Assessment and Future Directions of Nonlinear Model Predictive ...

8 E.F. Camacho and C. Bordonsat every sub-iteration (inside a sampling period) is used and a decrease in thecost function is achieved, optimisation can be stopped when the time is over andstability can still be guaranteed. It can be demonstrated that it is sufficient toachieve a continuous decrease in the cost function to guarantee stability.The main technique that uses this concept was proposed by Scokaert et al. [43],and consists of a dual-mode strategy which steers the state towards a terminalset Ω and, once the state has entered the set, a local controller drives the stateto the origin. Now, the first controller does not try to minimise the cost functionJ, but to find a predicted control trajectory which gives a sufficient reduction ofthe cost.Simultaneous Approach: In this approach [12] (also known as multiple shooting[6]) the system dynamics at the sampling points enter as nonlinear constraintsto the optimization problems, i.e. at every sampling point the following equalityconstraint must be satisfied:¯s i+1 =¯x(t i+1 , ¯s i , ū i )Here ¯s i is introduced as additional degree in the optimization problem and describesthe initial condition for the sampling interval i. This constraint requires,once the optimization has converged, that the state trajectory pieces fit together.Thus additionally to the input vector [ū 1 ,..., u¯N ] also the vector of the ¯s i appearsas optimization variables. For both approaches the resulting optimizationproblem is often solved using sqp techniques. This approach has different advantagesand disadvantages. For example the introduction of the initial states ¯s ias optimization variables does lead to a special banded-sparse structure of theunderlying qp problem. This structure can be taken into account to lead to afast solution strategy. A drawback of the simultaneous approach is, that only atthe end of the iteration a valid state trajectory for the system is available. Thusif the optimization cannot be finished on time, nothing can be said about thefeasibility of the trajectory at all.Use of Short Horizons: It is clear that short horizons are desirable froma computational point of view, since the number of decision variables of theoptimisation problem is reduced. However, long horizons are required to achievethe desired closed-loop performance and stability (as will be shown in the nextsection). Some approaches have been proposed that try to overcome this problem.In [46] an algorithm which combines the best features of exact optimisationand a low computational demand is presented. The key idea is to calculateexactly the first control move which is actually implemented, and to approximatethe rest of the control sequence which is not implemented. Therefore the numberof decision variables is one, regardless of the control horizon. The idea is thatif there is not enough time to calculate the complete control sequence, thencompute only the first one and approximate the rest as well as possible.An algorithm that uses only a single degree of freedom is proposed in [21] fornonlinear, control affine plants. A univariate online optimisation is derived byinterpolating between a control law which is optimal in the absence of constraints