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Global Neural Dynamic Surface Tracking Control of Strict-Feedback Systems With Application to Hypersonic Flight Vehicle Bin Xu, Chenguang Yang, Member, IEEE, and Yongping Pan, Member, IEEE Abstract— This paper studies both indirect and direct global neural control of strict-feedback systems in the presence of unknown dynamics, using the dynamic surface control (DSC) technique in a novel manner. A new switching mechanism is designed to combine an adaptive neural controller in the neural approximation domain, together with the robust controller that pulls the transient states back into the neural approximation domain from the outside. In comparison with the conventional control techniques, which could only achieve semiglobally uniformly ultimately bounded stability, the proposed control scheme guarantees all the signals in the closed-loop system are globally uniformly ultimately bounded, such that the conventional constraints on initial conditions of the neural control system can be relaxed. The simulation studies of hypersonic flight vehicle (HFV) are performed to demonstrate the effectiveness of the proposed global neural DSC design. Index Terms— Dynamic surface control, global stability, hypersonic flight vehicle, indirect and direct neural control, smooth switching, strict-feedback system.
I. I NTRODUCTION
H
YPERSONIC flight vehicles provide a promising and cost-effective means to fulfill the increasing commercial and military demands for space access. However, the design of flight control systems for HFVs faces a number of challenges [1], caused by the highly coupled and nonlinear nature of the vehicle dynamic behaviors. Different from the traditional flight vehicles, the high mach numbers and the high altitude make the hypersonic flight control extremely sensitive to changes in atmospheric conditions in addition to the physical and aerodynamic parameters. Thus, it is vital for the control system to ensure the flight safety. Manuscript received April 19, 2014, Revised November 5, 2014, February 22, 2015 and July 4, 2015, Accepted July 5, 2015. Date of publication February 19, 2015; date of current version September 16, 2015. This work was supported in part by the National Science Foundation of China under Grant 61304098, Grant 61134004, and Grant 61473120, in part by the Natural Science Basic Research Plan in Shaanxi Province under Grant 2014JQ8326 and Grant 2015JM6272, in part by the Fundamental Research Funds for the Central Universities under Grant 3102015AX001 and Grant 2015ZM065, and in part by the Beijing Natural Science Basic Research Plan under Grant 4142028. B. Xu is with the School of Automation, Northwestern Polytechnical University, Xi’an 710072, China (e-mail:
[email protected]). C. Yang is with the Center for Robotics and Neural Systems, Plymouth University, Devon PL4 8AA, U.K., and also with the College of Automation Science and Engineering, South China University of Technology, Guangzhou 510006, China (e-mail:
[email protected]). Y. Pan is with the Department of Biomedical Engineering, National University of Singapore, Singapore 119077 (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2456972
Schmidt [2] designed the classic and multivariable linear control for the longitudinal model of the flight dynamics developed in [3]. The robust velocity and altitude tracking in the presence of model uncertainties and varying flight conditions is addressed in [4], while in [5], the tracking control of a hypersonic aircraft with aerothermoelastic effects is studied. Recently, many advanced nonlinear and intelligent control techniques have been studied for complex nonlinear systems [6]–[8]. Similar idea has been applied to a hypersonic flight control, e.g., feedback linearization [9], [10], nonlinear dynamic inversion control [11], sliding mode control [12], [13], and intelligent control [14], [15]. Among various control design techniques, the back-stepping design [16] is one of the most popular design tools for a systematic nonlinear control synthesis, and has been successfully applied to the hypersonic flight control design. In [17], the hypersonic flight dynamics is formulated into the linearly parameterized form. The tracking of a nonminimum phase hypersonic vehicle model is analyzed in [18]. To deal with the well-known problem of explosion of the complexity associated with the back-stepping design, improved the dynamic surface control design [19] has been studied on the control-oriented model developed in [20], where the coupling effect of the engine to the airframe dynamics has also been considered. As a matter of fact, the linearly parameterized form is highly dependent on the structure of the nonlinearity, while currently, very little knowledge of hypersonic flight dynamics is available. Thus, the back-stepping design with partially model free control techniques has been widely studied. In [21], the discrete controller is designed using neural network (NN) to approximate the unknown dynamics. In [22], the control gain function gi is estimated by NN, and the indirect design is proposed for the longitudinal hypersonic flight dynamics. In the literature of intelligent control, the design with the fuzzy logic system (FLS)/NN has been widely studied for both continuous systems [23]–[28] and discrete systems [29]–[33] to achieve semiglobally uniformly ultimately bounded (SGUUB) of the closed-loop system. For the affine system x˙ = f (x) + g(x)u, the indirect design [23], [34], [35] is with the form u = 1/g(− ˆ fˆ + ν), ˆ where f and gˆ are the estimates of f and g, respectively, and ν is the signal to be determined. In the direct design [15], [36], the new function g −1 ( f − x˙d ) is defined and approximated by NN, so that there is no need to approximate the control gain function g.
