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This work was carried out at the department of computer aided design of aerospacemeasuring-measuring computing systems of the Saint Petersburg State Universityof Aerospace Instrumentation in English.Scientific supervisor:Candidate of Technical SciencesBrodsky Sergey AlexandrovichOfficial opponents:Doctor of Technical Science, professorTimofeev Adil VasilievichCandidate of Technical SciencesValentin Yurievich EmelianovLeading Organization:Samara State Aerospace UniversityDefense takes place with permission from the Higher Attestation Commission ofthe Russian Federation № 08.05/818-7 from 09.06.2011 in English on______________ 2011 at ___:___ at the session of dissertation council Д212.233.02 at the State University of Aerospace Instrumentation with the address190000, Saint Petersburg, Bolshaya Morskaya, 67.This thesis is available in the library of SUAIThe abstract was distributed on the ______ of _______, 2011.Scientific secretary ofDissertation councilDoctor of Technical Science, professorL.A. Osipov

MAIN CHARACTERISTIC OF WORKRelevance of topic. A successful designed control system should always be ableto maintain stability and performance level in spite of a priori uncertainty in systemdynamics and or in the working environment to a certain degree. Robustness is theability of a designed control system to maintain it performance and stabilitycharacteristics under the effect of all types of influences. The design of suchsystems using the methods of mathematical simulation substantially reduces thecost and time of design as well as increases the effectiveness of such designs. Thismakes it possible not to only set the mathematical models of the robust controlsystems, but also to get the performance using the quality control algorithms inaccordance with the stated goal.In classical single-input single-output control (SISO), robustness is achieved byensuring good gain and phase margins. Due to the increasing complexity ofphysical systems under control and rising demands on system properties, mostindustrial control systems are no longer single-input and single-output (SISO) butmulti-input and multi-output (MIMO) systems with a high interrelationship(coupling) between these channels. The number of (state) variables in a systemcould be very large as well. These systems are called multivariable systems. Whenmultivariable design techniques were first developed in the 1960s, the emphasiswas placed on achieving good performance, and not on robustness. Thesemultivariable techniques were based on quadratic performance index and Gaussiandisturbances, and proved to be successful in many aerospace applications whereaccurate mathematical models can be obtained, and descriptions for externaldisturbances/noise based on white noise are considered appropriate.However, application of such methods commonly referred to as the linearquadratic Gaussian (LQG) methods, to other aerospace problems where theaccurate mathematical models are unknown and the models are under the influenceof a priori uncertainties made apparent the poor robustness properties exhibited byLQG controllers. This led to a substantial research effort to develop a theory thatcould explicitly address the robustness issue in feedback design.The development in computer technology, especially in the area of softwarepackages, really contributed immensely to the development of control systems.This indirectly widened the field of application of control theory. Thisdevelopment in the Information Technology sector simplifies the way analysis andsimulations of complex control systems are carried out. The presence of thepackets of applications programs (MATLAB, MATHCAD, ANSYS, FLUENT,etc) facilitates the task of the practical realization of the theoretical solutions.The appearance of new and the development of the known methods of creatingthe robust laws of control under the conditions of a priori uncertainty, i.e., withincomplete information about the parameters or the characteristics of the object, all3

this testifies to the fact that the problem of robustness in control systems is not yetfully solved and is going on all the time. Main results on control theory under thecondition of a priori uncertainty is related to work by Nebylov, A.V., Dullerud,G.E., Paganini, F.G., Siouris, G.M., Petkov, P.H., Konstantinov, M.M., Sanchez-Pena, R., Sznaier, M., Yakubovich, S.V., Fradkov, A.L., Kalman, I.D., et al.The sounding rocket belongs to the class of flexible structure which isvulnerable to external disturbances, measurement noise and elastic deformation.The practice of design and construction of flying vehicles indicates that theflexibility construction of a sounding rocket makes the construction of a controlsystem more difficult. The effect of external disturbances and measurement noiseas well as the differences between mathematical models used for design and theactual system makes the construction of a control system a challenging task forcontroller design engineers. The differences between mathematical models usedfor the design and the actual system are called dynamic perturbations (Δ). The taskof the application of robust synthesis methodology for altitude stabilization of adynamic, nonlinear, non-stationary flexible object operating under the condition ofa priori uncertainty is relevance.The Aim of work consist of the development of robust control synthesismethodology for the attitude stabilization of a typical dynamic, nonlinear, nonstationaryflexible object operating under the condition of a priori uncertainty forexample - a sounding rocket.Main taskThe set goal was achieved by solving the following problem.1. Analyzing the existing robust control synthesis methodologies.2. Developing and investigating a full mathematical model of 3-dimensionalmotion of a flexible sounding rocket.3. Building a model of longitudinal bending vibrations of the rocket withthe use of experimental data corresponding to the dominant harmonics.4. Linearization of the full nonlinear model of the object at different points ofreference trajectories, determining the model of longitudinal motion.5. Representation of the object model in the M/∆form containing independentdescription of the nominal plant model and normalized uncertainties.6. Application of the robust control methodology based on H ∞ method incombination with μ - synthesis and analysis approaches.7. Selection and adjustment of weighting matrix functions and transfer function ofan ideal closed-loop system in the problem of synthesis of robust controllerusing LQG methods.8. Choice of an ideal transfer function of a closed-loop system with theuncertainty model parameters of elasticity, shapes and natural frequencies.4

