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PAGE 1 1 OPTIMIZED TRAJECTORY TRACKING FOR A HYPERSONIC VEHICLE IN VERTICAL FLIGHT By ERIK J. KITTRELL A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Erik J. Kittrell PAGE 3 3 To my loved ones PAGE 4 4 ACKNOWLEDGMENTS Foremost, I would like to thank my advisor Dr. Richard Lind for providing me the opportunity to study and research at the University of Florida Research and Engineering Education Facility (REEF) Without his guidance and insights this would not have been possible. Thanks are also extended to Dr. Crystal Pasiliao, who provided the bas is background and an abundance of support for this pro ject. I would also like to thank my colleagues and friends at the REEF and else ware, I am very appreciative. Without your support, both scholastic and social, this effort would have been much more difficult. Lastly, I acknowledge my family, whose constant love and support helped me overcome the challenges that arose both in scholastics and life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................ 4 LIST OF TABLES ........................................................................................................... 6 LIST OF FIGURES ........................................................................................................ 7 ABSTRACT .................................................................................................................... 9 CHAPTER 1 INTRODUCTION ................................................................................................... 11 Motivation for Research ......................................................................................... 11 Previous Applicable Research ............................................................................... 12 2 HYPERSONIC MODEL ......................................................................................... 14 Bolender Longitudinal Hypersonic Vehicle Model .................................................. 14 Modified Bolender Model ....................................................................................... 14 Trim Conditions ..................................................................................................... 18 3 CONTROL SYSTEM DESIGN AND RESULTS ..................................................... 22 General Pseudospectral Optimal Control Software (GPOPS) ................................ 22 GPOPS Example Results ................................................................................ 24 Maximum Optimized Vehicle Performance ...................................................... 28 Processing Time Verses Optimality ................................................................ 31 Open Loop System ................................................................................................ 35 Open Loop System Example ........................................................................... 36 Closed Loop System ............................................................................................. 40 Closed Loop System Example ........................................................................ 41 Effect of Optimizer Iterations on Closed Loop Performance ............................ 46 Complete System Performance ............................................................................. 48 4 CONCLUSIONS AND FUTURE WORK ................................................................ 50 Summary ............................................................................................................... 50 Future Work ........................................................................................................... 53 REFERENCE LIST ...................................................................................................... 54 BIOGRAPHICAL SKETCH ........................................................................................... 57 PAGE 6 6 LIST OF TABLES Table page 2 1 Trimmed flight state values produced by the model for full altitude and half altitude linearizations. ................................ ................................ ........................ 19 3 1 Maximum and minimum limits of states and inputs. ................................ ........... 23 3 2 Example initial state and control surface values. ................................ ............... 24 3 3 Maximum performance for changes in initial altitude. ................................ ........ 30 3 4 Maximum performance for changes in vehicle Mach number. ........................... 30 3 5 Maximum performance for changes in linearization altitude. ............................. 30 3 6 Optimization completion time and time stamp counts for various mesh iteration values. ................................ ................................ ................................ 32 3 7 Divergence time for different mesh interations. ................................ .................. 39 3 8 Closed loop response performance parameters for various mesh iteration amounts. ................................ ................................ ................................ ........... 48 3 9 Example full system performance results for various requested horizontal displacements. ................................ ................................ ................................ .. 49 PAGE 7 7 LIST OF FIGURES Figure page 2 1 State space control input gains for vehicle velocity for a range of altitudes. ....... 17 2 2 State space control input gains for vehicle pitch rate for a range of altitudes. .... 18 2 3 State spac e control input gains for vehicle angle of attack for a range of altitudes. ................................ ................................ ................................ ............ 18 2 4 Trimmed angle of attack for a range of altitud es. ................................ ............... 20 2 5 Trimmed pitch angle for a range of altitudes. ................................ ..................... 20 2 6 Trimmed control surface deflection angles for a range of altitudes. ................... 21 3 1 Example optimized control deflection inputs. ................................ ..................... 25 3 2 Example optimized horizontal displacement trajectory. ................................ ..... 26 3 3 Example optimized angle of attack. ................................ ................................ ... 26 3 4 Example optimized pitch angle. ................................ ................................ ......... 27 3 5 Example optimized altitude trajectory. ................................ ............................... 28 3 6 Example optimized downward velocity. ................................ ............................. 28 3 7 Comparison of optimized control deflection over time plots for various mesh iteration amounts. ................................ ................................ .............................. 33 3 8 Comparison of optimized ground location over time plots for various mesh iteration amounts. ................................ ................................ .............................. 34 3 9 Comparison of optimized angle of attack ov er time plots for various mesh iteration amounts. ................................ ................................ .............................. 35 3 10 Open loop system diagram. ................................ ................................ ............... 36 3 11 Comparison of optimized and interpolated control deflection inputs. ................. 37 3 12 Open loop and optimized horizontal deflection response. ................................ .. 37 3 13 Open loop and optimized angle of attack response. ................................ .......... 38 3 14 Open loop and optimized pitch response. ................................ .......................... 38 3 15 Closed loop system diagram. ................................ ................................ ............ 41 PAGE 8 8 3 16 Closed loop and optimized control system deflections. ................................ ...... 43 3 17 Wind gust control deflection plot. ................................ ................................ ....... 44 3 18 Closed loop and optimized horizontal displacement response. .......................... 44 3 19 Closed loop and optimized angle of attack res ponse. ................................ ........ 45 3 20 Closed loop and optimized pitch response. ................................ ....................... 46 3 21 Comparison of closed loop control deflection responses for various values of mesh iteration. ................................ ................................ ................................ ... 47 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science OPTIMIZED TRAJECTORY TRACKING FOR A HYPERSONIC VEHICLE IN VERTICAL FLIGHT By Erik J. Kittrell August 2013 Chair: Rick Lind Major: Aerospace Engineering Advancements in the development of hypersonic vehicles has led the United States Air Force to show interest in developing a high speed global strike weapon. Using the kinetic energy associated wi th hypersonic flight, an impactor style conventional weapon could be deployed and hit any target location on the globe in under two hours. This work is an initial study into the ability of a hypersonic vehicle to perform a horizontal displacement course c orrection in the terminal stage of flight. A linear, longitudinal, straight and level hypersonic vehicle model was modified for vertical flight and then used to track an optimized horizontal displacement trajectory using a closed loop system. System perf ormance of horizontal distance from target location, final angle of attack and pitch, and final flight path angle was simulated and compared. The ability of the system to handle external disturbances and internal system sensor errors was also taken into a ccount in this study. The simulation results shown in this study detail how the model linearization point changes the state space dynamics equation and the initial conditions affect the performance of the optimizer. The change of mesh size of the optimal trajectory were also shown to affect the performance of the PAGE 10 10 system at the cost of optimization processing time. The closed loop response results in this study successfully show the ability of a hypersonic vehicle to track an optimized horizontal displace ment trajectory in vertical flight with external and internal disturbances present. PAGE 11 11 CHAPTER 1 INTRODUCTION Motivation for Research Increasingly advanced development of hypersonic vehicles has led the United State s Air Force to show interest in developing a hypersonic impactor weapon. 51 A scramjet demonstrator ( 1 ) and the X 43A (2 ) could be used as rapid strike weapon s with the ability to hit any location on the globe in under two hours. One of the recurring issues with hypersonic flight is the difficulty of controlling the vehicle with the precision required to be used as a weapon. This paper details an initial study into the controllability and performance of a hypersonic vehicle in vertical flight. The study will focus on the ability of a hypersonic vehicle to perform a horizontal ground location displacement while tracking an optimized terminal trajectory in orde r to point of impact with a target location before impact. The ability of the vehicle to orient itself at into a vertical orientation and flight path angle will also be studied. Due to the lack of a clearly defined vehicle and mission profile, this study will use a linear longitudinal model developed by Bolender (22) that uses a wedge shaped vehicle with a body integrated scramjet engine and has elevators and canards for control surfaces. It will be assumed that the hypersonic impactor is in nose down fli ght with zero horizontal motion at the start of the manuver and maintains constant velocity throughout. PAGE 12 12 Previous Applicable Research The dynamics and controls of atmospheric hypersonic flight are very difficult to model. Air breathing hypersonic ve hicles are subject to engine and flight dynamics couplings, and the effects of bending body modes that can vary due to aerothermoelastic effects. Chavez and Schmidt (3), and Bolender and Doman (4, 5) developed simple, longitudinal dynamics models for vehi cles similar to the one being used in this study. Later on, the Bolender and Doman model was updated to include bending modes and aerothermal modeling (6 8). Using the above models, controls based studies were developed using near Parameter Varying (11), and other control strategies (12 21). A control oriented model (22) that included the interactions between the bending and rigid body modes was then developed which provided a full linearized steady state space model for the v ehicle. This model, developed by Parker, Serrani, Yurkovich, Bolender, and Doman is used as the base hypersonic model in this study. Optimal control involves the generation of control laws that reduce a provided cost function based on the dynamics equatio ns of the system. Most programs involve the direct transcription of an optimal control problem into a nonlinear program that is then solved using SNOPT (23), KNITRO (24), or Sparse (25) software packages. The numerical solution of optimal control problem s has become possible due to the development of pseudospectral methods (26 28) where the state and control are approximated using global polynomials and the system equations are collocated at orthogonal collocation points. Several mathematical optimizatio n methods such as the Gauss pseudospectral method (29 31) have been developed for various sets of PAGE 13 13 orthogonal collocation points. These were further developed to allow for multi phase control problem optimization using additional interior collocation poin ts (32). The optimizer used in this study is open sourced software that uses Gauss pseudospectral optimizations methods for multiphase control problems (33). PAGE 14 14 CHAPTER 2 HYPERSONIC MODEL The following chapter details the longitudinal hypersonic vehicle model used in this study and the process of developing a usable state space model and initial conditions. First the base model will be discussed followed by details on how the model was modified for vertical flight with an added horizontal displacement s tate. A discussion on the trends associated with different linearizing conditions including example state space equations and trim conditions will also be presented. Bolender Longitudinal