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Nonlinear Discrete-Time Reconfigurable Flight Control Law Using Neural Networks Dong-Ho Shin and Youdan Kim, Member, IEEE
Abstract—A neural-network-based adaptive reconfigurable flight controller is presented for a class of discrete-time nonlinear systems. The objective of the controller is to make the angle of attack, sideslip angle, and bank angle follow a given desired trajectory in the presence of control surface damage and aerodynamic uncertainties. The adaptive discrete-time nonlinear controller is developed using the backstepping technique and feedback linearization. Feedforward multilayer neural networks (NNs) are augmented to guarantee consistent performance when the effectiveness of the control decreases due to control surface damage. NNs learn through the recursive weight update rules that are derived from the discrete-time version of Lyapunov control theory. The boundness property of the error states and NN weight estimation errors is also investigated by the discrete-time Lyapunov analysis. The effectiveness of the proposed control law is demonstrated by applying it to a nonlinear dynamic model of the high-performance aircraft. Index Terms—Backstepping, discrete-time nonlinear systems, feedback linearization, Lyapunov, neural networks (NNs), reconfigurable flight controller.
I. INTRODUCTION
C
ONVENTIONAL flight control systems utilize the gain scheduling technique that combines the linear controller designed for a number of equilibrium points in the flight envelope. Gain scheduling design requires tedious work, especially for a high-performance aircraft with intrinsic highly nonlinear dynamics [1]. Usually, the controller designed by either conventional gain scheduling technique or nonlinear control methodology such as feedback linearization, may fail to maintain good performance in the presence of various faults, such as control surface damage and actuator/sensor faults. These failures are characterized by the variations of aerodynamic coefficients and/or the change of system dynamics by unexpected nonlinear behaviors, which may jeopardize the stability of the closed-loop system. Therefore, a reconfigurable flight control system is required to deal with the possible failures and maintain acceptable performance.
Manuscript received August 12, 2004; revised November 3, 2005. Manuscript received in final form November 28, 2005. Recommended by Associate Editor P. K. Menon. This work was supported by the Ministry of Science and Technology through National Research Laboratory (NRL) programs, Republic of Korea, under Contract M1-0318-00-0028. D.-H. Shin was with the Department of Aerospace Engineering, The Institute of Advanced Aerospace Technology, Seoul National University, Seoul 151-742, Korea. He is now with Hyundai Motor Company and Kia Motors Corporation, Kyunggi-Do 446–716, Korea (e-mail:
[email protected]). Y. Kim is with the Department of Aerospace Engineering, The Institute of Advanced Aerospace Technology, Seoul National University, Seoul 151-742, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCST.2005.863662
During the last decade, there has been a significant amount of research on reconfigurable flight control system design. The research has included the adaptive model following method [2], sliding mode control with two-time separation assumption [3], and feedback linearization with online parameter identification [4]. More recently, neural networks (NNs) have emerged as a potential tool for identifying and controlling unknown system dynamics because NNs can approximate any complex nonlinear function to the desired accuracy [5]. Various successful applications of NN-based adaptive control in reconfigurable flight control systems can be found in [6]–[11]. However, these applications were developed based on continuous-time systems, which resulted in differential equations of the NN update rule. On the other hand, the discrete-time implementation of controllers is still important. There are two methods for designing the digital controller. One method, called emulation, is a method which designs a controller based on the continuous-time system, and then discretizing the controller. The other method is to design the discrete controllers directly based on the discrete system. We adopted the second approach to design the NN-based nonlinear controller. In contrast to the emulation method, the discrete controller is designed in a discrete domain so that the performance of the controller may not depend on the sampling rate. The NN-based nonlinear controller, in general, has a highly complicated structure compared to the conventional controllers due to its own complexity of NN structure. In the emulation method, the differential equations for updating NN weight should be discretized using the Euler or Runge–Kutta method, which increases the computation load. Therefore, sample rates should be sufficiently faster so that the digital controllers emulated by the Euler or Runge–Kutta