## Abstract

Novel pressure gain combustion concepts invoke periodic flow disturbances in a gas turbine's last compressor stator row. This contribution presents studies of mitigation efforts on the effects of these periodic disturbances on an annular compressor stator rig. The passages were equipped with pneumatic active flow control (AFC) influencing the stator blade's suction side, and a rotating throttling disk downstream of the passages inducing periodic disturbances. For steady blowing, it is shown that with increasing actuation amplitudes $cμ$, the extension of a hub corner vortex deteriorating the suction side flow can be reduced, resulting in an increased static pressure rise coefficient Cp of a passage. The effects of the induced periodic disturbances could not be addressed intrinsically, by using steady blowing actuation, Considering a corrected total pressure loss coefficient $ζ*$, which includes the actuation effort, the stator row's efficiency decreases with higher $cμ$ due to the increasing costs of the actuation mass flow. Therefore, a closed-loop approach is presented to address the effects of the disturbances more specifically, thus lowering the actuation cost, i.e., mass flow. For this, a repetitive model predictive control (RMPC) was applied, taking advantage of the periodic nature of the induced disturbances. The presented RMPC formulation is restricted to a binary control domain to account for the used solenoid valves' switching character. An efficient implementation of the optimization within the RMPC is presented, which ensures real-time capability. As a result, Cp increases in a similar magnitude but with a lower actuation mass flow of up to 66%, resulting in a much lower $ζ*$ for similar values of $cμ$.

## Introduction

Ever since the first commercial flight, airlines and aircraft manufacturers have been encouraged to decrease the operating costs of airplanes. Rising fossil fuel prices in the past decades, as well as growing air pollution, intensify the requirement for clean and economic airliners. The fuel consumption of an aircraft strongly correlates with the aerodynamic design and the propulsion efficiency. Commercial aircraft usually operate under large subsonic Mach numbers in high altitudes, where the jet engine is the preferred propulsion concept. In modern aero engines, an axial compressor comprises up to 15 stages and takes up approximately 40% to 50% of the length of the engine [1]. The flow field in such compressors is dominated by highly complex secondary flow patterns that cause losses and limit the operating range of the component [2].

A promising way to increase the thermal efficiency in aero engines and industrial gas turbines is the transition to pressure gain combustion. Instead of constant pressure combustion, the fuel is burnt under nearly constant-volume conditions, leading to a total pressure increase. Two combustion concepts exploiting this advantage are pulsed detonation combustion and shockless explosion combustion (see Ref. [3]).

However, the additional discontinuities occurring with these nonstandard combustion concepts have to be damped to make the combustion concepts viable and controllable in a turbomachine. Without any measures, a high-pressure compressor located right upstream of the pressure gain combustion section would be periodically exposed to flow fluctuations at its outlet. This will result in a higher stalling risk and an efficiency drop, increasing the pressure loss leading to a worse operating point.

Applying active flow control (AFC) measures [1] is a promising solution to enable turbomachinery to operate with a pressure gain combustion system in the first place, thus increasing the overall efficiency of the machine. Actively controlling the secondary flow features, such as the passage vortex, may lead to higher static pressure recovery of a stator vane and possibly reduced flow losses [4]. An overview of actuator types used for AFC is given in Ref. [5]. An interesting approach of structural actuation using adaptive blade geometry has been recently demonstrated by Werder et al. [6]. By applying AFC through boundary layer suction at the end-wall region of a linear compressor stator cascade, an increased flow turning was reached [7]. As used by Nerger et al. [8], steady blowing actuators could be beneficial in terms of static pressure recovery. A more favorable AFC effect could be achieved when pulsed jet actuation is used [9]. The fluid pulses injected by the actuator create large eddy structures close to the wall [10]. The enhanced mixing reenergizes the boundary layer and delays flow separation [11]. An integration study on AFC applied to the suction side of compressor stators in a low-speed axial compressor was conducted by Culley et al. [12]. The authors concluded that the endwall losses typically found in hub-shrouded stators were not significantly reduced by the stator surface injection. Therefore, hub-side AFC seems to be a better solution for influencing the flow losses occurring there.

However, the literature provides contradictory statements for the efficiency of an AFC-enabled compressor stage. Pulsed blowing applied to a highly loaded linear compressor stator cascade was investigated by Hecklau et al. [13] and Staats et al. [1416]. The pulsed jet actuators were placed in the end walls and on the suction side of the compressor blades. Increased compressor stator efficiency, by 5%–6%, was feasible based on experimental and numerical investigations. A numerical comparison of three actuator types (steady blowing, plasma actuator, and synthetic jet actuator) applied to a compressor stator flow was conducted by Traficante et al. [17]. Their findings reveal that the plasma actuators' high power consumption led to a lower efficiency compared to the other two actuation concepts. However, a comparison of such studies is limited as not all authors utilize the same definition of efficiency.

In contrast, recent studies done by Steinberg and King [18] on a linear stator cascade have shown that by using the corrected total pressure loss coefficient as a figure of merit, an overall efficiency drop with increasing actuation amplitudes for blowing AFC is obtained. This is in agreement with Ref. [8]. In Ref. [18], the authors recommend that the AFC should only be used to maintain a necessary operating point, as no independent contribution to an efficiency improvement could be demonstrated. Apparently, the AFC's efficiency is strongly dependent on the test rig, its flow conditions, and the chosen figure of merit and should therefore be investigated specifically.

In this paper, results from an annular compressor stator cascade with AFC are discussed. The test rig is equipped with highly loaded, controlled diffusion airfoils and an end-wall actuator at the hub side of each passage. In the experiments, blowing actuation out of rectangular slots was performed using steady blowing and a closed-looped controller. For the latter, an optimization-based repetitive model predictive controller (RMPC) was implemented, which was first formulated by Lee et al. [1921] and successfully applied for the AFC of a linear stator cascade by Steinberg et al. [2224]. This approach combines the advantages of learning control, which can increase performance for periodic processes [2529], with those of classical optimization-based model predictive control (MPC). In recent years, the latter has been increasingly used in industrial applications [30,31], due to its versatile applicability and numerous further advantages. Especially of note here is the optimal control provided with regard to a definable cost function and the possibility to consider model and process constraints, such as control variable limitations. Also, limitations to binary control variables, as present in our case due to the use of solenoid valves, which are smaller, cheaper, and more suitable for a real application than proportional valves, can be considered in MPC readily. This is done by defining the optimization problem as a binary quadratic program (BQP) and, for example, by using a classical branch and bound (B&B) algorithm combined with an underlying quadratic program (QP) solver to obtain the optimal solution of the BQP in real-time [32,33].

The overall goal of this contribution is twofold. On the one hand, it aims to adapt the real-valued RMPC formulation presented in Refs. [2224] for effective damping of periodic disturbances in a stator cascade to the present binary control task invoked by the usage of fast solenoid valves. On the other hand, it aims to implement this closed-loop control concept in a more complex and realistic test rig, using an annular compressor stator cascade, and finally investigate the resulting flow features.

The paper is structured as follows. Section 2 presents the experimental setup, instrumentation, and methods for data acquisition used for this contribution. Section 3 carries out the model identification and introduces the RMPC. Section 4 discusses the implications of the model used in the AFC problem and, finally, Sec. 5 presents the concluding remarks.

## Experimental Setup and Procedure

All experimental investigations were carried out using a low-speed, open circuit wind tunnel at the Chair for Aero Engines of the TU Berlin. The wind tunnel is used for experimental investigations of the Collaborative Research Center (CRC) 1029 funded by the Deutsche Forschungsgemeinschaft, cf. Ref. [34]. Its main objective is the investigation of the influence of pressure gain combustion on compressor components. The annular design was chosen to create enhanced three-dimensional flow characteristics and thus enabled analyses of impacts similar to pressure gain combustion effects at a high spatial resolution.

Figure 1 shows a schematic depiction of the wind tunnel. A round inlet nozzle with an area contraction of $11:1$ was used to create smooth inflow conditions and low turbulence intensity, employing screens and honeycombs as flow straighteners. A variable inlet guide vane (VIGV) produced the swirl needed for the stator inlet conditions. A turning of 42 deg was achieved at midspan by using 19 turbine-shaped vanes. Moreover, these VIGVs provided the option to change the incidence to the stator by $±5 deg$.

