George Kopasakis and Joseph W. Connolly

Glenn Research Center, Cleveland, Ohio

Shock Positioning Controls Design 

for a Supersonic Inlet

NASA/TM—2010-216074

February 2010

AIAA–2009–5117
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George Kopasakis and Joseph W. Connolly

Glenn Research Center, Cleveland, Ohio

Shock Positioning Controls Design 

for a Supersonic Inlet

NASA/TM—2010-216074

February 2010

AIAA–2009–5117

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

45th Joint Propulsion Conference and Exhibit

cosponsored by the AIAA, ASME, SAE, and ASEE

Denver, Colorado, August 2–5, 2009
Acknowledgments

The authors would like to express their gratitude to the Supersonics Project of the 

NASA Fundamental Aeronautics Program for supporting this research effort.

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only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

This work was sponsored by the Fundamental Aeronautics Program 

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Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2010-216074 1 

Shock Positioning Controls Design for a Supersonic Inlet 

George Kopasakis and Joseph W. Connolly 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Under the NASA Fundamental Aeronautics Program, the Supersonics Project is working to overcome 

the obstacles to supersonic commercial flight. The supersonic inlet design that is utilized to efficiently 

compress the incoming air and deliver it to the engine has many design challenges. Among those 

challenges is the shock positioning of internal compression inlets, which requires active control in order 

to maintain performance and to prevent inlet unstarts due to upstream (freestream) and downstream 

(engine) disturbances. In this paper a novel feedback control technique is presented, which emphasizes 

disturbance attenuation among other control performance criteria, while it ties the speed of the actuation 

system(s) to the design of the controller. In this design, the desired performance specifications for the 

overall control system are used to design the closed loop gain of the feedback controller and then, 

knowing the transfer function of the plant, the controller is calculated to achieve this performance. The 

innovation is that this design procedure is methodical and allows maximization of the performance of the 

designed control system with respect to actuator rates, while the stability of the calculated controller is 

guaranteed. 

I. Introduction 

The supersonic inlet compresses incoming air supersonically and through the subsonic diffuser, in the 

process forming an oblique shock system and a normal shock in the vicinity where the flow changes from 

supersonic to subsonic, as shown for an axisymmetric inlet in Figure 1. If the normal shock travels 

slightly upstream of the throat area, the shock is quickly expelled from the inlet causing the inlet to 

unstart and creating large frictional forces in front of the inlet. On the other hand, if the normal shock is 

ingested by the engine that can result in component damage. Therefore, the normal shock position needs 

to be controlled at some distance downstream of the throat restriction to prevent such undesirable effects. 

However, propulsion efficiency increases as the shock position is controlled closer to this throat 

restriction, and thus a good control design that rejects disturbances can reduce this shock position margin 

and improve propulsion efficiency. 

Traditionally, proportional or proportional integral (PI) type control designs have been employed for 

inlet shock positioning control systems (Refs. 1 and 2). Predictive and optimal controls combined with PI 

are used in References 3 to 5. Quadratic Optimization is used in Reference 6 to minimize the expected 

frequency of inlet unstarts. These designs, however, do not take advantage of actuator speeds or 

bandwidths to maximize disturbance rejection, while maintaining desirable control system performance. 

The controls design that will be discussed in this paper is based on the loop shaping technique discussed 

in Reference 7, which shapes the closed loop gain of the system based on the actuator speeds, for 

disturbance rejection and stability performance criteria. This feedback control design technique is 

systematic, and allows for a control design that maximizes performance with respect to the speed of the 

plant and its actuation system(s). Even though satisfactory control may be achieved without a systematic 

design technique, there could be a sacrifice in shock position margin, which impacts the pressure recovery 

and thus, the inlet efficiency. 
NASA/TM—2010-216074 2 

A nonlinear inlet model named LAPIN (Large Perturbations Inlet) was utilized and perturbed to 

develop plant transfer functions (TF) for controls design. This inlet model is a quasi-1–D model based on 

the method of characteristics, developed at NASA Glenn Research Center (Ref. 8). This model allows for 

geometry manipulations and can be used to develop simulations of internal compression inlets of different 

geometries. 

The loop shaping feedback control design for inlet shock positioning follows the approach described 

in Reference 7. In this approach, the closed loop gain of the system is designed in the frequency domain 

to achieve maximum disturbance attenuation within the capability of the actuation system, while at the 

same time maintaining adequate gain and phase margins, as well as other performance criteria such as 

response and settling times. Then, knowing the TF of the plant, the plant is subtracted from the overall 

closed loop gain (in dB scale) which gives the desired feedback controller TF. The resulting controller TF 

is then fitted with poles and zeros in order to calculate the controller that will achieve the desired closed 

loop gain that meets the feedback control system performance requirements. The whole control design 

process is methodical, and also demonstrates how to trade between conflicting performance criteria. The 

innovation in the technique lies in its ability to directly tie actuator rates to the maximum performance 

attainable in the controls design. Also, by following the approach in Reference 7 of fitting the controller 

TF (magnitude and phase) with poles and zeros, explicit plant inversion is avoided, which guarantees 

stable controllers. 

