An Adaptive Reset Control System for Flight Safety in the Presence of Actuator Anomalies

Transcription

1 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 WeB15.2 An Adaptive Reset Control System for Flight Safety in the Presence of Actuator Anomalies Megumi Matsutani and Anuradha M. Annaswamy Abstract This paper addresses the effect of actuator anomalies on flight control design, with particular emphasis on loss of effectiveness and actuator saturation. An adaptive controller with an integral action and a resetting strategy is proposed to address the actuator anomalies. We show that the stability of the closed-loop system can be guaranteed and that the controller allows a graceful degradation of the system performance in the presence of saturation. Simulations of a nonlinear transport aircraft-model are carried out to validate the proposed adaptive controller. The results show that the adaptive reset controller leads to a significantly improved performance compared to a non-adaptive controller. I. INTRODUCTION Almost all real world control systems are subject to constraints, the most common of which is actuator saturation. As a result of input constraints, the actual plant input may be different from the output of the controller. When this happens, the controller does not drive the plant properly and as a result, the states of the controller are incorrectly updated. This effect, termed controller windup, causes significant performance deterioration, large overshoots in the output and sometimes instability. Several methods have been proposed in the literature to overcome the controller windup problem, which can be broadly classified into two categories, and their origins can be found in [1 and [2, respectively. The methods in [1 is referred to as back calculation and tracking, where the integral is recomputed so that its new value gives an output which is after being treated with the saturation limit. Over the years, this idea has been systematized and led to socalled observer-based designs. These studies together with further generalizations and extensions are usually termed as anti-windup designs. On the other hand, the methods in [2 led to a sub-field referred to as reset control, where a relevant signal in a controller such as an error-integral state is reset to zero under a certain condition. Earlier papers such as [3, [4 on reset controllers addressed improved feedback performance by providing more flexibility in linear controllers. This approach has achieved renewed attention during recent years and has been addressed both from a theoretic and application point of view in [5-[7. This work was supported through the NRA NNX8AC62A of the IRAC project of NASA. M. Matsutani is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, 2139, USA. megumim@mit.edu A. M. Annaswamy is with the Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 2139, USA. aanna@mit.edu In any system to be controlled, there always exist uncertainties of several kinds, majority of which are due to modeling errors, actuator degradation, and other unforeseen changes in the dynamics. An adaptive controller is one that automatically changes the controller gains to maintain satisfactory performance and stability even with such uncertainties present in the system. Over the past three decades, adaptive control has been developed extensively and its main performance and robustness properties have been established [8-[11. The effect of actuator saturation in the adaptive system was treated in a rigorous manner for the first time in [11, where a modified model reference adaptive controller was proposed so that it ensures the stability of the system with parametric uncertainties and input saturation nonlinearities. While the results in [11 guarantee stability for any system with uncertainties and actuator saturation for initial conditions inside a bounded set, aspects of performance and methods for performance improvement were not discussed. The question that we address in this paper is the impact of an adaptive controller with resetting on stability and performance improvement. In this paper, we develop an adaptive reset control system that deals with parametric uncertainties and actuator saturation using a novel architecture that contains parameter adaptation and a resetting control strategy. Using a reference model, a modified