Fig. 1
Fig. 1
Close modal

The annular measurement section consisted of a highly loaded compressor stator cascade. All 15 stator blades were installed hub shrouded so that they had no gap between the housing or the hub. That said, no tip clearance flow occurred during the experiments. The profiles are controlled diffusion airfoils, scaled down from the two-dimensional cascade presented by Kiesner and King [35], which was also part of the CRC 1029. These blades had been designed by Rolls Royce Deutschland to produce an axial outflow at the design point with a chord-based Reynolds number of $Re=6×105$. The axial mass flow through the wind tunnel at each time-step k was $m˙(k)≈9.3$ kg/s.

The axial distance between the VIGVs and the stator inlet was about three-chord lengths to ensure good mixing of the vane wakes, producing a stator inlet turbulence intensity of $Tu1=5%$. The degree of turbulence for the flow was measured utilizing hot wire anemometry at the leading edge (LE) of the stator profiles. Figure 2 depicts the geometric data and is complemented by Table 1. The wind tunnel features a rotating throttling disk equipped with two paddles $0.71·c$ downstream of the stator trailing edge (TE). The throttling disk generates a periodic disturbance to the stator, mimicking the effects of a pressure gain combustion. This simplified setup assumes that two pressure gain combustion tubes on opposite sides are permanently closed and that the firing process is in a circumferential sequence. As a consequence, each of the two paddles of the throttling disk blocks over 90% of the passage area at the maximum overlap. For all presented experiments, the throttling disk ran with a frequency of 3.7 Hz, which leads to a disturbance frequency of $fd=7.4$ Hz, due to the disk's double-sided paddle (Fig. 1). Similar to previous contributions [22,23], the corresponding Strouhal number was $Srd=fd·c/|v¯1|≈0.03$, with $|v¯1|$ being the magnitude of the stator's inlet velocity.

Fig. 2
Fig. 2
Close modal
Table 1

Geometric data of the annular test rig

NameParameterValue
Stator chord lengthc187.5 mm
Stator/passage heighth150 mm
SS coordinate lengthsmax198.42 mm
Stagger angleγ12 deg
Stator turning$Δα$42 deg
Stator inlet velocity$v¯1$$≈50.0$ m/s
Hub to tip ratio$rhub/rtip$0.5
Pitch to chord ratio midspant / c0.5
De Haller$|v¯2|/|v¯1|$0.65
AFC blowing angleΘ15 deg
AFC actuator positionsact25 mm
NameParameterValue
Stator chord lengthc187.5 mm
Stator/passage heighth150 mm
SS coordinate lengthsmax198.42 mm
Stagger angleγ12 deg
Stator turning$Δα$42 deg
Stator inlet velocity$v¯1$$≈50.0$ m/s
Hub to tip ratio$rhub/rtip$0.5
Pitch to chord ratio midspant / c0.5
De Haller$|v¯2|/|v¯1|$0.65
AFC blowing angleΘ15 deg
AFC actuator positionsact25 mm

### Instrumentation and Data Acquisition.

A Prandtl probe mounted in midspan position $1.0·c$ upstream of the stators leading edge was used to measure inflow conditions. Henceforth, these inflow measurements are denoted as upstream/inflow reference points. The flow at the inlet and wake of the stator passage was measured with a five-hole probe using differential pressure sensors [First Sensor: HDOM050] with a calibrated pressure range of –50 mbar to 50 mbar. The same pressure sensors were used for the Prandtl probe. The five-hole probe's head diameter was 3.18 mm (2.1% span), and the measured inlet plane was located $1.4·c$ upstream of the leading edge of the stator. The five-hole probe wake measurements were taken at $0.4·c$ downstream of the stators trailing edge (see Fig. 2). With the small head diameter of the five-hole probe, measurements could be performed in the range of 2%–98% span. To ensure that the mean values of the flow characteristics could be determined accurately enough, the measuring time at each point was 10 s, utilizing instrumentation amplifiers [DEWETRON DAQP-STF] with a subsequent 24-bit digitization [DEWETRON ORION-1624-200] featuring a delta-sigma ADC. To get a detailed pressure distribution and velocity profile, the five-hole probe was traversed to 400 points in a circumferential-based polar grid. The measurements were phase averaged at each position. The traversed grid had 20 equidistant radial lines. On each radial line, grid points were distributed equidistantly along the circumference. The sampling frequency was chosen to 1 kHz2 concerning the actuation valves' maximum operation frequency ($≈250$ Hz) to gain sufficient fidelity.

Using a traversable blade, the static pressure distribution was measured at 100 spanwise locations covering the whole passage in the z-direction. The traversable blade features 30 pressure taps at the suction side of the blade (Fig. 2). Thus, investigations of the static pressure distribution on the whole blade's suction surface in one passage of the cascade were possible. The same type of pressure sensors and instrument amplifiers were used for making the traversable blade measurements as for the five-hole probe.

An additionally manufactured control blade was used for closed-loop experiments. The blade featured 10 suction side embedded, highly sensitive miniature pressure transducers [KULITE XCS-062]. Their corresponding positions are marked in red in Fig. 2. The span-wise position of these sensors was chosen as $z/h=0.05$. For the closed-loop experiments, the signal processing and the RMPC algorithm were implemented on a real-time system featuring a multifunction I/O device [NATIONAL INSTRUMENTS PCIe 6259]. Here, the code generation was realized with matlab/simulink 2017a. For all closed-loop experiments, the sampling rate was set to $ns·fd=111$ Hz, with ns = 15 as the number of stator blades.

This choice implies that the blockage produced by the throttling disk moves one passage per sampling instant. As a result, the control signal applied to one passage could be used to actuate the next passage for the next time instant. Using this setup prevented having to individually measure the variables to be controlled in all passages, while still actuating all passages in a kind of closed-loop control. This assumes, however, that all passages behave exactly the same. If this were not true or was a too rough approximation, more sensors would be needed.

A pneumatic actuation system located on the hub side wall was used for conducting investigations during the experiments. The actuator system consisted of a rectangular outlet orifice, measuring $hact/c=0.053$ (slot height) and $dact/c=0.002$ (slot width) in relation to the chord length c (see Fig. 2). The outlet orifices had a blowing angle of $Θ=15 deg$ relative to the passage end-wall and were oriented perpendicular to the blade's surface.

With a mass flowmeter [BRONKHORST F-113AC-1MO-ABD-44-V], the inlet mass flow $m˙act(k)$ of the actuation tank at time-step k (Fig. 2) was measured. The tank's stagnation pressure $pact(k)$ was monitored with another pressure transducer [FESTO SPTW-P6R-G14-VD-M12] and closed-loop controlled to hold its pressure at the desired level to achieve constant actuation amplitudes. This was done by controlling a proportional valve [FESTO MPYE-5-3/8-010-B] at the tank's inlet. Furthermore, different pneumatic AFC actuation concepts were realized by controlling the solenoid valves [FESTO MHE2-MS1H-3/2G-QS-4-K] installed upstream of each actuator (Fig. 2).

Since in the experimental setup the actuation mass flow $m˙act(k)$ could only be measured at the inlet of the actuation pressure tank (see Fig. 2), the mass flow of an actuation jet has to be calculated
$m˙jet(k) ={m˙act(k)nact(k)nact(k)>00nact(k)=0$
(1)
with $nact(k)$ as the number of actuated passages. Further, the amplitude of actuation is defined as the momentum coefficient $cμ(k)$. The latter describes the momentum of a passage's actuation by means of the product of an actuation jet's mass flow $m˙jet(k)$ and its velocity $ujet(k)$ related to the momentum of a passage flow in the annular test rig3
$cμ(k)=m˙jet(k)·ujet(k)Apsg·q1,ref(k)=m˙jet2(k)Apsg·q1,ref(k)·ρ·dact·hact$
(2)

Here, Apsg represents the cross-sectional area of one passage, $q1,ref(k)$ the dynamic pressure measured upstream of the passage at the reference location, and ρ the density of the air. Note that $m˙jet(k)$ and so $cμ(k)$ are measures for the actuation effort of an actuated passage and not of a fixed one.

