具有不等式路径约束的微分代数方程系统的动态优化

Dynamic Optimization of Differential-algebraic Equations With Inequality Path Constraints

针对具有不等式路径约束的微分代数方程(Differential-algebraic equations, DAE)系统的动态优化问题,通常将DAE中的等式路径约束进行微分处理,或者将其转化为点约束或不等式约束进行求解.前者需要考虑初值条件的相容性或增加约束,在变量间耦合度较高的情况下这种转化求解方法是不可行的;后者将等式约束转化为其他类型的约束会增加约束条件,增加了求解难度.为了克服该缺点,本文提出了结合后向差分法对DAE直接处理来求解上述动态优化问题的方法.首先利用控制向量参数化方法将无限维的最优控制问题转化为有限维的最优控制问题,再利用分点离散法用有限个内点约束去代替原不等式路径约束,最后用序列二次规划(Sequential quadratic programming, SQP)法使得在有限步数的迭代下,得到满足用户指定的路径约束违反容忍度下的KKT (Karush Kuhn Tucker)最优点.理论上证明了该算法在有限步内收敛.最后将所提出的方法应用在具有不等式路径约束的微分代数方程系统中进行仿真,结果验证了该方法的有效性.

For dynamic optimization of differential algebraic systems with inequality path constraints, the equality constraint in differential-algebraic equations(DAE) is often differentiated or transformed into point constraints or inequality constraints to solve. As for the former, the compatibility of initial conditions needs to be checked or more constraints are added, making the original optimization problem infeasible in some cases. For the latter, the way that equality constraint is converted to other types of constraints increases the difficulty of solving. In order to overcome the above problems, a new method is proposed to solve the above problem directly based on the backward differentiation formula. Firstly, the control vector parameterization is used to convert the optimal control problem of infinite dimensions into a finite dimensional one. Then, a set of interior-time points by using the pointwise discretization method are used to replace the original inequality path constraints. Finally, the sequential quadratic programming(SQP) is used to locate the Karush Kuhn Tucker(KKT) points within finite-stepped iterations. Proof is provided for the finite convergence of the algorithm. The dynamic optimization algorithm is applied to the differential algebraic equation systems with inequality path constraint,and simulation studies are carried out to verify the effectiveness of the proposed method for the differential algebraic systems with inequality path constraints.