On-line parameter estimation for restructurable flight control systems

Aircraft Design 4 (2001) 19}50

On-line parameter estimation for restructurable #ight control systems Marcello R. Napolitano *, Yongkyu Song
, Brad Seanor Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506/6106, USA
School of Aerospace and Mechanical Engineering, Hankuk Aviation University, Goyangshi, South Korea

Abstract This paper describes the results of a study where an on-line parameter identi"cation (PID) technique is used for determining on-line the mathematical model of an aircraft that has sustained damage to a primary control surface. The mathematical model at post-failure conditions can then be used by a failure accommodation scheme to compute on-line the compensating control signal to command the remaining healthy control surfaces for a safe continuation and/or termination of the #ight. Speci"c criteria for the use of an on-line PID for these critical #ight conditions are "rst discussed. The methodology is illustrated through simulations of a "ghter jet at subsonic #ight conditions featuring a novel modeling procedure to characterize the post-failure/damage aerodynamic conditions. The simulations have shown the potential of this on-line PID within a fault tolerant #ight control system. The results have also highlighted the importance of conducting an &ad hoc' small amplitude and short-duration PID maneuver immediately following a positive failure detection to enhance the reliability of the on-line estimated parameters used in the accommodation scheme.  2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Research on fault tolerant #ight control systems is an important area in #ight controls. This interest is due to several factors. Within the DoD, the relatively low procurement of highperformance military aircraft along with the cancellation of plans for building new aircraft has induced an interest toward fault tolerant #ight control systems with capabilities for accommodating sensor and actuator failures.

* Corresponding author. Tel.: #1-304-293-4111x2346; fax: #1-304-293-6689. E-mail address: [email protected] (M.R. Napolitano). 1369-8869/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 9 - 8 8 6 9 ( 0 0 ) 0 0 0 2 3 - 9

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Nomenclature c K q S t x y

aerodynamic coe$cient numerical coe$cient pitch angular velocity, rad/s wing surface, ft time, s location on the longitudinal axis location on the lateral axis

Greek letters a D e g h d p u

angle of attack, rad or deg di!erence downwash angle, rad or deg dynamic pressure ratio pitch Euler angle, rad or deg control surface de#ection, rad or deg Standard deviation frequency, rad/s

Subscripts h L L

horizontal tail left side lift force

l M R S

rolling moment pitching moment right side stabilator

Acronyms AC AFA AFDI BLS CG EE EKF FDI FT FFT LS ML MIMO PID RLS WB

aerodynamic center actuator failure accommodation actuator failure detection and identi"cation batch least squares center of gravity estimation error extended Kalman "ltering failure detection and identi"cation Fourier transform "nite Fourier transform least squares maximum likelihood multi}input}multi-output parameter identi"cation recursive least squares wing body

The use of fault tolerant #ight control schemes is also very important for unmanned air vehicles (UAVs) used for both remote sensing and military purposes. Without a pilot on board and with only a very remote ground control the availability of a control scheme taking over in the event of a failure is a very attractive capability. The importance of a fault tolerant #ight control system following actuator failures is even more immediate for commercial aviation. An increase in the safety of commercial aviation has been declared a top priority by the US federal government; particularly there is a clearly de"ned mission to decrease the commercial aviation accident rate by 80% within the next 10 years. In general, a fault tolerant #ight control system is required to perform failure detection, identi"cation, and accommodation for sensor and actuator failures. This paper is related to the actuator failure problem, which also includes battle damage, that is damage sustained during a typical air-to-ground maneuver with ground "re or during an air combat situation.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

21

To implement a failure accommodation strategy a variety of control surfaces (speed brakes, wing #aps, di!erential dihedral canards, spoilers, etc.) and thrust mechanisms (di!erential thrust, thrust vectoring) can be used [1]. The need of introducing fault tolerant capabilities for a relaxed static stability aircraft with a #ight envelope up to non-linear angles of attack requires an increase in the number of independent control surfaces. The selection of the control mechanism to be used is a function of several factors: control e!ectiveness, increased aircraft complexity and costs, weight penalties, increased aerodynamic drag due to the increased wetted area, and applicability depending on aircraft type. For a fault tolerant control logic to be e!ective, the traditional #ight data, the actuator position for each surface, along with a fully operational #ight computer, are assumed to be available. Failures and/or battle damage of a control surface can be classi"ed in two categories, locked surface and missing surface. Generally, a locked surface is associated with a mechanical failure in the actuator; a battle damage, instead, mainly implies a missing surface or, more realistically, both missing and locked surface. Of particular concern is the dynamic coupling between longitudinal and lateral directional dynamics following any type of control surface failure in addition to non-linear dynamic and aerodynamic conditions. This may lead to a loss of stability and, possibly, to unrecoverable #ight conditions. Following an actuator failure the objectives of a fault tolerant #ight control system are to achieve, in increasing order of importance: (1) a lower damage-induced handling qualities degradation; (2) a lower mission abort rate; (3) a lower aircraft loss rate. This paper is organized as follows. The next section brie#y lists existing approaches for fault tolerant control systems for actuator failures and proposes an alternative overall scheme. Another section reviews an on-line PID scheme recently introduced followed by a section reviewing a detailed approach to aerodynamic modeling of an aircraft at post-failure or post-damage conditions. Then a section discusses the numerical simulation of the on-line PID following control surface damages typical of a combat situation for military aircraft under di!erent scenarios. A "nal section summarizes the paper with conclusions and recommendations.

2. Approaches for fault tolerant control laws for actuator failure Within fault tolerant control schemes one can distinguish between recon"gurable and restructurable approaches. A necessary condition for both classes of fault tolerant approaches is that the aircraft system is still controllable and, therefore, trimmable following the battle damage and/or generic failures. Therefore, failures for which the aircraft does not have an inherent redundancy over a dynamic mode } through multiple control surfaces and/or thrust mechanisms } are not considered. For both restructurable and recon"gurable approaches several methods } too numerous to be listed } have been proposed in recent years.

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Within recon"gurable #ight control systems a complete set of possible failures for di!erent control surfaces is de"ned a priori; the relative changes in control gains are then calculated o!-line } using one of several control approaches } and stored in the on-board memory, ready for on-line use whenever needed. A disadvantage of this approach is the required a priori extensive design to accommodate all possible control system failures at di!erent conditions in the aircraft #ight envelope. This, in turn, requires extensive memory space in the on-board computer. Restructurable fault tolerant systems are conceptually di!erent. Within these systems the reconstruction of the control laws is instead performed on-line. In this case the availability of computational power, as opposed to memory, could be a critical factor. Assuming the controllability and the trimmability of the aircraft at post-failure conditions, of direct interest to the topic in a subset of restructurable control systems where the accommodation control laws are formulated using on-line estimates of aircraft parameters from a real time PID scheme. In general, the battle damage and/or generic failure accommodation includes the two following tasks [1]: E actuator failure detection and identi"cation (AFDI); E actuator failure accommodation (AFA). In recent years a few experimental programs have been or are being conducted to demonstrate the feasibility of fault tolerant #ight control systems; a partial list is given by: E E E E

the the the the

self-designing controller (SDC) program [2], using a VISTA/F-16 aircraft; RESTORE program [3,4], using a tailless "ghter aircraft; ACTIVE program [5,6], using a F-15 aircraft; XV-15 program [7,8], using a tilt-rotor aircraft.