Since our definition of $cμ(k)$ (Eq. (2)) does not allow for an evaluation of the overall mass flow effort of a specific passage for pulsed actuation, another ratio describing the actuation mass flow of a specific passage $m˙act,psg(k)$ in relation to the passage's inlet mass flow $m˙psg(k)=m˙(k)/ns$ is defined
$μ(k)=m˙act,psg(k)m˙psg(k)$
(3)
Hereby, $m˙act,psg(k)$ is approximated with the general jet mass flow $m˙jet(k)$, weighted with the binary control input $uk$ for the valve at each time-step k of the considered passage
$m˙act,psg(k)=m˙jet(k)·uk$
(4)

It has to be emphasized that in this contribution, according to standard control engineering nomenclature, the binary, the dimensionless control input is named u, i.e., as well uk, as velocity, cf., ujet. For the oil-flow visualization, a mixture of light diesel oil and ultraviolet fluorescent dye powder was used. A removable blade plug-in assembly with two passages was used to ensure fast insertion and extraction of the blade used for the oil-flow visualization. The prepared annular compressor cascade was then exposed to the flow at the set operating point until the color mixture on the surface was completely dried. Finally, the plug-in passage assembly's individual blade was removed and digital images of the oil-flow patterns were taken under ultraviolet illumination. As the blade suction side surface is curved, there presents a challenge in relating the curved surface pattern to a surface coordinate system (s/smax). In order to rectify the pictures and eliminate camera parallax when photographing the surface pattern, the photographs were taken using a fixed photomount [37]. Using a previously taken calibration picture of the blades' suction side curvature, the distortion could be compensated for. Subsequently, the oil-flow images were evaluated and digitally vectorized to obtain the time-averaged surface wall shear stress directions.

## Closed-Loop Methods

In this section, a surrogate control variable yk is introduced to summarize the impact of the induced disturbances on the control blade's space-dependent static pressure coefficient $cp(z,s,k)$ at each time-step k into a scalar variable. The disturbances were produced by the throttling disk downstream of the stator row, mimicking the effect of closing combustion tubes. Furthermore, for model prediction within the control algorithm, a simple single-input single-output (SISO) model was identified, describing the influence of the actuation uk onto yk for the control blade. Finally, the formulation of the RMPC is derived and, additionally, an optimal estimator for state and disturbance reconstruction is presented.

### Model Identification.

For the definition of a control variable, the space-dependent static pressure coefficient
$cp(z,s,k)=p(z,s,k)−p1,ref(k)q1,ref(k)$
(5)

was used. Here, $p(z,s,k)$ is the static pressure at location z, s on the blade's suction side, $p1,ref(k)$ the measured static pressure of a reference location upstream of the blade's leading edge, and $q1,ref(k)$ the dynamic pressure of this reference location at time-step k.

Since every passage only had one actuator, a scalar surrogate control variable was needed to combine the space-dependent information of all pressure sensors of the control blade into a single, scalar signal. To this end, a principal component analysis [38] for all chord-wise positions was performed, for which all $cp(z,s,k)$ were stacked into a vector $c¯p(k)$. The analysis aimed to find the FIRST PRINCIPLE COMPONENT (PC) $p¯1∈ℝ$10 of the disturbance impact on the control blade. This vector can be interpreted as the direction of the highest disturbance's variance on $cp(z,s,k)$. The scalar product of $p¯1$ with the difference of $c¯p(k)$ and the time-averaged static pressure coefficient $c¯¯p$ can be understood as an amplitude of the disturbance influence. As the goal was to mitigate this influence, it was defined as the surrogate control variable
$yk=p¯1T·(c¯p(k)−c¯¯p)=p¯1T·c¯p(k)−ys$
(6)

with ys describing a kind of an operating point. As a consequence, when yk = 0, the variance of the disturbance's impact along $p¯1$ is zero. This situation would be the most desirable effect of control. Therefore, in RMPC, a desired reference rk = 0 will be formulated below.

In Eq. (6), $p¯1$ can be interpreted as a weighting vector for the influence of the sensors on yk. In Fig. 3(a), $p¯1$ is plotted over the sensor positions of the control blade. As the entries of $p¯1$ decrease over the suction side coordinate s, the first sensors have a much stronger influence on yk than the last ones. In Fig. 3(b), yk is shown over the cycle of one disturbance period.

Fig. 3
Fig. 3
Close modal
For the purpose of this study, we propose a linear, time-invariant discrete-time SISO state-space model of order nx
$x¯k+1=Ax¯k+b¯ (uk+vk)$
(7a)
$yk=c¯T x¯k+dk$
(7b)

Here4, $x¯k∈ℝnx$ is the state, $uk∈B$ the input from a binary domain $B, vk∈B$ a separate feed-forward control input, $yk∈ℝ$ the output signal, and $dk∈ℝ$ the output disturbance at time-step k.

Therefore, the following applies to the model:
$A∈ℝnx×nx, b¯∈ℝnx, and c¯∈ℝnx$

For simplicity, and as it is often done in learning control, the disturbance of the throttling disk is only considered as an output disturbance in Eq. (7b). For linear systems, this is a viable approach.

To identify the entries in $A, b¯$, and $c¯T$ of this black-box model, system excitations were performed for different actuation amplitudes and different blowing distributions for the actuated passages. Pseudo-random binary signals were used as valve control signals to identify $A, b¯$, and $c¯T$ for the physically possible frequency range of the solenoid valves. As according to Eq. (7) the influence of uk is only seen in the next time-step $yk+1$, the system is said to have a relative degree of one.

### Repetitive Model Predictive Control.

With an MPC, an optimization problem is solved to find an optimal trajectory of future control inputs so that a cost function is minimized. This cost function comprises future control errors, the spent control effort and penalizes control moves that are too large. For that, a prediction of the future evolution of the process has to be performed using the mathematical model of the process. As the model is uncertain and as future unknown disturbances cannot be accounted for, only the first entry of the found trajectory of future control inputs is applied to the process. At the next sampling instant, new measurement information is used to update the model state and the optimization starts again. This is repeated at every sampling instant to decrease the effect of uncertainties.

For cyclic processes as considered here, the performance can even be improved as the disturbance repeats itself. This leads to the RMPC, which will be outlined below. For the RMPC, an output error ek is defined as the difference between an output reference rk and the system's output yk
$ek=rk−yk$
(8)
Likewise, a state reference variable $ρ¯k$ and a corresponding state error $ε¯k$ is defined
$ε¯k=ρ¯k−x¯k$
(9)
In what follows, we assume that a separately designed feed forward vk achieves the state and output reference in the undisturbed case. Therefore,
$ρ¯k+1=Aρ¯k+b¯ v¯k$
(10a)
$rk=c¯T ρ¯k$
(10b)
applies with the same $A, b¯$, and $c¯T$ as in Eq. (7). Subtracting Eq. (7) from Eq. (10) and inserting Eqs. (8) and (9), we get the final state-space model, which describes the deviation from the reference
$ε¯k+1=Aε¯k−b¯ uk, ε¯0=ρ¯0−x¯0$
(11a)
$ek=c¯T ε¯k−dk$
(11b)
Since in this contribution the output reference is always zero, for the control task it follows that ideally, yk and, hence, ek should be zero as well. Nevertheless, note that the control problem is defined as a reference tracking formulation to increase the general validity for other control problems.
Propagating Eq. (11) forward in time to predict the evolution for a defined horizon, whereby the prediction and control horizon both have the same length n of the disturbance period, the equation for the predicted trajectory of the output error at time-step k reads
$e¯k+1|k =Fε¯k−d¯k+1|k−G u¯k|k$
(12a)
with
$F=[c¯TAc¯TA2⋮c¯TAn], G=[c¯Tb¯0…0c¯TAb¯c¯Tb¯…0⋮⋮⋱c¯TAn−1b¯ c¯TAn−2b¯ … c¯Tb¯]$
(12b)
The stacked, predicted output error, control input, and disturbance are
$e¯k+1|k=[ek+1|k, ek+2|k, …ek+n|k]T$
(12c)
$u¯k|k =[uk, uk+1|k, … uk+n−1|k]T$
(12d)
$d¯k+1|k =[dk+1|k, dk+2|k, …dk+n|k ]T$
(12e)

Hereby, the notation $(·)k+i|k$ describes a prediction at time-step k for the time-step k + i, using all information available at k. Likewise, $(·)k+i|k$ with a bold variable refers to all future values starting from k + 1 and based on information available at the actual time instant k. As the system has a relative degree of one, at time-step k, the first influenceable output is $ek+1|k$. For this reason, the first entry of the predicted trajectories Eqs. (12c) and (12e) is $ek+1|k$ or $dk+1|k$, while the first entry of Eq. (12d) is uk. In the literature, a stacked system notation as in Eq. (12) is often referred to as a “lifted system.”

