# Robust Adaptive Neural Control of Morphing Aircraft with Prescribed Performance.

1. IntroductionMorphing aircraft has received considerable interest, since it possesses distinct advantages, which is capable of altering autonomously its aerodynamic configuration to obtain optimal flight performance, adapting different flight environments and high efficiency executing multiple types of missions [1]. As regards the control issue of morphing vehicle, the key point is to design a flight control system that has capability to guarantee the stability of the aircraft [2, 3]. The difficulty associated with the control design of such system arises from the fact that morphing aircraft manifests time-varying characteristic of aerodynamic forces, moments, and mass distribution as well as strong nonlinear nature [4, 5]. In the literature, several effective methods have been presented to tackle the control problem of folding-wing and swept-back wing aircraft. A multiloop control structure comprised of linear inner-loop and outer-loop controller is proposed for a kind of folding-wing aircraft [4]; similarly, the idea can also be found in [1] depending on gain self-scheduled [H.sub.[infinity]] technique. Subsequently, a finite-time boundedness control approach [2] and a switching linear parameter varying method [6] are investigated for swept-back wing aircraft, respectively. The common feature in [1, 2, 4, 6] is that the controllers are capable of ensuring the aircraft flight steady subject to the wing shape changes, but those designs are highly dependent on the precise prior knowledge of the dynamic model. However, the aerodynamic forces and moments are also quite difficult to model accurately. Moreover, general aircraft dynamics possess strong nonlinearities and uncertainties, which have necessitated the use of nonlinear control methods. Therefore, designing a nonlinear control method independent of prior knowledge of the aerodynamic model for the morphing aircraft is still an interesting yet challenging problem.

Adaptive back-stepping method has been widely studied in tracking control designs for nonlinear systems in strict-feedback or pure-feedback form, because it owns capability of systematically manipulating mismatched uncertainties [7-10]. Lately, some significant works regarding adaptive neural/fuzzy control for nonlinear systems with totally unknown or nonlinearly parameterized nonlinearities are investigated [11-13], and the noticeable problem of "explosion of items" is elegantly avoided by virtue of dynamic surface control or nonlinear differentiator technique [7, 14-16]. More specifically, adaptive neural/fuzzy approaches are also extensively utilized to the control problem of flight vehicles accompanied with aerodynamic parameters uncertainty in [17-20]. Although the previous adaptive neural control methods have the ability to guarantee the steady-state performance converging to a small residual set, few results consider the transient performance related to overshoot and convergence rate [21]. More recently, the prescribed performance control (PPC) scheme which denotes that the tracking error should converge to a predefined bound with convergence rate no less than a certain value has been an active research area [21-30]. Based on an error transformation technique that incorporates the desirable performance, Bechlioulis and Rovithakis firstly presented an adaptive PPC method for strict-feedback nonlinear systems [22]. Afterwards, an observer based fuzzy adaptive prescribed performance tracking approach is investigated for nonlinear stochastic systems subject to input saturation [26]. Subsequently, the PPC approach has been extended to deal with MIMO state and output feedback nonlinear control problems [27-30]. Moreover, this tool has also been successfully applied to flight control area [24, 31].

Despite the recent progress in the neural networks control of unknown nonlinear systems, certain issues still remain open. In practice, even though the stability analysis of aforementioned adaptive neural control schemes is proven, it relies on the condition that the approximation ability of NN must be effective all the time (i.e., the NN should be permanently working in the neural active region). Therefore, the deterioration of the tracking performance or even instability may happen provided that the transient states overstep the neural active region. Additionally, such a condition is also difficult to verify beforehand in real applications [32]. This difficulty can be naturally eliminated in case the invariance of the neural active region is guaranteed. In [33], by introducing n error transformation functions into the back-stepping design, a priori guaranteed evolution within the NN approximation set methodology is proposed to conquer this issue. However, the method developed in [33] constitutes an overparameterized solution which is hard to implement in practical application. Another alternative approach is designing a smooth switching function, which has the capability of switching the normal neural controller to a robust controller for pulling back the escaped transient into the neural effective regions. Several smooth switching function based neural adaptive back-stepping control schemes are proposed to solve aforementioned problem, where each virtual controller contains a neural controller and a robust controller effectively working inside or outside the active region, respectively [32, 34-37]. However, numerous adaptive parameters still need to be updated online and the transient performance problem is also omitted in those methods.

