CAJ | 학술논문

化工过程的动态优化控制命题大多可以写成微分代数混合方程组(differential algebraic equations)的形式,联立法是求解该类命题的一种重要的数值方法.目前联立法中常用的离散方法是局部法,其中有限元正交配置(orthogonal collocation on finite elements)具有精度高、计算量小、稳定性好等优点.然而伪谱法(pseudo-spectral method)作为一种全局法,在离散中也有独特的优点,特别是具有指数级的收敛速度和较高精度,而且产生的NLP规模较小.本文分别以有限元正交配置法和伪谱法代表局部法和全局法比较其原理,并讨论离散配置点以及其在两种方法上的不同应用,针对离散后两种方法产生的NLP,分别提出判据以保证足够的自由度,最后用连续DAEs与不连续优化控制两个例子进一步比较这两种方法得出,如果命题平滑采用PS方法具有更好的收敛速度.
화공과정적동태우화공제명제대다가이사성미분대수혼합방정조(differential algebraic equations)적형식,련입법시구해해류명제적일충중요적수치방법.목전련입법중상용적리산방법시국부법,기중유한원정교배치(orthogonal collocation on finite elements)구유정도고、계산량소、은정성호등우점.연이위보법(pseudo-spectral method)작위일충전국법,재리산중야유독특적우점,특별시구유지수급적수렴속도화교고정도,이차산생적NLP규모교소.본문분별이유한원정교배치법화위보법대표국부법화전국법비교기원리,병토론리산배치점이급기재량충방법상적불동응용,침대리산후량충방법산생적NLP,분별제출판거이보증족구적자유도,최후용련속DAEs여불련속우화공제량개례자진일보비교저량충방법득출,여과명제평활채용PS방법구유경호적수렴속도.
Most of the dynamic optimal control problems in chemical engineering can be written in the form of differential algebraic equations (DAEs).Simultaneous approach is an important method for solving these problems.The discretization strategies often used in this approach are local methods, for instance, orthogonal collocation on finite elements (OCFE), which has many advantages.Pseudo-spectral (PS), as a global method, has its own unique properties.It may offer a rapid convergence rate for the approximation of analytic functions and has high precision and low computational effort with a simple structure.As the representations of local method and global method respectively, OCFE and PS were compared.The collocation points and their different distributions were presented, and the degree of freedom (DOF) of non-linear programming (NLP) after discretization was discussed, as well as the criteria were offered to ensure the DOF of the NLPs.At last, a continuous case and a discontinuous case were studied, and it was concluded that if the problem was smooth enough then the convergence of PS was better than that of OCFE.