Hypersonic Vehicle Model A longitudinal hypersonic vehicle model d eveloped by Bolender (22) will be used for this system. The model produces a linearized eleven state and four input steady state space model for straight and level flight of a hypersonic vehicle The linearization is performed around a provided altitude and Mach number which are used to determine the velocity in feet per second and local air density for force and moment calculations. pitch angle ng mode states ( as well as the inputs of elevator deflection canard deflection diffuser area ratio, and fuel flow ratio. Modified Bolender Model For the purpose of this study only the five rigid body states, and elevator and canar d deflection inputs will be used. It will also be assumed that the vehicle maintains a constant velocity throughout the flight and that the air density is constant. The governing equations behind the Bolender model must be modified in such a manner that the vehicle is in a nose down orientation and the acceleration due to gravity is along the x axis rather than the z axis of the vehicle. The model determines the vehicle lift PAGE 15 15 and drag by summing the forces along the x and z axis of the vehicle and then ap plying those forces geometrically to produce the overall lift and drag. In modifying the equations, the force due to gravity is removed from the lift and processed as x and z axis force components as shown in equations 2 1, 2 2, 2 3, and 2 4 The equatio ns also take into account pressure on the lower fore body ( ), exhaust pressure on the aft body ( ), the forces due to air entering the engine inlet ( ), the forces due to the elevators ( ), and canards ( ), the pressure on the upper body ( ), pressure on the lower body ( ), mass of the vehicle ( ), acceleration due to gravity ( ), and angle of attack ( ). ( 2 1) ( 2 2) ( 2 3) ( 2 4) With the Bolender model modified for a nose down oriented vehicle, a Mach and altitude can be provided and a modified state space model with provided trim values for states and inputs is produced as shown in equation 2 5 ( 2 5) PAGE 16 16 In the modified state space equation, the original altitude state ( ) now corresponds to the position north or south (P) of the initial ground track point and a new altitude state must be added with a correlation to the downwa rd velocity of the vehicle as shown in equation 1 6 ( 2 6) Two example modified state space equations generated by the modified Bolender model are presented below. Both equations, as well as a majority of the results presented in this paper, use the initial conditions of an altitude of 75,000 feet and a veloc ity of Mach 8. Equation 2 7 shows the modified state space equation for these initial conditions and a linearization altitude around the initial point. Equation 2 8 shows a modified state space equation with the same initial conditions but linearized at one half of the initial altitude. ( 2 7) ( 2 8) PAGE 17 17 When comparing the two equations, the gains of the vehicle angle of attack on other states and the effectiveness of the control surfaces have significantly increased for the state space linearized at lower altitudes. This increasing effectiveness of the control surfaces is expected as the vehicle passes through denser air at lower altitudes resulting in larger forces imposed on the control surfaces. Figures 2 1, 2 2, and 2 3 show the linearized equation gains of control surfaces for vehicle velocity, pitch angle, and angle of attack (B11 B32) between altitudes of 100 ,000 and zero feet and traveling at Mach 8. Although the gains approach zero at high altitudes due to the lack of air density, the overall smoothness in the changing of the gains as the altitude increases begins to unravel above 40,000 feet. This could be at tributed to a combination of the multiple low cost options available. Figure 2 1. State space control input gains for vehicle velocity for a range of altitudes. PAGE 18 18 Figure 2 2. State space control input gains for vehicle pitch rate for a range of altitudes. Figure 2 3. State space control input gains for vehicle angle of attack for a range of altitudes. Trim Conditions The trimmed flight state values produced by th e model for full altitude and half altitude linearizations are shown in table 2 1 As expected with lower altitudes and higher density air, the vehicle velocity for Mach 8 flight has decreased. The angle of PAGE 19 19 attack required to provide lift has also signif icantly decreased and the equally angled pitch provides for straight and level flight. The pitch rate also indicates straight and level flight with a value of zero for both altitudes. Figures 2 4 and 2 5 show the state values for angle of attack and pitc h for lineariza tion altitudes from zero to 100,000 feet. As expected from a straight and level model, the angle of attack and pitch angles are equal for all altitudes. Although rather chaotic at higher altitudes due to the minimization function, the gene ral trend towards larger angles is shown, and accounts for the reduction in force of air on the nose of the vehicle. The magnitude of deflection of the control surfaces has decreased to for the lower linearization altitude to account for the larger forces enacted on them by the lower altitude air. Table 2 1. Trimmed flight state values produced by the model for full altitude and half altitude linearizations. Altitude (ft) 75000 37500 Velocity (ft/s) 7792.83 7743.34 Angle of Attack ( deg ) 3.00 0.48 Pitch Rate (deg/s) 0.00 0.00 Pitch ( deg ) 3.00 0.48 Elevator ( deg ) 10.88 7.72 Canard ( deg ) 5.20 4.80 PAGE 20 20 Figure 2 4. Trimmed angle of attack for a range of altitudes. Figure 2 5. Trimmed pitch angle for a range of altitudes. The control surface straight and level deflection angles for a range of altitudes are shown in figure 2 6 The plots show a general trend of reducing towards zero as the altitude decreases for values less than 40,000 feet. The higher altitude portion of results shows a chaotic mixture of control surface angles. This is attributed to of the PAGE 21 21 lower density air allowing for various potential low values of the Bolender model cost function. Figure 2 6. Trimmed control surface deflection angles for a range of altitudes. The higher altitude elevator and canard angle peaks and valleys also appear to mirror one another. This is because to the elevator and canard affecting the same states. A positive elevator produces a pitch down moment while a negative canard deflection produces the same result. Therefore, as the elevator is transitioned to smaller deflections, the angle of the canard must negatively increase to compensate for the reduced nose down moment. The mirror effect is shadowed by the stable, h igher air density associated trends at the lower altitudes. The linearized state equation and trimmed flight state values will be used as the dynamics equations and initial conditions of the control system. PAGE 22 22 CHAPTER 3 CONTROL SYSTEM DESIGN AND RESULTS In the following chapter the horizontal displacement trajectory and control input optimizer will be discussed. Analysis of an example optimized state response will be detailed and optimized system maximum performance will be presented. An open loop system and a response example will be