Showing papers in "Journal of Guidance Control and Dynamics in 1987"
TL;DR: In this article, an algorithm for the direct numerical solution of an optimal control problem is given, which employs cubic polynomials to represent state variables, linearly interpolates control variables, and uses collocation to satisfy the differential equations.
Abstract: An algorithm for the direct numerical solution of an optimal control problem is given. The method employs cubic polynomials to represent state variables, linearly interpolates control variables, and uses collocation to satisfy the differential equations. This representation transforms the optimal control problem to a mathematical programming problem which is solved by sequential quadratic programming. The method is easy to program for a very general trajectory optimization problem and is shown to be very efficient for several sample problems. Results are compared with solutions obtained with other methods.
1,100 citations
TL;DR: An algorithm is created that can be used to predict the behavior of the beam when the base undergoes general three-dimensional motions, and fundamental flaws in certain multibody computer programs currently under development or already in use are drawn attention to.
Abstract: The behavior of a cantilever beam built into a rigid body that is performing a specified motion of rotation and translation is studied with two objectives in mind. First, because the subject is of interest in connection with spacecraft antennae, helicopter rotor blades, robot arms, and other systems that perform complex motions, we create an algorithm that can be used to predict the behavior of the beam when the base undergoes general three-dimensional motions. Effects such as centrifugal stiffening and vibrations induced by Coriolis forces are accommodated automatically, rather than with the aid of ad hoc provisions. The second objective is to draw attention to fundamental flaws in certain multibody computer programs currently under development or already in use. To this end, we construct a second simulation algorithm, one that embodies the procedure apparently employed in the programs in question, and then compare simulation results produced by computer programs based on the two algorithms. Conflicts between the two approaches that thus come to light are discussed in detail.
710 citations
TL;DR: In this paper, the synthesis of nonlinear flight test trajectory controllers for a fixed-wing aircraft is described using singular perturbation theory and the recently developed theory of prelinearizing transformations.
Abstract: Flight test trajectory control systems are designed to enable the pilot to follow complex trajectories for evaluating an aircraft within its known flight envelope and to explore the boundaries of its capabilities. Previous design approaches were baed on linearized aircraft models necessitating a large amount of data storage along with gain schedules. In this paper, the synthesis of nonlinear flight test trajectory controllers for a fixed-wing aircraft is described. This approach uses singular perturbation theory and the recently developed theory of prelinearizing transformations. These controllers do not require gain scheduling for satisfactory operation, can be used in arbitrarily nonlinear maneuvers, and are mechanized with a direct, noniterative analytic solution.
195 citations
TL;DR: In this paper, a fixed-duration, fuel-optimal rendezvous problem for the constant-mass case through the solution for the primer vector is investigated and the original differential equations that describe the spacecraft motion during unpowered flight.
Abstract: Based on the linearized equations of motion of a spacecraft near a satellite in general Keplerian orbit and the assumptions of a bounded thrust magnitude and constant exhaust velocity, a fixed-duration, fuel-optimal rendezvous problem is formulated and is investigated for the constant-mass case through the solution for the primer vector, which is shown to satisfy the original differential equations that describe the spacecraft motion during unpowered flight. It is shown that there are no singular solutions to this rendezvous problem for noncircular Keplerian orbits and that consequently all optimal solutions for orbit eccentricities greater than zero are constructed from a finite number of intervals of full thrust and coast where the switches are determined from the primer vector.
187 citations
TL;DR: In this article, a perturbation approach is presented in which the quantities defining the rigid-body maneuver are regarded as the unperturbed motion and the elastic motions and deviations from the rigid body motions are considered as the perturbed motion.
Abstract: This paper is concerned with the derivation of the equations of motion for maneuvering flexible spacecraft both in orbit and in an earth-based laboratory. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is presented in which the quantities defining the rigid-body maneuver are regarded as the unperturbed motion and the elastic motions and deviations from the rigid-body motions are regarded as the perturbed motion. The perturbation equations are linear, non-self-adjoint, and with time-dependent coefficients. A maneuver force distribution exciting the least amount of elastic deformation of the spacecraft is developed. Numerical results highlight the vibration caused by rotational maneuvers.
117 citations
TL;DR: In this paper, the authors examined the robust, multivariable control of large space structures by controllers designed on a reduced-order model using positivity concepts, and compared three different controller methodologies: the familiar multivariability control, individual modal control, and individual sensor control.
