Research Article

Optimal Control of Diesel Engines: Numerical Methods, Applications, and Experimental Validation

Table 2

Performance of the NLP solvers on test cycle 1, number of outer iterations, and overall time required for the solution (in brackets). For the discretisation with order 5*, a piecewise-constant control was imposed by additional linear constraints. This variant imitates multiple shooting by resolving the state variables finer than the control inputs. The last row is the pure pseudospectral method. Symbols used are (length of collocation intervals), (collocation order), (number of NLP variables), (degrees of freedom in the NLP), and QN/EN (quasi/exact Newton method). For WORHP, only the exact Newton method is shown. For the IP method using a direct solver in KNITRO (KN-IPDIR), only the QN method could solve the problem, which is shown here. The IP-CG method always performed worse and thus is not shown. An accuracy of is requested with respect to optimality and feasibility. If this accuracy was not achieved within 200 iterations, the optimisation was terminated, which is indicated by italic script. Bold script indicates the fastest solution for each case.

SNOPT IPOPT, QN IPOPT, EN WORHP KN-SQP, QN KN-SQP, EN KN-IPDIR

0.25 1 0 225 61 20 (3.5) 39 (6.4) 13 (4.5) 18 (6.7) 66 (11.3) 58 (29.6) 54 (7.8)
0.25 3 10 657 190 69 (15.2) 40 (9.9) 15 (9.8) 18 (11.9) 151 (49.0) 65 (64.9) 70 (19.3)
0.25 5* 0 1,089 60 23 (7.1) 30 (10.7)15 (14.3) 21 (23.2) 136 (82.2) 112 (167.5) 137 (86.8)
3.00 15 1 279 78 42 (6.8) 62 (10.2) 12 (4.8) 21 (8.5) 200 (43.8) 200 (107.1) 144 (32.3)
6.00 30 0 279 80 154 (27.0) 82 (21.9) 28 (10.7) 27 (13.7) 200 (45.9) 200 (118.2) 200 (45.9)