Wayne C. Durham

Constrained control allocation

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.

Computationally efficient control allocation

A computationally efficient method for calculating near-optimal solutions to the three-objective, linear control allocation problem is disclosed. The control allocation problem is that of distributing the effort of redundant control effectors to achieve some desired set of objectives. The problem is deemed linear if control effectiveness is affine with respect to the individual control effectors. The optimal solution is that which exploits the collective maximum capability of the effectors within their individual physical limits. Computational efficiency is measured by the number of floating-point operations required for solution. The method presented returned optimal solutions in more than 90% of the cases examined; non-optimal solutions returned by the method were typically much less than 1% different from optimal and the errors tended to become smaller than 0.01% as the number of controls was increased. The magnitude of the errors returned by the present method was much smaller than those that resulted from either pseudo inverse or cascaded generalized inverse solutions. The computational complexity of the method presented varied linearly with increasing numbers of controls; the number of required floating point operations increased from 5.5 i, to seven times faster than did the minimum-norm solution (the pseudoinverse), and at about the same rate as did the cascaded generalized inverse solution. The computational requirements of the method presented were much better than that of previously described facet-searching methods which increase in proportion to the square of the number of controls.

Multiple Control Effector Rate Limiting

The effect of the choice of control allocation scheme upon individual control effector rate demands is examined. Three previously reported and one new variation of control allocation schemes are described and compared. The new allocation scheme exploits the maximum attainable moment rates of a given control effector configuration and is called moment-rate allocation. The bases of comparison are single-axis sinusoidal and triangular sawtooth, and multiaxis helical time-varying moment demands placed upon the allocation schemes. It is shown that 1) the choice of allocation scheme greatly affects the onset of effective rate limiting for a particular time-varying input but not consistently for all inputs; 2) widely varying results are obtained depending on whether the input is singleor multiaxis and on the amplitude and shape of the input; 3) none of the observed behavior (except as regards generalized inverses) was easily predictable for arbitrary time-varying inputs; and 4) from the point of view of moment and moment-rate generating capabilities, moment-rate allocation clearly yielded best results.

Unified development of lateral-directional departure criteria

Several frequently used departure prediction indicators for both open- and closed-loop control of flight are developed using a unified, rigorous analytical approach applied to a linear version of the aircraft model. These criteria are for departure caused by aerodynamic disturbances only. It is shown that these indicators are limited in their accuracy because of restrictive assumptions and terms omitted. A second approach is presented that leads to the same results as the first, but is more applicable to the nonlinear problem. Some ideas concerning the application of the linear methods to the nonlinear problem are presented.

Control Stick Logic in High-Angle-of-Attack Maneuvering

The relationships between pilot control stick inputs and control effector deflections are examined. Specifically, we address multiply redundant control effector arrangements and command-driven control laws. During high-angleof-attack, low-dynamic-pressure maneuvering, there is both a control power and control coordination problem. Control effector deflections are not one to one with pilot inputs, and the maximum capabilities of effectors to respond to pilot inputs varies dynamically with the state of the airplane. The problem is analyzed in the context of a generic control law that continuously regulates sideslip. A means is presented to relate the fixed control effector limits to the dynamically varying control response limits. This information may be used to re-establish the one-to-one correspondence of pilot inputs to control capabilities. Nomenclature F = force G = gearing ratio / = moment of inertia L = transformation matrix L, D, C = aerodynamic lift, drag, and sideforce, wind axes L,M,N = rolling, pitching, and yawing moments m = mass m = vector of moments (or moment coefficients) p,q,r = rolling, pitching, and yawing rates T = thrust u = vector of control effectors V = velocity a = angle of attack ft = sideslip angle

Dynamic inversion and model-following control

The similarities and differences of dynamic inversion control and model-following control laws are examined. For the forms of these control laws assumed in this paper it is shown that dynamic inversion may be considered a special case of model-following. For any given dynamic inversion control law there is a model-following control law that achieves exactly the same response and therefore is in every way equivalent to it. This same model-following control law may be modified in its error dynamics without changing the desired response implied by the dynamic inversion law. The modification in error dynamics may be used to improve the tracking of the desired response in the presence of modeling errors. (Author)