error similar to that in [11, and the resetting strategy, it is shown that a stable adaptive controller can be designed. The simplicity of the underlying control design together with its adaptive and resetting feature has led to a successful implementation in several simulation studies [9, [12 using a high-fidelity nonlinear model of the Generic Transport Model and resulted in superior performance and therefore a significant improvement of flight safety against adverse conditions. A brief description of the results obtained is included in this paper for the sake of completeness. II. ADAPTIVE RESET CONTROLLER FOR A FIRST-ORDER PLANT In this section, we discuss the developments of an adaptive reset control system that deals with parametric uncertainties and actuator saturation by integrating a resetting strategy into the basic model reference adaptive controller. We begin with a simple problem, where a first-order system with a single input is considered, and is given by ẋ p (t) = a p x p (t)+b p λu(t) /1/$ AACC 138

2 where b p is known, a p and λ are unknown, but the sign of λ is assumed to be known. The input u(t) is additionally subject to the magnitude constraint u(t) u where u is unknown constant. We include an integral controller in order to achieve better tracking performance, and is of the form ẋ c (t) = b c (hx p (t) r(t)) where b c and h is known, and r(t) is a piecewise-continuous bounded function. We set h = 1 without any loss of generality. The overall plant to be controlled is obtained by combining the dynamics of two states x p,x c as [ẋp = ẋ c ẋ [ ap b c A An adaptive controller is chosen as u(t) = [ [ [ xp bp + λu+ r. x c b c x B 1 B 2 v(t) = k p x p +k c c { v(t) if v(t) u, u sgn(v(t)) if v(t) > u. where k p and k c are time varying. c is the outcome of the error integrator after treated with the virtual saturation limit. These saturation limits are given by R 1 (x p ) = max(, u k p x p, u k p x p ) (1) k c k c R 2 (x p ) = min(, u k p x p, u k p x p ) (2) k c k c and x c if R 2 (x p ) x c R 1 (x p ), c(t) = R 1 (x p ) if R 1 (x p ) x c, (3) R 2 (x p ) if R 2 (x p ) x c. R 1 and R 2 can be viewed as virtual saturation limits on the error integral states and are somewhat similar to the proportional bands defined in [13. A resetting action taken at time t ri is defined as x c (t + ri ) = c(t ri) (4) where t ri is the time instant at which (a) v(t ri ) u and (b) ẋ c (t ri ) =. These conditions imply that resetting occurs at an instant of which the sign of ẋ c changes simultaneously with actuator saturation. (See Fig. 1) The overall closed loop can be described as (5) ẋ = Ax+B 1 λ(kx+k c c+ u)+b 2 r (6) K = [ k p k c, u(t) = u(t) v(t) c(t) = c(t) x c (t) Fig. 1. Resetting Strategy A reference model for this closed-loop system can be constructed as [ [ [ [ẋpm ap +b = p Λkp b p Λkc xpm + r ẋ cm b c x cm b c ẋ m A m x m B 2 where k p,k c are ideal gains which ensure perfect tracking performance and are such that A m is Hurwitz. Defining the output error as e = x x m and K = K K, we obtain the error equation as ė = A m e+b 1 λ( Kx+k c c+ u) To remove the effect of u and c, we generate a signal e (t) as the output of a differential equation ė = A m e +B 1ˆλ(kc c+ u) For e u (t) = e(t) e (t), we obtain that ė u = A m e u +B 1 Λ Kx+B 1 λ(kc c+ u) where λ = λ ˆλ. This equation is in a standard error model form for which we can use the adaptive laws K = ΓB T 1 Pe u x T ˆλ = Γ λ (k c c+ u)b T 1 Pe u where Γ >,Γ λ >. This results in a Lyapunov function V = e T upe u + Tr( K T Γ 1 λsign(λ) K)+ λ T Γ 1 λ λ (7) since V = e T u(pa m + A T mp)e u = e T uqe u. Hence k p (t), k c (t), λ(t), and e u (t) are bounded t t. We now prove the boundedness of x. For efficiency of notation we define the following: q min = min(eig(q)),p min = min(eig(p)),p max = max(eig(p)) pmax ρ =,γ max = max(eig(γ),eig(γ λ )) p min 139