Based on this approach, first the feedback controls of the bypass door actuation system utilized for the 

shock control is designed to have high bandwidth with high disturbance attenuation and adequate stability 

margins. This is followed by the feedback design of the shock positioning controls themselves in the 

presence of atmospheric disturbances, as covered in Reference 9. 

The paper is organized as follows. First, the feedback control systems loop shaping design approach 

(Ref. 7) is described, followed by a description of the linear inlet model derived by perturbing the 

nonlinear LAPIN model. Next, the bypass door actuator model and its feedback control design are 

described, followed by a description of atmospheric disturbances. Then the design of the shock position 

controller is covered, followed by a description of its performance attenuating atmospheric disturbances. 

Finally, some concluding remarks are presented. 
NASA/TM—2010-216074 3 

II. Feedback Control Systems Loop Shaping Design Methodology 

In this section a brief description of the loop shaping design methodology will be given. For more 

detail on this approach the reader is referred to Reference 7. This loop shaping design methodology is a 

systematic feedback controls systems approach, which is designed to simultaneously meet a variety of 

control performance criteria or specifications. The approach is to design the overall closed loop gain of 

the control system based on the performance criteria and the speed of the actuation system first. This is 

followed by designing the controller so that the closed loop system meets the desired closed loop gain. 

This is accomplished with an emphasis on maximizing disturbance rejection while meeting adequate 

stability margins, emphasizing how to trade-off these competing requirements. 

A traditional feedback control system diagram is shown in Figure 2, with the controller the plant and 

disturbance TFs shown as Gc, Gp, and Gd respectively. The closed loop gain (CLG) is defined as the 

product of the TFs in the closed loop system, which includes the feedback. For distinction, the CLG is the 

gain that appears in the denominator of a feedback control system TF. Whereas, the open loop gain is the 

gain that appears in the numerator of that TF, with those gains being the same when the feedback gain is 

unity. Thus, for unity feedback, as will be the case for the shock position control design presented, the 

CLG of the control system is 

 L(s) = Gc(s)Gp(s) (1) 

It is this CLG that is shaped in this approach based on the design criteria or requirements. The first step is 

to choose the maximum bandwidth of the feedback control system. This translates to how fast the system 

can be driven without saturation or damage to the hardware. From Reference 7, the crossover frequency 

or the bandwidth of the control system (i.e., the frequency where the CLG crosses the 0 dB axis) in rad/s, 

can be computed as 

mrlrco ω = raC (2) 

where rm is the maximum step input of the system, arl is actuator rate limit in units of magnitude per 

second, and Cr is the ratio of system output response to actuator input. Typically, for a unit step input and 

for Cr=1, the cross-over frequency is equal to the actuation rate limit. The cross over frequency of the 

CLG, ωco, is also associated with the response or time constant of the system (i.e., the time at which the 

time response reaches 63 percent of its final value) as τtc=1/ ωco. The input to output frequency response 

of the control system can be expressed as 

)(1

)(

)(

)(

sL

sL

sR

sC

+ = (3) 
NASA/TM—2010-216074 4 

The disturbance response TF, for disturbance coming in at the plant output, can be expressed as 

)(1

)(

)(

)(

sL

sG

sD

sC d

+ = (4) 

From Equation (3) it can be seen that the CLG is directly associated with the stability characteristics of 

the closed loop system and from Equation (4) it can be seen that the CLG is also directly associated with 

the ability of the system to reject disturbances. Since, the time constant(s) of the system are also a part of 

the CLG, designing for the CLG addresses all the requirements associated with the performance of 

classical feedback control systems. 

Figure 3 shows an example CLG design, with the associated time response of the closed loop system 

shown in Figure 4, based on the following CLG transfer function 

( )

( )( ) 110001100

125 13.353)(

+ +

+ = ssss sL (5) 