One of the innovative concepts from this research e!ort is to envision an original general restructurable fault tolerant scheme where the failure accommodation is divided into the following phases: Phase C1: Following a positive actuator failure detection and identi"cation (AFDI) and an on-line PID } as discussed in the next sections } a restructurable control law } of MIMO nature and capable of handling non-linear dynamics } is formulated on-line to regain control of the aircraft before the aircraft enters unrecoverable #ight conditions. Phase C2: Once back to stable and linear conditions, a linear restructurable control law is formulated for designing the control laws } using the results from an on-line PID scheme } to continue the mission with the least amount of handling qualities degradation. The scheme is shown with details in Fig. 1. The "rst author has previously introduced approaches for the AFDIA scheme and for the Phase C1-on-line learning neural controllers for AFA in Fig. 1 [1,9,10].

3. Frequency-based PID method for on-line real-time PID The estimation of aircraft aerodynamic parameters, namely stability and control derivatives, from #ight data is a well-known discipline. In the majority of the applications this process is

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

23

Fig. 1. General block diagram of the fault tolerant #ight control system.

conducted o!-line using #ight data relative to speci"cally designed parameter identi"cation (PID) maneuvers. The maximum likelihood (ML) method has been the most utilized approach for this task starting in the mid-1960s [11}13]. However, its computational complexity, its sensitivity to the presence of noise in the #ight data, along with the need for a "ne tuning of the &a priori' values, has precluded any attempt to on-line real-time applications. On-line PID, particularly within an adaptive control scheme, is a more challenging and demanding task. Ultimately, the e!ectiveness of an adaptive algorithm depends on the amount of unmodeled dynamics and on the presence of system and measurement noise. A basic assumption for on-line PID is a modeling of the dynamic system with time-varying parameters, which can be aerodynamic parameters (stability and control derivatives), #ight conditions (dynamic pressure), or inertial parameters (weight, moments, and products of inertia). Like o!-line PID approaches, on-line PID methods can be formulated either in the time-domain or in the frequency domain. Within time-domain on-line PID techniques least-squares (LS) algorithms, based on the use of the gradient, are used in lieu of techniques based on the use of the gradient and Hessian because of their convergence robustness and lower computational e!ort. Therefore on-line time-domain PID techniques mainly include variations of the LS regression method, such as recursive least square (RLS) [14,15], RLS with a forgetting factor [16], a modi"ed

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

sequential least square (MSLS) [2], a real-time batch least squares (BLS) [17,18], and extended Kalman "ltering (EKF) [19]. The real-time applications of any of these methods presents substantial challenges due to a combination of the unavoidable presence of system and measurement noise, lack of information for PID purposes in the #ight data (such as a prolonged steady-state #ight condition), and potential unavailability of independent control inputs } a necessary condition for an accurate PID } due to the interactions with the closed-loop control laws. Analytical mechanisms to handle some of the above problems include the use of temporal and spatial constraints (such as forgetting factors and/or the use of short set of #ight data). Another potential problem with time-domain PID techniques may be the lack of a reliable parameter for an on-line assessment of the accuracy of the estimates in the presence of unmodeled noise. In trying to overcome some or all the problems described above, frequency-based PID techniques, using Fourier transforms, have been introduced. Of particular interest is a simple singlestep technique based on discrete Fourier transform [20}22] using previous work described in [23]. A brief description of the method is provided below. First, the aircraft dynamics is modeled using the conventional continuous-time state variable model given by x (t)"Ax(t)#Bu(t),

(1)

y(t)"Cx(t)#Du(t).

(2)

For a generic signal x(t), its "nite Fourier transform (FFT) is given by



x (u)"

2 x(t)e\ x dt. 

(3)

An approximation for the FFT [22] is given by ,\ x (u)"*t x e\ SRG "X(u)*t, G 

(4)

where t "i*t. G

(5)

It should be noted that this approximation is accurate if the sampling rate is much higher than the frequencies of interest. The application of the Fourier transform (FT) to the state variable model will provide jux (u)"Ax (u)#Bu (u),

(6)

y (u)"Cx (u)#Du (u).

(7)

As for the LS regression method, the measurements of the vector x, u, and y can be used to set up a cost function having the coe$cients of A, B, as argument. In particular, consider the generic kth

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

25

state equation over a frequency range:

   ju x (u ) x 2(u ), u 2(u )  I    ju x (u ) x 2(u ), u 2(u )  I    " ...... ....... ......

.......

ju x (u ) K I K

x 2(u ), u 2(u ) K K

 

A2 I , B2 I

(8)

where A and B are the kth row of the state and control matrices A and B, respectively; I I x (u ), u (u ) are the FT of the kth state and control variable relative to the frequency u . I L I L L If we denote the above equation as >"XH#e with complex equation error e and traditional de"nitions for >, X, and H, the problem can be formulated as a standard LS regression problem with complex data. Thus, with the following cost function 1 K J " " ju x (u )!A x (u )!B u (u )" L I L I L I L I 2 L "(>!XH)H(>!XH), 

(9)

the minimizing parameter vector estimate is immediately given as HK "[Re(XHX)]\Re(X*>)

(10)

where H indicates a complex conjugate transpose. It should be emphasized that the cost function is made of a summation over m frequencies in a range of frequencies of interest. In addition the covariance matrix of the estimates of H is computed as cov(HK )"E+(HK !H)(HK !H)H,"p[Re(X*X)]\,

(11)

where p is the equation error variance and it can be estimated on-line using 1 [(>!XHK )H(>!XHK )]. p( " (m!p)

(12)

Furthermore, the standard deviation of the estimation error for the lth unknown of the p parameters in H can be evaluated as the square root of the l, lth coe$cient (main-diagonal coe$cient) of the covariance matrix. This particular parameter allows an on-line assessment of the accuracy of the estimates of the parameter. After the overall description of the scheme the type of required on-line calculations should be described for an assessment of the computational e!ort. For a given frequency u , the discrete L

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fourier transform (FT) at the ith time step is related to the discrete FT at the (i-1)th time step as follows: X (u )"X (u )#x e\ SL G R, G L G\ L G where e\ SL G R"e\ SL Re\ SL G\ R.