Due to the periodic nature of the disturbance produced by the throttling disk, we now assume that the actually predicted disturbance trajectory equals the disturbance trajectory of the last period at kn
$d¯k+1|k=d¯k−n$
(13)
Now, taking the output error trajectory of the last period
$e¯k−n=Fε¯k−n−d¯k−n−Gu¯k−n$
(14)
subtracting Eq. (14) from Eq. (12a), and considering Eq. (13), we obtain the final formulation for the error prediction, in which the disturbance no longer appears5
$e¯k+1|k=e¯k−n+FΔε¯k+Gu¯k−n︸s¯k+1|k−Gu¯k|k$
(15)
with $Δε¯k=ε¯k−ε¯k−n$. The RMPC's cost function can now be composed of a part considering the running or so-called stage costs $Il(u¯k|k)$ and another part considering end costs $Ie(u¯k|k)$ needed for stability reasons
$I(u¯k|k)=Il(u¯k|k)+Ie(u¯k|k)$
(16)
Here, $Il(u¯k|k)$ is defined as follows:
$Il(u¯k|k)=e¯k+1|kTWe e¯k+1|k+u¯k|kTWu u¯k|k+Δu¯k|kTWΔ Δu¯k|k$
(17)
where the positive and positive semidefinite diagonal matrices $We ∈ℝ>0n×n, Wu ∈ℝ≥0n×n$ weight the predicted error and the control input effort, respectively. Additionally, the diagonal matrix $WΔ ∈ℝ>0n×n$ accounts for the change of the control input trajectory relative to the last cycle $Δu¯k|k=u¯k|k−u¯k−n$ and can be used to tune the convergence rate of the RMPC. Finally, the end costs can be set to
$Ie(u¯k|k)=12ek+n|kT We ek+n|k$
(18)
with a scalar weight $We ∈ℝ≥0$. According to the last row in the vector-valued Eq. (15)
$ek+n|k=ek+FˇΔε¯k+Gˇu¯k−n︸t¯k+n|k−Gˇu¯k|k$
(19a)
follows6 with $(·)ˇ$ as the last row of a matrix:
$Fˇ=c¯TAn , Gˇ=[c¯TAn−1b¯ c¯TAn−2b¯ … c¯Tb¯]$
(19b)

As our identified system is stable, we can ensure stability for the closed-loop system with an adequate set of $We, Wu, WΔ$, and $We$.

The total cost now can be written as a quadratic function with the Hessian matrix $H$. Thus, the optimization problem to calculate the optimal control trajectory $u¯k|k*$ can be summarized as
$I(u¯k|k)=u¯k|kTH u¯k|k+f¯kTu¯k|k+αk$
(20)
$u¯k|k*=argminu¯k|k{I(u¯k|k)}$
(21)
subject  to
$u¯k|k ∈Bn$
(22)
with
$H=GTWe TG+Wu+WΔ+12GˇTWe Gˇ$
(23)
$f¯kT=−2(s¯k+1|kTWe G+u¯k−nTWΔ+12t¯k+n|kTWe Gˇ)$
(24)
$αk=s¯k+1|kTWe s¯k+1|k+u¯k−nTWΔ u¯k−n+12t¯k+n|kTWe t¯k+n|k$
(25)
and
$s¯k+1|k=e¯k−n+FΔε¯k+Gu¯k−n$
(26)
$t¯k+n|k=ek+FˇΔε¯k+Gˇu¯k−n$
(27)

As solenoid valves were used in the test rig, the design variables of this optimization problem are binary (see Eq. (22)).

After solving the BQP, the first input of the optimal control trajectory $u¯k|k*(1)$ is used as control input for the control blade's passage valve and, at the next time-step k +1, the optimization problem has to be solved again, and so on. Also, at k +1, the former control input $u¯k|k*(1)$ is used for the next passage following the rotational influence of the throttling disk and its corresponding frequency fd. As a result, the RMPC is set to control the entire annular stator cascade.

Finally, the setup for the learning weight $WΔ$, with equal diagonal entries of $wΔ$ is briefly discussed here. As suggested by Schäpel et al. [39], $wΔ$ is implemented as an adaptive parameter to account for peculiarities of the binary domain. In contrast to the case of real-valued control signals, for a fixed $wΔ≫0$, the binary solution for $u¯k|k*$ is likely to get stuck in a suboptimal solution. To overcome this, at every time-step, it is checked whether the control signal trajectory has changed compared to the previous cycle. If this is not the case, $wΔ$ is reduced by 5%, to allow a possible higher change for the new optimal control trajectory $u¯k|k$. This adaption of the weight is done every time the control trajectory has not changed anymore until $wΔ$ falls below a small threshold of 0.01, where the RMPC is likely to be converged. After that, $wΔ$ is set back to the initial $wΔ,0=100$. The described heuristic allows for smooth learning behavior without much influence of noncyclic disturbances, such as measurement noise. The BQP is solved in real-time by the use of a classical depth-first search Branch and Bound algorithm [40] combined with a fast active-set QP-solver (see Ref. [41]).

### State Estimation.

Since one cannot generally assume that the state vector is directly measurable, a state estimator has to be used to calculate the estimated state $x̂¯k$ and, by this, the estimated change of the state error $Δε̂¯k$. The latter is necessary to solve the BQP. For that, a Kalman filter, an optimal state estimator, is used in this study.

Here, a problem arises due to the disturbance dk. For the RMPC, the disturbance could be eliminated in Eq. (15). For the state estimation, this is not possible. Therefore, to improve the quality of the estimation, we introduce a disturbance model. Considering the similarity of the induced disturbances with a harmonic oscillation (compare Fig. 3(b)), we assume an oscillating second-order system with zero damping and state vector $z¯k∈ℝ2$. Furthermore, a constant bias δk is added to the output of the disturbance model for offset estimation of the modeled disturbance $d´k$. So the disturbance model reads as
$(z¯k+1δk+1) =[A´OO1](z¯kδk)$
(28a)
$d´k =[101](z¯kδk)$
(28b)
Here O is a zero matrix/vector of the corresponding dimension and $A´$ is chosen so that the eigenfrequency of the second-order system is equal to fd. Finally, with the state-space model of Eq. (7), we create an augmented model and assume the latter to be stochastically disturbed through a zero-mean system white noise $w¯k$ and a zero-mean white measurement noise ωk
$(x¯k+1z¯k+1δk+1) =[AOOOA´OOO1](x¯kz¯kδk)+[b¯000](uk+vk)+w¯k$
(29a)
$yk =[c¯T101](x¯kz¯kδk)+ωk$
(29b)

The final augmented state-space model (Eq. (29)) is used in a Kalman filter to calculate the augmented state vector. The equations and settings for a Kalman filter are well documented in the literature (e.g., see Ref. [42]), so they will not be discussed here. The estimate of $x̂¯k$ is needed to calculate the state error estimation $ε̂¯k$ for the RMPC formulation (see Eq. (9)). Additionally, the output estimation $ŷk$ is used to calculate the estimated output error $êk$, which is also used in the RMPC, instead of the measured ek, to lower the effect of measurement noise.

## Results

In the experiments, either the control blade or the traversable blade could be used. To still allow for a more detailed comparison of open- and closed-loop control, we proceeded as follows. First, a closed-loop controller was designed and applied with the control blade. In this section, the outcome of these closed-loop experiments is presented first. The results were then used to determine an optimal RMPC trajectory for open-loop experiments in order to compare RMPC actuation with steady blowing with regard to overall passage and wake characteristics. The results of the open-loop passage experiments include time-averaged $c¯p$-distributions for the different actuation cases acquired with the traversable blade. In addition, oil-flow visualizations were created to indicate the mean flow conditions on the suction side of a blade. Finally, the effect of the actuation on the passage's wake was determined and typical characteristics for a stator passage, like static pressure rise and total pressure loss coefficients, were processed and are compared in this section.