In Figure 3 the symbol DA signifies disturbance attenuation at the mid-frequency band, and φM signifies 

phase margin for stability. Closed loop gain designs can be easily realized based on the control system 

performance requirements, following the process described in Reference 7. In this design the gain margin 

is infinite and it is not shown. The gain margin is the difference in gain between the 0 dB and the gain at 

the frequency at which the phase crosses –180°. In this design the closed loop system has two dominant 

time constants; the initial time constant that is related to the cross-over frequency as discussed before and 

associated with the fast second order type response, followed by a time constant associated with the 

frequency of the zero in Equation (5) with a first order response. These time constants are evident in the 

response of Figure 4; first the initial rapid underdamped response, followed by the longer response before 

the system settles. The inset in Figure 4 shows a longer time response at the neighborhood of the 

amplitude that is dominated by the first order response of the zero. The advantages of these two time 

constants or dominant natural responses together in the design are increased disturbance attenuation and 

enhanced stability (Ref. 7), as will be discussed later. 
NASA/TM—2010-216074 5 

In brief, the design of the CLG starts with deciding on the bandwidth or the cross-over frequency 

based on Equation (2). This is followed by placing a pole at the origin for zero steady-state error, a zero at 

some low frequency to boost the phase for stability and for high disturbance attenuation at the midfrequency range as well as for settling time requirements (not all of these objectives are in concert), 

followed by a pole at approximately one decade below the desired cross-over frequency. Typically, such a 

design with two poles and one zero can be made to achieve approximately 20 dB disturbance attenuation 

at the mid-frequency range, up to one decade before the cross-over frequency. Designing for 20 dB 

disturbance attenuation at the mid-frequency range is near the maximum achievable considering the 

actuator bandwidth. Slightly more disturbance attenuation is possible, however, in that case the design of 

the CLG would end up being higher order and trade the stability margins for disturbance attenuation, as 

evident with the CLG example in Figure 3. 

Following the design of the CLG based on the feedback control system performance requirements, the 

process TF is subtracted from the designed CLG in the frequency domain (dB scale) to come-up with the 

controller design frequency response. This frequency response is then fitted with poles and zeros as 

detailed in Reference 7, in order to come-up with the controller design. As a final step, the designed CLG 

based on the specification is compared to the actual CLG obtained from this design process. Since the 

publishing of Reference 7, the process of calculating the controller frequency response was automated, by 

forming the complex vectors of the CLG and the process TFs and subtracting these vectors in dB scale to 

come up with the controller frequency response. The last step of fitting the controller TF with poles and 

zeros is still being performed manually, following the process outlined in Reference 7. 

III. Linear Supersonic Inlet Model 

For the supersonic inlet model, LAPIN was utilized, which simulates a finite quasi 1–D internal 

compression inlet, utilizing a modeling methodology associated with the method of characteristics. This 

model also includes upstream volume elements starting from the freestream and thus, it includes a 

representative simulation of oblique shock(s) bouncing off the inlet forebody. The accuracy of this model 

has been verified against experimental data obtained for internal compression inlets (Ref. 8). This 

simulation allows the user to define the inlet geometry in the 1–D sense. Since the emphasis of the NASA 

Supersonics Project is fundamental research, the project hasn’t baselined an inlet geometry. Thus, the 

two-dimensional bifurcated (2DB) supersonic inlet, previously utilized in the NASA High Speed 

Research program will also be used here for the controls design studies. 
NASA/TM—2010-216074 6 

Closed loop control of the shock position downstream of the inlet throat is primarily accomplished by 

adjusting the bypass door exit area to bleed air out of the inlet. An upstream movement of the shock position 

will cause the controller to open the bypass door, and vice versa. The center body for an axisymmetric inlet or 

the forward ramps for a square shaped inlet can also translate to control the shock, but these actuation systems 

have much slower response times and are often scheduled open or closed loop. In this control design, the inlet 

operating point will be at cruise (M~2.35) and the control design will be examined at this operating point, 

without control of the slower inlet translating center body or the ramps. 

For a linear control design, the representative TFs are obtained by applying frequency sweeps to the 

LAPIN model to generate TF for the variables of interest for both the disturbances and the control 

variables. The disturbance TFs were fitted with poles and zeros by using MATLAB(The MathWorks, 

Inc.) System ID. The TF for the control variables were fitted by hand, because even though System ID 

does a good job fitting the magnitude, that was not the case for the phases, which are very important for 

controls design. The accuracy of the sinusoidal sweeps to generate accurate TFs was always verified by 

examining the coherence. 

A. Inlet Plant Transfer Functions 

The overall inlet feedback control system for this study consists of the inlet bypass door actuator, the 

actuator position (corresponding to the bypass door area opening) to the shock position response, and the 

feedback controller that controls the actuator to position the shock. The plant TF for the inlet system will 

then be a product of the actuator TF and the actuator position to the shock response TF. Disturbances 

come in at the output of this control system, at the shock position, similar to the feedback control diagram 

shown in Figure 2. The center body for an axisymmetric inlet or the ramps for a two-dimensional type 

inlet are also controlled either open or closed loop and this control also effects the shock position. 

However, even closed loop control for these geometries is relatively slow compared to the bypass door 

actuation response. Therefore, centerbody or ramp control for the shock is not anticipated to significantly 

add to the overall disturbance rejection of the system, especially at higher frequencies where it is most 

important, and thus this type of control will not be included in this study. 