method may match the performance of the continuous controller. Also, the NN-based controllers with lots of neurons require more computation time, which results in decreasing the sampling rate and, thus, the degradation of the performance of a discretized controller. On the other hand, in the discrete design method, the differential equations representing adaptive NN weight update rules are replaced by the difference equations; and the performance of the NN-based nonlinear controller is guaranteed irrelevant to the sampling period. In summary, the direct digital design using discrete models is meaningful especially for NN-based nonlinear controllers consisting of a lot of neurons. Another merit of direct digital design is that the upper bounds of the NN weight update rates guaranteeing the convergence can be estimated analytically while emulation method is otherwise. Discrete-time adaptive control design is much more complicated than the continuous-time adaptive control design since
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the discrete-time Lyapunov derivatives tend to have pure and coupling quadratic terms in the states and/or NN weights. Recently, Yeh and Kokotovic [12], [13] introduced the adaptive control of a class of single-input single-output (SISO) nonlinear discrete-time systems with linearly parameterized uncertainties. Jagannathan and Lewis [14] developed a discrete-time neural net adaptive controller. The NN structures used in [15] are confined to linearly parameterized feedforward NN (LPNN) for the convenience of analysis, and the considered discrete system is of the Brunovsky form. Ge et al. [16] presented adaptive NN control for a class of strict-feedback discrete-time SISO nonlinear systems by using higher order NN, which can be considered as a type of LPNNs. Note that they transformed a one-step ahead system description into (system dimension)-step ahead predictor description to overcome the causality contradiction. Lee [17] developed a direct adaptive NN controller for second-order discrete-time nonlinear systems. In this paper, an NN-based adaptive controller without offline training is presented for a class of discrete-time multi-input multi-output (MIMO) nonlinear systems, which are a more general description form compared with the systems considered in the previous works. The NN controller is developed by using the backstepping and feedback linearization techniques. The proposed controller avoids the causality contradiction problem encountered in the designing of the backstepping controller by managing the NN inputs appropriately. All the subsystems of the discrete-time nonlinear dynamics are assumed to have parametric uncertainties, as well as modeling error. NNs with a single hidden layer are augmented and adjusted adaptively to accommodate the control surface damage and parameter uncertainties. The NN weights are adjusted recursively using the stable online learning algorithm derived from discrete-time Lyapunov derivatives. It is also shown that the states of error dynamics and weight parameter estimation errors are bounded by using the discrete version of the LaSalle theorem [18]. This paper is outlined as follows. Section II deals with the preliminaries of the NNs and mathematical background. Section III is devoted to describing the system dynamics considered in this study, followed by a statement of the control objective. Section IV describes the control design procedure. Stability analysis for the boundness of error states and NN weight estimation errors are also performed. Section V validates the feasibility and performance of the proposed control law by numerical simulations on the nonlinear dynamic system of the high-performance aircraft. Finally, Section VI presents the concluding remarks. II. BACKGROUND In this section, the structures and the properties of NNs and stability notions required to show the boundness of NN weight estimation error, are introduced. A. NNs To approximate nonlinear continuous functions, an NN with a three-layer network structure represented by the following equation is considered: (1)
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is an NN input vector, is an where is a weight matrix beNN output vector, tween the hidden layer and the output layer, is a weight matrix between the input layer and the hidden layer, with a hidden-layer and . In this study, the following sigmoidal activation function activation function is adopted: (2) NNs can approximate any continuous functions to the desired accuracy over a compact set [19]. That is, for any given , there exists hidden layer neurons ( , , ) such that
in some input space.
(3)
Let and be the estimated values of the ideal weight values, , and , respectively, and let the weight deviation or and weight estimation errors be defined as . Then, the following equation can be easily obtained:
(4) where . It is assumed that the ideal weight matrices are bound as follows: (5) where
denotes the Frobenius norm, and with the trace
.