3 where P = [ p1 p 3 p 3 p 2 P B is defined by using induced norms, so that property is x T PB 1 λ P B x Also we pick some constants β,r so that they satisfy < β < q min K { < r 2 < min 1, q min β K } 2P pmax B q min β K 2P B r p max K max = 2P B +β From (7), we can deduce that Also we define, K max = max(sup K ) u β x max = q min 2P B K +2P B r pmin 2P B r p min x max x min = q min 2P B K max βk max β K r max = r pmin x max Theorem 1: The adaptive system has bounded solutions if Further, 1 (i) x(t ) < x max ρ, (ii) V(t ) < K 1 max, and γ max (iii) r(t) < r max x(t) < x max t t. Proof: Due to space limitations, the proof of this theorem is only roughly outlined here. The complete proof can be found in [14. Let W(x) = x T Px and define a level set, B, of W as { } B : x W(x) = p min x 2 max Now define the annulus region A as } A = {x x min x x max. (8) From condition (ii), it follows that K max < K max. It leads to From (8), Also from (8) and (9), ρx min < x max (9) x x max x B x min < 1 ρ x max x x B From the definition of A we conclude therefore that B A. The proof is completed in two steps. As the first step, it can be proven that Ẇ < x A when t {t ri } t. In the second step we show that any t {t ri }, W(x(t + ri )) < W(B). In the latter step, the following proposition is used. Proposition : When resetting occurs, the post state satisfies if W(x(t + ri )) < W(B) i) W(x(t ri )) < W(B), and ii) x p (t ri ) < r p min x max. Due to space limitations, the proof of this proposition is also omitted here. Condition (i) from Theorem 1 implies that W(x(t )) < W(B). (1) Therefore the results of steps 1 and 2 imply that proving the theorem. W(x(t)) < W(x(t )) t t Theorem 1 stated above shows that the boundedness of the closed-loop system is indeed guaranteed with the usage of c and u. In addition, the resetting strategy is structured in such a way that the use of u and c, which become nonzero when saturation begins, enable the system to return to an unsaturated state in a quicker and smoother manner. Fig. 2 shows the structure of the adaptive reset control system constructed in this section. As seen in the figure and (1)-(5), the resetting strategy denoted as R in the figure, treats both control signals from the baseline controller and the adaptive controller. To ensure stability, the modified error e u is calculated using the signals u and c and replaces the original error e in the adaptive laws. The similar idea of this modification can be found in [11. In the figure this strategy is represented as AW (anti-windup) block. Note that both u and c become nonzero and have the potential to lead to improved behavior when the control signal undergoes saturation. When these signals are zero, it can be seen that the system is reduced to the case without 131

4 The actuators saturate symmetrically in this fomulation, though asymmetric saturation limits can be treated straightforwardly with the same approach. Also, the system has error integral states to achieve command tracking, as ẋ c (t) = Hx p (t) r(t) Fig. 2. Adaptive Reset Control System any resetting strategy or anti-windup compensation. In other words, the system does not get altered until the actuator begins to saturate. This shows that this adaptive reset control system is not overly conservative. Remark 1 Condition (iii) of Theorem 1 implies that we have an additional constraint on the reference input. Its magnitude is upper-bounded by r max to ensure the boundedness of the states in the system, a single component of which may jump due to resetting. This constraint restricts the region in the state-space where resetting can occur, as a consequence of which we can prove that every time after resetting happens, the initial condition (1) is still satisfied. III. ADAPTIVE RESET CONTROLLER FOR HIGHER-ORDER SYSTEMS In this section, we derive the adaptive reset controller for higher dimensional systems. The focus is on an nth order system with an mth order input and an lth order reference command, which is given by ẋ p (t) = A p x p (t)+b p ΛE s (u(t)) where B p is known, A p and Λ are unknown, but the signs of elements of Λ is assumed to be known. The input u(t) is additionally subject to the magnitude constraint where the function E s (.) is an elliptical multi-dimensional saturation function defined by { u if u g(u) E s (u) = ū if u > g(u) where g(u) is give by ( m g(u) = i=1 [ êi u maxi 2 ) 1/2 ê = u u denotes the unit vector in the direction of u, u maxi is the absolute value of the saturation limit for the i th actuator, and ū is given by ū = êg(u) where H is known, and r(t) is a piecewise-continuous bounded function. Defining e i = HT i H i i = 1,...,m, we can assume that e T i e j = for i j and e T i e i = 1 without loss of generality. An adaptive controller which adopts the resetting strategy is chosen as u(t) = K p x p +K c c (11) where K p