Non‐quadratic exponential stabilisation of non‐linear hyperbolic partial differential equation systems

Abstract

In this study, a new systematic approach is proposed to design the fuzzy controller for a class of Takagi-Sugeno fuzzypartial differential equation (TS fuzzy-PDE) systems which describe the non-linear distributed parameter system formulated by first-order semi-linear hyperbolic PDEs. In this study, non-quadratic Lyapunov function is utilised and some slack matrices are introduced to derive stability conditions in terms of linear matrix inequalities (LMIs). The proposed approach has three main features. First, stability conditions are not derived in the form of spatial differential LMI. Second, conservativeness of LMI conditions is reduced. Third, there is no restriction on the form of semi-linear hyperbolic PDE systems and therefore more semi-linear systems classes can be stabilised. Also, the proposed approach is more suitable for practical implementation compared with the recently published papers.

Distributed Saturated Control for a Class of Semilinear PDE Systems: An SOS Approach

“…N ×n y and ∆x = l2−l1 X , then the closed-loop system (19) is locally exponentially stable with decay rate ρ in the region (14), with controller (17) fulfilling bounds (18). The controller gains can be obtained as…”

“…Finally we will prove that (25) ensures (21) in the region Π, i.e., it ensures (21) with the bound (18). Recalling the Lyapunov functional (20) and the closed-loop system (19), the stability condition (21) yields:…”

“…Hence, the objective of this paper is finding a suitable state feedback controller (17) fulfilling the bounds (18) such that, in closed loop with (9), ensures exponential stability with a decay rate ρ > 0 starting from initial conditions y 0 (x) ⊂ Ω(y,…”

“…On the other hand, multipliers ϑ together with conditions (40) make (33) hold also in Π, as follows. We need to ensure the bound (18) inside each level set {(y, x q ) :…”

“…Hence, on the basis of the existing SDLMI technique, [4] addressed the distributed H ∞ control considering input constraints, [15] proposed an output feedback based on distributed-fuzzy observers, [16] addressed the tracking problem with variable-structure controllers ensuring exponential or practical stability and, recently, the static H ∞ output feedback with Markovian actuator faults was presented in [17]. Furthermore, the quadratic framework was left in [18] using fuzzy Lyapunov functions.…”