Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
FrC07.1
Robust Fault-Tolerant Flight Control using a New Failure Parameterization Jovan D. Boˇskovi´c, Joshua Redding and Raman K. Mehra Scientific Systems Company, Inc. 500 West Cummings Park, Suite 3000 Woburn, MA 01801 Email:
[email protected],
[email protected],
[email protected] Abstract— Most available approaches for adaptive accommodation of failures in flight control actuators result in a large number of parameters that need to be adjusted on-line. In this paper we propose a new failure parameterization that models a large class of failures in terms of a single parameter. The class of failures includes lock-in-place (LIP), float, hardover and loss of effectiveness (LOE). It is shown that the new parameterization accurately models this class of failures, and that the resulting model can be used for observer design to estimate the uncertain parameters on-line. The use of the resulting estimates in the adaptive reconfigurable control law at every instant is shown to result in a stable closed-loop system. The estimation and control algorithms are integrated within the θ-FLARE (θ-parameterized Fast on-Line Actuator Reconfiguration Enhancement) architecture. Properties of θFLARE and convergence properties of the failure parameter estimates are illustrated through simulations of F/A-18 aircraft dynamics under multiple flight-critical failures.
I. I NTRODUCTION In the last several years many different approaches have been proposed for accommodation of actuator failures in flight control [1], [2], [3], [5], [7], [15], [16], [17]. In most of the cases the failures are parameterized as parametric uncertainty, and either direct or indirect adaptive control techniques are used to assure the closed-loop stability. Indirect adaptive control strategies estimate A and B matrices on-line [1], [16], [17], while direct adaptive control adjust the controller parameters directly based on the response of the closed-loop system [15]. These approaches result in a large number of parameters that need to be adjusted online and complicate the V&V procedure for the resulting adaptive system. In addition, accurate failure identification is not feasible due to the lack of persistent excitation of commands commonly encountered in flight control. In [3], [5] a strategy is proposed that models each failure in terms of only two parameters σ and k. σ models lockin-place (LIP) failures, while k models loss of effectiveness (LOE). It was shown that, besides LIP and LOE, float and hard-over failures can also be modeled in this way. The resulting algorithms have been integrated with those for damage and disturbance compensation within the FLARE (Fast on-Line Actuator Reconfiguration Enhancement) architecture shown in Figure 1. It is seen that the FLARE system consists This research was supported by the NASA Langley Research Center under contract No. NNL06AA26P to Scientific Systems Company.
1-4244-0989-6/07/$25.00 ©2007 IEEE.
FLARE System
Global FDIR & Disturbance Estimation
Observer for Actuator 1 Observer for Actuator 2
Adaptive Retrofit Reconfigurable Controller
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Observer for Actuator N
Decentralized FDI
Baseline Control Signal
Add−on Signal
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Actuator Commands
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Baseline Flight Control System
Fig. 1. Structure of the FLARE (Fast on-Line Actuator Reconfiguration Enhancement) system