discussed. A closed loop system and example system response will then be detailed and the overall optimized closed loop system performance for an example with various horizontal deflections will be presented. Throughout the chapter the interaction between the optimizer mesh iterations and optimizer processing time, open loop stability, and closed loop response will be discussed. General Pseudospectral Optimal Control Software (GPOPS) GPOPS (33) is a MATLAB based control opti mizer that uses provided state based differential algebraic equations, initial conditions, cost function, and state limits to produce an optimized set of open loop control inputs. The optimizer attempts to minimize the cost function while using a discreti zed mesh of the dynamics equations while maintaining the states and controls within the maximum and minimum limits provided. In the case of this study, the dynamics equation and initial conditions are the state space equation and trim conditions produced by the Bolender Model. The cost function shown in equation 3 1 is designed to ensure that the vehicle reaches the requested ground position ( ), is oriented directly into the ground with an angle of attack and pitch angle of zero, and completes the man euvers before a commanded termination altitude ( ). In this study, a positive ground position change is in the direction of the top of the vehicle or positive Z axis while a negative change is towards PAGE 23 23 the belly or negative Z axis. The gains were chose n to ensure that the small angle of attack angles, pitch angles, and relatively small ground position were not outweighed by the large altitude values. ( 3 1) Table 3 1 shows the maximum and minimum values for the states and inputs of the system within the GPOPS program. Since GPOPS requires limits for every state, several of the values such as the limits for altitude and velocity are excessively large in order to provi de a limit without realistically constraining the system. The more realistic limits for angle of attack, pitch, and control surface deflections were selected to constrain the values to realistic values. For example, a hypersonic vehicle in the lower atmo sphere is be thermally compromised if in an excessive angle of attack orientation. Table 3 1. Maximum and minimum limits of states and inputs. State Min Max 50 ft 100000 ft 25000 ft 25000 ft 5 d eg 5 d eg 150 d eg 150 d eg 50000 ft 50000 ft 15 d eg 15 d eg 45 d eg 45 d eg 45 d eg 45 d eg PAGE 24 24 GPOPS Example Results The example state and control surface position plots used in this paper are based off of a vehicle traveling at Mach 8 at an altitude of 37,500 feet. The starting conditions used for the trajectory optimization are a velocity of Mach 8 and an initial altitude of 75,000 with a final altitude of 5,000 feet. The starting conditions were chosen to represent a realistic maneuver initiation point that would allo w enough flight time to produce results for both long and short distance ground location changes. The linearization point at half of the initial altitude was chosen to provide additional control surface effectiveness that the vehicle would experience as i t moved lower into the atmosphere as shown previously in figure 1 1 The initial conditions for the states and control surface angles, identical to the straight and level state and control deflection values, are shown in table 3 2 Table 3 2. Example initial state and control surface values. Altitude (ft) 75000 Velocity (ft/s) 7743.34 Angle of Attack ( deg ) 0.48 Pitch Rate (deg/s) 0.00 Pitch ( deg ) 0.48 Elevator ( deg ) 7.72 Canard ( deg ) 4.80 For this example, the vehicle is requested to make a 12,500 foot horizontal displacement; a value that is near the upper limit for a vehicle with these linearization points. The time of transition from initial altitude to final altitude is 9.04 seconds. Figure 3 1 shows the optimized open loop control inputs that are generated by the GPOPS program for such a requested maneuver. Both the elevator and canard positions initially change rapidly to move the vehicle into an orientation that moves it PAGE 25 25 towards t he requested ground position. The control surfaces then rapidly move again at the end of the maneuver to properly orient the aircraft to meet the termination requirements of zero pitch and angle of attack. Figure 3 1. Example o ptimized control deflection inputs. The plot of the horizontal displacement shown in figure 3 2 appears to be a smooth transition of position starting at zero with steady vertical flight and leveling off at the requested 12,500 feet before the optimized run terminates. T he angle of attack state output shown in figure 3 3 demonstrates the ability of GPOPS to optimize between a set of limits. As noted in table 3 1 the angle of attack is limited to plus or minus five degrees to reduce the chances of the vehicle burning up at such high speeds and low altitudes. The plot shows the angle of attack running along the upper limit of the bandwidth during the initial portion of the flight without exceeding the limit while still providing a favorable vehicle trajectory. The requir ement for a small angle of attack at maneuver termination is also accomplished. PAGE 26 26 Figure 3 2. Example optimized horizontal displacement trajectory. Figure 3 3. Example optimized angle of attack. The vehicle orientation requirement for pitch is also met by the GPOPS optimized state output as shown in figure 3 4 As expected, the vehicle moves into a positive pitch orientation to produce positive horizontal movement. The pitch is then returned to zero near the end of the maneuver as requested by the cost function used in the PAGE 27 27 optimization. It can also be noted that the pitch limit of plus or minus 15 degrees is maintained throughout the flight. Figure 3 4. Example optimized pitch angle. The full horizontal displacement maneuver is required to be completed before the vehicle hits 5,000 feet. Figure 3 5 shows the altitude of the aircraft as a function of time. The altitude changes at a nearly constant rate until the termination line is reached. A slight variation is the vehicle velocity, shown in figure 3 6 is due to the additional drag produced by the increased control surface deflections and increased magnitude angle of attack throughout the flight. The vehicle reaccelerates to a velocity past the initial condition due to the slightly more aerodynamic orientation of the aircraft near the termination point of the maneuver. PAGE 28 28 Figure 3 5. Example optimized altitude trajectory. Figure 3 6. Example optimized downward velocity. Maximum Optimized Vehicle Performance The maximum horizontal displacement for the vehicle relies on the initial altitude, Mach number, and linearization altitude. In determining the maximum performance, a horizontal displacement was requested for a static set of model para meters in increments starting at zero and increasing by one thousand. Once a GPOPS error was PAGE 29 29 displayed or the user determined the results to be unrealistic, the request was decreased back by steps of 250 feet until a successful and realistic optimization was requirements. One thing of noted during the maximum performance simulations was tha t the GPOPS code would display a numerical