ℓ₁-control using linear programming for systems with asymmetric bounds

Control of systems with asymmetric bounds using linear programming: Application to a hydrogen reformer

“…The following result, which is an extension to MIMO systems of the theorem proposed in Naib et al [11], gives necessary and sufficient conditions for a vector of signals to avoid the constraints, that is, ϕ(z) ≤ Z for any bounded vector of inputs ϕ(w) ≤ W such that z = Mw.…”

“…This section presents the objective functional proposed by the authors (Naib et al [11]), which replaces the peak-to-peak-norm normally used in the 1 analysis when asymmetric signals are used. For simplicity reasons, the method is presented for the singleinput/single-output case, although it is very simple to extend it to the multivariable case, as it is discussed later on.…”

“…To solve this drawback, the following objective functional (denoted by ϕ(w)) was proposed by the authors (Naib et al [11]), to give information on the signal amplitude, retaining information on the asymmetricity, and being valid for multivariable systems:…”

“…Then, this paper studies a novel extension of the 1 control theory, to directly handle the asymmetrical aspect of bounded signals. The authors introduced this extension from a purely theoretical point of view in a previous paper (Naib et al [11]), following the methods proposed for positive invariance (Dambrine et al [6]; Benzaouia and Burgat [1]). This proposed extension to asymmetric signals is necessary because when the considered signal u is constrained to evolve in an asymmetrical domain (i.e., signal u is such that −u min ≤ u(k) ≤ u max , where u min ≥ 0; u max ≥ 0; u min = u max ), in this case the peakto-peak norm of the signal u (by definition, u ∞ = max k (|u(k)|) = max(u max ,u min )) gives a quite conservative information on the signal amplitude, so a new functional must be defined that takes separately into account the positive and negative deviations of the signals.…”

“…This proposed extension to asymmetric signals is necessary because when the considered signal u is constrained to evolve in an asymmetrical domain (i.e., signal u is such that −u min ≤ u(k) ≤ u max , where u min ≥ 0; u max ≥ 0; u min = u max ), in this case the peakto-peak norm of the signal u (by definition, u ∞ = max k (|u(k)|) = max(u max ,u min )) gives a quite conservative information on the signal amplitude, so a new functional must be defined that takes separately into account the positive and negative deviations of the signals. This paper presents how the functional defined in (Naib et al [11]) can be extended to a multiblock synthesis problem for a real engineering problem: the control of a hydrogen reformer plant in petrochemical industry, which prompted us the development of this technique.…”