Abstract: This paper examines the robust, multivariable control of large space structures by controllers designed on a reduced-order model using positivity concepts. Controllers are designed using the DRAPER I and DRAPER II structures. Three different controller methodologies are compared: the familiar multivariable control, individual modal control, and individual sensor control. Controller robustness is measured qualitatively from the plots of the minimum singular value of the return difference matrix as a function of the frequency. All controllers, when designed to give the same total average control cost, have a very similar line-of-sight response. In addition, closed-loop stability can be maintained in the event of sensor and/or actuator failure.
97 citations
TL;DR: In this article, the authors defined the semimajor axis of satellite orbit, in units of the Earth's equatorial radius r a = semi-major axis, n.mi.
Abstract: Nomenclature A = semimajor axis of satellite orbit, in units of the Earth's equatorial radius r a = semimajor axis, n.mi. e = eccentricity h = satellite altitude, n.mi. J2 coefficient of the second harmonic of the Earth's potential function i = inclination, deg M =mean anomaly, deg P = semiparameter, = a ( 1 e) p = perpendicular distance from Earth center to satellite plane, n.mi. r = satellite radius, n.mi. Re =mean radius of Earth, =3442 n.mi. Amax = radial perturbation, ft Sl9S2,S3,S4 = satellite designator Si ,$2,£3,£4 = satellite surborbital points T = constellation period, h a = satellite visibility angle ("look angle") above horizon, deg co = argument of perigee, deg cb = rotation of perigee, deg/yr Q = right ascension of ascending node, deg Q = regression of line of nodes, deg/yr H = Earth's gravitational constant, = 1.4077xl0ftVs
92 citations
TL;DR: In this paper, the feasibility of a moving-bank multiple model adaptive estimator/controller was examined and five different adaptive controller algorithms were developed and compared to each other and to a benchmark filter with artificial knowledge of the "true" parameter value.
Abstract: The feasibility of a moving-bank multiple model adaptive estimator/controller is examined. Compared to a conventional full-bank multiple model adaptive algorithm, a significant reduction in the number of required elemental filters is accomplished through a dynamic re-declaration of the positions in parameter space that the elemental filters occupy. Critical to the performance of the moving-bank estimator is the decision method that governs movement of the bank of elemental filters. Five such methods are developed and their performances are compared to each other and to a benchmark filter with artificial knowledge of the "true" parameter value. Three adaptive controller algorithms are also generated and evaluated. A simple but physically motivated example is used to gain insights into relative performance potential of the proposed algorithms.
91 citations
TL;DR: The state-space techniques of modern control theory are used to develop a methodology for the design of autopilots for bank-to-turn missiles that accommodates the gyroscopic and coriolis cross-coupling between the pitch and the yaw axes that result due to the high roll rates that can be present.
Abstract: The state-space techniques of modern control theory are used to develop a methodology for the design of autopilots for bank-to-turn missiles. The methodology accommodates the gyroscopic and coriolis cross-coupling between the pitch and the yaw axes that result due to the high roll rates that can be present. The design uses the assumption that the roll rate is constant, but not zero, and results in an autopilot structure in which there are cross-couplings between the pitch and yaw channels that are dependent on the roll rate. The autopilot gains are also scheduled as functions of the dynamic pressure. A reduced-order extended Kalman filter, with fixed gains, is used to estimate the actuator states and the commanded acceleration. The performance of an autopilot designed by this methodology was evaluated in a six-degree of freedom simulation using the dynamics of a typical high-performance tactical missile. Excellent performance was obtained in teres of low miss distance and small side-slip.
91 citations
TL;DR: In this article, a real-time algorithm that estimates the mass-property parameters commonly used in spacecraft control laws is developed based upon a stochastic estimation viewpoint. But the method is limited to a single maneuver and the rate of convergence of each estimate depends strongly upon the particular maneuver being performed.
Abstract: Real-time algorithms that estimate the mass-property parameters commonly used in spacecraft control laws are developed based upon a stochastic estimation viewpoint. The elements of the inverse inertia matrix and the center-of-mass location vector are estimated from noisy measurements of the angular velocity using a secondorder filter, while estimates of the mass reciprocal are generated from noisy linear velocity measurements using a Kalman filter. Simulation results show that the rate of convergence of each estimate depends strongly upon the particular maneuver being performed, but that the mass properties can be estimated to within 1% error.