of local observers for each of the actuators, a global observer for damage and disturbances, and a retrofit adaptive reconfigurable controller. FLARE was developed, implemented and successfully tested through piloted simulation of the F/A-18 aircraft [6]. The FLARE system estimates σ and k on-line, and uses these estimates in the reconfigurable control law assuring overall system stability. However, even when each actuator failure is modeled using these two parameters, the number of parameters to be estimated is large. In the case of m control surfaces, the number of parameters to be estimated is 2m, where, for instance, m = 12 for F/A-18 aircraft. In this paper we propose a new failure parameterization that covers the same class of failures as the σ −k parameterization, but models the failures in terms of a single parameter θ. The resulting parameterization is referred to as the θparameterization, and the overall adaptive reconfigurable flight control system is referred to as the θ-FLARE. It will be shown that the use of suitably generated on-line estimates of θ in the reconfigurable control law results in a stable closedloop system. The adaptive reconfigurable control system with the new θ-parameterization (θ-FLARE) retains the favorable features
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FrC07.1 of FLARE, including the capability to deal effectively with both multiple failures and failure recoveries. In addition, it addresses a limitation of FLARE to distinguish between LIP and LOE, which is important for on-line diagnostics and prognostics. The results of simulations using an F/A-18 model are included to illustrate the performance obtained with the θFLARE system, and show that the convergence properties of the parameter estimates are substantially improved compared to the case of σ − k parameterization. II. P LANT DYNAMICS In this paper our focus is on a class of nonlinear models of aircraft dynamics, and linear actuator dynamics. The overall model is of the form: x˙ 1 x˙ 2
= x2 , = f (x) + g(x)u,
u˙ = −Λ(u − uc ),
(1) (2) (3)
where x ∈ IRn and u ∈ IRm denote respectively the state and control input vectors, uc ∈ IRm is the signal generated by the controller, u is a vector of actuator outputs, f (x) and g(x) are sufficiently smooth, and Λ = diag[λ1 λ2 ... λm ], where λi > 0. The above model corresponds to the case when the components of the vector x1 are Euler angles, while their derivatives, rather than the angular rates, are a part of vector x2 . Remaining components of x2 are the angle-of-attack, side-slip angle and total velocity. The transformation from the standard aircraft model to the above form is given previously [5]. In this paper the focus will be on aircraft models characterized by the following assumption: Assumption 1: (a) m > n; (b) g(x)g(x)T is invertible on a domain D; (c) Elements of g(x) are bounded for all x ∈ D. (d) Up to m − n effectors can undergo total LOE failure (e) All effectors can undergo partial LOE failure. (f) λi >> 1, i = 1, 2, ..., m. In many cases encountered in flight control, actuator dynamics are characterized by similar equations using λi ≥ 30. Hence, the above assumption is justified in such a case. III. C ONTROL O BJECTIVE
AND
BASELINE C ONTROLLER
In this section it will be shown that, in the case of fast actuator dynamics, the design of a baseline control strategy is straightforward. This strategy is based on model following as discussed below. Reference Model: The objective of the control design is to arrive at a strategy that assures that the states of the plant follow those of a reference model. The reference model is chosen in the form: x˙ ∗1 = x∗2 , x˙ ∗2 = Am x∗ + Bm r,
(4)
where x∗ is the state of the reference model, the matrix: 0 I A¯m = − − −− (5) Am
is asymptotically stable, and r is a vector of bounded piecewise continuous reference inputs. Control Objective: The objective is to design a control law uc (t) such that the error x(t) − x∗ (t) tends to zero asymptotically even in the presence of different control effector failures. Baseline Control Strategy: To achieve the control objective in the ideal case (i.e. the case without control effector failures), the following baseline control law, referred to as the Inverse Dynamics Control Law (IDCL), is chosen: uc = g(x)T (g(x)g(x)T )−1 η,