Using these sinusoidal sweeps with the LAPIN model, the bypass door to shock position TF was 

derived as 

)13707/3.13707/)(13519/3.13519/(

1

*

)13330/3.13330/)(13142/1.13142/)(1911/5.1911/)(1118/(

)12513/)(1653/(325.3

)(

22 22

22 22 22

+ + + +

+ ++ + + + +

+ +

=

s s s s

ss ss s s s

ssK

sG

cr

BS

 (6) 

where Kcr is a geometric scale property of the cowl lip radius utilized in the LAPIN model. 

A Moog, Inc., valve was used for the bypass door actuator, based on past experimental setups utilized 

for the 2DB supersonic inlet. This valve with the vendor controller (inner loop) and the feedback 

controller designed in this development (outer loop) is shown in Figure 5. The transfer function of the 

valve and the various gains shown for the valve feedback are taken from vendor literature. The inner 

valve feedback control design is set by the vendor and cannot be changed. The outer loop feedback 

controller is designed here to provide a control bandwidth of about 175 Hz, again based on vendor 

specifications for the valve. The rest of the controller design (i.e., the outer loop shown in Figure 5) is 

used to provide high disturbance attenuation of 20 dB or greater at the mid-frequency range and a phase 

and gain margin of greater than 60° and 10 dB respectively. This design will not be detailed here, but it 

follows along the same design procedures discussed in the previous section (Ref. 7) and the design of the 

shock controller that will be discussed later. The overall input/output TF of the valve and its feedback 

controller is approximated as 
NASA/TM—2010-216074 7 

)14753/6.04753/)(11136/8.01136/)(160/(

)163/( )( 22 22 + + + + ++ = ss s s ss

A sG (7) 

This approximation is computed to match the actual TF (both magnitude and phase) of the valve feedback 

control system from low frequency up to a little over the cross-over frequency of its CLG. 

B. Inlet Disturbance Transfer Functions 

The main objective of the inlet shock control system is to maintain the shock at a desired position 

downstream of the inlet throat in the presence of upstream disturbances (i.e., atmospheric, 

aeroservoelastic, yaw, and angle of attack) and downstream disturbances (coming from the engine). For 

the upstream disturbances, how the inlet itself propagates these disturbances in terms of Mach number, 

static pressure and static temperature to effect the shock position needs to be taken into account. For the 

control analysis presented in this paper, only the atmospheric disturbances will be included. Thus, the 

atmospheric disturbance models for these flow quantities are also needed in this analysis. 

Perturbing the LAPIN model with sinusoidal sweeps and using MATLAB System ID to estimate the 

TF of the inlet disturbances, the static pressure to shock position TF is computed as 

1786.61562.81366.11078.1707.1309.3

1475.81416.21019.8737.5408.255.1 )( 6 5 4 3 2

5 4 3 2

eseseseseses

eseseseses

PS sG

+ + + + + +

− + − + − = (8) 

As mentioned before, the phase of these disturbance TFs, which would be associated with the time 

delay as a function of frequency that it would take for a disturbance to be felt at the shock, was not 

matched with MATLAB System ID and it was offset by 360°, as a phase lead. Thus, it is assumed that a 

phase lead would be worst case compared to the actual time delay of the disturbances. But in the future 

these disturbance TFs will be more accurately fitted to incorporate better accuracy for the phase delay as 

well. Similarly, the static temperature to shock TF was computed as 

2447.32265.21997.41602.7

26.62093.21712.11420.2

1369.51046.3751.1387.3

1128.1769.5478.189.4 )(

3 2

3 2

8 7 6 5 4

7 6 5 4

esesese

esesese

seseseses

seseses

TS sG

+ + + +

− + − −

+ + + +

+− − + =

 (9) 

The Mach number to shock TF was computed as 
NASA/TM—2010-216074 8 

2121.11931.11623.2

2169.41952.11669.2

1388.31016.2742.1312.3

1301.1916.9624.3288.4 )(

2

2

7 6 5 4 3

6 5 4 3

esese

esese

seseseses

sesesese

MS sG

+ ++

+ ++

+ + + +

− + − =

 (10) 

C. Atmospheric Disturbances 

Derivations for atmospheric disturbance spectral densities for variations in acoustic velocity, 

temperature, and pressure are shown in Reference 9. In this Reference, atmospheric disturbance models 

were created by scaling an existing model and by fitting these atmospheric models, which are fractional 

order, with a product of first order poles and zeros. Then time domain atmospheric disturbances can be 

obtained by forming a combination of unit magnitude sinusoids, distributed in frequency starting from 

low sub hertz frequencies up to about 200 Hz (approximately the bandwidth limitation for the type of 

actuators used for bypass doors), which are used as inputs to the TFs representing these atmospheric 

disturbances. 

