Noncertainty Equivalent Adaptive Backstepping Control for Advanced Fighter Subject to Unsteady Effects and Input Constraints

This paper presents a noncertainty equivalent adaptive backstepping control scheme for advanced fighter attitude tracking, in which unsteady effects, parameter uncertainties, and input constraints are all considered which increase the design difficulty to a large extent. Based on unsteady attitude dynamics and the noncertainty equivalent principle, a new observer is first developed to reconstruct the immeasurable and time-varying unsteady states. Afterwards, the unsteady aerodynamics is compensated in the backstepping controller where the command filter is introduced to impose physical constraints on actuators. In order to further enhance the robustness, the noncertainty equivalent adaptive approach is again used to estimate the uncertain constant parameters. Moreover, stability of the closed-loop system that includes the state observer, parameter estimator, and backstepping controller is proven by the Lyapunov theorem in a unified architecture. Finally, simulation results show that performance of the deterministic control system can be captured when attractive manifolds are achieved. The effectiveness and robustness of the proposed control scheme are verified by the Herbst maneuver.


Introduction
Flight envelopes are substantially extended by the fourth generation fighter which directly leads to urgent demands for higher performance control laws. Since full envelope dynamics of aircraft is nonlinear and involves wide variations in aerodynamic parameters, traditional linear control methods are incapable of addressing such challenges [1]. Therefore, to ensure adequate stability and tracking performance in extreme flight regimes, a number of nonlinear control techniques have been extensively investigated, such as dynamic inversion [2,3], fuzzy logic [4], neural network [5], backstepping [6], and sliding mode control [7]. In particular, Lyapunov-based backstepping control is among the most widely studied of these methods. Due to the cascaded structure of aircraft dynamics, many efforts have been made to develop a flight control system via the combination of backstepping theory and other control technologies, such as disturbance observer [8], radial basis function neural network [9], and adaptive control method [10,11].
Generally, adaptive backstepping provides a systematic approach to solve the tracking or regulation problems of uncertain nonlinear systems. However, it must be noted that the adaptive backstepping control methods mentioned above have not considered input constraints which cannot be ignored in practice due to the existence of physical limits on actuators [12]. When aircraft maneuvers at a high angle of attack (AOA), drastic decreases of control authority often causes aerodynamic surface saturation which may severely degrade control performance by giving undesirable inaccuracy or leading to instability [13]. In some applications, this problem is crucial, especially in combination with online approximation-based control, which tends to be aggressive in seeking desired tracking performance [14]. For the purpose of circumventing this dilemma, it is necessary to take the input constraints into account at the level of the control design. In constraint adaptive control, the key issue is how to analyze the constraint effects on a closed-loop system. Toward this end, an auxiliary system of which states are used for feedback control is proposed in [15,16]. Recently, it was incorporated into backstepping architecture to solve trajectory tracking problems for reentry vehicles and airbreathing hypersonic vehicles in the presence of actuator constraints [17][18][19]. In addition, the command filter adaptive backstepping control proposed in [14,20] is also a wise choice to tackle input constraints, in which the command filter is used to calculate the derivatives of virtual controls and impose constraints on states and inputs. This control scheme is presented for the F-16/MATV fighter in [21], where tracking errors in adaptive laws are replaced by compensated errors; hence, the adaption process can be isolated, i.e., not affected by saturation of virtual controls or actuators.
