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The optimal control of structures which consist of composite materials or of perforated materials is discussed. Asymptotic formula, derived from the so-called homogenization theory, are presented which allow the replacement of very complicated problems by much simpler ones.
Product Distribution (PO) theory was recently developed as a broad framework for analyzing and optimizing distributed systems. Here we demonstrate its use for adaptive distributed control of Multi-Agent Systems (MASS), i.e., for distributed stochastic optimization using MAS s. First we review one motivation of PD theory, as the information-theoretic extension of conventional full-rationality game theory to the case of bounded rational agents. In this extension the equilibrium of the game is the optimizer of a Lagrangian of the (Probability dist&&on on the joint state of the agents. When the game in question is a team game with constraints, that equilibrium optimizes the expected value of the team game utility, subject to those constraints. One common way to find that equilibrium is to have each agent run a Reinforcement Learning (E) algorithm. PD theory reveals this to be a particular type of search algorithm for minimizing the Lagrangian. Typically that algorithm i s quite inefficient. A more principled alternative is to use a variant of Newton's method to minimize the Lagrangian. Here we compare this alternative to RL-based search in three sets of computer experiments. These are the N Queen s problem and bin-packing problem from the optimization literature, and the Bar problem from the distributed RL literature. Our results confirm that the PD-theory-based approach outperforms the RL-based scheme in all three domains.
An extension of classical methods of optimal control theory for systems described by ordinary differential equations to distributed-parameter systems described by partial differential equations is presented. An application is given involving the minimum-mass design of a simply-supported shear plate with a fixed fundamental frequency of vibration. An optimal plate thickness distribution in analytical form is found. The case of a minimum-mass design of an elastic sandwich plate whose fundamental frequency of free vibration is fixed. Under the most general conditions, the optimization problem reduces to the solution of two simultaneous partial differential equations involving the optimal thickness distribution and the modal displacement. One equation is the uniform energy distribution expression which was found by Ashley and McIntosh for the optimal design of one-dimensional structures with frequency constraints, and by Prager and Taylor for various design criteria in one and two dimensions. The second equation requires dynamic equilibrium at the preassigned vibration frequency.
Optimal control of distributed parameter systems using multilevel techniques
The Space Station Freedom (SSF) is planned with a wireless communication system in place for the transmission of information between crew members on board. The clarity of transmission is paramount to an effective system of communication. A short overview is presented of the system including the requirements of interest, and a statement of the problem. The theory used to solve the problem is explored. The results given are for the experiments performed on a mockup of the proposed structure at NASA-Marshall. The requirements on the signal level are that there is a 45 dB signal to noise ratio from end to end, and that coverage over 99 pct. of the volume be maintained. The Rice probability distribution function, a simple extension of the Rayleigh distribution, is used to estimate the field strength inside a volume, where a significant line of sight from the transmitter to the receiver exists. For the SSF, this distribution will correspond to the summation of a coherent line of sight path between the transmitter and the receiver and an incoherent portion. The incoherent portion is the sum of reflections from the walls and the equipment inside the SSF. The Rice distribution was found to be the optimal distribution from the results.
This paper examines the globally optimal maneuver of distributed systems undergoing large overall rigid-body motion and small relative elastic motion. Using floating coordinates the rigid-body motion and the elastic motion are decoupled thereby allowing the globally optimal maneuver problem to be separated into components associated with the rigid-body and elastic motions. The maneuvers are performed using distributed maneuvering forces based on modal measurements. The modal measurements are extracted from the physical measurements using modal filters. Rest-to-rest maneuvers of a uniform beam illustrate the decentralized nature of the globally optimal solutions.
Research on control, design and programming of kinematically redundant robot manipulators (KRRM) is discussed. These are devices in which there are more joint space degrees of freedom than are required to achieve every position and orientation of the end-effector necessary for a given task in a given workspace. The technological developments described here deal with: kinematic programming techniques for automatically generating joint-space trajectories to execute prescribed tasks; control of redundant manipulators to optimize dynamic criteria (e.g., applications of forces and moments at the end-effector that optimally distribute the loading of actuators); and design of KRRMs to optimize functionality in congested work environments or to achieve other goals unattainable with non-redundant manipulators. Kinematic programming techniques are discussed, which show that some pseudo-inverse techniques that have been proposed for redundant manipulator control fail to achieve the goals of avoiding kinematic singularities and also generating closed joint-space paths corresponding to close paths of the end effector in the workspace. The extended Jacobian is proposed as an alternative to pseudo-inverse techniques.