Thus, atmospheric disturbance TF for the longitudinal acoustic velocity is expressed as (Ref. 9) 

)11.1593/)(17.85/)(11.30/)(146.1/(

)15.335/)(10.55/)(12.9/(74.8 )(

+ + + +

+ + + = sss s

ss s

LA sG (11) 

where ε stands for the eddy dissipation rate, which has a worst case value of 8.6×10–5 (m2

/s3) based on 

data collected at altitudes of 25 to 40 k ft; L stands for the integral length scale with a typical value of 

762 m, which is related to outer length scale that signifies the length of the atmospheric turbulence patch 

(Ref. 9). The eddy dissipation rate has values that increase in magnitude with decreasing altitude below 

25 k ft, and it is orders of magnitude larger near ground level. For this study, supersonic operation implies 

that the shock control will be needed only at relatively high altitudes. The atmospheric temperature 

disturbance TF is expressed as (Ref. 9) 

)13.816/)(18.109/)(11.25/)(11.1/(

)14.602/)(16.45/)(10.33/(75.41 )(

+ + + +

+ + + = sss s

s s s

T sG (12) 

Temperature fluctuations will cause acoustic velocity disturbances through the speed of sound change, 

and this velocity disturbance due to temperature is expressed as 

)13.816/)(18.109/)(11.25/)(11.1/(

)14.602/)(16.45/)(10.33/(88.20 )(

+ + + +

γ + + + = ss s s

s s s

a

RM sG o

TA (13) 

where M signifies the Mach number, γ is the ratio of specific heats, R is the universal gas constant in units 

of (N*m)/(kg*°K), and ao is the local speed of sound. Close examination of Equation (13) will show that 

acoustic disturbances due to temperature fluctuations are considerably larger than those due to pure 

acoustic velocity fluctuations, Equation (11). The atmospheric pressure disturbance TF is expressed as 

(Ref. 9) 

)13.816/)(18.109/)(11.25/)(11.1/(

)14.602/)(16.45/)(10.33/(96.37 )(

+ + + +

+ + + = ss s s

s s s

P sG (14) 

These atmospheric TFs have corresponding units of m/s (Eqs. (11) and (13)), °K (Eq. (12)), and Pa (Eq. (14)). 

The input to these TFs are unitless sinusoids of unit amplitude, uniformly distributed in frequency. 
NASA/TM—2010-216074 9 

IV. Inlet Shock Position Control Design 

Having derived the inlet linear plant model for shock position control at cruise and the bypass door 

actuation system TF as well as the atmospheric disturbance models and inlet disturbance TFs, the control 

design can begin. Of course, the cruise operating point is just one point in the plant model and the plant 

model can differ for different operating points. However, the Supersonics Project has not yet baselined an 

inlet design and also, the objective of this control design is to demonstrate a design methodology, rather 

than an encompassing control design for the entire operating envelope. 

The approach for the feedback control design has been discussed briefly in Section II and in more 

detail in Reference 7. The approach that has been taken for this control design is to maximize disturbance 

rejection based on the actuation system bandwidth capability listed in Vendor’s specifications, as discuss 

in Section III, while providing sufficient stability margins (i.e., about 60° phase margin or greater and at 

least 10 dB gain margin). The next step is to conduct simulations and analysis to understand how this 

control design handles these disturbances (for now just atmospheric disturbances). For the shock position 

control or in general, this design approach allows the designer to maximize the performance of the control 

system and it also provides for real justification to ask for additional hardware capability in case the 

design doesn’t meet the prescribed requirements. 

A block diagram of the inlet shock position feedback control system via bypass door control with the 

atmospheric disturbances is shown in Figure 6. In the actual shock control system, the actuator feedback 

control (Fig. 5) takes the place of GA(s). The command schedule block (Cmnd Sch) in this figure is a 

control schedule for the actuator, which is utilized to move the bypass door to the desired position 

required to initialize the shock control system and it’s a constant in this study. 

As discussed before, the first step of the loop shaping control design methodology (Ref. 7) is to 

decide on the control bandwidth (i.e., the cross-over frequency) based on the speed or bandwidth of the 

actuation system. Since the Moog valve was used before for actuating the bypass door, and its 

specification called for a bandwidth up to about 175 Hz, the same bandwidth was selected for the actuator 

feedback control system. Disturbance attenuation for the actuator position, for disturbances coming from 

the vehicle structural vibration modes, was also considered important for this design. Also since this 

control design is multi-loop, disturbance rejection capability of the actuator position control loop would 

be expected to contribute to the overall shock position disturbance rejection performance. The details of 

the actuator feedback control design shown in Figure 5 will not be discussed in this paper, but are similar 

to the design that will be described for the shock position feedback control system. Based on this 

bandwidth of the actuator feedback control system, the shock position feedback control system bandwidth 
NASA/TM—2010-216074 10 

was chosen to be about 145 Hz. Normally, significant separation would be designed for these control 

bandwidths to avoid coupling, but such a design would significantly compromise disturbance attenuation. 

