非线性控制系统平行计算的基础

众所周知,多重子结构法中子结构的凝聚与次序无关,由计算结构力学与最优控制的模拟理论自然可证明时段的消元也与消元次序无关,这就为非线性控制系统进行平行计算打下了基础.采用结构力学方法计算最优控制问题,其关键步骤是时段的消元.本文考虑的非线性控制系统,其动力方程为

On Parallel Computation in Nonlinear Control System

It is well known that the substructure condensation in multi-level substructural method is indepen- dent of substructure order.From the analogy between computational structural mechanics and optimal control1],we know that the time-interval condensation is also independent of time-interval order.Thus the author deems that parallel computation in nonlinear control system becomes possible. The key to making method of structural mechanics applicable to the problem of optimal control is the time-interval condensation.The nonlinear control system now to be considered has the following dy- namic equation: (?)=f(x,u,t) Its cost functional is J=(?)X(x)+U(u)]dt+P(x_f) where the meanings of all symbols are as given in a paper by the author published in 19932]. The author takes the first term of the Taylor series of f(x,u,t) and the first two terms of the Taylor series of J and thus transforms the problem into linear qudratic (LQ) control problem.In order to estab- lish the condensation formulas of two consecutive time-intervals,it is necessary to consider the specific characteristics of the nonlinear problem in order to obtain the expression of mixed-energy of time-inter- val.Then in accordance with the generalized variational principle the author obtains the following con- densation formulas: Q_c=Q_1+F_1~T(I+G_1Q_2)~(-T)Q_2F_1,;G_c=G_2+F_2(I+G_1Q_2)~(-1)G_1F_2~T] F_c=F_2(I+G_1Q_2)~(-1)F_1 r_(qc)=r_(q2)+F_2(I+G_1Q_2)~(-1)(r_(q1)-G_1r_(p2)) r_(pc)=r_(p1)+F_1~T(I+G_1Q_2)~(-1)(Q_2r_(q1)-r_(p2)) The next step is to condense the mixed-energy of time-interval recursively until the global interval is obtained.Then the author performs the back solving of the state and control vectors for all the interme- diate time steps.This corresponds to the travel of pre-order structural tree in the multi-level substructural method in structural mechanics. Parallel computation is effective for condensing the mixed-energy of time-interval recursively until the global interval is reached.For example,64 time-intervals are condensed into the global interval only by 6 condensations.If the traditional algorithm is used,the number of condensations is the much higher 63.