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It is noted that the above-mentioned intelligent design is on the condition that the approximation should remain valid all the time. However, such a condition is difficult to verify beforehand [37]. Since the approximation ability of NNs is only valid in a compact domain, the NN-based hypersonic control can only guarantee the SGUUB stability. As a result, the deterioration of the tracking performance or even instability may occur in real applications. In [38], using a robust stabilization controller and an indirect fuzzy controller, the global design is proposed for a class of systems in the controllable canonical form. In [39] and [40], a global tracking stability has also been achieved by mainly approximating the unknown nonlinearity in terms of the reference signals. In [37] and [41], the global neural design for strict-feedback systems is presented using the switching mechanism. However, the stability analysis of these two design methods is rather complicated, because the derivative of virtual control and the nth-order smoothness of the switching function are required. The above-mentioned problems make the neural control design with the global stability really complex. Inspired by the aforementioned work of the neural control design, in this paper, we propose to incorporate the dynamic surface design into the global neural control design, and then develop both indirect and direct controllers for strictfeedback systems. In comparison to the previous results, the contributions of this paper are summarized as follows. 1) In view of intelligent control, this paper investigates the indirect and direct neural DSC design combining both neural controller and robust controller to achieve globally uniformly ultimate boundedness. Compared with the method developed in [37] and [41], the dynamic surface design is incorporated into the scheme, and the switching signal is not required to be the nth-order smooth. These have greatly relaxed the constraints associated with the conventional NN approximation design. The GUUB stability of the closed-loop system is rigorously established by the Lyapunov approach. 2) In view of hypersonic flight control, the longitudinal dynamics is formulated into the velocity subsystem and the attitude subsystem. Toward the third-order system with flight path angle (FPA), pitch angle, and pitch rate, different from the previous neural design [14], [22] achieving SGUUB, the global DSC could ensure the flight safety. This paper is organized as follows. Section II describes the longitudinal dynamics of a generic HFV. The radial basis function (RBF) NNs and the useful lemmas are presented in Section III. Section IV presents the global indirect neural adaptive controller design, whereas the global direct design is analyzed in Section V. The simulation result is included in Section VI. Section VII presents several comments and the final remarks. II. M ODEL DYNAMICS AND P ROBLEM F ORMULATION A. Hypersonic Flight Dynamics The control-oriented model of the hypersonic longitudinal dynamics considered in this paper is given in [20]. This model
is comprised of five state variables X h = [V, h, α, γ, q]T and two control inputs U = [δe , ]T , where V is the velocity, γ is the FPA, h is the altitude, α is the attack angle, q is the pitch rate, δe is elevator deflection, and is the fuel equivalence ratio. The differential equations are as follows: T cos α − D − g sin γ V˙ = m h˙ = V sin γ L + T sin α g cos γ γ˙ = − mV V α˙ = q − γ˙ M yy q˙ = I yy
(1) (2) (3) (4) (5)
where g, m, and I yy represent the acceleration due to gravity, the mass of aircraft, and the moment of inertia about pitch axis. T , D, L, and M yy represent thrust, drag, lift-force, and pitching moment, respectively, and have the following expressions: T = T (α) + T0 (α) ≈ [β1 + β2 ]α 3 + [β3 + β4 ]α 2 + [β5 + β6 ]α + [β7 + β8 ] 2 D ≈ q¯ S C αD α 2 + C αD α + C 0D L = L 0 + L α α ≈ q¯ SC L0 + q¯ SC Lα α M yy = MT + M0 (α) + Mδe δe α2 2 δe α 0 ≈ z T T + q¯ S c¯ C M α + CM α + CM δe + q¯ S cC ¯ M 1 h − h0 q¯ = ρV 2 , ρ = ρ0 exp − . 2 hs The detailed explanation of the parameter values can be found in [19] and [20]. B. Dynamics Transformation For the velocity subsystem (1), it is formulated as V˙ = gv + f v