However, designing the controller with a hybrid or composite dominant natural frequency will help to 

mediate (to some degree) the coupling problem for the outer loop of the shock control feedback. Still, in 

the final control system design, this bandwidth may need to be lowered if the speed of the outer loop 

causes the valve position control loop to be overdriven. Thus, the approach is to design the control 

bandwidth as high as possible, without taking into consideration the disturbance frequencies, in order to 

enable maximizing disturbance attenuation. More on this subject will be discussed later. 

After determining the control system bandwidth, the CLG can be shaped to meet these specifications. 

For a mid-frequency gain of 20 dB (for disturbance rejection) and for the CLG to cross-over at 145 Hz, a 

pole needs to be placed one decade below this frequency, at 14.5 Hz (remember the gain slope due to a 

pole is –1 or –20 dB/decade). Next, select the frequency where the low frequency zero will be placed. The 

main purpose of the low frequency zero is to preserve the gain of the CLG at the mid-frequency range for 

disturbance attenuation and to boost the phase for better phase margin at the cross-over frequency. But the 

low frequency zero of the CLG will also influence the response time, especially the settling time (Ref. 7). 

For the phase boost, it is advantageous to place the zero lower in frequency, and for the settling time it is 

beneficial is to place the zero at higher frequency. So some compromise is in order. With no hard 

requirement for the settling time, the frequency of the zero was placed at 5 Hz. Also, for zero steady state 

error, a pole needs to be placed at the origin. For disturbance attenuation, a mid-frequency gain of 20 dB 

is chosen for the CLG, which maximizes the mid-frequency gain for maximum disturbance attenuation. 

Actually, it is possible to obtain a little more mid-frequency gain with a little more complicated design 

(Ref. 7). Based on the disturbance attenuation design of 20 dB at the mid-frequency range (starting at 

5 Hz) and with only the pole at zero influencing the magnitude of the CLG up to that point, the gain of the 

CLG at the frequency of the zero can be calculated as 

dB

s

K jL L )( 20

)5(2

=ω =

π=ω

 (15) 

where KL is the proportional gain of the CLG, L. Based on Equation (15), this gain can be calculated as 

KL =2π(5)1020/20 = 310 (i.e., the antilog based on 20log10(x)). In actuality, the gain will break 3 dB above 

at the frequency of the zero and 3 dB below at the frequency of the pole. In general the proportional gain 

of the CLG is 

20/ 10Lmfr K ω= zL (16) 

where ωz is the frequency where the zero is placed and Lmfg is the gain of the CLG desired or specified at 

the mid-frequency range. This completes the design of the CLG as 

)1)5.142/((

)1)52/(( 310)(

+π

+π = sss sL (17) 

Barring the fact that the shock controller in this design is a multi-loop controller with almost 

overlapping loop bandwidths, a single loop control system with this CLG design will have a phase 

margin of φM = 180 – 90 + 90 – 90= 90° (more than enough) and an infinite gain margin. A bode plot of 

this design is shown in Figure 7. 

In order for the CLG of the feedback control system to approximate the design of Equation (17), it 

means that the controller times the plant TFs will also need to equate to this CLG as 

= BSAC sGsGsGsL )()()()( (18) 

Therefore, using Equations (7), (6), and (17) (see also Fig. 6), the controller TF can be calculated as 
NASA/TM—2010-216074 11 

)()(

)( )( sGsGsL sG

BSA

C = (19) 

Or in terms of magnitude in dB scale, where it would be desirable to display GC(s) in order to fit the 

approximation with poles and zeros 

)()(log20)(log20)( 10 10 C sG = sL − BSA sGsG (20) 

And the phase can be computed as 

φ C = φ − φ BSA sGsGsLsG ))()(())(())(( (21) 

Figure 8 shows the calculation of the controller TF GC(s) and its fitting with poles and zeros. This fitting 

was done by hand, which resulted in the control TF as 

)175002/4.1)75002/()(170002/4.1)70002/((

)16502/60.0)6502/()(15502/0.1)5502/((

)165002/4.1)65002/()(160002/4.1)60002/((

)14502/0.1)4502/()(13502/36.1)3502/((

)155002/4.1)55002/()(150002/4.1)50002/((

)12502/3.1)2502/()(11452/98.0)1452/((

)11042/)(15.142/(

)18.182/)(152/(310 )(

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

+π+π+π+π

+π+π+π+π

+π+π+π+π

+π+π+π+π

+π+π+π+π

+π+π+π+π

+π+π

+π π + =

s s s s

s s s s

s s s s

s s s s

s s s s

s s s s

ss s

ss

C sG

 (22) 