### Closed-Loop Experiments.

Figure 4 shows the results of closed-loop experiments performed with the control blade. In Figs. 4(a) and 4(c), the converged RMPC trajectory $u¯∞$ and averaged output trajectory $y¯¯∞$ over the time steps k of a disturbance cycle j are presented. In comparison to that, the cases of steady actuation (I) and no actuation (O) are added. As discussed above, $y∞=0$ means that the influence of the disturbance generator on the first principle component of the pressure readings on the suction side disappears. However, without control (O), $y∞$ clearly deviates from zero indicating the disturbance's influence. In contrast, the RMPC, which opened the actuation valves of the control blade in the time frame k = {4, 5,…,8}, led to smaller absolute values of $y∞$ for almost the complete cycle. For open-loop control with steady actuation (I), the absolute value of $y∞$ could only be decreased for very few time instants while in other phases, the results were even worse than without actuation. In Figs. 4(b) and 4(d), the convergence behavior of the RMPC with respect to the duty cycle DCj of the actuation and the relative two-norm of the output error trajectory $e¯j$ of a period j
$||e¯̃j||2=||e¯jn||2$
(30)

can be seen. The RMPC was activated at j =0, and after ten cycles, it had converged and lowered the error from $||e¯̃j||2≈0.68$ to $||e¯̃j||2≈0.51$, an improvement of about 25%. For that, DCj was at 33%, so the solenoid valve of the control blade was opened for 33% of the time (compare Fig. 4(a)). In contrast, the result for steady actuation shows that with $||e¯̃j||2≈0.75$ the performance was worse than in the case with no actuation at all.7

Fig. 4
Fig. 4
Close modal

The closed-loop experiment presented in Fig. 4 was repeated four times with very similar outcomes, and the converged RMPC trajectories were averaged and used for the following open-loop passage and wake experiments.

### Open-Loop Passage Experiments.

In this section, the results of open-loop experiments showing the influence of the two different actuation concepts on the passage flow are presented. First, the traversable blade was used in experiments to obtain detailed static pressure information for the suction side. Second, oil-flow visualizations were created to get a deeper insight into the corresponding flow structures.

Figure 5(a) shows the time-average of the static pressure coefficient $c¯p(z,s)$ of the base flow. Especially in the lower part, it can be seen that the isolines, starting from $0.2·s/smax$, are curved heavily up to the trailing edge at $1.0·s/smax$. This was caused by a corner vortex growing over the suction side up to $0.5·z/h$ and thus blocking the passage flow. This deteriorating effect could be effectively reduced with steady blowing due to permanent momentum input into the main flow, as shown in Fig. 5(b) showing the averaged $c¯p(z,s)$-distribution for $cμ≈2.3%$. Here, the isolines are much less curved, and the $c¯p(z,s)$-values near the trailing edge are higher, indicating an increased static pressure. In comparison, the RMPC actuation had a lower effect due to nonpermanent momentum input. But still, it was slightly able to straighten the isolines of the averaged $c¯p(z,s)$-distribution, therefore lowering the influence of the corner vortex (Fig. 5(c)).

Fig. 5
Fig. 5
Close modal

#### Oil-Flow Visualization.

A surface oil-flow technique was used for time-averaged qualitative streakline visualization on the suction side of the stator vane under the different flow conditions considered for this paper. In Figs. 5(d)5(f), the results showing the time-averaged surface wall shear stress directions on the suction side are presented for the base flow, the steady blowing case, and the RMPC actuation. These results provide a comparative depiction of the changes in the time-averaged flow topology for the different flow control effects. The oil-flow visualization of the base flow (Fig. 5(d)) shows a large area of separated flow. In addition to an area of reverse flow that propagates from the trailing edge against the flow direction, two vortex focus points could be identified. Similar to the examinations of the traversable blade measurements, the separation line starts at a distance of about $0.2·s$/smax downstream of the leading edge and extends to nearly $0.5·z$/h in a span-wise direction at the trailing edge. The blockage resulting from the loss cores is clearly shown, causing a span-wise convergence of the stream tube as the flow passes from the blade's leading edge to the trailing edge. This observation is similar to those in our previous studies, presented in Ref. [43]. However, in these studies, there were other flow conditions since no periodic disturbances were introduced into the flow. Figure 5(e) shows that the secondary flows on the suction side of the blade were significantly affected when introducing steady blowing. The corner vortex was reduced in its spanwise extension, and the areas of reversed flow were diminished. The reverse flow propagation from the trailing edge moved in a span-wise direction toward the hub side. As a result, a lower blocking of the passage could be observed. The same observation holds true for the oil-flow visualization presented in Fig. 5(f) for RMPC actuation. However, the oil-flow visualization indicates a somewhat reduced effect on the stream tube convergence when the passage was actuated using the RMPC-enabled AFC. Here, the reverse flow propagation from the trailing edge has moved in a spanwise direction toward the casing again. Nonetheless, the effect of both AFC strategies on the suction side secondary flow features was still remarkably eminent.

### Open-Loop Wake Experiments.

Since the overall purpose of a stator vane is to convert dynamic pressure into static pressure, the static pressure rise coefficient $Cp(z,y,k)$ of a passage is a significant parameter to evaluate the operation of a stator vane. It is defined as the difference between the local static pressure downstream $p2(z,y,k)$ and the area-averaged static pressure upstream $p¯1(k)$ of the passage relative to the mass-averaged upstream local dynamic pressure $q¯1(k)$
$Cp(z,y,k)=p2(z,y,k)−p¯1(k)q¯1(k)$
(31)
Note that in this contribution, as suggested by Cumpsty and Hurlock [44], area-averaging is used for static pressure while for total and dynamic pressure mass-averaging is used. In comparison to $Cp(z,y,k)$, the total pressure loss coefficient $ζ(z,y,k)$ takes the difference of the mass-averaged total pressure upstream $g¯1(k)$ and downstream $g2(z,y,k)$ of the passage into account
$ζ(z,y,k)=g¯1(k)−g2(z,y,k)q¯1(k)$
(32)
This common characteristic is a measure for a passage's efficiency, but it does not consider the actuation effort. Hence, for stator vane experiments with flow actuation, a corrected total pressure loss coefficient was suggested by Nerger et al. [8]
$ζ*(z,y,k)=g¯1*(k)−g2(z,y,k)q¯1(k)$
(33)
with the corrected inlet total pressure of the passage
$g¯1*(k)=m˙psg(k)·g¯1(k)+m˙act,psg(k)·pact(k)m˙psg(k)+m˙act,psg(k)$
(34)

considering the measured passage's actuation mass flow $m˙act,psg(k)$ and the total pressure of the actuation, defined here as the pressure $pact(k)$ in the actuation stagnation tank (see Fig. 2).

In Figs. 5(g)5(i), the time-averaged magnitude of the velocity vector $|v¯2(z,y)|$ in the wake of one and a half passages (ymax) for base flow, steady actuation, and RMPC actuation is shown. In the case of the base flow (Fig. 5(g)), the impact of each corner vortex on the suction side (SS) near the hub is visible in terms of a large area of significantly lowered velocity. In contrast, on the blade's pressure side, no major velocity drops through the corner vortex can be seen.

In comparison, a relatively small impact of each corner vortex near the casing can be seen, which is characteristic of an annular stator rig with a radially expanding flow area. Similar to the traversable blade results of Fig. 5(a), with the usage of steady blowing, the impact of the corner vortex near the hub could be lowered accordingly, producing a much more homogeneous wake (see Fig. 5(h)). In contrast to the traversable blade investigations, it is visible that the vortex expansion along the y-axis was significantly reduced, resulting in a diminished passage blockage. In comparison, the flow field effect was lower with RMPC actuation (see Fig. 5(i)) due to nonpermanent momentum input with actuation mass flow, but still remarkable.

In Fig. 6, all characteristics defined above are area or mass-averaged with respect to their physical characteristics (see Refs. [44] and [8]) and for one passage. The base flow, steady, and RMPC actuation cases are then shown averaged over the disturbance phase angle $φ$. One can see an overall increase of $C¯p(φ)$ and decrease of $ζ¯(φ)$ through steady actuation, which means an improvement in static pressure rise as well as the total pressure loss. Although, in consideration of $ζ¯*(φ)$, it is shown that the corrected total pressure loss is heavily increased when taking the actuation effort into account. This confirms the efficiency analysis results mentioned above on a linear stator cascade by Steinberg and King [18].

Fig. 6
Fig. 6
Close modal

Therefore, the actuation mass flow should be used economically, as an increased value of $ζ¯*$ causes reduced stator vane efficiency. Thus, it seems beneficial that the RMPC actuation is only active between $φ≈[110 deg , 230 deg]$, resulting in a much lower actuation mass flow consumption as shown in Fig. 6(d). This also resulted in an increase of $ζ¯*(φ)$ during the active actuation, while during the rest of the period, $ζ¯*(φ)$ was nearly as low as without actuation (see Fig. 7(c)).

Fig. 7
Fig. 7
Close modal

Finally, RMPC-enabled AFC led to a relatively similar improvement of $C¯p(φ)$ and a lower improvement of $ζ¯(φ)$ during the influenced range of $φ≈[160 deg , 360 deg]$. Outside this range, the impact on $C¯p(φ)$ and $ζ¯(φ)$ was lower because of the inactive actuation on the considered passage. However, one could see some improvements during this phase, probably because of the overall actuation of the entire annular stator stage.

