Optimal-stopping control for airborne collision avoidance and return-to-course flight

This paper considers an aircraft collision avoidance design problem that also incorporates design of the aircraft’s return-to-course flight. This control design problem is formulated as a non-linear optimal-stopping control problem; a formulation that does not require a prior knowledge of time taken to perform the avoidance and return-to-course manoeuvre. A dynamic programming solution to the avoidance and return-to-course problem is presented, before a Markov chain numerical approximation technique is described. Simulation results are presented that illustrate the proposed collision avoidance and return-to-course flight approach.

On the statistical determination of optimal camera configurations in large scale surveillance networks

The selection of optimal camera configurations (camera locations, orientations etc.) for multi-camera networks remains an unsolved problem. Previous approaches largely focus on proposing various objective functions to achieve different tasks. Most of them, however, do not generalize well to large scale networks. To tackle this, we introduce a statistical formulation of the optimal selection of camera configurations as well as propose a Trans-Dimensional Simulated Annealing (TDSA) algorithm to effectively solve the problem. We compare our approach with a state-of-the-art method based on Binary Integer Programming (BIP) and show that our approach offers similar performance on small scale problems. However, we also demonstrate the capability of our approach in dealing with large scale problems and show that our approach produces better results than 2 alternative heuristics designed to deal with the scalability issue of BIP.

Development of optimal designs of insulated rail joints

Proper functioning of Insulated Rail Joints (IRJs) is essential for the safe operation of the railway signalling systems and broken rail identification circuitries. The Conventional IRJ (CIRJ) resembles structural butt joints consisting of two pieces of rails connected together through two joint bars on either side of their web and the assembly is held together through pre-tensioned bolts. As the IRJs should maintain electrical insulation between the two rails, a gap between the rail ends must be retained at all times and all metal contacting surfaces should be electrically isolated from each other using non-conductive material. At the gap, the rail ends lose longitudinal continuity and hence the vertical sections of the rail ends are often severely damaged, especially at the railhead, due to the passage of wheels compared to other continuously welded rail sections. Fundamentally, the reason for the severe damage can be related to the singularities of the wheel-rail contact pressure and the railhead stress. No new generation designs that have emerged in the market to date have focussed on this fundamental; they only have provided attention to either the higher strength materials or the thickness of the sections of various components of the IRJs. In this thesis a novel method of shape optimisation of the railhead is developed to eliminate the pressure and stress singularities through changes to the original sharp corner shaped railhead into an arc profile in the longitudinal direction. The optimal shape of the longitudinal railhead profile has been determined using three nongradient methods in search of accuracy and efficiency: (1) Grid Search Method; (2) Genetic Algorithm Method and (3) Hybrid Genetic Algorithm Method. All these methods have been coupled with a parametric finite element formulation for the evaluation of the objective function for each iteration or generation depending on the search algorithm employed. The optimal shape derived from these optimisation methods is termed as Stress Minimised Railhead (SMRH) in this thesis. This optimal SMRH design has exhibited significantly reduced stress concentration that remains well below the yield strength of the head hardened rail steels and has shifted the stress concentration location away from the critical zone of the railhead end. The reduction in the magnitude and the relocation of the stress concentration in the SMRH design has been validated through a full scale wheel – railhead interaction test rig; Railhead strains under the loaded wheels have been recorded using a non-contact digital image correlation method. Experimental study has confirmed the accuracy of the numerical predications. Although the SMRH shaped IRJs eliminate stress singularities, they can still fail due to joint bar or bolt hole cracking; therefore, another conceptual design, termed as Embedded IRJ (EIRJ) in this thesis, with no joint bars and pre-tensioned bolts has been developed using a multi-objective optimisation formulation based on the coupled genetic algorithm – parametric finite element method. To achieve the required structural stiffness for the safe passage of the loaded wheels, the rails were embedded into the concrete of the post tensioned sleepers; the optimal solutions for the design of the EIRJ is shown to simplify the design through the elimination of the complex interactions and failure modes of the various structural components of the CIRJ. The practical applicability of the optimal shapes SMRH and EIRJ is demonstrated through two illustrative examples, termed as improved designs (IMD1 & IMD2) in this thesis; IMD1 is a combination of the CIRJ and the SMRH designs, whilst IMD2 is a combination of the EIRJ and SMRH designs. These two improved designs have been simulated for two key operating (speed and wagon load) and design (wheel diameter) parameters that affect the wheel-rail contact; the effect of these parameters has been found to be negligible to the performance of the two improved designs and the improved designs are in turn found far superior to the current designs of the CIRJs in terms of stress singularities and deformation under the passage of the loaded wheels. Therefore, these improved designs are expected to provide longer service life in relation to the CIRJs.