9. Synthesis of reduced controllers of fixed dimension, followed by interpolatingthe parameters for an arbitrary point of the trajectory, with dissenting points oflinearization.Methods of Research. In the course of this dissertation research, the followingmethods are used: the methods of system analysis, optimal and robust controltheory, identification theory of dynamic systems, and optimization theory andmathematical programming.Scientific novelty of research of the concluded researched consist of thefollowing:1. The flexibility model of a sounding rocket suitable for the synthesis of controlsystem under the influence of various forms of uncertainties in the systemmodel was developed.2. The method and procedure for the synthesis of robust control algorithms fornonlinear non-stationary flexible dynamic object under the influence of a prioriuncertainties of natural shapes and frequencies was developed.3. The procedure of selection of weighting functions and transfer function of anideal model of the object, ensuring the feasibility of robust synthesis wasdeveloped.4. The procedures for the identification of distributed parameters of the model ofelastic object corresponding to the first experimental dominant harmonics,taking into account a priori information the distributed mass and stiffness wasdeveloped.Practical valueThe results of dissertation work are obtained in the course of the research mayserve as a basis for construction the general method of synthesis of robust controlsystem of dynamic elastic aerospace objects taking into account various a prioriuncertainties in the object model and external influences.The results obtained in the thesis allows for the following.1. To carry out effective processing of experimental data to determine theparameters of the model of elasticity, using the proposed methods for theidentification of distributed parameters.2. Reduce the complexity of investigations and reduce the computationalcomplexity of the procedures for modeling and synthesis using developedmethods of reducing the model of aeroservoelasticity.3. Ensure the feasibility of robust control synthesis methodology, using in theselection of weighting parameters results from LQG synthesis.4. Ensure smooth change in parameters of the control system of non-stationaryflexible objects by applying the proposed method of interpolation parameters ofreduced controller of given order.5

5. Improve the quality and reliability of the control system by the optimalplacement of additional sensors to determine the elastic components of motionof the object.The practical value also lies in the software implementation of the proposedmethods of robust control synthesis using the simulation packageMATLAB/SIMULINK.At the defense the following positions would be carried out:1. A generalized structural model of a flexible sounding rocket, designed for theproblem of robust synthesis of control systems containing uncertainties of keyparameters.2. Method of the synthesis of the algorithms of the robust control system fornonlinear non-stationary flexible dynamic system working under the conditionof a priori uncertainty.3. Analysis of the results of the developed method of synthesis of robust controlsystems.Implementation of Results. The mathematical model and proposed methods ofsynthesis of robust control systems have been used in scientific research carriedout the International Institutes for Advanced Aerospace Technologies (IIAAT).Methodological developments have been used in learning process at department№11 of SUAI in preparation of master technicians from National Space Researchand Development Agency (NASRDA) of Nigeria. Scientific research results havebeen used in a program for creating sounding rockets at the Nigerian SpaceAgency.Approbation of workMain results of dissertation work where presented and discussed at the 16 th and17 th Saint Petersburg International Conference on Integrated Navigation Systems(Saint Petersburg, 2009, 2010), at the Annual Scientific Sessions of the StateUniversity of Aerospace Instrumentation Scientific Session (Saint Petersburg,2009, 2010, 2011), at the 2 nd IFAC International conference on Intelligent ControlSystems and Signal Processing, ICON Istanbul, 2009, at the IFAC Workshop onAerospace, Guidance, Navigation and Control Systems (Samara 2009) at the 5 thInternational Conference on Recent Advances in Space Technologies, Istanbul,2011 and at the International conference on Scientific and TechnologicalExperiments on Automatic Space Vehicles and Small Satellites SPEXP, Samara,Russia. 2011.Publications. The content of dissertation work is stated in 12 publications whichconsist of 2 publications and 1 accepted for publication in peer review journals andin the Russian Federation and 9 articles worldwide.In the works, published in co-authorship, the following scientific and practicalresults belong to the dissertator: Intelligent Approach to Experimental Estimation6