87 citations
TL;DR: In this article, an extension of the idea of using analytical redundancy to design a match between m components of the observation error space instead of using state estimates has been presented, which eliminates the need for state-space computations, thus producing an effective real-time fault monitor for fly-by-wire aircraft.
Abstract: A new method of analyzing faults in the m measurements of an nth-order system is presented. The proposed approach uses the estimation error space of each observer in a bank of observers to detect and isolate sensor faults. The designs are applied to a nonlinear model of an unmanned aircraft that has been described in previous publications. The reconfigurability of the aircraft sensor system is demonstrated, and the results show rapid recovery from a faulty sensor. The use of the observation error eliminates the need for state-space computations, thus producing an effective real-time fault monitor for fly-by-wire aircraft. N an earlier paper,1 a comparison of two techniques of instrument fault diagnosis (IFD) was made. This work is an extension of Patton and Willcox's idea of using analytical redundancy to design a match between m components of the observation error space instead of using state estimates di- rectly as discussed by Clark,3'4 Clark and Setzer,5 Frank and Keller,6 and Watanabe and Himmelblau.7 IFD in dynamic systems has received a significant amount of attention recently.2"11 Most methods described in the liter- ature discuss the analytical redundancy approach in prefer- ence to the use of redundant hardware. Analytical redundancy provides redundant (estimate) information from different measurements of a process, usually with observer or Kalman filter schemes. The commonly discussed state estimate solu- tion to IFD is based on the principle of generating estimates of part or all of the system state vector from subsets of the measurements, which when compared with similar estimates from other observers can be used to monitor the health of an instrument. The problem with the state estimate solution to IFD arises as the observer requires a good linear model of the process, and it must also be assumed that the disturbances on the system are well modeled or else have an insignificant effect on plant parameter variations. These limitations cause the state estimate approach to be inadequate for many real en- gineering applications. Sensitivity to input-induced parameter variations causes uncertain errors between redundant state estimate vectors, and in an IFD scheme these errors could cause false signaling of an instrument fault. It becomes clear that the bandwidth of uncertain signals should be estimated prior to the IFD system design. The use of frequency domain sensitivity information in this way enables a robust approach to the observer design to be made. The conjecture used is that the "innovations" or prediction error signals contain all the information concerning the parameter variations of the pro- cess being identified and controlled. Attention is thus turned toward the use of an innovations-based approach to system fault diagnosis that has wide potential applications. By using a weighting of the measurement estimation error as a parity
TL;DR: In this article, a novel approach to the dynamics of satellites with flexible multibody components is proposed, where the property of invariance under superposed rigid-body motions of geometrically-exact structural theories is employed to refer the dynamic motion directly to the inertial frame.
Abstract: A novel approach to the dynamics of satellites with flexible multibody components is proposed. The property of invariance under superposed rigid-body motions of geometrically-exact structural theories is employed to refer the dynamics of motion directly to the inertial frame. To avoid numerical ill conditioning, the dynamics of the far field and the near field are treated separately by introducing a rotationally-fixed floating frame, which is a parallel translate of the inertial frame. Constraint conditions to determine the orientation of floating frames proposed in the past are thus entirely bypassed. The proposed formulation can accommodate an unrestricted class of maneuvers under the action of follower forces and gravitational force, and is particularly suited for the dynamics of flexible multibody systems undergoing a broad range of deformations.
TL;DR: Guidance strategies for near-optimum performance in a wind shear are examined in this paper, where the takeoff problem is considered with reference to flight in a vertical plane; the presence of a downdraft is assumed.
Abstract: Guidance strategies for near-optimum performance in a wind shear are examined The takeoff problem is considered with reference to flight in a vertical plane; the presence of a downdraft is assumed Trajectories for optimum performance in a wind shear are determined for different wind shear models and intensities Numerical experiments with the optimum control approach lead to the conclusion that, for weak to moderate shear/downdraft combinations, the optimal trajectory is characterized by a monotonic climb, and for severe shear/downdraft combinations, it is characterized by an initial climb, followed by a nearly horizontal flight, followed by renewed climbing after the aircraft has passed through the shear region An acceleration guidance scheme based on relative acceleration is presented in both analytical form and feedback control form Numerical results with this scheme result in trajectories close to the optimum and considerably superior to those arising from alternative guidance schemes
TL;DR: In this paper, a closed-loop, nonlinear optimal guidance law for three-dimensional flight for both the mid-course and terminal phases is presented, which can quickly modify the missile trajectory during midcourse guidance when the target direction changes.