Optimal designs for evaluating a series of treatments

Several articles in this journal have studied optimal designs for testing a series of treatments to identify promising ones for further study. These designs formulate testing as an ongoing process until a promising treatment is identified. This formulation is considered to be more realistic but substantially increases the computational complexity. In this article, we show that these new designs, which control the error rates for a series of treatments, can be reformulated as conventional designs that control the error rates for each individual treatment. This reformulation leads to a more meaningful interpretation of the error rates and hence easier specification of the error rates in practice. The reformulation also allows us to use conventional designs from published tables or standard computer programs to design trials for a series of treatments. We illustrate these using a study in soft tissue sarcoma.

Optimal irrigation planning in river basin development: The case of the Upper Cauvery river basin

The study deals with the irrigation planning of the Cauvery river basin in peninsular India which is extensively developed in the downstream reaches and has a high potential for development in the upper reaches. A four-reservoir system is modelled on a monthly basis by using a mathematical programming (LP) formulation to find optimum cropping patterns, subject to land, water and downstream release constraints, and applied to the Cauvery basin. Two objectives, maximizing net economic benefits and maximizing irrigated cropped area, considered in the model are analysed in the context of multiobjective planning and the trade-offs discussed.

Active vibration suppression of non-linear beams using optimal dynamic inversion

Euler–Bernoulli beams are distributed parameter systems that are governed by a non-linear partial differential equation (PDE) of motion. This paper presents a vibration control approach for such beams that directly utilizes the non-linear PDE of motion, and hence, it is free from approximation errors (such as model reduction, linearization etc.). Two state feedback controllers are presented based on a newly developed optimal dynamic inversion technique which leads to closed-form solutions for the control variable. In one formulation a continuous controller structure is assumed in the spatial domain, whereas in the other approach it is assumed that the control force is applied through a finite number of discrete actuators located at predefined discrete locations in the spatial domain. An implicit finite difference technique with unconditional stability has been used to solve the PDE with control actions. Numerical simulation studies show that the beam vibration can effectively be decreased using either of the two formulations.

A generalized Linear Programming based approach to optimal divisible load scheduling

In this paper we propose a general Linear Programming (LP) based formulation and solution methodology for obtaining optimal solution to the load distribution problem in divisible load scheduling. We exploit the power of the versatile LP formulation to propose algorithms that yield exact solutions to several very general load distribution problems for which either no solutions or only heuristic solutions were available. We consider both star (single-level tree) networks and linear daisy chain networks, having processors equipped with front-ends, that form the generic models for several important network topologies. We consider arbitrary processing node availability or release times and general models for communication delays and computation time that account for constant overheads such as start up times in communication and computation. The optimality of the LP based algorithms is proved rigorously.