of Flexible Aerospace Vehicle Parameters, Nigeria Space Programs, ActualProblems of Aviation and Aerospace Systems and Application of robust synthesismethodology for sounding rocket attitude stabilization.Structure and volume of dissertationThe dissertation work consists of introduction, five chapters, conclusion,references and appendix. The work is presented in 129 pages of basic text, 60figures, 7 tables, 82 references and 17 appendixes.SUMMARIZED CONTENT OF WORKIntroduction. In the introduction the relevance and practical significance of theresearch was justified, the aim of work and basic research task were identified. Abrief description of the chosen robust methodology used in this research work aswell as the advantages of the chosen method is presented. The relevance ofrobustness in control system design, different types of perturbation that isencountered in control systems, the effect of elasticity and aeroservoelasticity onthe dynamics of the control system is also discussed. The effect of the emergenceof powerful software packages in control system design and analysis is presentedin this section of dissertation work. . The scientific and practical results for thedefense were formulated.Chapter 1 of the dissertation provides an overview of the models of structuraldynamic system, the effect of aeroelasticity and aeroservoelasticity on the controlsystem of a flexible aerospace vehicle. The chapter also gives a brief survey of thetraditional and modern robust control methods and the classification and modelingof uncertainties in control systems. In this research the input multiplicativeperturbation configuration is described as the most appropriate form ofperturbation representation that is associated with the design and simulation of thecontrol system of the sounding rocket. One of the main reasons for applyingfeedback control rather than feedforward is to reduce the effect of uncertainties incontrol system design. Uncertainties pose limitations on achievable feedbackcontrol performance. This chapter also gives a brief survey of the traditional andmodern robust control methods in used in control engineering, the pros and cons ofeach method is described.Chapter 2 of the dissertation work gives a description of a full mathematicalmodel of an elastic sounding rocket, which is functionally divided intointerconnected subsystems. Each subsystem has inputs and outputs; therelationship between them is determined by systems of differential equations,7

functional or temporary relationships in analytical or tabular form. Each subsystemis also represented by block diagram.As a simulation environment, the software package Matlab containing the toolfor building and simulating dynamic system – Simulink was chosen. Using theexisting building blocks in it, the various functions and utilities such as AerospaceBlockset and Robust Control Toolbox, reduces the time of building models andallow to focus on its characteristics.As a base for the standard model, the 6-DoF variable mass missile airframe waschosen. Translational and rotational motion of an object in the associatedcoordinate system is described by a system of nonlinear differential equations. Invector form these equations are:dVF= − Ω× V(1)dt mdΩ−= I 1( M − Ω× ( I ⋅Ω))dt(2)where V is velocity vector of the center of mass, Ω - angular velocity vector, F andM are vectors of forces and moments acting on the body, m and I – the mass andinertia tensor of the solid body. F and M are the forces and moments acting on thebody are formed as a result of aerodynamics, engine thrust and gravity.The equations of elastic flexural vibrations caused by the influence of distributedforces and moments in the discrete form described by a system of ordinarydifferential equations in vector form is:2Φ ′ MΦ ξ + Φ′ΞΦ ξ+ Ω Φ′MΦ ξ = Φ′f(3)where the state variables are the modes ξ of natural elastic vibrations, lateraldisplacement q of the selected anchor points of elastic line due to the modes of alinear transformation is:q = Φξ(4)where matrix Φ consist of column vectors of the natural shapes of flexuralvibrations, the matrix Φ′ MΦ is diagonal matrix of generalized masses { Mi}, Ω -diagonal matrix of local masses { m i }.Generalized forces Φ′ f correspond to a distributed transverse load f, which isformed taking into account the distributed aerodynamic forces, as well asconcentrated forces and moments at the points x i of application controls.Generalized damping matrix can be taken as diagonal matrix:Φ ′ΞΦ = diag(2ζiM iωi) . (5)where ζicorresponds to the damping coefficients of separate modes.8

