Abstract-In order to deal with the computational burden of optimal control, it is common practice to reduce the degrees of freedom by fixing the input or its derivatives to be constant over several time-steps. This policy is referred to as "move blocking". This paper will address two issues. First, a survey of various move blocking strategies is presented and the shortcomings of these blocking policies, such as the lack of stability and constraint satisfaction guarantees, will be illustrated. Second, a novel move blocking scheme, "Moving Window Blocking" (MWB), will be presented. In MWB, the blocking strategy is time-dependent such that the scheme yields stability and feasibility guarantees for the closed-loop system. Finally, the results of a large case-study are presented that illustrate the advantages and drawbacks of the various control strategies discussed in this paper.

Abstract-In order to deal with the computational burden of optimal control, it is common practice to reduce the degrees of freedom by fixing the input or its derivatives to be constant over several time-steps. This policy is referred to as "move blocking". This paper will address two issues. First, a survey of various move blocking strategies is presented and the shortcomings of these blocking policies, such as the lack of stability and constraint satisfaction guarantees, will be illustrated. Second, a novel move blocking scheme, "Moving Window Blocking" (MWB), will be presented. In MWB, the blocking strategy is time-dependent such that the scheme yields stability and feasibility guarantees for the closed-loop system. Finally, the results of a large case-study are presented that illustrate the advantages and drawbacks of the various control strategies discussed in this paper.

Summary. In this contribution, explicit model predictive control is applied to the benchmark problems. Emphasis is put on the reduction of complexity both for the off-line and on-line computation. For the pendulum, a control-invariant set is computed first. By using this set in the subsequent optimal control problem, a simple solution can be attained which guarantees invariance of the resulting closed-loop system by construction. The computation of such invariant sets for systems with a partially constrained state-space is discussed.For the reactor, a different approach was chosen. First, a simple model predictive control problem is solved explicitly which results in a low-complexity controller. Invariance is established a posteriori by analysing the explicit control law. Both approaches employ an analysis method to prove stability.

The electrical Variable Speed Drive (VSD) system presented is designed for installation on the sea floor to drive nearby electric motors for pumps and gas compressors. A modular concept of the VSD is developed and intended to operate a wide range of subsea motors of powers from 0.5 to 18 MVA, with voltages from 2.0 kV to 7.2 kV or more, and fundamental frequencies up to 300 Hz. Step-out distances from a few km to over 600 km can be accommodated. The pressure compensated design effectively removes limits as to the depth of deployment. Pressure compensation is achieved by submerging the drive hardware including the drive transformer in a dielectric liquid which also acts as coolant. The electric power components, including capacitors, semiconductors, and the control electronics are designed with increased margins and redundant hardware, pressure resistance, and materials chosen for compatibility with the dielectric liquid, to achieve a highly reliable design of the overall VSD. The drive was deployed into shallow water in a harbor in Vaasa Finland for testing. A top side station was built implementing a "Power-In-the-Loop" approach, where the VSD output energy is recovered back into the drive input such that the grid supply only provides the lost power, but not the much higher circulated power. The drive operated more than 1000 h at 22 kV input and 6.9 - 7.2 kV output voltage at different power levels. We conclude from this first shallow water test, that all components of the VSD system work properly together up to 1000 A output current. Different operation conditions reflecting the envisioned application, including redundancy capability were successfully tested. The thermal performance was extensively verified, including an optional external heat exchanger to achieve high ratings even in warm waters. To our knowledge this is the first time a medium voltage drive is operated at 9 to 12 MVA for an extended time submerged in a sea water environment. All its modules are designed to operate down to depths of 10’000 ft / 3000 m or more and are concluding qualification according to API17F and SEPS 1002.