B. Stability Notation Consider the linear time-varying discrete-time system given by
(6) where , , and are appropriate dimensional constant matrices. be the state-transiLemma 1 [16]: Let for system (6), i.e., tion matrix corresponding to . If , , then system (6) is (i) globally exponen), and is (ii) tially stable for the unforced system (i.e., bounded-input bounded-output (BIBO) stable. III. SYSTEM DESCRIPTION Nonlinear dynamic equations of an aircraft in body axes for the flat earth, can be represented in the space-state form as follows: (7) (8) (9) (10)
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where
, , , , is the total velocity, and are the angle of attack and sideslip angle, respectively, , , and are the roll, pitch, and yaw rate about the body-fixed frame, respectively, , , and are the elevator, aileron, and rudder deflection angle, respectively, and , , and are the roll, pitch, and yaw angle, respectively. By discretizing (7)–(10) through Euler approximation [20], discrete-time nonlinear dynamics can be expressed as follows: (11) (12) (13) (14) where is a sampling period. To introduce an affine approximation of nonlinear aircraft dynamics, which is considered as nominal aircraft dynamics, the aerodynamic forces and moments are assumed to be nonlinearly parameterized with the angle of attack and side-slip angle, and linearly parameterized with the angular rates and the control
TABLE I CONTROL VARIABLES RANGES
surface deflections [21]. The six-degree-of-freedom nonlinear flight dynamics considered in this study are [11] (see (15)–(17) at the bottom of the page), where is the gravity accelerais the aircraft mass, is the air density, is the wing tion, area, is the mean aerodynamic chord length, is the span, are the moments of inertia. Equations and , (15)–(17) are the aircraft dynamics considered in designing the controller. Note that even though the velocity equation is not considered in deriving the controller, it is included in the numerical simulation to verify the performance of the proposed controller. The ranges of the control surface deflections are summarized in Table I.
(15)
(16)
(17)
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The resulting (15)–(17) can be written in a compact form as follows [11]:
(18) (19) (20)
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could be available two steps ahead. Two-step time delay of can be used for the implementation. In the next section, the discrete-time adaptive NN control design methodology will be proposed to satisfy the above control objective. IV. NN-BASED ADAPTIVE FLIGHT CONTROL SYTEM DESIGN A. Backstepping Controller Design
where , and represents a nominal function matrix. Furthermore, (18) can be approximated as follows:
Consider the dynamics of the error state,
subsystem. By defining the , we have
(21)
(26)
where denote approximated dynamics. Usually, the control surface deflections have relatively small effects on lift, drag, and side force. This property is used to design the controllers of fast and slow dynamics independently on the two-time separation principle. Similarly, we assume that the control surfaces do not produce any force that has a direct influence on and as in (21). Even with these assumptions, proposed NN-based adaptive discrete control laws using the backstepping approach will guarantee the stability of the overall system without separating the fast dynamics and the slow dynamics [1], [22], [23]. By discretizing the nominal aircraft dynamics, (19)–(21) via Euler approximation, the following discrete-time nonlinear equations can be obtained:
Using (12) and (22), the above equation can be rewritten as follows:
(27) where the term
is defined as
(28) The term may be estimated using the feed-forward NN with one hidden layer as follows:
(22)
(29) where
(23) (24) The aerodynamic coefficient model of nonlinear aircraft dynamics usually includes the nonlinear terms with control inputs, which result in nonaffine dynamic equations. In this study, the higher order terms of control input variables are neglected, and only the linear terms of control input variables are considered for approximated discrete-time nonlinear equations given by (23). The modeling errors caused by the assumption given by (22) and (23), will be compensated by NNs in the proposed control design methodology. In this study, the objective of the controller track the given deis to make the state variables trajectory, i.e., sired
with . Note that may be chosen adequately to make the order of NN inputs almost the same, which results in better estimation. Substituting (29) into (27) yields
(30) To apply the backstepping technique, should be considered as a virtual control. The stabilizing controller for subsystem with as an input would be chosen as the
for some specific constant (25) where is assumed to be available for two-step ahead, , . Remark 1: The proposed NN-based adaptive flight controller is developed under the assumption that the desired trajectories
(31) is a diagonal gain matrix with the magnitude of a where diagonal element less than one. Note that the estimated weight
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matrices are used in the above equation, since the ideal weight matrices are unknown. By defining the error variable and using (4) and (31) in (30), the closed-loop can be obtained as error dynamics of the slow subsystem
Define the control input
(38)
