Solving optimal control problems with state constraints using nonlinear programming and simulation tools

This paper illustrates how nonlinear programming and simulation tools, which are available in packages such as MATLAB and SIMULINK, can easily be used to solve optimal control problems with state- and/or input-dependent inequality constraints. The method presented is illustrated with a model of a single-link manipulator. The method is suitable to be taught to advanced undergraduate and Master's level students in control engineering.

Optimal kinematic control of an autonomous underwater vehicle

This note investigates the motion control of an autonomous underwater vehicle (AUV). The AUV is modeled as a nonholonomic system as any lateral motion of a conventional, slender AUV is quickly damped out. The problem is formulated as an optimal kinematic control problem on the Euclidean Group of Motions SE(3), where the cost function to be minimized is equal to the integral of a quadratic function of the velocity components. An application of the Maximum Principle to this optimal control problem yields the appropriate Hamiltonian and the corresponding vector fields give the necessary conditions for optimality. For a special case of the cost function, the necessary conditions for optimality can be characterized more easily and we proceed to investigate its solutions. Finally, it is shown that a particular set of optimal motions trace helical paths. Throughout this note we highlight a particular case where the quadratic cost function is weighted in such a way that it equates to the Lagrangian (kinetic energy) of the AUV. For this case, the regular extremal curves are constrained to equate to the AUV's components of momentum and the resulting vector fields are the d'Alembert-Lagrange equations in Hamiltonian form.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Optimal discrimination and classification of THz spectra in the wavelet domain

In rapid scan Fourier transform spectrometry, we show that the noise in the wavelet coefficients resulting from the filter bank decomposition of the complex insertion loss function is linearly related to the noise power in the sample interferogram by a noise amplification factor. By maximizing an objective function composed of the power of the wavelet coefficients divided by the noise amplification factor, optimal feature extraction in the wavelet domain is performed. The performance of a classifier based on the output of a filter bank is shown to be considerably better than that of an Euclidean distance classifier in the original spectral domain. An optimization procedure results in a further improvement of the wavelet classifier. The procedure is suitable for enhancing the contrast or classifying spectra acquired by either continuous wave or THz transient spectrometers as well as for increasing the dynamic range of THz imaging systems. (C) 2003 Optical Society of America.

Comparison of subspace and ARX models of a waveguide's terahertz transient response after optimal wavelet filtering

A quasi-optical deembedding technique for characterizing waveguides is demonstrated using wide-band time-resolved terahertz spectroscopy. A transfer function representation is adopted for the description of the signal in the input and output port of the waveguides. The time-domain responses were discretized and the waveguide transfer function was obtained through a parametric approach in the z-domain after describing the system with an AutoRegressive with eXogenous input (ARX), as well as with a state-space model. Prior to the identification procedure, filtering was performed in the wavelet domain to minimize both signal distortion, as well as the noise propagating in the ARX and subspace models. The optimal filtering procedure used in the wavelet domain for the recorded time-domain signatures is described in detail. The effect of filtering prior to the identification procedures is elucidated with the aid of pole-zero diagrams. Models derived from measurements of terahertz transients in a precision WR-8 waveguide adjustable short are presented.

Optimal 3D surface reconstruction from a small number of conventional 2D X-ray images

This paper describes a method for reconstructing 3D frontier points, contour generators and surfaces of anatomical objects or smooth surfaces from a small number, e. g. 10, of conventional 2D X-ray images. The X-ray images are taken at different viewing directions with full prior knowledge of the X-ray source and sensor configurations. Unlike previous works, we empirically demonstrate that if the viewing directions are uniformly distributed around the object's viewing sphere, then the reconstructed 3D points automatically cluster closely on a highly curved part of the surface and are widely spread on smooth or flat parts. The advantage of this property is that the reconstructed points along a surface or a contour generator are not under-sampled or under-represented because surfaces or contours should be sampled or represented with more densely points where their curvatures are high. The more complex the contour's shape, the greater is the number of points required, but the greater the number of points is automatically generated by the proposed method. Given that the number of viewing directions is fixed and the viewing directions are uniformly distributed, the number and distribution of the reconstructed points depend on the shape or the curvature of the surface regardless of the size of the surface or the size of the object. The technique may be used not only in medicine but also in industrial applications.