To finally compare the effects of different actuation momentum coefficients $cμ$, all discussed characteristics were additionally phase-averaged and, except for μ, divided by the corresponding characteristic value for the case without actuation. The outcome of that is shown in Fig. 7, where the total averaged and normalized values $C̃p$, $ζ̃$, $ζ̃*$, and $μ̃$ are presented over different $cμ$ values. The plots' courses show that with increasing $cμ$, the positive effect of RMPC-enabled actuation was just moderately lower than with steady actuation. However, for the latter, the consumption of actuation mass flow was up to three times higher, while the corrected total pressure loss coefficient was up to two times higher than with RMPC actuation. Consequently, using RMPC-enabled AFC, the efficiency drop was much lower while producing a comparable static pressure rise.

## Conclusion

An annular low-speed compressor stator rig with hub side pneumatic AFC was investigated. Downstream the stator vanes, a rotating throttling disk was installed to emulate the impact of periodic disturbances as they would occur with the usage of pressure gain combustion concepts. Previous investigations within the CRC 1029 for a linear compressor stator rig indicated that although the flow field in a stator row can be manipulated effectively, the overall efficiency of the cascade will be lowered with higher actuation amplitudes.

Hence, an RMPC—a model predictive control approach especially suited for periodic closed-loop tasks—was applied to address the impact of the periodic disturbances more specifically and lower the effort of actuation mass flow compared to steady actuation. For that, a scalar surrogate control variable was defined for a control-blade equipped with high-bandwidth pressure sensors, and a simple input–output model was identified to predict the influence of the binary8 actuation on the defined control variable. With that, an RMPC was formulated and solved in a real-time manner using a Branch and Bound algorithm combined with a QP solver. During operation, the calculated input trajectory was time-shifted with respect to the throttling disk's fixed rotating speed to expand the control from the control blade to the whole annular compressor stator rig. In closed-loop experiments, it was shown that the RMPC was converging relatively fast while lowering the considered two-norm of the disturbance impact on the control variable up to 25%. For this control task, the performance with steady actuation was insufficient and not suited to lower the disturbance's effects satisfyingly.

With several RMPC runs, an optimal input trajectory was obtained and henceforth used for further open-loop experiments (traversable blade measurements, oil-flow visualization, and wake measurements) in order to obtain a more detailed insight into the influence of the passage flow and the wake of the considered passage. In comparison, a steady blowing approach was used. While the closed-loop experiments clearly showed that the periodic disturbances induced by the throttling disk were approached more specifically with RMPC actuation, the open-loop experiments with steady actuation revealed an advantage concerning the reduction of the hub sided corner vortex. However, although the latter was reduced less with the RMPC actuation, a similar static pressure rise of the considered passage was achieved, while the actuation mass flow effort was up to 66% lower depending on the actuation amplitude. This is particularly advantageous since it could be confirmed for the annular test rig that the efficiency of a passage decreases with increasing actuation amplitude. In summary, AFC can improve an operating condition but should only be used selectively due to actuation costs. For this purpose, in a periodic operation, an RMPC can be a promising option to specifically lower periodic disturbances induced, for example, by pressure gain combustion, while simultaneously keeping the actuation effort at a necessary minimum level. Another advantage of the binary control strategy realized here is the possibility of using solenoid switching valves in a closed-loop approach. These valves are smaller, cheaper, and have a larger bandwidth than conventional proportional valves, making them more relevant for industrial application.

In the future, it would be of some interest to adapt the presented control approach to a high-speed stator cascade in the compressible flow regime. Furthermore, to exploit new findings regarding optimal firing patterns of the combustion tubes [45], a more realistic disturbance generator could be used.

## Acknowledgment

The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) as part of Collaborative Research Centre CRC 1029 “Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamics” in Projects B01 and B06.

## Funding Data

• Deutsche Forschungsgemeinschaft (SFB 1029; Funder ID: 10.13039/501100001659).