(6)
where η = −Ax + Am x + Bm r. It can be readily shown that, under assumption 1(f), the lower-order model of actuator dynamics is simply u = uc . Substituting this expression into (1) one obtains: x˙ 2 = f (x) + Buc ,
(7)
which describes the approximate lower order dynamics of the system consisting of the plant and the actuators. Upon substituting the control law (6) into the above equation, one obtains x ¨1 = Am x+Bm r. Hence the reducedorder dynamics of the closed-loop system coincides with that of the reference model (modulo initial conditions), and the IDCL achieves the objective in an approximate sense. In this paper the case of different actuator failures will be considered, and the way in which the controller should be designed will be discussed to assure the stability and desired performance of the overall system. IV. ACTUATOR FAILURE PARAMETERIZATIONS Flight control actuator failures can be broadly divided into two categories: (i) Failures that result in a total loss of effectiveness of the control effector; and (ii) Failures that cause partial loss of effectiveness. The former includes Lock-In-Place (LIP), float, and Hard-Over Failure (HOF), while the latter is referred to as the Loss-Of-Effectiveness (LOE) type of failure. Uncertainty associated with each of the actuator models is due to: (i) Unknown time of failure tF i ; (ii) Unknown LOE coefficient ki ; and (iii) Unknown value at which the control effector locks. In this section the focus is on first-order actuator dynamics. Failure Model: In the case of first-order actuator dynamics the failure model is of the form: u˙ = −ΛΣ(u − Kuc ),
(8)
where K = diag[k1 k2 ... km ] Σ = diag[σ1 σ2 ... σm ], and where ki ∈ [k , 1], k θ¯2 . Then |ϕ| = | ((θ¯1 −θ¯2δ()+ | ≤ 1. The same θ¯2 +δ)(θ¯2 +δ) ¯ ¯ can be easily shown for θ1 < θ2 . Theorem VI.1: Adaptive algorithms: ˙ θˆi ξi
δ θ2 + δ
δ θ1 + δ
···
δ ), θm + δ
(9) (10)
and 0 < δ 0, u˙ i ∼ = −λi (ui − θ¯i uci ) since θiδ+δ ∼ = 0. Hence the above model has the desired properties of covering both the LIP and LOE cases for a sufficiently small δ. VI. PARAMETER E STIMATION First-order actuator dynamics with failures is rewritten as: u˙ i = −λi ui + λi (θi uci +
δui ), i = 1, 2, ..., m, θi + δ
(11)
where λi >> 1. Observer: The adaptive observer is chosen as: δui u ˆ˙ i = −λi ui + λi (θˆi uci + ) − τi (ˆ ui − ui ) + ξi , (12) ˆ θi + δ where τi > 0 is the observer gain, and ξi is to be designed to assure the stability of the overall system. Error Model: Let φθi = θˆi −θi . Upon subtracting the failure model (11) from (12) one obtains: eˆ˙ i = −τi eˆi + λi (uci −
δui )φθi + ξi . ˆ (θi + δ)(θi + δ)
The main problem encountered here is that θˆi appears nonlinearly in the observer equation. While some results for nonlinear parameterizations are available [11], [12], this problem is highly dependent on the type of nonlinear parameterization. Here it will be shown that proper design of the signal ξi assures that the estimation error eˆi = u ˆ i − ui ∈ L∞ ∩ L2 , which is an important step in demonstrating the stability of the overall system. One intermediate result, discussed in the following assertion, is used in the proof. Assertion 1:
The following holds for all (θˆi , θi ) ∈ [0, 1]:
|ϕ(θˆi , θi )| = |
δ(θˆi − θi ) (θˆi + δ)(θi + δ)
| ≤ 1,
(13)
(14) (15)
where γi > 0, assure that eˆi ∈ L∞ ∩ L2 . In the above adaptive law, the Projection Algorithm (see e.g. [5]) is used to project the estimates θˆi (t) to the interval [0, 1] at every instant. Proof: The equation (13) is expressed in the form: eˆ˙ i = −τi eˆi + λi uci φθi −
where: ∆ = diag(
= Proj[0,1] {−γi λi uci eˆi }, θˆi (0) = 1, = −λi ui sign(ˆ ei ui ),
δui φθi (θˆi + δ)(θi + δ)
+ ξi .
