Multicriteria Optimization for Design of Structures with Active Control

Abstract

This paper describes a method to design a structure and control system simultaneously using a multiobjective optimization approach based on global criteria. The control system was based on a modified linear quadratic regulator (LQR) with bounds placed on the control forces to simulate limitations of real actuators. In the design of the control system, the state space equations were integrated using a Runge-Kutta method for a specified initial boundary condition. The structural weight, the weight of the actuators, the time required to suppress an initial disturbance, and the performance index were considered as different objective functions to be optimized. The design variables were the bounds on the maximum values of the control force, the cross­ sectional areas of the structural elements, and elements of the weighting matrices in the control design. As an example to illustrate the application of the approach, a box beam idealized by rod elements was used. The actuators and sensors were collocated and assumed to be embedded in structural elements. The results are presented for optimum designs obtained by changing different parameters in the definition of the global criteria. During the past several years, researchers in the field of optimum design have focused on developing algorithms to de­ sign structure in the multidisciplinary domain so as to take advantage of the interaction between different disciplines. The principal goal of this approach was to achieve an optimum design satisfying distinct requirements of various disciplines simultaneously so that the overall system would have peak performance. In this paper, the two disciplines considered are structural design and control design. The traditional approach to the design of a controlled structural system is to design the structure first, by satisfying its requirements, and then to de­ sign the control system. The structure is designed with con­ straints on weight, allowable stresses in the elements, general instability, frequency distribution, etc. When the selection of the geometry, cross-sectional areas of the members, and ma­ terial are completed for a structure, its structural frequencies and vibration modes become input to the control design. Be­ cause the cross-sectional areas of the structural elements influ­ ence the structural frequencies and the distribution of frequen­ cies affects the control design, it is appropriate to integrate these two disciplines in the design algorithm and determine optimum values of the design variables in both systems. In the design of actively controlled structures, most often the objective function is the weight of the structure wherein constraints are imposed on the structural and control response functions, such as structural frequencies, closed-loop damping, control performance defined by the linear quadratic control cost, closed-loop frequency distribution, efficiency require­ ments, and robustness parameters. Hale et aI. (1985) consid­ ered minimizing the sum of the weight of the structure and the quadratic control cost to satisfy specific constraints with structural parameters as the design variables. Haftka et al. (1985) studied the effect of structural changes on the design of the structure and control system. Rew et aI. (1987) used multiple optimization criteria for designing the structural con­ trol system. The problem of integrated design was solved by Khot (1988), using analytical expressions for the sensitivities and imposing constraints on the closed-loop damping and fre- 'Res. Aerosp. Engr., Struct. Div., Air Vehicles Directorate, Air Force Res. Lab., Wright-Patterson AFB, OH 45433-7542.