Longitudinal SAS

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Longitudinal Stability Augmentation Systems

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A Case Study

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Consider an example of Lockheed F-104 Starfighter

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A Case Study

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Consider Longitudinal transfer function of F-104

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Handling Qualities For an aircraft of class IV, operating in flight phase

category C, assuming Level 1 flying qualities are desired

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Pitch Feedback

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Pitch Feedback

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Pitch Feedback

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Pitch Feedback

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Thus the phugoid damping is increased by about 14 times andits frequency remains nearly constant.

The short period mode damping is approximately halved whilstits frequency is increased by about 50%. Obviously thephugoid damping is the parameter which is most sensitive tothe feedback gain by a substantial margin.

A modest feedback gain of say, 𝐾𝐾𝜃𝜃 = −0.1 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟 wouldresult in a very useful increase in phugoid damping whilstcausing only very small changes in the other stabilityparameters.

However, the fact remains that pitch attitude feedback toelevator destabilizes the short period mode by reducing thedamping ratio from its open loop value. This then, is not thecure for the poor short period mode stability exhibited by theopen loop F-104 aircraft at this flight condition.

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Can you propose a solution?

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Pitch Rate Feedback The correct solution is to augment pitch damping by

implementing pitch rate feedback to elevator (velocityfeedback in servo mechanism terms).

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Pitch Rate Feedback

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Lets consider same aircraft

For an aircraft of class IV, operating in flight phase category C, assuming Level 1 flying qualities are desired

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Pitch Rate Feedback

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Pitch Rate Feedback

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Pitch Rate Feedback

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Clearly, pitch rate feedback to elevator is ideal since it causes the damping of both the phugoid and short period modes to be increased although the short period mode is most sensitive to feedback gain.

Further, the frequency of the short period mode remains more-or-less constant through the usable range of values of feedback gain 𝐾𝐾𝑞𝑞. For the same range of feedback gains the frequency of the phugoid mode is reduced.

At test point a 𝐾𝐾𝑞𝑞 = −0.3 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠 which is the smallest feedback gain required to bring the closed loop short period mode into agreement with the flying qualities boundaries.

Allowing for a reasonable margin of error and uncertainty a practical choice of feedback gain might be 𝐾𝐾𝑞𝑞 = −0.5 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠.

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Pitch Rate Feedback

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Pitch Rate Feedback

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This augmentation system is the classical pitch damperused on many airplanes from the same period as theLockheed F-104 and typical feedback gains would be inthe range – 0.1 ≤ 𝐾𝐾𝑞𝑞 ≤ −1.0 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠.

It is not known what value of feedback gain is used in theF-104 at this flight condition but the published descriptionof the longitudinal augmentation system structure is thesame as that shown below

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Pitch Rate Feedback

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At 𝐾𝐾𝑞𝑞 = −0.5 rad/rad/s

For an aircraft of class IV, operating in flight phase category C, assuming Level 1 flying qualities are desired

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Pitch Rate Feedback

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Clearly, at this value of feedback gain the flying qualitiesrequirements are met completely with margins sufficientto allow for uncertainty.

The closed loop system thus defined provides the basisfor further analytical studies concerning theimplementation architecture and safety issues.

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How a feedback affect System dynamics

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Lets start with a simple example of mass spring system

𝑚𝑚�̈�𝑥 𝑡𝑡 + 𝑐𝑐�̇�𝑥 𝑡𝑡 + 𝑘𝑘𝑥𝑥 𝑡𝑡 = 𝐹𝐹(𝑡𝑡)

𝐺𝐺 𝑠𝑠 =𝑋𝑋(𝑠𝑠)𝐹𝐹(𝑠𝑠)

= 𝐾𝐾𝜔𝜔𝑛𝑛2

𝑠𝑠2 + 2𝜁𝜁𝜔𝜔𝑛𝑛 + 𝜔𝜔𝑛𝑛2

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How a feedback affect System dynamics

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Position feedback:

𝐺𝐺(𝑠𝑠)

𝐾𝐾𝑥𝑥

𝑚𝑚�̈�𝑥 𝑡𝑡 + 𝑐𝑐�̇�𝑥 𝑡𝑡 + 𝑘𝑘𝑥𝑥 𝑡𝑡 = 𝐹𝐹 𝑡𝑡 − 𝐾𝐾𝑥𝑥𝑥𝑥(𝑡𝑡)

𝑚𝑚�̈�𝑥 𝑡𝑡 + 𝑐𝑐�̇�𝑥 𝑡𝑡 + (𝑘𝑘 + 𝐾𝐾𝑥𝑥)𝑥𝑥 𝑡𝑡 = 𝐹𝐹 𝑡𝑡

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How a feedback affect System dynamics

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So position feedback basically affects “k” (Stiffness) ofthe system thus the natural frequency.