Besides model uncertainties and input constraints, unsteady effects are also potential issues that should be considered for an advanced fighter. Unsteady aerodynamic states are immeasurable in practice and may change rapidly during maneuvers which pose additional challenges for control [22]. Due to the difficulties in unsteady aerodynamic modelling, unsteady effects are usually treated as a part of model uncertainties which increase robust requirements for control system design. Based on an unsteady aerodynamic model, an alternative way is to estimate the unsteady effects via state observation and then make compensation in the controller design. Generally, it is not difficult to merely handle state observation, but the complexity increases drastically when uncertainties and input constraints are all considered. In recent years, a novel approach for stabilization and adaptive control of uncertain nonlinear systems based on immersion and invariance (I&I) methodology has been proposed in [23] and then further developed in [24,25]. This method gives a noncertainty equivalent adaptive (NCEA) law which is different from the traditional certainty equivalent adaptive (CEA) method. The main feature of this approach lies in the construction of an estimator, which is a sum of a partial estimate generated by an update law and a judiciously chosen nonlinear function. The essential idea is to create a manifold in the extended space of states and parameters to which trajectories are attracted. As the trajectory evolves on this manifold, the closed-loop system captures the behavior of a deterministic system [26]. Due to the advantages in prescribing estimate error dynamics and separately synthesizing controller and estimator, this method shows great potential for uncertain nonlinear systems with complex structures. However, it relies on solving a partial differential equation (PDE), which is difficult for multivariable systems [27]. To overcome this difficulty, auxiliary state filter and regressor matrix filter are introduced [28][29][30]. Recently, its applications to the control of missiles, quad-rotor UAVs, and satellites are studied successively [26,31,32]. Apart from flight control, the I&I approach is also applied in aeroelastic control [33] and mechatronic systems [34].
In this research, a novel adaptive backstepping control scheme based on noncertainty equivalent principle is proposed for advanced fighter attitude tracking. Compared with our preceding study in [35], the prior hypothesis about the upper bounds of the uncertainties is removed, and the input constraints are dealt with in a simpler way. Moreover, the coupling relationship of the observer and estimator is considered fully in the closed-loop design. Therefore, the main contributions of this paper can be summarized as follows: (1) The I&I method is used to estimate not only immeasurable states but also unknown parameters; it is further modified with different forms in these two situations and is closely associated with the characteristics of an advanced fighter (2) The mutual effects of unsteady states and parameter uncertainties make the closed-loop control design complex. The observer and estimator are coupled together, but the coupling effects can be weakened by selecting appropriate parameters; in other words, the observer and estimator can be designed to satisfy the spectrum separation principle (3) The proposed control scheme involves a state observer, a constraint backstepping controller, and two parameter estimators which are designed separately, but the stability of the closed-loop system is proven in a unified architecture The rest of this paper is organized as follows. In the next section, unsteady attitude dynamics is modeled. The third section details the derivation of the state observer. In the fourth section, the constraint adaptive backstepping control design and stability analysis are presented. Simulations and conclusions are given in fifth and sixth sections, respectively.

