Comparison of On-Line Parameter Estimation Techniques Within a Fault Tolerant Flight Control System By:

This paper describes the results of a study where two on-line parameter identification (PID) methods are compared for application within a fault tolerant flight control system. One of the PID techniques is time-domain based while the second is featured in the frequency domain. The time domain method was directly suitable for the on-line estimates of the dimensionless aircraft stability derivatives. The frequency domain method was modified from its original formulation to provide direct estimates of the stability derivatives. This effort was conducted within the research activities of the NASA IFCS F-15 program. The comparison is performed through dynamic simulations with a specific procedure to model the aircraft aerodynamics following the occurrence of a battle damage/failure on a primary control surface. The two PID methods show similar performance in terms of accuracy of the estimates, convergence time, and robustness to noise. However, the frequency domainbased method outperforms the time domain-based method in terms of computational requirements for on-line real time applications. The study has also emphasized the advantages of using “ad-hoc” short pre-programmed maneuvers to provide enough excitation following the occurrence of the actuator failure to allow the parameter estimation process. # Associate Professor, School of Aerospace and Mechanical Engineering, Hankuk Aviation University, 200-1 Hwajondong, Goyangshi, Kyonggido, 412-791,South Korea, AIAA Member ∗ Research Assistant Professor, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506/6106, AIAA Member + Professor, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506/6106 % Graduate Research Assistant, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506/6106, AIAA Member Symbols English c Aerodynamic coefficient K Numerical coefficient L Lift force q Pitch angular velocity, rad/sec S Wing surface, ft t Time, sec x Longitudinal axis y Lateral axis z Vertical axis Greek α Angle of attack, rad or deg ε Downwash angle, rad or deg η Dynamic pressure ratio θ Pitch Euler angle, rad or deg δ Control surface deflection, rad or deg σ Standard deviation ω Frequency, rad/sec Subscripts h Horizontal tail L Left side l Rolling moment M Pitching moment R Right side S Stabilator Acronyms AC Aerodynamic Center BLS Batch Least Squares CG Center of Gravity DTFT Discrete Time Fourier Transform EE Estimation Error EKF Extended Kalman Filtering FDI Failure Detection and Identification FTR Fourier Transform Regression FFT Finite Fourier Transform LS Least Squares LWR Locally Weighted Regression ML Maximum Likelihood PID Parameter Identification RLS Recursive Least Squares S&CD Stability and Control Derivatives Introduction Aircraft parameter estimation from flight data has been extensively conducted as a post flight analysis for several years. Several statistical methods have been used for this purpose, with the Maximum Likelihood method being one of the most widely used approaches [1,2,3]. In recent years, drastic increases in the available on-board computational power have allowed the flight control community to consider the application of on-line parameter estimation techniques. In particular, the on-line extension of the PID process has immediate and potentially very important applications for control of time varying aircraft systems, such as an aircraft subjected to substantial changes in the dynamic and aerodynamic characteristics. On a parallel path, research on fault tolerant flight control systems has been an important issue in flight controls. A fault tolerant flight control system is required to perform failure detection, identification, and accommodation following a battle damage and/or failure to a critical control surface. To implement a failure accommodation strategy, a variety of control surfaces (speed brakes, wing flaps, differential dihedral canards, spoilers, etc.) and thrust mechanisms (differential thrust, thrust vectoring) can be used [4]. Recent experimental research programs – the Self Designing Controller (SDC) program [5], the RESTORE program [6,7], and the IFCS F-15 program (previously known as the F-15 ACTIVE program) [8,9] – have proposed specific fault tolerant control laws formulated using on-line estimates of aircraft parameters obtained from a real time PID scheme. The research effort described in this study was conducted within the NASA IFCS F-15 program. A block diagram of the