Asymmetric State Feedback for Linear Plants With Asymmetric Input Saturation

Abstract

We consider a linear plant with decentralized input saturation, whose limits are not necessarily symmetric. We propose a nonlinear static state feedback stabilizer that is asymmetric, in such a way that the non-symmetric nature of the saturation is fully exploited in the control design, for larger regions of attraction. We show by example that the proposed technique provides significantly larger regions of attractions as compared to the symmetric solution results, in a case where the positive and negative saturation limits are different.

An Asymmetric Stabilizer Based on Scheduling Shifted Coordinates for Single-Input Linear Systems With Asymmetric Saturation

“…For the shifted dynamics (10), it is evident that the same result as that of Corollary 1 applies. This fact is stated in the following corollary, where a more convenient expression of u is deduced from (10b) exploiting the identities (4) and…”

“…This implies that performance is locally preserved, which we state for completeness in the following corollary. This result could not be attained by the asymmetric solution proposed in [10], due to the convex scaling performed therein, which reduced by one half the convergence rate (see [3,Thm 3.3]).…”

“…Finally, more sophisticad stabilizers have been proposed in [13], where a switching dynamical controller is designed, capable of exploiting the available range of the control action on both sides of the saturation levels. In an attempt to provide enlarged regions of attraction, we recently proposed in [10] the design of an asymmetric stabilizer, based on the convex scaling proposed in [3] for the shifted stabilizer. That solution provides significantly larger guarantees but has the drawback of 1) restricting the achievable region of attraction with a conservative point inclusion condition required to apply the technique in [3] and 2) reducing the local performance of the symmetric solution.…”

“…8], and continuously driving that equilibrium back to the origin by relying on the solution of a parametric optimization problem. As compared to [10], this solution is not based on convex scalings and therefore preserves the convergence rate given by the symmetric stabilizer in its guaranteed ellipsoidal set. Moreover, it does not require any point inclusion conditions, therefore obtaining an estimate of the basin of attraction containing all the contractive ellipsoids that can be constructed in shifted coordinates.…”

Solving Nonlinear Algebraic Loops Arising in Input-Saturated Feedbacks

“…An example of control design providing well-posed selections of the gains K and L in (2) is the solution of the LMIs in the next proposition, which is a classical result (see, e.g., [37], [25] but also the more recent works [22], [20] for its proof). Proposition 2.…”