Using the Assertion 1 and the definition of ξi , it follows that: eˆi eˆ˙ i = −τi eˆ2i + λi uci eˆi φθi −
λi δui eˆi φθi − λi |ui eˆi | ˆ (θi + δ)(θi + δ)
≤ −τi eˆ2i + λi uci φθi . Let ωθi = uci , and let a tentative Lyapunov function be: V (ˆ ei , φθi ) =
1 2 φ2θi [ˆ e + ]. 2 i γi
The following property of the adaptive algorithms with projection is used next (see e.g. [5]): if the adaptive law is of the form ζ˙ = Proj[−ζ, ¯ ζ] ¯ {−eω}, then ζ ζ˙ ≤ −eζω. Since ˙ θi is constant for t ≥ tF i , it follows that φ˙ θi = θˆi , which ˙ implies that φθi φθi ≤ −λi eˆi φθi ωθi . Now, using Assertion 1, the first derivative of V along the solutions of the system is: φθi φ˙ θi V˙ = −τi eˆ2i + λi φθi eˆi ωθi + ≤ −τi eˆ2i ≤ 0. γi Hence each eˆi is bounded (φθi is bounded by the use of the projection algorithm). Upon integrating V˙ from 0 to ∞, one R∞ 2 obtains: V (0) − V (∞) ≥ τi 0 eˆi (τ )dτ. Since the term on the left hand side is bounded, it follows that eˆi ∈ L∞ ∩ L2 . Remarks: • It is assured that the derivative of the Lyapunov function is negative semi-definite by the choice of ξi that cancels the effect of the uncertainty arising from the nonlinear parameterization of the problem. • Since the use of the sign function in ξi may lead to chattering of the signals in the system, the sign function can ζ be approximated by sign(ζ) ∼ = |ζ|+ , where > 1, from (11) one has:
Substituting the control law (20) and using (21) yields:
δui ∼ 1 u˙ i = −ui + θi uc + = 0, i = 1, 2, ..., m, (16) λi θi + δ resulting in u ∼ u. Upon substituting this expression = Θuc +∆¯ into (2) one obtains:
Let x = [xT1 xT2 ]T , ei = xi − x∗i , e = [eT1 eT2 ]T , and eˆ = [ˆ e1 eˆ2 ... eˆm ]T . Now, using (5) the overall closed-loop system can be expressed as: 0 M eˆ, e˙ = A¯m e + g(x)
x˙ 1 = x2 , x˙ 2 = f (x) + g(x)[Θuc + ∆¯ u].
(17)
Now the following assertion is considered: u i − ui ) ∼ Assertion 2: If θi ∈ [0, 1], then θiδ+δ (¯ = 0. δ ∼ Proof: When θi ∈ (0, 1], θi +δ = 0. When θi = 0, which happens at tF i , one has that ui (tF i ) = u ¯i . Based on Assertion 2, the plant equation is rewritten as: x˙ 1 = x2 , x˙ 2 = f (x) + g(x)[Θuc + ∆u].
(18)
The ideal reconfigurable controller is now chosen as: u = (g(x)Θ)T (g(x)Θ2 g(x)T )−1 (η − g(x)∆u),
ˆ T (g(x)Θ ˆ 2 g(x)T )−1 u = (g(x)Θ) m X δui ·(η − gi (x)[ + ui sign(ˆ ei ui )]). (20) ˆ θi + δ i=1 Theorem VII.1: If τi is of the same order of magnitude as λi , then the control law (20) for the plant (1)-(3), in which the parameter estimates are generated using (12), (14) and (15), assures that all the signals in the system are bounded and, in addition, that limt→∞ [x(t) − x∗ (t)] = 0. Proof: Since τi and λi are of the same order of magnitude, the error model (13) can be written as: τi δui 1 ˙ eˆi = − eˆi + (uci − )φθi −ui sign(ˆ e i ui ) ˆ λi λi (θi + δ)(θi + δ) ∼ = 0, from where one obtains: λi δui eˆi ∼ )φθi − ui sign(ˆ ei ui )]. (21) = [(uci − ˆ τi (θi + δ)(θi + δ)
x˙ 2
= f (x) + g(x)[Θuc + ∆u] m X δui = f (x) + gi (x)[θˆi uci + ˆi + δ θ i=1 δui φθi ]. −(uci − ˆ (θi + δ)(θi + δ)
m X
gi (x)
i=1
τi eˆi . λi
]. Since each eˆi has been where M = diag[ λτ11 λτ22 ... λτm m shown previously to satisfy eˆi ∈ L∞ ∩ L2 , and since Am is asymptotically stable, using the assumption 1(c) and BIBO stability arguments, it follows that e ∈ L∞ ∩ L2 . Since e = x − x∗ is bounded, and x∗ is a bounded signal, it follows that x is bounded as well. Bounded x implies that uc is bounded as well, which in turn implies that u is bounded. It now follows that each eˆ˙ i is bounded, and, from Barbalat’s lemma, limt→∞ eˆi (t) = 0. It can now be concluded that limt→∞ e(t) = 0. VIII. S IMULATIONS
(19)
where η = −f (x) + Am x + Bm r. Substituting (19) into (18) yields the equation of the reference model. Hence the ideal reconfigurable controller achieves the control objective. Adaptive Reconfigurable Controller: Now the adaptive reconfigurable controller is chosen in the form:
The plant equation (18) is now expressed as:
x˙ 2 = Am x + Bm r +
