Optimization of the Integrated Guidance and Control for a Dual Aerodynamic Control Missile

Las futuras misiones para misiles aire-aire operando dentro de la atmósfera requieren la interceptación de blancos a mayores velocidades y más maniobrables, incluyendo los esperados vehículos aéreos de combate no tripulados. La intercepción tiene que lograrse desde cualquier ángulo de lanzamiento. Una de las principales discusiones en la tecnología de misiles en la actualidad es cómo satisfacer estos nuevos requisitos incrementando la capacidad de maniobra del misil y en paralelo, a través de mejoras en los métodos de guiado y control modernos. Esta Tesis aborda estos dos objetivos simultáneamente, al proponer un diseño integrando el guiado y el control de vuelo (autopiloto) y aplicarlo a misiles con control aerodinámico simultáneo en canard y cola. Un primer avance de los resultados obtenidos ha sido publicado recientemente en el Journal of Aerospace Engineering, en Abril de 2015, [Ibarrondo y Sanz-Aranguez, 2015]. El valor del diseño integrado obtenido es que permite al misil cumplir con los requisitos operacionales mencionados empleando únicamente control aerodinámico. El diseño propuesto se compara favorablemente con esquemas más tradicionales, consiguiendo menores distancias de paso al blanco y necesitando de menores esfuerzos de control incluso en presencia de ruidos. En esta Tesis se demostrará cómo la introducción del doble mando, donde tanto el canard como las aletas de cola son móviles, puede mejorar las actuaciones de un misil existente. Comparado con un misil con control en cola, el doble control requiere sólo introducir dos servos adicionales para accionar los canards también en guiñada y cabeceo. La sección de cola será responsable de controlar el misil en balanceo mediante deflexiones diferenciales de los controles. En el caso del doble mando, la complicación añadida es que los vórtices desprendidos de los canards se propagan corriente abajo y pueden incidir sobre las superficies de cola, alterando sus características de control. Como un primer aporte, se ha desarrollado un modelo analítico completo para la aerodinámica no lineal de un misil con doble control, incluyendo la caracterización de este efecto de acoplamiento aerodinámico. Hay dos modos de funcionamiento en picado y guiñada para un misil de doble mando: ”desviación” y ”opuesto”. En modo ”desviación”, los controles actúan en la misma dirección, generando un cambio inmediato en la sustentación y produciendo un movimiento de translación en el misil. La respuesta es rápida, pero en el modo ”desviación” los misiles con doble control pueden tener dificultades para alcanzar grandes ángulos de ataque y altas aceleraciones laterales. Cuando los controles actúan en direcciones opuestas, el misil rota y el ángulo de ataque del fuselaje se incrementa para generar mayores aceleraciones en estado estacionario, aunque el tiempo de respuesta es mayor. Con el modelo aerodinámico completo, es posible obtener una parametrización dependiente de los estados de la dinámica de corto periodo del misil. Debido al efecto de acoplamiento entre los controles, la respuesta en bucle abierto no depende linealmente de los controles. El autopiloto se optimiza para obtener la maniobra requerida por la ley de guiado sin exceder ninguno de los límites aerodinámicos o mecánicos del misil. Una segunda contribución de la tesis es el desarrollo de un autopiloto con múltiples entradas de control y que integra la aerodinámica no lineal, controlando los tres canales de picado, guiñada y cabeceo de forma simultánea. Las ganancias del autopiloto dependen de los estados del misil y se calculan a cada paso de integración mediante la resolución de una ecuación de Riccati de orden 21x21. Las ganancias obtenidas son sub-óptimas, debido a que una solución completa de la ecuación de Hamilton-Jacobi-Bellman no puede obtenerse de manera práctica, y se asumen ciertas simplificaciones. Se incorpora asimismo un mecanismo que permite acelerar la respuesta en caso necesario. Como parte del autopiloto, se define una estrategia para repartir el esfuerzo de control entre el canard y la cola. Esto se consigue mediante un controlador aumentado situado antes del bucle de optimización, que minimiza el esfuerzo total de control para maniobrar. Esta ley de alimentación directa mantiene al misil cerca de sus condiciones de equilibrio, garantizando una respuesta transitoria adecuada. El controlador no lineal elimina la respuesta de fase no-mínima característica de la cola. En esta Tesis se consideran dos diseños para el guiado y control, el control en Doble-Lazo y el control Integrado. En la aproximación de Doble-Lazo, el autopiloto se sitúa dentro de un bucle interior y se diseña independientemente del guiado, que conforma el bucle más exterior del control. Esta estructura asume que existe separación espectral entre los dos, esto es, que los tiempos de respuesta del autopiloto son mucho mayores que los tiempos característicos del guiado. En el estudio se combina el autopiloto desarrollado con una ley de guiado óptimo. Los resultados obtenidos demuestran que se consiguen aumentos