Regulation of relaxed static stability aircraft

Abstract

We formulate and solve a regulator problem for nonlinear parameter dependent dynamics. It is shown that the problem is solvable except at parameter values associated with bifurcation of the equilibrium equations and that such bifurcations are inherently linked with the system zero dynamics. These results are applied to the study of the regulation of the longitudinal dynamics of aircraft. It is shown how the ability to regulate aircraft flight path is dependent on system parameters. Computational studies focus on changes in the center of gravity location. We indicate how design choices affect the ability to regulate over a range of parameter values.

Constructing linear families from parameter-dependent nonlinear dynamics

“…1, adapted from [2]. Notice that the surface, although smooth, may have folds, and consequently we cannot expect globally valid functions x(); u() to define equilibria.…”

“…If x0; u0; 0 corresponds to an equilibrium point, that is, f (x 0 ; u 0 ; 0 ) = 0 and y 0 = h(x 0 ; u 0 ; 0 ) = 0, the linear perturbation equations are _ x = @f(x0; u0; 0) @x x + @f(x0; u0; 0) @u u y = @h(x 0 ; u 0 ; 0 ) @x x + @h(x 0 ; u 0 ; 0 ) @u u: (2) Suppose, however, that we wish to construct a family of linear models in which the parameter is considered to be an independent variable. Then, in principle, we need to solve the algebraic equilibrium equations for x 0 (); u 0 () in order to obtain _ x = @f(x 0 (); u 0 (); ) @x x + @f(x 0 (); u 0 (); ) @u u = A()x + B()u y = @h(x0(); u0(); ) @x x + @h(x0(); u0(); ) @u u = C()x + D()u: The need to characterize the dependence of equilibria on the parameters is the essential and difficult aspect of the linearization problem.…”

Aircraft Accident Prevention: Loss-of-Control Analysis

“…The paper extends results previously introduced by the authors in. 4,5 There we showed that the ability to regulate a system was lost at points associated with bifurcation of the trim equations; ordinarily indicating stall in an aircraft. Such a bifurcation point is always associated with a degeneracy of the zero structure of the system linearization at the bifurcation point.…”

“…4 At the bifurcation point an arbitrarily small perturbation of parameters changes the zero structure of the system thereby requiring a fundamental change in the controller. 6 The point we wish to emphasize is that losing the capacity to regulate nonlinear flight dynamics is intimately connected to the bifurcation structure of the trim equations of the aircraft.…”

“…[24][25][26] Only in the past two decades have formal methods of bifurcation analysis been applied to aircraft. 4,[27][28][29][30][31][32][33][34][35][36] Bifurcation analysis has been employed to identify the conditions for occurrence of undesirable behaviors, to investigate recovery methods from dangerous post bifurcation modes and to formulate feedback control systems that modify bifurcation behavior.…”