The high frequency poles in Equation (22) (those in the 1000’s of Hz) are inserted in order to make the 

controller TF proper without impacting the phase at the frequency range of interest for the fit, but not too 

high in frequency to significantly slow down the simulation. Figure 9 shows how the design CLG with 

the computed controller TF matches the desired CLG of Equation (17) (Fig. 7) when inserted into the 

shock control feedback system of Figure 6 with the actual valve feedback controller. As can be seen, the 

actual phase of the control system design deviates somewhat at the cross-over and beyond, but there is 

still plenty of phase margin. Also, the gain after the cross-over needs to keep on decreasing (especially 

near the cross-over), which it does, but it is not fully shown in this figure. 
NASA/TM—2010-216074 12 
NASA/TM—2010-216074 13 

Figure 10 shows a step response starting from zero initial conditions for the shock response. It turns 

out that the valve position is overdriven in this design, generating relatively large excursions (not shown) 

as evident by the chopping response, because of the tightly coupled control bandwidths of the multi-loop 

controllers. In terms of the large overshoot in this design, two points can be made. One, the shock 

position reference will not be commanded in a step fashion, so there is nothing that will move the shock 

instantaneously. Thus, that will not be a problem as long as the system is stable. Secondly, under an 

ordinary design, this much overshoot will render the control system oscillatory and virtually unstable. 

However, as discussed before, the composite natural response in this design, with the low frequency zero 

dominating the response after the initial response takes place, has a stabilizing effect on the control 

system by changing the control system response from second order to a first order type. It is beneficial to 

have a fast initial response, indicative of a high bandwidth control design, because such a design allows 

for a high disturbance attenuation design. 

The oscillations in the response shown in Figure 10, due to overdriving the valve position control, can 

be improved by decreasing the controller gain in Equation (22) until a satisfactory response is obtained 

that includes the valve position. By decreasing the controller gain, the bandwidth will decrease, but 

disturbance attenuation will decrease proportionally as well. Based on that, the controller gain in Equation 

(22) was decreased to 100, which equates to approximately 10 dB reduction in gain. Thus, roughly from 

Figure 9, subtracting 10 dB from the gain will cause the CLG to cross-over at about 35 Hz. The step 

response of the shock control system with this reduction in the controller gain is shown in Figure 11. 

Once the desired cross-over frequency or bandwidth (that sufficiently reduces coupling) in this multiloop control system is found, the proper design procedure would be to repeat the whole controller design 

process in order to maximize disturbance rejection. Again, the frequencies of the disturbances are not 

taken into account in the design of the bandwidth of the control system. The reason is that if the highest 

possible control bandwidth is designed, and if it turns out that this is not good enough for disturbance 

rejection (in terms of let’s say of the command tracking performance or control margin reduction), the 

option left is to request more capability from the hardware designs, such as actuation speeds. Repeating 

the control design process, with the newly found bandwidth, will not be done here and for the studies that 

follow a controller gain of 100 is used. 

Atmospheric disturbances were applied utilizing Equations (11) to (14) with unit amplitude sinusoidal 

inputs distributed from very low frequencies up to 200 Hz as shown in the feedback control diagram of 

Figure 6. A time domain slice of these disturbances is shown in Figure 12. A control disturbance response 

was assumed with worst case atmospheric disturbance conditions at high altitudes as those shown in 

Figure 12 with an eddy dissipation rate ε=8.6×10–5, with a longitudinal acoustic wave velocity 
NASA/TM—2010-216074 14 

disturbance along the axis of the inlet accompanied by a temperature fluctuation that also produces 

acoustic wave velocity disturbances, together with a pressure fluctuation. The combined disturbances 

influencing the shock position, and the feedback control responses are shown in Figure 13. It can be seen 

from the control response that this feedback control system design does a good job attenuating anticipated 

worst case atmospheric disturbances, down to about 0.15 in. peak-to-peak. 
NASA/TM—2010-216074 15 

Checking the actuation rates of the valve position (not shown in detail in Fig. 13), it turns out that the 

actuation rates are well below that corresponding to the valves 175 Hz bandwidth, which satisfies the 

design. The low actuation rates are the result of the atmospheric disturbances whose spectral densities 

decrease with frequency as well as the volume of the inlet which also attenuates these disturbances. 

Therefore, based only on these results and if needed, the bandwidth of the actuator position controller can 

be increased with a corresponding increase on the shock position controller bandwidth without violating 

the actuator bandwidth limit. However, in order to increase the bandwidth of the valve controller outer 

loop, the inner loop design (Fig. 5) would also need to be taken into account. 

