Multivariable flight control design using root locus techniques

Abstract

Heavy dependency upon flight control to achieve multiple integrated objectives in production vehicles motivates the need for a technique which exploits the benefits of multidimensional crossfeeds, as in the modern approach, yet preserves the graphical, insightful aspects of the conventional approach. Extensions of the scalar Evans root locus technique to multivariable settings is therefore addressed. For a two channel system, the governing root loci relationships are developed in terms of an overall system gain and relative gains between the channels. The most interesting result involves the closed-loop poles tracking moving which are themselves migrating towards the transmission zeros as the system gain increases. This result also applies to L;! optimal root loci. These relationships allow the flight control designer to step away from the loop perspective towards an overall systems viewpoint, while still retaining the thread between a specific feedback gain and it's role and influence upon various closed-loop dynamic properties. Nevertheless, multivariable flight control is a challenging task, and familiarity with vehicle dynamic characteristics is an essential element, as always. Introduction Increased reliance upon flight control in the attainment of multiple, coupled desi n requirements has been the trend for several decades? For example, highly maneuverable aircraft undergoing extreme angular rates at excessive angle of attack and side slip will experience severs coupling in the rolYyaw axes, yet the vehicle must exhibit crisp, well-damped coordinated responses in these axes which are acceptable to the pilot, having been excited by multiple control inputs. Limited multiple closed-loop objectives are achieved in practice by applying conventional techniques in a sequential ~ n a n n e r . ~ > ~ The graphical nature of these conventional techniques make clear the effects of a particular feedback upon the closed-loop properties. This approach has worked well in the past, but does not take advantage of the full capabilities offered by multivariable control (i.e., two way crossfeed compensation is usually avoided). On the other hand, the theoretical control community has been concentrating on multivariable * Assistant Professor. Senior Member AIAA. techniques which close all feedback loops simultaneously for over three decades. Powerful methods such as LQRLQGLTR and H, have been created to take full advanta e of multidimensional crossfeed ~ o m ~ e n s a t i o n . ~ ~ ~ Recent activity has concentrated on tuning existing methods to provide closed-loop robustness properties. Two obstacles limiting the use. of these-methods in practice are the multivariable nature of the algorithms masking specific, individual behavior and theoreticaVmathematica1 sophistication linking relevant closed-loop properties to the design knobs. These complications limit insight into the control law's stabilization and/or augmentation of the vehicle, which reduces confidence in the implemented control law. Further, if results suggest the need for modifications, it is often unclear as to how to proceed. As an example, no technique is available to graphically predict how the closed-loop poles will be-augmented ki th changin compensation and gain, except for asymptotic cases. 6f0 This paper tries to chip away at this separation between conventional and modern methods4 by considering an extension of the Evans root locus technique1 l to multivariable control settings. Although this concept has undoubtedly entertained the minds of most control system engineers, new useful interpretations have been developed which do not appear in the literature. The technique fosters development of truly multivariable control by conventional means. Further, the results shed light on the complete root loci behavior of multivariable optimal L2 control systems. The latter result opens a window for insight into linear optimal control system design, as well as offers a conventional means of modifying an optimal controller. Conventional Two Channel System Consider a 2x2 system whose dynamics are to be augmented with feedback such as an aircraft with inherently poor flying qualities, or where gij denotes the transfer function between response yi and input uj. Using sequential mot locus concepts.2 typically the y l to u l would be closed initially, or Ul = k l l ( ~ ; c ~ 1 ) (2) where k.represents compensation and y;c denotes a 'J command signal. For this loop, the root locus corresponds to l+kl l g l l = 0 and the augmented system