A FRAMEWORK FOR THE ERROR ANALYSIS OF DISCONTINUOUS FINITE ELEMENT METHODS FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS AND APPLICATIONS TO C-0 IP METHODS

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of C-0 interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings.

QoS Constrained Optimal Sink and Relay Placement in Planned Wireless Sensor Networks

We are given a set of sensors at given locations, a set of potential locations for placing base stations (BSs, or sinks), and another set of potential locations for placing wireless relay nodes. There is a cost for placing a BS and a cost for placing a relay. The problem we consider is to select a set of BS locations, a set of relay locations, and an association of sensor nodes with the selected BS locations, so that the number of hops in the path from each sensor to its BS is bounded by h(max), and among all such feasible networks, the cost of the selected network is the minimum. The hop count bound suffices to ensure a certain probability of the data being delivered to the BS within a given maximum delay under a light traffic model. We observe that the problem is NP-Hard, and is hard to even approximate within a constant factor. For this problem, we propose a polynomial time approximation algorithm (SmartSelect) based on a relay placement algorithm proposed in our earlier work, along with a modification of the greedy algorithm for weighted set cover. We have analyzed the worst case approximation guarantee for this algorithm. We have also proposed a polynomial time heuristic to improve upon the solution provided by SmartSelect. Our numerical results demonstrate that the algorithms provide good quality solutions using very little computation time in various randomly generated network scenarios.

Near-Optimal Detection Thresholds for Bayesian Spectrum Sensing Under Fading

This paper considers the problem of energy-based, Bayesian spectrum sensing in cognitive radios under various fading environments. Under the well-known central limit theorem based model for energy detection, we derive analytically tractable expressions for near-optimal detection thresholds that minimize the probability of error under lognormal, Nakagami-m, and Weibull fading. For the Suzuki fading case, a generalized gamma approximation is provided, which saves on the computation of an integral. In each case, the accuracy of the theoretical expressions as compared to the optimal thresholds are illustrated through simulations.

Physical constraints, fundamental limits, and optimal locus of operating points for an inverted pendulum based actuated dynamic walker

The inverted pendulum is a popular model for describing bipedal dynamic walking. The operating point of the walker can be specified by the combination of initial mid-stance velocity (v(0)) and step angle (phi(m)) chosen for a given walk. In this paper, using basic mechanics, a framework of physical constraints that limit the choice of operating points is proposed. The constraint lines thus obtained delimit the allowable region of operation of the walker in the v(0)-phi(m) plane. A given average forward velocity v(x,) (avg) can be achieved by several combinations of v(0) and phi(m). Only one of these combinations results in the minimum mechanical power consumption and can be considered the optimum operating point for the given v(x, avg). This paper proposes a method for obtaining this optimal operating point based on tangency of the power and velocity contours. Putting together all such operating points for various v(x, avg,) a family of optimum operating points, called the optimal locus, is obtained. For the energy loss and internal energy models chosen, the optimal locus obtained has a largely constant step angle with increasing speed but tapers off at non-dimensional speeds close to unity.

Optimal input cross-power spectra in shake table testing of asymmetric structures

The study considers earthquake shake table testing of bending-torsion coupled structures under multi-component stationary random earthquake excitations. An experimental procedure to arrive at the optimal excitation cross-power spectral density (psd) functions which maximize/minimize the steady state variance of a chosen response variable is proposed. These optimal functions are shown to be derivable in terms of a set of system frequency response functions which could be measured experimentally without necessitating an idealized mathematical model to be postulated for the structure under study. The relationship between these optimized cross-psd functions to the most favourable/least favourable angle of incidence of seismic waves on the structure is noted. The optimal functions are also shown to be system dependent, mathematically the sharpest, and correspond to neither fully correlated motions nor independent motions. The proposed experimental procedure is demonstrated through shake table studies on two laboratory scale building frame models.

Selection of Optimal Time-to-go in Generalized Vector Explicit Guidance

A new method of selection of time-to-go (t(go)) for Generalized Vector Explicit Guidance (GENEX) law have been proposed in this paper. t(go) is known to be an important parameter in the control and cost function of GENEX guidance law. In this paper the formulation has been done to find an optimal value of t(go) that minimizes the performance cost. Mechanization of GENEX with this optimal t(go) reduces the lateral acceleration demand and consequently increases the range of the interceptor. This new formulation of computing t(go) comes in closed form and thus it can be implemented onboard. This new formulation is applied in the terminal phase of an surface-to-air interceptor for an angle constrained engagement. Results generated by simulation justify the use of optimal t(go).

Fuel-optimal G-MPSP Guidance for Powered Descent Phase of Soft Lunar Landing

A fuel optimal nonlinear sub-optimal guidance scheme is presented in this paper for soft landing of a lunar craft during the powered descent phase. The recently developed Generalized Model Predictive Static Programming (G-MPSP) is used to compute the required magnitude and angle of the thrust vector. Both terminal position and velocity vector are imposed as hard constraints, which ensures high position accuracy and facilitates initiation of vertical descent at the end of the powered descent phase. A key feature of the G-MPSP algorithm is that it converts the nonlinear dynamic programming problem into a low-dimensional static optimization problem (of the same dimension as the output vector). The control history update is done in closed form after computing a time-varying weighting matrix through a backward integration process. This feature makes the algorithm computationally efficient, which makes it suitable for on-board applications. The effectiveness of the proposed guidance algorithm is demonstrated through promising simulation results.

Campaigning in Heterogeneous Social Networks: Optimal Control of SI Information Epidemics

We study the optimal control problem of maximizing the spread of an information epidemic on a social network. Information propagation is modeled as a susceptible-infected (SI) process, and the campaign budget is fixed. Direct recruitment and word-of-mouth incentives are the two strategies to accelerate information spreading (controls). We allow for multiple controls depending on the degree of the nodes/individuals. The solution optimally allocates the scarce resource over the campaign duration and the degree class groups. We study the impact of the degree distribution of the network on the controls and present results for Erdos-Renyi and scale-free networks. Results show that more resource is allocated to high-degree nodes in the case of scale-free networks, but medium-degree nodes in the case of Erdos-Renyi networks. We study the effects of various model parameters on the optimal strategy and quantify the improvement offered by the optimal strategy over the static and bang-bang control strategies. The effect of the time-varying spreading rate on the controls is explored as the interest level of the population in the subject of the campaign may change over time. We show the existence of a solution to the formulated optimal control problem, which has nonlinear isoperimetric constraints, using novel techniques that is general and can be used in other similar optimal control problems. This work may be of interest to political, social awareness, or crowdfunding campaigners and product marketing managers, and with some modifications may be used for mitigating biological epidemics.

Optimal Guidance for Accurate Lunar Soft Landing with Minimum Fuel Consumption using Model Predictive Static Programming

In this paper the soft lunar landing with minimum fuel expenditure is formulated as a nonlinear optimal guidance problem. The realization of pinpoint soft landing with terminal velocity and position constraints is achieved using Model Predictive Static Programming (MPSP). The high accuracy of the terminal conditions is ensured as the formulation of the MPSP inherently poses final conditions as a set of hard constraints. The computational efficiency and fast convergence make the MPSP preferable for fixed final time onboard optimal guidance algorithm. It has also been observed that the minimum fuel requirement strongly depends on the choice of the final time (a critical point that is not given due importance in many literature). Hence, to optimally select the final time, a neural network is used to learn the mapping between various initial conditions in the domain of interest and the corresponding optimal flight time. To generate the training data set, the optimal final time is computed offline using a gradient based optimization technique. The effectiveness of the proposed method is demonstrated with rigorous simulation results.