Discrete-time model-based output regulation of fluid flow systems
Introduction
Flow control and manipulation play an important role in the realm of aerodynamics and hydrodynamics especially when it comes to drag reduction, lift enhancement and turbulence suppression. Generally speaking, there are mainly three approaches to cope with general fluid dynamics problems, namely, theoretical analysis, numerical simulation, and experimental study. Considering that experimental study often requires a prohibitive amount of time and cost, while numerical simulation heavily relies on advanced computational technology, computing capacity and most importantly on the availability of accurate, robust and flexible dynamic models, this manuscript seeks to propose an computationally efficient, implementable and scalable modelling method for fluid flow output regulation and manipulation.
Differently from the dynamics of lumped parameter systems, fluid dynamics often take place in both time and space domains and their states evolve on infinite-dimensional Hilbert spaces, which requires relatively complex spatiotemporal modelling techniques. Mathematically stated, most fluid dynamics models are governed by partial differential equations (PDEs) and/or delay equations, leading to general distributed parameter systems (DPSs). For instance, the complex Ginzburg-Landau equation (CGLE) involves a first-order temporal derivative, and first- and second-order spatial derivatives with complex model coefficients, which dramatically increases the difficulty of accurate modelling and corresponding control designs. In addition, one may need to address multiple spatial variables (e.g. three spatial components in the Navier-Stokes equation) and even higher-order derivatives and nonlinear terms (e.g. fourth-order spatial derivative and nonlinear multiplication term in the Kuramoto–Sivashinsky equation (KSE)). Hence, these considerations stated above provide a strong motivation to seek advanced modelling and control techniques for effective and implementable flow control of fluid dynamics systems.
Among the aforementioned fluid dynamics processes, vortex shedding has attracted increasing attention, due to their wide existence manifested by vortex formation when flows pass submerged obstacles with Reynolds numbers larger than the critical values. More specifically, a schematic diagram illustrating the vortex shedding phenomenon in a 2D flow behind a cylinder is given in Fig. 1, where it clearly shows an unstable vortex shedding and its evolution [1]. Additionally, there is also strong interest in falling thin film phenomena described by the Kuramoto–Sivashinsky equation, which as a representative PDE flow model accounts for a wide range of other complex phenomena, such as unstable flame fronts evolution, phase turbulence in Belousov–Zhabotinsky reaction-diffusion systems and interfacial instabilities between multiple viscous phases [38], [43], [53]. As shown in Fig. 2, a two-phase annular falling flow in a vertical tube is illustrated using a schematic [18]. For the sake of brevity, this manuscript considers vortex shedding phenomena and falling thin film processes as two representative examples, and review some existing work on modelling and control of CGLE and KSE sequentially.
Regarding vortex shedding analysis and suppression, plenty of studies have been carried out experimentally and using numerical simulation. From an experimental perspective, it has been revealed that the laminar Kármán vortex can be suppressed within a certain range of Reynolds number by several distinct approaches, including: external oscillating the cylinder normal to the mean flow [9], feedback control through suction and blowing treatment on the surface [27], [33], [51], and acoustic feedback of signals collected from hot-wires in the wake of the cylinder [58]. In addition, the Navier–Stokes equation has been explored to model the dynamics of the cylinder wake theoretically [12], [31], [47]. However, owing to the inherent complexity of the Navier–Stokes equation, most of the related work has been conducted numerically. On the other hand, the complex Ginzburg–Landau equation with appropriate coefficients was suggested as a simplified model to describe vortex shedding processes in [34]. Along the line of controller designs, a proportional feedback controller was proposed for Kármán vortex shedding suppression with Reynolds numbers close to the critical value ( based on the cylinder diameter) [52]. In addition, a non-linear one-dimensional Ginzburg–Landau wake model at above the critical Reynolds number was controlled using a conventional proportional-integral-derivative (PID) controller and a non-linear fuzzy controller [15]. For feedback boundary control, the backstepping approach has been extensively utilized for stabilization of 1D and 2D CGLE systems [1], [2], [3], [4]. The developed controllers were validated using computational fluid dynamics (CFD) simulations [44], [46] and extended to 3D scenarios [45]. A hybrid method and evolution strategies were deployed to study 2D and 3D vortex evolution of cylinder wakes in [49], [50]. From an optimal control perspective, a model predictive controller (MPC) was proposed to solve the problem of CGLE stabilization under consideration of input and state constraints [35]. These studies on CGLE are oriented on stabilizing control while work related to the output regulation of CGLE is limited, which motivates this contribution.