and K c are time varying. c is the outcome of the error integrator after treated with the virtual saturation limit. Specifically, R 1i (x p ) = max(, u i k pi x p k ci x c + x ci, u i k pi x p k ci x c + x ci ) R 2i (x p ) = min(, u i k pi x p k ci x c + x ci, u i k pi x p k ci x c + x ci ) where k p1 k c1 u 1 k p2 k c2 u 2 K p =..,K c =..,ū =..... k pm k cm u m and (12) (13) c(t) = [ T c 1 c 2... c m, where (14) x ci if R 2i (x p ) x ci R 1i (x p ), c i (t) = R 1i (x p ) if R 1i (x p ) x ci, (15) R 2i (x p ) if R 2i (x p ) x ci. In addition, resetting action is taken as x ci (t + rj ) = c i(t rj ) (16) where t rj is the time instant at which (a) ẋ ci (t rj ) =, (b) u i (t rj ) u i and (c) H x (t rj ) s 1, H s x p (t rj ) s 2. (17) H,x,H s are defined using the following; We assume that e 1 corresponds to the direction of the error integral state in which the resetting occurs at t = t rj, i.e. e T 1 x p = x ci. Let H s = [ e 2,...,e m. In order to define H, let us further define e i as e T i = [ e T i.... Starting with this e i s, we can find matrix E so that [e 1,...,e m,e spans the whole R n+m n+m space. Then orthonormal bases [e 1,...,e m 1,e m,...,e n+m can be achieved from [e 1,...,E 1311

5 by using Gram-Schmitt scheme. H R n n+m 1 can be defined as H = [ e 2,...,e n+m. Also, xs is the subset of the state x c from [ which x ci is excluded. Then x can be defined as x xp =. x s A reference model can be constructed by the following form [ẋpm [ Ap +B = p ΛKp B p ΛKc ẋ cm H ẋ m A m [ xpm x cm x m [ + I B 2 r (18) where K p,k c are ideal gains which ensure the perfect tracking performance. Since this is a reference model, A m is Hurwitz. As before, we define the output error as e = x x m and generate e (t) as ė = A m e +B 1 diag(ˆλ)(k c c+ u) (19) where λ is a vector, the terms of which are the estimates of the diagonal terms of the unknown matrix Λ. With e u (t) = e(t) e (t), the adaptive laws are K = ΓB T 1 Pe u x T ˆΛ = Γ λ diag(k c c+ u)b T 1 Pe u (2) where Γ >,Γ λ > and B T 1 = [ B T p... The stability of the resulting adaptive reset controller for higher-order plants can be shown in the same manner as for the first-order case, and is omitted here due to space limitations. Remark 2 To ensure the stability of the higher order system, an additional condition (17)(c) is necessary. This is added to ensure that the remaining states other than the one resetting are in an allowable region. IV. SIMULATION RESULTS Here we briefly present simulation results of the full nonlinear dynamic model of Generic Transport Model(GTM), which is a dynamically-scaled experimental aircraft developed by NASA Langley Research Center, that are obtained from the application of the suggested adaptive reset controller to flight control problem against adverse conditions. The details can be found in [12, [15. The overall model is given by where X = [V T Ẋ = F(X,ΛU) (21) α β p q r x y h φ θ ψ T which corresponds to true aerodynamic speed, angle of attack, sideslip angle, roll rate, pitch rate, yaw rate, longitude, latitude, altitude, and the three Euler angles, respectively. Also U = [δ elo δ eli δ ero δ eri δ al δ ar δ ru δ rl δ tl δ tr T denotes the left and right, outer and inner elevator inputs, left and right aileron inputs, lower and upper rudder inputs, left and right throttle inputs. Λ=diag [ λ elo λ eli λ ero λ eri λ al λ ar λ ru λ rl λ tl λ tr describes the control effectiveness in U, and is equal to identity under nominal conditions. For the purpose of control, Eq. (21) is linearized about a trim condition (X,U ) as LTI systems of the form where x p = A p x p +B p u+g(x p,u) x p = X X, u = U U A p = F(X,U) X, B p = F(X,U) U X, U X, U and g(x p,u) is higher order terms. For this linearized plant dynamics, we implement an adaptive reset controller as in (11)-(17). In this example, the aircraft is initially trimmed for a level flight with an aerodynamic speed 8 knots at altitude of ft. The simulation is conducted for 4 seconds except for the case in which the aircraft hits the ground. As a failure, we assume that center of gravity is moved backward for 5 % of the mean aerodynamic chord from its nominal location and the system is tested for a doublet-command in α. Figures 3(a), 3(b), 4(a), and 4(b) show the closed-loop responses corresponding to the baseline and adaptive controllers with resetting strategy. The commands