David F. Chichka

Peak-seeking control with application to formation flight

A new controller is proposed for linear systems with unknown optimal operating points. It is assumed that the desired operating point represents the maximum of a function of the states of the system. The controller uses a modified Kalman filter to estimate the derivatives of this function, and the controller drives the system to the optimum. The controller is derived in the context of the formation flight of aircraft for drag reduction, and a simulation is used to provide preliminary results.

Peak-Seeking Control for Drag Reduction in Formation Flight

Formation flight is a known method of improving the overall aerodynamic efficiency of a pair of aircraft. In particular, one craft flying in the correct position in the vortex wake of another can realize substantial reductions in drag, with the amount of the reduction dependent on the relative positions of the two craft. This paper looks at such a pair, with one craft flying behind and to the side of the lead plane. The precise position of the second craft relative to the first to maximize the drag reduction is to be determined online, leading to a peak-seeking control problem. A new method of speak-seeking control, using a Kalman filter to estimate the characteristics of the drag reduction, is derived and discussed. A simple model of the two-plane formation using horseshoe vortices is defined, and the peakseeking controller is applied to this model. The method is demonstrated in simulation using this simplified model. S an airplane flies, it causes an upwash ahead of the wing and leaves a wake behind. This wake is characterized by the downwash behind the wing and by an accompanying upwash in the area on either side of the downwash region. By flying in the area of upwash, a second aircraft can gain a substantial efficiency boost because of the reduction in induced drag it will experience. This leads to the well-known fact that two aircraft flying in an appropriate formation can achieve overall efficiency much greater than were they flying separately. 1 This effect is analyzed using inviscid aerodynamic assumptions and lifting-line theory in Ref. 2, where it is noted that the effects were considered by Munk as early as 1919. The theory was put to test in actual aircraft by Hummel, 3 who established a fifteen per cent reduction on the second of a pair of civilian aircraft. Because of the gains in efficiency, formation flight has been investigated as a way of increasing the range and duration of autonomous aerial vehicles. In Refs. 4 and 5, formations of several aircraft are considered, with the object of creating a solar-powered formation that could cruise at high altitude for arbitrarily long times. In Ref. 4, decentralized controllers are derived for a formation of five highaspect-ratio craft and are shown to be capable of maintaining a prescribed formation despite the nonlinear, destabilizing moments induced on each plane by the aircraft ahead of it in the formation. The formation maintenance problem for a pair of F-16 class aircraft is considered in Ref. 6, though that paper relegates the rolling moments on the trailing craft to an inner-loop controller and considers only the lift and side force in designing an autopilot for the trailing plane. In this paper, only a pair of aircraft is considered. The two craft can be thought of as a leader and a follower. The leader flies straight

An Instrumentation System Applied to Formation Flight

As part of a NASA dryden autonomous formation flight program for improved drag reduction of multiple F/A-18 aircraft, a new instrument, the formation flight instrumentation system (FFIS), for the precise estimation of the relative position, velocity, and attitude between two moving aircraft without the aid of ground-based instruments, was developed. The FFIS uses a global position system (GPS) receiver and an inertial navigation sensor (INS) instrumentation package on each aircraft combined with a wireless communication system for sharing measurements between vehicles. An extended Kalman filter structure blends the outputs of each GPS/INS in a distributed manner so as to maximize the accuracy of the relative state estimates. Differential carrier phase GPS measurements are used to provide high accuracy relative range measurements to the filtering algorithm. A multiple hypothesis Wald test for estimating the integer ambiguity between the two moving vehicles was developed as part of this project. The FFIS was tested in a hardware-in-the-loop simulation (HIL Sim) before being tested in actual F-18 flight tests. Test results validated the FFIS performance. Flight test results showed that the Wald test accurately estimates the integer ambiguity and that relative range estimates using least squares provide accurate position estimates with a mean of approximately 7 cm and a standard deviation of 13 cm

