Optimal Parameter Trajectory Estimation in Parameterized SDEs: An Algorithmic Procedure

We consider the problem of estimating the optimal parameter trajectory over a finite time interval in a parameterized stochastic differential equation (SDE), and propose a simulation-based algorithm for this purpose. Towards this end, we consider a discretization of the SDE over finite time instants and reformulate the problem as one of finding an optimal parameter at each of these instants. A stochastic approximation algorithm based on the smoothed functional technique is adapted to this setting for finding the optimal parameter trajectory. A proof of convergence of the algorithm is presented and results of numerical experiments over two different settings are shown. The algorithm is seen to exhibit good performance. We also present extensions of our framework to the case of finding optimal parameterized feedback policies for controlled SDE and present numerical results in this scenario as well.

Planar pursuit-evasion with variable speeds, part 1, extremal trajectory maps

Pursuit evasion in a plane is formulated with both players allowed to vary their speeds between fixed limits. A suitable choice of real-space coordinates confers open-loop optimality on the game. The solution in the small is described in terms of the individual players'' extremal trajectory maps (ETM). Each map is independent of role, adversary, and capture radius. An ETM depicts the actual real-space trajectories. A template method of generating constant control arcs is described. Examples of ETM for an aircraft flying at a constant altitude with fixed and varying speeds are presented.

Determination of optimal trajectory under design constraints for a satellite launch vehicle

An optimal pitch steering programme of a solid-fuel satellite launch vehicle to maximize either (1) the injection velocity at a given altitude, or (2) the size of circular orbit, for a given payload is presented. The two-dimensional model includes the rotation of atmosphere with the Earth, the vehicle's lift and drag, variation of thrust with time and altitude, inverse-square gravitational field, and the specified initial vertical take-off. The inequality constraints on the aerodynamic load, control force, and turning rates are also imposed. Using the properties of the central force motion the terminal constraint conditions at coast apogee are transferred to the penultimate stage burnout. Such a transformation converts a time-free problem into a time-fixed one, reduces the number of terminal constraints, improves accuracy, besides demanding less computer memory and time. The adjoint equations are developed in a compact matrix form. The problem is solved on an IBM 360/44 computer using a steepest ascent algorithm. An illustrative analysis of a typical launch vehicle establishes the speed of convergence, and accuracy and applicability of the algorithm.

Trajectory sensitivity modification in optimal linear systems

A new procedure for reducing trajectory sensitivity for the optimal linear regulator is described. The design is achieved without increase in the order of optimization and without the feedback of trajectory sensitivity. The procedure is also used in the input signal design problem for linear system identification by interpreting it as increasing trajectory sensitivity with respect to parameters to be estimated.

Trajectory controllability of nonlinear integro-differential system

A stronger concept of complete (exact) controllability which we call Trajectory Controllability is introduced in this paper. We study the Trajectory Controllability of an abstract nonlinear integro-differential system in the finite and infinite dimensional space setting. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Optimal explicit guidance of multistage launch vehicle along three-dimensional trajectory

Details of an efficient optimal closed-loop guidance algorithm for a three-dimensional launch are presented with simulation results. Two types of orbital injections, with either true anomaly or argument of perigee being free at injection, are considered. The resulting steering-angle profile under the assumption of uniform gravity lies in a canted plane which transforms a three-dimensional problem into an equivalent two-dimensional one. Effects of thrust are estimated using a series in a recursive way. Encke's method is used to predict the trajectory during powered flight and then to compute the changes due to actual gravity using two gravity-related vectors. Guidance parameters are evaluated using the linear differential correction method. Optimality of the algorithm is tested against a standard ground-based trajectory optimization package. The performance of the algorithm is tested for accuracy, robustness, and efficiency for a sun-synchronous mission involving guidance for a multistage vehicle that requires large pitch and yaw maneuver. To demonstrate applicability of the algorithm to a range of missions, injection into a geostationary transfer orbit is also considered. The performance of the present algorithm is found to be much better than others.

