An Explicit Reference Governor for the robust constrained control of nonlinear systems

Embedding Constrained Model Predictive Control in a Continuous-Time Dynamic Feedback

“…where is a repulsion term that points away from the constraint boundary. As discussed in [42], η > 0 is an arbitrarily small radius which ensures that ργ(r, γ) gradually goes to zero in r = γ. The scalars δi > 0 are the static safety margins used to define the set R, whereas the scalars ζi > δi are influence margins that ensure that the contribution of the i-th constraint is non-zero if and only if ai ξr + bi(r) > −ζi.…”

Explicit Reference Governor for the Constrained Control of Linear Time-Delay Systems

“…Finally, convergence to the desired reference r is ensured by the fact that, for any strictly steady-state admissible reference v, the condition ∆(xτ , vτ ) = 0 cannot hold indefinitely due to (21d), meaning that v(t) will eventually converge to the equilibrium point of the vector field ρ(v, r), which is r. Detailed proofs for the ideas behind these argumentations can be found in e.g. [23]. The following subsections will illustrate how to construct a suitable dynamic safety margin and attraction field for LTD systems subject to linear constraints.…”

“…Proof: For what concerns the time window θ ∈ [0, T ], it is sufficient to note that (23) captures the minimum distance between the trajectory of ( 17)- (19) and the boundary of constraints (2). As for the time window θ > T , it follows from (18) that v(θ) = v for θ ∈ [T − τ, ∞).…”

“…It is worth noting that, even though Figures 3-4 and Figures 5-6 present feature slightly different responses (likely due to model mismatch), constraint satisfaction is still guaranteed. As discussed in [23], this is due to the inherent robustness of the Lyapunov-based ERG.…”

A Fast Reference Governor for the Constrained Control of Linear Discrete-Time Systems with Parametric Uncertainties

“…Another class of solutions for constrained systems is the Reference Governor (RG), which is an add-on unit capable of ensuring constraint satisfaction by manipulating the reference of a pre-stabilized system [7], [8]. Robust reference governor schemes have been developed using both trajectory prediction tubes [9]- [11] and invariance-based considerations [12], [13]. Applications have been reported in [14], [15].…”