The effective use of multiple passive and active members in adaptive structures necessitates that these members be optimally distributed throughout the structure. In truss structures, the problem falls into the class of combinatorial optimization for which the solution becomes exceedingly intractable as the problem size increases. This is overcome by using the simulated annealing algorithm to obtain near optimal locations for passive and/or active members. The maximization of the rate of energy dissipation over a finite time period as the measure of optimality is adopted. The selection of optimal locations for both passive and active members is consistently treated through the use of the energy dissipation rate criterion within the simulated annealing algorithm. Numerical examples are used to illustrate the effectiveness of the methodology for large truss structures.
We compare Genetic Algorithms (GA's) with Probability Collectives (PC), a new framework for distributed optimization and control. In contrast to GA's, PC-based methods do not update populations of solutions. Instead they update an explicitly parameterized probability distribution p over the space of solutions. That updating of p arises as the optimization of a functional of p. The functional is chosen so that any p that optimizes it should be p peaked about good solutions. The PC approach works in both continuous and discrete problems. It does not suffer from the resolution limitation of the finite bit length encoding of parameters into GA alleles. It also has deep connections with both game theory and statistical physics. We review the PC approach using its motivation as the information theoretic formulation of bounded rationality for multi-agent systems. It is then compared with GA's on a diverse set of problems. To handle high dimensional surfaces, in the PC method investigated here p is restricted to a product distribution. Each distribution in that product is controlled by a separate agent. The test functions were selected for their difficulty using either traditional gradient descent or genetic algorithms. On those functions the PC-based approach significantly outperforms traditional GA's in both rate of descent, trapping in false minima, and long term optimization.
A two-time scale approximation for the linear quadratic optimal output feedback regulator program is examined. Necessary conditions for optimality, as well as an algorithm for computing locally near-optimal gains are derived. If it is assumed that the slow and fast subsystem initial conditions are uniformly distributed, optimal gains for the two-time-scale problem provide a second-order approximation to optimal closed-loop performance in the unperturbed system. This is verified with a numerical example.
Results are presented from detailed analytical studies made to define methods for obtaining improved multisegment lining performance by taking advantage of relative placement of each lining segment. Properly phased liner segments reflect and spatially redistribute the incident acoustic energy and thus provide additional attenuation. A mathematical model was developed for rectangular ducts with uniform mean flow. Segmented acoustic fields were represented by duct eigenfunction expansions, and mode-matching was used to ensure continuity of the total field. Parametric studies were performed to identify attenuation mechanisms and define preliminary liner configurations. An optimization procedure was used to determine optimum liner impedance values for a given total lining length, Mach number, and incident modal distribution. Optimal segmented liners are presented and it is shown that, provided the sound source is well-defined and flow environment is known, conventional infinite duct optimum attenuation rates can be improved. To confirm these results, an experimental program was conducted in a laboratory test facility. The measured data are presented in the form of analytical-experimental correlations. Excellent agreement between theory and experiment verifies and substantiates the analytical prediction techniques. The results indicate that phased liners may be of immediate benefit in the development of improved aircraft exhaust duct noise suppressors.
Distributed prognostics architecture design is an enabling step for efficient implementation of health management systems. A major challenge encountered in such design is formulation of optimal distributed prognostics algorithms. In this paper. we present a distributed GPR based prognostics algorithm whose target platform is a wireless sensor network. In addition to challenges encountered in a distributed implementation, a wireless network poses constraints on communication patterns, thereby making the problem more challenging. The prognostics application that was used to demonstrate our new algorithms is battery prognostics. In order to present trade-offs within different prognostic approaches, we present comparison with the distributed implementation of a particle filter based prognostics for the same battery data.