Stability/control derivatives

Can you relate?

𝜕𝜕𝐹𝐹𝜕𝜕𝑥𝑥

= 𝑘𝑘𝜕𝜕𝐹𝐹𝜕𝜕�̇�𝑥

= 𝑐𝑐

𝑀𝑀𝛼𝛼 ,𝑀𝑀𝑞𝑞, 𝑀𝑀�̇�𝛼

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Modes Approximation Phugoid Mode

Short Period Mode

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Influence of Stability Derivatives

Stability Derivative Mode affected

𝑀𝑀𝑞𝑞 + 𝑀𝑀�̇�𝛼Damping of short-period mode of

motion

𝑀𝑀𝛼𝛼Frequency of short-period mode of

motion

𝑋𝑋𝑢𝑢Damping of the phugoid or long-

period mode of motion

𝑍𝑍𝑢𝑢Frequency of phugoid mode of

motion

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Longitudinal Stability Augmentation

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In the previous case study it has been shown how negativefeedback using a single variable can be used to selectivelyaugment the stability characteristics of an airplane.

It has also been shown how the effect of single variablefeedback may readily be evaluated with the aid of a root locusplot.

The choice of feedback variable is important in determiningthe nature of the change in the stability characteristics of theairplane since each variable results in a unique combination ofchanges.

Provided that the aircraft is equipped with the appropriatemotion sensors various feedback control schemes arepossible and it then becomes necessary to choose thefeedback variable(s) best suited to augment the deficiencies ofthe basic airframe.

It is also useful to appreciate what effect each feedbackvariable has on the stability modes when assessing a FCSdesign.

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Longitudinal Stability Augmentation

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However complex the functional structure of the systemthe basic augmentation effect of each feedback variabledoes not change.

Feedback is also used for reasons other than stabilityaugmentation, for example, in autopilot functions. In suchcases augmentation will also occur and it may not bedesirable, in which case a thorough understanding of theeffects of the most commonly used feedback variables isinvaluable.

In order to evaluate the effect of feedback utilizing a particularresponse variable it is instructive to conduct a survey of all thesingle loop feedback options.

In every case the feedback loop is reduced to a simple gaincomponent only. By this means the possible intrusive effects ofother loop components, such as noise filters, phasecompensation filters, sensor and actuator dynamics, areprevented from masking the true augmentation effects.

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Longitudinal Stability Augmentation

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The survey is conducted by taking each motion variablein turn and evaluating its influence on the closed loopstability characteristics as a function of the loop gain K.

The root locus plot enables the relative influence on, andthe relative sensitivity of, each of the stability modes tobe assessed simultaneously.

As the detailed effect of feedback depends on the aircraftand flight condition of interest it is not easy to generalizeand is best illustrated by example.

Consequently the following survey, is based on a typicalaircraft operating at a typical flight condition and theobservations may be applied loosely to the longitudinalstability augmentation of most aircraft.

Example

McDonnell Douglas A-4D Skyhawk

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Case Study: McDonnell Douglas A-4D Skyhawk

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Longitudinal Stability Augmentation

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Longitudinal Stability Augmentation

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Transfer function data for the McDonnell Douglas A-4DSkyhawk aircraft was obtained from Teper (1969). The flightcondition chosen corresponds with an all up weight of17,578 lb at an altitude of 35,000 ft at Mach 0.6. Infactorized form the longitudinal characteristic equation is

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Longitudinal Stability Augmentation

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Longitudinal Stability Augmentation

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These stability mode characteristics would normally beconsidered acceptable with the exception of the short periodmode damping ratio which is too low.

The Skyhawk is typical of combat airplanes of the 1960s inwhich modest degrees of augmentation only are required torectify the stability deficiencies of the basic airframe.

This, in turn, implies that modest feedback gains only arerequired in the range say, typically, 0 ≤ 𝐾𝐾 ≤ 2.0.

In modern FBW aircraft having unstable airframes ratherlarger gain values would be required to achieve the samelevels of augmentation.

In general, the greater the required change in the stabilitycharacteristics the greater the feedback gains needed to effectthe change.

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Pitch attitude feedback to elevator

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Pitch attitude feedback to elevator

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As 𝐾𝐾𝜃𝜃 is increased the phugoid damping increases rapidlywhilst the frequency remains nearly constant. The shortperiod mode frequency increases whilst the dampingdecreases, both characteristics changing relatively slowly.