Unsteady Attitude Dynamics
The model adopted in this research is a certain advanced fighter which is developed to investigate the flight dynamics during high AOA maneuvers. The fighter attitude dynamics is formulated as The aerodynamic model is extracted from wind tunnel tests which are conducted on sufficiently close points to capture the nonlinear behavior of the aerodynamics. Based on linear superposition principle, the aerodynamic forces and moments are expressed by where η Y , η Z , η l , η m , η n represent the unsteady states, and the expressions of C i , i = X, Y, Z, l, m, n, are given in [36]. Unsteady effects on the axial force are neglected since the axial force measured in the test is small in contrast to other aerodynamic forces [36].
Usually, the quasisteady model in which unsteady states are equal to zero can be employed to describe the aerodynamics in normal flight condition. However, owing to notable effects caused by separated and vortical flow, it becomes inadequate to describe the aircraft dynamics at high AOA.
Thus, unsteady aerodynamics are introduced and modeled in the following form [37,38]: where are fitting polynomials given in [36]. Note that the unsteady effects will converge to zero exponentially as the maneuver ends since (4) is stable.

Unsteady State Observation
The main objective of this section is to develop a nonlinear state observer which precisely reconstructs the unsteady states in the presence of parameter uncertainties. To this end, the attitude dynamics used for state observation and control design are rewritten as where σ = ½α, β T is the incidence angle, ω = ½p s , q s , r s T represents the stability axis angular rate, x = ½σ T , ω T T denotes the measurable state, G 1 = diag ½1,−1, θ 1 = ½ΔC X , ΔC Y , ΔC Z T and θ 2 = ½ΔC l , ΔC m , ΔC n T are constant parameter uncertainties, ω = ½q s , r s T represents the intermediate control variable, u = ½δ a , δ e , δ r , δ yl , δ yr , δ zl , δ zr T denotes the control input, and f 1 ðxÞ, f 2 ðxÞ, W 1 ðxÞ, W 2 ðxÞ, G 2 ðσÞ, D 1 ðσÞ, D 2 ðσÞ are vectors or matrices with appropriate dimensions (see appendix).
First of all, related assumptions used in the subsequent developments are given below.  3 International Journal of Aerospace Engineering Assumption 1. The desired trajectories α d , β d , p sd and their derivatives are bounded. The compact set is defined as follows: where ϖ is a known positive constant and j ⋅ j stands for the absolute value of a scalar.
with compact subset Ω 2 containing the origin, and k ⋅ kmeans the two norms of a vector or matrix.
To achieve precise reconstruction of unsteady states, a state filter is first introduced: where K 1 , K 2 are positive definite matrices to be designed, b η denotes the estimate of η, − ω are filter errors, and b θ 1 , b θ 2 are estimates of θ 1 , θ 2 , respectively. According to (5) and (7), dynamics of z 1 , z 2 take the form as The state observer based on I&I theory is proposed as follows: where φ 0 denotes the observer state, μ 0 ½z, WðxÞ is a smooth nonlinear function, Taking the time derivative ofη and using (8), we can get The nonlinear function μ 0 ½z, WðxÞ is selected so thatη has a stable behavior. So we choose with γ 0 > 0. Therefore, the nonlinear function is selected as Then, substituting (12) into (10), we obtain In (13), _ W i ðxÞ contains the immeasurable state η and uncertain parameters θ 1 , θ 2 which cannot be used directly. To overcome these difficulties, a command filter is employed to provide filtered derivative of W i ðxÞ; the filter dynamics is expressed by in which the initial condition satisfies W i ½xð0Þ = W i ½xð0Þ, _ W i ½xð0Þ = 0. Based on (13) and (14), the adaptation law is designed as Substituting (15) into (13), we can get According to Assumption 2, there exists a positive constant κ which depends on τ 21 , τ 31 , τ 22 , τ 32 , ς, ω n such that kΔ _ W 1 ðxÞk ≤ κ, kΔ _ W 2 ðxÞk ≤ κ [39].
Consider the following Lyapunov function candidate International Journal of Aerospace Engineering Using (8) and (16), the time derivative of V 1 yields Using Young's inequality, we can get where a 1 ∈ ð0, 1Þ, a 2 , a 3 > 0. Then, (18) is modified as with kWðxÞηk 2 = kW 1 ðxÞηk 2 + kW 2 ðxÞηk 2 and λ min ð⋅Þ is the minimum eigenvalue of matrix. Note if D 1 ðσÞ e θ 1 = D 2 ðσÞ e θ 2 = 0 and the following inequalities hold then _ V 1 ≤ 0, which implies that the state observation process is asymptotically stable. In the following parameter estimation, b θ 1 , b θ 2 will be determined to regulate D 1 ðσÞ e θ 1 = D 2 ðσÞ e θ 2 = 0.
Remark 3. The differences between the preceding studies [26,27,[29][30][31][32][33] and this research lie in the following two aspects: (1) The I&I approach adopted in this section is used to reconstruct the immeasurable and time-varying states (2) The filtered derivative of the regressor matrix, instead of filtered regressor matrix itself, is employed to construct the observer to avoid solving PDE