muy importantes en las actuaciones frente a misiles con control canard o control en cola, y que la interceptación, cuando se lanza cerca del curso de colisión, se consigue desde cualquier ángulo alrededor del blanco. Para el misil de doble mando, la estrategia óptima resulta en utilizar el modo de control opuesto en la aproximación al blanco y utilizar el modo de desviación justo antes del impacto. Sin embargo la lógica de doble bucle no consigue el impacto cuando hay desviaciones importantes con respecto al curso de colisión. Una de las razones es que parte de la demanda de guiado se pierde, ya que el misil solo es capaz de modificar su aceleración lateral, y no tiene control sobre su aceleración axial, a no ser que incorpore un motor de empuje regulable. La hipótesis de separación mencionada, y que constituye la base del Doble-Bucle, puede no ser aplicable cuando la dinámica del misil es muy alta en las proximidades del blanco. Si se combinan el guiado y el autopiloto en un único bucle, la información de los estados del misil está disponible para el cálculo de la ley de guiado, y puede calcularse la estrategia optima de guiado considerando las capacidades y la actitud del misil. Una tercera contribución de la Tesis es la resolución de este segundo diseño, la integración no lineal del guiado y del autopiloto (IGA) para el misil de doble control. Aproximaciones anteriores en la literatura han planteado este sistema en ejes cuerpo, resultando en un sistema muy inestable debido al bajo amortiguamiento del misil en cabeceo y guiñada. Las simplificaciones que se tomaron también causan que el misil se deslice alrededor del blanco y no consiga la intercepción. En nuestra aproximación el problema se plantea en ejes inerciales y se recurre a la dinámica de los cuaterniones, eliminado estos inconvenientes. No se limita a la dinámica de corto periodo del misil, porque se construye incluyendo de modo explícito la velocidad dentro del bucle de optimización. La formulación resultante en el IGA es independiente de la maniobra del blanco, que sin embargo se ha de incluir en el cálculo del modelo en Doble-bucle. Un típico inconveniente de los sistemas integrados con controlador proporcional, es el problema de las escalas. Los errores de guiado dominan sobre los errores de posición del misil y saturan el controlador, provocando la pérdida del misil. Este problema se ha tratado aquí con un controlador aumentado previo al bucle de optimización, que define un estado de equilibrio local para el sistema integrado, que pasa a actuar como un regulador. Los criterios de actuaciones para el IGA son los mismos que para el sistema de Doble-Bucle. Sin embargo el problema matemático resultante es muy complejo. El problema óptimo para tiempo finito resulta en una ecuación diferencial de Riccati con condiciones terminales, que no puede resolverse. Mediante un cambio de variable y la introducción de una matriz de transición, este problema se transforma en una ecuación diferencial de Lyapunov que puede resolverse mediante métodos numéricos. La solución resultante solo es aplicable en un entorno cercano del blanco. Cuando la distancia entre misil y blanco es mayor, se desarrolla una solución aproximada basada en la solución de una ecuación algebraica de Riccati para cada paso de integración. Los resultados que se han obtenido demuestran, a través de análisis numéricos en distintos escenarios, que la solución integrada es mejor que el sistema de Doble-Bucle. Las trayectorias resultantes son muy distintas. El IGA preserva el guiado del misil y consigue maximizar el uso de la propulsión, consiguiendo la interceptación del blanco en menores tiempos de vuelo. El sistema es capaz de lograr el impacto donde el Doble-Bucle falla, y además requiere un orden menos de magnitud en la cantidad de cálculos necesarios. El efecto de los ruidos radar, datos discretos y errores del radomo se investigan. El IGA es más robusto, resultando menos afectado por perturbaciones que el Doble- Bucle, especialmente porque el núcleo de optimización en el IGA es independiente de la maniobra del blanco. La estimación de la maniobra del blanco es siempre imprecisa y contaminada por ruido, y degrada la precisión de la solución de Doble-Bucle. Finalmente, como una cuarta contribución, se demuestra que el misil con guiado IGA es capaz de realizar una maniobra de defensa contra un blanco que ataque por su cola, sólo con control aerodinámico. Las trayectorias estudiadas consideran una fase pre-programada de alta velocidad de giro, manteniendo siempre el misil dentro de su envuelta de vuelo. Este procedimiento no necesita recurrir a soluciones técnicamente más complejas como el control vectorial del empuje o control por chorro para ejecutar esta maniobra. En todas las demostraciones matemáticas se utiliza el producto de Kronecker como una herramienta practica para manejar las parametrizaciones dependientes de variables, que resultan en matrices de grandes dimensiones. ABSTRACT Future missions for air to air endo-atmospheric missiles require the interception of targets with higher speeds and more maneuverable, including forthcoming unmanned supersonic