Distributed aerodynamic loads depend on the local angles of attack and theparameters of angular and linear velocity of the solid. Current parameters of therotational and translational motion of the solid part of the equation (1,2) acting onthe body forces and moments, and influence on the aerodynamic and otherparameters of a non-stationary model of the object. Block diagram of the effect ofaeroelasticity is presented in Figures 1 and 2.Changes in the distribution of mass of the object is associated with fuelconsumption, determines the change of coordinates of center of mass, which leadsto a change in the point of application of the moments of the aerodynamic forcesand engine thrust, center of mass.Dependence of model parameters on the parameters of motion and time arepredictable and can be represented in functional or tabular form. Uncertainty in thevalue of parameters derived from the natural or computational experiment can beconsidered as a range of parameter deviations from the nominal value. Theaerodynamic coefficients of sounding rocket c x , c z α , c y β , m z α , m y β , m y δy , m zδzareobtained using the finite element method (FEM). Computational experiment wascarried out using the software package FLUENT for different Mach numbers.Varying the angles of attack α, sideslip angles β and roll angles γ at differentangular velocities p, q, r and angles of fin deflection δ x δ y δ z . They were differentfrom the experimental data obtained during test at the Nigerian Space Agency.On this basis, it is assumed that the dependence of aerodynamic coefficients,shown in Figure 3 and Figure 4 contain the multiplicative uncertainty is within10% (0.9 – 1.1 of the nominal value).Similarly, by comparing the results of field and numerical experiments usingprogram ANSYS defined uncertainty factors associated with elasticity.Multiplicative uncertainty in the values of natural frequencies and shapes of theflexural vibrations at certain points is also conventionally assumed to be 10%, asshown in Figure 1.9

Figure 1. Block diagram of the aeroelasticity modelChapter 3 of the dissertation work describes and analysis of the effect offlexibility on the design of the control system. The relationship between the rigidbody model of the sounding rocket and the model of elasticity via distributedaerodynamic forces is discussed. The so-called strip theory as a first approximationis used. In this theory, it is assumed that the local force is proportional to the localangle of attack.10

Figure 2. Block diagram of the model of local aerodynamic forcesFigure 3. Structure of the uncertainties in aerodynamic coefficients of the finsThe local angle of attack α ∗ i at a point with coordinate x i on the longitudinal axisof the vehicle taking into account flexible oscillation is given as:α ∗ i = α + X c−X iθ̇ − q ı ∂q+̇ i(6)V i V ı ∂x iwhere, ∂q iis the slope of elastic line in current time,∂xq̇i is the velocity of shape ofielastic line. V i is the local air velocity, α is the angle of attack of solid body, ρ isthe air density, θ̇ is the angular velocity.11

Figure 4. Structure of the uncertainties in main aerodynamic coefficientsThe distributed aeroelastic force (F z (x)) along the longitudinal axis of thevehicle is evaluated as:F zi = ρ V i 2c ∗2ziα i (7)The effect of the appearance of elastic vibration in the feedback loop of thecontrol system is connected with elastic translational and rotational movement ofthe measuring system, the aerodynamic control surfaces and engine nozzle. Amathematical model of sensors, taking into account the flexural vibrations ofairframe (Elastic Sensors Subsystem) is shown in Figure 5.The sensors system includes models of measuring devices installed on the rocketbody, such as gyroscopes and accelerometers. They are presented in the form oftransfer functions. If the elastic components of the input sensors (structuralfeedback) is ignored or not taken into account properly, there are many instances,when self-exiting oscillation occurs on the body of the rocket. Increasing12

oscillation might lead to the destruction of the structure. Thus, the effect of elasticoscillations in the feedback loop must be addressed properly.Problems of dynamic interaction between the control system and elasticstructure depend on the location of the sensors, the distributed parameters ofelasticity, the natural forms and frequencies, and damping characteristics. To solvethem, various attempts are used such as:1. Notch filters are used to suppress the resonant frequencies.2. Selection of favorable sensors locations.3. Increasing the numbers of sensorsMeasuring system occupies a key place in the development of control systemsand angular stabilization of elastic aircraft. The correct choice and placement of thesensors used are of great importance to improve the accuracy of regulation. Theuncertainty of the natural forms of the elastic component of the measuredparameters of motion in the longitudinal plane is taken into account in the model ofthe sensors as shown in Figure 5.Figure 5. Model of the measuring system taking into account elasticityControl of sounding rocket is carried out by synchronous rotations of fouraerodynamic control surfaces that make up its fins. Model of the servo rudder ispresented in the form of transfer function (Fig. 6). The deflection of aerodynamiccontrol surfaces give rise to aerodynamic forces and moments, which mustcompliance with the specified control law.Deflection and rotation of the elastic line at the point of rotation of the rudders,as well as at the application point of the thrust vector of the solid motor leads toadditional aerodynamic loads on the rocket body, which in turn leads to itsdeformation. Effect of bending strain in the longitudinal plane on the action of therudders and thrust with the uncertainty of natural forms is shown in Figure 6.13

