Explicit solution methods

5.4.1.1 Redistributed Pseudo-Inverse (RPI)

The redistributed pseudo-inverse (RPI) is a multi-step method which starts with the unconstrained least square (i.e. pseudo-inverse) solution of effective matrix ۰ as presented by Eq. (5-19). If the resulted control inputs, ܝ, are within the bounds (limitations), no further steps are needed and the solution stops. Otherwise, the components of the control vector that exceed the limits are set to their limitations, and the pseudo-inverse is recomputed with the actuators that are still within the limits. The procedure is repeated until all components have saturated, or until the solution of the reduced least-squares problem satisfied (i.e. the control error becomes null).

More specifically, the algorithm first obtains the optimal control vector by solving the weighted pseudo-inverse of۰:

ܝ = ۰#_ૌ

If some elements of the allocated control vector exceed their limits, the control input vector and the control effectiveness matrix are decomposed into unsaturated and saturated groups as:

ૌ= [۰௦ ۰௥] ቈ ܝ௦ ܝ௥቉

(5-21)

where ܝ_௦ are the elements of control input that exceed their limits and ܝ_௥ are the rest of actuation elements which are within their limits, and ۰_௦ and ۰_௥ are their corresponding effectiveness matrix, respectively. The magnitude of ܝ_௦ exceed their bounds, so their value is set to their limitations,ܝ_ୱ, i.e.

and the remaining value of demanded force or moments (corresponding to control error) which should be generated by the rest of the actuators is:

ૌ୰= ૌ− ۰ୱܝୱ (5-23)

The redistributed control input for the unsaturated group of actuators can be calculated as:

ܝ୰= ۰௥#(ૌ− ۰ୱܝୱ) (5-24)

The algorithm repeats until a solution within the limits is obtained or all the controls are saturated.

The redistributed algorithms is simple, fast and have been employed in many applications (Zhang & Jiang, 2008). However, it is shown in (Bodson, 2002) through an example that the method might not lead to an optimal solution in all cases.

5.4.1.2 Daisy-chain

The daisy-chain approach assumes a hierarchy of control effectors therefore the actuator control inputs ܝ are decomposed into two or more groups. The method allocates redundant groups of controls in the following prioritised manner: elements of the second groups are not used until at least one element of the first group is saturated. The same procedure repeats for the rest groups of the actuators (Buffington & Enns, 1996). The algorithm starts with the pseudo-inverse solution for the first group of actuators. If the requested control commands (i.e. virtual force and moments) are not satisfied by the first group of actuators because of the actuator saturation, the control input are set to the saturation limits, therefore, there is an error existing between the commanded values and those produced by the control effectors. In the next step, the remaining demands (control error) are passed to the second group of actuators. If there are still virtual control demands that are not satisfied, those are passed to the third group and so on. The algorithms end when the control error becomes null or no more control freedom is available.

ૌ= ۰ܝ ܝ ≤ ܝ ≤ ܝ (5-25)

can be decomposed into several control effector groups such as:

ૌ= [۰ଵ ۰ଶ ۰ଷ] ൦ ܝଵ ܝଶ ܝଷ ൪ ൞ ܝଵ≤ ܝଵ≤ ܝଵ ܝଶ≤ ܝଶ≤ ܝଶ ܝଷ≤ ܝଷ≤ ܝଷ (5-26)

The daisy-chain algorithm starts by solving the CA problem for the first group of effectors (۰_ଵ). Employing (unconstrained) weighted pseudo-inverse method leads to:

ܝෝଵ= ۰ଵ#ૌ (5-27)

If ܝෝ_ଵ is within its allowable values (i.e. ܝ_ଵ≤ ܝ_ଵ≤ ܝ_ଵ) this means that all the requested forces or moments can be generated by the first set of actuators: the rest of the actuators never utilized and the solution stops. Otherwise, the value of the first control inputs (ܝ_ଵ) clipped at their limits, i.e.

ܝଵ= ܝෝଵ if ܝଵ< ܝଵ< ܝଵ ܝଵ= ܝଵ or ܝଵ= ܝଵ otherwise

(5-28)

The remaining value of requested force or moments which should be provided by the rest of the actuators is:

ૌଶ= ૌ− ۰ଵܝଵ (5-29)

Therefore, the control input for the second group of actuators can be calculated from:

ܝଶ= ۰ଶ#(ૌ− ۰ଵܝଵ) (5-30)

and the algorithm repeats. The solution ends when the control error becomes null or no more control freedom is available. Figure 5-3 shows a schematic diagram of daisy-chain allocation scheme with 3 groups of effectors. However, this process can be extended to any number of control effectors. From the above explanation, one can conclude that the maximum number of iterations in

the daisy-chain method is equal to the number of effector groups (which is three for the example shown in Figure 5-3).

Figure 5-3: Example of daisy-chain allocation

Both daisy-chain and redistributed pseudo-inverse methods are denoted as sub-optimal or approximately optimal solutions, because the allocated actuator controls are not always obtained from the entire attainable set of actuators (Bodson, 2002). Moreover, in the daisy-chain method if, for example, the first group of actuators can provide the total amount of the requested forces or moments, the rest of the actuators never utilized, and the solution is more sub- optimal if utilisation of all the actuators are in concern.

In fact, the main difference between daisy-chain and redistributed pseudo- inverse methods is in the way that they deal with actuator saturation. In redistributed method, the control authority is distributed among all the existing actuators in each sequence, but the actuator which has the tightest limitations (i.e. has the lowest capacity) is saturated first (and therefore clipped at its limits) and the actuator which has the highest capacity will be saturated last. On the other hand, in daisy-chain method, the preference of actuators has been set in advance, based on “a priori” knowledge of the system. This feature is criticised as a drawback of daisy-chain method (Oppenheimer, Doman, & Bolender, 2011). However, this could be beneficial in some applications and makes the method attractive from a practical point of view. In this dissertation, we take this advantage to set the vehicle dynamics actuators preference (here brake over

steering actuations), to satisfy the system requirements (see section 3-2 for more detail).