address this very issue, as demonstrated by Reinschke & Smith (2003). Loop shaping design is a (temporal) frequency domain approach to **control** systems design. However, it should be noted that frequency does not unambiguously distinguish large structure moving quickly from small structure moving slowly, and it is the former that makes the greater contribution to skin friction. In the present work, we have assumed, for simplicity, that the walls are densely populated with sensors. As a result, the linearised system is rendered observable. Similarly, if in the **control** objective, small structure plays an important part in generating skin-friction drag then the techniques described in this paper ensures mesh reﬁnement to an appropriate level by increasing the spatial ﬁdelity of the model. Hence, the model reﬁnement process indicates the appropriate degree of reﬁnement required to meet the **control** objective. For more realistic conﬁgurations, future research should also address the consequences of non-conservative domains, i.e. those in which the nonlinearity of the disturbance ﬁeld may be taken to be signiﬁcant, and the extent to which it may be accommodated by the disturbance rejection framework. For ﬂows exhibiting a more broadband forcing, such as in a turbulent mixing layer for example, our approach would require a high bandwidth controller to reject high frequency disturbances, which would likely necessitate the use of fast actuation, which may or may not be possible. The leads onto the ﬁnal point that despite the potential eﬀectiveness of the linear controllers developed in this paper, it is possible that some form of nonlinear **control** may provide enhanced performance by selectively exploiting the nonlinearity of the ﬂow in some desirable fashion, and designing such controllers could be an interesting avenue of future research.

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the micro-scale, it is necessary to use kinetic theory to get a more detailed descrip- tion in terms of the probability distribution function of particles [Bird, Armstrong, and Hassager (1987)]. Thus, rheological properties at the macroscopic level can be solved by a multi-scale strategy consisting in searching for the information on the microstructures of the fluids. The information is then used to solve the macro- scopic governing equations. This continuum-microscopic (also known as macro- micro) multi-scale approach does not require closed form constitutive equations [Ottinger (1996)]. The approach is an attempt to emulate the situation in real liq- uids, where the full information about the stress is contained in the configuration of molecules which results from the micro-scale deformation history. The main idea of these techniques is that the polymer contribution to the stress is directly calcu- lated from a large ensemble of microscopic configurations without having to derive a closed form constitutive equation, which is a powerful feature for the **modelling** of materials [Engquist, Lötstedt, and Runborg (2000)]. On the computational side, several numerical techniques have been developed for the continuum-microscopic multi-scale approach [Laso and Ottinger (1993); Hulsen, van Heel, and van den Brule (1997); Somasi and Khomami (2001); Jourdain, Lelièvre, and Bris (2002); Tran-Canh and Tran-Cong (2002); Keunings (2004); Tran, Phillips, and Tran-Cong (2009)].

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at zero Reynolds number, and both agree with the Navier–Stokes system regarding the critical Reynolds number for the onset of instability. However, the structures of these models differ significantly, in particular the number of degrees of freedom. The Benney equation is a single evolution equation for the interface height h(x, t), while the weighted-residual model comprises coupled equations for h(x, t), and the independently- evolving down-slope flux q(x, t), and of course the Navier–Stokes equations at finite Reynolds number allow evolution of h(x, t) together with evolution of the flow velocity at every point within the **fluid**. The robustness of **control** strategies to changes in the model is one of the major themes of this paper; we seek to understand what features of the system state need to be measured to deliver effective **control**, and whether the **control** system can be designed without needing detailed knowledge of the system state and underlying dynamics.

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Where Q and R are the weight matrices, Q is required to be high quality precise or high quality semi-definite symmetry matrix. R is required to be positive definite symmetry matrix. One practical method is to Q and R to be diagonal matrix. The value of the factors in Q and R is related to its contribution to the cost function J. The **feedback** manage regulation that minimizes the value of the cost is:

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There were candidate approaches to Sturm-Liouville dynamics **control**. Some are based on order-truncated plants, for examples [16-20], which can employ the small-gain theorem in Class C3. Thereof, the spatiotemporally **robust** performance is not guaranteed at the stage of finite-dimensional syntheses. Another two types are involved in **modelling** with infinite-order transfer-functions [21-24] and identification with fraction-order transfer-functions [25-29], respectively, which can apply the small-gain theorem in Class C2 toward **robust** performance with merely well-posedness and dissipativity. Besides, they were not developed originally for distributed sensing and actuation. The other is nD state-space **robust** **control** [30-35] extended from 1D **robust** state-space synthesis to nD version in space-time domain. It can merely employ the small-gain theorem in Class C2 to guarantee well-posedness and dissipativity. For these reasons, this work show how the small-gain theorem in Class C4 leads to modal-spectral loopshaping of Sturm-Liouville controllers to guarantee Hurwitz, dissipativity, passivity and well-posedness, which is beyond the capacities of the above approaches.

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Among the few earlier efforts relying on streaming measurements from a few sensors, a trained neural network using surface measurements is employed to re- duce the drag of a turbulent channel flow with an opposition **control** strategy in [7]. In [8], pressure sensors and an auto-regressive model (AutoRegression with eXoge- nous inputs, ARX) are used to reduce flow-induced cavity tones. An autoregressive approach is also followed in [9] and [10], while a genetic programming technique is adopted in [11] to **control** a separated boundary layer. Interested readers may refer to [12] for a comprehensive review of the topic. The present approach aims at deriving an efficient, yet **robust**, nonlinear closed-loop **control** method compliant with actual situations. Among its distinctive features compared to other methods is a combination of both performance and fast learning.

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We have successfully implemented reduced-order based modeling and **feedback** **control** for the long wave pertubations for the viscous lm ows, particularly, the Kuramoto- Shivashinsky equation. The associated diculties are due to innite dimensionality and the nonlinearity, which resembles the convection of many uid applications, such as the Navier-Stokes equations. We discussed two reduced-order methods, the Approximate In- ertial Manifold and the Proper Orthogonal Decomposition. We have shown that both methods can convert an innite dimensional nonlinear system to that of smaller dimen- sions, which optimal **feedback** controls can be designed and eciently synthesized. For the **feedback** **control** methodologies, we apply the linear and nonlinear quadratic regulators, which are the rst- and second-order solutions of the Hamilton-Jacobi-Bellman equation, respectively. Numerical solutions of the controlled problem are presented and the results justify the robustness of the nonlinear **feedback** **control** over the linear one.

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In practice, it is impossible to fully avoid packet reordering or loss, and distributed traffic monitoring has to be **robust** to such network noise. Unlike distribution of static policies [3]– [7], distributed execution of a stateful task needs additional means to acquire such robustness. Open-loop and closed- loop **control** constitute two general approaches to dealing with network noise. While a closed-loop design can adapt its operation to the current level of network noise, the **feedback**- driven robustness increases latency, which is undesirable for real-time monitoring. Also, asymmetric routing, restrictions on generation of new packets in a network element, and other factors might make it infeasible – or at least very difficult – to provide **feedback** to a previous element on the path of a unidirectional traffic flow [8]. Hence, the problem of **robust** distributed monitoring of traffic **flows** is more amenable to the open-loop approach that can communicate flow state in-band and keep latency low.

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are LQR, and H . It is known that if the controller is not **robust** enough, the uncertainties of the system may destroy the efficiency of the controller. The H , **control** provides better robustness than LQR [12-14]. The aim of this work is to design an H , **robust** controller for a beam bonded with piezoelectric sensors and actuators and to investigate the behaviour of the controlled beam. First, a detailed shear-deformable (Timoshenko) model for a laminated beam structure is developed. A finite element formulation is presented for the model. Quad- ratic Hermitian polynomials are used for the transverse and rotational displacements, respectively. The differen- tial equations are based on the Timoshenko beam theory [15]. The governing state equation is established and used for the design of the **control**. The numerical simula- tions carried out on the laminated beam shows that the vibration of the system is significantly suppressed within the permitted actuator voltages. Herein the integration of **control** into a home-made finite element code developed in MATLAB is presented [16]. The numerical solution of the H **feedback** controller has been done by using a nonconvex, non differentiable optimization approach with the usage of HIFOO software within MATLAB [14,17]. For the numerical results wind type loads are taken into account. The effectiveness of the technique in the model- ling of the standard uncertainties is also presented.