(13)

(14)

Since the quantity e\ SL R is constant for a given u and a given *t, the on-line computation of L X (u ) requires a reasonably low computational e!ort. In addition, a very important attribute of G L this technique is that time-domain data from previous #ight maneuvers } containing `good informationa for PID purposes } can still be used by simply iterating the calculation of the FT. Thus, this discrete FT approach allows retaining all the PID results from previous time step and, at the same time, can provide the necessary #exibility to follow changes in the system dynamics. A potential problem of each on-line real time PID technique is a slow drift of the estimates at steady-state conditions. The simulation results from the next section will show that this technique is not sensitive to this problem. In general, this problem could be alleviated by the introduction of a switching logic turning o! the on-line PID (and therefore freezing the estimates at current levels) during extended periods of #ight at steady state conditions. In terms of frequency range, the m frequencies over which the cost function is evaluated can be selected as evenly spaced between u and u . Typically, the rigid body dynamics frequency    range for the considered aircraft can be selected allowing, therefore, "ltering out higher frequency noise and/or structural interference.

4. Aerodynamic modeling of actuator failure and/or battle damage As outlined in a previous section, the overall problem of fault tolerance following actuator failures has been approached from many di!erent points of view with a variety of control tools. In a worst-case scenario, an actuator failure and/or battle damage may imply a missing surface or, more realistically, a missing surface with the actuator jammed at a given position. Within most of the previous e!orts failures have been modeled as jammed actuators without missing surface; another approach to somewhat include the e!ect of the missing surface has been to introduce a robust controller accounting for minor/major changes in the post-failure stability characteristics of the aircraft. The approach in this paper emphasizes the importance of an accurate post-failure aerodynamic model for simulation purposes. From conventional aerodynamic modeling [24,25], the aerodynamic characteristics of a surface can be expressed in terms of normal force, axial force and moment around some "xed points or axes. For the purpose of this paper we will refer to a longitudinal failure only; this failure is believed to be more critical than aileron and rudder failures because of the unavoidable coupling between the longitudinal and lateral-directional dynamics. In general, a true assessment of the criticality of a failure involves additional considerations such as the availability of independent control channels for symmetric and asymmetric surfaces (stabilators/elevators and ailerons) as well as the variety of control surfaces and thrust control mechanisms available for the recon"guration process.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

27

A control surface damage with a missing portion induces instantaneous changes in its aerodynamic characteristics. A simplifying assumption in the aerodynamic modeling is that the axial forces exerted by control surface de#ections (in this case longitudinal surfaces) are negligible. Thus, the net e!ect of any control surface failure with a missing portion is a change (a reduction) in the relative normal force coe$cient. The aerodynamic moments around the di!erent axes can then be considered proportional to this normal force coe$cient through the aircraft geometric parameters. Therefore, to test the capabilities of the previously introduced PID scheme to evaluate on-line the post-failure mathematical model, the objective is to obtain closed-form expressions of the non-dimensional aerodynamic stability and control derivatives as function of the normal force coe$cient relative to the control surface object of failure and/or battle damage. Using conventional aerodynamic modeling these closed-form relationships can be obtained for &conventional' subsonic aerodynamic con"gurations. A more detailed e!ort involving higher level modeling (using wind tunnel analysis or the use of CFD codes) would be required for supersonic conditions. It should be emphasized that, without any loss of generality, the outlined analysis is performed at open-loop conditions. In fact, the main modeling issues remain unchanged in the presence of control laws for stability augmentations and autopilot systems. The described aerodynamic modeling was performed using the mathematical model of an F-4 aircraft [24] at subsonic #ight conditions. A state variables simulation code was built using linearized aerodynamics and non-linear dynamics. The #ight conditions, the trim conditions, and the inertial data for the F-4 aircraft are shown in Table 1 while Table 2 shows the values of the dimensionless stability and control derivatives at nominal conditions. The objective is to obtain closed-form expressions for the following derivatives: c ? , c ? , c ? , c ? , c O , c O * * * in terms of the c B of the left and right side of the longitudinal control surface, in this case * stabilators. A similar expression is also needed for the induced rolling moment, de"ned as . Using aerodynamic modeling for subsonic #ight conditions [24,25], it is known that *c " J $  Re S c . (15) c ? "c ? g & 1! * &S * Ra *?&






Also, for the stabilator we would have S c c B1 "c B1}0 #c B1}* "g & c ?& Nc ?& " *B1 . (16) * * * &S * * g S /S & & Therefore, breaking down the c B1 contribution from the left and right stabilator we would have the * following expression for c ? : * Re Re c ? "c ?5 # 1! c B1}0 # 1! c . (17) * * * Ra Ra *B1}*











Using aircraft data [25] a numerical value for the down-wash e!ect was found for the given #ight conditions. Using such value and knowing the value for c ? at nominal conditions it was possible to * solve for a value for c ?5 . As a check, assuming c ?5 c ?5 , a value for c ?5 was also found using an * * * *

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Table 1 F-4 aircraft general data (subsonic cruise condition) Flight conditions data Altitude"35,000 ft. Dyn. pressure"283.2 lb/ft Reference geometry Wing surface"530 ft Wing span"38.7 ft Inertial data Weight"39,000 lbs I "139,800 slug ft XX

Mach number" 0.9 CG location (chord fraction)"0.29

Airspeed"876 ft/s Angle of attack"2.63

Mean aerodynamic chord"16 ft

I "25,000 slug ft VV I "2200 slug ft VX

I "122,200 slug ft WW

Table 2 F-4 aircraft aerodynamic data (subsonic cruise condition) c  "0.26, c  "0.03, c 6 "0.03, c  "0.0, c 2 "0.0 * " 2

c  "0.0205, c S "0.027, c ? "0.3, c B1 "!0.1, c 6S "!0.064 " " " " 2 c  "0.1, c S "0.27, c ? "3.75, c ? "0.86, c O "1.8, c B1 "0.4 * * * * * * c  "0.025, c S "!0.117, c ? "!0.4, c ? "!1.3, c O "!2.7, c 2S "c 2? "0, c B1 "!0.58

K K

K c @ "!0.08, c N "!0.24, c P "0.07, c B "0.042, c B0 "0.006 J J J J J c @ "!0.68, c N "0.0, c P "0.0, c B "!0.016, c B0 "0.095 W W W W W c @ "0.125, c N "!0.036, c P "!0.27, c B "!0.001, c B0 "!0.066 L L L L L

empirical method [25]; the numerical results were within 3% of each other con"rming the goodness of the modeling. Therefore, numerical values were found for the c ? relationship above * leading to the expression: c ? "K #K c B1}0 #K c B1}* , (18) *   *  * where the numerical values for the K's coe$cients are reported in Table 3. A similar approach was used for the modeling of c ? . Starting from the following expression:

Re S c ? "c ?5 #c ?& "c ?5 (x !x 5 )!g & 1! (x !x )c a& (19)

* !% ! &S !% * Ra !&






"c B1 S/S g we would have * & & Re Re c ? "c ?5 ! 1! (x !x )c B1}0 ! 1! (x !x )c B1}* .

!% * !% * Ra !& Ra !&

and using c

*?&











(20)

Knowing x from the available data [24] and evaluating (x & !x ) from the aircraft geometry !% ! !% it was possible to solve for c ?5 starting from the nominal value for c ? . As a check this value was

compared with the value of c ?5 "c ?5 (x !x 5 ) with x 5 0.27; these two values for * !% ! !