IFCS F-15 fault tolerant scheme is shown in Figure 1 [9]. The scheme features a set of Baseline Neural Networks acting as look-up tables for 26 stability and control derivatives (SCDs) of the IFCS F-15. At nominal flight conditions these 26 SCDs are provided as inputs to the IFCS F-15 controller featuring control laws designed through on-line solution of the Riccati equations. Following a failure on a primary control surface a PID scheme is tasked with providing on-line estimates of the same 26 SCDs to an On-Line Neural Network (OLNN). The OLNN’s task is to interpolate throughout the flight IFCS F-15 flight envelope the correction terms for the 26 SCDs which, added to the values from the Baseline Neural Networks, are provided as inputs to the controller. The current effort focuses on the selection of a suitable method for the on-line PID task. Like off-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 are used in lieu of techniques based on the use of the gradient and Hessian because of their lower computational effort and better convergence characteristics. Therefore, on-line time domain PID techniques mainly include variations of the LS regression method, such as Recursive Least Square (RLS) [10,11], RLS with a forgetting factor [12], a Modified Sequential Least Square (MSLS) [5], a real-time Batch Least Squares (BLS) [13,14], and Extended Kalman Filtering (EKF) [15]. The real-time application of any of these methods is challenging due to a combination of the unavoidable presence of system and measurement noise, possible lack of information for PID purposes in the flight data (such as a prolonged steady state flight 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 sets of flight data). A potential problem with the previous time-domain PID techniques is the lack of a reliable parameter for an on-line assessment of the accuracy of the estimates in the presence of unmodeled noise. Another less known problem for real time application of PID techniques is the presence of time skews and/or delays from the instant flight data are acquired by the flight control system and the instant the data are supplied to the PID algorithm. In trying to overcome some or all the problems described above, two techniques, one time-domain based and one frequency-domain based, have recently been introduced. The primary objective of this effort was to conduct a detailed comparison of the performance of these two PID techniques through simulations with a detailed aerodynamic modeling. Due to the realtime nature of the application of the PID process, the comparison was performed using the following parameters: convergence time for the parameters to be estimated; CPU requirements; robustness to measurement and system noise; amount of excitation required for an effective estimation; on-line computed variance of the estimation error. The paper is organized as follows. The next section briefly reviews a detailed approach of the aerodynamic modeling for an aircraft at post-failure and/or damage conditions. Two following sections review the two on-line PID methods. Then a section discusses the comparison of the two methods following control surface damages typical of a combat situation for a military aircraft under different scenarios. A final section summarizes the paper with conclusions. Aerodynamic Modeling of Actuator Failure and/or Battle Damage In a worst-case scenario, 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. Using a simplified modeling failure and/or battle failure can be modeled as jammed actuators without missing surface. In this effort emphasis has been placed in the accurate post-failure aerodynamic model for simulation purposes. From conventional aerodynamic modeling [16,17], the aerodynamic characteristics of a surface can be expressed in terms of normal force, axial force and moment force around some fixed points or axes. In this paper we will examine failures in the longitudinal axis only. This failure is believed to be more critical than aileron and rudder failures due to the coupling between the longitudinal and lateral-directional dynamics. A control surface damage with a missing portion induces instantaneous changes in its aerodynamic characteristics. An assumption in the aerodynamic modeling is that the axial forces exerted by a control surface deflection (in this case longitudinal surfaces) are negligible. Thus, the net effect of any control surface failure with a missing portion is a change (a reduction) in the relative normal force coefficient. The aerodynamic