(32) where . From the definition of dynamics of -subsystem can be described by
, the error
as
is a diagonal gain matrix with the magnitude of diwhere is a correction term to agonal element less than one, and be designed later. By substituting (38) into (37) and using (4) in the resulting equation yield the closed-loop error dynamics of as the fast subsystem
(33) If the future values of a virtual control , i.e., , are known, then a controller can be designed by augmenting is a virtual control of the fuNNs. Unfortunately, ture, hence, this controller cannot be implemented in practice. This problem is known as causality contradiction, which is one of the major problems that are encountered in designing a backstepping controller for discrete-time systems [16]. In this study, the versatile mapping capability of NN is used to overcome the causality contradiction. defined by Consider the term
(39) where . To apply feedback linearization technique, the following assumption and Lemma are required. Assumption 1: The Frobenius norm of is upper-bounded by a known value for all time index . and are Lemma 2: invertible. Proof: See [10] for details. Fig. 1 shows the schematic block diagram of the proposed NN control system based on the discrete-time adaptive controller. B. Stability Analysis
(34) The term can be estimated using the feed-forward NN with one hidden layer as follows:
In this section, using discrete-time Lyapunov stability analysis, an NN weight-tuning algorithm is proposed to guarantee the boundness of error states and the NN weight estimation error. Let us consider the following Lyapunov candidate function:
(35) Note from (34) that the value of is unknown. Howis a ever, from (31) with (11)–(14), we can infer that , , , , , , and function of . Therefore, the inputs of second NNs representing the error term can be chosen as follows:
(40) The difference between two successive Lyapunov candidate functions is
(36) By choosing (36) as an NN input variable, the causality contrais diction can be avoided bacause the information of not required any more. In (36), is a given desired trajectory, for is always available. and, therefore, From (34) and (35), (33) can be rewritten as follows:
where
(41) (42)
(43)
(37)
(44)
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Fig. 1. NN-based adaptive controller architecture.
(45)
(49)
(46) Substituting (49) into (48) yields
(47) For the convenience of notations, the time index will be denotes the Euclidian norm when is a omitted. Also, vector, or a Frobenius norm when is a matrix. Substituting the error dynamics of slow subsystem (32) into (42) gives
(50) Let the error dynamics in NN weight
be taken as
(51) Using (51), the expression of
in (43) can be rewritten as
(48) is the maximum singular value of and . From Assumption 1 and the fact that and are bounded because the activation function is defined by (2), the following boundness conditions can be obtained: where
(52)
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where ,
,
where , .
, and Similarly, let us choose the error dynamics in NN weight
,
, and . Let the error dynamics in NN weight
be taken as
as
(58)
(53) Using (58), the expression of where (53) into (44), the expression of as
in (46) can be rewritten as
. Substituting in (44) can be expressed
(54) where ,
, , and
. Now, let us analyze the discrete-time Lyapunov derivatives of the fast subsystem by following the analysis procedure of the slow subsystem. Substituting the error dynamics of the fast subsystem (39) into (45), gives the following inequality equation:
(59) ,
where , ,
, and . Similar to the discrete-time Lyapunov derivatives analysis of the error dynamics in NN weight , choose the error dynamics of as
(60)
(55) where is the maximum singular value of . To eliminate the coupling terms between and from (50) and (52), the correction term in (55) is chosen as
where can be expressed as follows:
. Then, (47)
where (61) (56) ,
where Substituting (56) into (55) yields
,
, and
. The sum of (50), (52), (54), (57), (59), and (61) gives the following expression:
(57)
(62)
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where
(65.c) (65.d) and
(65.e) (65.f)
To obtain the condition under which is negative, the completion of squares for , , or ( ,2) is performed. The completing the squares with respect to error variyields ables except for
(66) where . One can easily show that as long as (65) is satisfied with Assumption 1 and the boundness property of sigmoidal activation function, and the in (66) is negative, which is guaranteed quadratic term of when Using and by the definition of , , and the coupling term in (62) can be bounded as follows:
, (67) Similarly, the bounds of other error variables guaranteeing can be found as follows:
(63) and, therefore,
(68) where (64) where each coefficient defined as follows, should be positive:
(65.a)
(65.b)
From (67) and (68), is guaranteed as long as (65) is satisfied and also, either (67) or any one of (68) holds. In other
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words, if any of state errors and weight errors are beyond the corresponding bounded value, the error variables go to some com. Consequently, it follows that the state pact set since tracking error and the errors in weight estimates are uniformly and ultimately bounded and will converge to the compact set de, where noted by , via discrete-time version of LaSalle theorem [18]. This completes the stability analysis.