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

29

Table 3 Post-failure aerodynamic modeling (stabilator failure) c ? NK "3.53, K "0.55, K "0.55 *    c ? NK "0.07, K "!1.17, K "!1.17

   c ? NK "0.093, K "1.917, K "1.917    * c ? NK "0.333, K "!4.083, K "!4.083   

c O NK "0.096, K "4.26, K "4.26    * c O NK "1.292, K "!9.982, K "!9.982

   *cl " N(y & /b/2)"0.023 $  

c ?5 were almost identical. Thus, numerical values were found for the c ? relationship above

leading to the expression (21) c ? "K #K c B1}0 #K c B1}* ,   *  *

where the numerical values for the K 's coe$cients are reported in Table 3. Next, the modeling of c ? was performed starting from the following expression: * S Re c ? "c ? 5 #c ? & "c ? 5 #2c ? (x & !x )g & . (22) * * * * * & ! !% & S Ra




Using c ? "c B1 S/S g we would have & & *& * Re Re c ? "c ? 5 #2(x & !x ) c #2(x & !x ) c * * ! !% Ra *B1}0 ! !% Ra *B1}*







(23)

with the values for (x & !x ), (Re/Ra) previously determined, the value for c ? 5 can be found !% * ! using the provided nominal value for c ? . As before, values were found for the following * c ? relationship: * (24) c ? "K #K c B1}0 #K c B1}* ,   *  * * where the numerical values for the K 's coe$cients are reported in Table 3. Next, the derivative c ? was analyzed. Starting from the expression K S Re (25) c ? "c ? 5 #c ? & "c ? 5 !2c ?& (x & !x )g &

* ! !% & S Ra




using c ? "c B1 S/S g we would have *& * & & Re Re c !2(x & !x ) c . c ? "c ? 5 !2(x & !x )

! !% Ra *B1}0 ! !% Ra *B1}*







The value for c ? 5 is found using the provided nominal c ? leading to

c ? "K #K c B1}0 #K c B1}* .   *  *

(26)

(27)

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

A very similar approach is used for the remaining longitudinal derivatives c O , c O . For c O , starting * * from S c O "c O5 #c O& "c O5 #2c ? (x & !x )g & * * * * * & ! !% & S "c O5 #2(x & !x )c B1}0 #2(x & !x )c B1}* . * ! !% * ! !% *

(28)

With the value for c O5 found using the nominal value for c O , the following expression is achieved: * * c O "K #K c B1}0 #K c B1}* .   *  * *

(29)

Similarly, for c O , starting from

S c O "c O5 #c O& "c O5 !2.2c ?& (x & !x )g &

* ! !% & S

"c O5 !2.2(x & !x )c B1}0 !2.2(x & !x )c B1}* .

! !% * ! !% *

(30)

With the value for c O5 found using the nominal value for c O , the following expression is

determined: c O "K #K c B1}0 #K c B1}* .   *  *

(31)

To complete the aerodynamic modeling a coe$cient is introduced to model the failure-induced rolling moment. Using the standard sign convention for the rolling moment and the stabilator de#ection, the following coe$cient is introduced: "!c B1}0 (y & /b/2)d } #c B1}* (y & /b/2)d } , *cl " * $  * A 10 A 1*

(32)

where y & represents the lateral location of the tail mean aerodynamic chord. A A state variable model of the F-4 dynamics at nominal #ight conditions was derived using

 a

Z ? X ? M ? 0

Z S X S M S 0

Z O X OO M O 1

0

0

p

0

r

UQ

u q

HQ bQ

ZH

0

0

0

0

XH

0

0

0

0

MH

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

> @ ¸ @ N @ 0

> N ¸ N N N 1

" > >U P ¸ 0 P N 0 P tg(H ) 0 ,



 a u q

H b p r

U

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Z 1*0 B X 1*0 B M 1*0 B 0

0

0

0

0

0

0

0

0

0

>  B ¸  B N  B 0

# 0 0 0

> 0 B ¸  B N  B 0 ,

 

d 1*0 d  ,

d 0

31

(33)

Similarly, with the given modeling for the stability derivatives, an additional state variable model of the F-4 dynamics at post-failure #ight conditions was derived using

 a

Z ? Z ? M ? 0

Z S Z S M S 0

Z O X O M O 1

0

0

p

0

r

UQ

u q

HQ bQ

ZH

0

0

0

0

XH

0

0

0

0

MH

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

> @ ¸ @ N @ 0

> N ¸ N N N 1

"

Z 1}* B X 1}* B M 1}0 B 0

Z 1}0 B X 1}0 B M 1}0 B 0

0

0

0

0

0

0

0

0

0

0

¸ 1}* B 0

¸ 1}0 B 0

0

0

>  B ¸  B N  B 0

#

> >U P > 0 P N 0 P tg(H ) 0 $ 

 a u q

H b p r

U

 d d

1*

10 . > 0 d B  ¸ 0 d $ B 0 N 0 B 0 $

(34)

It should be emphasized that at post-failure conditions the coe$cients of the "rst and second column of the control matrix will have di!erent values due to the loss of symmetry associated with a missing surface. This e!ect will lead to a coupling between longitudinal and lateral directional dynamics. Expressions for the coe$cients in the above state and control matrices in terms of conventional dimensional stability derivatives [24] are shown in Table 4.

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M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Table 4 Coe$cients of the state and control matrices at nominal conditions Z "(g sin c #Z )/(; !Z  ), Z "Z /(; !Z  ), Z "(; #Z )/(; !Z  ), ?  ?  ? S S  ? O  O  ? ZH "!g sin c /(; !Z  ), Z 1*0 "Z 1*0 /(; !Z  )   ? B B  ? X "g cos c #X , X "X , XH "!g cos c , X 1*0 "X 1*0  B B ?  ? S S M "M  Z #M , M "M  Z #M , M "M  Z #M ? ? ? ? S ? S S O ? S O MH "M  ZH , M 1*0 "M  Z 1*0 #M 1*0 ? B ? B B > "> /; , > "> /; , >"(> /; )!1, >U"g cos H /; , > ">  /; , > 0 "> 0 /; @ @  N N  P P    B B  B B  ¸ "(¸ #c N )/(1!c c ), ¸ "(¸ #c N )/(1!c c ), ¸ "(¸ #c N )/(1!c c ), @ @  @   N N  N   P P  P   ¸  "(¸  #c N  )/(1!c c ), ¸ 0 "(¸ 0 #c N 0 )/(1!c c ) B B  B   B B  B   N "(N #c ¸ )/(1!c c ), N "(N #c ¸ )/(1!c c ), N "(N #c ¸ )/(1!c c ), @ @  @   @ N  N   P P  P   N  "(N  #c ¸  )/(1!c c ), N 0 "(N 0 #c ¸ 0 )/(1!c c ) B  B   B B  B   B c "I /I , c "I /I  VX VV  VX XX