Abstract: This paper presents a combined midcourse and terminal guidance law design for missiles to achieve range enhancement with excellent intercept performance. We derive analytic solutions of a closed-loop, nonlinear optimal guidance law for three-dimensional flight for both the midcourse and terminal phases. This combined guidance law can quickly modify the missile trajectory during midcourse guidance when the target direction changes. Zero heading error is achieved at handover from the midcourse to the terminal phase. The guidance algorithm is in a feedback form, either in inertial coordinates or in seeker coordinates. It is sufficiently simple for onboard implementation and has been applied successfully for on-line operation.
TL;DR: In this paper, the authors considered the combined structural and control optimization problem for flexible systems, where the sum of structural mass and controlled system energy terms is minimized simultaneously in structural and controlling parameters using gradient searches.
Abstract: This paper considers the combined structural and control optimization problem for flexible systems. The sum of structural mass and controlled system energy terms is minimized simultaneously in structural and control parameters using gradient searches. The purpose of control is to effectively suppress structural vibrations due to initial excitations. Starting with a baseline structural design, the objectives of the combined structural/control optimization are twofold: 1) to reduce the structure's mass, and 2) to reduce total system strain, kinetic and control energies observed when the structure is excited. The appropriate weighting of energy and structural mass for balanced design and the dependence of optimal designs on initial conditions are discussed. The ideas presented are illustrated through numerical simulations using a 10-bar cantilevered truss.
TL;DR: In this paper, the authors consider the problem of designing control laws for linear systems with time-varying uncertainty and present a method for determining a linear feedback control which stabilizes the system for all possible uncertainty.
Abstract: This paper considers the problem of designing control laws for linear systems with time-varying uncertainty. A method for determining a linear feedback control which stabilizes the system for all possible uncertainty is presented. This control is robust in the sense that it guarantees asymptotic stability regardless of the disturbance. The results are applied to several aircraft examples .
TL;DR: In this article, a Space Station attached tethered system for micro-g/variable-g applications is described, which consists of three platforms: the Space Station, an end mass anchored at the end of a 10 km long kevlar tether and a micro g/variable g laboratory with the capability of crawling along the tether.
Abstract: This paper describes a Space-Station attached tethered system for micro-g/variable-g applications. The system consists of three platforms: the Space Station, an end mass anchored at the end of a 10 km long kevlar tether and a micro-g/variable-g laboratory with the capability of crawling along the tether. Control strategies are devised for performing both the deployment and the station-keeping maneuvers of the system. Effective algorithms are identified for damping out the major vibrational modes.
TL;DR: In this article, it was shown that the assumption that only directional changes contribute to eigenvector sensitivity is incorrect and the resulting equations are incorrect, except for the classical and most commonly encountered case of a self-adjoint problem having orthogonal eigenvectors.
Abstract: Analytical expressions for eigenvector derivatives for general non-self-adjoint systems using a modal expansion approach have not been correctly derived in several papers and books that address this problem. A common mistake in several developments on deriving eigenvector derivatives (or its perturbation forms) has been to ignore the eigenvector in question, in the expansion of its derivative, although the remaining eigenvectors forming the basis are generally non-orthogonal. This assumption is based upon explicit or implicit heuristic arguments that only directional changes contribute to eigenvector sensitivity. It is shown in this paper that the above assumption and the resulting equations are incorrect, except for the classical and most commonly encountered case of a self-adjoint problem having orthogonal eigenvectors. For the non-self-adjoint case, certain basis coefficients in the eigenvector derivative expansion have not been resolved correctly in the literature. A careful re-examination of the eigenvalue problem reveals that the two independent sets of normalizations (required for uniquely specifying the right and left eigenvectors) can be used to uniquely determine the basis coefficients. The solution derived herein for the eigenvector derivatives is shown to have generally nonzero projections onto all eigenvectors. It is also shown that basis coefficients for left and right eigenvector derivatives are related by a simple expression. A numerical example is included to demonstrate the present formulation for eigenvector derivatives with respect to a scalar parameter. We also extend Nelson's algebraic approach (for self-adjoint eigenvalue problems) to the general non-self-adjoint problem and the modal truncation approach to approximate eigenvector derivatives for the non-self-adjoint case.