When it comes to flow control of falling thin film processes modelled by the Kuramoto–Sivashinsky equation, many control methods have been developed based on the recent advances in the area of control of distributed parameter systems. Among these, one important contribution lies in the stabilization of the KSE model, including: global stabilization of KSE by in-domain output feedback control [5], [14] and through boundary control [37], [39]. A single-input-single-output (SISO) and multiple-input-single-output (MISO) boundary model predictive controllers were presented for KSE stabilization in the presence of input and state constraints using a truncated modal decomposition [18], [19], [60]. A zero dynamics inverse design method was proposed for tracking regulation of a nonlinear KSE with two boundary actuators [11]. For sampled-data control of KSE, a spatially distributed controller was constructed for local stabilization of KSE by using a time-delay approach and a descriptor method [36]. Recently, a delayed boundary controller was designed for global stabilization of a linear KSE by means of the spectral decomposition and the Artstein transform [28]. Although these contributions have provided elegant solutions and controller designs to guarantee the exponential stability of the closed-loop system, most of them are conducted in a continuous-time setting, which at the implementation level needs to be realized in a computationally feasible setting and in addition brings another layer of complexity and questions to be addressed. Therefore, a realizable sampled-data servo-control design is needed for manipulation of various fluid dynamics systems represented by the complex Ginzburg-Landau equation and the Kuramoto–Sivashinsky equation. Moreover, the realization of continuous-in-time designs in the sampled-data setting with finite computational resources has not been fully addressed since temporal and spatial approximations (and/or model reduction) need to be performed for control algorithm realization with the hope that approximate controllers account for the infinite-dimensional nature of underlying distributed parameter models. Hence, the motivation behind this work is not to take the path of continuous designs first, then approximate at the realization level, but to discretize the model in time by application of structure-preserving Cayley–Tustin discretization (linear system properties including stability, controllability and observability are preserved [29], [30]) and conduct discrete-in-time regulator design for complex fluid flow systems without any model lumping or spatial approximation. In this way, both regulator design and control law computation can be accomplished in the natural discrete-time (sampled-data) setting of modern microchips.
Therefore, this manuscript is devoted to the development of effective, computationally realizable and scalable servo controllers in the discrete-time setting for fluid flow output regulation of linear CGLE and KSE systems. In particular, the Cayley–Tustin transform is deployed for infinite-dimensional model discretization in the time domain and it does not induce any spectral decomposition or spatial approximation. Then, the internal model principle [21] is revisited and extended to the output regulation of infinite-dimensional discrete-time systems. By further establishing a finite-dimensional discrete-time exogenous system, discrete-time regulator equations are formulated and utilized for solving a state feedback regulation problem. As for model stabilization, a pole-shifting approach is employed for continuous PDE models and linked to their discrete counterparts. Additionally, an output feedback regulator design is completed by the development of finite-dimensional (exo-system) and infinite-dimensional (fluid flow-plant) observers. Furthermore, we emphasize that proposed design is applicable for a general class of Riesz-spectral distributed parameter systems, and can be extended to DPS models other than CGLE and KSE systems.
The remainder of the paper is organized as follows. Section 2 presents a necessary preliminary, including: general continuous-time PDE model description, and time discretization with the aid of the Cayley–Tustin approach. In Section 3, a finite-dimensional discrete-time exogenous system is provided and a discrete output regulator design framework is given. To demonstrate the feasibility and effectiveness of the proposed method, two representative PDE flow models, i.e. CGLE and KSE systems, are analyzed in detail. The model formulation, spectrum analysis, analytic resolvent determination, and simulation studies for two models are shown in Sections 4 and 5, respectively. Finally, a conclusion is drawn in Section 6.
Section snippets
Notations
In this manuscript, the following notations are used. Suppose that and are two Hilbert spaces and is a linear operator. denotes the set of linear bounded operators from to . If we simply write . The domain, spectrum, resolvent set and resolvent operator of a linear operator are denoted by: and with respectively. We denote the space as the space with the norm and the space as the completion of
Output regulation
In this section, an output regulator design method is developed for output regulation of a class of infinite-dimensional systems in a discrete-time setting. In particular, a discrete-time output regulator design is proposed for self-adjoint Riesz-spectral PDE systems, based on the discretized distributed parameter flow plant and discrete exogenous system.
Complex Ginzburg Landau flow model
In this section, a linearized complex Ginzburg–Landau equation, which takes form of a complex parabolic partial differential equation, is considered. More specifically, a discrete-time CGLE model is generated without spatial approximation or model reduction using the Cayley–Tustin discretization method. Additionally, a resolvent operator is found in a closed analytic form and deployed in the realization of the discrete CGLE model.
Kuramoto–Sivashinsky Equation
In this section, a continuous-time nonlinear Kuramoto–Sivashinsky equation (KSE) is introduced to describe the falling thin film dynamics. For the sake of simplicity, a linear KSE is achieved by performing linearization. In the same manner, the Cayley–Tustin transformation is utilized for time discretization for the linear KSE model. Moreover, an explicit closed-form solution is obtained for the corresponding resolvent operator, which is exploited for the discrete-time regulator design.
Conclusion
In this work, discrete-time output regulators are designed for PDE model-based fluid flow manipulation and regulation. To model the vortex shedding process and falling thin film dynamics, the linearized complex Ginzburg–Landau and Kuramoto–Sivashinsky models are utilized. For realistic implementation of regulators in digital computer systems, the Cayley–Tustin time discretization method is utilized for discrete-in-time analysis with system properties preserved and no spatial approximation.
Declaration of Competing Interest
None.
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