NONLINEAR MODEL-FOLLOWING CONTROL APPLICATION TO AIRPLANE CONTROL

Nonlinear model-following control design is applied to the problem of control of the six degrees of freedom of an airplane that lacks direct control of lift and side force. The nonlinear expressions for the error dynamics of the model-following control are examined using Lyapunov stability analysis. The analysis results in nonlinear feedforward and feedback gains that are functions of the airplane and model states. As a consequence, gain scheduling requirements for the implementation of the model-following control are reduced to only those involving the estimation of stability and control derivatives of the airplane. The use of these gains is shown through an example application to the control of a nonlinear aerodynamic and engine model provided by NASA Ames-Dryden Flight Research Facility. The model being followed is based on a trajectory generation algorithm, and represents a form of dynamic inversion. HE design methodology to be used is based on the applica- tion of nonlinear model-following to the problem of the control of the six degrees of freedom of an airplane. This methodology is related to nonlinear inverse model theory. It is a more complete approach in that it provides a means for analysis of the dynamics of the errors involved in model-follow- ing. The particular approach has been successfully applied to the control of a nonlinear aerodynamic model of a high-angle- of-attack research vehicle (HARV) through large attitude and angle of attack changes. In general, model-following control attempts to make an actual airplane behave similarly to a prescribed mathematical model of an airplane with different force and moment character- istics than the actual airplane. The model behavior may be based on desirable flying qualities, and the matching of those flying qualities is taken to be the design objective. In this case the pilot controls are applied to the model (either conceptually or literally, to a simulation) and the airplane controls are deter- mined. Alternatively, the mathematical model may be a simplified representation of the actual airplane being controlled, in which case model-following control becomes a solution to the inverse problem. Here the state trajectory of the model is determined from a specification of a particular flight path or maneuver, and the airplane controls required to follow it are determined. Perfect, explicit model-following solutions to the inverse prob- lem provide more than the open-loop controls required to fly a maneuver, since this formulation allows control of the errors between the airplane and model during the maneuver. It is this application of model-following control that is used in this paper. To develop the nonlinear model-following controller, we will first review the model-following concepts used here. Initially a standard form of the airplane and model equations is presented with the conditions for perfect dynamic matching presented. Associated with the conditions for perfect dynamics matching are differential equations for the error. In many cases these error equations are linearized and standard linear control ideas applied to guarantee stability (i.e., they tend to go to zero in time). Hence one is led to a gain scheduling scheme. In the method presented, however, using an approach based on the stability theory of Lyapunov, a set of gains which insure stability of the nonlinear error dynamics can be found. These require no updates but are functions of the current state. The result of this analysis is illustrated through application to the nonlinear airplane simulation provided by NASA Ames- Dryden Flight Research Facility. In this application, the model being followed is a simplified description of the airplane being controlled. The model is not, however, directly flown by exter- nally applied (pilot) controls. Rather, it represents the states and state rates required to execute some prescribed maneuver.

CONTROL ALLOCATION WITH ADAPTIVE FAILURE CONTROL

In the age of modern aircraft and fly-by-wire control systems, the inclusion of mechanical backup systems for handling instances of control failures is becoming more uncommon. As a result, pilots are forced to fully rely on the failure immunities designed into these systems and be assured that, should such failures occur, the aircraft maintain adequate flying qualities long enough for a safe ejection or an emergency landing. This paper demonstrates the control reconfiguration aspects of a control allocation with rate limiting algorithm to adapt to various control failures while still achieving desired moments, thus providing a safe environment for the pilot.

PERFECT EXPLICIT MODEL-FOLLOWING CONTROL SOLUTION TO IMPERFECT MODEL-FOLLOWING CONTROL PROBLEMS

For cases in which perfect model-following is not possible for a particular desired model, a class of candidate models is defined that can be followed perfectly by the given plant. A candidate model that most closely matches the dynamics of the desired model is then determined through constrained parameter optimization. The result is perfect model-following of a model that has an eigenstructure which resembles that of the desired model. In the development of this method, a new variation on perfect model-following control law development is shown. This method explicitly displays the feed-forward and feedback gains that determine the system error dynamics, which may be artibrarily selected by conventional pole placement methods if the plant is completely controllable. The method is applied to a problem involving the linearized lateral-directional equations of motion of the B-26 airplane. The results show that a candidate model can be found that has virtually the same dynamic behavior as the desired model, and that it can be followed perfectly by the original plant with arbitrarily assigned error dynamics.

Flight simulation in flight dynamics education

Thii paper describes the use of manned Sight simulation for flight dynamics education. The uses of a simulator for instruction and some of the capabilities that make these uses possible are discussed. Specific examples of the use of simulation in classes in the Virginia Tech Aerospace Engineering curriculum are presented. Lessons learned from these examples included the diiculty in scheduling student use of the simulator, the importance of sufficient preparation before demonstrations, and the need for students to be involved in preand post-experiment briefings and demonstrations.