摘要
文献[1,2]分别说明了计算结构力学与最优控制之间在理论上存在相似性原理,可以在数学上得到统—.本文在此基础上进一步讨论受约束控制系统的最优解问题. 相似性原理的关键是结构力学中的一维坐标对应于最优控制中的时间坐标,相应地,子结构对应于时段.由此,最优控制中的动力问题就可以用结构静力学中的方法进行处理.
In a previous paper[2], Zhong et al presented their discovery of the analogy between optimal control and computational structural mechanics. The present paper, starting from the definition of mixed-energy given in Ref.[2], goes further to make it applicable when constraint is present. Even more important than making mixed-energy applicable to the constrained problem, the author develops further the embryonic idea in Ref.[2]. It is now believed to be clear that the key to the above mentioned analogy is the two correspondences of the one-dimensional coordinate and the substructure in structural mechanics to the time coordinate and the time-interval in optimal control respectively. So the dynamic problems in optimal control can be treated by the methods in structural static mechanics. 2~N type algorithm is a special method in multi-level substructural method, and can be naturally transplanted to the optimal control problem. In structural mechanics, the characteristics of a substructure can be expressed by the energy expressions. For the non-constrained control system, its dual equations are given in Refs.[1,2]: Now the mixed-energy of the time-interval can be expressed by Q. G and φ. The mathematical expression of mixed-energy of the time-interval (k,k+1) is Space does not permit more than a brief description of the application of the author's idea to the constrained LQ control system. Based on Refs.[1,2],we first introduce the concept of mixed-energy into the constrained system, give the definition of mixed-energy of time-interval, then use the generalized variational principle and total variational expression to formulate the condensation formulas of time-interval (corresponding to those of substructure in structural mechanics). Finally, we establish the two Riccati equations, called the first and second Riccati equations, for the constrained problem. The two equations can then be solved by the 2~N type algorithm. Take the following example. We divide the time-interval (0, t_f) into 10~6 sub-time-intervals. The computation time increases in direct proportion. When we use the 2~N type algorithm, 10~6 sub-time-intervals are only equivalent to 20 computation steps; moreover the total errors will also be reduced. The above algorithm has been successfully applied to the computation of the nonlinear control prob-lem[4], and also applied to the constrained nonlinear control system in a paper to be published.
出处
《西北工业大学学报》
EI
CAS
CSCD
北大核心
1993年第3期387-388,共2页
Journal of Northwestern Polytechnical University
基金
国家自然科学基金(19242001)
关键词
2^N类算法
结构力学
控制系统
约束
2~N type algorithm
mixed-energy
analogy
Riccati equation
constraint