Motivated by the aforementioned discussion, a switching strategy based composite adaptive neural control scheme is proposed for a swept-wing morphing aircraft. NNs are employed to approximate unknown functions; thus a priori knowledge of the aerodynamic parameters is not necessary. It is proven that all the signals in the closed-loop systems are bounded. The main contributions of this work are shown as follows:

(1) Different from the works [32, 34-37] which completely neglect the transient performance related to overshoot and convergence rate, by introducing an error transformation, the proposed controller is capable of allowing attributes such as a lower bound on the convergence rate and steady error to be specified. The MLP and FOSD are incorporated into the neural back-stepping design to reduce the online updating parameters of NN and to conquer the problem of "explosion of the items," thus deriving a low-computational scheme.

(2) In contrast to traditional neural control schemes [21-24, 26-30] whose stability analysis relies on the condition that the NNs should always be kept within the neural active region, a smooth switching function based composite control scheme, presented to manipulate the exchange of control authorities between normal neural controller working in the active region and a robust controller out of this scope, is constructed to relax this constraint, and a new kind of adaptive laws is proposed. Note that it is also the first low-computational neural control scheme which can efficiently address the prescribed performance issue and relax the constraint on NN simultaneously.

2. Model Dynamics and Problem Formulation

2.1. Morphing Aircraft Model. The longitudinal dynamics of a morphing aircraft considered in this study are derived from [5, 38]. This model includes state variables (V, h, [sigma], [gamma], q) and control inputs ([[delta].sub.e], T), where V denotes the velocity, h is the altitude, [alpha] denotes angle of attack, [gamma] represents the flight path angle(FPA), and q is the pitch rate; [[delta].sub.e] and T represent elevator deflection and thrust, respectively.

[mathematical expression not reproducible], (1)

[??] = V sin [gamma], (2)

[??] = L + T sin [alpha] - mg cos [gamma] - [F.sub.IKz]/mV (3)

[??] = -L- T sin [alpha] + mg cos [gamma] + [F.sub.IZ]/mV + q, (4)

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible], (6)

where D, L, and [M.sub.A] denote drag force, lift force, and pitch moment, respectively. m, [I.sub.y], and g denote the mass of aircraft, moment of inertia about pitch axis, and gravity constant. [F.sub.IX], [F.sub.IZ], [F.sub.IKz] and [M.sub.Iy] are the inertial force and moment caused by morphing process. [Z.sub.T] is the position of engine in the body axis. [S.sub.x] is the static moment caused by wing sweep. The related definitions are given as follows:

[mathematical expression not reproducible], (7)

where [zeta] is the sweep angle; the detailed explanation of the other parameters can be found in [5].

2.2. Model Transformation and Control Objective. Thrust T mainly affects velocity V, and elevator deflection [[delta].sub.e] has a dominant contribution to altitude h change; thus the dynamics model is reasonably decomposed into two subsystems including altitude and velocity subsystems.

2.2.1. Altitude Subsystem. Define [x.sub.1] =h, [x.sub.2] = [gamma], [x.sub.3] = [theta], and [x.sub.4] = q, where [theta] = [alpha] + [gamma] and [bar.x] = ([x.sub.2], [x.sub.3], [x.sub.4]). Therefore, the altitude subsystem can be converted into the following formulation:

[mathematical expression not reproducible], (8)

where y is the output signal of altitude subsystem (8); [mathematical expression not reproducible] are unknown functions.

2.2.2. Velocity Subsystem. Velocity subsystem is transformed into the following formulation:

[??] = [f.sub.v] ([x.sub.v], T) + T, (9)

where [f.sub.v]([x.sub.v], T) = (-D + T cos [lambda] - mg sin [gamma] + [F.sub.Ix])/m -T is an unknown function.

Remark 1. In order to transform the altitude system into pure-feedback system, [F.sub.Ikz] in (3) is regarded as an unmodeled term. Since we only consider the cruise phase in this paper, [gamma] is quite small and we can take sin [gamma] [approximately equal to] [gamma] in (2) to simplify the system.

Control Objective. The control objective in this study is to design adaptive controllers [[delta].sub.e] and T such that

(1) the altitude and velocity can track the desired trajectory [h.sub.d] and [V.sub.d], while guaranteeing that all the signals in the closed loop are bounded;

(2) the corresponding altitude and velocity tracking errors achieve prescribed transient and steady-state performance.