combat vehicles. The interception will need to be achieved from any angle and off-boresight launch conditions. One of the most significant discussions in missile technology today is how to satisfy these new operational requirements by increasing missile maneuvering capabilities and in parallel, through the development of more advanced guidance and control methods. This Thesis addresses these two objectives by proposing a novel optimal integrated guidance and autopilot design scheme, applicable to more maneuverable missiles with forward and rearward aerodynamic controls. A first insight of these results have been recently published in the Journal of Aerospace Engineering in April 2015, [Ibarrondo and Sanz-Aránguez, 2015]. The value of this integrated solution is that it allows the missile to comply with the aforementioned requirements only by applying aerodynamic control. The proposed design is compared against more traditional guidance and control approaches with positive results, achieving reduced control efforts and lower miss distances with the integrated logic even in the presence of noises. In this Thesis it will be demonstrated how the dual control missile, where canard and tail fins are both movable, can enhance the capabilities of an existing missile airframe. Compared to a tail missile, dual control only requires two additional servos to actuate the canards in pitch and yaw. The tail section will be responsible to maintain the missile stabilized in roll, like in a classic tail missile. The additional complexity is that the vortices shed from the canard propagate downstream where they interact with the tail surfaces, altering the tail expected control characteristics. These aerodynamic phenomena must be properly described, as a preliminary step, with high enough precision for advanced guidance and control studies. As a first contribution we have developed a full analytical model of the nonlinear aerodynamics of a missile with dual control, including the characterization of this cross-control coupling effect. This development has been produced from a theoretical model validated with reliable practical data obtained from wind tunnel experiments available in the scientific literature, complement with computer fluid dynamics and semi-experimental methods. There are two modes of operating a missile with forward and rear controls, ”divert” and ”opposite” modes. In divert mode, controls are deflected in the same direction, generating an increment in direct lift and missile translation. Response is fast, but in this mode, dual control missiles may have difficulties in achieving large angles of attack and high level of lateral accelerations. When controls are deflected in opposite directions (opposite mode) the missile airframe rotates and the body angle of attack is increased to generate greater accelerations in steady-state, although the response time is larger. With the aero-model, a state dependent parametrization of the dual control missile short term dynamics can be obtained. Due to the cross-coupling effect, the open loop dynamics for the dual control missile is not linearly dependent of the fin positions. The short term missile dynamics are blended with the servo system to obtain an extended autopilot model, where the response is linear with the control fins turning rates, that will be the control variables. The flight control loop is optimized to achieve the maneuver required by the guidance law without exceeding any of the missile aerodynamic or mechanical limitations. The specific aero-limitations and relevant performance indicators for the dual control are set as part of the analysis. A second contribution of this Thesis is the development of a step-tracking multi-input autopilot that integrates non-linear aerodynamics. The designed dual control missile autopilot is a full three dimensional autopilot, where roll, pitch and yaw are integrated, calculating command inputs simultaneously. The autopilot control gains are state dependent, and calculated at each integration step solving a matrix Riccati equation of order 21x21. The resulting gains are sub-optimal as a full solution for the Hamilton-Jacobi-Bellman equation cannot be resolved in practical terms and some simplifications are taken. Acceleration mechanisms with an λ-shift is incorporated in the design. As part of the autopilot, a strategy is defined for proper allocation of control effort between canard and tail channels. This is achieved with an augmented feed forward controller that minimizes the total control effort of the missile to maneuver. The feedforward law also maintains the missile near trim conditions, obtaining a well manner response of the missile. The nonlinear controller proves to eliminate the non-minimum phase effect of the tail. Two guidance and control designs have been considered in this Thesis: the Two- Loop and the Integrated approaches. In the Two-Loop approach, the autopilot is placed in an inner loop and designed separately from an outer guidance loop. This structure assumes that spectral separation holds, meaning that the autopilot response times are much higher than the guidance command