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A new framework is presented for estimation and control of instabilities in wall-bounded shear flows described by the linearised Navier-Stokes equations. The control design considers the use of localised actuators/sensors to account for convective instabilities in an H 2 optimal control framework. External sources of disturbances are assumed to enter the control domain through the inflow. A new inflow disturbance model is proposed for external excitation of the perturbation modes that contribute to transition. This model allows efficient estimation of the flow perturbations within the localised control region of a conceptually unbounded domain. The state-space discretisation of the infinite dimensional system is explicitly obtained, which allows application of linear control theoretic tools. A reduced order model is subsequently derived using exact balanced truncation that captures the input/output behaviour and the dominant perturbation dynamics. This model is used to design an H 2 optimal controller to suppress the instability growth. The 2-D non-periodic channel flow is considered as an application case. Disturbances are generated upstream of the control domain and the resulting flow perturbations are estimated/controlled using point wall shear measurements and localised unsteady blowing and suction at the wall. The controller is able to cancel the perturbations and is robust to both unmodelled disturbances and sensor inaccuracies. For single frequency and multiple frequency disturbances with low sensor noise nearly a full cancellation is achieved. For stochastic forced disturbances and high sensor noise an energy reduction in perturbation wall shear stress of 96% is shown.

Selective Frequency Damping (SFD) is a popular method for the computation of globally unstable steady-state solutions in fluid dynamics. The approach has two model parameters whose selection is generally unclear. In this article, a detailed analysis of the influence of these parameters is presented, answering several open questions with regard to the effectiveness, optimum efficiency and limitations of the method. In particular, we show that SFD is always capable of stabilising a globally unstable systems ruled by one unsteady unstable eigenmode and derive analytical formulas for optimum parameter values. We show that the numerical feasibility of the approach depends on the complex phase angle of the most unstable eigenvalue. A numerical technique for characterising the pertinent eigenmodes is presented. In combination with analytical expressions, this technique allows finding optimal parameters that minimise the spectral radius of a simulation, without having to perform an independent stability analysis. An extension to multiple unstable eigenmodes is derived. As computational example, a two-dimensional cylinder flow case is optimally stabilised using this method. We provide a physical interpretation of the stabilisation mechanism based on, but not limited to, this Navier-Stokes example.

A new methodology is presented for model reduction of linear parabolic partial differential equations (PDEs) on general geometries using multivariate splines on triangulations. State-space descriptions are derived that can be used for control design. This method uses Galerkin projection with B-splines to derive a finite set of ordinary differential equations (ODEs). Any desired smoothness conditions between elements as well as the boundary conditions are flexibly imposed as a system of side constraints on the set of ODEs. Projection of the set of ODEs on the null space of the system of side constraints naturally produces a reduced-order model that satisfies these constraints. This method can be applied for both in-domain control and boundary control of parabolic PDEs with spatially varying coefficients on general geometries. The reduction method is applied to design and implement feedback controllers for stabilisation of a 1-D unstable heat equation and a more challenging 2-D reaction-convection-diffusion equation on an irregular domain. It is shown that effective feedback stabilisation can be achieved using low-order control models.

High performance flight control systems based on the nonlinear dynamic inversion (NDI) principle require highly accurate models of aircraft aerodynamics. In general, the accuracy of the internal model determines to what degree the system nonlinearities can be canceled; the more accurate the model, the better the cancellation, and with that, the higher the performance of the controller. In this paper a new control system is presented that combines NDI with multivariate simplex spline based control allocation. We present three control allocation strategies which use novel expressions for the analytical Jacobian and Hessian of the multivariate spline models. Multivariate simplex splines have a higher approximation power than ordinary polynomial models, and are capable of accurately modeling nonlinear aerodynamics over the entire flight envelope of an aircraft. This new method, indicated as SNDI, is applied to control a high performance aircraft (F-16) with a large flight envelope. The simulation results indicate that the SNDI controller can achieve feedback linearization throughout the entire flight envelope, leading to a significant increase in tracking performance compared to ordinary polynomial based NDI.