right-hand sides of (72.a) and (72.b) be positive can be easily obtained as follows: or (73.a) or
C. Design Parameters
(73.b)
In this section, let us discuss how to choose the design parameters satisfying the conditions given by (65). If there are hidden-layer neurons and the maximum value of the each hidden-node output is taken as unity (as for sigmoidal activais given by [28] tion function), then the bound of (69) and To satisfy the conditions of (65), the adaptation gains should be chosen considering the value of . If adaptation gains are chosen inversely proportional to the maximum bound [conservatively given by (69)], then the adaptation gains of become smaller than necessary. This may degrade the performance of the NN weight update rate and compensation ability. This property has been studied in [27]. To overcome this major drawback, a projection algorithm was suggested [28]. Inspired by the projection algorithm, we choose the adaptation and as follows: gains (70) , , , and are positive constant values. Subwhere stituting (70) into (65.a), (65.b) with Assumption 1 yields the following equations:
(73.c) (73.d) and for a fixed sampling time It is easy to choose and are always smaller because than one. Note that and are positive small values. Remark 2: In error dynamics of NN weight estimates (51), (53), (58), and (60), considering the fact that ideal weights are , constant (i.e., in case of discrete version, ) and and are bounded by and , respectively, one can easily prove by using Lemma 1 and and are bounded in some compact sets without (73) that the requirement of persistent excitation (PE) condition. The -modification or -modification is usually adopted to avoid the PE condition in a continuous system [25], [26]. The proposed adaptation rules of NN weight, can be considered as a discrete version of -modification. By summarizing aforementioned stability analysis, the following theorem can be obtained. Theorem 1: For the discrete-time system given by (11)–(14), all of the error states of closed-system and the estimation errors of NN weights are uniformly and ultimately bounded and will converge to the compact set if the control law (38) is used with the virtual control (31) and the weight adaptation rules summarized as follows:
(71.a) (71.b) Also, the following conditions for from (65.c) and (65.d):
and
(74.a)
can be obtained
(72.a) (74.b) (72.b) It is a trivial task to choose design parameters such that the righthand sides of (71.a) and (71.b) are positive since the number of design parameters is more than that of inequality conditions and most terms in the right sides of (71.a) and (71.b) are multiplied by a second or fourth order term with respect to the samis pling time . Note from (71.b) that if the magnitude of ), the condition that the very large (roughly speaking, right-hand side of (71.b) be positive cannot be guaranteed, however, this is usually not the case in the aircraft dynamics. Also, the conditions satisfying (65.d) and (65.f) and guaranteeing the
(74.c)
(74.d) and if the conditions given by (65) hold.
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Remark 3: The degradation of the NN accuracy results in making the ultimate bounds increase. Many researches dealing with the design of the adaptive controller using NNs based on a continuous-time system have shown that the ultimate bounds could be made arbitrarily small by adjusting the design parameters of the controller. The previous works on the discrete-time control systems using NN have not shown that the ultimate bounds could be arbitrarily small since the controller design of nonlinear discrete-time system is very complicated and requires tedious work due to the coupling terms in the states and/or NN weights resulting from the stability analysis of the discrete-time Lyapunov derivatives. If the number of neurons in the hidden layer is sufficiently large, the bound of the NN accuracy can be made sufficiently smaller for a larger domain of inputs to the NN. If the initial and NN weight estimation errors values of states errors and are not too big, then the proposed boundness property of error variables are valid and meaningful. The latter assumption is valid since initial tracking errors can be made small by generating the smooth reference desired trajectory between the initial values of states and the command. Also, the NNs in this work approximate the errors between nominal functions and the functions composing of aircraft dynamics after control surface damage, which may not be changed drastically in comparison with nominal dynamics. To achieve the smaller bounds of the errors, in view of practical implementation, the closed-loop poles of tracking error dynamics given by (32) and (39) should be inside the unit circle [which are at least guaranteed by the conditions given in (65.a) and (65.b)] and near the origin. This can be achieved by adjusting the maximum singular and of NN weight values and , and the parameters estimation error dynamics in (51), (53), (58), and (60) considering that the definition of weight estimation errors be chosen as well. In addition, note from (65.c)–(f), and (68.b)–(e) that the are, the smaller smaller the design parameters the bounds of NN weight estimation errors are. V. NUMERICAL SIMULATION The proposed discrete-time adaptive NN controller is applied to an F-16 aircraft [29]. Simulation environments including the variations of aerodynamic parameters, control objective, and the scenario of control surface damage are adopted from [10]. Note that full nonlinearities of the other aerodynamic coefficients related to states and control inputs are considered and the effects were of control deflection on force coefficients included in the simulation. A. Nominal Simulation Let us consider the ideal case in which actuator dynamics, time delay, and sensor noise are not included in the simulation. To verify the performance of the proposed control law, the following command values of the angle of attack, sideslip angle, and bank angle were applied. This maneuver is composed of a pitch-up and a roll with zero sideslip angle s s s
