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Maximal Output Admissible Sets (MOAS) are the set of all initial states and references such that the output response is always constraint admissible. Introduced in [1], the MOAS holds a special place in constrained control theory and is often used in strategies such as Model Predictive Control and Reference Governors to guarantee constraint satisfaction.

As desirable as it is to find the Maximal Output Admissible Set, it is often very difficult to compute it in closed form. Depending on the system and the constraints, the MOAS may not even be convex, bounded, closed, etc. For discrete-time LTI systems subject to linear state and input constraints, [1] provides an algorithm for computing the MOAS. In our latest paper [2], we show that it is possible to compute a provably larger set by treating input saturations as a nonlinearity as opposed to a constraint.

This intuition gives rise to the Input-Saturated Output-Admissible Set (ISOAS), which is the MOAS for systems with a saturated control law.

INPUT-SATURATED OUTPUT-ADMISSIBLE SETS

The traditional MOAS for an LT-DTI system is computed by using a prestabilizing linear controller.  Since the MOAS depends on the prestabilizing controller, it is possible to find a larger set (ISOAS) by using a saturated control law, rather than a linear one. The computation of ISOAS is done by leveraging the piecewise affine nature of the prestabilized system. We partition the state\reference space into three regions: Non-Saturated, Upper-Saturated, and Lower-Saturated.

The above animation shows the step-by-step computation of the ISOAS for a linear system with output constraints and input saturation. The specific system is described in [2, Section VI.C]. The Non-Saturated Region is shown in green, whereas the Upper and Lower-Saturated Regions are shown in yellow. The computation process of the ISOAS is as follows:

• First, the constraints are propagated within each region, causing each polyhedron to evolve independently from the others;
• Second, the constraints are shared between regions, thereby causing each reagion to be affected by the others.

This process is repeated until the final ISOAS is computed. 

EROSION PREVENTION 

One of the difficulties in the computation of ISOAS during the constraint sharing phase of the computation algorithm is that the constraints for one region might be harmful or redundant for the other one. To solve this issue, we introduced a erosion prevention step in our algorithm, whereby sharing constraints between regions is done only after eliminating any redundant constraints. 

The above animations show the difference between the computation of the ISOAS with and without erosion prevention for the example detailed in [2, Section VI.A]. The first image has erosion prevention, while the second does not. As can be seen, the first set is both bigger and faster to compute than the second set.

EMPTY SET PREVENTION

Another complication that can arise in the saturated regions is that the constraint propagation may lead to an empty set if performed incorrectly. To solve this issue, we introduce the concept of empty set prevention, whereby we omit any constraint such that the 1-step violation requirement is fully redundant with respect to the 2-step violation requirements. Below is an example of computation of the ISOAS with and without empty set prevention algorithm for the system featured in [2, Section VI.B].

The first animation shows the computation with the empty set prevention algorithm, while the second does not have that feature. It can be observed that the saturated regions in the second animation eventually reduce to an empty set and the computation of the ISOAS can not continue further. However, the first computation shows the actual  ISOAS, which is a polyhedron with non-empty saturated regions. 

REFERENCES

[1] E. G. Gilbert and K. T. Tan, “Linear systems with state and control constraints: The theory and application of maximal output admissible sets,” IEEE Transactions on Automatic Control, vol. 36, no. 9, pp. 1008–1020, 1991.

[2] Y. Gautam and M. M. Nicotra, "Finite-Time Computation of Polyhedral Input-Saturated Output-Admissible Sets", IEEE Transactions on Automatic Control (under review)