5. Parameter estimation at post-failure conditions One of the objectives of this study was to analyze the interface of the PID scheme with the given aerodynamic modeling of the post-failure conditions. In general, for a PID scheme to be successful within a fault tolerant #ight control system the following requirements must apply: E fast convergence time for the estimates of interest (in the range of a few sec., to avoid excessive deterioration of handling qualities and transition to unrecoverable #ight conditions); E low CPU and memory requirements for implementation on the on-board computer; E robustness to measurement and system noise; E amount of excitation required for an e!ective estimation; E availability of an on-line assessment of the reliability of the estimates. Therefore, the objective was to use the listed criteria to evaluate the PID method introduced above applied to a realistic simulation of an actuator failure and/or battle damage. This evaluation was conducted in three distinct and sequential studies. For each of these studies the availability of a failure detection, and identi"cation (FDI) scheme was assumed. Furthermore, it was assumed that the FDI scheme requires a 1.5 s computational time to clearly identify the failed/damaged control surface. The other assumption was the availability of independent commands to the left and right stabilators, and the availability of the angular de#ection data for all control surfaces. During the simulations the presence of noise was simulated by adding gaussian noise on the generic j-th state, except control surface de#ections [20], such that %noise}level[MAX!MIN] }   H , p }"   H 3

(35)

5, 10 and 20% noise levels were considered for our purposes. In the "rst study the application of the scheme to a failure simulation was conducted in the best-possible scenario for PID purposes with di!erent noise levels. A longitudinal simulation with on-line PID was started at nominal conditions with a doublet, a typical PID maneuver. After a few seconds during which the estimates converged to steady-state values a failure involving a stuck left

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

33

elevator at a mid range position with a 67% e!ectiveness reduction was simulated along with a second PID maneuver for the remaining healthy right stabilator. The idea was to obtain benchmark results for the on-line PID scheme in the best-possible scenario for PID purposes. The results of this study are shown in Figs. 2}10 for the case with a 5% noise level on the dynamic variables. Fig. 2 shows the stabilator maneuver with the same de#ections for both sides until the failure/damage occurrence followed by a stuck left stabilator. Fig. 3 shows instead the longitudinal states a and h while Fig. 4 shows the pitch rate and the failure induced rolling moment. The direct results of the on-line PID scheme are illustrated in Figs. 5}10. The failure is simulated to occur at t"15 s; following the failure it is assumed that an actuator FDI scheme is performing a detection and a positive identi"cation of the failed surface and that this process is performed within 1.5 s. From the outcome of the FDI process a logic scheme provides the PID scheme with the information that the left and right stabilators have to be considered separate outputs. Using this information the PID scheme modi"es the relative mathematical model by adding an additional control input; it is clear that the PID scheme at nominal conditions could not start using separate left and right stabilators since at nominal conditions these two control surfaces are completely correlated. Fig. 5 shows the on-line estimates of the parameter Z at nominal and post-failure conditions ? along with the on-line calculated standard deviation (p) of the estimation error for Z . The plot has ? been zoomed to highlight the failure/damage e!ects showing a slight decrease for Z at post-failure ? conditions. The PID scheme shows very desirable performance for the estimate of this critical

Fig. 2. Study C1 } time histories of longitudinal control surface de#ections.

34

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fig. 3. Study C1 } time histories of angle of attack and pitch angle.

Fig. 4. Study C1 } time histories of pitch and roll rates.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

35

Fig. 5. } Study C1 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise ? level along with$p of the estimation error.

Fig. 6. Study C1 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

36

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fig. 7. Study C1 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise ? level along with$p of the estimation error.

Fig. 8. Study C1 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

37

Fig. 9. Study C1 } time history of the estimates of parameters M 1*0 , M 1}0 , and M 1}* at nominal and post-failure B B B conditions with a 5% noise level along with$p of the estimation error.

Fig. 10. Study C2 } time histories of longitudinal control surface de#ections.

38

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Table 5 Study C1. Statistics of the PID estimates at nominal conditions for di!erent noise levels Param True value

Z ?

!53

Z S

!1.27e-4

Z O

0.99

0% Noise level

5% Noise level

Estimate (% error)

p ##

!0.5341 (0.02%)

7.46e-4

!0.5286 (1.04%)

1.94e-2

!33e-4 (4.1%)

3.67e-5

!1.2223e-4 (4.08%)

9.494e-4

0.9966 (0%)

2.45e-4

1.0001 (0.35%)

6.4e-3

Z 1 B

!0.056

X ?

27.24

X S

!0.012

0.0013 (110%)

M ?

!7.74

M S

Estimate (% error)

p ##

10% Noise level

20% Noise level

p ##

Estimate (% error)

p ##

!0.5740 (7.45%)

4.05e-2

!0.5057 (5.33%)

7.7e-2

!5.06e-4 (297%)

2.0e-3

!1.457e-4 (14.34%)

4.0e-3

1.0068 (1.03%)

1.33e-2

1.0069 (1.03%)

2.6e-2

Estimate) (% error

!0.0565 (0%)

1.1e-3

!0.0676 (19.64%)

0.0295

!0.0472 (16.46%)

6.13e-2

!0.1110 (96.46%)

0.118

27.1898 (0.19%)

2.070

27.3337 (0.33%)

2.1829

25.5487 (6.22%)

3.3358

27.7658 (1.92%)

3.899

0.1086

0.0074 (159.86%)

0.1144

0.0106 (186%)

0.1762

0.0165 (234.4%)

0.201

!7.7414 (0.06%)

0.0245

!7.7527 (0.21%)

6.69e-2

!7.7424 (0.08%)

0.1280

!7.7707 (0.44%)

0.264

!2.6e-3

!3.2e-3 (23.53%)

1.2e-3

!2.0e-3 (22.63%)

3.3e-3

!8.294e-4 (68.04%)

6.3e-3

2.1e-3 (181.9%)

1.3e-2

M O

!0.7173

!0.7135 (0.53%)

8.0e-3

!0.7063 (1.53%)

2.21e-2

!0.6604 (7.93%)

4.21e-2

!0.6835 (4.71%)

8.8e-2

M 1 B

!11.3853

!11.3337 (0.45%)

3.72e-2

!11.1710 (1.88%)

0.1016

!11.2016 (1.61%)

0.1934

!10.6869 (6.13%)

0.402

Note: p

##

indicates the standard deviation of the estimation error.

parameter at both nominal and post-failure conditions. In fact, it is capable of estimating the correct value with a small error; furthermore, the p of the estimation error converges at both nominal and post-failure conditions to a reasonably small value within 4}5 s. Fig. 5 also shows $p of the estimation error along with the estimate of Z to visualize the reliability of the estimate. ? A similar trend is shown in Fig. 6 relative to the parameter Z . Once again the plot has been O zoomed to improve the visualization of the e!ects of the failure/damage. Somehow the PID scheme overestimates Z at nominal #ight conditions and underestimates it at post-failure conditions. O However, the error is in both cases within a small range (approx. 1%). Furthermore the p values for the estimation error are in a low desirable range. An even more desirable trend for the results is shown in Figs. 7 and 8 relative to the parameters M and M , respectively. The "rst important remark is that the parameters experience substantial ? O deterioration (38 and 47% drops from their nominal values) at post-failure conditions. Nevertheless, the PID scheme is capable of estimating the true values for both parameters with similar accuracy at both nominal and post-failure conditions; in addition, the estimates can be considered reliable given the low values of the estimation error p for both parameters.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

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Table 6 Study C1. Statistics of the PID estimates at post-failure conditions for di!erent noise levels Param True value

0% Noise level

5% Noise level

10% Noise level p ##

Estimate (% error)

p ##

Estimate (% error) !0.5187 (1.01%)

3.86e-2

!0.5277 (0.72%)

4.5e-2

!0.5028 (4.03%)

7.82e-2

Estimate (% error)

p ##

20% Noise level Estimate (% error)

p ##

Z ?