Thus, as might be expected since pitch attitude is adominant variable in the phugoid mode, this mode isconsiderably more sensitive to the loop gain than the shortperiod mode.

Since this feedback option further destabilizes the shortperiod mode its usefulness in a SAS is very limited indeed.

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Pitch attitude feedback to elevator

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However, it does improve phugoid stability, the modebecoming critically damped at a gain of 𝐾𝐾𝜃𝜃 = −0.37 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟 . A practical gain value might be 𝐾𝐾𝜃𝜃 = −0.1 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟 which would result in a good level of closed loopphugoid stability without reducing the short period modestability too much.

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Pitch rate feedback to elevator

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Pitch rate feedback to elevator

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As 𝐾𝐾𝑞𝑞 is increased the short period mode damping increasesrapidly whilst the frequency remains nearly constant. Thephugoid frequency and damping decrease relatively slowly.

More typically, a slow increase in phugoid damping would beseen. Thus, as might be expected since pitch rate is a dominantvariable in the short period mode, this mode is considerablymore sensitive to the loop gain than the phugoid mode.

As discussed in previous example this feedback option describesthe classical pitch damper and is found on many airplanes. It alsoexactly describes the longitudinal stability augmentationsolution used on the Skyhawk.

Its dominant effect is to artificially increase the magnitude of thederivative 𝑀𝑀𝑞𝑞, it also increases the magnitude of the derivatives𝑋𝑋𝑞𝑞 and 𝑍𝑍𝑞𝑞 but to a lesser degree. The short period mode becomescritically damped at a gain of 𝐾𝐾𝑞𝑞 = −0.53 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠.

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Pitch rate feedback to elevator

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A practical gain value might be 𝐾𝐾𝑞𝑞 = −0.3 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠 whichwould result in an adequate level of closed loop shortperiod mode stability whilst simultaneously increasing thefrequency separation between the two modes.

However, at this value of feedback gain the changes in thephugoid characteristics would be almost insignificant. Asbefore, these observations are in good agreement with thefindings of previous Example.

This feedback directly affects the 𝑀𝑀𝑞𝑞 derivative, thus mainlythe damping of short period mode.

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Velocity feedback to elevator

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Velocity feedback to elevator

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As 𝐾𝐾𝑢𝑢 is increased the short period mode frequency increasesquite rapidly whilst, initially, the damping decreases. However, atvery large gain values the damping commences to increase againto eventually become critical.

Whereas, as 𝐾𝐾𝑢𝑢 is increased both the frequency and damping ofthe phugoid mode increase relatively rapidly. Thus, at this flightcondition both modes appear to have similar sensitivity tofeedback gain.

The stabilizing influence on the phugoid mode is much as mightbe expected since velocity is the dominant variable in the modedynamics.

The dominant effect of the feedback is therefore to artificiallyincrease the magnitude of the derivative 𝑀𝑀𝑢𝑢 and since 𝑀𝑀𝑢𝑢 isusually small it is not surprising that even modest values offeedback gain have a significant effect on phugoid stability.

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Velocity feedback to elevator

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It also increases the magnitude of the derivatives 𝑋𝑋𝑢𝑢 and 𝑍𝑍𝑢𝑢but to a lesser degree. A practical gain value might be 𝐾𝐾𝑢𝑢 =0.001 𝑟𝑟𝑟𝑟𝑟𝑟/𝑓𝑓𝑡𝑡/𝑠𝑠 which would result in a significantimprovement in closed loop phugoid mode stability whilstsimultaneously decreasing the stability of the short periodmode by a small amount.

However, such values of feedback gain are quiteimpractically small and in any event this feedback optionwould not find much useful application in a conventionallongitudinal SAS.

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Angle of attack feedback to elevator

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Angle of attack feedback to elevator

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As 𝐾𝐾𝛼𝛼 is increased the short period mode frequency increases veryrapidly whilst, initially, the damping decreases slowly. However, as thegain increases further the damping slowly starts to increase toeventually become critical at an impractically large value of feedbackgain.

At all practical gain values the damping remains more-or-less constant. As 𝐾𝐾𝛼𝛼 is increased both the frequency and damping of the phugoid are

reduced, the mode becoming unstable at a gain of 𝐾𝐾𝛼𝛼 = −3.5 𝑟𝑟𝑟𝑟𝑟𝑟/𝑟𝑟𝑟𝑟𝑟𝑟 in this example.

Incidence feedback to elevator is a powerful method for augmenting thelongitudinal static stability of an airplane and finds extensiveapplication in unstable FBW aircraft.