Motivated by advances in signal processing technology that support more complex algorithms, a new look is taken at the problem of estimating the phase and other parameters of a periodic waveform in additive Gaussian noise. The general problem was introduced and the maximum a posteriori probability criterion with signal space interpretation was used to obtain the structures of optimum and some suboptimum phase estimators for known constant frequency and unknown constant phase with an a priori distribution. Optimal algorithms are obtained for some cases where the frequency is a parameterized function of time with the unknown parameters and phase having a joint a priori distribution. In the last section, the intrinsic and extrinsic geometry of hypersurfaces is introduced to provide insight to the estimation problem for the small noise and large noise cases.
Warm rain collision-coalescence has been persistently difficult to parameterize in bulk microphysics schemes. We use a flexible bulk microphysics scheme with bin scheme process parameterizations, called AMP, to investigate reasons for the difficulty. AMP is configured in a variety of ways to mimic bulk schemes and is compared to simulations with the bin scheme upon which AMP is built. We find that an important limitation in traditional bulk schemes is the use of separate cloud and rain categories. When the drop size distribution is instead represented by a continuous distribution, the simulation of cloud-to-rain conversion is substantially improved. We also find large sensitivity to the threshold size to distinguish cloud and rain in traditional schemes; substantial improvement is found by decreasing the threshold from 40 to 25 μm. Neither the use of an assumed functional form for the size distribution nor the choice of predicted distribution moments has a large impact on the ability of AMP to simulate rain production. When predicting four total moments of the liquid drop size distribution, either with a traditional two-category, two-moment scheme with a reduced size threshold, or a four-moment single-category scheme, errors in the evolution of mass and the cloud size distribution are similar, but the single-category scheme has a substantially better representation of the rain size distribution. Optimal moment combinations for the single-category approach are investigated and appear to be linked more to the information content they provide for constraining the size distributions than to their correlation with collision-coalescence rates.
An approximate method is presented for the synthesis of an optimal control of distributed parameter systems, based on a combination of the ideas of Lyapunov's direct method and those of Bellman's successive approximation in policy space. The method is such that the approximate equation is improved after each iteration in the sense of the performance index being used.
The objective is to simulate and optimize distributed cooperation among a network of robots tasked with cooperative excavation on an extra-terrestrial surface. Additionally to examine the concept of directed Emergence among a group of limited artificially intelligent agents. Emergence is the concept of achieving complex results from very simple rules or interactions. For example, in a termite mound each individual termite does not carry a blueprint of how to make their home in a global sense, but their interactions based strictly on local desires create a complex superstructure. Leveraging this Emergence concept applied to a simulation of cooperative agents (robots) will allow an examination of the success of non-directed group strategy achieving specific results. Specifically the simulation will be a testbed to evaluate population based robotic exploration and cooperative strategies while leveraging the evolutionary teamwork approach in the face of uncertainty about the environment and partial loss of sensors. Checking against a cost function and 'social' constraints will optimize cooperation when excavating a simulated tunnel. Agents will act locally with non-local results. The rules by which the simulated robots interact will be optimized to the simplest possible for the desired result, leveraging Emergence. Sensor malfunction and line of sight issues will be incorporated into the simulation. This approach falls under Swarm Robotics, a subset of robot control concerned with finding ways to control large groups of robots. Swarm Robotics often contains biologically inspired approaches, research comes from social insect observation but also data from among groups of herding, schooling, and flocking animals. Biomimetic algorithms applied to manned space exploration is the method under consideration for further study.
A particular type of three-impulse transfer between two circular orbits is analyzed. The possibility of three plane changes is recognized, and the problem is to optimally distribute these plane changes to minimize the sum of the individual impulses. Numerical difficulties and their solution are discussed. Numerical results obtained from a conjugate gradient technique are presented for both the case where the individual plane changes are unconstrained and for the case where they are constrained. Possibly not unexpectedly, multiple minima are found. The techniques presented could be extended to the finite burn case, but primarily the contents are addressed to preliminary mission design and vehicle sizing.
A method based on the Navier-Stokes equations was developed for determining analytically the three-dimensional unsteady flow patterns around oscillating wings. The Helmholz vorticity transport equations were discretized in three-dimensional finite element form from a variational formulation and integrated numerically. At each time step of the numerical integration the velocity field was calculated from the representation of the three-dimensional wing by a system of optimized distribution of vortices in space. During the numerical integration of the vorticity transport equations the time-dependent boundary conditions on the wing were specified as external constraint conditions. Examples of obtained results describing the three-dimensional unsteady flow around a wing were presented.