Abstract:This paper addresses the problem of designing low-order and linear robust feedback controllers that provide a priori guarantees with respect to stability and performance when applied to a fluid flow. This is challenging since whilst many flows are governed by a set of nonlinear, partial differential-algebraic equations (the Navier-Stokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are firstly: linear, secondly: described by ordinary diff… Show more
“…However, the question remains, what effect do these differences have when using the system formulations for feedback control design? We again stress at this point that models that are suitable for simulation are not necessarily suitable for control, and vice-versa [5,12]. In order to gain insight into this, actuation and sensing are now defined for the 2D channel flow in order to determine the system zeros, and compare the frequency responses from actuation to sensing.…”
Section: In the Cases Of [Ppe2]-[ppe4]mentioning
confidence: 99%
“…As in many previous studies (see, e.g. [48,12,13]), we assume actuation in the form of wall transpiration, in this case at the upper wall. The actuator is modelled as a first-order system:…”
SUMMARYMotivated by the need to efficiently obtain low-order models of fluid flows around complex geometries for the purpose of feedback control system design, this paper considers the effect on system dynamics of basing plant models on different formulations of the linearised Navier-Stokes equations. We consider the dynamics of a single computational node formed by spatial discretisation of the governing equations in both primitive variables (momentum equation & continuity equation) and pressure Poisson equation (PPE) formulations. This reveals fundamental numerical differences at the nodal level, whose effects on the system dynamics at the full system level are exemplified by considering the corresponding formulations of a two-dimensional (2D) channel flow, subjected to a variety of different boundary conditions.
“…However, the question remains, what effect do these differences have when using the system formulations for feedback control design? We again stress at this point that models that are suitable for simulation are not necessarily suitable for control, and vice-versa [5,12]. In order to gain insight into this, actuation and sensing are now defined for the 2D channel flow in order to determine the system zeros, and compare the frequency responses from actuation to sensing.…”
Section: In the Cases Of [Ppe2]-[ppe4]mentioning
confidence: 99%
“…As in many previous studies (see, e.g. [48,12,13]), we assume actuation in the form of wall transpiration, in this case at the upper wall. The actuator is modelled as a first-order system:…”
SUMMARYMotivated by the need to efficiently obtain low-order models of fluid flows around complex geometries for the purpose of feedback control system design, this paper considers the effect on system dynamics of basing plant models on different formulations of the linearised Navier-Stokes equations. We consider the dynamics of a single computational node formed by spatial discretisation of the governing equations in both primitive variables (momentum equation & continuity equation) and pressure Poisson equation (PPE) formulations. This reveals fundamental numerical differences at the nodal level, whose effects on the system dynamics at the full system level are exemplified by considering the corresponding formulations of a two-dimensional (2D) channel flow, subjected to a variety of different boundary conditions.
“…In this section, we show that robust control tools can quantify the model uncertainty associated with these choices. A related approach was used by Jones et al (2015) in a grid refinement study.…”
Section: Controller Robustness and Model Uncertaintymentioning
confidence: 99%
“…Maximising b(G w , K ∞ ) can be shown (Vinnicombe 2001;Jones et al 2015) to maximise the robust stability of the system with respect to perturbations to the normalised coprime factors of the model, which can be used to represent a large set of realistic model uncertainties. In fact, it can be shown (Vinnicombe 2001) that the norms of both sets of transfer functions are equal:…”
mentioning
confidence: 99%
“…Hence, maximising b(G w , K ∞ ) minimises the influence of the worst-case disturbances entering the model from all positions in the feedback loop and can be seen as giving a measure of robust performance as well as robust stability (Jones et al 2015).…”
Obtaining low-order models for unstable flows in a systematic and computationally tractable manner has been a long-standing challenge. In this study, we show that the Eigensystem Realisation Algorithm (ERA) can be applied directly to unstable flows, and that the resulting models can be used to design robust stabilising feedback controllers. We consider the unstable flow around a D-shaped body, equipped with body-mounted actuators, and sensors located either in the wake or on the base of the body. A linear model is first obtained using approximate balanced truncation. It is then shown that it is straightforward and justified to obtain models for unstable flows by directly applying the ERA to the open-loop impulse response. We show that such models can also be obtained from the response of the nonlinear flow to a small impulse. Using robust control tools, the models are used to design and implement both proportional and H ∞ loop-shaping controllers. The designed controllers were found to be robust enough to stabilise the wake, even from the nonlinear vortex shedding state and in some cases at off-design Reynolds numbers.
The paper considers control system design for linearized three-dimensional perturbations about a nominal laminar boundary layer over a flat plate (the Blasius profile). The objective is prevention of the laminar to turbulent transition using appropriate inputs, outputs, and feedback controllers. They are synthesized with a view to reducing transient energy growth, a known precursor to important transition scenarios. The linearized Navier–Stokes equations are reduced to the Orr–Sommerfeld and Squire equations with wall-normal velocity actuation entering through the boundary conditions on the wall. The sensor output is taken to be the wall-normal derivative of the wall-normal vorticity measured on the plate. Several multivariable output controllers are examined, including simple constant gain output feedback, loop transfer recovery, and $$H_{\infty }$$
H
∞
loop shaping. Reduced order compensators are developed using balanced truncation and analyzed for robustness using the gap metric between reduced order models and full order models. It is demonstrated that the level of minimum transient energy growth that can be achieved is similar for these diverse controller methodologies but falls short of that which can be achieved using optimal state feedback.
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