TL;DR: In this paper, the authors considered a roll moment that can be induced by a trim angle and can cause resonant lock-in spin and derived the existence and stability conditions for equilibrium spin.
Abstract: : A slightly asymmetric missile is a basically symmetric missile with a nonzero pitch moment at zero angle of attack. In flight this moment causes a trim angle that rotates with the missile and has a maximum when the spin is near resonance with the missile's natural pitch frequency. This report considers a roll moment that can be induced by this trim angle and can cause resonant lock- in spin. Simple expressions for this induced rolled moment are given and the existence and stability conditions for equilibrium spin are derived. The special case of the induced roll moment caused by a radial center of mass offset is considered and a number of different types of possible equilibrium spin combinations are shown. The possibility of resonant lock-in spin in the opposite sense to the expected steady-state spin is indicated. It is further shown that there are induced roll moments for which the design steady-state spin will not occur under any launch conditions. If several stable equilibrium spins are possible, the one that occurs in flight can be determined by the orientation of the initial pitch angular velocity. Finally the form of an induced pitch moment is given and its possible effect on the angular motion is discussed.
TL;DR: In this paper, the authors introduce the concept of a gyroelastic spacecraft, which is a vehicle comprising not only a continuous distribution of mass and elasticity but also of gyricity (stored angular momentum) as well.
Abstract: This paper introduces the concept of a gyroelastic spacecraft—a vehicle comprising not only a continuous distribution of mass and elasticity but a continuous distribution of gyricity (stored angular momentum) as well. It is assumed that the spacecraft has a number of gyroelastic appendages and that the (constrained) mode shapes are known for each of these appendages. General vehicle deformations are expanded in terms of these mode shapes. The eigenproblem for the (unconstrained) vehicle reveals a significant departure from the modal behavior of nongyric elastic vehicles. In particular, a gyroelastic vehicle can exhibit a "pseudorigid" mode in which the vehicle rotates uniformly in a deformed state. Although the frequency of this models zero, the associated strain energy is nonzero, and it is therefore not a "rigid" mode.
TL;DR: In this paper, the eigenstructure assignment flight control design methodology is extended to include dynamic compensator synthesis and damping ratio sensitivity reduction, where sensitivity measures are developed that relate the perturbation of the damping ratios to perturbations in the stability derivatives.
Abstract: The eigenstructure assignment flight control design methodology is extended to include dynamic compensator synthesis and damping ratio sensitivity reduction. Dynamic compensators may be designed via eigenstructure assignment by utilizing a composite system structure. The success of this design methodology depends upon proper choice of the desired eigenvectors. Sensitivity measures are developed that relate the perturbation of the damping ratio to perturbations in the stability derivatives. A damping ratio sensitivity plot is introduced that allows the damping ratio sensitivity to be reduced without altering the nominal damping ratio. Examples of the lateral dynamics of an L-1011 aircraft are presented to illustrate the design methods.
TL;DR: In this paper, the authors present an algorithm that processes a sequence of pairs of measured vectors to obtain a minimum variance estimate of the Euler angles which describe the attitude between two coordinate systems.
Abstract: This work presents an algorithm that processes a sequence of pairs of measured vectors to obtain a minimum variance estimate of the Euler angles which describe the attitude between two coordinate systems. The measurement equation is nonlinear and, unlike the direction cosine matrix and the quaternion estimation problems, the dynamics of the estimated angles are nonlinear as well. The algorithm can also handle singular cases, thus extending the use of Euler angles to all-attitude vehicles. Results of Monte-Carlo simulations are presented that demonstrate the efficiency of the algorithm. This work is a natural extension of the algorithms that were recently developed for estimating the direction cosine matrix and the quaternion, therefore it completes the set of recursive minimum variance algorithms for estimating the three common forms of expressing attitude.
TL;DR: In this paper, a closed-form necessary and sufficient condition for the stabilizability of linear time-invariant systems by a controller of chosen order is established by solving a constrained optimization problem and results in a system of nonlinear matrix equations.
Abstract: Constant-gain, fixed-order controllers for linear time-invariant systems are considered. A closed-form necessary and sufficient condition for the stabilizability of such a system by a controller of chosen order is established. This criterion is obtained by solving a constrained optimization problem and results in a system of nonlinear matrix equations. A method based on homotopy is proposed and studied to solve this system of nonlinear equations. The numerical implementation of the homotopy is discussed and various properties of the optimizing feedback control as a function of the homotopy parameter are established.