2.3. Some Preliminaries

2.3.1. Prescribed Performance. To achieve the control objective, the tracking error [z.sub.i] (t), i = 1,V should satisfy the following prescribed performance bounds [22, 39]:

-[[lambda].sub.i] (t) < [z.sub.i] (t) < [[lambda].sub.i] (t), (10)

where [[lambda].sub.i] (t) > 0 named performance function is defined as

[[lambda].sub.i] (t) = ([[lambda].sub.i0] - [[lambda].sub.i[infinity]]) exp (- [l.sub.i]t) + [[lambda].sub.i[infinity]], (11)

where [[lambda].sub.i0] - [[lambda].sub.i[infinity]], and [l.sub.i] are positive constants; [[lambda].sub.i0] - [[lambda].sub.i](0) and [[lambda].sub.i[infinity]] = [lim.sub.t[right arrow][infinity]] [[lambda].sub.i](t); [l.sub.i] denotes the minimum speed of convergence and [[lambda].sub.i[infinity]] is the maximum steady-state error.

To transform the constrained tracking error condition (10) into an equivalent unconstrained one, the following state transformation is employed. So we have

[[mu].sub.i] (t) = [R.sub.i] ([z.sub.i](t)/[[lambda].sub.i](t)), (12)

where [[mu].sub.i] (t) is the transformed error and [R.sub.i] (*) is an increasing transformation function shown as follows:

[mathematical expression not reproducible]. (13)

The derivative of (12) is shown as

[mathematical expression not reproducible], (14)

where [mathematical expression not reproducible]

Using (13), the following inequalities can be obtained:

[r.sub.i] > [r.sub.imin] > 0, (15)

where [r.sub.min] = 2/([[lambda].sub.i](0)).

2.3.2. Useful Function and Key Lemmas

Definition 2 (see [35]). The boundaries of the compact subsets [[OMEGA].sub.i] are defined by several prescribed constants 0 < [r.sub.i]1 < [r.sub.i2]> i = 2,4; meanwhile some useful switching functions are described as

[mathematical expression not reproducible], (16)

where [[omega].sub.k] > 0 and [b.sub.1] >0 are positive constants.

Lemma 3 (see [40]). The following inequality holds for any [[omega].sub.0] > 0 and [eta] [member of] R;

0 [less than or equal to] [absolute value of [eta]] - [eta] tanh ([eta]/[[omega].sub.0]) [less than or equal to] [k.sub.0][[omega].sub.0], (17)

where [k.sub.0] is a constant satisfying [mathematical expression not reproducible]; that is, [k.sub.0] = 0.2785.

Lemma 4 (see [41]). The "first-order sliding mode differentiator (FOSD)" is designed as

[mathematical expression not reproducible], (18)

where [mathematical expression not reproducible] are the states of system (18), [[bar.[micro]].sub.0] and [bar.[micro]].sub.1] are the designed parameters of FOSD, and l(t) is an input function. [[??].sub.0] can estimate [??](t) to an arbitrary precision in case the initial values [mathematical expression not reproducible] are bounded.

3. Controller Design

In order to process the derivation, motived by [8,12], filtered signals are used to circumvent algebraic loop problems encountered in the following design; thus we define

[mathematical expression not reproducible] (19)

where [x.sub.3j], [u.sub.f], and [T.sub.f] are the filtered signals defined by [8]:

[mathematical expression not reproducible], (20)

where [H.sub.L](s) is a Butterworth low-pass filter. The corresponding filter parameters of Butterworth filters can be obtained in [8].

Assumption 5. In this paper, we assume that all of the system states are measurable.

Assumption 6. The functions [f.sub.i](*), i = 2,4, are unknown and are bounded by [absolute value of [f.sub.i](*)] [less than or equal to] [f.sup.u.sub.i](*), where [f.sup.u.sub.i](*) are known nonnegative smooth functions. Meanwhile, it is also assumed that [DELTA][f.sub.i] are bounded.