## Nomenclature

### Roman

Roman

• $A$ =

dynamic matrix of state space model

•
• $A´$ =

dynamic matrix of disturbance state space model

•
• Apsg =

flow cross area of a passage entrance

•
• $b¯$ =

input vector of state space model

•
• c =

chord length

•
• $c¯$ =

output vector of state space model

•
• $cp(z,s,k)$ =

static pressure coefficient on the SS at k

•
• $Cp(z,y,k)$ =

static pressure rise coefficient in yz-plane at k

•
• $c¯p(k)$ =

vector of $cp(z,s,k)$ of the CB at k

•
• $cμ(k)$ =

momentum coefficient amplitude at k

•
• dact =

actuator outlet width

•
• dk =

disturbance at k

•
• $d´k$ =

output of disturbance model at k

•
• $dk+i|k$ =

prediction of $dk+i$ at k

•
• $d¯k+1|k$ =

disturbance trajectory prediction at k

•
• $d¯k−n$ =

disturbance trajectory of the last cycle

•
• DCj =

duty cycle of the jth period

•
• ek =

output error at k

•
• $e¯j$ =

output error of the jth period

•
• $ek+i|k$ =

prediction of $ek+i$ at k

•
• $e¯k+1|k$ =

output error trajectory prediction at k

•
• $e¯k−n$ =

output error trajectory of the last cycle

•
• $F$ =

state error/output error matrix of the lifted system

•
• $f¯d$ =

frequency of the disturbance

•
• $f¯k$ =

linear vector of QP at k

•
• $g(z,y,k)$ =

total pressure in yz-plane at k

•
• $g*(z,y,k)$ =

corrected total pressure in yz-plane at k

•
• $G$ =

input/output error matrix of the lifted system

•
• h =

•
• $H$ =

Hessian matrix of QP

•
• hact =

actuator outlet height

•
• $I(u¯k|k)$ =

total costs of RMPC

•
• $Ie(u¯k|k)$ =

end costs of RMPC

•
• $Il(u¯k|k)$ =

running costs of RMPC

•
• j =

period counter

•
• k =

time step k

•
• $m˙(k)$ =

overall main mass flow at k

•
• $m˙act(k)$ =

overall actuation mass flow at k

•
• $m˙act,psg(k)$ =

mass flow of a specific passage actuation at k

•
• $m˙jet(k)$ =

mass flow of an active actuation jet at k

•
• $m˙psg(k)$ =

passage mass flow at k

•
• $n$ =

disturbance time steps / RMPC horizont

•
• $nact(k)$ =

number of open valves at k

•
• ns =

number of stator vanes

•
• nx =

order of state space model

•
• $p(z,s,k)$ =

static pressure on the SS at k

•
• $p(z,y,k)$ =

static pressure in yz-plane at k

•
• $p1,ref(k)$ =

upstream reference static pressure at k

•
• $p¯1$ =

first principle component

•
• $p¯act$ =

total pressure in actuation tank

•
• $q(z,y,k)$ =

dynamic pressure in yz-plane at k

•
• $q1,ref(k)$ =

upstream reference dynamic pressure at k

•
• rk =

output reference at k

•
• Re =

Reynolds number

•
• s =

suction side length coordinate

•
• smax =

suction side length

•
• $s¯k+1|k$ =

of $u¯k|k$ independent part of $e¯k+1|k$

•
• Srd =

Strouhal number of the disturbance

•
• $t¯k+n|k$ =

of $u¯k|k$ independent part of $e¯k+n|k$

•
• Tu =

degree of turbulence

•
• uk =

control signal at k

•
• $ujet(k)$ =

velocity of an active actuation jet at k

•
• $uk+i|k$ =

prediction of $uk+i$ at k

•
• $u¯k|k$ =

control signal trajectory prediction at k

•
• $u¯k|k*$ =

optimal control signal trajectory prediction at k

•
• $u¯k−n$ =

control signal trajectory of the last cycle

•
• $u¯∞$ =

converged realized control trajectory

•
• $v¯$ =

velocity vector

•
• vk =

feed forward control signal at k

•
• $wΔ$ =

diagonal element of $WΔ$

•
• $wΔ,0$ =

initial diagonal element of $WΔ$

•
• $We$ =

weighting matrix for $e¯k+1|k$

•
• $We$ =

weighting scalar for $ek+n|k$

•
• $Wu$ =

weighting Matrix for $u¯k|k$

•
• $WΔ$ =

weighting matrix for change of $u¯k|k$ to the last cycle

•
• $wΔ$$WΔ$$wΔ,0$$WΔ$x =

x coordinate

•
• $x¯k$ =

state vector at k

•
• y =

y coordinate

•
• ymax =

maximum width of wake measurements

•
• yk =

output at k

•
• ys =

output signal operating point

•
• $y¯¯∞$ =

converged mean output trajectory

•
• z =

z coordinate

•
• $z¯k$ =

disturbance model state vector at k

### Greek Symbols

Greek Symbols

• αk =

offset of QP at k

•
• δk =

disturbance model offset at k

•
• $Δu¯k|k$ =

change of $u¯k|k$ to the last cycle

•
• $Δεk$ =

change of $εk$ to the last cycle

•
• $ε¯k$ =

state error vector at k

•
• $ζ(z,y,k)$ =

total pressure loss coefficient in yz-plane at k

•
• $ζ*(z,y,k)$ =

corrected total pressure loss coefficient in yz-plane at k

•
• ρ =

density

•
• ρk =

state reference at k

•
• $φ$ =

phase of disturbance's period

### Operators/Special Accents and Subscripts

Operators/Special Accents and Subscripts

• $(·)0$ =

upstream of the VIGVs

•
• $(·)1$ =

•
• $(·)2$ =

•
• $(·)¯$ =

averaged value

•
• $(·)̂$ =

estimated value

•
• $(·)̃$ =

normalized value

•
• $(·)ˇ$ =

last row of matrix

•
• $|·|$ =

magnitude of vector

•
• $||·||̃2$ =

relative Euclidean norm

•
• $argmin{·}$ =

argument which minimizes the cost function

### Abbreviations

Abbreviations

• AFC =

active flow control

•
• B&B =

branch and bound

•
• BQP =

•
• CB =

•
• CRC =

Collaborative Research Center

•
• LE =

•
• MPC =

model predictive control

•
• PC =

principal component

•
• PS =

pressure side

•
• QP =

•
• RMPC =

repetitive model predictive control

•
• SISO =

single-input single-output

•
• SS =

suction side

•
• TB =

•
• TE =

trailing edge

•
• VIGV =

variable inlet guide vane

## Footnotes

2

This results in 135 phase increments for a disturbance period.

3

For a more accurate determination of ujet, the jet was investigated with a miniature pitot probe for different pressures in the actuator chamber and different actuation trajectories, see Ref. [36].

4

For a better readability in this section, variables at a specific time-step have a corresponding time-step index.

5

$s¯k+1|k$ is the part of Eq. (15), which is independent of $u¯k|k$.

6

$t¯k+n|k$ is the part of Eq. (19), which is independent of $u¯k|k$.

7

Note: The case without actuation can be seen at the RMPC course before activation (j <0).

8

The solenoid valves used for AFC just can be opened or closed.