updates. The developed nonlinear autopilot is linked in the study to an optimal guidance law. Simulations are carried on launching close to collision course against supersonic and highly maneuver targets. Results demonstrate a large boost in performance provided by the dual control versus more traditional canard and tail missiles, where interception with the dual control close to collision course is achieved form 365deg all around the target. It is shown that for the dual control missile the optimal flight strategy results in using opposite control in its approach to target and quick corrections with divert just before impact. However the Two-Loop logic fails to achieve target interception when there are large deviations initially from collision course. One of the reasons is that part of the guidance command is not followed, because the missile is not able to control its axial acceleration without a throttleable engine. Also the separation hypothesis may not be applicable for a high dynamic vehicle like a dual control missile approaching a maneuvering target. If the guidance and autopilot are combined into a single loop, the guidance law will have information of the missile states and could calculate the most optimal approach to the target considering the actual capabilities and attitude of the missile. A third contribution of this Thesis is the resolution of the mentioned second design, the non-linear integrated guidance and autopilot (IGA) problem for the dual control missile. Previous approaches in the literature have posed the problem in body axes, resulting in high unstable behavior due to the low damping of the missile, and have also caused the missile to slide around the target and not actually hitting it. The IGA system is posed here in inertial axes and quaternion dynamics, eliminating these inconveniences. It is not restricted to the missile short term dynamic, and we have explicitly included the missile speed as a state variable. The IGA formulation is also independent of the target maneuver model that is explicitly included in the Two-loop optimal guidance law model. A typical problem of the integrated systems with a proportional control law is the problem of scales. The guidance errors are larger than missile state errors during most of the flight and result in high gains, control saturation and loss of control. It has been addressed here with an integrated feedforward controller that defines a local equilibrium state at each flight point and the controller acts as a regulator to minimize the IGA states excursions versus the defined feedforward state. The performance criteria for the IGA are the same as in the Two-Loop case. However the resulting optimization problem is mathematically very complex. The optimal problem in a finite-time horizon results in an irresoluble state dependent differential Riccati equation with terminal conditions. With a change of variable and the introduction of a transition matrix, the equation is transformed into a time differential Lyapunov equation that can be solved with known numerical methods in real time. This solution results range limited, and applicable when the missile is in a close neighborhood of the target. For larger ranges, an approximate solution is used, obtained from solution of an algebraic matrix Riccati equation at each integration step. The results obtained show, by mean of several comparative numerical tests in diverse homing scenarios, than the integrated approach is a better solution that the Two- Loop scheme. Trajectories obtained are very different in the two cases. The IGA fully preserves the guidance command and it is able to maximize the utilization of the missile propulsion system, achieving interception with lower miss distances and in lower flight times. The IGA can achieve interception against off-boresight targets where the Two- Loop was not able to success. As an additional advantage, the IGA also requires one order of magnitude less calculations than the Two-Loop solution. The effects of radar noises, discrete radar data and radome errors are investigated. IGA solution is robust, and less affected by radar than the Two-Loop, especially because the target maneuvers are not part of the IGA core optimization loop. Estimation of target acceleration is always imprecise and noisy and degrade the performance of the two-Loop solution. The IGA trajectories are such that minimize the impact of radome errors in the guidance loop. Finally, as a fourth contribution, it is demonstrated that the missile with IGA guidance is capable of performing a defense against attacks from its rear hemisphere, as a tail attack, only with aerodynamic control. The studied trajectories have a preprogrammed high rate turn maneuver, maintaining the missile within its controllable envelope. This solution does not recur to more complex features in service today, like vector control of the missile thrust or side thrusters. In all the mathematical treatments and demonstrations, the Kronecker product has been introduced as a practical tool to handle the state dependent parametrizations that have resulted in very high order matrix equations.