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Our objective is to solve the problem of directly controlling the unsteady flow separation using real-time velocity measurements, which are available in realistic applications, see, for instance, [15]. We propose this flow separation problem as a practical application of the new theoretical results in [25]. In particular, the aim of this paper is to show how, despite the high complexity of the system, a very sim- ple **robust** output regulator is sufficient to effectively suppress the flow separation along an aerofoil, using a single DBD plasma actuator. First, a novel **control**- oriented ROM of the flow/actuator dynamics, whose state variables have a clear and consistent physical meaning, is proposed. The method combines DMD, as an alternative to linearisation, and balanced POD, as a way to select the most observ- able and controllable DMD modes. The high-order DMD model is projected using the balanced POD modes, thus yielding a balanced, stable, linear ROM. Further- more, on the basis of the so-obtained model, we extend the recent results in [25] to a wider class of **control** systems and propose their application to this specific problem, which is of practical interest. Accurate finite element simulations of the full-order nonlinear equations are performed in order to test the **control** effectiveness and validate the **modelling** assumptions: they illustrate the **robust** performance, with respect to both parameter variations (i.e., geometry of the domain and Reynolds number) and model uncertainties, of the proposed **control** algorithm.

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We explore the visualization of violent wave dynamics and erosion by waves and jets in laser-cut reliefs, laser engravings, and three-dimensional printing. For this purpose we built table-top experiments to cast breaking waves, and also explored the creation of extreme or rogue waves in larger wave channels. Surprisingly, there are nano-scale analogues of these wave patterns in surface engineering with ion beams instead of water waves. Insights in applied mathematics and **fluid** dynamics, materials, fabrication and aesthetics informed our explorations. The resulting patterns give us not only new ways to communicate to specialist and general audiences about mathematics and **fluid** dynamics on different scales, they also provide new, abstract imagery which can be used in architectural and design applications.

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Despite the potential of modern **control** techniques with different structures, power system utilities still prefer the conventional lead-lag controller design. The gain settings of these stabilizers are determined based on the linea- rized model of the power system around a nominal oper- ating point. Since power system are highly non-linear and the operating conditions can vary over a wide range, conventional power systems performance is degraded when the operating point changes from one to another because of fixed parameters of the stabilizers. Also con- ventional techniques are time consuming as they are iter- ative and require complex computation procedures and show convergence.

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The present study describes the effect of viscous dissipation and joule heating on magneto hydrodynamic stag- nation-point flow towards permeable stretching surface. To analyze the new analytic solution of this problem, we apply homotopy analysis method. By this powerful and newly developed technique, the convergence series solutions are obtained. To validate the analytic solutions of velocity distribution and temperature distribution using HAM method, we have computed the convergence regions. The HAM solutions have an excellent agree- ment. A novel result of this problem is that the temperature increases with the increase in Hartmann Number H at a certain distance from the stretching surface in the presence of suction parameter. Heat **flows** from the sur- face to the **fluid** at near stagnation point on the surface and on the hand far from the stagnation-point, heat **flows** from the **fluid** to surface due to combining effect of ohmic dissipation and strain energy inside the boundary layer.

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6390 A nonlinear **feedback** controller for the (HR) response during treadmill exercise is introduced, where **robust** techniques and tracking **control** are applied for nonlinear model [12]. This **control** design does not rely on linear approximation. The Lyapunov-type stability arguments are applied to design the continuous, model-based, nonlinear **feedback** laws for treadmill speed to ensure that an ideal (HR) profile is tracked in an exponentially fast manner.

