DOTcvpSB: a Matlab Toolbox for Dynamic Optimization in Systems ...

CHAPTER 3Control vector parametrizationThe basis of the CVP method rests only in the parameterization of control trajectories, the state trajectories remaincontinuous. The original problem of dynamic optimization is transformed into the finite dimensional problem(NLP) – static optimization. Further, a suitable gradient method with a NLP type algorithm is needed. Thegradients can be obtained by finite difference, adjoint or sensitivity approach. Each of the gradient method hasadvantages and disadvantages, the summary is shown in [25]. The algorithm has an iterative character and whenthe optimality conditions are satisfied the computation stops. Finally, it is good to emphasize that the CVP methodis only of a local nature and for this reason it is more effective to use a combination of the deterministic andstochastic method to get the vicinity of a global optimum. This combination is directly used in the hybrid methods.Another option is to optimize a process with several different initial conditions, what is called as multistart method.This option is possible to choose in the DOTcvp toolbox.3.1 NLP FORMULATIONAs mentioned at the beginning of this chapter the dynamic optimization problem is transformed into thestatic optimization problem. It follows that the original continuous control trajectory can be approximated on thefinite number of the time intervals N as piecewise constantu(t) = u i , t i−1 ≤ t < t i , i = 1,N (3.1)or piecewise linearu(t) = u i1 +u i2 t (3.2)where the different ∆t i = t i − t i−1 is denoted as the time interval length. The further constraints are defined aslower and upper boundaries of the optimized variables∆t i ∈ [∆t mini ,∆t maxi ] (3.3)u i ∈ [u mini ,u maxi ] (3.4)p ∈ [p min ,p max ] (3.5)Subsequently can be defined the vector of decision variables y ∈ R ny which contains information aboutlengths of the time intervals (∆t i ), control variables (u i ), and time-independent parameters (p)y T = [∆t 1 ,...,∆t N ,u T 1,...,u T N,p] (3.6)The vector of decision variables (3.6) together with corresponding lower and upper bounds (3.3-3.5) issuitable to handle with any MI/NLP solver which is able to minimize the cost function (2.3) with respect to