(6)
with fv = (T0 cos α − D)/m − g sin γ, gv = (T cos α)/m. Assumption 1: Since γ is quite small during the cruise phase, we take sin γ ≈ γ in (2) for simplification. The thrust term T sin α in (3) can be neglected, because it is generally much smaller than L. For the altitude subsystem (2)–(5), the tracking error of the altitude is defined as h˜ = h − h r , and the FPA command is chosen as ˜ + h˙ r −kh h˜ − ki hdt . (7) γd = V If kh > 0 and ki > 0 are chosen and the FPA is controlled to follow γd , the altitude tracking error is regulated to zero exponentially [15].
XU et al.: GLOBAL NEURAL DYNAMIC SURFACE TRACKING CONTROL OF STRICT-FEEDBACK SYSTEMS
Fig. 1.
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Control scheme.
Define X = [x 1 , x 2 , x 3 ]T , x 1 = γ, x 2 = θ p , x 3 = q where θ p = α + γ. The attitude subsystem can be formulated as x˙1 = f 1 (x 1 ) + g1 x 2 x˙2 = x 3 x˙3 = f 3 (X) + g3 u u = δe y = x1
(8) (9)
where f 1 = (L 0 − L α γ )/(mV )−g/V cos x 1 , g1 = L α /(mV ), f 3 = (MT + M0 (α))/I yy , and g3 = Mδe /I yy . Assumption 2: There exists positive constants g¯ i and gi , such that gi ≤ |gi | ≤ g¯ i , i = 1, 3. With the time-scale conclusion in [42] and [43], the velocity is considered as slow dynamics. For the controller design of attitude subsystem, velocity is approximated as constant, and its derivative is zero. Remark 1: With Assumption 2, γ˙d is calculated as −ki h˜ − kh (V sin γ − h˙ r ) + h¨ r −ki h˜ − kh h˙˜ + h¨ r = . γ˙d ≈ V V (10) From (7) and (10), the task of altitude tracking is now on controlling the attitude subsystem (8) to follow FPA commands γd and γ˙d . Remark 2: It is further derived that g1 = Wg1 θg1 and g3 = Wg3 θg3 with Wg1 = (1/m)C Lα , θg1 = q¯ S/V , δe Wg3 = (1/I yy )cC ¯ M , and θg3 = q¯ S. It is clear that gi , i = 1, 3 are the functions of velocity. With Assumption 2, gi are considered as unknown constants and g˙ i = 0. Remark 3: Attitude subsystem (8) is a third-order system with the strict-feedback form. Thus, the controller designs and the main results in this paper could be easily extended to the following nth-order general strict-feedback systems: x˙i = fi (x¯i ) + gi (x¯i )x i+1 , 1 ≤ i ≤ n − 1 x˙n = fn (x¯n ) + gn (x¯n )u, n ≥ 2 y = x1
of altitude subsystem. For the velocity subsystem, we simply employ a PID controller. With the command transformation (7), the goal pursued in this paper is to design a global neural dynamic controller δe to steer FPA from a given set of initial values to desired trimmed conditions with signal γd , so that the altitude will follow the tracking reference h r . III. P RELIMINARIES A. RBF Neural Networks In this paper, the RBF NNs [44] are employed to approximate the unknown nonlinearity f with the following form: fˆ(X in ) = ωˆ T θ (X in )
where X in ∈ R M is the input vector of the RBF NNs, fˆ ∈ R is the NN output, ωˆ ∈ R L N is the adjustable parameter vector, θ (·) : R M → R L N is a nonlinear vector function of the inputs, and L N is the number of NN nodes. The components of vector θ are the basis functions denoted as i . A commonly used basis function is the so-called Gaussian function of the following form: 1 X in − ξi 2 exp − (13) i (X in ) = √ 2σi2 2πσi where ξi is an M-dimensional vector representing the center of the i th basis function and σi is the variance representing the spread of the basis function. Given any real continuous function f on a compact set X in ∈ R M and an arbitrary εm > 0, there exist RBF NNs in the form of (14) and an optimal parameter vector ω∗ such that f (X in ) = ω∗ T θ (X in ) + ε
(14)
| ε |< εm
(15)
sup
X in ∈ X in
(11)
where x¯i = [x 1, . . . , x i ]T .
