A closed‐loop nonlinear control and sliding mode estimation strategy for fluid flow regulation

Abstract

Summary Novel sliding mode observer (SMO) and robust nonlinear control methods are presented, which are shown to achieve finite‐time state estimation and asymptotic regulation of a fluid flow system. To facilitate the design and analysis of the closed‐loop active flow control (AFC) system, proper orthogonal decomposition–based model order reduction is utilized to express the Navier‐Stokes partial differential equations as a set of nonlinear ordinary differential equations. The resulting reduced‐order model contains a measurement equation that is in a nonstandard mathematical form. This challenge is mitigated through the detailed design and analysis of an SMO. The observer is shown to achieve finite‐time estimation of the unmeasurable states of the reduced‐order model using direct sensor measurements of the flow field velocity. The estimated states are utilized as feedback measurements in a closed‐loop AFC system. To address the practical challenge of actuator bandwidth limitations, the control law is designed to be continuous. A rigorous Lyapunov‐based stability analysis is presented to prove that the closed‐loop flow estimation and control method achieves asymptotic regulation of a fluid flow field to a prescribed state. Numerical simulation results are also provided to demonstrate the performance of the proposed closed‐loop AFC system, comparing 2 different designs for the SMO.

Finite-Time State Estimation for an Inverted Pendulum under Input-Multiplicative Uncertainty

“…Note that, based on Assumption 2, the observer gains β i,j , for i, j = 1, 2 can be selected to achieve finite-time state estimation using only approximate knowledge of the system parameters. Provided inequalities Equations (30), (31) (36) and (37) are satisfied, the solution to Equation (46) can be upper bounded as˙X…”

Sliding mode estimation and closed‐loop active flow control under actuator uncertainty

“…T ∈ R n contains the unmeasurable coefficients resulting from Galerkin projection, g(x) ∈ R n×m denotes an input gain matrix, u(t) ≜ [u 1 (t), … , u m (t)] ∈ R m is a subsequently defined virtual control input generated by an array of SJAs, and y(t) ∈ R is the sensor measurement equation (ie, the direct measurements of flow field velocity or pressure 27 ).…”

“…for t ≥ t 1 . It follows from (18) and (27) that h 2 (x) = 2 (t). The estimation error dynamics forx 2 can then be expressed as…”

“…The contribution of this article is a SMO and SJA-based closed-loop AFC system, which achieves local finite-time state estimation and locally asymptotic flow field velocity regulation in the presence of the parametric uncertainty that is inherent in the SJA actuator model. In an extension of our recent work, 27 a SMO is designed, which formally incorporates a compensator for the input-multiplicative parametric uncertainty in the SJA actuator model. A robust nonlinear control method is then developed, which incorporates a feedback element to compensate for an uncertain and time-varying input gain matrix.…”