The proposed θ-FLARE system was implemented in simulation using the linearized dynamics of the F/A-18 aircraft during a 30o lateral doublet. The simulation consists of linear A and B matrices for a straight flight regime, and actuator dynamics and position and rate limits on the control effectors. The states of the model are: Total velocity V , pitch rate q, pitch angle θ, angle-of-attack α, altitude h, side-slip angle β, roll rate p, yaw rate r, roll angle φ, and yaw angle ψ. The control surfaces include: Left and right Leading-Edge Flaps (LEF); Left and right Trailing-Edge Flaps (TEF); Left and right Ailerons (AIL); Left and right Stabilators (STAB); and Left and right Rudders (RUD). Control inputs also include left and right engine (PLA). The following failure scenario was chosen: • All right-wing surfaces lock at t = 4 seconds. The resulting lateral σ matrix is Σ = diag[0 0 0 0 0 1], • All left-wing surfaces undergo Loss-of-Effectiveness (LOE) at t = 4 seconds. The resulting lateral K matrix is K = diag[0.35 0.65 0.01 0.6 0.8 1], • Right TEF and AIL recover from failure at t = 12 seconds. Simulations under the original FLARE system as well as with the proposed θ-FLARE system are included and compared. The use of the σ-k parameterization in the original FLARE system stabilizes the closed-loop and results in the performance comparable to that achieved with the baseline controller in the no-failure case. The resulting response of FLARE to the lateral doublet under failures is shown in Figures 2 and 3. It is important to note that, despite achieving the desired performance, the σ-k parameter estimates obtained using the FLARE system are far from their true values. This is due to the fact that there is not enough persistent excitation
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in the system. However, since the FLARE system is based on indirect adaptive control, it still results in the convergence of the tracking error to zero despite the lack of persistent excitation. The next set of simulations show the response obtained using the θ-FLARE system under identical flight commands and failure conditions. The resulting performance is shown in Figures 4 and 5. It is seen that the tracking performance is close to that in the case of the σ-k parameterization, with the exception that the θi parameters converge closer to their true values despite the limited persistent excitation from a single doublet. IX. C ONCLUSIONS In this paper we propose a new failure parameterization that models a large class of failures in terms of a single parameter. The class of failures includes lock-in-place (LIP), float, hard-over and loss of effectiveness (LOE). It is shown that the new parameterization accurately models this class of failures, and that the resulting model can be used for observer design to estimate the uncertain parameters on-line. The
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use of the resulting estimates in the adaptive reconfigurable control law at every instant is shown to result in a stable closed-loop system. The estimation and control algorithms are integrated within the θ-FLARE (θ-parameterized Fast onLine Actuator Reconfiguration Enhancement) architecture. Properties of θ-FLARE and convergence properties of the failure parameter estimates are illustrated through simulations of F/A-18 aircraft dynamics under multiple flightcritical failures. Future work in this area will focus on the use of the new θ parameterization for parameter estimator design for diagnostics and prognostics, and their integration with reconfigurable control within the FLARE architecture.
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R EFERENCES [1] M. Bodson and J. Groszkiewicz, “Multivariable Adaptive Algorithms for Reconfigurable Flight Control”, IEEE T-CST, 5(2), p. 217, 1997. [2] Boeing Phantom Works, ”Reconfigurable Systems for Tailless Fighter Aircraft - RESTORE (First Draft)”, Contract No. F33615-96-C-3612, System Design Report, No. A007, St. Louis, M0, May 1998.