Control of transient thermal response during sequential open-die forging: A trajectory optimization approach

A trajectory optimization approach is applied to the design of a sequence of open-die forging operations in order to control the transient thermal response of a large titanium alloy billet. The amount of time tire billet is soaked in furnace prior to each successive forging operation is optimized to minimize the total process time while simultaneously satisfying constraints on the maximum and minimum values of the billet's temperature distribution to avoid microstructural defects during forging. The results indicate that a "differential" heating profile is the most effective at meeting these design goals.

NETGEM: Network Embedded Temporal GEnerative Model for gene expression data

Background: Temporal analysis of gene expression data has been limited to identifying genes whose expression varies with time and/or correlation between genes that have similar temporal profiles. Often, the methods do not consider the underlying network constraints that connect the genes. It is becoming increasingly evident that interactions change substantially with time. Thus far, there is no systematic method to relate the temporal changes in gene expression to the dynamics of interactions between them. Information on interaction dynamics would open up possibilities for discovering new mechanisms of regulation by providing valuable insight into identifying time-sensitive interactions as well as permit studies on the effect of a genetic perturbation. Results: We present NETGEM, a tractable model rooted in Markov dynamics, for analyzing the dynamics of the interactions between proteins based on the dynamics of the expression changes of the genes that encode them. The model treats the interaction strengths as random variables which are modulated by suitable priors. This approach is necessitated by the extremely small sample size of the datasets, relative to the number of interactions. The model is amenable to a linear time algorithm for efficient inference. Using temporal gene expression data, NETGEM was successful in identifying (i) temporal interactions and determining their strength, (ii) functional categories of the actively interacting partners and (iii) dynamics of interactions in perturbed networks. Conclusions: NETGEM represents an optimal trade-off between model complexity and data requirement. It was able to deduce actively interacting genes and functional categories from temporal gene expression data. It permits inference by incorporating the information available in perturbed networks. Given that the inputs to NETGEM are only the network and the temporal variation of the nodes, this algorithm promises to have widespread applications, beyond biological systems. The source code for NETGEM is available from https://github.com/vjethava/NETGEM

Ascent Phase Trajectory Optimization of a Generic Launch Vehicle

Trajectory optimization of a generic launch vehicle is considered in this paper. The trajectory from launch point to terminal injection point is divided in to two segments. The first segment deals with launcher clearance and vertical raise of the vehicle. During this phase, a nonlinear feedback guidance loop is incorporated to assure vertical raise in presence of thrust misalignment, centre of gravity offset, wind disturbance etc. and possibly to clear obstacles as well. The second segment deals with the trajectory optimization, where the objective is to ensure desired terminal conditions as well as minimum control effort and minimum structural loading in the high dynamic pressure region. The usefulness of this dynamic optimization problem formulation is demonstrated by solving it using the classical Gradient method. Numerical results for both the segments are presented, which clearly brings out the potential advantages of the proposed approach.

Simultaneous attitude control and trajectory tracking of a micro quadrotor: a SNAC aided nonlinear dynamic inversion approach

This paper presents an advanced single network adaptive critic (SNAC) aided nonlinear dynamic inversion (NDI) approach for simultaneous attitude control and trajectory tracking of a micro-quadrotor. Control of micro-quadrotors is a challenging problem due to its small size, strong coupling in pitch-yaw-roll and aerodynamic effects that often need to be ignored in the control design process to avoid mathematical complexities. In the proposed SNAC aided NDI approach, the gains of the dynamic inversion design are selected in such a way that the resulting controller behaves closely to a pre-synthesized SNAC controller for the output regulation problem. However, since SNAC is based on optimal control theory, it makes the dynamic inversion controller to operate near optimal and enhances its robustness property as well. More important, it retains two major benefits of dynamic inversion: (i) closed form expression of the controller and (ii) easy scalability to command tracking application even without any apriori knowledge of the reference command. Effectiveness of the proposed controller is demonstrated from six degree-of-freedom simulation studies of a micro-quadrotor. It has also been observed that the proposed SNAC aided NDI approach is more robust to modeling inaccuracies, as compared to the NDI controller designed independently from time domain specifications.