Constraint Adaptive Backstepping Control
In this section, a constraint adaptive backstepping controller based on I&I method is developed. The control objective is to track the predefined trajectories in the presence of input constraints and parameter uncertainties.
4.1. Constraint Backstepping Control. The constraint backstepping control design procedure is initiated by defining the following tracking errors: where σ r = ½α r , β r T is the reference trajectory generated by the desired command σ d = ½α d , β d T and ω r = ½q sr , r sr T is the virtual control law produced by ω d = ½q sd , r sd T . To eliminate the "explosion of term" problem, ω d is filtered via a command filter to provide ω r and _ ω r . The difference between ω r and ω d is evaluated by an auxiliary filter where A 1 > 0, G 1 = ð0 2×1 , G 1 Þ, ξ 1 is the filter state, and ξ 2 will be defined later. The compensated errors are defined by where e 2 = ðp s − p sr , e T 2 Þ T , and p sr denotes the reference trajectory of stability axis roll rate generated by p sd . Taking time derivative of e 1 and using (5), we obtain Therefore, the nominal virtual control law ω d is given by Combining (23), (24), (25), and (26), the dynamics of ε 1 takes the form as Similar to (26), the nominal control law u d for angular rate subsystem is designed as where G + 2 ðσÞ is pseudoinverse of G 2 ðσÞ, _ ω r = ð _ p sr , _ ω T r Þ T , _ p sr can be obtained by filtering p sd .
To impose physical constraints on actuators, u d is filtered via a command filter to provide the physical limited u. The command filter integrated with magnitude and rate limits is described as [14] 5 International Journal of Aerospace Engineering where S M ð⋅Þ, S R ð⋅Þ are magnitude and rate saturation function, respectively. The impact of (29) can be evaluated by where A 2 > 0. Combining (5), (24), (28), and (30), the dynamics of ε 2 is expressed by Then, the following Lyapunov function candidate is considered: Taking the time derivative of V 2 along (27) and (31) and using Young's inequality, we have where b 1 , b 2 , b 3 , b 4 > 0.

Uncertain Parameter
Estimation. Now the design of the parameter estimator is considered. This process is completed by deriving the nonlinear functions and adaptation laws for partial estimates. The estimator of θ 1 based on the I&I approach is constructed as where φ 1 is the partial estimate of θ 1 and μ 1 ½ε 1 , D 1 ðσÞ denotes the nonlinear function of ε 1 and D 1 ðσÞ. Taking the time derivative of b θ 1 and combining (27), we can get To ensure that e θ 1 has a stable behavior, the nonlinear function μ 1 ½ε 1 , D 1 ðσÞ is chosen as where γ 1 > 0. In view of (35) and (36), if φ 1 is updated by then substituting (37) into (35), we can get For angular rate subsystem, the estimator of θ 2 is given as follows with where γ 2 > 0. Combining (39), (40), and (41), the dynamics of e θ 2 can be expressed by Define the Lyapunov function candidate Differentiating V 3 and invoking (38) and (42), we have Remark 4. Assume that the state observer performs perfectly in reconstructing the unsteady states, thenη = 0 can be achieved. Hence, according to (38), the dynamics of e θ 1 can be treated as a linear time-varying system given by _ e θ 1 ðtÞ = −γ 1 D T 1 ½σðtÞD 1 ½σðtÞ e θ 1 ðtÞ. Note that if at some instant t 1 , e θ 1 ðt 1 Þ = 0, then e θ 1 ðtÞ = 0 for t ≥ t 1 . That is, the manifold defined by Ω 3 = fðx, η, tÞjφ 1 + μ 1 ½ε 1 , D 1 ðσÞ − θ 1 = 0g is an invariant manifold. Therefore, the parameter estimate φ 1 + μ 1 ½ε 1 , D 1 ðσÞ remains frozen at its actual value for t ≥ t 1 . This invariant characteristic also holds for e θ 2 dynamics. 6 International Journal of Aerospace Engineering Remark 5. Note that availability of _ α, _ β is assumed in the estimator design; otherwise, we will need to solve PDE. Although _ α, _ β cannot be measured directly, we could attempt to compute via velocity components and incidence angles. Remark 6. According to (38) and (42), we can find that the state reconstruction errorη may affect the estimator performance, that is to say, the observer and estimator are coupled together. To weaken the effects between each other, the design parameters of the observer and estimator need to be chosen appropriately. In practice, the reconstruction errorη should converge much more quickly than the estimate errors e θ 1 , e θ 2 to satisfy the spectrum separation principle.