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TABLE II AVERAGE AERODYNAMIC MODELING ERROR FOR SLOW SUBSYSTEM (%)
TABLE III AVERAGE AERODYNAMIC MODELING ERROR FOR FAST SUBSYSTEM (%)
The following third-order linear command filters were used to generate the smooth desired trajectories:
where rad/s, rad/s, and , respectively. The proposed controller is developed based on a discrete-time nonlinear system obtained by Euler approximation, which may cause approximation errors between the actual system and the approximated discrete-time system. The proposed discrete-type controller was applied to a continuous-time nonlinear aircraft system model to show the feasibility of the proposed controller. s is adopted. For discrete controller, sampling time Numerical studies were performed for three cases. For Case I, aerodynamic modeling error and control surface damage are considered in the simulation. The control surface damage is modeled by control effectiveness reduction of the elevator, aileron, and rudder of 50%, 50%, and 30%, respectively, and the time of fault occurrences was 5 s for all control channels. Aerodynamic modeling errors considered in slow dynamics and fast dynamics are summarized in Tables II and III. Average aerodynamic modeling error (AME) given in this study is defined as follows:
where is a sampling index, and simulation time s per time step s . One hundred neurons were assigned to a hidden layer for both first NNs used in the stabilizing function and the second NN in the controller.
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Fig. 2. Time histories of controlled states and control surface deflections: nominal simulation. (a) Controlled states (dash and dash-reference, solid-achieved). (b) Control variables.
The initial values of the elements in the NN weight matrices were chosen randomly between 0.0001 and 0.0001, and the control design parameters were chosen as follows:
For our particular aircraft model, numerical studies show that the upper bound defined in Assumption 1 is 1.9678 in the , , ranges of states, , and . By inserting the control design parameters and the upper bound into (71), (72), the right-hand sides of (71.a) and (71.b) have the values of 0.0968 and 0.0967, respectively. And both the numerators at right-hand sides of (72.a) and (72.b) become 0.75, which vali-
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date the condition that the right-hand sides of (71.a) and (71.b) and . Similarly, by substituting are positive with positive the control design parameters into (73), one can show that the control design parameters satisfy the inequality equations given by (73). By using this approach, appropriate control design parameters can be selected. For Case II, all simulation environments are identical to Case I and except for control design parameters chosen as . For Case III, aerodynamic modeling errors are not considered to examine the time histories of control surface deflections when actuation faults occur in a steady-state phase after pitch-up maneuver. Except for that, all simulation scenarios and control design parameters are the same as those in Case I. Numerical results are shown in Fig. 2. In Fig. 2(a), the solid line corresponds to the desired angle of attack, and the circle and solid line, the cross and solid line,andthe triangle and solid line indicate the response of the controlled states for each cases, respectively. The angle of attack, side-slip angle, and bank angle responses for Case II show that the tracking performance is better than Case I after the transient stage of adaptive learning. These results can be explained as follows. Considering the discrete closed-loop error dynamics of subsystems given by (32) and (39), if the maximum singular values of and , and are small, then the values will be located in the closer region of the origin of the unit circle, which results in good tracking performance and robustness. Note that more control efforts may be required for the case that the and are small. maximum singular values of Sideslip angle fluctuates but remains near zero during the period of high roll rate. The control histories are shown in Fig. 2(b). Although the simulation scenario is very strict, Fig. 2(b) demonstrates that the control surface deflections are in the range of the control surface. For Case I and Case II, numerical simulations show that the elevator deflection, which is the main control surface in the considered maneuvers, is almost unchanged after the fault during maneuver. It is not easy to analyze the effects of control surface damage by investigating the control input history because the simulation also includes the aerodynamic uncertainties. Numerical simulation is performed for the case that the same fault occurs without aerodynamic uncertainties to show the nature of elevator deflection after the fault. As shown in Case III of Fig. 2, the steady-state value of to . the elevator deflection is changed from To show the controller performance in the highly nonlinear regimes of high angle of attack, numerical simulation is also performed under high pitch-up commands of angle of attack, 35 and 40 . The control surface damage is considered by 50% control effectiveness decrease in elevator at 5 s. Fig. 3 shows that the proposed reconfigurable controller maintains relatively good tracking performance in spite that elevator surface becomes severely saturated due to a high maneuver command. However, numerical investigations show that when the actuator dynamics are considered, the performance deteriorates rapidly and eventually instability may occur when the damaged control surface is saturated during the maneuver. This is because the actuator cannot follow the high command input immediately. Therefore, to guarantee the closed-loop stability especially in the highly nonlinear regimes, actuator dynamics should be considered in designing the controller.