Active vibration suppression of beams with discrete actuators using optimal dynamic inversion

Most of the structural elements like beams, cables etc. are flexible and should be modeled as distributed parameter systems (DPS) to represent the reality better. For large structures, the usual approach of 'modal representation' is not an accurate representation. Moreover, for excessive vibrations (possibly due to strong wind, earthquake etc.), external power source (controller) is needed to suppress it, as the natural damping of these structures is usually small. In this paper, we propose to use a recently developed optinial dynamic inversion technique to design a set of discrete controllers for this purpose. We assume that the control force to the structure is applied through finite number of actuators, which are located at predefined locations in the spatial domain. The method used in this paper determines control forces directly from the partial differential equation (PDE) model of the system. The formulation has better practical significance, both because it leads to a closed form solution of the controller (hence avoids computational issues) as well as because a set of discrete actuators along the spatial domain can be implemented with relative ease (as compared to a continuous actuator).

Robust/Optimal Temperature Profile Control of a High-Speed Aerospace Vehicle Using Neural Networks

An approximate dynamic programming (ADP)-based suboptimal neurocontroller to obtain desired temperature for a high-speed aerospace vehicle is synthesized in this paper. A I-D distributed parameter model of a fin is developed from basic thermal physics principles. "Snapshot" solutions of the dynamics are generated with a simple dynamic inversion-based feedback controller. Empirical basis functions are designed using the "proper orthogonal decomposition" (POD) technique and the snapshot solutions. A low-order nonlinear lumped parameter system to characterize the infinite dimensional system is obtained by carrying out a Galerkin projection. An ADP-based neurocontroller with a dual heuristic programming (DHP) formulation is obtained with a single-network-adaptive-critic (SNAC) controller for this approximate nonlinear model. Actual control in the original domain is calculated with the same POD basis functions through a reverse mapping. Further contribution of this paper includes development of an online robust neurocontroller to account for unmodeled dynamics and parametric uncertainties inherent in such a complex dynamic system. A neural network (NN) weight update rule that guarantees boundedness of the weights and relaxes the need for persistence of excitation (PE) condition is presented. Simulation studies show that in a fairly extensive but compact domain, any desired temperature profile can be achieved starting from any initial temperature profile. Therefore, the ADP and NN-based controllers appear to have the potential to become controller synthesis tools for nonlinear distributed parameter systems.

An integrated model for optimal reservoir operation for irrigation of multiple crops

An integrated model is developed, based on seasonal inputs of reservoir inflow and rainfall in the irrigated area, to determine the optimal reservoir release policies and irrigation allocations to multiple crops. The model is conceptually made up of two modules, Module 1 is an intraseasonal allocation model to maximize the sum of relative yields of all crops, for a given state of the system, using linear programming (LP). The module takes into account reservoir storage continuity, soil moisture balance, and crop root growth with time. Module 2 is a seasonal allocation model to derive the steady state reservoir operating policy using stochastic dynamic programming (SDP). Reservoir storage, seasonal inflow, and seasonal rainfall are the state variables in the SDP. The objective in SDP is to maximize the expected sum of relative yields of all crops in a year. The results of module 1 and the transition probabilities of seasonal inflow and rainfall form the input for module 2. The use of seasonal inputs coupled with the LP-SDP solution strategy in the present formulation facilitates in relaxing the limitations of an earlier study, while affecting additional improvements. The model is applied to an existing reservoir in Karnataka State, India.

Active vibration suppression of beams with discrete actuators using optimal dynamic inversion

Most of the structural elements like beams, cables etc. are flexible and should be modeled as distributed parameter systems (DPS) to represent the reality better. For large structures, the usual approach of 'modal representation' is not an accurate representation. Moreover, for excessive vibrations (possibly due to strong wind, earthquake etc.), external power source (controller) is needed to suppress it, as the natural damping of these structures is usually small. In this paper, we propose to use a recently developed optimal dynamic inversion technique to design a set of discrete controllers for this purpose. We assume that the control force to the structure is applied through finite number of actuators, which are located at predefined locations in the spatial domain. The method used in this paper determines control forces directly from the partial differential equation (PDE) model of the system. The formulation has better practical significance, both because it leads to a closed form solution of the controller (hence avoids computational issues) as well as because a set of discrete actuators along the spatial domain can be implemented with relative ease (as compared to a continuous actuator)