!524

!0.523 (0.18%)

3.27e-2

Z S

!1.27e-4

2.15e-4 (269%)

2.907e-4

2.8e-4 (318%)

3.43e-4

3.57e-4 (380%)

4.03e-4

4.16e-4 (453%)

7.02e-4

Z O

0.99

1.35e-2

0.9915 (0.62%)

1.59e-2

0.999 (0.14%)

1.86e-2

0.9766 (144.24%)

3.2e-2

0.9954 (0.22%)

Z 1} B

!2.83e-2

!3.8e-2 (35.35%)

3.1e-2

!4.5e-2 (59.37%)

3.7e-2

Z 1}* B

!9.4e-3

!0.19 (1,918%)

0.21

!0.21 (2,165%)

0.24

!5.4e-2 (92.06%) !0.25 (2,577%)

4.39e-2 !6.9e-2 (96.46%)

7.59e-2

0.287

0.494

!0.2549 (2,605%)

X ?

27.24

X S

!0.012

!0.13 (939%)

0.20

!0.13 (939.4%)

0.20

!0.13 (941.8%)

0.20

!0.13 (934.7%)

0.202

M ?

!4.72

!4.74 (0.52%)

0.11

!4.73 (0.21%)

0.12

!4.71 (0.33%)

0.166

!4.68 (0.91%)

0.20

M S

!2.6e-3

!3.2e-3 (22.02%)

9.56e-4

!3.1e-3 (18.26%)

1.1e-3

!3.0e-3 (16.15%)

1.5e-3

2.8e-3 (7.45%)

M O

!0.38

!0.36 (5.30%)

4.4e-2

!0.383 (0.39%)

4.87e-2

!0.3291 (13.7%)

6.86e-2 !0.448 (17.52%)

8.26e-2

M 1}0 B

!5.70

!5.59 (1.87%)

0.105

!5.66 (0.63%)

0.115

!5.63 (1.14%)

0.162

!5.88 (3.17%)

0.196

M 1}* B

!1.899

!1.66 (12.47%)

0.68

!1.73 (8.84%)

0.75

!1.76 (7.48%)

1.059

!1.924 (13.6%)

1.275

¸ 1}0 B

!1.07

!0.925 (13.55%)

6.66e-2

!0.925 (13.56%)

6.74e-2

!0.928 (13.32%)

6.76e-2 !0.9244 (13.6%)

7.23e-2

!0.0243 (106.8%)

0.3129e-3 !0.0235 (106.59%)

0.317

!0.0287 (108.0%)

0.317

0.34

¸ 1}* B

0.357

26.88 (1.32%)

31.79

27.16 (0.29%)

31.79

25.87 (5.05%)

31.49

28.34 (4.03%)

!0.0211 (105.93%)

31.93

1.8e-3

In terms of estimation of the control derivatives, the most important result is shown in Fig. 9 relative to the parameters M 1 at nominal conditions, followed by the parameters M 1}* and B B M 1}0 at post-failure conditions. In the "rst part of the simulation (t(15 s) the PID scheme B provides a fairly accurate estimate of the parameter M 1 . As outlined above, the PID scheme at B post-failure conditions provides two di!erent parameters. An important remark is that the on-line estimate for M 1}0 is more accurate than the one for M 1}* ; this is due to the fact that the right B B stabilator is &healthy' and providing a substantial excitation while the left damaged stabilator is

40

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fig. 11. Study C2 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise ? level along with$p of the estimation error.

stuck at its position. All the estimates are reasonably accurate; additionally the values of the estimation error p are fairly low. The results for the "rst study at di!erent noise levels are summarized in Tables 5 and 6 relative to nominal and post-failure conditions, respectively. In the second study the application of the scheme to a failure simulation was conducted in the worst possible conditions for PID purposes, that is assuming that no signi"cant excitation of the aircraft dynamics, other than the excitation due to the failure/damage itself, was available for PID purposes. In other words these conditions replicate the case when no control surface de#ections on the remaining healthy control surfaces are commanded for the purpose of aiding the post-failure PID process. The results of this study are summarized in Figs. 10}16. Fig. 10 shows the de#ections of the both sides of the stabilators with the failure on the left side. The di$culty by the PID scheme to converge to reliable estimates in the absence of PID excitations, other than the excitation provided by the failure itself, is clearly shown in Figs. 11}16 for the estimates of Z , Z , M , M , and M 1}* respec? O ? O B tively. Note that estimations for M 1}0 are not available since this control surface is never de#ected B in this study. However, even if the estimates are not as accurate and reliable as in the previous study, the PID scheme still provides fairly accurate outputs for the main derivatives under the worst-possible conditions, that is in absence of any PID excitation. A complete summary of the statistical results for di!erent levels of noise is provided in Table 7.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

41

Fig. 12. Study C2 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

From the results of the second study the need is clear for some excitation at post failure/damage condition. This important issue has been discussed with some details in [26]; in fact, at post-failure conditions the de#ections of multiple control surfaces within some form of control allocation scheme for an over-actuated aircraft can substantially increase the complexity of the PID problem. As a solution to this problem two PID stages were formulated [26,27]; the "rst phase is for the estimates of the main derivatives without a speci"c PID maneuver while the second phase is for the estimates of the control derivatives of the individual surfaces } deployed for accommodation purposes } to be used in the on-line control allocation algorithm. In this paper a proposed solution to this problem is for the #ight control system itself to initiate, following a positive failure detection, identi"cation, and accommodation, a short duration preprogrammed speci"c PID maneuver. Clearly, this is not a trivial matter since starting a PID process following a failure/damage is a potentially very critical issue. Nevertheless, since the post-failure/damage handling qualities may be already seriously compromised, it can be understandable that taking an additional small hazard may lead to a successful failure accommodation following a successful PID allowed by some form of PID excitation. Therefore, a third scenario is envisioned where a short PID maneuver is simulated, featuring the remaining healthy stabilator, following a positive failure detection and identi"cation. In other words, assuming that the failure detection, and identi"cation (FDI) scheme in the #ight control system is capable of clearly identify

42

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fig. 13. Study C2 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise ? level along with$p of the estimation error.

