A new Saturation function to convert an output constraint into an input constraint

Abstract

The main result of our new method is to convert an output constraint into an input constraint for a nonlinear system. Then, standard saturated input design can be used as an anti-windup design for instance which is known to be efficient to enlarge a closed-loop system stability domain in presence of control saturations. We only address the single input single 'saturated' output problem in this paper. I. INTRODUCTIONOutput and Input hard constraints problems arise in most of the control applications because achieving good performance often requires to reach the inherent physical and safety limitations of a concrete system. Even if one has to cope with possible future destabilizing effects, the application of a saturation placed just after an input is a simple and sure strategy to handle an input constraint whereas there does not exist an equivalent element for an output limitation. That's probably why the input constraint problem has received more attention in the past few decades (see for instance [4]). Existing solutions to these problems can be divided into two groups : those who predict the future of the closed loop trajectory and check if the constraints will be violated and the other, which at each time 'do their best' to avoid the constraints. Concerning Linear systems, not only future prediction is easier but we can also apply some dedicated LMI-based methods [6], [2]. Moreover, state and input constraints problems are very close since it is possible to apply a relationship between a constrained output and an induced constrained input (whose constraints are state dependent) when the system is perfectly known and discretized [3], [1]. In this paper, we propose to generalize this idea to nonlinear systems by transforming an output constrained nonlinear system into an input constrained nonlinear system whose constraints are state dependent. As we will see our method can still be applied when some disturbances are applied to the nonlinear system and does not require to discretize the system but contrary to the linear case it can be slightly conservative. By comparison to existing results based on nonlinear model predictive control [5], [9], [7], [10] (to cite a few), our method does not use prediction and can thus be easily implemented ; however, it is fair to say that we do not address several output constraints problems (where the number of inputs is inferior to the number of constrained outputs) in this preliminary work. Other possible and very useful methods [8], [11], [12] address output constraints problems for nonlinear systems of special form and/or try to escape the constraints while

Output to input saturation transformation: Demonstration and application to disturbed linear systems

“…where 1) . Since e and its derivatives are not known, it is not possible to simplify α (j−1) (t) in the expression of Ω j (t).…”

“…, (1) , e (1) +Ω 1 (38) where Ω 0 > 2g (e, e). Then the values of the κ i 's are deduced κ1 =κ 1 +2g e (1) ,e (1) Ω 0 −2g(e,e) κ2 =κ 2 −Ω 1 +2g e (2) ,e (2) (1) ,e (1) (39) whereκ = κ 1κ2 = 30 10 . Finally, the expressions of the saturations to apply to the controller output are inspired from Eq.…”

“…The output-to input-saturation transformation (OIST) theory was first detailed in [1], [2] and proposes to find a remedy to this by reformulating 'saturations' (or expected bounds) on the regulated variable α into saturations on the controller output u. Other strategies include [3], [4] which use anti-windup loops to constrain the state or outputs in the time-domain.…”

Longitudinal Manoeuvre Load Control of a Flexible Large-Scale Aircraft

“…General result: The general method which consists in converting an interval constraint on an output z(t) into a saturation on the control law u(t) is described in Burlion (2012). The first step consists in computing the relative degree (denoted d rel u (z)) of the constrained output z(t) with respect to the control value u(t).…”

“…In the simple case of relative degree 0, the Output to Input Saturation Transform (OIST) (Burlion (2012), Burlion and de Plinval (2013)) boils down saying that…”