(12)
where εm > 0 denotes the supremum value of the reconstruction error ε that is inevitably generated.
C. Control Goal
B. Useful Functions and Key Lemmas
Based on the functional decomposition, as shown in Fig. 1, the design is mainly on the global dynamic surface design
Definition 1: Given the parameter pi in subset p with upper bound piU and lower bound piL , for estimation pˆ i ,
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the smooth parameter projection Proj(τi ) is with the following form: ⎧ τi if ( pˆ i ∈ p ) ⎪ ⎪ ⎪ ⎨ or pˆ i = piL , τi ≥ 0 (16) Proj(τi ) = ⎪ or pˆ i = piU , τi ≤ 0 ⎪ ⎪ ⎩ 0 otherwise. Definition 2: Given constants 0 < ri1 < ri2 , i = 1, 3 being the constants defining the boundaries of the compact subsets i , the set of switching functions is given as
m i (x¯i ) =
i
Bk (x k )
(17)
and if |x k | < rk1
= g1 x 2 + f 1 − x˙1d
(21)
where x˙1d = γ˙d . Using NN to approximate the unknown function f 1 on the compact set x1 , we have (22)
where ω∗f 1 is the optimal NN weights vector, θ1 (x 1 ) is the basis function vector, ε1 is the NN construction error, and |ε1 | ≤ ε1m . Taking x 2 as the virtual control variable, x 2c is designed as (23)
with k1 > 0 as the gain parameter, gˆ 1 = Wˆ g1 θg1
if |x k | > rk2 (18)
with k > 0 and b ≥ 1 being the spread and the order of the function Bk (x k ), respectively. Remark 4: In [41], it is claimed that the function B(k) in [37] cannot be exactly zero or one, and therefore, the neural controller and the robust controller must work together inside the neural active region. This inevitably results in a certain degree of a waste of control energy. However, the switching functions m i designed in [41] using the information of all states, e.g., x¯i , make the design not applicable in some cases when x i is out of the region but x¯i is within the design region. Therefore, in this paper, to make the design more realistic, the switching function defined in (17) considers the neural active region of each variable while retaining the advantage as stated in [41]. Lemma 1 [45]: The following inequality holds for any w0 > 0 and η ∈ R : η 0 ≤ |η| − η tanh (19) ≤ κw0 w0 where κ is a constant satisfying κ = e−(κ+1), i.e., κ = 0.2785. IV. G LOBAL N EURAL I NDIRECT C ONTROL OF ATTITUDE S UBSYSTEM The recursive design procedure breaks down into three steps. From Step 1 to Step 2, a virtual control x ic , i = 2, 3 is designed at each step, and an overall control law δe is constructed in Step 3. During the controller design, RBF NN is employed for a nonlinear function approximation of f i , i = 1, 3. Assumption 3: The functions f i (x¯i ), i = 1, 3 are unknown and bounded by
where f iU are known nonnegative smooth functions.
