The paper proposes a multi-domain approach to the optimization of the dynamic response of an underactuated vibrating linear system through eigenstructure assignment, by exploiting the concurrent design of the mechanical properties, the regulator and state observers. The approach relies on handling simultaneously mechanical design and controller synthesis in order to enlarge the set of the achievable performances. The underlying novel idea is that structural properties of controlled mechanical systems should be designed considering the presence of the controller through a concurrent approach: this can considerably improve the optimization possibilities. The method is, first, developed theoretically. Starting from the definition of the set of feasible system responses, defined through the feasible mode shapes, an original formulation of the optimality criterion is proposed to properly shape the allowable subspace through the optimal modification of the design variables. A proper choice of the modifications of the elastic and inertial parameters, indeed, changes the space of the allowable eigenvectors that can be achieved through active control and allows obtaining the desired performances. The problem is then solved through a rank-minimization with constraints on the design variables: a convex optimization problem is formulated through the “semidefinite embedding lemma” and the “trace heuristics”. Finally, experimental validation is provided through the assignment of a mode shape and of the related eigenfrequency to a cantilever beam controlled by a piezoelectric actuator, in order to obtain a region of the beam with negligible oscillations and the other one with large oscillations. The results prove the effectiveness of the proposed approach that outperforms active control and mechanical design when used alone.

本文提出了一种多域方法，通过利用机械特性，调节器和状态观测器的并行设计，通过特征结构分配优化欠驱动振动线性系统的动态响应。该方法依赖于同时处理机械设计和控制器综合，以扩大可实现性能的集合。潜在的新颖思想是，应通过并行方法考虑控制器的存在来设计受控机械系统的结构特性：这可以大大提高优化的可能性。首先，该方法是理论上开发的。从通过可行模式形状定义的可行系统响应集的定义开始，提出了最优性准则的原始表述，以通过对设计变量进行最优修改来适当地塑造可允许的子空间。弹性和惯性参数的修改的适当选择，实际上，变化可以通过主动控制来实现和允许获得所需的性能容许特征向量的空间。然后，通过对设计变量施加约束的秩最小化来解决该问题：通过“半限定嵌入引理”和“跟踪启发式”来拟定凸优化问题。最后，通过将模式形状和相关的本征频率分配给由压电致动器控制的悬臂梁来提供实验验证，为了获得光束的区域具有可忽略不计的振荡而另一个区域具有较大的振荡。结果证明了所提出的方法的有效性，该方法优于单独使用时的主动控制和机械设计。