Stability Analysis.
The stability analysis of closed-loop system is initiated by the following theorem.

Proof. The Lyapunov function is chosen as
To ensure stability of the closed-loop system, we select appropriate matrices A 1 , A 2 , K 1 , K 2 and adaptation gain γ 0 to satisfy the following inequalities: and inequalities (21), then _ V 4 ≤ 0 can be achieved. Thus, we can conclude that all closed-loop signals are bounded based on Assumptions 1 and 2. Further, because V 4 is lowerbounded and monotonic by the negative-semidefiniteness of _ V 4 , we know that Ð ∞ 0 _ V 4 ðtÞdt exists and is finite, which in turn implies Since WðxÞ, D 1 ðσÞ, D 2 ðσÞ and their derivatives are all bounded, according to Barbalat's lemma, it follows that Thus, WðxÞη,η, z 1 , z 2 , ε 1 , ε 2 , D 1 ðσÞ e θ 1 , D 2 ðσÞ e θ 2 can be guaranteed to converge to zero.
According to Theorem 7, WðxÞη, D 1 ðσÞ e θ 1 , D 2 ðσÞ e θ 2 asymptotically converges to zero. Thus, the manifold defined by with is an attractive manifold. Along any trajectories evolving on Ω 7 , the dynamics of ε 1 , ε 2 can be described by Apparently, (52) can be obtained if a controller based on attitude dynamics with measurable unsteady states and known aerodynamic parameters is implemented. Hence, which implies that ε 1 , ε 2 will converge to zero as expected.
Remark 8. For the closed-loop stability analysis, two scenarios are considered: (1) When actuator constraints are not in effect, note the fact that ω r − ω d and u − u d are bounded, then the auxiliary filter states ξ 1 , ξ 2 converge to a sufficient small domain around origin by selecting appropriate natural angular frequency ω n and control matrices A 1 , A 2 . Therefore, from (24), it is obvious that the tracking errors e 1 , e 2 can be guaranteed to be bounded. Consequently, the stability of the closedloop system can be ensured (2) When input saturation occurs, the primary control object is to maintain the stability of online approxi-mation process [14]. It is obvious that the tracking errors may increase while the compensated errors still converge to zero when saturation happens. Through replacing the tracking errors by compensated errors in the parameter estimation, the stability of the adaptation process can be ensured not affected by the saturation In conclusion, the proposed control scheme consists of three modules: unsteady states observation, constraint backstepping control, and parameter estimation. The overview of the design architecture is depicted in Figure 2.

Numerical Simulations and Discussions
This section presents several numerical simulations which are carried out to investigate the performance of the proposed control scheme. The comparisons of the NCEA and CEA laws are completed in the first simulation. The second simulation is performed to examine the effectiveness of the proposed control scheme via the Herbst maneuver. Both the simulations are conducted in the MATLAB/Simulink environment, where the step size is set to 5 ms. The parameters of the fighter are given in [36].  To illustrate the essence of the NCEA law, adaptive backstepping control approach proposed in [14] is adopted for comparison, the CEA laws are given as where γ 3 , γ 4 are adaptation gains to be assigned. Consider that the transient control performance is highly dependent on adaptation gains, four sets of gains with different adaptation rates are selected, that is, In order to compare the performance of different adaptive laws, the root-mean-square (RSME) of tracking errors and the energy consumption (EC) are chosen as the evaluation indexes which are defined as where the superscript i means the i th sampling point, n denotes the number of samples, t f denotes the final time of simulation, and L c , M c , N c are the commands of total control moment in each channel.
Simulation results are shown in Figures 3-6 and Table 2. It should be pointed out that the unsteady states are perfectly reconstructed with fast converge rates for both adaptive laws; therefore, the results are not presented herein. From Figures 3 and 4 and Table 2, it is apparent that the NCEA law outperforms the three CEA laws both in command tracking and energy consumption. The track performance of the

