Design of State-Feedback Controllers for Linear Parameter Varying Systems Subject to Time-Varying Input Saturation

Abstract

All real-world systems are affected by the saturation phenomenon due to inherent physical limitations of actuators. These limitations should be taken into account in the controller’s design to prevent a possibly severe deterioration of the system’s performance, and may even lead to instability of the closed-loop system. Contrarily to most of the control strategies, which assume that the saturation limits are constant in time, this paper considers the problem of designing a state-feedback controller for a system affected by time-varying saturation limits with the objective to improve the performance. In order to tie variations of the saturation function to changes in the performance of the closed-loop system, the shifting paradigm is used, that is, some parameters scheduled by the time-varying saturations are introduced to schedule the performance criterion, which is considered to be the instantaneous guaranteed decay rate. The design conditions are obtained within the framework of linear parameter varying (LPV) systems using quadratic Lyapunov functions with constant Lyapunov matrices and they consist in a linear matrix inequality (LMI)-based feasibility problem, which can be solved efficiently using available solvers. Simulation results obtained using an illustrative example demonstrate the validity and the main characteristics of the proposed approach.

Design of shifting state‐feedback controllers for constrained feedback linearized systems: Application to quadrotor attitude control

“…Note that the region (x) is an asymmetric region, even if constraint limits in (3) are symmetric, and that the region bounds may potentially have the same sign, owing to the appearance of terms M(x) −1 [l] b(x) on both sides of the inequality (18).…”

“…Considering a mapping as the one proposed in Reference 18, the relationship that links $$$ \varphi \in \Phi \subset {\mathbb{R}}^{n_{\varphi }} $$$ with the possible instantaneous values of $$$ {\sigma}_l{(x)}^2 $$$ is defined as follows: $$$$ {\varphi}_l\triangleq \frac{{\overline{\sigma}}_l^2-{\sigma}_l{(x)}^2}{{\overline{\sigma}}_l^2-{\underset{\_}{\sigma}}_l^2},\kern1em {\varphi}_l\in \left[0,1\right],\kern1em \forall l\in {\mathcal{I}}_{\left[1,m\right]}, $$$$ allowing the definition of $$$ \Phi $$$ as a hypercube with $$$ {2}^{n_{\varphi }} $$$ vertices. Additionally, the time‐variability of $$$ {\sigma}_l{(x)}^2 $$$ in (22) can be described as a function of $$$ {\varphi}_l $$$ by the following expression: …”

“…The shifting paradigm concept allows us to exploit the properties of polytopes and LMIs in order to design controllers that adapt the system's closed‐loop performance online depending on the chosen performance criteria, for example, guaranteed decay rate, pole clustering or robust bounds as $$$ {\mathscr{H}}_{\infty } $$$. In References 18 and 19, a shifting state‐feedback controller has been designed so that the closed‐loop system convergence speed could be regulated in concordance with the instantaneous saturation limit values. Furthermore, a shifting controller with the capability to adapt its closed‐loop system response to avoid saturation has been developed for an N‐DoF robotic manipulator via the pole clustering criterion 20 …”

“…Motivated by the preceding discussion, the aim of this work is to integrate the shifting paradigm concept and the FBL technique to regulate the closed‐loop performance of a full linearized system subject to state‐dependent input saturation. To this end, an LMI‐based methodology via a parameter‐independent quadratic Lyapunov function (QLF) has been developed for designing a shifting state‐feedback controller, extending the methodology presented in Reference 18 to the case where an input‐output FBL technique is used on multiple‐input multiple‐output (MIMO) nonlinear systems with input saturation. The main contributions are as follows:A shifting control strategy for guaranteeing the stabilization of constrained linearized systems under saturation avoidance is proposed, such that the combination of the linearizing law and the shifting state‐feedback control law remains within the limits of the actuator. Unlike the previously mentioned works that employed the FBL technique, the proposed shifting state‐feedback controller adjusts the convergence rate of the closed‐loop system response according to the instantaneous values of the time‐varying system's linearity region. The demonstration of the performance and potential of the developed approach is carried out through an experimental application to the Quanser 3‐DoF hover 21 …”

LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of 𝒟R-regions

“…This approach allows designing the controller gain in such a way that different values of the varying parameters imply different regions where the closed-loop poles are situated. By means of the shifting paradigm, the online modification of the performance can be achieved, as demonstrated for example by Ruiz, Rotondo, and Morcego (2019) and Ruiz, Rotondo, and Morcego (2020), who have applied this concept to saturated system showing that it is possible to schedule the closed-loop performance according to changes in the saturation function.…”