choosing Ca = 1 and by strating from an arbitrary trial value Et0> the vector Cbi0> is then determined by solving the linear system... [Pg.273]


See other pages where Linearized system is mentioned: [Pg.472]    [Pg.2334]    [Pg.2341]    [Pg.418]    [Pg.423]    [Pg.264]    [Pg.429]    [Pg.486]    [Pg.720]    [Pg.724]    [Pg.1530]    [Pg.99]    [Pg.110]    [Pg.234]    [Pg.687]    [Pg.11]    [Pg.232]    [Pg.120]    [Pg.16]    [Pg.53]   
See also in sourсe #XX -- [ Pg.151 ]

See also in sourсe #XX -- [ Pg.21 , Pg.22 , Pg.24 ]




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A System of m Linearly Arranged Subunits

A non-linear biochemical reaction system with concentration fluctuations

Algebra and Systems of Linear Equations

Approximate non-linear lumping in systems with time-scale separation

Biphasic systems linear crystalline polymers and their properties

Classification of Linear Systems

Conditionally Linear Systems

Conjugate Gradient method linear algebraic systems

Conjugated systems linear

Controllability linear system

Coordination systems, linear

DQMOM linear system for source terms

Defining the Linear System

Detector/system linearity

Detector/system linearity calculation

Differential transformer system, linear variable

Dynamic system linear modeling

Dynamical systems, linear

Elimination methods for solving linear systems

Equality linear system, residuals

Equivalent linear system

Excited states, linear system

Explosive instability of a linear system

First-order linear system

For linear systems

Full linear system

Fusion Linear systems

Fusion linear systems, application

General approach to linear systems of reactions

Gravimetric system linearity

Heat flow linear systems

Identifiability problem linear systems

Identification of linear systems

Iterative large linear system solution

Iterative methods to solve the linear system

Kriging system of linear equations

LINEAR SCALING IN MANY-BODY SYSTEMS

Lagrangian linear systems

Large linear system solution, with iterative

Large linear system solution, with iterative methods

Linear -conjugated molecular systems

Linear 96-channel systems

Linear Conjugated n Systems

Linear Isotherm Systems—Solution to the General Model

Linear Isotherm System—Simple Models

Linear Lumping in Systems with Timescale Separation

Linear Nernstian systems

Linear Passive System

Linear Quadratic Gaussian control system design

Linear Reaction Systems

Linear System Solution with Iterative Methods

Linear Systems of Binding Sites

Linear algebraic systems

Linear algebraic systems Gaussian elimination)

Linear algebraic systems range

Linear and Nonlinear Systems

Linear chain conducting systems

Linear compensation systems

Linear equations systems

Linear flow system

Linear least-square systems

Linear lumping in systems with time-scale separation

Linear multi-variable systems

Linear pi systems

Linear polyenes, numbering system

Linear programming canonical system

Linear programming modeling systems

Linear reaction diffusion system, stationary solution

Linear reaction systems, oscillatory chemical

Linear sparse systems

Linear system analysis

Linear system graph model

Linear system identifiability

Linear system limitations

Linear system membrane transport model

Linear system response

Linear system solution

Linear system state space form

Linear system, three-term

Linear system-size scaling

Linear systems

Linear systems

Linear systems and the principle of superposition

Linear systems approach

Linear systems approach basic concepts

Linear systems approach errors

Linear systems approach mathematical elements

Linear systems approach methods

Linear systems models

Linear systems theory

Linear systems, irreversible processes

Linear systems, partial solution

Linear systems, underdimensioned

Linear thermal expansivity polymeric systems

Linear time invariant system

Linear tt Systems

Linear unique system conditioning

Linearization of nonlinear systems

Linearized System Equations

Matrices and Systems of Linear Equations

Membranes linear systems

Model systems linear additive models

Modeling of Response in Linear Systems

Modeling, linear control system

Molecular Orbital Theory for Linear Pi Systems

Non-linear dynamic systems

Non-linear dynamical systems

Non-linear least-square systems isochrons

Non-linear system

Non-linear systems, parameter

Non-linear systems, parameter estimation

Nonhomogeneous Linear Systems

Nonlinear system linear differential equations

Numeric calculation linear equation system

On the Extremum Properties of Thermodynamic Steady State in Non-Linear Systems

Polymerization systems linear step-growth

Power feed system, linear

Process control, automatic linear systems

Row Reduction and Systems of Linear Equations

Second-order linear system

Small Increments Make the System Linear

Solution Methods for Linear Algebraic Systems

Solution of Linear Equation Systems

Solving Systems of Linear Algebraic Equations

Solving Systems of Linear Equations

Square linear system

State Space Form of Linear Constrained Systems

System Peaks in Linear Chromatography

System identification linearity tests

System linearization

System of implicit non-linear equations the Newton-Raphson method

System of linear differential equations

Systems of linear algebraic equations

Systems of linear equations

Systems of linear equations and their general solutions

Systems of non-linear algebraic equations

Systems of non-linear equations

The conformation of linear systems

Third-order linear system

Three-Phase System and Linear Approximation

Transformation of linear systems

Under dimensioned Linear Systems

Underdetermined system of linear equations

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