Partially integrated guidance and control of an air-dropped guided munition with explicit trajectory fixing

An innovative partially integrated guidance and control (PIGC) technique is developed for trajectory fixing by considering six degree-of-freedom (Six-DOF) nonlinear engagement dynamics for successful interception of ground targets by guided munitions. This trajectory fixing algorithm gives closed form solution, where two different trajectories are designed in x - h and x - y planes separately using simple quadratic equations. In order to follow designed trajectories commanded pitch and yaw rates are generated in outer loop using dynamic inversion technique. In inner loop these body rates are tracked using faster dynamic inversion loop by generating the necessary control surface deflections. Simulation studies with actuator dynamics have been carried out to account for three dimensional (3D) engagement geometry to demonstrate the usefulness of PIGC technique.

Optimal trajectory planning for path convergence in three-dimensional space

This article addresses the problem of determining the shortest path that connects a given initial configuration (position, heading angle, and flight path angle) to a given rectilinear or a circular path in three-dimensional space for a constant speed and turn-rate constrained aerial vehicle. The final path is assumed to be located relatively far from the starting point. Due to its simplicity and low computational requirements the algorithm can be implemented on a fixed-wing type unmanned air vehicle in real time in missions where the final path may change dynamically. As wind has a very significant effect on the flight of small aerial vehicles, the method of optimal path planning is extended to meet the same objective in the presence of wind comparable to the speed of the aerial vehicles. But, if the path to be followed is closer to the initial point, an off-line method based on multiple shooting, in combination with a direct transcription technique, is used to obtain the optimal solution. Optimal paths are generated for a variety of cases to show the efficiency of the algorithm. Simulations are presented to demonstrate tracking results using a 6-degrees-of-freedom model of an unmanned air vehicle.

Generative Maximum Entropy Learning for Multiclass Classification

Maximum entropy approach to classification is very well studied in applied statistics and machine learning and almost all the methods that exists in literature are discriminative in nature. In this paper, we introduce a maximum entropy classification method with feature selection for large dimensional data such as text datasets that is generative in nature. To tackle the curse of dimensionality of large data sets, we employ conditional independence assumption (Naive Bayes) and we perform feature selection simultaneously, by enforcing a `maximum discrimination' between estimated class conditional densities. For two class problems, in the proposed method, we use Jeffreys (J) divergence to discriminate the class conditional densities. To extend our method to the multi-class case, we propose a completely new approach by considering a multi-distribution divergence: we replace Jeffreys divergence by Jensen-Shannon (JS) divergence to discriminate conditional densities of multiple classes. In order to reduce computational complexity, we employ a modified Jensen-Shannon divergence (JS(GM)), based on AM-GM inequality. We show that the resulting divergence is a natural generalization of Jeffreys divergence to a multiple distributions case. As far as the theoretical justifications are concerned we show that when one intends to select the best features in a generative maximum entropy approach, maximum discrimination using J-divergence emerges naturally in binary classification. Performance and comparative study of the proposed algorithms have been demonstrated on large dimensional text and gene expression datasets that show our methods scale up very well with large dimensional datasets.

A fractional order proportional integral controller for path following and trajectory tracking of miniature air vehicles

In this paper, a fractional order proportional-integral controller is developed for a miniature air vehicle for rectilinear path following and trajectory tracking. The controller is implemented by constructing a vector field surrounding the path to be followed, which is then used to generate course commands for the miniature air vehicle. The fractional order proportional-integral controller is simulated using the fundamentals of fractional calculus, and the results for this controller are compared with those obtained for a proportional controller and a proportional integral controller. In order to analyze the performance of the controllers, four performance metrics, namely (maximum) overshoot, control effort, settling time and integral of the timed absolute error cost, have been selected. A comparison of the nominal as well as the robust performances of these controllers indicates that the fractional order proportional-integral controller exhibits the best performance in terms of ITAE while showing comparable performances in all other aspects.