Obviously, there exist ideal weight vectors [W.sup.*.sub.2], [W.sup.*.sub.4], and [W.sup.*.sub.v] such that

[mathematical expression not reproducible], (21)

where [[epsilon].sub.i] and [[epsilon].sub.iM] denote the approximation errors and their upper bounds, respectively. [W.sup.*.sub.i] is the weight of NN. [[PHI].sub.i] (*) = [[[phi].sub.1](*),..., [[phi].sub.n](*)] is the basis function vector with [[epsilon].sub.i] (*) Exp[(-(* - [c.sub.i]).sup.T] (* - [c.sub.i]/[b.sup.2.sub.i]), wherein [c.sub.i] and are the centers and widths of [[epsilon].sub.i] (*). Obviously, the ideal weights [[W.sup.*.sub.2], [W.sup.*.sub.4], and [W.sup.*.sub.v] are completely unknown. Thus, the MLP technique is employed to estimate the norm of [W.sup.*.sub.2], [W.sup.*.sub.4], and [W.sup.*.sub.v] to reduce the computation burden. Those parameters are defined as [[phi].sub.i] = [parallel][W.sup.*.sub.i][parallel].sup.2] (i = 2,4,V). In the following, we replace [[phi].sub.i](*) with [[phi].sub.i] to simplify the expression.

3.1. Velocity Controller Design. Define velocity tracking error as

[z.sub.v] = V - [V.sub.d]. (22)

The time derivative of [z.sub.v] can be described as

[mathematical expression not reproducible]. (23)

According to (14) and (23), the time derivation of the transformed error [[mu].sub.V] (t) is shown as

[mathematical expression not reproducible], (24)

where [mathematical expression not reproducible]

By employing MLP technique, the controller [T.sub.d] is designed as

[mathematical expression not reproducible], (25)

where [k.sub.v1] and [[omega].sub.v1] are positive design parameters. [mathematical expression not reproducible] denote the estimation of [[phi].sub.V] and [d.sub.VM], respectively. [d.sub.v] = [DELTA][f.sub.v] + [[epsilon].sub.v] is the lump approximation error with [absolute value of [d.sub.v] [less than or equal to] [d.sub.VM] and [[??].sub.v] denotes the estimation of [[??].sub.v] by means of sliding mode differentiator. According to Lemma 4, we can easily obtain [mathematical expression not reproducible].

Consider the following adaptive laws for [mathematical expression not reproducible]:

[mathematical expression not reproducible], (26)

where [p.sub.vl], [p.sub.v2], [[sigma].sub.V1], and [[sigma].sub.V2] denote positive design parameters.

Theorem 7. Suppose that the velocity subsystem (9) satisfies Assumption 5; if the adaptive controller is selected as (25) and updating laws are selected as (26), the signals including [mathematical expression not reproducible] are ensured to be bounded.

Remark 8. The velocity design is partially derived from [24]. Note that the FOSD is used to estimate unknown item [r.sub.v]. By introducing [mathematical expression not reproducible]), the stability analysis problem in [24] is overcome.

3.2. Altitude Controller Design. The following coordinate change is constructed to facilitate the control design:

[mathematical expression not reproducible], (27)

where [[sigma].sub.1], [[sigma].sub.2], and [[sigma].sub.3] are the virtual controllers to be designed at Steps 1,2, and 3, respectively. [y.sub.d] = [h.sub.d] is the reference signal. The control scheme for the altitude subsystem is developed in the framework of back-stepping technique, which contains 4-step recursive design procedure.

Step 1. The time derivative of [z.sub.1] = [x.sub.1] - [y.sub.d] is expressed as

[mathematical expression not reproducible]. (28)

By using (15) and (28), the time derivative of the transformed altitude error pi(t) is shown as follows:

[mathematical expression not reproducible], (29)

where [mathematical expression not reproducible].

The virtual controller [[alpha].sub.i] is designed as

[mathematical expression not reproducible] (30)

where [k.sub.1] is positive parameter. It is worth noticing that [[??].sub.1] can be easily obtained via system states.

Invoking (28) and (30), one has

[mathematical expression not reproducible]. (31)

In order to avoid the tedious computation of [[alpha].sub.1], the following FOSD is adopted to estimate it:

[mathematical expression not reproducible], (32)

where [mathematical expression not reproducible] are the states of FOSD (32) and [[mu].sub.10] and [[mu].sub.11] are the positive design constants.

Then, we have

[mathematical expression not reproducible], (33)

where [[tau].sub.1] is the estimation error of the FOSD with [[tau].sub.1] [less than or equal to] [bar.[tau]].sub.1].