A single frequency disturbance of amplitude of one peak-to-peak was applied at a 10 Hz frequency to 

check the mid-frequency disturbance attenuation of this design as shown in Figure 14. The disturbance 

attenuation shown with the shock position is approximately 20 dB. This disturbance attenuation is 

achieved despite the gain reduction discussed before of about 10 dB, which was expected to reduce midfrequency disturbance attenuation by 10 dB (i.e., reduced down from the original design of 20 dB). The 

reason is that the shock position control loop is working in concert with the actuator control loop to 

enhance disturbance rejection. 
NASA/TM—2010-216074 16 

V. Conclusion 

In this paper a systematic inlet shock position control system design based on feedback loop shaping was 

demonstrated with emphasis on disturbance rejection. Results presented with atmospheric disturbances show 

that the shock position control is able to sufficiently reject these disturbances. More work remains to be done 

to also add inlet upstream flow field disturbances generated from the vehicle aeroservoelastic modes, also 

disturbances from the vehicle pitch and yaw maneuvers, as well as downstream disturbances coming from the 

rest of the propulsion system. In addition, linear models of the inlet need to be generated at other flight 

conditions, from transonic all the way to cruise, and similar control design and analysis would need to be 

conducted. Finally, the control design would need to be applied to the nonlinear inlet model as well to the 

integrated propulsion system, and finally to the aeropropulsoservoelastic system of the integrated vehicle. 

However, so far it is encouraging to see that the inlet shock position control system has no trouble attenuating 

expected atmospheric disturbances. If this trend of tight shock position control can be maintained throughout 

the operating envelope and disturbances, it could allow a decrease in the shock position margin and thus an 

increase in the propulsion efficiency. 

References 

1. Cole, G.L., “Atmospheric Effects on Inlets for Supersonic Cruise Aircraft,” NASA TM X–73647, 

AIAA–77–874, Thirteenth Propulsion Conference, Jul. 11–13, 1977. 

2. Chang, J., “Observers Design for Shock Position of Supersonic Inlet Based on Equilibrium 

Manifold,” AIAA, 42nd Joint Propulsion Conference, Jul. 9–12, 2006. 

3. Ahsun, U., “Design of a Near-Isentropic Supersonic Inlet Using Active Control,” Journal of 

Propulsion and Power, Vol. 21 No. 2, Mar.–Apr. 2005. 

4. MacMartin, D.G., “Dynamics and Control of Shock Motion in a Near-Isentropic Inlet,” Journal of 

Aircraft, Vol. 41, No. 4, July-August, 2004. 

5. Yarng, J.B., “Optimal Control of Supersonic Inlet/Engine Combination,” Journal of Guidance Control 

and Dynamics, Vol. 11, No. 2, Mar.–Apr. 1988. 

6. Lehtinen, B., “Application of Quadratic Optimization to Supersonic Inlet Control,” NASA TM X–

67905, Fifth Congress of the International Federation of Automatic Control,” June 12–17, 1972. 

7. Kopasakis, G., “Feedback Control Systems Loop Shaping Design With Practical Considerations,” 

NASA/TM—2007-215007, Sept. 2007. 

8. Varner, M.O., Martindale, W.J., Phares, K.R., Kneile, K.R., Adams, J.C., “Large Perturbation Flow 

Field Analysis and Simulation for Supersonic Inlets Final Report,” NASA CR–174676. 

9. Kopasakis, G., “Atmospheric Turbulence Modeling for Aero Vehicles—Fractional Order Fits,” 

NASA TM submitted for publication. 
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4. TITLE AND SUBTITLE

Shock Positioning Controls Design for a Supersonic Inlet 

5a. CONTRACT NUMBER

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6. AUTHOR(S)

Kopasakis, George; Connolly, Joseph, W. 

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WBS 984754.02.07.03.20.02 

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National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

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E-17122 

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14. ABSTRACT

Under the NASA Fundamental Aeronautics Program, the Supersonics Project is working to overcome the obstacles to supersonic 

commercial flight. The supersonic inlet design that is utilized to efficiently compress the incoming air and deliver it to the engine has many 

design challenges. Among those challenges is the shock positioning of internal compression inlets, which requires active control in order to 

maintain performance and to prevent inlet unstarts due to upstream (freestream) and downstream (engine) disturbances. In this paper a novel 

feedback control technique is presented, which emphasizes disturbance attenuation among other control performance criteria, while it ties 

the speed of the actuation system(s) to the design of the controller. In this design, the desired performance specifications for the overall 

control system are used to design the closed loop gain of the feedback controller and then, knowing the transfer function of the plant, the 

controller is calculated to achieve this performance. The innovation is that this design procedure is methodical and allows maximization of 

the performance of the designed control system with respect to actuator rates, while the stability of the calculated controller is guaranteed.

15. SUBJECT TERMS

Supersonic inlet control; Inlet control; Inlet feedback control; Atmospheric disturbances; Atmospheric turbulence 

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