Figure 6. Block diagram of the effect of elasticity in the model of servo actuator ofaerodynamic rudders a) and models of engine thrust b)The set values of uncertainties, which correspond to the results of comparison ofindividual natural and computational experiments, can be changed at the designcontrol system, for example to compensate for the effect of manifestations in thefeedback loop. In this case, the nominal value of their own forms of flexuralvibrations and the range of variation defined by the synthesis problem.Chapter 4 of the dissertation work describes the procedure for the design of therobust controller for the sounding rocket. The procedure of linearization of themodel of the object at the given points of nominal trajectory was described. Theselection from the full model of the spatial motion only the rotational motion in thepitch plane taking into account flexural vibrations was carried out. Presentation ofa linear model in the form containing the nominal system model and a unifiedmodel of all the uncertainties was addressed. The question of the application of H ∞method in combination with μ - synthesis and analysis was addressed.A complete description of open-loop system that includes a model oflongitudinal motion of the rocket, taking into account the elasticity model, servoand sensor model is defined as the upper linear transformation (LFT).Parameters of the nominal model G are calculated for a series of points ofreference trajectories. State vector of the nominal model contains the variables inthe equation of rigid body dynamics, actuator state variables and state variablesmodel of elasticity (elastic vibration modes). For special conditions such as zerospeed or angle of attack, some variables of the state vector and certain parametersof uncertainty does not appear and can be excluded from the model. Aerodynamicmodel for the longitudinal motion of a rigid body contains five undeterminedcoefficients:c i = c x , c α δz , cyz, m ωy y , m δy y , i = 1 … 5αwhere c x is the aerodynamic drag force coefficient of the body and fins, c z is theδaerodynamic lift force coefficients due to the angle of attack, cyzis the14

aerodynamic lift force coefficient due to the deflection of the fin in the y-direction,m y ωy is the aerodynamic damping moment coefficient due to yaw rates ω y andm y δy is the aerodynamic control moment coefficient due to the fin deflection in they-direction.For the simulation of the flexible model considering only the first three modes offlexible oscillation, 12 coefficients of variation are added to the model namely,c i = c ω1 , c ω2 , c ω3 , c a , c g , c F , c T , i = 6 … 12where c ω1 , c ω2 , c ω3 are coefficients of variation of the natural frequencies of thefirst, second and third modes respectively, c a , c g , c F and c T are the coefficients ofvariation of the accelerometer data, gyroscope data, angle of desired fin deflectionand magnitude of thrust respectively due to structural feedback phenomena.These coefficients can be represented in the form:c = c̅ (1 + p c δ c ) (8)where p c is the relative uncertainty, c̅ is the nominal value of c and δ c is the normbound of the uncertainty coefficient. In this research the relative uncertainty isassigned as 10% and the norm bound:−1 ≤ δ c ≤ 1 (9)The uncertainty coefficient can also be presented in the form of upper linearfractional transformation:c = F U (M c , δ c ) (10)where M c = 0 cp c c (11)Figure 7. Structure of uncertainties in the form of LFTUncertainty coefficients can be isolated from the model in a separate uncertaintyblock ∆. The uncertainty block Δ is the diagonal of all the coefficients of variationof the model under investigation and consists of normalized parameter uncertaintywith zero nominal value and range of change from -1 to +1.Δ = diag(δ ci ) (12)The model of a system in the form of upper LFT F U (G, ∆) is shown in Figure 8.15