An optimal dynamic inversion approach for controlling a class of one-dimensional nonlinear distributed parameter systems

Combining the principles of dynamic inversion and optimization theory, a new approach is presented for stable control of a class of one-dimensional nonlinear distributed parameter systems, assuming the availability a continuous actuator in the spatial domain. Unlike the existing approximate-then-design and design-then-approximate techniques, here there is no need of any approximation either of the system dynamics or of the resulting controller. Rather, the control synthesis approach is fairly straight-forward and simple. The controller formulation has more elegance because we can prove the convergence of the controller to its steady state value. To demonstrate the potential of the proposed technique, a real-life temperature control problem for a heat transfer application is solved. It has been demonstrated that a desired temperature profile can be achieved starting from any arbitrary initial temperature profile.

Optimal control of arrivals to queues with delayed queue length information

We consider discrete-time versions of two classical problems in the optimal control of admission to a queueing system: i) optimal routing of arrivals to two parallel queues and ii) optimal acceptance/rejection of arrivals to a single queue. We extend the formulation of these problems to permit a k step delay in the observation of the queue lengths by the controller. For geometric inter-arrival times and geometric service times the problems are formulated as controlled Markov chains with expected total discounted cost as the minimization objective. For problem i) we show that when k = 1, the optimal policy is to allocate an arrival to the queue with the smaller expected queue length (JSEQ: Join the Shortest Expected Queue). We also show that for this problem, for k greater than or equal to 2, JSEQ is not optimal. For problem ii) we show that when k = 1, the optimal policy is a threshold policy. There are, however, two thresholds m(0) greater than or equal to m(1) > 0, such that mo is used when the previous action was to reject, and mi is used when the previous action was to accept.

Optimal allocation of carbon credits to emitting agents in a carbon economy

Reduction of carbon emissions is of paramount importance in the context of global warming. Countries and global companies are now engaged in understanding systematic ways of achieving well defined emission targets. In fact, carbon credits have become significant and strategic instruments of finance for countries and global companies. In this paper, we formulate and suggest a solution to the carbon allocation problem, which involves determining a cost minimizing allocation of carbon credits among different emitting agents. We address this challenge in the context of a global company which is faced with the challenge of determining an allocation of carbon credit caps among its divisions in a cost effective way. The problem is formulated as a reverse auction problem where the company plays the role of a buyer or carbon planning authority and the different divisions within the company are the emitting agents that specify cost curves for carbon credit reductions. Two natural variants of the problem: (a) with unlimited budget and (b) with limited budget are considered. Suitable assumptions are made on the cost curves and in each of the two cases we show that the resulting problem formulation is a knapsack problem that can be solved optimally using a greedy heuristic. The solution of the allocation problem provides critical decision support to global companies engaged seriously in green programs.

Optimal local polynomial regression of noisy time varying signals

We address the problem of local-polynomial modeling of smooth time-varying signals with unknown functional form, in the presence of additive noise. The problem formulation is in the time domain and the polynomial coefficients are estimated in the pointwise minimum mean square error (PMMSE) sense. The choice of the window length for local modeling introduces a bias-variance tradeoff, which we solve optimally by using the intersection-of-confidence-intervals (ICI) technique. The combination of the local polynomial model and the ICI technique gives rise to an adaptive signal model equipped with a time-varying PMMSE-optimal window length whose performance is superior to that obtained by using a fixed window length. We also evaluate the sensitivity of the ICI technique with respect to the confidence interval width. Simulation results on electrocardiogram (ECG) signals show that at 0dB signal-to-noise ratio (SNR), one can achieve about 12dB improvement in SNR. Monte-Carlo performance analysis shows that the performance is comparable to the basic wavelet techniques. For 0 dB SNR, the adaptive window technique yields about 2-3dB higher SNR than wavelet regression techniques and for SNRs greater than 12dB, the wavelet techniques yield about 2dB higher SNR.