Distributed Parameter Control Strategies for the Wave Equation with Fractional Order Derivative

In this paper, the control strategies are studied for the fractional order uncertain wave equation subject to persistent external disturbances in Hilbert spaces. The twisting and super-twisting fractional order sliding mode controllers (SMCs) are designed for the infinite dimensional setting and they are applied for addressing the asymptotic state tracking of the fractional order perturbed wave equation. Furthermore, by introducing the adaptive control law to the twisting controller, the bound of the external disturbances which is unknown is dealt with, and for the design of the super-twisting SMC, a fractional order sliding mode manifold is utilized which results in a continuous input control and a chattering free signal. Both of the controllers are associated with the fractional order parameter, which influences the convergence rate of the proposed control algorithms. In addition, the relative theorem involved in the paper for the proof of the stability is proved. Then, the control algorithms are extended to globally asymptotically stabilize the fractional order uncertain wave equation by choosing the appropriate Lyapunov functional. Finally, numerical simulations are presented to verify the viability and efficiency of the proposed fractional order controllers.

Using functional gains for effective sensor location in flow control: a reduced-order modelling approach

The problem of active feedback control of fluid flows falls into a class of problems in the area of distributed parameter control. Distributed parameter systems are typically defined by partial differential equations that model the time and spatial evolution of the process. We consider the problem of locating sensors for effective feedback control of a fluid flow problem described by the Navier–Stokes equations. In this setting, the state of the system is the velocity field$\boldsymbol{v}(t,x)$, and hence all feedback laws are a function of this velocity field or, in most practical settings, a function of sensor outputs. In many designs, the feedback control law can be represented as a linear function of the state defined by an integral operator with a kernel function called the functional gain. In this paper we show that these functional gains can be used to determine effective sensor placement in complex flow control applications. The approach is to choose measurements of the state that would provide good quadrature points for the integral operator. We provide a computational validation of this approach by controlling the vortex shedding in a two-dimensional cylinder flow using a pair of fluid actuators on the cylinder surface. This model is linearized about the mean flow and a feedback control is designed by pole placement. Distributed parameter control theory yields the existence and form of the functional gains which are used to locate sensors. In particular, we use the location of the supports of the functional gains to determine two sets of four sensor locations in the wake. One of these measurement sets coincides with large magnitudes of the gain and the other set coincides with small magnitudes. Numerical experiments with a reduced-order model confirm superior performance of the closed-loop (CL) system using the former sensor set. We also show that choosing sensor locations associated with small magnitudes of the functional gains actually destabilizes the CL system.