difficulties error for specific ground position changes that were much less than successful maneuvers performed during program debugging. Attributed to the complexity of the model straining the internal GPOPS pro gram parameters, additional simulations would be performed up to 1000 feet beyond the error producing maneuver to ensure that the error was due to a true performance limit. Table 3 3 shows the maximum vehicle change of ground location as the initial alti tude changes but the Mach number and linearization point remain 8 and 37,500 feet respectively. Table 3 4 shows the maximum performance for changes in Mach number with an initial altitude of 75,000 feet and a linearization altitude of 37,500 feet. Table 3 5 shows the performance for changes in linearization altitude with an initial altitude of 75,000 feet and a Mach number of 8. Most of the maximum performance points were determined by observation of the control surface deflections, angle of attack, or p itch exceeding the provided limits. PAGE 30 30 Table 3 3. Maximum performance for changes in initial altitude. Initial Altitude (ft) Maximum Horizontal Displacement (ft) 75000 14000 60000 9000 45000 7250 30000 2750 Table 3 4. Maximum performance for changes in vehicle Mach number. Mach Number Maximum Horizontal Displacement (ft) 10 12500 9 12500 8 14000 7 13750 Table 3 5 Maximum performance for changes in linearization altitude. Linearization Altitude (ft) Maximum Horizontal Displacement ( ft) 75000 5000 60000 5000 45000 15250 37500 14000 30000 16000 PAGE 31 31 As expected, the maximum performance decreased as the initial altitude was lowered and maneuvering space was diminished. The effects of changing the linearization altitude were also expected as effectiveness of the control surfaces decreased as linearization altitude increased. This decreased effectiveness reduces the ability of the vehicle to maneuver and results in decreased ground position change performance. The lack of large va riation of performance for changes in Mach number suggests that the diminished time of flight associated with increased velocity is counteracted by the increase of aerodynamic force on the control systems. Processing Time Verses Optimality The GPOPS softwa re is a computer memory and processor intensive program. The program allows the user to determine the number of iterations, or mesh reductions, that the program will perform in order to produce a more optimized or refined result. As the number of iterati ons increases, the time step of the optimized output decreases resulting in a higher number of control position updates This in principal will allow for a tighter control of the trajectory and produces a more optimized result. Unfortunately the increase of iterations adversely affects the computational time of the software due to the increased number of equations to process and large amounts of variables to maintain in memory as the number of system time stamps increases. For this study, the maximum all owable computation time was 30 minutes or 1800 seconds. Table 3 6 shows the optimization completion time and time stamp count for a ground location change of 2500 feet. A change of 2500 feet is used as an example because it was one of the few distances that could be optimized with a high number of iterations and remain within the maximum calculable time. As discussed, the number of time stamps and optimization time increases by large factors as the number of iterations requested increases. PAGE 32 32 Table 3 6. Optimization completion time and time stamp counts for various mesh iteration values. Iteration s Optimization Time (s) Time Stamps 0 4.229 9 1 7.909 41 2 236.535 233 3 369.613 957 4 757.533 1701 Figure 3 7 shows how the introduction of additional time stamps through iteration can affect the optimization results. As the number of iterations increases, optimized control positions are refined until the plots appear to normalize. PAGE 33 33 Figure 3 7. Comparison of optimized control deflection over time plots for various mesh iteration amounts. The additional time stamps allow for more frequent control inputs producing a more optimized ground pos ition path and angle of attack outputs as shown in figures 3 8 and 3 9. By inspection, the variation between the optimized responses diminishes when two or more iterations are used. The additional frequency of control inputs may also offer additional stability to an open loop response of the Bolender model using the optimized control inputs. 0 Iterations 1 Interation 2 Iterations 3 Iterations PAGE 34 34 Figure 3 8. Comparison of optimized ground location over time plots for various mesh iteration amounts. 0 Iterations 1 Interation 2 Iterations 3 Iterations PAGE 35 35 Figure 3 9. Comparison of optimized angle of attack over time plots for various mesh iteration amounts. Open Loop System An open loop response of the modified Bolender linearized state space model can be generated using the GPOPS optimized control surface deflections interpolated into higher frequency time stamps as the inputs. A diagram of the open loop system is shown in figure 3 10 The optimized and interpolated controller inputs are fed into the modified Bolender model state space equation and the updated states are outputted. 0 Iterations 1 Interation 2 Iterations 3 Iterations PAGE 36 36 Should the optimized inputs be accurate enough, the vehicle is expected to maintain the proposed flight path. Figure 3 10. Open loop system diagram. Open Loop System Example Figure 3 11 shows the interpolated control inputs compared to the GPOPS optimized control surface deflections for a vehicle model linearized at 37,500 feet, traveling at Mach 8 from 75,000 feet altitude, a reques ted horizontal displacement of 2,500 feet, and optimiz ed using two iterations. The interpolated control inputs match the GPOPS control deflections perfectly but instead of updating at an average frequency of 36 Hz, the interpolated inputs update at 1000 Hz. PAGE 37 37 Figure 3 11. Comparison of optimized and interpo lated control deflection inputs. Using the high frequency control surface inputs, an open loop response is generated. The ground position, angle of attack, and pitch state responses are compared to the GPOPS optimized state responses in figures 3 1 2 3 1 3 and 3 1 4 Figure 3 12. Open loop and optimized horizontal deflection response. PAGE 38 38 Figure 3 13. Open loop and optimized angle of attack response. Figure 3 1 4 Open loop and optimized pitch response. Indicating an unstable system response, all three states diverge from the expected path without any external wind gusts or simulated sensor errors. The usage of a more refined set of optimized control surface deflections has the potential to allow the sys tem to complete the maneuver. This was tested for several additional iterations PAGE 39 39 with no success. Table 3 7 shows the time of divergence for open loop responses using optimized deflection inputs from various iteration counts. The point of divergence was selected as the moment that the percent error between the optimized and open loop ground locations exceeds 10 percent. A stipulation requiring a total difference to be greater than 10 feet is also included to offset large percent errors produced when the ground