Exponential dissipative control for conformable nonlineardynamical systems

This paper addresses the design of a controller that exponentially stabilizes Conformable Nonlinear Dynamical Systems (CNDSs) with time delay. To address the nonlinear dynamics of the system, we employ a Polynomial Fuzzy (PF) modeling approach. This method provides an exact representation of the original CNDS and serves as a generalized framework that extends the classical Takagi-Sugeno Fuzzy (TSF) model. The controller is designed to ensure not only exponential stability, which implies a prescribed convergence rate compared to asymptotic stability, but also the strictly $(\mathcal{U},\mathcal{V},\mathcal{W})$-$\sigma$-dissipativity of the closed-loop system. Moreover, the controller design explicitly accounts for partial state measurements by employing an observer to estimate the unmeasured states. The proposed conditions reduce conservatism for several reasons. A decoupling technique is employed that alleviates the limitations of the singular value decomposition approach, as the Lyapunov matrix is not restricted to a specific structure. In addition, the decision variables, essentially the controller gains, are not constant but are allowed to be polynomial functions, computed using the SOSTOOLS framework rather than the standard LMI toolbox. Furthermore, recently proposed relaxed conditions for parameterized Linear Matrix Inequalities (LMIs) in double-sum form are extended to parameterized Sum-of-Squares (SOS) constraints. The effectiveness of the proposed results is shown through a numerical example.