Figure 8. Structure of model of a system in the form of LFTInput signal u corresponds to the control signal, applied to the servo actuator,output signal θ′ - the measured angular velocity in pitch plane, output signal n z -measured vertical acceleration associated with the coordinate system.The designed control system must ensure the angular stabilization in the pitchplane and testing of the given control law according to the normal acceleration inthe presence of random external disturbances and noises.Block diagram of the closed-loop system in Figure 9 including feedback controllerK as well as blocks reflecting the uncertainty of the model and weighting functionsassociated with the implementation of performance requirements for closed-loopsystem.Figure 9. Block diagram of the closed-loop system for H ∞ synthesisThe system has a reference signal r, noises η a and η g on measurement of n y andθ̇, respectively, and two weighted outputs e p , and e u that characterise performancerequirements. The transfer functions W a and W g represent the dynamics of the16

accelerometer measuring n y , and the dynamics of the rate gyro measuring θ̇respectively. The coefficient K a is the accelerometer gain.The high frequency noises η a and η g occurred in measuring the normalacceleration and the derivative of the pitch angle, and satisfy the conditions:‖η a ‖ 2 ≤ 1, η g ≤ 1. Weighting function W p is chosen for amplification of the2mismatch signal e p in the low frequencies.Initially, to find the transfer function of the controller K the suboptimal H ∞synthesis procedure is used. The standard configuration of the system for H ∞synthesis is shown in Figure 9, where w denotes the external input signals, z –output signals, which must be minimized, y – the vector of available measuredcontroller K and u – vector of control signals.а) b) с)Figure 10. Standard system configuration for H ∞ synthesis (a) and for μ - synthesis(b) and (c)Block P(s) is called the generalized plant or interconnected system, when ∆ = 0(Figure 9), taking into consideration that:z = col(e p , e u ), w = col(r, η a , η g ), y= col(y1, y2) (13)The generalized model of system can be presented in the form:P(s) = P 11(s) P 12 (s)P 21 (s) P 22 (s) (14)z = [P 11 + P 12 K(I − P 22 K) −1 P 21 ] w = : F l (P, K)w (15)where F l (P, K) is the lower linear fractional transformation of P and K.The design objective is to find a stabilizing controller K such that the H ∞ -normof the closed-loop transfer function is less than a given positive number, i.e.,‖F l (P, K)‖ ∞ < γ (16)where17

γ > γ 0 ≔ min Kstabilizing ‖F l (P, K)‖ ∞ (17)The optimal solution is to find during iterations decrement of the given numberγ. The result of the synthesis of the H ∞ controller is given in chapter 5 of thiswork. Experience has shown that the H ∞ controller guarantees robust stability andnot robust performance. Such is the case in this research; analysis of the robustcharacteristics of the designed controller confirms the above statement. To obtainbetter result, µ synthesis is combined with H ∞ infinity.Stability of the system with fixed controller and various kinds of uncertaintiescan be investigated using μ-analysis. For the systems with M/∆-configuration forall perturbations ‖∆(s)‖ ∞ < γ of a given structure, robust stability holds if andonly ifsup ω≥0 µ(M(jω)) ≤ 1 γ(18)where μ(M) is structured singular value for the system M = F l (P, K).Solution of the problem of choosing the optimal controller, which ensures thatthe condition (18) is found by the μ-synthesis. Considering the requirements for thedynamic characteristics of a closed-loop system in the synthesis allows the additionof a fictitious uncertainty block ∆ F to the block of parametric uncertainties in themodel (Figure 10.c).Thus, the problem boils down to the choice of the regulator, providing thefulfillment of the condition (18), and the problem of maximum robustness (i.e.finding the maximum γ , for which robust stability can be ensured) to minimumM(P, K)(jω) for К:inf K(s) sup ωϵR µ[M(P, K)(jω)] (19)There is no direct method for solving this optimization problem. In practice, themethod of solution is called DK iteration. Although the DK iteration proceduredoes not guarantee convergence, but in most cases, converges to the minimumvalue of μ.The procedures of H ∞ and μ-synthesis are implemented for the extended modelof sounding rockets, taking into account all the above uncertainties andrequirements for dynamic characteristics of the system using the standard featuresof the MATLAB Robust Toolbox package.Chapter 5 gives the analysis of the results of the design of the controller usingthe H ∞ method in combination with μ-synthesis. Firstly a controller is designed forthe rigid rocket body using the H ∞ sub-optimal method and μ-synthesis approach.The design and simulation the controller using the H ∞ sub-optimal method incombination with µ synthesis is carried for forty second of flight duration. Theresults obtained for the controller design using for the rigid body model H ∞ sub-18

