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Finite dimensional optimal nonlinear state estimators are derived for bilinear systems evolving on nilpotent and solvable Lie groups. These results are extended to other classes of systems involving polynomial nonlinearities. The concepts of exact differentials and path-independent integrals are used to derive optimal finite dimensional estimators for a further class of nonlinear systems.
Approximations to optimal nonlinear filters for problems involving signal and noise processes
A problem in optimal nonlinear control
The effect of stationary, finite-amplitude, linear and nonlinear optimal perturbations on the modal disturbance growth in a Mach 6 axisymmetric flow over a 7 deg. half-angle cone with 0:126 mm nose radius and 0:305 m length is investigated. The freestream parameters (M = 6, Re(exp 1) = 18 x 10(exp. 6) /m) are selected to match the flow conditions of a previous experiment in the VKI H3 hypersonic tunnel. Plane-marching parabolized stability equations are used in conjunction with a partial-differential equation based planar eigenvalue analysis to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode and first-mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone; however, subharmonic first-mode instabilities, which are destabilized by the presence of the streaks, do reach N = 6 near the end of the cone. The highest stabilization is observed at streak amplitudes of approximately 20 percent of the freestream velocity. Because the use of initial disturbance profiles based on linear optimal growth theory may yield suboptimal control in the context of nonlinear streaks, the computational predictions are extended to nonlinear optimal growth theory. Results show that by using nonlinearly optimal perturbation leads to slightly enhanced stabilization of plane Mack mode disturbances as well as reduced destabilization of subharmonic first-mode disturbances.
Nonlinear optimal guidance algorithms for space missions
Some computational issues associated with the calcualtion of optimal feedback controls for nonlinear systems in a tensor setting are described. The specific issues addressed pertain to the combinatorial nature of the loading of the elements into tensors used to represent the system, cost, and feedback, and the subsequent calculations involving these elements. Particular attention is given to: the symmetric tensor algebra which is a natural setting for representing polynomials; the conversions between symmetric and nonsymmetric tensors; the general nature of the calculations required; and the solution equation for nonlinear optimal feedback control. It is concluded that nonlinear tensor feedback can improve performance both in terms of system responses and in terms of system stability region.
An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, and approximate problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second, the Ginzburg-Landau equations of superconductivity, and the third, the Navier-Stokes equations for incompressible, viscous flows.
Multidisciplinary design optimization (MDO) gives rise to nonlinear optimization problems characterized by a large number of constraints that naturally occur in blocks. We propose a class of multilevel optimization methods motivated by the structure and number of constraints and by the expense of the derivative computations for MDO. The algorithms are an extension to the nonlinear programming problem of the successful class of local Brown-Brent algorithms for nonlinear equations. Our extensions allow the user to partition constraints into arbitrary blocks to fit the application, and they separately process each block and the objective function, restricted to certain subspaces. The methods use trust regions as a globalization strategy, and they have been shown to be globally convergent under reasonable assumptions. The multilevel algorithms can be applied to all classes of MDO formulations. Multilevel algorithms for solving nonlinear systems of equations are a special case of the multilevel optimization methods. In this case, they can be viewed as a trust-region globalization of the Brown-Brent class.
Nonlinear optimization problems that are encountered in science and industry are examined. A method of projecting the gradient vector onto a set of linear contraints is developed, and a program that uses this method is presented. The algorithm that generates this projection matrix is based on the Gram-Schmidt method and overcomes some of the objections to the Rosen projection method.
Method for reducing sensitivity of optimal nonlinear systems to parameter uncertainty
The formulation and basic theorems of singularly perturbed nonlinear optimal control problems are discussed and the solution of such problems using matched asymptotic expansions is described.
Optimal nonlinear feedback control derived from quartic and higher order performance criteria
Open-loop solutions of dynamical optimization problems can be numerically computed usingexisting software packages. The computed time histories of the state and control variables, formultiple sets of end conditions can then be used to train a neural network to 'recognize' the optimal,nonlinear feedback relation between the states and controls of the system. The 'learned' network canthen be used to output an approximate optimal control given a full set (or a partial set) of measuredsystem states. With simple neural networks, we have successfully demonstrated the efficacy of theproposed approach using a minimum-time orbit injection problem. The usefulness and limitations ofthis novel approach on real-life optimal guidance and control problems, with many state and control variables as well as path inequality constraints, remain to be seen.
Generalized Lur coordinates used in canonical forms for controllable systems, with applications to optimal nonlinear feedback
The general code described is capable of solving difficult nonlinear optimal control problems by using finite elements and a symbolic manipulator. Quick and accurate solutions are obtained with a minimum for user interaction. Since no user programming is required for most problems, there are tremendous savings to be gained in terms of time and money.
A relatively large amount of computer time was used for the calculation of a optimal trajectory, but it is subject to reduction with moderate effort. The Deterministic, Nonlinear, Optimal Control algorithm yielded excellent aircraft performance in trajectory tracking for the given microburst. It did so by varying the angle of attack to counteract the lift effects of microburst induced airspeed variations. Throttle saturation and aerodynamic stall limits were not a problem for the case considered, proving that the aircraft's performance capabilities were not violated by the given wind field. All closed loop control laws previously considered performed very poorly in comparison, and therefore do not come near to taking full advantage of aircraft performance.
For optimization problems associated with engineering design, parameter estimation, image reconstruction, and other optimization/simulation applications, low accuracy function and gradient values are frequently much less expensive to obtain than high accuracy values. Here, researchers investigate the computational performance of trust region methods for nonlinear optimization when high accuracy evaluations are unavailable or prohibitively expensive, and confirm earlier theoretical predictions when the algorithm is convergent even with relative gradient errors of 0.5 or more. The proper choice of the amount of accuracy to use in function and gradient evaluations can result in orders-of-magnitude savings in computational cost.
We present a simple multi-dimensional exhaustive search method to obtain, in a reasonable time, the optimal solution of a nonlinear programming problem. It is more relevant in the present day non-mainframe computing scenario where an estimated 95% computing resources remains unutilized and computing speed touches petaflops. While the processor speed is doubling every 18 months, the band width is doubling every 12 months, and the hard disk space is doubling every 9 months. A randomized search algorithm or, equivalently, an evolutionary search method is often used instead of an exhaustive search algorithm. The reason is that a randomized approach is usually polynomial-time, i.e., fast while an exhaustive search method is exponential-time i.e., slow. We discuss the increasing importance of exhaustive search in optimization with the steady increase of computing power for solving many real-world problems of reasonable size. We also discuss the computational error and complexity of the search algorithm focusing on the fact that no measuring device can usually measure a quantity with an accuracy greater than 0.005%. We stress the fact that the quality of solution of the exhaustive search - a deterministic method - is better than that of randomized search. In 21 st century computing environment, exhaustive search cannot be left aside as an untouchable and it is not always exponential. We also describe a possible application of these algorithms in improving the efficiency of solar cells - a real hot topic - in the current energy crisis. These algorithms could be excellent tools in the hands of experimentalists and could save not only large amount of time needed for experiments but also could validate the theory against experimental results fast.