Implicit (iterative) Solution Methods

Another class of solutions for the constrained CA problem is obtained by converting the CA problem into the “constraints optimisation formulation” and employ one of the standard optimisation solutions to solve the problem (Rao, 2009; Nocedal & Wright, 2006). Employing ݈_ଵ norm21 for the cost function, will convert the CA into linear programming (LP) formulation whereas, with ݈_ଶ norm a quadratic programming (QP)22formulation is derived.

5.4.2.1 Linear Programming

By using ݈_ଵ norm (which is the sum of the absolute values of the component of the vector) for the cost function in (5-7), the optimisation problem statement is defined as follows:

݉ ݅݊ ܬ= ‖܅ఛ(۰ܝ − ૌ)‖ଵ+ ߝฮ܅௨൫ܝ − ܝ௣൯ฮ_ଵ ݏݑܾ݆݁ܿݐݐ݋ ܝ ≤ ܝ ≤ ܝ

ܝ (5-31)

By introducing some auxiliary variables, the above formulation can be converted to a standard linear programming (LP) formulation (Bodson, 2002; Johansen & Fossen, 2012) such as:

݉ ݅݊ ܬ= ܋்_{ܠ ݏݑܾ݆݁ܿݐݐ݋ ۯܠ = ܊ , ܠ ≥ ૙}

ܠ (5-32)

which can be solved with one of the already developed numerical LP algorithms including simplex or interior point methods (Nocedal & Wright, 2006).

21

The݈_ଵ(1-norm) of a vectorܝ is defined as:

‖ܝ‖ଵ= ෍ |ݑ௜| ௜

and the݈_ଶ(2-norm) ofܝ is

‖ܝ‖ଶ= ൬෍ ݑ௜ଶ ௜ ൰

ଵ/ଶ

22

Quadratic programming also denoted as constrained least square problem in the literatures (Rao, 2009).

Simplex methods are usually the most practical and efficient algorithms to solve LP problems (Bodson, 2002). They belong to a general class of algorithms for constrained optimisation known as active set methods in which the actuator controls are divided into a saturated (active) set and an unsaturated (free) set. The principle of active set approach to a LP problem is based on the explicit estimation of the active and free sets which are being updated at each step of the algorithm. In other words, the search for optimality in the active set approach is done by visiting the vertices of the polytope described by the constraints of the problem (Nocedal & Wright, 2006).

Interior point methods, where the optimality search is done from the interior and/or exterior of the constraint polytope, has better theoretical convergence properties, and are often preferred for large scale problem (Nocedal & Wright, 2006). Moreover, when “warm” state initialisation (based on the previous time step solution) is not stored or available; the interior point method provides better results than active set method and is preferable even for small scale problems. One of the disadvantages of using LP approach for solving CA problem is in the fact that when the original mixed ݈_ଵ norm optimisation problem is converted to the LP problem, the size of the problem increased significantly. For example, if we want to allocate ݉ virtual control vector, ૌ, to ݌ control inputs , ܝ, then the resultant matrices for the converted LP problem will be ۯ ∈ ℝ௠ ×(ଶ௠ ାଶ௣) , ܋∈ ℝ(ଶ௠ ାଶ௣)×ଵ_{, ܊ ∈ ℝ}(ଶ௠ ାଶ௣)×ଵ _{(Wang, 2007). This expansion of the LP} problem dimension increases the required computational time and cost which is undesirable. As a rough estimation, the time required to solve a linear program may be exponential in the size of the problem (Nocedal & Wright, 2006).

5.4.2.2 Quadratic Programming

The quadratic programming formulation for mixed optimisation CA problem23 is derived by employing ݈₂ norm (which is the minimum energy control effort) for the cost function in (5-7):

23

݉ ݅݊ ܬ= ‖܅ఛ(۰ܝ − ૌ)‖ଶଶ+ ߝฮ܅௨൫ܝ − ܝ௣൯ฮ_ଶଶ ݏݑܾ݆݁ܿݐݐ݋ ܝ ≤ ܝ ≤ ܝ ܝ

(5-33)

which can be solved by one of the standard QP solutions such as active set, interior point or gradient projection methods (Nocedal & Wright, 2006).

Several active set methods, including sequential quadratic programming (SQP) (Ono, Hattori, Muragishi, & Koibuchi, 2006), weighted least square (WLS) (Harkegard, 2002) and minimal least square (MLS) (Lötstedt, 1984) methods, has been proposed for solving CA problems, which among them, the weighted least square method was shown to be the most efficient (Harkegard, 2002). Similar to simplex method for LP problem, the active set solutions for quadratic programming starts by dividing the actuator controls into a saturated (active) set and an unsaturated (free) set, but the updates of the free sets are calculated based on the pseudo-inverse solution and the active set is reflected by calculating Lagrangian parameters (Oppenheimer, Doman, & Bolender, 2011). Note that the active set solution, as described above, is very similar to the concept of redistributed pseudo inverse (RPI) method presented in section 5.4.1.1. The difference is that an active set algorithm is more sensitive regarding which variable to saturate, and that an active set algorithm has the ability to free a variable that was saturated in a previous time step24 (which did not happen in RPI method). The active set algorithm always converges to the optimum solution in a finite number of steps and is shown to be efficient for problems of small to medium size, but an upper bound in the number of iteration can be very large (Harkegard, 2002).

Similar to LP problems, the interior point (IP) method can also be employed to solve the quadratic CA formulations. The advantage of the IP method is uniform convergence and knowledge of the relative distance to the optimal solution. In (Petersen & Bodson, 2006) a prim-dual Interior point method, based on (Vanderbei & Shanno, 1999), is implemented in order to exactly solve a quadratic program. The method is compared with a fixed point, (Burken, Lu, &

24

Wu, 1999), and an active set method, (Harkegard, 2002). Active set and interior point methods are now available as two standard solution algorithms within the Matlab® Control Optimization Toolbox™ (The MathWorks, 2013).

The fixed-point method is a recursive algorithm similar to a gradient search, classified as derivative free optimisation (DFO) algorithms (Nocedal & Wright, 2006). A fixed-point method was used by (Burken, Lu, & Wu, 1999) to solve a mixed optimisation problem as stated in (6-27) for aerospace applications. The method is also employed for solving IVDC problem by (Wang, 2007). Fixed- point method is very easy to code, fast for most achievable commands and reaches to the exact optimal solution within a finite number of iterations. Although, the algorithm has a theoretically proven global convergence, but it is quite slow in practice if the command values are large (unattainable) (Bodson & Frost, 2011). For that reason, the fixed-point method is usually implemented with a bounded number of iterations; this means that we accept some level of sub-optimality in the solution (Harkegard, 2002).