Step 2. The differentiation of [z.sub.2] is obtained as follows:

[mathematical expression not reproducible]. (34)

The virtual controller [[alpha].sub.2] is designed as

[mathematical expression not reproducible] (35)

with

[mathematical expression not reproducible], (36)

where [k.sub.2], [[omega].sub.21], and [[omega].sub.22] are positive design parameters. [d.sub.2] = [DELA][f.sub.2] + [[epsilon].sub.2] is bounded with [mathematical expression not reproducible] denote the estimations of [[phi].sub.2] and [d.sub.2M], respectively.

The structure of adaptive control laws is expressed as follows:

[mathematical expression not reproducible]. (37)

Substituting (35) into (34), (34) can be rewritten as

[mathematical expression not reproducible]. (38)

The following FOSD is adopted to estimate [[alpha].sub.2]:

[mathematical expression not reproducible], (39)

where [mathematical expression not reproducible] are the states of system (32) and [[mu].sub.20] and [[mu].sub.21] are positive design constants.

From (39) and Lemma 3, we have

[mathematical expression not reproducible], (40)

where [[tau].sub.2] is the estimation error with [absolute value of [[tau].sub.2]] [less than or equal to] [[bar.[tau].sub.2].

Step 3. The differentiation of [z.sub.3] is obtained as follows:

[mathematical expression not reproducible]. (41)

The virtual control law [[alpha].sub.3] is designed as

[mathematical expression not reproducible], (42)

where [k.sub.3] is a positive design parameter.

Substituting (42) into (41) yields

[mathematical expression not reproducible]. (43)

As done previously, the following FOSD is employed to estimate [[??].sub.3]:

[mathematical expression not reproducible], (44)

where [[mu].sub.20] are the states of the system and [[mu].sub.30] and [[mu].sub.31] are the positive design constants.

Thus, we have

[mathematical expression not reproducible], (45)

where [[tau].sub.3] is an estimation error with [absolute value of [[tau].sub.3] [less than or equal to] [bar.[tau]].sub.3].

Step 4. In this step, the actual controller u will be developed. The differentiation of [z.sub.4] can be obtained as follows:

[mathematical expression not reproducible]. (46)

The controller u is designed as

[mathematical expression not reproducible] (47)

with

[mathematical expression not reproducible], (48)

where [k.sub.4], [[omega].sub.41], and [[omega].sub.42] are the positive design constants. [d.sub.4] = [DELTA][f.sub.4] + [[epsilon].sub.4] is the lump approximation error with [mathematical expression not reproducible] and [[??].sub.4] denote the estimations of [[phi].sub.4] and [d.sub.4M], respectively. [[??].sub.4] and [[??].sub.4] are updated as

[mathematical expression not reproducible]. (49)

Thus, (46) can be rewritten as

[mathematical expression not reproducible]. (50)

Theorem 9. Consider the altitude subsystem (8) with Assumptions 5 and 6; if the switching adaptive neural prescribed performance control scheme is selected as (30), (35), (42), and (47), adaptive laws are selected as (37) and (49), and FOSD is selected as (32), (39), and (44), the signals [mathematical expression not reproducible] in the closed-loop system are bounded.

Remark 10. The altitude controller, composed of a normal adaptive neural controller working in the neural active region, a robust controller being in charge outside the neural approximation region, and a switching strategy supervising the exchange of the former two controllers, is constructed.

Remark 11. In this paper, in order to estimate the derivative of virtual controllers [[alpha].sub.1], [[alpha].sub.2], and [[alpha].sub.3], the FOSD (first-order sliding mode differentiator) is employed. Using (33) as an example, [??] is the estimation of [??] and [[tau].sub.i] is the estimation error between actual [mathematical expression not reproducible]. It must be noted that [[tau].sub.1] is not used in the controller design but is just employed for stability analysis (please see (B.6)).