Fig. 14. Study C2 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

43

Fig. 15. Study C2 } time history of the estimates of parameters M 1*0 , M 1}0 , and M 1}* at nominal and post-failure B B B conditions with a 5% noise level along with $p of the estimation error.

Fig. 16. Study C3 } time histories of longitudinal control surface de#ections.

44

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Table 7 Study C2. Statistics of the PID estimates at post-failure conditions for di!erent noise levels Param True value

0% Noise level Estimate (% error)

Z ?

!0.524

Z S

!1.27e-4

Z O

0.998

2.06 (493.49%) 1.7e-3 (1,452%) 1.45 (45.68%)

p ##

5% Noise level Estimate (% error)

10% Noise level p ##

2.06

1.24 (336.6%)

1.75

1.2e-3

1.0e-3 (896%)

1.1e-3

1.044

0.375 (62.46%)

Estimate (% error) 3.89e-2 (1074%)

p ##

20% Noise level Estimate (% error)

p ##

1.23

0.135 (125.7%)

1.106

2.77e-4 (316.9%)

9.4e-4

1.57e-4 (223.3%)

8.41e-4

0.72

!0.202 (120.2%)

0.578

!0.291 (129.1%)

0.347

Z 1} B

!9.4e-3

0.928 (9.947%)

0.863

0.447 (4.840%)

0.741

!0.181 (1.821%)

0.502

!2.48e-2 (163%)

0.609

Z 1}* B

!9.4e-3

0.928 (9.947%)

0.863

0.447 (4.840%)

0.741

!0.181 (1.821%)

0.502

!2.48e-2 (163%)

0.609

X ?

27.24

!278.17 403.3 (1.121%)

X S

!0.012

!0.466 (3.684%)

0.442

!0.45 (3.560%)

0.434

!0.453 (3.583%)

0.485

!0.324 (2.532%)

0.34

M ?

!4.72

!3.51 (25.74%)

0.933

!1.55 (67.2%)

1.85

0.657 (113.9%)

1.756

3.46 (173.1%)

3.53

M S

!2.6e-3

!1.7e-3 (33.16%)

5.4e-4 !6.98e-3 (73.2%)

1.1e-3

6.27e-3 (124.0%)

1.3e-3

1.5e-3 (158.0%)

2.7e-3

M O

!0.38

!0.17 (55.56%)

0.473

!0.128 (66.3%)

0.761

!0.029 (92.5%)

0.828

!0.229 (160%)

1.11

M 1}* B

!1.90

!1.46 (23.31%)

0.391

!0.715 (62.31%)

0.772

0.111 (105.8%)

0.725

1.49 (178.4%)

1.941

!1.1e-3 (100.3%)

3.7e-2 !1.1e-3 (100.3%)

3.73e-2

!1.3e-3 (100.4%)

3.73e-2 !1.2e-3 (100.3%)

0.373

¸ 1}* B

0.357

!261.13 (1.058%)

391.82

!247.96 418.08 (1.010 %)

!145.73 (634.9%)

300.97

the failed surface, a short pre-programmed PID maneuver for the `healthya stabilator side is executed. The only requirements for this `pre-programmeda PID maneuver is to be brief and to induce just enough excitation for PID purposes without further endangering the safety of the aircraft. Within this e!ort no speci"c studies for the selection of this maneuver were conducted; in fact the only condition was for this PID maneuver to induce a *g-response in the range of$0.5 g. Additional research should focus on how to design and pre-program these post-failure/damage PID maneuvers such that they provide a dynamic excitation with a clear and distinct signature with respect to the failure/damage dynamic excitation. The results of this study are summarized in Figs. 16}21. Fig. 16 shows the de#ections of the both sides of the stabilators with the failure on the left side and the post-failure PID maneuver for the right side. The improvements in the results, with respect to the previous study, for the estimates of Z , Z , S , M , M 1}* , and M 1}0 , are shown in B ? O ? O B

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

45

Fig. 17. Study C3 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise ? level along with $p of the estimation error.

Fig. 18. Study C3 } time history of the estimate of parameter Z at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

46

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Fig. 19. Study C3 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise ? level along with$p of the estimation error.

Fig. 20. Study C3 } time history of the estimate of parameter M at nominal and post-failure conditions with a 5% noise O level along with$p of the estimation error.

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

47

Fig. 21. Study C3 } time history of the estimates of parameters M 1*0 , M 1}0 , and M 1}* at nominal and post-failure B B B conditions with a 5% noise level along with $p of the estimation error.

Figs. 17}21. A complete summary of the statistical results for di!erent levels of noise is provided in Table 8. The clear conclusion for this study is that a short and small-magnitude post-failure PID phase could be extremely useful in allowing the on-line PID scheme to provide accurate and reliable estimates of all the principal stability and control derivatives at post-failure conditions.

6. Conclusions The study in this paper has shown the application of a recently developed frequency-based on-line PID technique to provide on-line estimates of the aircraft parameters following failure/ damage to a primary control surface. The performance of the PID process has been demonstrated through a simulation study featuring a detailed modeling of the post-failure conditions. The study has also emphasized the advantages of using &ad hoc' small and short pre-programmed PID maneuvers to help the PID process following the failure/damage occurrence. In addition to its accuracy the proposed scheme has the advantage of providing an on-line quanti"cation of the statistical reliability of its estimates; this particular feature would be very appealing for the purpose of interfacing this technique with a fault accommodation control logic.

48

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

Table 8 Study C3. Statistics of the PID estimates at post-failure conditions for di!erent noise levels Param True value

0% Noise level Estimate (% error)

Z ?

!0.524

Z S

!1.27e-4

Z O

0.998

Z 1} B

!2.83e-2

Z 1}* B

!9.4e-3

X ?

27.24

X S

!1.23e-2

M ?

!4.724

M S

!2.6e-3

M O

!0.381

M 1}0 B

!5.70

M 1}* B

!1.90

¸ 1}0 B

!1.07

¸ 1}* B

0.357

!0.511 (2.53%) 2.56e-4 (300.8%) 0.994 (0.4%) !7.52e-2 (166.1%)

5% Noise level p ## 0.119 3.22e-4 4.96e-2 0.108

!0.163 0.203 (1,629%) 12.17 97.83 (55.32%) !0.158 0.222 (1,187%) !4.71 5.46e-2 (0.15%) !2.4e-3 1.48e-4 (6.73%) !0.383 2.3e-2 (0.38%) !5.71 4.98e-2 (0.27%) !1.97 9.36e-2 (3.64%) !0.926 6.93e-2 (13.5%) 4.74e-5 9.7e-2 (100.0%)

Estimate (% error) !0.49 (7.09%) 3.67e-4 (387.9%) 1.01 (0.94%) !4.86e-2 (71.9%) !0.177 (1,781%) 12.55 (53.9%) !0.158 (1,183%) !4.65 (1.5%) !2.2e-3 (15.6%) !0.416 (9.1%) !5.54 (2.72%) !2.079 (9.5%) !0.924 13.6(%) 8.35e-4 (99.8%)