TL;DR: In this article, it is shown that once per-orbit modulations of the length of an orbiting tether can be used to modify the eccentricity, energy, and line of apsides orientation of its center of mass while maintaining a constant angular momentum and orbital parameter.
Abstract: It is shown that once-per-orbit modulations of the length of an orbiting tether can be used to modify the eccentricity, energy, and line of apsides orientation of its center of mass while maintaining a constant angular momentum and orbital parameter. Appropriate length variation laws are developed and the effects are quantified and interpreted. Several applications are discussed, including modifications to the orbit of a space station, reduction by 30% of the A V to launch a satellite to 12 h orbit, and the relocation of the apogee for observation satellites. HE common observation of children "pumping" themselves up at a swing by rhythmically shifting their legs' weight has prompted speculation about the possibility of performing fuelless orbital changes using conformable spacecraft.1'2 Particularly intriguing are the possibilities opened by the advent of ultralong, thin tethers that can be conveniently reeled in and out from a satellite while carrying another body at its end. Obviously, any such reel-unreel operation will set up Coriolis forces that will induce in-plane libration of the tether. To conserve angular momentum, equal and opposite variations will appear in the center of mass orbital angular momentum. If these can be made to resonate with the orbital motion, it is conceivable that the system could bootstrap itself and either increase or decrease the orbital energy without setting up unbounded librations (and getting wrapped up in the process). A clear limitation of any such scheme is the need to supply mechanical power onboard. Another one is the impossibility of modifying the overall angular momentum in this manner, since operation in a central force field is assumed and no internal angular momentum is allowed to accumulate. The question of whether the center of mass motion can be affected by means of forces internal to the mass-tether system is more subtle; although it is true that the acceleration of the center of mass is due to only the net external force, nonrigid changes to the configuration of the system can affect the net external force for a given center of mass location and hence can affect the center of mass motion. In this paper we show how this can be accomplished using a rewindable tether. The practical application of the concept to several space transportation missions is also discussed. The analysis presented here is restricted to the small oscillation of a "dumbbell" satellite whose cable length is assumed fully controllable by means of powered winches on board one or both of the connected masses.
TL;DR: Through analytical evaluations and simulation studies, it is shown that the proposed algorithm is superior to the conventional nearest neighbor algorithm.
Abstract: A track correlation algorithm in a multi-sensor integration system for a surveillance mission is proposed. The performance of such track correlation processing is strongly dependent not only on the target state estimation error distribution but also on the target spatial density distribution. Therefore, a track correlation problem is formulated as the likelihood ratio test problem which can take both target state estimation error distribution and target state spatial density distribution into consideration. From this formulation, the correlation algorithm for on-line processing is derived by modifications and approximations. Through analytical evaluations and simulation studies, it is shown that the proposed algorithm is superior to the conventional nearest neighbor algorithm.
TL;DR: Basic error equations of inertial navigation which apply to any properly constructed inertial navigator are derived by a vectorial analysis, finding the conventional velocity and position errors are shown to be transfer errors.
Abstract: This paper derives basic error equations of inertial navigation which apply to any properly constructed inertial navigator. The equations are deduced from the integral equations of inertial navigation by a vectorial analysis. A major result of this analysis is a set of fundamental error propagation equations that has apparently been missed. These equations regard the absolute navigational errors. The conventional velocity and position errors are shown to be transfer errors.
TL;DR: A two-time scale discrete control system is considered and a composite, closed-loop suboptimal control is created from the sum of the slow and fast feedback optimal controls.
Abstract: A two-time scale discrete control system is considered. The closed-loop optimal linear quadratic (LQ) regulator for the system requires the solution of a full-order algebraic matrix Riccati equation. Alternatively, the original system is decomposed into reduced-order slow and fast subsystems. The closed-loop optimal control of the subsystems requires the solution of two algebraic matrix Riccati equations of order lower than that required for the full-order system. A composite, closed-loop suboptimal control is created from the sum of the slow and fast feedback optimal controls. Numerical results obtained for an aircraft model show a very close agreement between the exact (optimal) solutions and computationally simpler composite (suboptimal) solutions. The main advantage of the method is the considerable reduction in the overall computational requirements for the closed-loop optimal control of digital flight systems.