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[8] J. D. Boˇskovi´c and R. K. Mehra, ”A Decentralized Scheme for Autonomous Compensation of Multiple Simultaneous Flight-Critical Failures”, in Proc. 2002 AIAA Guidance, Navigation and Control Conf., Monterey, CA, August 2002.
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[9] J. D. Boˇskovi´c and R. K. Mehra, ”Intelligent Adaptive Control of a Tailless Advanced Fighter Aircraft under Wing Damage”, AIAA J. of Guidance, Control & Dynamics, 23(5), 876-884, Sept. 2000.
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Fig. 4. k parameterization: Lateral state and actuator response using the θ-FLARE system in the case of failures
[3] J. D. Boˇskovi´c and R. K. Mehra, ”A Multiple Model-based Decentralized System for Accommodation of Failures in Second-Order Flight Control Actuators”, Proc. 2006 ACC, Minneapolis, MN, June 2006. [4] J. D. Boˇskovi´c, S. E. Bergstrom and R. K. Mehra, ”Adaptive Accommodation of Failures in Second-Order Flight Control Actuators with Measurable Rates”, Proc. 2005 ACC, Portland, OR, June 2005. [5] J. D. Boˇskovi´c and R. K. Mehra, ”Robust Fault-Tolerant Control Design for Aircraft Under State-Dependent Disturbances”, AIAA J. Guidance, Control & Dynamics, 28(5), 902-917, Sept. 2005. [6] J. D. Boˇskovi´c et al, ”Fast on-Line Actuator Reconfiguration Enabling (FLARE) System”, Proc. 2005 AIAA Guidance, Navigation and Control Conference, San Francisco, CA, August 15-18, 2005. [7] J. D. Boˇskovi´c and R. K. Mehra, ”A Multiple Model Adaptive Flight Control Scheme for Accommodation of Actuator Failures”, AIAA J. Guidance, Control & Dynamics, 25(4), pp. 712-724, 2002.
[10] J. D. Boˇskovi´c, S.-H. Yu, and R. K. Mehra, ”A Stable Scheme for Automatic Control Reconfiguration in the Presence of Actuator Failures”, Proc. American Control Conf., Philadelphia, PA, June 1998 [11] A. P. Loh, A. M. Annaswamy and F. P. Skanze, ”Adaptation in the Presence of a General Nonlinear Parameterization: An Error Model Approach”, IEEE T-AC, 44, 1634–1652, 1999. [12] J. D. Boˇskovi´c, ”Adaptive Control of a Class of NonlinearlyParametrized Plants”, IEEE T-AC, 43(7), 930–934, 1998. [13] J. Brinker and K. Wise, “Flight Testing of a Reconfigurable Flight Control Law on the X-36 Tailless Fighter Aircraft”, Proc.1998 AIAA GNC Conference, Denver, CO, August 2000. [14] J. Brinker and K. Wise, “Reconfigurable Flight Control of a Tailless Advanced Fighter Aircraft”, Proc. 1998 AIAA GNC Conference, Vol. 1, pp. 75-87, Boston, MA, August 1998. [15] A. Calise, S. Lee and M. Sharma, “Direct Adaptive Reconfigurable Control of a Tailless Fighter Aircraft”, Proc. 1998 AIAA GNC Conference, Vol. 1, pp. 88-97, Boston, MA, August 1998. [16] P. Chandler, M. Pachter and M. Mears, “System Identification for Adaptive and Reconfigurable Control”, Journal of Guidance, Control & Dynamics, Vol. 18, No. 3, pp. 516-524, May-June 1995. [17] J. D. Monaco et al, ”Implementation and Flight Test Assessment of an Adaptive, Reconfigurable Flight Control System”, Proc. AIAA GNC Conference, New Orleans, LA, August 1997. [18] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice Hall Inc., Englewood Cliffs, New Jersey, 1988.
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