We consider an artificial swarm mechanical system consisting of multiple agents. The agents are composed of mechanical components. The ideal kinematic performance includes mutual attractions and repulsions. This kinematic performance is embedded into the dynamics by being treated as a constraint. The Udwadia–Kalaba theory is then used to generate the required servo constraint force to assure the constraint is met for the nominal system. The system also includes uncertainty. The uncertainty in the swarm mechanical system is time-varying, whose value falls within a prescribed fuzzy set. For the robust control design, a creative $\beta$-measure-based approach is introduced. The robust control guarantees uniform boundedness and uniform ultimate boundedness regardless of the actual value of the uncertainty. For the optimal choice of a control design parameter, a fuzzy-theoretic performance index is introduced. The resulting optimization problem is proven to be tractable, with the global solution to be existent and unique. Furthermore, the analytic expression of this solution is obtained. As a result, the optimal design problem is completely solved. To further demonstrate its effectiveness, we compare the performances of the swarm mechanical system under the robust control and linear–quadratic regulator control through simulation results with an illustrative example.

中文翻译：

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控制不确定的群体机械系统：一种基于β-度量的方法

我们考虑由多个代理组成的人工群机械系统。代理由机械部件组成。理想的运动性能包括相互吸引和排斥。这种运动学性能通过被视为约束而嵌入到动力学中。然后使用 Udwadia - Kalaba 理论来生成所需的伺服约束力，以确保满足标称系统的约束。该系统还包括不确定性。群机械系统中的不确定性是随时间变化的，其值落在规定的模糊集合内。对于稳健的控制设计，引入了一种创造性的基于 $\beta $-measure 的方法。无论不确定性的实际值如何，鲁棒控制都保证统一有界和统一最终有界。对于控制设计参数的最佳选择，引入了模糊理论性能指标。由此产生的优化问题被证明是易于处理的，全局解决方案是存在且唯一的。进而得到该解的解析表达式。这样，优化设计问题就完全解决了。为了进一步证明其有效性，我们通过仿真结果和说明性示例比较了群机械系统在鲁棒控制和线性-二次调节器控制下的性能。优化设计问题完全解决。为了进一步证明其有效性，我们通过仿真结果和说明性示例比较了群机械系统在鲁棒控制和线性-二次调节器控制下的性能。优化设计问题完全解决。为了进一步证明其有效性，我们通过仿真结果和说明性示例比较了群机械系统在鲁棒控制和线性-二次调节器控制下的性能。