An O(N log N) algorithm for shape modeling.

We present a shape-recovery technique in two dimensions and three dimensions with specific applications in modeling anatomical shapes from medical images. This algorithm models extremely corrugated structures like the brain, is topologically adaptable, and runs in O(N log N) time, where N is the total number of points in the domain. Our technique is based on a level set shape-recovery scheme recently introduced by the authors and the fast marching method for computing solutions to static Hamilton-Jacobi equations.

A fast marching level set method for monotonically advancing fronts.

A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

Sistemas integrables discretos, polinomios matriciales ortogonales y problemas de Riemann-Hilbert

El propósito de esta tesis doctoral es el estudio de la conexión, mediante el problema de Riemann-Hilbert, entre sistemas discretos y la teoría de polinomios matriciales ortogonales. La investigación de los modelos integrables se originó en la Mecánica Clásica, en relación a la resolución de las ecuaciones de Newton [2]. Los trabajos de Liouville, Hamilton, Jacobi y otros sentaron las bases de los sistemas integrables como prototipos modelos resolubles por cuadraturas, v.g., por integración directa [7]. Hay una cantidad importante de investigación dedicada a los aspectos geométricos de los sistemas clásicos integrables y superintegrables [66], [82], especialmente en relación a la separación de variables de la ecuación de Hamilton-Jacobi [75]. Fue la aplicación, en la segunda mitad del siglo pasado, de la transformada espectral inversa para la resolución del problema de Cauchy de la ecuación de Korteweg-de Vries [42, 43] la que marcó el inicio de una nueva etapa en este campo, el del estudio de sistemas integrables con un número infinito de grados de libertad, que generalmente se expresan en términos de jerarquías de ecuaciones no lineales en derivadas parciales. Particularmente reseñable, por su aplicación en la hidrodinámica y en la óptica cuántica, es la aparición de las soluciones a un número de solitones arbitrario. En las últimas tres décadas ha habido un importante interés por el estudio de modelos discretos, v.g., sistemas dinámicos de nidos en un retículo de puntos, y expresados en términos de ecuaciones no lineales en diferencia parciales. Muchas de las técnicas encontradas en el mundo continuo se extendieron a este nuevo contexto discreto. Hay dos razones fundamentales para este interés...

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.