The effect of the feedback is equivalent to increasing the pitch stiffnessof the aircraft which artificially increases the magnitude of thederivative 𝑀𝑀𝑤𝑤 (≡ 𝜕𝜕𝜕𝜕𝑚𝑚/𝜕𝜕𝛼𝛼) and to a lesser degree it also increases themagnitude of the derivatives 𝑋𝑋𝑤𝑤 and 𝑍𝑍𝑤𝑤.

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Angle of attack feedback to elevator

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Thus the increase in short period mode frequency together with theless significant influence on damping is entirely consistent with theaugmentation option. Since phugoid dynamics are typically very nearlyincidence constant, the expected effect of the feedback on the mode isnegligible. This is not the case in this example, probably due toaerodynamic effects at the relatively high subsonic Mach number.

This is confirmed by the fact that the phugoid roots do not evenapproximately cancel with the complex pair of numerator roots, whichmight normally be expected. It would therefore be expected to see someincidence variation in the phugoid dynamics.

It mainly affects 𝑀𝑀𝛼𝛼

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Normal Acceleration Transfer Function

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Normal acceleration feedback to elevator

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Normal acceleration feedback to elevator

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Since the transfer function is not proper care must be exercisedin the production of the root locus plot and in its interpretation.However, at typically small values of feedback gain itsinterpretation seems quite straightforward.

Since an accelerometer is rather more robust than an incidencesensor, normal acceleration feedback to elevator is commonlyused instead of, or to complement, incidence feedback. Bothfeedback variables have a similar effect on the phugoid and shortperiod stability mode at practical values of feedback gain.However, both modes are rather more sensitive to feedback gainsince very small values result in significant changes to the modecharacteristics.

As 𝐾𝐾𝑎𝑎𝑧𝑧 is increased the short period mode frequency increasevery rapidly whilst, initially, the damping decreases slowly.However, as the gain increases further the damping slowly startsto increase to eventually become critical at an impractically largevalue of feedback gain.

It has similar effect of 𝛼𝛼 feedback, thus it mainly affects and 𝑀𝑀𝛼𝛼

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Longitudinal Stability Augmentation (Another Case Study)

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At all practical gain values the damping remains more-or-less constant. The full short period mode branch of thelocus is not shown in Figure since the gain range requiredexceeded the capability of the computational software usedto produce the plots.

As 𝐾𝐾𝑎𝑎𝑧𝑧 is increased both the frequency and damping of thephugoid are reduced, the mode becoming unstable at a gainof 𝐾𝐾𝑎𝑎𝑧𝑧 = 0.0026 𝑟𝑟𝑟𝑟𝑟𝑟/𝑓𝑓𝑡𝑡/𝑠𝑠2 in this example. Since thenormal acceleration variable comprises a mix of incidence,velocity and pitch rate (see Section 5.5, M. V. Cook) then theaugmentation it provides in a feedback control system maybe regarded as equivalent to the sum of the effects offeedback of the separate variables.

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Longitudinal Stability Augmentation (Another Case Study)

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Thus at moderate gains the increase in pitch stiffness issignificant and results in a rapid increase in short periodmode frequency. The corresponding increase in shortperiod mode damping is rather greater than that achievedwith incidence feedback alone due to the effect of implicitpitch rate feedback. Since the incidence dependent termdominates the determination of normal acceleration it isnot surprising that normal acceleration feedback behaveslike incidence feedback.

It is approximately equivalent to artificially increasing themagnitude of the derivative 𝑀𝑀𝑤𝑤 and to a lesser degree italso increases the magnitude of the derivatives 𝑋𝑋𝑤𝑤 and 𝑍𝑍𝑤𝑤.

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A Typical Stability Augmentation System

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Objective Feedback Side effect Derivative

Short perioddamping 𝑞𝑞 𝑀𝑀𝑞𝑞

Phugiod damping 𝑢𝑢𝜃𝜃

Short period damping reduced 𝑀𝑀𝑢𝑢

Short period frequency

𝛼𝛼𝑟𝑟𝑧𝑧

𝑀𝑀𝛼𝛼

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Stability Derivative Mode affected

𝑀𝑀𝑞𝑞 + 𝑀𝑀�̇�𝛼Damping of short-period mode of

motion

𝑀𝑀𝛼𝛼Frequency of short-period mode of

motion

𝑋𝑋𝑢𝑢Damping of the phugoid or long-

period mode of motion

𝑍𝑍𝑢𝑢Frequency of phugoid mode of

motion