Reference trajectory

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

Accuracy, however, in biomolecular trajectories, must be defined somewhat subjectively. In the absence of exact reference data (from experiment or from an analytical solution), the convention has been to measure accuracy with respect to reference trajectories by a Verlet-like integrator [18, 19] at a timestep of 1 or 0.5 fs (about one tenth or one twentieth the period, respectively, of the fastest period an 0-H or N-H stretch). As pointed out by Deufihard et al. [20], these values are still larger than those needed to... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

A detailed examination of LN behavior is available [88] for the blocked alanine model, the proteins BPTI and lysozyme, and a large water system, compared to reference Langevin trajectories, in terms of energetic, geometric, and dynamic behavior. The middle timestep in LN can be considered an adjustable quantity (when force splitting is used), whose value does not significantly affect performance but does affect accuracy with respect to the reference trajectories. For example, we have used Atm = 3 fs for the proteins in vacuum, but 1 fs for the water system, where librational motions are rapid. 

The speedup is relative to the reference trajectory, which takes 14 hours. 

The performance index is expressed in terms of future control moves and the predicted deviations from the reference trajectory. 

We want to examine the relative behaviors of the two neighboring phase space trajectories, x (t) and x(t), starting from the initial conditions x (0) and x(0) = x (0) -f-Jx(0), respectively. The time evolution of their separation, x(i), may be approximated by linearizing the equations about the reference trajectory, x (t) ... 

The channel for state estimation opened first and after some period of time (when a good reference trajectory was obtained) the two channels for the parameters were opened simultaneously. 

A reference trajectory is used to represent the desired output response over the prediction horizon. 

The prediction of the control input is computed via an optimization method that minimizes a suitably defined objective function, usually composed by two terms the first one is related to the deviation of the predicted output from the reference trajectory (i.e., the tracking error), while the second term takes into account control input changes. Hence, the optimization problem has the form... 

Operator acceptance of MPC is reported to be very good. Displaying the long-term predicted closed-loop behavior of the process convinces the operator that the input moves, which might appear unusual in the short term, are reasonable. Tuning is accomplished via the prescription of a reference trajectory or the adjustment of a filter, both of which are related directly to the speed of the closed loop s response. 

Here again, we take a classical trajectory as a reference path in a reaction tube that passes across the transition region between two basins a and b with the flow direction b —> a. Set the time origin t = 0 at just the moment of transition. At a given time t, we take a sphere of a radius rt in 30-dimensional phase space, the center of which proceeds along the reference trajectory. Pick random points in this sphere, and let them run backward in time. Some of them will go back to the basin b if the sphere still lies inside the same tube, and the others will move to some other basins if this sphere is already out of the tube. Should the latter happen, a similar procedure is to be redone with a smaller radius r,. Repeating... 

Figure 9. Time-dependent behavior of a reaction tube in terms of Pj a, where a = PBP, and COCT, 1ST, and SKEW are at the energy of — 11,208e. P a has been measured with use of a sphere of the radius rflxed = 0.05. Since the reference trajectory runs from COCT to PBP, Pcoct pbp = 1-0 in the initial stage. (Reproduced from Ref. 10 with permission.)...

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

If fl = 1, every atom in the slider has the same velocity at every instant of time, once steady state (not necessarily smooth sliding) has been reached. Hence the problem is reduced to the motion of a single particle, for which one obtains Fj = 1. This provides an upper bound of Fj for arbitrary a. If the walls are incommensurate or disordered, one can again make use of the argument that the motion of all atoms relative to their preferred positions is the same up to temporal shifts once steady state has been reached. Owing to the incommensurability, the distribution of these temporal shifts with respect to a reference trajectory cannot change with time in the thermodynamic limit, and the instantaneous value of Fk is identical to Fk at all times. This gives a lower bound for Fj for arbitrary a. The static friction for arbitrary commensurability and/or finite systems lies in between the upper and the lower bound. 

The first is often referred to as reference trajectory and can be thought of as a filter on setpoint changes. The filtered setpoint in essence becomes a trajectory—the controller attempts to move the MVs so that the CV follows their individual trajectories. Increased filtering produces slower controller action. 

If we, as before, expand the potential in a power series around the reference trajectory, R(t), and equate terms of the same power we obtain a set of coupled equations for the expansion coefficients e (f). It has recently been demonstrated that inclusion of these higher-order correction terms does improve the agreement between the exact quantum and the semiclassical transition probabilities for inelastic collision problems (14). 

Recently, a ZPE-preserving method based on taking the classical limit of the Hamiltonian has been proposed (49). The trajectory so formed is referred to as the reference trajectory. The trajectories using the full Hamiltonian are then shifted with respect to the reference, and cannot, by construction, lose ZPE. 

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