Sex allocation and local mate competition in Old World non-pollinating fig wasps

The populations of many species are structured such that mating is not random and occurs between members of local patches. When patches are founded by a single female and all matings occur between siblings, brothers may compete with each other for matings with their sisters. This local mate competition (LMC) selects for a female-biased sex ratio, especially in species where females have control over offspring sex, as in the parasitic Hymenoptera. Two factors are predicted to decrease the degree of female bias: (1) an increase in the number of foundress females in the patch and (2) an increase in the fraction of individuals mating after dispersal from the natal patch. Pollinating fig wasps are well known as classic examples of species where all matings occur in the local patch. We studied non-pollinating fig wasps, which are more diverse than the pollinating fig wasps and also provide natural experimental groups of species with different male morphologies that are linked to different mating structures. In this group of wasps, species with wingless males mate in the local patch (i.e. the fig fruit) while winged male species mate after dispersal. Species with both kinds of male have a mixture of local and non-local mating. Data from 44 species show that sex ratios (defined as the proportion of males) are in accordance with theoretical predictions: wingless male species < wing-dimorphic male species < winged male species. These results are also supported by a formal comparative analysis that controls for phylogeny. The foundress number is difficult to estimate directly for non-pollinating fig wasps but a robust indirect method leads to the prediction that foundress number, and hence sex ratio, should increase with the proportion of patches occupied in a crop. This result is supported strongly across 19 species with wingless males, but not across 8 species with winged males. The mean sex ratios for species with winged males are not significantly different from 0.5, and the absence of the correlation observed across species with wingless males may reflect weak selection to adjust the sex ratio in species whose population mating structure tends not to be subdivided. The same relationship is also predicted to occur within species if individual females adjust their sex ratios facultatively. This final prediction was not supported by data from a wingless male species, a male wing-dimorphic species or a winged male species.

Learning and natal host influence host preference, handling time and sex allocation behaviour in a pupal parasitoid

The host choice and sex allocation decisions of a foraging female parasitoid will have an enormous influence on the life-history characteristics of her offspring. The pteromalid Pachycrepoideus vindemiae is a generalist idiobiont pupal parasitoid of many species of cyclorrhaphous Diptera. Wasps reared in Musca domestica were larger, had higher attack rates and greater male mating success than those reared in Drosophila melanogaster. In no-choice situations, naive female R vindemiae took significantly less time to accept hosts conspecific with their natal host. Parasitoids that emerged from M. domestica pupae spent similar amounts of time ovipositing in both D. melanogaster and M. domestica. Those parasitoids that had emerged from D. melanogaster spent significantly longer attacking M. domestica pupae. The host choice behaviour of female P. vindemiae was influenced by an interaction between natal host and experience. Female R vindemiae reared in M. domestica only showed a preference among hosts when allowed to gain experience attacking M. domestica, preferentially attacking that species. Similarly, female parasitoids reared on D. melanogaster only showed a preference among hosts when allowed to gain experience attacking D. melanogaster, again preferentially attacking that species. Wasp natal host also influenced sex allocation behaviour. While wasps from both hosts oviposited more females in the larger host, M. domestica, wasps that emerged from M. domestica had significantly more male-biased offspring sex ratios. These results indicate the importance of learning and natal host size in determining R vindemiae attack rates. mating success, host preference and sex allocation behaviour, all critical components of parasitoid fitness.

Quasi-optimal assimilation of satellite retrievals for incremental 4D-Var

An interface between satellite retrievals and the incremental version of the four-dimensional variational assimilation scheme is developed, making full use of the information content of satellite measurements. In this paper, expressions for the function that calculates simulated observations from model states (called “observation operator”), together with its tangent linear version and adjoint, are derived. Results from our work can be used for implementing a quasi-optimal assimilation of satellite retrievals (e.g., of atmospheric trace gases) in operational meteorological centres.

Solving complex optimal control problems at no cost with PSOPT

This paper introduces PSOPT, an open source optimal control solver written in C++. PSOPT uses pseudospectral and local discretizations, sparse nonlinear programming, automatic differentiation, and it incorporates automatic scaling and mesh refinement facilities. The software is able to solve complex optimal control problems including multiple phases, delayed differential equations, nonlinear path constraints, interior point constraints, integral constraints, and free initial and/or final times. The software does not require any non-free platform to run, not even the operating system, as it is able to run under Linux. Additionally, the software generates plots as well as LATEX code so that its results can easily be included in publications. An illustrative example is provided.