Optimal investment under stochastic volatility and power type utility function

2000 Mathematics Subject Classification: 37F21, 70H20, 37L40, 37C40, 91G80, 93E20.

Desenvolvimento de equações generalizadas para estimativa da coordenação motora em crianças e adolescentes portadores de deficiencia mental

Universidade Estadual de Campinas . Faculdade de Educação Física

Fracionamento dos carboidratos pelas equações do Cornell Net Carbohydrate and Protein System de três cultivares de girassol na presença ou não de irrigação

Objetivou-se quantificar as frações de carboidratos pelas equações do Cornell Net Carbohydrate and Protein System (CNCPS) de três cultivares de girassol (Helianthus annuus L.) cultivados na presença ou não de irrigação. A utilização de uma preparação fibrosa, denominada parede celular (PC), nas equações da CNCPS, em substituição à fibra em detergente neutro (FDN) não promoveu diferenças nas frações de carboidratos B1 e C, mas influenciou as frações A e B2. Como os valores da fração B1, obtidos pelo modelo CNCPS foram menores que os teores de amido e pectina determinados em laboratório, supõe-se que a pectina e outros oligossacarídeos da parede celular, solubilizados pela solução de detergente neutro (fibra solúvel), nunca fizeram parte da fração B1, e sim da fração A. Apesar de os carboidratos da fibra solúvel apresentarem elevadas taxas de degradação, não parece adequada a caracterização da fibra solúvel na fração A. Parece mais adequado que a fibra solúvel (que inclui a pectina) seja alocada a uma fração exclusivamente sua, que pode ser a fração B2, e que seja criada uma nova fração, a B3, para os carboidratos digeríveis da parede celular. Assim, a fração B1 seria composta apenas de amido. A equação da fração C, que estima os carboidratos indigeríveis da parede celular, pode ser simplificada, relacionando a fração indigerível ao teor de lignina na matéria seca, e não à FDN isenta de cinzas e proteína, como atualmente utilizado. Esta proposta tem implicações práticas, uma vez que a fração indigerível da parede celular tem sido expressa em relação à FDN, e não na MS, com base no fato de que os efeitos inibitórios da lignina ocorrem sobre os componentes fibrosos da parede celular vegetal, e não sobre o conteúdo celular.

Antecedentes do relacionamento e da performance em empresas da cadeia de suprimentos: estruturação e aplicação de modelos de equações estruturais

The relationship between companies is an important issue in the management of supply chains. Several aspects relating to the flow and exchange of information along the chain are considered as having a decisive influence on the success of this relationship. The main objective of this work was to structured and test models that link aspects of this nature with performance and the purchaser-supplier relationship in the supply chain. Aspects relevant to communication and the use do IT in relationships between companies were investigated. The importance of performance in this relationship was also investigated. The research were based on empirical data obtained by means of structural equation modeling. The results show that some aspects contribute in a significant way to the success of this relationship while others that, a priori, are considered important make no contribution.

Hughie Hamilton, Alex McDonald during Ky visit demonstration

Group of people including Hughie Hamilton, Alex McDonald and Vilma Ward during visit of former South Vietnamese vice president Nguyen Cao Ky to Brisbane, Australia in January 1967.

Maximal sets of Hamilton cycles in Kn,n

In this article, we prove that there exists a maximal set of m Hamilton cycles in K-n,K-n if and only if n/4 < m less than or equal to n/2. (C) 2000 John Wiley & Sons, Inc.

On the Hamilton-Waterloo problem

The Hamilton-Waterloo problem asks for a 2-factorisation of K-v in which r of the 2-factors consist of cycles of lengths a(1), a(2),..., a(1) and the remaining s 2-factors consist of cycles of lengths b(1), b(2),..., b(u) (where necessarily Sigma(i)(=1)(t) a(i) = Sigma(j)(=1)(u) b(j) = v). In thus paper we consider the Hamilton-Waterloo problem in the case a(i) = m, 1 less than or equal to i less than or equal to t and b(j) = n, 1 less than or equal to j less than or equal to u. We obtain some general constructions, and apply these to obtain results for (m, n) is an element of {(4, 6)1(4, 8), (4, 16), (8, 16), (3, 5), (3, 15), (5, 15)}.