chosen combined with the CG change are such that the elevators saturate fairly quickly, as seen in Fig. 3(b). The improved performance resulting from the adaptation is apparent. In Fig. 4 we also note that the adaptive reset controller suppresses the high frequency oscillations after a short transient at 1(sec). It should be noted that resetting occurs 3 times during the first 1 seconds. While the functioning elevators and throttle inputs corresponding to the baseline controller saturate repeatedly, those corresponding to the adaptive controller only do so during this transient. And it is during this transient that most of the adaptation takes place. Moreover, it was observed that due to the resetting strategy, the system leaves saturation earlier than the case without having the strategy. V. CONCLUSION In this paper, we proposed an adaptive reset control system that deals with parametric uncertainties and actuator saturation by using a novel architecture that integrates parameter adaptation and a resetting control strategy. The stability of the overall system, with appropriate resetting laws, was established in this paper. The results show that the adaptive reset controller achieves superior performance and leads to a significant improvement of the flight safety against adverse conditions. 1312

6 α q (deg/s) tas (knot) alt (ft) θ β p (deg/s) r (deg/s) φ ψ closed loop command α q (deg/s) tas (knot) alt (ft) θ β p (deg/s) r (deg/s) φ ψ closed loop command reference Model (a) States (a) States δ tl (%) δ ru δ al δ eli δ elo δ ero δ eri δ ar δ rl δ tr (%) δ elo δ eli δ al δ ru δ tl (%) δ ero δ eri δ ar δ rl δ tr (%) closed loop reference Model (b) Inputs (b) Inputs Fig. 3. Flight Performance with Nominal Controller with Resetting Strategy Fig. 4. Flight Performance with Adaptive Reset Controller VI. ACKNOWLEDGMENTS The authors would like to acknowledge Dr. Luis G. Crespo of National Institute of Aerospace for his support and encouragement. REFERENCES [1 Fertik, H.A., Ross, C.W., Direct digital control algorithm with anti-windup feature. ISA transactions. [2 Clegg, J.C., A nonlinear integrator for servomechanisms. Trans. A.I.E.E. [3 Krishnan, K.R., Horowitz, I.M., Synthesis of a non-linear feedback system with significant plant-ignorance for prescribed system tolerances Source. International Journal of Control. [4 Horowitz, I., Rosenbaum, P., Non-linear design for cost of feedback reduction in systems with large parameter uncertainty. International Journal of Control. [5 Beker, O., Hollot, C.V., and Chait Y., 21. Plant with an integrator: an example of reset control overcoming limitations of linear feedback. IEEE Transactions Automatic Control. [6 Beker, O., Hollot, C.V., Chait, Y., and Han. H., 24. Fundamental properties of reset control systems. Automatica. [7 Chait, Y., and Hollot, C.V., 22. On Horowitzs contributions to reset control. International Journal of Robotics and Nonlinear Control. [8 Ioannou, P. A. and Sun, J., Robust adpaptive control. Prentice Hall, Upper Saddle River, NJ. [9 Matsutani, M., Gibson, T., Jang, J., Crespo, L. G., Annaswamy, A. M., 29. An adaptive control technology for safety of a GTM-like aircraft. In Proceedings of the American Control Conference, St. Louis, MO. [1 Jang, J., Annaswamy, A. M., Lavretsky, E., 27. Adaptive flight control in the presence of multiple actuator anomalies. In Proceedings of the American Control Conference, New York, NY. [11 Karason, S. P., Annaswamy, A. M., Adaptive control in the presence of input constraints. IEEE Transactions on Automatic Control. [12 Crespo, L. G., Matsutani, M., Jang, J., Gibson, T., Annaswamy, A. M., 29. Design and verification of an adaptive controller for the Generic Transport Model. In Proceedings of AIAA Guidance, Navigation and Control Conference, Chicago, IL. [13 Astrom, K.J., Rundqwist, L., Integrator Windup and How to Avoid It. In Proceedings of the American Control Conference, Pittsburgh, PA. [14 Matsutani, M., Annaswamy, A. M., 29. An Adaptive Reset Control System for Flight Safety in the Presence of Actuator Anomalies. Technical report 93, Active-Adaptive Control Laboratory, MIT, September 29. [15 Matsutani, M., Crespo, L. G., Annaswamy, A. M., 21. Application of a Novel Adaptive Reset Controller to the GTM. In Proceedings of AIAA Guidance, Navigation and Control Conference, Toronto, Canada, to appear. 1313