location change is initially very small. Table 3 7. Divergence time for different mesh interations. Iteration s Divergence Time (s) 0 0.786 1 0.975 2 1.043 3 1.045 4 1.045 The table shows that as the number of iterations increases, the open loop response diverges later in the maneuver but still diverges never the less. The divergence time for the 2500 foot position change normalizes at 1.045 seconds within the first 2 iter ations. It can be assumed that optimizations using more iterations would not affect the divergence time significantly. Due to the open loop divergence and overall open loop susceptibility to external noise such as wind gusts, a close loop system is devel oped to provide system stability throughout the flight. PAGE 40 40 Closed Loop System A feedback system can be used in conjunction with the optimized and interpolated GPOPS control inputs to combat external interactions as well as any situations in which the vehicl e moves off of the intended path. In this study, the feedback gains are generated using an LQR algorithm designed to track the ground position of the vehicle. LQR generates the gains for a continuous time system by determining the value for P such that e quation 3 2 is true and then using that value in equation 3 3 to find the gains, K. In these equations is the state weighting matrix and to determine the feedback gains. (3 2) (3 3) In order to produce the proper gains, the state space equation is modified to move the tracked state, in this case the ground position, into the top row of the matrices and add an additional integral of error state to allow for the horizontal displacement error gain to be calculated. The overall system shown in figure 3 1 5 is laid out such that the optimized ground position ( ) is compared to the current ground position (P(1)) and fed through the feed forward gain (k) and an integrator (I). The resulting inputs are modified by subtracting the results of the full state feedback gain (K), the GPOPS optimized control inputs (U), and an external disturbance (N). This combined signal is inputted into the GPOPS modified system plant and a full state output is p roduced that PAGE 41 41 will then be used in the feedback gains after a simulated sensor error (E) is added. The output of the system (y) is the closed loop response to the requested ground position change maneuver. In this study, the external disturbance, in this case a wind gust, is generated by adding additional control surface deflections into the system. The simulated sensor error is constant for all six states and is generated as a small percentage of the maximum value of the optimized state output. Figure 3 1 5 Closed loop system diagram. Closed Loop System Example With linearization conditions of Mach 8 vehicle velocity and 37,500 foot altitude, the state weighting matrix and control weight matrix are shown in equations 3 4 and 3 5 respectively. The weights of the matrices were generated using a high fidelity study of the final closed loop response vehicle position error for a single change of position request across a variety of weights from 0 to 100. The weighting matrices that produced the smallest final error are shown below. PAGE 42 42 (3 4) (3 5) As discussed previously, the modified linearized Bolender state space equation must be modified a second time. This modifica tion will add an extra state for the integral of error ( ) and move the tracked state, ground location ( ) into the top row. The new state space equation is shown in equation 3 6 Using the modified A and B matrices and the Q and R weight matrices, th generate the closed loop feedback gains which are shown in equation 3 7 (3 6) (3 7) The closed loop system response for a requested change of ground posi tion of 5,000 feet is shown below. Figure 3 1 6 compares the closed loop control surface deflections to the optimized deflections. Rather than matching the optimized control surface deflections the closed loop system uses a set of deflections designed to match PAGE 43 43 the vehicles trajectory to the inputted o ptimized trajectory. The wavering in the closed loop control positions near the three and four second marks is a simulated wind gust disturbance that provides a one degree deflection to both the canard and elevator. A plot of the wind gust in terms of re lative control surface deflection is shown in figure 3 1 7 The wind gust is designed to act as if the vehicle is passing through an altitude band with higher winds than at other altitudes. Based on the lack of a large notch in the control surface deflect ions in the region of the wind gust, the closed loop system appears to have nullified the effects of the external disturbance as designed Figure 3 1 6 Closed loop and optimized control system deflections. PAGE 44 44 Figure 3 1 7 Wind gust control deflection plot The horizontal displacement state response for the closed loop system is shown in figure 3 1 8 The closed loop displacement lags slightly behind the commanded optimized path but upon hitting the 5,000 foot altitude maneuver termination point, the closed loop response is only off by 6.09 feet which is equivalent to a 0.12 percent error. Figure 3 1 8 Closed loop and optimized horizontal displacement response. PAGE 45 45 The closed loop pitch and angle of attack responses are compared to the optimized respons es in figures 3 1 9 and 3 20 respectively. The closed loop angle of attack follows the optimized plot loosely and fails to track the quick return to zero degrees angle of attack near the end of the maneuver resulting in a 2.57 degree final angle. The clo sed loop pitch follows the optimized pitch much tighter but also fails to return to zero at the end of the maneuver resulting in a 2.20 degree final pitch. Since the flight path angle is calculated based off of the difference of the angle of attack and pitch, the final flight path angle for this example is 0.37 degrees or nearly straight towards the impact point on the ground. Figure 3 1 9 Closed loop and optimized angle of attack response. PAGE 46 46 Figure 3 20 Closed loop and optimized pitch response. Ef fect of Optimizer Iterations on Closed Loop Performance Due to the usage of the GPOPS optimized horizontal displacement and control surface deflections as inputs, the accuracy and stability of the closed loop system is effected by the number of optimizer i terations are performed before the data is used. Figure 3 2 1 shows the effects of optimizer iterations on the closed loop control surface deflections throughout a horizontal displacement of 2500 feet. Although the figure shows high frequency and high re lative amplitude control movements for the zero iterations plot, the plots representing two or more iterations have smooth control movements that genuinely follow the optimized control surface deflections. A comparison of the vehicle state responses assoc iated with the control surface plots is shown in table 3 8 The table shows that while the calculation time for each optimization increases dramatically at the second iteration, the accuracy of the maneuver also increases dramatically. The comparison of the angle of attack and pitch also show that the flight path angle becomes more vertical as more optimizations are PAGE 47 47 performed. Based on the lack of response improvement but continued increase in calculation time for three or more iterations, as well the re sults from the GPOPS optimality discussion, the most efficient number of iterations for this system is determined to be two. Figure 3 2 1 Comparison of closed loop control deflection responses for various values of mesh iteration. 