4. Simulations

In this section, two comparative cases are presented to illustrate the effectiveness of the switching functions based adaptive neural control for longitudinal model of the morphing aircraft. The aerodynamic coefficients and model parameters are the same as [5]. The initial conditions are set as [X.sub.0] = [[gamma].sub.0], [[alpha].sub.0], [h.sub.0][V.sub.0] = [0[degrees], 0.99512[degrees], 0[degrees]/s, 1000m, 30m/s]. The control parameters are selected as [k.sub.1] = 0.3, [k.sub.2] = 0.3, [k.sub.3] = 0.05, [k.sub.4] = 3.75, and [K.sub.v1] = 3. Gains for the adaptive laws are set as [p.sub.21] = 40, [[sigma].sub.21] = 0.01, [p.sub.22] = 10, [[sigma].sub.22] = 0.1, [p.sub.41] = 250, [[sigma].sub.41] = 0.01, [p.sub.42] = 100, [[sigma].sub.42] = 0.01, [p.sub.v1] = 10, [[sigma].sub.v1] = 0.1, [p.sub.v2] = 100, [[sigma].sub.v2] = 0.1, [[omega].sub.21] = 1, [[omega].sub.22] = 1, [[omega].sub.41] = 1, and [[omega].sub.42] = 5. The aforementioned transient and steady output error bounds are prescribed by the performance functions [mathematical expression not reproducible]. The corresponding neural active regions are defined as V [member of] [30m/s 40m/s], [x.sub.2] [member of] [-4deg 4deg], [x.sub.3] [member of] [-6deg 6deg], and [x.sub.4] [member of] [-6 deg/s 6 deg/s]. The centers such as [c.sub.2], [c.sub.4], and [c.sub.V] including 50 nodes are evenly spaced in their bounds. The widths of Gaussian functions are chosen as [b.sub.i2] = 2, [b.sub.i4] = 1.5, and [b.sub.iv] = 4. The parameters of switching function are selected as [r.sub.21] = 4 deg, [r.sub.22] = 7 deg, [r.sub.31] = 6 deg, [r.sub.32] 10 deg, [r.sub.41] = 5 deg/s, [r.sub.42] = 10 deg/s, [[omega].sub.i=123] = 10, and [b.sub.1] = 1. Reference commands are smoothened via several second-order filters which are given in (51).

[mathematical expression not reproducible] (51)

Case 1. In this simulation, we assume that the aircraft is cruising at trim states, and only the morphing process is considered. The initial tracking errors are assumed to be [z.sub.1](0) = 0.1 m and [z.sub.v](0) = 0.3 m/s. For comparison purposes, the switching function based adaptive neural control (SAN which means the PPC technique is not employed in the control design) and adaptive neural prescribed performance control (PPC described in Remark 10) are utilized; meanwhile the control gains are kept fixed to the values used in the proposed control scheme (SPPC). Simulation results are presented in Figures 1-4. Specifically, the output tracking errors are presented in Figure 2. Moreover, the required control input [[delta].sub.e], T, system states, and the evolution of NNs' weight are provided in Figures 3 and 4. Notice that during the morphing process the input states ([x.sub.2], [x.sub.3], [x.sub.4]) of NNs always stay in the active region which can be explained in Figure 1(b); thus the proposed SPPC scheme is equivalent to PPC scheme. As expected, output tracking with prescribed performance as well as states boundedness is achieved with reasonable control effort. In contrast to PPC or SPPC approach, although the SAN control scheme can keep the stability of the aircraft, it owns relatively poor system performance as shown in Figure 2. Moreover, the superiority of the proposed SPPC scheme will be further revealed in Case 2 simulation.

Case 2. In this simulation, the control gains are kept the same as Case 1; meanwhile the SAN and PPC approaches are also used as comparison. The upper bounds of [f.sup.u.sub.2] and [f.sup.u.sub.4] are set as [f.sup.u.sub.2] = 0.5 and [f.sup.u.sub.4] = 1. Simulation results are presented in Figures 5-9. The output tracking for SAN and SPPC is presented in Figures 5(a) and 5(c) and the corresponding tracking errors are presented in Figures 5(b) and 5(d) along with their performance bounds. Particularly, the control inputs and system states as well as the value of the switch functions are provided in Figures 6, 7, and 8. As expected, output tracking with prescribed performance and states boundedness are both achieved by using the proposed SPPC scheme. Unfortunately, the altitude and velocity tracking errors of SAN transcend the prescribed bounds [[lambda].sub.1] and [[lambda].sub.V]. It is worth noting that, different from Case 1, the switch values are not always equal to one as depicted in Figure 8, which means that the NNs are out of the neural approximation regions. The curve of [[delta].sub.e] is obtained as shown in Figure 7(a). After about 11 seconds, the robust controller pulls back the escaped transient to the neural effective regions. Some corresponding responses for PPC scheme are pictured in Figure 9. From this figure, we can conclude that the altitude error oversteps the prescribed bounds without the utilization of switching strategy, thus leading to the instability of the system. In all, compared with simulation results, the superiority of SPPC scheme is obvious.