10% Noise level p ## 0.141 3.8e-4 6.0e-2 0.13

0.242

Estimate (% error) !0.53 (1.18%) 5.29e-4 (515%) 1.0 (0.39%) !6.61e-2 (133.9%)

!0.338 (3,484%) 98.37 11.02 (59.54%) 0.222 !0.161 (1,207%) 0.194 !4.54 (3.93%) 5.23e-4 !2.5e!3 (5.64%) 8.25e-2 !0.435 (14.18%) 0.176 !5.49 (4.3%) 0.331 !1.80 (4.94%) 7.0e-2 !0.927 (13.4%) 9.79e-2 3.1e-3 (99.1%)

20% Noise level p ##

Estimate (% error)

0.17

!0.59 (12.5%) 4.51e-4 9.21e-5 (172.2%) 7.35e-2 0.965 (3.24%) 0.157 !0.103 (265.9%) 0.293

!0.294 (3,022%) 100.57 !1.21 (104.4%) 0.224 !0.167 (1,257 %) 0.303 !4.44 (6.07%) 8.03e-4 !1.9e-3 (26.7%) 0.131 !0.307 (19.55%) 0.272 !6.07 (6.53%) 0.51 !2.20 (15.6%) 7.07e-2 !0.906 (15.6%) 9.91e-2 1.0e-3 (99.7%)

p ## 0.248 6.83e-4 0.103 0.226

0.43 94.52 0.222 0.67 1.8e-3 0.279 0.617 1.17 7.0e-2 9.82e-3

Acknowledgements Support for the "rst and third author has been provided by a grant from the Institute for Software Research (ISR), Fairmont, WV. Support for the second author has been provided by the Korea Science and Engineering Foundation Grant No. 1999-1-305-001-5.

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References [1] Napolitano MR, Naylor S, Neppach C, Casdorph V. On-line learning non-linear direct neurocontrollers for restructurable control systems. Journal of Guidance, Control and Dynamics 1995;18(1):170}6. [2] Monaco J, Ward D, Barron R, Bird R. Implementation and #ight test assessment of an adaptive, recon"gurable #ight control system. Proceedings of the 1997 AIAA Guidance Navigation and Control Conference, AIAA Paper 97-3738, New Orleans, LA, August 1997. [3] Brinker JS, Wise KA. Nonlinear simulation analysis of a tailless advanced "ghter aircraft recon"gurable #ight control law. Proceedings of the 1999 AIAA Guidance Navigation and Control Conference, AIAA Paper 99-4040, Portland, OR, August 1999. [4] Calise AJ, Lee S, Sharma M. Direct adpative recon"gurable control of a tailless "ghter aircraft. Proceedings of the 1998 AIAA Guidance Navigation and Control Conference, AIAA Paper 98-4108, Boston, MA, August 1998. [5] Totah J. Adaptive #ight control and on-line learning. Proceedings of the 1997 AIAA Guidance Navigation and Control Conference, AIAA Paper 97-3537, New Orleans, LA, August 1997. [6] Jorgensen CC. Direct adaptive aircraft control using dynamic cell structure neural networks. NASA TM 112198, May 1997. [7] Rysdyk R, Nardi F, Calise AJ. Robust adaptive nonlinear #ight control applications using neural networks. Proceedings of the 1999 American Control Conference, San Diego, CA, June 1999. [8] Rysdyk R, Calise, AJ. Fault tolerant #ight control via adaptive neural network augmentation. Proceedings of the 1998 AIAA Guidance Navigation and Control Conference, AIAA Paper 98-4483, Boston, MA, August 1998. [9] Napolitano MR, Chen CI, Naylor S. Aircraft failure detection and identi"cation using neural networks. AIAA Journal of Guidance, Control and Dynamics 1993;16(6):999}1009. [10] Napolitano MR, Younghwan A, Seanor B. A fault tolerant #ight control systems for sensor and actuator failures using neural networks, Aircraft Design 2000;3(2):103}28. [11] Ili! KW, Taylor LW. Determination of stability derivatives from #ight data using a Newton-Raphson minimization technique. NASA TN D-6579, 1972. [12] Ili! KW, Maine RE. Practical aspects of using a maximum likelihood estimation method to extract stability and control derivatives from #ight data. NASA TN D-8209, 1976. [13] Maine RE, Ili! KW. Identi"cation of dynamic systems: theory and formulation. NASA RF 1168, June 1986. [14] Mendel JM. Discrete techniques of parameter estimation. New York, NY: Marcel Dekker, 1973. [15] Ljung L. System identi"cation: theory for the user. Englewood Cli!s, NJ: Prentice-Hall, 1987. [16] Bodson M. An information-dependent data forgetting adaptive algorithm. Proceedings of the 1995 American Control Conference, Seattle, WA, June 1995. [17] Smith L, Chandler PR, Patcher M, Regularization techniques for real time identi"cation of aircraft parameters. Proceedings of the 1997 AIAA Guidance Navigation and Control Conference, AIAA Paper 97-3740, New Orleans, LA, August 1997. [18] Chandler P, Patcher M, Mears M. System identi"cation for adaptive and recon"gurable control. AIAA Journal of Guidance, Control and Dynamics 1995;18(3):516}24. [19] Gelb A, et al. Applied optimal estimation. Cambridge, MA: MIT Press, 1973. [20] Morelli EA. Real-time parameter estimation in the frequency domain. Proceedings of the 1999 AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 99-4043, Portland, OR, August 1999. [21] Morelli EA. In #ight system identi"cation. Proceedings of the 1998 AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 98-4261, Boston, MA, August 1998. [22] Morelli EA. High accuracy evaluation of the "nite Fourier transform using sampled data. NASA TM 110340, June 1997. [23] Klein V. Aircraft parameter estimation in frequency domain. Proceedings of the 1978 AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 78-1344, Palo Alto, CA, August 1978. [24] Roskam J. Airplane #ight dynamics and automatic #ight controls: Part I. Roskam Aviation and Engineering Coorporation, Ottawa, KA, 1994.

50

M.R. Napolitano et al. / Aircraft Design 4 (2001) 19}50

[25] Hoak DE, Finck RD. USAF stability and control DATCOM. AFWAL-TR-83-3048, October 1960, Revised 1978. [26] Patcher M, Chandler PR, Nelson E, Rasmussen S. Parameter estimation for over-actuated aircraft with non linear aerodynamics. Proceedings of the 1999 AIAA Guidance Navigation and Control Conference, AIAA Paper 99-4046, Portland, OR, August 1999. [27] Bu$ngton J, Chandler PR, Patcher M. Integration of on-line system identi"cation and optimization-based control allocation. Proceedings of the 1998 AIAA Guidance Navigation and Control Conference, AIAA Paper 98-4487, Boston, MA, August 1998.