0 Iterations 1 Interation 2 Iterations 3 Iterations PAGE 48 48 Table 3 8. Closed loop response performance parameters for various mesh iteration amounts. Iteration Calculation Time (s) Distance from Target (ft) Percent Error Angle of Attack ( deg ) Pitch ( deg ) Flight Path Angle ( deg ) 0 4.23 25.086 1.003 4.016 2.883 1.133 1 7.91 21.805 0.872 3.143 2.073 1.07 2 236.54 6.822 0.273 3.605 3.172 0.433 3 369.61 6.821 0.273 3.605 3.172 0.433 4 757.53 6.821 0.273 3.605 3.172 0.433 Complete System Performance The full system performance results for a model and system using an initial altitude of 75,000 feet, a linearization point of 37,500 feet, and a velocity of Mach 8 are shown in Table 3 9 for a range of requested horizontal displacements changes The system has a horizontal displacement range between 14,000 and 10,000 feet. The time to provide a full closed system response from initiation of the modified Bolender model state space generation to the closed loop simulation completion ranges from 84 to 847 seconds and appears to be independent of the requested ground position change. The final distance from the target correlated with the requested horizontal displacement with small movements resulting in errors of less than one foot but displacem ents changes having 40 foot discrepancies. The percent error for distance from target remained less than two tenths of a percent for positive movement and less than half a percent for negative locations. The percent errors for small horizontal displaceme nts are larger due to the small requested movement in relation to the final distance from target. The final angle of attack and pitch angle also increase in magnitude as the requested movement distance increases. For larger position changes, the magnitud e of angle of attack exceeds the GPOPS limit but does not PAGE 49 49 exceed ten degrees. A true angle of attack limit cannot be known without thermal modeling and testing. The pitch angle does not exceed a magnitude of 7 degrees and when combined with the angle of attack the overall flight path angle remains under a magnitude of two and a half degrees with a majority of final angles being less than one degree. Table 3 9. Example full system performance results for various requested horizontal displacements. Request ed Horizontal Displacement (ft) Run Time (s) Closed Loop Position (ft ) Distance from Target (ft) Percent Error Angle of Attack ( d eg) Pitch ( d eg) Flight Path Angle ( d eg) 10000 163.482 9960.001 39.999 0.400 8.842 6.692 2.151 7500 240.640 7476.439 23.561 0.314 7.404 6.040 1.365 5000 127.560 4983.456 16.544 0.331 6.840 5.826 1.015 2500 236.535 2493.178 6.822 0.273 3.605 3.172 0.433 1000 141.014 999.314 0.686 0.069 1.199 1.146 0.053 500 Numerical Difficulties GPOPS Error 50 56.316 50.682 0.682 1.364 0.511 0.543 0.032 0 132.668 0.748 0.748 N/A 0.484 0.519 0.036 50 84.393 49.130 0.870 1.741 0.433 0.476 0.043 500 253.909 498.700 1.300 0.260 0.173 0.243 0.070 1000 399.983 998.174 1.826 0.183 0.498 0.026 0.524 2500 529.436 2496.580 3.420 0.137 1.044 0.841 0.203 5000 433.760 4993.909 6.091 0.122 2.572 2.202 0.370 7500 251.915 7491.257 8.743 0.117 4.094 3.558 0.537 10000 188.782 9988.585 11.415 0.114 5.623 4.919 0.704 12500 846.724 12483.980 16.020 0.128 7.281 6.302 0.980 14000 517.065 13968.999 31.001 0.221 8.124 6.400 1.724 PAGE 50 50 CHAPTER 4 CONCLUSIONS AND FUTURE WORK In this study, a method for optimized trajectory tracking for a hypersonic impactor performing a requested horizontal displacement was developed and discussed. A linearized longitudinal state space model for a hypersonic vehicle was modified from straight and level flight horizontal flight to vertical flight. The modified linearized st ate space model and trim conditions were then used in a General Pseudospectral Optimal Control Software to produce an optimi zed state response and control surface deflections for a requested change of ground position of the vehicle in flight. The optimize d control outputs and states were used in individual open loop and closed loop systems to produce system realistic system responses for a requested horizontal displacement of a hypersonic impactor. Summary The modified Bolender model is highly dependent on the altitude of the linearization point, specifically the air density. As air density increased, plots and showed that the gains, or effectiveness, of the control surfaces increased for velocity, pitch rate, and angle of attack. Plots of the steady stat e conditions showed that at altitudes about 40,000 feet, the steady state control surface deflections, steady state pitch, and steady state angle of attack greatly varied at different but similar altitudes. This was attributed to the minimization algorith m having multiple low value configurations and the small changes in altitude caused it switch between which configuration was the lowest. The resulting state space equations also showed that change of ground position is effected solely by angle of attack and vehicle pitch while the control positions did not directly affect that state. PAGE 51 51 Using the Bolender modified state space model in the GPOPS software a variety of optimized state responses were generated for a multitude of initial conditions, linearizatio n points, and ground position change distances. The plots of a large distance example showed how the ground track trajectory would be a smooth transition while additional plots showed how the optimized maintained the states and controls within limits and returned specific states to the requested maneuver termination parameters. The values for maximum ground maneuver performance were tabulated for various initial altitudes, Mach numbers, and linearization altitudes. For reducing initial altitudes the perf ormance was also reduced as the vehicle ran out of altitude to maneuver in. For a reduction in linearization altitude, the maximum performance increases as the extra air density at low altitudes allows the control surfaces to be more effective. The reduc tion in Mach number showed mixed results as the reduced velocity resulted in a longer flight time but lower magnitude aerodynamic forces on the control surfaces reducing their effectiveness. A discussion on how the number of optimizer interations effected the processing time and accuracy of the optimization showed that the optimized response normalized by the second iteration but the processing time exponentially increased with each additional iteration. An optimized state and control response example gene rated by GPOPS was placed into an open loop system to and proved to be highly unstable. Although additional iterations were used, which would reduce the time steps in between optimizer control inputs, the open loop system normalized to a divergence time o f 1.045 seconds. It was noted that even if the system was open loop stable, external disturbances such as wind gusts could easily cause the system to diverge. PAGE 