5. Conclusion

A composite switching neural prescribed performance control scheme has been proposed for the longitudinal dynamic model of the morphing aircraft. In the control design, by using neural networks to approximate the unknown functions, the prior information of the aerodynamic parameters is unnecessary. By introducing the performance function, the proposed controller is able to permit attributes such as a lower bound on the convergence rate and maximum allowable steady error to be specified. A switching mechanism supervising the exchange of control authorities between the normal neural controller and a robust controller is used to relax the constraint that NN should be kept in the active regions all the time. Two comparative simulations have revealed the superiority of this control scheme.

Appendix

A. Proof of Theorem 7

Proof. Invoking (24) and (25) yields

[mathematical expression not reproducible]. (A.1)

Consider the following candidate Lyapunov function:

[mathematical expression not reproducible], (A.2)

where [mathematical expression not reproducible].

Based on (26) and (A.1), the time derivative of [L.sub.V] is given by

[mathematical expression not reproducible]. (A.3)

Note that the following inequalities hold:

[mathematical expression not reproducible]. (A.4)

By considering (A.4), [[??].sub.v] can be reformulated as

[mathematical expression not reproducible], (A.5)

where [mathematical expression not reproducible].

If [mathematical expression not reproducible], define the following compact sets:

[mathematical expression not reproducible]. (A.6)

It is obvious that [[??].sub.v] is negative [mathematical expression not reproducible] and [mathematical expression not reproducible]. Therefore, the signals in [mathematical expression not reproducible], in the closed-loop system are bounded.

B. Proof of Theorem 9

Proof. Select the candidate Lyapunov function as follows:

L = [L.sub.1] + [L.sub.2] + [L.sub.3] + [L.sub.4], (B.1)

where [mathematical expression not reproducible] denotes the upper bound of V.

On the basis of (31), the time derivative of [L.sub.1] is given by

[mathematical expression not reproducible]. (B.2)

Differentiating [L.sub.2] with respect to time, invoking (37) and (38), we have

[mathematical expression not reproducible]. (B.3)

Employing (43), the time derivative of [L.sub.3] is obtained as

[mathematical expression not reproducible]. (B.4)

Using (49) and (50) results in the time derivative of [L.sub.4]:

[mathematical expression not reproducible]. (B.5)

Consider the following facts:

[mathematical expression not reproducible]. (B.6)

We have

[mathematical expression not reproducible], (B.7)

where the corresponding design parameters should be chosen such that [k.sub.1]-0.5/[k.sub.11] > 0, [k.sub.2]-0.5-0.5 [k.sub.11] > 0, [k.sub.3]-0.5/[k.sub.12] > 0, ([k.sub.4] - 0.5) > 0, and [[sigma].sub.ij] > 0, i = 2,4, j = 1,2.

[mathematical expression not reproducible]. (B.8)

Define the following compact sets:

[mathematical expression not reproducible]. (B.9)

If [mathematical expression not reproducible], we know that [??] will be negative. Therefore, the signals [mathematical expression not reproducible] in the closed-loop system are bounded.

https://doi.org/10.1155/2017/1401427

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant nos. 61374032 and 61573286).

References

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Zhonghua Wu, (1) Jingchao Lu, (1) Jingping Shi, (1) Yang Liu, (1) and Qing Zhou (2)

(1) School of Automation, Northwestern Polytechnical University, Xi'an 710072, China

(2) Xian Aeronautics Computing Technique Research Institute, AVIC, Xi'an 710068, China

Correspondence should be addressed to Zhonghua Wu; 463897575@qq.com

Received 14 October 2016; Revised 28 March 2017; Accepted 11 April 2017; Published 17 May 2017

Academic Editor: Asier Ibeas

Caption: Figure 1: Sweep reference and switch signal.

Caption: Figure 2: Altitude and velocity tracking errors.

Caption: Figure 3: System states and NN weights.

Caption: Figure 4: Control inputs.

Caption: Figure 5: Altitude and velocity tracking.

Caption: Figure 6: System states and NN weights.

Caption: Figure 7: Control inputs.

Caption: Figure 8: Switching functions.

Caption: Figure 9: Partial response of PPC scheme.

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Title Annotation: | Research Article |
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Author: | Wu, Zhonghua; Lu, Jingchao; Shi, Jingping; Liu, Yang; Zhou, Qing |

Publication: | Mathematical Problems in Engineering |

Article Type: | Report |

Date: | Jan 1, 2017 |

Words: | 6355 |

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