Nonlinear model-following control application to airplane control

Abstract

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.