## References

1.
Lord
,
W.
,
MacMartin
,
D.
, and
Tillman
,
G.
,
2000
, “
Flow Control Opportunities in Gas Turbine Engines
,”
AIAA
Paper No. 2000-2234.10.2514/6.2000-2234
2.
Hah
,
C.
, and
Loellbach
,
J.
,
1999
, “
Development of Hub Corner Stall and Its Influence on the Performance of Axial Compressor Blade Rows
,”
ASME J. Turbomach.
,
121
(
1
), pp.
67
77
.10.1115/1.2841235
3.
Bobusch
,
B. C.
,
Berndt
,
P.
,
Paschereit
,
C. O.
, and
Klein
,
R.
,
2014
, “
Shockless Explosion Combustion: An Innovative Way of Efficient Constant Volume Combustion in Gas Turbines
,”
Combust. Sci. Technol.
,
186
(
10–11
), pp.
1680
1689
.10.1080/00102202.2014.935624
4.
Zander
,
V.
, and
Nitsche
,
W.
,
2013
, “
,”
Proc. Inst. Mech. Eng. Part A
,
227
(
6
), pp.
674
682
.10.1177/0957650913495538
5.
Cattafesta
,
L. N.
, III
,., and
Sheplak
,
M.
,
2011
, “
Actuators for Active and Flow Control
,”
Annu. Rev. Fluid Mech.
,
43
(
1
), pp.
247
272
.10.1146/annurev-fluid-122109-160634
6.
Werder
,
T.
,
Liebich
,
R.
,
Neuhäuser
,
K.
,
Behnsen
,
C.
, and
King
,
R.
,
2021
, “
Active Flow Control Utilizing an Adaptive Blade Geometry and an Extremum Seeking Algorithm at Periodically Transient Boundary Conditions
,”
ASME J. Turbomach.
,
143
(
2
), p.
021008
.10.1115/1.4049787
7.
Liu
,
Y.
,
Sun
,
J.
, and
Lu
,
L.
,
2014
, “
Corner Separation Control by Boundary Layer Suction Applied to a Highly Loaded Axial Compressor Cascade
,”
Energies
,
7
(
12
), pp.
7994
8007
.10.3390/en7127994
8.
Nerger
,
D.
,
Saathoff
,
H.
,
,
R.
,
Gümmer
,
V.
, and
Clemen
,
C.
,
2012
, “
Experimental Investigation of Endwall and Suction Side Blowing in a Highly Loaded Compressor Stator Cascade
,”
ASME J. Turbomach.
,
134
(
2
), p.
021010
.10.1115/1.4003254
9.
Seifert
,
A.
,
Bachar
,
T.
,
Koss
,
D.
,
Shepshelovich
,
M.
, and
Wygnanski
,
I.
,
1993
, “
Oscillatory Blowing: A Tool to Delay Boundary-Layer Separation
,”
AIAA J.
,
31
(
11
), pp.
2052
2060
.10.2514/3.49121
10.
Greenblatt
,
D.
, and
Wygnanski
,
I.
,
2000
, “
The Control of Flow Separation by Periodic Excitation
,”
Prog. Aerosp. Sci.
,
36
(
7
), pp.
487
545
.10.1016/S0376-0421(00)00008-7
11.
Amitay
,
M.
, and
Glezer
,
A.
,
2002
, “
Role of Actuation Frequency in Controlled Flow Reattachment Over a Stalled Airfoil
,”
AIAA J.
,
40
(
2
), pp.
209
216
.10.2514/2.1662
12.
Culley
,
D. E.
,
Bright
,
M. M.
,
Prahst
,
P. S.
, and
Strazisar
,
A. J.
,
2004
, “
Active Flow Separation Control of a Stator Vane Using Embedded Injection in a Multistage Compressor Experiment
,”
ASME J. Turbomach.
,
126
(
1
), pp.
24
34
.10.1115/1.1643912
13.
Hecklau
,
M.
,
Gmelin
,
C.
,
Nitsche
,
W.
,
Thiele
,
F.
,
Huppertz
,
A.
, and
Swoboda
,
M.
,
2011
, “
Experimental and Numerical Results of Active Flow Control on a Highly Loaded Stator Cascade
,”
Proc. Inst. Mech. Eng. Part A
,
225
(
7
), pp.
907
918
.10.1177/0957650911410156
14.
Staats
,
M.
, and
Nitsche
,
W.
,
2016
, “
Active Control of the Corner Separation on a Highly Loaded Compressor Cascade With Periodic Nonsteady Boundary Conditions by Means of Fluidic Actuators
,”
ASME J. Turbomach.
,
138
(
3
), p.
031004
.10.1115/1.4031934
15.
Staats
,
M.
, and
Nitsche
,
W.
,
2018
, “
Active Flow Control on a Non-Steady Operated Compressor Stator Cascade by Means of Fluidic Devices
,”
New Results in Numerical and Experimental Fluid Mechanics XI
,
A.
Dillmann
,
G.
Heller
,
E.
Krämer
,
C.
Wagner
,
S.
Bansmer
,
R.
, and
R.
Semaan
, eds., Vol.
136
,
Springer
, Berlin, p.
337
.
16.
Staats
,
M.
,
Nitsche
,
W.
,
Steinberg
,
S. J.
, and
King
,
R.
,
2017
, “
,”
CEAS Aeronaut. J.
,
8
(
1
), pp.
197
208
.10.1007/s13272-016-0232-1
17.
Traficante
,
S.
,
Giorgi
,
M. G. D.
, and
Ficarella
,
A.
,
2016
, “
Flow Separation Control on a Compressor-Stator Cascade Using Plasma Actuators and Synthetic and Continuous Jets
,”
J. Aerosp. Eng.
,
29
(
3
), p.
04015056
.10.1061/(ASCE)AS.1943-5525.0000539
18.
Steinberg
,
S. J.
, and
King
,
R.
,
2019
, “
Efficiency of Active Flow Control in an Unsteady Stator Vane Flow Field
,”
In Fundamentals of High Lift for Future Civil Aircraft
,
Springer
, Berlin, p.
631
.
19.
Lee
,
K. S.
, and
Lee
,
J. H.
,
1998
, “
Model-Based Predictive Control Combined With Iterative Learning for Batch or Repetitive Processes
,”
Iterative Learning Control Analysis, Design, Integration and Applications
,
Z.
Bien
, and
J.-X.
Xu
, eds.,
Springer
, Berlin, p.
313
.
20.
Lee
,
K. S.
,
Chin
,
I. S.
,
Lee
,
H. J.
, and
Lee
,
J. H.
,
1999
, “
Model Predictive Control Technique Combined With Iterative Learning for Batch Processes
,”
AIChE J.
,
45
(
10
), pp.
2175
2187
.10.1002/aic.690451016
21.
Lee
,
J. H.
,
Natarajan
,
S.
, and
Lee
,
K.
,
2001
, “
A Model-Based Predictive Control Approach to Repetitive Control of Continuous Processes With Periodic Operations
,”
J. Process Control
,
11
(
2
), pp.
195
207
.10.1016/S0959-1524(00)00047-0
22.
Steinberg
,
S. J.
,
Staats
,
M.
,
Nitsche
,
W.
, and
King
,
R.
,
2016
, “
Constrained Repetitive Model Predictive Control Applied to an Unsteady Compressor Stator Vane Flow
,”
ASME
Paper No. GT2016-56002.10.1115/GT2016-56002
23.
Steinberg
,
S. J.
,
Staats
,
M.
,
Nitsche
,
W.
, and
King
,
R.
,
2017
, “
Comparison of Conventional and Repetitive MPC With Application to a Periodically Disturbed Compressor Stator Vane Flow
,”
IFAC-PapersOnLine
,
50
(
1
), pp.
11107
11112
.10.1016/j.ifacol.2017.08.948
24.
Steinberg
,
S. J.
, and
King
,
R.
,
2018
, “
Closed-Loop Active Flow Control of Repetitive Disturbances in a Linear Stator Cascade
,”
AIAA
Paper No. 2018-3689.10.2514/6.2018-3689
25.
Phan
,
M. Q.
, and
Longman
,
R. W.
,
1988
, “
A Mathematical Theory of Learning Control for Linear Discrete Multivariable Systems
,”
Proceedings of Astrodynamics Conference No. 88-4313-CP
, Minneapolis, MN, Aug. 15–17, p.
4313
. 10.2514/6.1988-4313
26.
Arimoto
,
S.
,
Kawamura
,
S.
, and
Miyazaki
,
F.
,
1984
, “
Bettering Operation of Robots by Learning
,”
J. Robot. Syst.
,
1
(
2
), pp.
123
140
.10.1002/rob.4620010203
27.
Wang
,
Y.
,
Gao
,
F.
, and
Doyle
,
F. J.
,
2009
, “
Survey on Iterative Learning Control, Repetitive Control, and Run-to-Run Control
,”
J. Process Control
,
19
(
10
), pp.
1589
1600
.10.1016/j.jprocont.2009.09.006
28.
Bristow
,
D. A.
,
Tharayil
,
M.
, and
Alleyne
,
A. G.
,
2006
, “
Survey of Iterative Learning Control
,”
IEEE
Control Syst. Mag.
,
26
(
3
), p.
96
.10.1109/MCS.2006.1636313
29.
Longman
,
R. W.
,
2000
, “
Iterative Learning Control and Repetitive Control for Engineering Practice
,”
Int. J. Control
,
73
(
10
), pp.
930
954
.10.1080/002071700405905
30.
Qin
,
S. J.
, and
,
T. A.
,
2003
, “
A Survey of Industrial Model Predictive Control Technology
,”
Control Eng. Pract.
,
11
(
7
), pp.
733
764
.10.1016/S0967-0661(02)00186-7
31.
Mayne
,
D. Q.
,
2014
, “
Model Predictive Control: Recent Developments and Future Promise
,”
Automatica
,
50
(
12
), pp.
2967
2986
.10.1016/j.automatica.2014.10.128
32.
Camacho
,
E. F.
,
Ramirez
,
D. R.
,
Limon
,
D.
,
Muñoz de la Peña
,
D.
, and
Alamo
,
T.
,
2010
, “
Model Predictive Control Techniques for Hybrid Systems
,”
Annu. Rev. Control
,
34
(
1
), pp.
21
31
.10.1016/j.arcontrol.2010.02.002
33.
Axehill
,
D.
,
Besselmann
,
T.
,
Raimondo
,
D. M.
, and
Morari
,
M.
,
2014
, “
A Parametric Branch and Bound Approach to Suboptimal Explicit Hybrid MPC
,”
Automatica
,
50
(
1
), pp.
240
246
.10.1016/j.automatica.2013.10.004
34.
King
,
R.
,
2019
,
Active Flow and Combustion Control 2018
, Vol.
141
,
Springer
, Berlin.
35.
Kiesner
,
M.
, and
King
,
R.
,
2015
, “
Closed-Loop Active Flow Control of the Wake of a Compressor Blade by Trailing-Edge Blowing
,”
ASME
Paper No. GT2015-42026.10.1115/GT2015-42026
36.
Fietzke
,
B.
,
Mihalyovics
,
J.
,
King
,
R.
, and
Peitsch
,
D.
,
2021
, “
A Comparison of Optimal, Binary Closed-Loop Active Flow Control Applied to an Annular Compressor Stator Cascade With Periodic Disturbances
,”
Proceedings of Act Flow Control Combust Conference
, Berlin, Germany, Sept. 28–29, p. 335.
37.
Eck
,
M.
,
Rückert
,
R.
,
Tüzüner
,
E.
, and
Mihalyovics
,
J.
,
2021
, “
Surface Flow Visualization Techniques
,”
Proceedings of Workshop Near-Wall Flow Turbomachinery Bl Rows 2021
, Dresden, Berlin, Jan.
28
29
.
38.
Shlens
,
J.
,
2014
, “
A Tutorial on Principal Component Analysis
,” arXiv Preprint arXiv:1404.1100.
39.
Schäpel
,
J. S.
,
King
,
R.
,
Yücel
,
F.
,
Völzke
,
F.
,
Paschereit
,
C. O.
, and
Klein
,
R.
,
2018
, “
Fuel Injection Control for a Valve Array in a Shockless Explosion Combustor
,”
ASME
Paper No. GT2018-75295. 10.1115/GT2018-75295
40.
Morrison
,
D. R.
,
Jacobson
,
S. H.
,
Sauppe
,
J. J.
, and
Sewell
,
E. C.
,
2016
, “
Branch-and-Bound Algorithms: A Survey of Recent Advances in Searching, Branching, and Pruning
,”
Discrete Optim.
,
19
, pp.
79
102
.10.1016/j.disopt.2016.01.005
41.
Schmid
,
C.
, and
Biegler
,
L. T.
,
1994
, “
Quadratic Programming Methods for Reduced Hessian SQP
,”
Comput. Chem. Eng.
,
18
(
9
), pp.
817
832
.10.1016/0098-1354(94)E0001-4
42.
Bar-Shalom
,
Y.
,
Li
,
X.-R.
, and
Kirubarajan
,
T.
,
2004
,
Estimation With Applications to Tracking and Navigation: Theory Algorithms and Software
,
Wiley
, New York.
43.
Mihalyovics
,
J.
,
Brück
,
C.
,
Peitsch
,
D.
,
Vasilopoulos
,
I.
, and
Meyer
,
M.
,
2018
, “
Numerical and Experimental Investigations on Optimized 3D Compressor Airfoils
,”
ASME
Paper No. GT2018-76826.10.1115/GT2018-76826
44.
Cumpsty
,
N. A.
, and
Horlock
,
J. H.
,
2006
, “
Averaging Nonuniform Flow for a Purpose
,”
ASME J, Turbomach.
,
128
(
1
), pp.
120
129
.10.1115/1.2098807
45.
Wolff
,
S.
, and
King
,
R.
,
2019
, “
Optimal Control for Firing Synchronization in an Annular Pulsed Detonation Combustor Mockup by Mixed-Integer Programming
,”
AIAA
Paper No. 2019-1742.10.2514/6.2019-1742