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Fig. 3. Time histories of controlled states and control surface deflections: high pitch-up maneuver.
B. Simulation With Time Delay In the presence of unmodeled dynamics and time delay, the performance of an adaptive controller can be degraded. To investigate the robustness property of the proposed controller, actuator dynamics and time delay effects are considered in the numerical simulation. Actuator dynamics are modeled for all actuation channels by the second-order linear dynamics as follows:
where rad/s and . For the time delay effects, command control input is replaced by with , and denotes the simulation time step s and s. (0.01 s). Two cases are considered: Control parameters for each case are the same as those in Case I (nominal simulation). Fig. 4 shows the simulation results. As shown in Fig. 4(a), the tracking performance is degraded and the time histories of the controlled states (especially for bank s than in case angle) are more fluctuated in case s. Numerical study shows that robustness to time delay of the proposed control law may be guaranteed in respect of s. tracking performance under about
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Fig. 4. Time histories of controlled states and control surface deflection: simulation with time delay (solid-h = 0:05 s, Dash and Dot-h = 0:1 s). (a) Controlled states (dash and dash-reference). (b) Control variables.
C. Simulation With Sensor Noise In this simulation, sensor noise is considered. Sensor noises are assumed as zero mean uncorrelated white noise. The state covariance matrix used in the simulation is as follows:
Control parameters are chosen the same as those in Case I (nominal simulation). Fig. 5 shows the simulation results. Numerical results show that the proposed controller exhibits satisfactory tracking performance in spite of the measurement noise. VI. CONCLUSION
where
.
An NN-based adaptive control law is proposed for a class of discrete-time MIMO nonlinear flight systems assuming all associated system dynamics have modeling errors and
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Fig. 5. Time histories of controlled states and control surface deflection: simulation with sensor noise. (a) Controlled states (dash and dash-reference, solid-achieved). (b) Control variables.
uncertainties. The conventional backstepping technique and feedback linearization with augmentation of NNs are employed to compensate for the effects of the modeling errors and parameter uncertainties in the presence of control surface damage. The noncausal problem encountered in backstepping controller for discrete-time systems has been solved by selecting the inputs of NNs properly. Numerical simulations are performed for an F-16 aircraft to show that the proposed reconfigurable flight control system succeeds in maintaining good performance even in the event of control surface damage. REFERENCES [1] S. A. Snell, D. F. Enns, and W. L. Garrard, “Nonlinear control of a supermaneuverable aircraft,” in Proc. AIAA Guid., Nav., Control Conf., 1989, pp. 519–531.
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Dong-Ho Shin was born in Seoul, Korea, on December 5, 1971. He received the B.S., M.S., and Ph.D. degrees in aerospace engineering from Seoul National University, Seoul, Korea, in 1995, 1997, and 2004, respectively. He is currently a Senior Research Engineer at Hyundai Motor Company and Kia Motors Corporation, Kyunggi-Do, Korea. His current research interests include nonlinear control using neural networks, sliding mode control, adaptive control and their application to flight systems, and ground vehicle systems.
Youdan Kim (M’94) received the B.S. and M.S. degrees in aeronautical engineering from Seoul National University, Seoul, Korea, in 1983 and 1985, respectively, and the Ph.D. degree in aerospace engineering from Texas A&M University, College Station, TX, in 1990. From 1990 to 1991, he was a Research Associate at Texas A&M University. In 1992, he joined the faculty of Seoul National University, where he is currently a Professor in the Department of Aerospace Engineering. His current research interests include control system design for aircraft and spacecraft, reconfigurable flight control system, missile guidance, trajectory optimization, and flexible structure control.