optimal approach is compared with the results of controller design using the µ-synthesis approach for the same rigid body dynamics without taking the effect offlexibility and aeroservoflexibility into consideration. This is to prove that the µ-synthesis controller gives a better robust performance for the same set of boundeduncertainties than a H ∞ sub-optimal controller (Figure 11) although bothcontrollers guarantees robust stability criteria (Figure 12).2.5ROBUST PERFORMANCE0.45ROBUST PERFORMANCE0.4mu21.510.5mu0.350.30.250.20.150.10.05010 -3 10 -2 10 -1 10 0 10 1 10 2Frequency (rad/s)010 -3 10 -2 10 -1 10 0 10 1 10 2Frequency (rad/s)Fig. 11 Robust performance characteristics of H ∞ controller-left, µ -controller right10 0 ROBUST STABILITY10 0 ROBUST STABILITY10 -210 -110 -410 -2mumu10 -610 -310 -810 -410 -1010 -3 10 -2 10 -1 10 0 10 1 10 2Frequency (rad/s)Fig. 12 Robust stability characteristics of H ∞ controller-left, µ -controller rightNext, the simulation of the sounding rocket model taking into consideration theflexibility and aeroservoflexibility effects of the first 3 modes of flexibleoscillations is carried out using the µ-synthesis method.The results obtained for theµ-controller designed for the rigid body dynamics and µ-controller designed forflexible body dynamics are then compared (Figure 13, 14, 15).Emphases are placed on the comparisons of the following characteristics, thetransient responses of the closed loop system with the design µ controller for a step1910 -510 -3 10 -2 10 -1 10 0 10 1 10 2Frequency (rad/s)

signal with magnitude ±15g which corresponds to a change in the normalacceleration n z with ±15g, the angle of deflection δ z of control fins in the closedloop system and frequency responses of the closed loop system with respect to thedesigned μ controller.From Figure 11 it is clear that the robust performance characteristics of theclosed loop system is not guaranteed with the design H-infinity controller(maximum value of the structured singular value frequency response, SSV isgreater than 1), but the mu-controller guarantees robust performance for the sameset of bounded uncertainties, SSV is less than 1. Since the mu controller givesbetter robust performance for the same set of uncertainties, the mu synthesis andanalysis approach is further used in the design of the controller for the flexibilitymodel of the sounding rocket. The first 3 modes of flexible oscillation weresimulated and the results of the simulation are compared with the results for the mucontroller designed for the rigid body model.From Figure 13 the transient response of the closed loop system with thedesigned mu controller for both the rigid model (right) and flexible model (left) isillustrated. The overshoot for the same reference signal for both controllers isunder 25% and the settling time about 3 seconds.20Response in n y25Response in n y152010151055n y(g)0n y(g)0-5-5-10-10-15-15-20-200 1 2 3 4 5 6 7 8 9Time (secs)-250 1 2 3 4 5 6 7 8 9Time (secs)Fig. 13 Transient response in n y for RM –left and for FM – rightFigure 14 illustrate the closed loop angle of control fin deflection δ z . The angleis about 18 degrees for the rigid body model and 8 degrees for the flexible model.20

0.25Closed-loop control and Fins deflection0.4Closed-loop control and Fins deflectionControl δ o0.20.15Control δ oFins δ0.30.2Fins δδ o (rads)0.10.050-0.05δ o (rads)0.10-0.1-0.1-0.15-0.2-0.2-0.3-0.250 1 2 3 4 5 6 7 8 9Time (secs)-0.40 1 2 3 4 5 6 7 8 9Time (secs)Fig. 14 Closed loop control and fin deflection for RM-left and for FM-rightThe closed loop magnitude plot shown in Figure 15 shows that over the lowfrequency region the magnitude response is close to 1 i.e. the reference signal canbe followed accurately. The closed loop system bandwidth is about 3 rad/sec. Theresults of the simulation obtained attest to the validity of the uncertainty modelused.20Closed-loop magnitude plot20Closed-loop magnitude plot00-20-20Magnitude (dB)-40Magnitude (dB)-40-60-60-80-80-10010 -1 10 0 10 1 10 2 10 3Frequency (rad/s)-10010 -1 10 0 10 1 10 2 10 3Frequency (rad/s)Fig. 15 Closed loop magnitude plot for RM-left and for FM-right21

Fig. 16 Fragment of the GUI software developed for simulation of rocket dynamicsIn this research a graphical user interface (GUI) software was developed usingMatlab (Figure 16). The GUI software allows for easy simulation and analysis ofthe rocket dynamics for different models. The flow chart of the overall simulationprocess is presented in Fig. 17. It will be of great assistance for control designengineering students in various institutions of higher learning, particular studentsof the State University of Aerospace Instrumentation in Saint Petersburg. Thedeveloped software would also be of immense benefit to entire technical staff ofthe National Space Research and Development Agency in Nigeria.22

