Harold Stalford

Robust control of uncertain systems in the absence of matching conditions: Scalar input

Robust control is derived for general linear uncertain systems with scalar control. Matching conditions are not assumed to hold. The solution to this general class is made possible by transforming the original system to canonical controllable form. A constant switching surface Fx = 0 is constructed by designing the closed-loop characteristics as a function of the uncertainties so that the uncertainties in the positive definite solution of Lyapunovs equation cancel the uncertainties in the canonical controllable transformation. The derived control law u(x) has a linear negative feedback term -Kx and a nonlinear term ¿u(x) that switches on the switching surface Fx = 0. The nonlinear component ¿u(x) of the robust control law is shown to depend on the product of the canonical controllable transformation and the difference between the closed-loop and open-loop characteristics. Examples are presented.

Accurate Modeling of Nonlinear Systems using Volterra Series Submodels

We investigate the problem of accurately modeling nonlinear systems (such as aircraft flight in high angle-of-attack/sideslip flight) using simple low-order Volterra submodels. First, we apply this technique to a simplified nonlinear stall/post-stall aircraft model for the case of a longitudinal limit cycle. Our simulation study demonstrates that the responses of the Volterra submodels accurately match the responses of the original nonlinear model, whereas the responses of a piecewise-linear model do not. Next, we apply the technique to a simplified high a nonlinear model of wing rock. Our simulation study demonstrates that the second-order Volterra approximation predicts the wing rock limit cycle, while a linear approximation does not. Third-, fourth- and fifth-order Volterra approximations are observed to give wing rock amplitudes that converge quadratically to the nonlinear value.

On Robust Control of Wing Rock using Nonlinear Control

Nonlinear control theory of uncertain systems is applied to the high angle-of-attack flight dynamic phenomenon of wing rock. Wing rock, an unsteady aerodynamic effect, is an undamped oscillation primarily in the roll axis and is exhibited by many modern combat aircraft. It causes maneuver limitations ranging in severity from degradation in tracking effectiveness to loss of control. Robust control is investigated for a generic nonlinear wind tunnel model of a modern combat aircraft that exhibits wing rock. Feedback control is derived using the nonlinear control approach of uncertain systems. A typical example is presented.

A necessary and sufficient condition in Lyapunov robust control

We consider Lyapunovs equationPA+ATP+Q=0, whereQ is symmetric positive definite andA is in controllable companion form. We prove that a necessary and sufficient condition thatA be stable is that the first rowP1 of theP-matrix be a stablen−1 coefficient vector. This result is related to the minimum phase property of linear systems and is useful in designing robust controllers.

Necessary and Sufficient Conditions for Matching Conditions in Uncertain Systems: Scalar Input

The uncertain transformation of scalar input uncertain systems to controllable canonical form is used to derive necessary and sufficient conditions for the satisfaction of matching conditions. In particular, it is shown that the matching conditions imply that the uncertain system is completely controllable for all uncertainties. They also imply that the controllable canonical dynamics matrix with zero eigenvalues when transformed back to the original state space by the uncertain transformation is a constant dynamics matrix. Equations are given for computing the matching conditions matrices in terms of the uncertain coefficients of the characteristic polynomial and the uncertain transformation. In addition, the necessary and sufficient conditions provide insight in the development of robust control for uncertain systems in the absence of matching conditions.

On the robustness of linear stabilizing feedback control for linear uncertain systems: multi-input case

The robust stabilization of linear systems with constant uncertainties against structured perturbations using Lyapunovs theory is investigated. The only information needed on the uncertainties is the knowledge of their boundaries. The matching conditions of the uncertain systems are not required to be satisfied. It is first shown that, under some assumptions, the system can be transformed into a certain canonical controllable companion form. Then, under some additional assumptions, the existence of a linear controller which stabilizes the system based on Lyapunovs theory is shown.

Classical turning performance of a fighter aircraft revisited

I n this paper, we consider a n analytical model of a high angle of attack fighter aircraft. We relate our 6-degree-offreedom dynamics with the type of dynamics treated previously by other authors. We evaluate classical turning performance characteristics. Finally we explore simple turning tactics based on multiple time scales. The results show that turning performance with constant energy along the trajectory is only a small improvement over a classical index of turning performance defined by constant altitude and cons tant speed. Thrust vectoring was found to have a n insignificant effect on turning performance for constant altitude turns and for turns in which energy is constant along the trajectory.

Fast half-loop maneuvers for a high alpha fighter aircraft using a singular perturbation feedback control law

Singular perturbation analysis is used to derive an outer layer feedback control law for a high alpha fighter aircraft to perform the half-loop maneuver. Pitch rate and angle of attack are treated as fast variables in the derivation. Bang-bang controls are derived to transfer the aircraft state from trim to the outer layer and from the outer layer to specified final half-loop values. The pitch rate is treated as a varibale faster than the angle of attack in the transfer of the state to and from the outer layer. A simulation of the derived control law is conducted at Mach 0.6 and 15,000 feet altitude. The half-loop was performed in 13.12 seconds. It is compared with a NASA pilot simulated half-loop maneuver which took 22.42 seconds for the same initial conditions.

On the Robustness of Linear Stabilizing Feedback Control for Linear Uncertain Systems

We investigate linear time-invariant scalar-input systems with constant uncertainties that are not required to satisfy matching conditions. In a previous paper, the existence of stabilizing discontinuous controllers is established under three assumptions for such systems. The first assumption requires controllability of the system for each uncertainty. The second is a condition on an uncertain Lyapunov equation, and the third is a boundedness condition related to the controllability matrix. In this paper, using the same assumptions, we show the existence of a linear stabilizing control. Our result is related to the high-gain theorem of classical control.

Recent work using volterra series as a methodology to analyze nonlinear aircraft dynamic properties

Preliminary results are discussed which demonstrate the accuracy of the Volterra series approach and its potential for defining nonlinear flying quality parameters. Simulations are presented which show that two-term Volterra submodels predict nonlinear responses, such as limit cycles, that are not predicted using piecewise linear techniques. Incorporating higher order terms in the approximation is shown to result in more accurate predictions.