Satellite Clusters with Constant Apparent Distribution

Theproblemofcreatingaclusterofsatellitessuchthatthearrangementmaintainsitsapparentrelativeformation as viewed from the planetary surface, without active station keeping, is considered. It is shown that it is possible to create a cluster with nearly constant circular form, but that rotates about its center. The amount of variation from the perfect circle is characterized in terms of the eccentricity of the satellite orbits. Because of the nature of the cluster, it is possible to put a large number of satellites into a cluster, with no danger of satellite collisions. An illustrative example is included. I. Introduction T HE work begun in this paper was motivated by the problem of creating a cluster of satellites that would maintain a constant, or nearly constant, shape and size when viewed from the Earth. Such a cluster might be said to have constant apparent (as opposed to physical) distribution. In such a formation, all satellites would have a clear e eld of view of the surface, and would remain in sufe ciently close formation to share their information. Such formations are of interest for large distributed-aperture sensing, for example. Another possibility is that of forming clusters from many small, inexpensive satellites, each with a particular type of sensor and some computing power. Remaining in a close formation would allow the satellites to share information and computing ability among themselves. Forsuch missions,it mightbe unnecessary tomaintain extremely precise relative positions; rather, it might be sufe cient to know the relative position accurately, and to remain in close enough proximity to allow intercraft communication. In addition, an effect of constant apparent distribution is that the angular dispositions of the satellites relative to their ground targets and each other would be

Decentralized Control of Satellite Clusters Under Limited Communication

Two decentralized control algorithms, each employing a unique information exchange scheme, are considered. One of these algorithms employs a communication paradigm in which control and local state estimate information is exchanged between members of a formation after any one of the members executes a control. The other algorithm requires only that control information be exchanged between members; this is thought to be the least information exchange that can sustain an affine control law. The performance of these algorithms when applied to a satellite formation control problem is evaluated. To this end, the algorithms are applied in the context of a nonlinear satellite cluster simulation, where Earth oblateness effects and measurement uncertainties are taken into account. The cluster is required to achieve a specified formation at a defined point on an elliptic reference orbit. Simulation results are obtained for several different orbits and compared to a baseline case involving a centralized controller design.

Extremum seeking loops with assumed functions

Extremum seeking (also peak-seeking) controllers are designed to operate at an unknown set-point that extremizes the value of a performance function. This performance function is approximated by an assumed function with a finite number of parameters. These parameters, which are estimated online, are assumed to change slowly compared to the plant and compensator dynamics. Philosophically, the approach of assuming a function is in contrast with traditional approaches that use time scale separation between gradient computation and function minimization and the system dynamics. To analyze our current scheme, quadratic functions or exponentials of quadratic functions are assumed as approximations to the performance function. This allows the peak-seeking control loop to be reduced to a linear system. For this loop, compensators can be designed and robust performance and stability analysis of the loop due to parameter uncertainty in the assumed performance functions can be obtained.

Functional feedback in an extremum seeking loop

We analyze the effect of functional feedback in an extremum seeking loop. Our study is motivated by a formation flight problem where the trailing aircraft experiences an outward rolling moment as the vortex effects of the leading aircraft come into play. Viewed as the extremized functional, this rolling moment feeds itself back to the lateral dynamics of the aircraft. In the initial half of the paper we impose restrictions on the curvature of the extremized functional and examine the effect under the assumption of a dynamic time-scale separation between the tracking loops of the linear dynamical system and the extremum seeking loop; in the latter half we remove both these assumptions and instead impose assumed functional forms on the extremized functional.

An efficient design algorithm for optimal fixed structure control

An extremely flexible design method for developing optimal fixed structure controllers is to place the cost of subsystem interconnections in the cost function. The optimal gain is then obtained via a numerical search procedure. Previous work with this class of cost functions has only made use of steepest-descent searches. In this paper, a descent function search algorithm is presented that significantly decreases the number of iterations needed to converge. An example shows that the descent function algorithm has superior convergence properties.

An adaptive controller based on disturbance attenuation

This paper discusses the control of linear systems with uncertain parameters in the control coefficient matrix, under the influence of both process and measurement noise. A disturbance attenuation approach is used, and from this a multiplayer game problem is generated. First, the minimax formulation is presented, which represents an upper bound on the game cost criterion. Second, a dynamic programming approach is used to solve the game. It is shown that for this class of problems, the controller determined from the dynamic programming approach is equivalent to the minimax controller. Therefore the minimax controller is also a saddle point strategy for the differential game. Controller development appears to be much simpler from the dynamic programming standpoint. A simple scalar example is presented.<<ETX>>