A recent passive mechanical controller concept, which has been success- fully utilised in vehicle suspension, Sharma and Limebeer (2012), and mo- torcycle steering, Evangelou et al. (2007), might offer a ‘best-of-both-worlds’ solution. This technique does not require a power supply, but offers the benefits of actively controlled flaps. The controller is a mechanical network consisting of springs, dampers and inerters that offers good performance in- cluding such things as fast response, robustness and so on. Our recent work Limebeer et al. (2011) has shown that controlled conventional flaps, hinged at the trailing edge of an aerofoil, or thin bridge deck section, and driven by a mechanical controller, can substantially reduce buffet loads induced by in- cident unsteady flow and can also be used to raise the critical flutter speed. Because the direction of the incident wind may be uncertain, controllable flaps may have to be fitted on both deck edges. This arrangement means that leading- and trailing-edge flaps will be available for **control** whatever the wind direction. Mechanical **control** systems, which are insensitive to wind direction, are proposed in Zhao et al. (2015). Controllable leading edge flaps, which are unusual in aircraft, are shown to be capable of making a sig- nificant contribution to buffet load reduction and to raising the flutter speed. This flap arrangement may, however, produce adverse effects. A conventional trailing edge flap, in order to be efficient, will usually have a ‘sharp’ trailing edge. When the wind direction is reversed such a flap, now on the upwind side, presents a sharp leading edge to the wind that can produce early-onset flow separation.

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There are two main approaches for accommodating all of these issues in network- based **control** system design. One way is to design the **control** system without regard to the packet delay and loss but design a communication protocol that minimizes the likelihood of these events. For example, various congestion **control** and avoidance algorithms have been proposed [13, 14] to gain better performance when the network traffic is above the limit that the networks can handle. The other approach is to treat the network protocol and traffic as given conditions and design **control** strategies that explicitly take the above mentioned issues into account. For example, to handle time delay, one might formulate **control** strategies based on the study of delay-differential equations [15]. In this thesis, an adaptive fuzzy logic modulation scheme is proposed to compensate for the network-induced time delay in the network-based PI **control** systems. This scheme is based on modulating the **control** signal provided by the PI controller and applied to the remote controller via the network.

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And then, Ronnenberg [80] and Escudier [81] took some experiments to observe the flow produced in cylindrical container by a rotating end wall, in which they found the formation of a concentrated vortex core along the center axis. Based on their research work, many experimental and numerical studies have been carried out later. Spohn et al. [82] experimentally studied the vortex breakdown bubbles which appeared in steady-state flow in a closed cylindrical container with the rotating bottom. Details of the flow were visualized by means of the electrolytic precipitation technique, whereas a particle tracking technique was used to characterize the whole flow field. Later, Sotiropoulos et al. [83] carried out the first experiment to verify their early numerical findings which show the existence of chaotic behavior in the Lagrangian transport with the vortex-breakdown bubbles for **flows** are steady. In the computations, numerous cylindrical problems were investigated. Lopez [77,84,85] published three papers to investigate the axisymmetric vortex breakdown from 1990 to 1992. Sotiropoulos et al. [86,87] studied three-dimensional structure of confined swirling **flows** with vortex breakdown in 2001. Apart from the one side rotation, people find two sides rotation can give new insight to the problems and give some new ideas on how to **control** the vortex. The influence of co- and counter-rotation of the other end wall of the cylinder on vortex breakdown was studied experimentally by Bar-Yoseph et al. [88] Gautier et al. [89] and Fujimura et al. [90]. In computations, Valentine and Jahnke [91] and Lopez [92] studied the case of co-rotating end walls with the same angular velocity for steady and unsteady swirl flow.

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Abstract. We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase **flows** in a porous region, and also a class of compressible three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling two- phase **flows** with mass transfer.

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In this paper, we are interested in investigating the problem of **robust** output **feedback** **control** of positive switched systems with time-varying delays. Firstly, the definition of exponential stability is given. Secondly, by using the ADT approach and copositive Lyapunov-Krasovskii functional method, a static output **feedback** law is designed and sufficient conditions are obtained to guarantee that the closed-loop system is exponentially stable, such conditions can be easily solved by linear programming.

A finite horizon predictive **feedback** scheme was proposed with an end constraint on the **control** horizon. In this formulation a direct **feedback** action is included in the optimization that allows the uncertainties and disturbances to be considered in a simple way. The resulting **control** law is time varying and achieves the **robust** performance of the system although constraints are present. Furthermore, it has been shown that the addition of the end constraint to the optimal **control** problem introduces strong stability guarantees, even if there are uncertainties in the system. The results obtained by simulating a linear model and a continuous stirred tank reactor, with important non-linearities, show the effectiveness of the proposed controller. The robustness and performance achieved in the presence of constraints was demonstrated.

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