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International Journal of Aerospace Engineering CEA laws is sensitive to the adaptation gains. When compared with the case 1, the CEA law in case 3 achieves higher estimate precision and faster converging rate at the expense of sacrificing the energy consumption and transient control performance. It can be observed from Figure 5 that D 1 ðσÞ e θ 1 , D 2 ðσÞ e θ 2 are converged to zero; thus, the closed-loop trajectory eventually confines to the manifold Ω 7 , and the NCEA law recovers the performance of the deterministic backstepping controller. Figure 6 shows the time history of parameter estimation; it is obvious that the parameters estimated by CEA laws drift from their true values, especially in case 1. For the NCEA law, ΔC m can be precisely estimated while this is not the case for ΔC Z . The reason is that the rank of D 2 ðσÞ equals to e θ 2 , then D 2 ðσÞ e θ 2 = 0 means e θ 2 = 0; therefore, the invariant manifold is achieved and θ 2 can be estimated accurately. However, the rank of D 1 ðσÞ is less than e θ 1 ; hence, e θ 1 = 0 cannot be guaranteed via D 1 ðσÞ e θ 1 = 0. In addition, it can be seen that b θ 1 is frozen after t = 15 s at which instant the attractive manifold D 1 ðσÞ e θ 1 = 0 is achieved.

Robust Analysis under Parameter Uncertainties and Unsteady Effects.
To justify the effectiveness of the proposed control scheme, three different cases are considered for comparison. In the first case, the controller proposed in Section 4 is adopted in which the effects of unsteady aerodynamics and uncertain parameters are both compensated. For the second case, the effects of uncertain parameters are not considered in the controller design. The third case is the worst case in which both the unsteady effects and uncertain parameters are not compensated. The design parameters of the controller and initial condition are all kept the same as that in the previous subsection. The uncertainties of aerodynamic forces and moments in longitudinal and lateral channels are all considered in this simulation. Moreover, the Herbst maneuver is implemented to validate the control performance; the commands of α d , β d , p sd are defined in [35]. The commands tracked in these three cases are compared in Figures 7 and 8, and the control performance of the proposed adaptive backstepping controller in the first case is presented in Figures 9-14. In Figure 14, e 1i , ε 1i , ξ 1i , i = 1, 2, are 13 International Journal of Aerospace Engineering components of e 1 , ε 1 , ξ 1 , respectively; e 2i , ε 2i , ξ 2i , i = 1, 2, 3 are components of e 2 , ε 2 , ξ 2 , respectively. It can be observed from Figures 7 and 8 that the tracking performance of the first case outperforms the other two cases. It is apparent that smaller steady tracking errors and better transient tracking performances can be achieved since the effects of unsteady aerodynamics and uncertain parameters are both compensated. It also can be concluded that the unsteady aerodynamics seriously affects the aircraft dynamics and the compensation of unsteady effects could significantly improve the control performance. Note that the thrust force presented in Figure 9 increases drastically in the maneuver; it is quite rational for the pilot to increase the throttle to provide a larger thrust force when the fighter operates at high AOA. In addition, the aerodynamic surfaces are all set to zero when α > 55°a nd the attitude control was completed via thrust vector merely in this deep stall area. From Figures 10-12, it can be seen that the tracking errors increase while the compensated errors still converge to zero even though thrust vector nozzles are saturated. Moreover, the proposed observer is effective in state reconstruction with strong robustness against parameter uncertainties, especially in the lateral channel. Meanwhile,