Start6-DoF rigid body modelWeighting functionsselectionFlexibilitymodel needed ?Robust controller designFlexibility model of thefirst three modesMeetrequirements ?Rigid modeluncertainty blockController reductionFlexibility modeluncertainty blockMeetrequirements ?Local linearizationFigure 17 Flow chart of the simulation processConclusions1. In modeling of the uncertainties, it is necessary to simplify the model as muchas possible in order to avoid errors in linear approximation during theapplication of linear of linear robust control system design techniques for anonlinear plant.2. The uncertainty model derived for the flexible sounding rocket system containsreal parametric uncertainties in a highly structured form. Such a model appealto the application of µ-synthesis and analysis method having been usedextensively to synthesize a number of control laws for several different flexiblestructure experiments with great success.3. Both the H-Infinity and µ-controllers guarantees robust stability of thecorresponding closed loop system but the H-Infinity controller does notguarantee robust performance for a set of bounded uncertainties, which makes23End

it necessary to combine it with µ-synthesis to achieve better results in terms ofrobust performance for the same set of uncertainty bound.4. The nonlinear simulation results confirm the ability of the µ-controller toachieve disturbance attenuation and good responses to reference signals. Thesimulation confirms the validity of the uncertainty model used.5. The µ-controller may be used for various trajectories and Mach numbers.However, in order to control the rocket efficiently through the whole flightenvelope it may be necessary to implement several controller design fordifferent flight conditions.6. It is possible to reduced the controller order to as low as 10 without distortingthe dynamics of the nonlinear closed loop system.7. Better characteristics in terms of frequency and time responses of the closedloop system for the designed µ-controller can be achieved by tuning theweighting functions.Publications1. Аро Х.О. Применение методологии робастного синтеза к системеугловой стабилизации метеорологической ракеты. Известия вузов –Приборостроение, Санкт-Петербург, №11, 2011 (рецензируемыйнаучный журнал из Перечня ВАК, принято к публикации).2. Аро Х.О. Метод робастного синтеза аэрокосмического аппарата.Гироскопия и навигация, №3 (66), 2009, с.86 (рецензируемый научныйжурнал из Перечня ВАК).3. Аро Х.О. Разработка низкостоимостной интегрированнойнавигационной системы для метеорологической ракеты. Гироскопияи навигация, №3 (70), 2010, с.97 (рецензируемый научный журнал изПеречня ВАК).4. Aro H.O., Adetoro L.M. Review of Nigerian Space Programs. Proceedings ofthe annual Scientific Conference of SUAI. Saint Petersburg, Russia. 2009.5. Aro H.O., Adetoro L.M. Nigeria Space Programs. International ScientificRussian-American Journal, Actual Problems of Aviation and AerospaceSystems. Kazan, Daytona Beach, vol. 2(29). 2009.6. Aro H.O. Robust Control Approach of Aerospace Vehicle. Proceedings of the16 th Saint Petersburg International Conference on Integrated NavigationSystems. Saint Petersburg, Russia. 2009.7. Aro H.O. Investigating the Robust methods used in Controlling Non-stationary,Nonlinear Flexible Object. Proceedings of the IFAC Workshop on Aerospace,Guidance, Navigation and Control Systems. Samara, Russia. 2009.24

8. Aro H.O., Brodsky .S. A., Adetoro L.M. Intelligent Approach to ExperimentalEstimation of Flexible Aerospace Vehicle Parameters. Proceedings of 2 nd IFACInternational conference on Intelligent Control Systems and Signal Processing,ICON, Istanbul. 2009.9. Aro H.O. Algorithms for Robust Control of Flexible Flying Vehicle.Proceedings of the annual Scientific Conference of SUAI. Saint Petersburg,Russia. 2011.10. Aro H.O. Development of a Low-Cost Strap-down Integrated INS/GPS forSounding Rocket. Proceedings of the 17 th Saint Petersburg InternationalConference on Integrated Navigation Systems. Saint Petersburg, Russia. 2010.11. Аро Х.О. Влияние нежесткости корпуса на систему управленияметеорологической ракетой. Научная сессия ГУАП: Сб. докл.: В 3 ч. Ч.I.Технические науки/ СПб.: ГУАП, 2011. (принято к публикации).12. Aro H.O. Application of Robust Control Methodology for Sounding RocketAttitude Stabilization. Proceedings of the 5 th International Conference onRecent Advances in Space Technologies, Istanbul, 2011.25