52 52 A closed loop system was developed using LQR generated gains that are designed to track the grou nd position throughout the maneuver. The closed loop system uses a tracked state and full state feedback as well as the provided optimized control surface deflections. A wind gust that affects the control surface deflections and a state sensor error are also included to ensure that the system can handle external and internal disturbances. Plots of an example system response show that the control surfaces do not closely track the optimized deflections. The angle of attack and pitch angle do closely track the optimized plots but fail to return to zero before the vehicle hits the maneuver floor altitude. The flight path angle for the vehicle, based on the difference between pitch and angle of attack, is less than one degree at the final time stamp. The ex ample closed loop response also closely follows but slightly lags the optimized ground trajectory with a final location that is just over 6 feet off of the target for a 5,000 foot horizontal displacement The effects of optimizer iterations were plotted a nd tabulated to show that the flight control surface deflections and final ground position, angle of attack, and pitch angle all normalize for iterations of two or more. Based on the previous results involving the Bolender model linearization point, the GP OPS iteration affects, the open loop response, and the closed loop response, an example response for the complete system is tabulated for various changes of ground position amounts. Using a low altitude linearization point and two iterations during optim ization, the full system responses show that the design is able to successfully simulate the feasibility and performance of a hypersonic vehicle performing a horizontal displacement in vertical flight. The example system performance showed maneuverability between 10,000 and 14,000 feet with less than a half percent error of PAGE 53 53 ground location, flight final path angle s under two and a half degrees and final angle s of attack less than ten degrees. Future Work While this study successfully developed a method t o have a hypersonic impactor track a trajectory for a change of ground location while maintaining specific limits, there are several assumptions that could be address in future studies. The assumption of a constant velocity may not be realistic with the t emperament of current SCRAM jet technology. The usage of a vehicle model that assumes zero thrust from the engine could show the effects of reduction of velocity due to air resistance on the maneuverability of the vehicle and performance of the system. Th e GPOPS software allows the usage of phase based optimized that could allow for additional controllability of the vehicle. In this study the maneuver was requested to be completed at an altitude of 5,000 feet. A secondary and must faster maneuver system could be setup to fine tune the ground position and vehicle orientation. The assumption of constant air density at all altitudes of flight can also be removed with further work. GPOPS allows the usage of state dependent dynamics equations that could produ ce an optimized response that uses a state space equation specific to the current altitude of the vehicle at each time step. Closed loop gain scheduling and a state dependent dynamics equation could then be used to ensure the best response for the closed loop system. The results of this work show the feasibly of changing the ground position of a hypersonic impactor in vertical flight. With future work the system accuracy in both performance and realism can only be improved. PAGE 54 54 REFERENCE LIST [1] United States Air Force. (n.d.). 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[21] Sigthorsson,D.,O., Serrani,A., Yurkovich,S., Bolender,M.,A., and Doman,D.,B., Steady State Constraints," August 200 6, AIAA Paper 2006 6558 [22] Parker,T.,J., Serrani,A., Yurkovich,S., Bolender,A.,M., and Doman,B.,D., Oriented Modeling of an Air Breathing Hypersonic Vehicle," Journal of Guidance, Control, and Dynamics Vol. 30,No. 3, 2007, pp. 856 869. PAGE 56 56 [23] Gill P. E ., Murray W., AND Saunders M. A. 2002. SNOPT: An SQP algorithm for large scale constrained optimization. SIAM J. Optimiz. 12 4, 979 1006. [24] Byrd R. H., Nocedal J., AND Waltz R. 2006. KNITRO: An Integrated Package for Nonlinear Optimization University of Colorado Boulder, CO, and Northwestern University, Evanston, IL. [25] Betts J. T. AND Frank P. D. 1994. A sparse nonlinear optimization algorithm. J. Optimiz. Theor. Appl. 82 2, 193 207. [26] Canuto C., Hussaini M. Y., QUARTERONI, A., AND ZANG, T. A. 1988. Spectral Methods in Fluid Dynamics Spinger Verlag, Heidelberg, Germany. [27] Fornberg B. 1998. A Practical Guide to Pseudospectral Methods Cambridge Monographs on Applied and Computational Mathematics. Camb ridge University Press, Cambridge, U.K. New York, NY. [28] Trefethen L. N. 2001. Spectral Methods in MATLAB SIAM Press, Philadelphia, PA. [29] Benson D. A. 2004. A gauss pseudospectral transcription for optimal control. Ph.D. dissertation. MIT, Cambridge, M A. [30] Benson D. A., Huntington G. T., Thorvaldsen T. P., AND Rao A. V. 2006. Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Contr. Dynam. 29 6, 1435 1440. [31] Huntington G. T. 2007. Advancement and analysis of a gauss pseudospectral transcription for optimal control. Ph.D. dissertation. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA. [32] Rao A. V. 2003. Extension of a pseudospectral legendre method fo r solving non sequential multiple phase optimal control problems. Proceedings of the AIAA Guidance, Navigation, and Control Conference (Austin, TX). AIAA paper 2003 5634. [33] Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I. and Huntinton, G. T. 2010. Algorithm 902: GPOPS, aMATLAB software for solving multiple phase optimal control problems using the Gauss pseudospectral method. ACM Trans.Math. Softw. 37, 2, Article 22 (April 2010) PAGE 57 57 BIOGRAPHICAL SKETCH Erik Kittrell was born in 1988 in Hollywood, Florida. He has a younger sister who engineering grew from the many Space Shuttle launches that were witnessed at moonlit hours from School in 2006. Accepted to the University of Florida, Erik spent the next five years in Gainesville, Florida pursuing a dual degree in Mechanical and Aerospace Engineering. During the few hours of free time available to engineering students, Erik enjoyed playing intramural racquetball and softball as well as attending the Florida Gator football, basketball, and softball games. In the fall of his second year at UF, Erik was accepte d into the United Space Alliance Co Op Internship program. He spent three nonconsecutive semesters working as a mechanical flight structures engineer for the Space Shuttle Solid Rocket Boosters aft controls was sparked during his last year at UF in the Stability and Controls, and Senior Design classes taught by Dr. Richard Lind. In the spring of 2011, he graduated with a B.S. in both oject on hypersonic trajectory tracking the following fall with Dr. Lind as his advisor. Prior to completing his engineer at Eglin Air Force Base in Valparaiso, Florida. With the completion of his and potentially pursue a doctoral degree in the near future. 