Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briey reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice.

. In recent years, general-purpose sequential quadratic programming (SQP) methods have been developed that can reliably solve constrained optimization problems with many hundreds of variables and constraints. These methods require remarkably few evaluations of the problem functions and can be shown to converge to a solution under very mild conditions on the problem. Some practical and theoretical aspects of applying general-purpose SQP methods to optimal control problems are discussed, including the influence of the problem discretization and the zero/nonzero structure of the problem derivatives. We conclude with some recent approaches that tailor the SQP method to the control problem. Key words. large-scale optimization, sequential quadratic programming (SQP) methods, optimal control problems, multiple shooting methods, single shooting methods, collocation methods AMS subject classifications. 49J20, 49J15, 49M37, 49D37, 65F05, 65K05, 90C30 1. Introduction. Recently there has been c...

and loving encouragement. Contents 1 Introduction 3 1.1 The mathematical problem formulation : : : : : : : : : : : : : : : : : : : : 7 1.2 Notational conventions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2 The Collocation Discretization 11 2.1 Collocation for two point BVP in ODE : : : : : : : : : : : : : : : : : : : : 11 2.1.1 Choice of collocation points : : : : : : : : : : : : : : : : : : : : : : 14 2.1.2 The polynomial representation : : : : : : : : : : : : : : : : : : : : : 14 2.1.3 A tempting combination : : : : : : : : : : : : : : : : : : : : : : : : 15 2.2 Collocation for BVP in DAE with invariants : : : : : : : : : : : : : : : : : 17 2.2.1 DAE models from mechanics : : : : : : : : : : : : : : : : : : : : : : 17 2.2.2 Collocation discretization of two point BVP in DAE : : : : : : :

ly, the set of orientation matrices SO(3) ae R 3\Theta3 is a compact 3-dimensional C 1 -submanifold which has no global 3-parameter representation without singularities. Consider a point p fixed on a moving body. Its inertial position and velocity are p 0 (t) = r(t) + R(t)p and p 0 (t) = r(t) + R(t)p, respectively. The rotation part can be written Rp = ! 0 \Theta (Rp) = (R!) \Theta (Rp) = R(! \Theta p); where ! and ! 0 are the angular velocities in the body frame and inertial frame, respectively. On the other hand, we have ! \Theta p = ~ !p where ~ ! := 0 @ 0 \Gamma! 3 ! 2 ! 3 0 \Gamma! 1 \Gamma! 2 ! 1 0 1 A = \Gamma ~ ! 2 A(3) is an antisymmetric matrix; the corresponding inertial angular velocity matrix is ~ ! 0 = R~!R . (Asterisk superscripts denote transposition throughout the paper.) The above relations define a sequence of canonical linear isomorphisms between R 3 and the tangent space of SO(3) at R (the matrix velocity space), TRSO(3) = RA(3) ¸ = A(3) ¸ =...

. This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by time-dependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differential-algebraic equations (DAEs) with a package for large-scale optimization based on sequential quadratic programming (SQP). DASOPT is intended for the computation of the optimal control of time-dependent nonlinear systems of PDEs in two (and eventually three) spatial dimensions, including possible inequality constraints on the state variables. By the use of either finite-difference or finite-element approximations to the spatial derivatives, the PDEs are converted into a large system of ODEs or DAEs. Special techniques are needed in order to solve this very large optimal control problem. The use of DASOPT is illustrated by its application to a nonlinear parabolic PDE boundary control problem in two spatial dimensions. Computational resu...

this paper we investigate more closely the structures of two application examples which can be considered representative for the rather wide class of large-scale optimization problems in so far as they stem from discretizations:

The growing interest in model predictive control for nonlinear systems is motivated by the fact that today's processes need to be operated under tighter performance specifications to guarantee profitable and environmentally safe production. Nonlinear model predictive control (NMPC) does allow the direct consideration of a nonlinear process model as well as state and input constraints. Thus NMPC seems to be well suited for these kind of processes. Many theoretical issues in NMPC have been attacked and solved in recent years. Despite this progress there are a number of problems that have to be solved before NMPC can be applied in practice.

Despite many control theoretic and numerical advances, up to now there is no realistic feasibility study of modern nonlinear model predictive control (NMPC) schemes for the real-time control of large-scale processes. In this paper the application of NMPC to a nontrivial process control example, namely the control of a high-purity binary distillation column, is considered. Using models of di erent complexity and different control schemes, the computational load, resulting closed loop performance and the effort needed to design the controllers is compared. It is shown that a real-time application of modern NMPC schemes is feasible with existing techniques, even for a 164 order model with a sampling time of 30 s, if a state of the art dynamic optimization algorithm and an efficient NMPC scheme are used.

Introduction and Problem Statement Interest in dynamic simulation and optimization of chemical processes has increased significantly during the last two decades. Common problems include control and scheduling of batch processes; startup, upset, shutdown and transient analysis; safety studies and the evaluation of control schemes. Chemical processes are modeled dynamically using differentialalgebraic equations (DAEs). The DAE formulation consists of differential equations that describe the dynamic behavior of the system, such as mass and energy balances, and algebraic equations that ensure physical and thermodynamic relations. The general dynamic optimization problem can be stated as follows: min z(t);y(t);u(t);t f ;p '(z(t f ); y(t f ); u(t<F8

In space based robotics, one of the most important problems is the disturbance to the satellite attitude and to the satellite microgravity environment caused by satellite mounted robot operation. This paper reports on computer-aided motion planning strategies to overcome this problem. Point to point motion designs are generated which not only connect prescribed start and end points of the robot motion, but also simultaneously return the satellite to its original attitude. Theoretical characterizations of some of those motion designs are presented, as well as numerical results. The computation time required to produce such motion designs is 1 or 2 minutes on a workstation. Thus, it can be practical to use these motion plans in space. Introduction Satellite mounted robots can aid in the automated assembly of large structures in space and sometimes eliminate the need for extra-vehicular activity or "space walks", which are time-consuming and dangerous for astronauts. Robots in space can ...