moments around the different axes can then be considered proportional to this normal force coefficient through the aircraft geometric parameters. Therefore, to test the capabilities of the PID schemes to evaluate on-line the postfailure mathematical model, the objective is to obtain closed-form expressions of the nondimensional aerodynamic stability and control derivatives as function of the normal force coefficient 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 configurations. A more detailed effort involving higher level modeling would be required for supersonic conditions. The described aerodynamic modeling was performed using the mathematical model for an F-4 aircraft [16] at subsonic flight conditions. Since a simulation code for the IFCS F-15 was not readily available, the F-4 model was selected because of general similarities with the F-15 model in the size of the control surfaces. A simulation code was built using linearized aerodynamics and non-linear dynamic equations. F-4 data are summarized in Table 1. For fault tolerance and PID purposes it is assumed that the F-4 has stabilators which can be decoupled as two independent surfaces at post failure conditions. The objective is to obtain closed-form expressions for the following derivatives: q q m L m L m L c c c c c c , , , , , α α α α ! ! in terms of the δ L c 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, defined as Failure l c ∆ . Using aerodynamic modeling for subsonic flight conditions [16,17], it is known that: H L H H WB L L c S S c c α α α α ε η       ∂ ∂ − + = 1 (1) Also, for the stabilator we would have: S S c c c S S c c c H H L H L H L H H L L L S L S R S S η η δ α α δ δ δ = ⇒ = + = _ _ (2) Therefore, breaking down the S L c δ contribution from the left and right stabilator we would have the following expression for α L c : L S R S L L WB L L c c c c _ _ 1 1 δ δ α α α ε α ε       ∂ ∂ − +       ∂ ∂ − + = (3) Using aircraft geometric data [17] a numerical value for the down-wash effect was found for the given flight conditions. Using such value and knowing the value for α L c at nominal conditions it was possible to solve for a value for WB L c α . Therefore, numerical values were found for the α L c relationship above leading to the expression: L S R S L L L c K c K K c _ _ 3 , 1 2 , 1 1 , 1 δ δ α + + = (4) where the numerical values for the K’s coefficients are reported in Table 2. A similar approach was used for the modeling of α m c . Starting from the following expression: H L CG AC H H AC CG WB L m m m c x x S S x x c c c c H WB H WB α α α α α α ε η ) ( 1 ) ( −       ∂ ∂ − − − = + = (5) and using H H L H L S S c c S η δ α = we would have: L S H R S H WB L CG AC L CG AC m m c x x c x x c c _ _ ) ( 1 ) ( 1 δ δ α α α ε α ε −       ∂ ∂ − − −       ∂ ∂ − − = (6) Knowing CG x from the available data [16] and evaluating ) ( CG AC x x H − from the aircraft geometry it was possible to solve for WB m c α starting from the nominal value for α m c . Thus, numerical values were found for the α m c relationship above leading to the expression: L S R S L L m c K c K K c _ _ 3 , 2 2 , 2 1 , 2 δ δ α + + = (7) where the numerical values for the K’s coefficients are reported in Table 2. Next the modeling of α! L c was performed starting from the following expression:       ∂ ∂ − + = + = α ε η α α α α α S S x x c c c c c H H CG AC H L L L L L H WB H WB ) ( 2 ! ! ! ! (8) Using H H L H L S S c c S η δ α = we would have: L S H R S H WB L CG AC L CG AC L L c x x c x x c c _ _ ) ( 2 ) ( 2 δ δ α α α ε α ε       ∂ ∂ − +       ∂ ∂ − + = ! ! (9) with the values for ( ) α ε ∂ ∂ − ), ( CG AC x x H previously determined, the value for WB L c α! can be found using the provided nominal value for α! L c . As before, values were found for the following α! L c relationship: L S R S L L L c K c K K c _ _ 3 , 3 2 , 3 1 , 3 δ δ α + + = ! (10) where the numerical values for the K’s coefficients are reported in Table 2. Next, the derivative α! m c was analyzed. Starting from the expression:       ∂ ∂ − − = + = α ε η α α α α α S S x x c c c c c H H CG AC H L m m m m H WB H WB 2 ) ( 2 ! ! ! ! (11) using H H L H L S S c c S η δ α = we would have: L S H R S H WB L CG AC L CG AC m m c x x c x x c c _ _ 2 2 ) ( 2 ) ( 2 δ δ α α α ε α ε       ∂ ∂ − −       ∂ ∂ − − = ! ! (12) The value for WB m c α! is found using the provided nominal α! m c leading to: L S R S L L m c K c K K c _ _ 3 , 4 2 , 4 1 , 4 δ δ α + + = ! (13) The same approach is used for the remaining longitudinal derivatives . , q q m L c c For q L c , starting