Integrated control/structure optimization by multilevel decomposition

A method for integrated control/structure optimization by multilevel decomposition is presented. It is shown that several previously reported methods were actually partial decompositions wherein only the control was decomposed into a subsystem design. One of these partially decomposed problems was selected as a benchmark example for comparison. The present paper fully decomposes the system into structural and control subsystem designs and produces an improved design. Theory, implementation, and results for the method are presented and compared with the benchmark example. INTRODUCTION Over the last decade, increasing attention has been given to control/structure interaction (CSI) problems and the integrated design of a structure and its controller. In general, the approaches to integrated design can be categorized as either sequential or simultaneous [1]. A sequential method is one in which one disciplinary design iteration precedes the other. The interactions between the structure and control are examined in an analysis step but are not rigorously included in the determination of design changes in the next iteration step. A simultaneous design method is one in which the control and structure design problems are combined into a single design problem. Interactions are considered at the outset and the effects of the structural and control design variables are considered together. References I through 19 give examples of simultaneous structure and control design approaches, each of them optimization-based. Study of these papers shows that there are a variety of possible approaches to the problem of simultaneous structure and control design even within the limits of being based in optimization * Staff Engineer. Member, AIAA ** Aerospace Engineer, Senior Member. MAA methods. Often, the objective function is s_'uctural weight plus some weighted function of controlled response. Typical constraints might be on closedloop frequencies or structural deflections. Usually, the methods proposed have stated the design problem as a single optimization problem wherein detailed structural design variables (member sizes) and control gains are of equal status as problem design variables. The problems on which these methods have been exercised have been relatively small, consisting of a simple structure with a few structural design variables and simple controllers with a few gains. While these methods have worked well, they could easily require extremely large design problems for a Iarge, complex structure. To make the problem size more manageable, the number of design variables might have to be reduced to ineffectual levels. There is also an organizational problem associated with combining the synthesis of structure and controller so directly. A typical aerospace design organization is highly segregated along disciplinary lines. Obviously, sufficient communication exists between disciplinary design groups to effect the design of successful aerospace systems. However, there are cases of actual flight hardware with problems that might well have been caught before production had the design process more closely coordinated structure and control design functions. Multilevel decomposition is an alternative approach to large multidisciplinary system design that has been proposed [20]. In this approach, a large multidisciplinary system is broken down along disciplinary and hierarchial lines into subsystem designs that are smaller and more easily managed than the complete, integrated system. The subsystem designs are coordinated at a higher level where the influences from each subsystem are integrated. At this top level, the design variables quantify the influence of the subsystems on the total system, but are not at the level of detailed design. Desired changes in these variable are passed to the appropriate subsystem design functions wherein detailed designs are accomplished treating the quantities specified from above as parameters to be held constant. An important element of this process is the return, from the subsystem designs to the upper level, of the values of sensitivity derivatives with respect to the parameters held constant. Thesesensitivities areusedatthetop level to constructgradientsfor use in an optimization.Thesensitivitiesmaybeof any quantifies thatinfluence theobjective andconstraint functionsbeingusedat thetop level. If the subsystemdesigns arethemselves accomplished through optimization, then the sensitivities would be the so-called "optimum sensitivity derivatives" [21] or, more correctly, "sensitivities of the optimum". Thus these sensitivities account for how the optimum subsystem design will change as the parameters are changed. Further, since the controlling influence upon the disciplinary designs exists at the top level where the design variables are related to each of the contributing subsystems, the top level design is a simultaneous one. Several of the simultaneous structure and control design methodologies cited in this paper [5, 13, 15'17] have actually been set, knowingly or not, in this multilevel format. In these papers, the structure was represented completely at the top level with no decomposition. The structural design variables were either the detailed structural sizes or more global structural parameters (such as stiffness characteristics) without regard to the detailed structural design. All of them, however, also incorporated the optimal, steady-state, Linear Quadratic Regulator ('LQR) as controller. At each iteration step, a LQR design is obtained and sensitivities of this optimum are computed with respect to the top level variables (structural sizes and, in some cases, elements of the weighting matrices used in the LQR synthesis), and used at the top level in constructing gradients of the objective and constraint functions for the next iteration. Sensitivity equations for the LQR and LQG (Linear Quadratic Gaussian for systems incorporating Gaussian white noise) control have been well developed over the past years [22]. Sensitivities and even multilevel decompositions for purely structural systems under static loading have also been developed [23, 24]. However, to date, decomposition of both structure and control in the controlled structure design problem has not been achieved. It is the purpose of this paper to report on a simultaneous structure and control optimization wherein the total system has been decomposed into both structural and control subsystems.