x˙˜1 = x˙1 − x˙1d
gˆ 1 x 2c = −k1 x˜ 1 + x˙1d − m 1 (x 1 )u 1N − (1 − m 1 (x 1 ))u r1
if rk1 ≤ |x k | ≤ rk2
| f i (x¯i )| ≤ f iU (x¯i ) ∀x¯i ∈ R i
Step 1: Define x˜1 = x 1 − x 1d where x 1d = γd . The dynamics of the FPA tracking error x˜1 is written as
f 1 = ω∗T f 1 θ1 (x 1 ) + ε1
k=1
Bk (x k ) ⎧ 1 ⎪ 2b ⎪ ⎪ 2 ⎨ x k2 −rk1 − 2 −x 2 = rk2 2 −r 2 k rk2 k k1 ⎪ 2 −r 2 e ⎪ r ⎪ ⎩ k2 k1 0
A. Controller Design
(20)
u 1N = fˆ1 (x 1 ) = ωˆ Tf 1 θ1 (x 1 ) x˜1 f 1U (x 1 ) r U u 1 = f 1 (x 1 ) tanh w1
(24) (25)
where ωˆ f 1 is the estimation of ω∗f 1 , Wˆ g1 is the estimation of Wg1 , and w1 is the positive parameter. Remark 5: If FLS/NN is employed to approximate control gain function g1, the bound of the NN weights is required [38] to avoid the potential singularity. From Remark 2, we know that g1 can be written into the linearly parameterized form as a single item. Therefore, there is no need to construct the complex form using many basis functions and in the indirect adaptive design, the projection function (16) is used to restrict the estimation Wˆ g1 into the predefined compact. Let us introduce a new state variable x 2d , which can be obtained by the following first-order filter: α2 x˙2d + x 2d = x 2c , x 2d (0) = x 2c (0)
(26)
where α2 > 0 is the filter parameter. Define y2 = x 2d − x 2c , x˜2 = x 2 − x 2d . Then, (21) is calculated as x˙˜ 1 = g1 x 2 + f 1 − x˙1d = g1 (x 2 − x 2c ) + g1 x 2c − gˆ 1 x 2c + gˆ 1 x 2c + f 1 − x˙1d = g1 (x 2 − x 2c ) + g˜ 1 x 2c + m 1 (x 1 )( f˜1 + ε1 ) + (1 − m 1 (x 1 ))( f 1 − u r1 ) − k1 x˜1 = g1 x˜2 + g1 y2 + g˜ 1 x 2c − k1 x˜1 + m 1 (x 1 )( f˜1 + ε1 ) + (1 − m 1 (x 1 )) f 1 − u r1
(27)
where f˜1 = ω˜ Tf 1 θ1 , g˜ 1 = W˜ g1 θg1 , ω˜ f 1 = ω∗f 1 − ωˆ f 1 , and W˜ g1 = Wg1 − Wˆ g1 . The adaptation laws of the estimated parameters are designed as ω˙ˆ f 1 = f 1 (θ1 x˜1 m 1 (x 1 ) − δ f 1 ωˆ f 1 ) ˙ Wˆ g1 = Proj(g1 (θg1 x˜1 x 2c − δg1 Wˆ g1 ))
(28) (29)
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where f 1 is a symmetric positive definitive matrix, g1 , δ f 1 , and δg1 are positive design parameters. Step 2: The dynamics of the pitch angle tracking error x˜2 is written as x˙˜2 = x˙2 − x˙2d = x 3 − x˙2d .
(30)
Take x 3 as virtual control and design x 3c as x 3c = −k2 x˜ 2 + x˙2d
(31)
where k2 > 0 is the gain parameter. Introduce a new state variable x 3d , which can be obtained by the following first-order filter: α3 x˙3d + x 3d = x 3c , x 3d (0) = x 3c (0)
(32)
where α3 > 0 is the filter parameter. Define y3 = x 3d − x 3c , x˜3 = x 3 − x 3d . Then, we have x˜˙ 2 = x 3 − x˙2d = x 3 − x 3d + x 3d − x 3c + x 3c − x˙2d = x˜3 + y3 − k2 x˜2 .
Fig. 2.
(33)
Step 3: The dynamics of the pitch rate tracking error x˜3 is written as x˜˙3 = x˙3 − x˙3d = g3 u + f 3 − x˙3d .
(34)
Using NN to approximate the unknown function f 3 on the compact set X , we have f 3 = ω∗T f 3 θ3 (X) + ε3
(35)
where ω∗f 3 is the optimal NN weights vector, θ3 (X) is the basis function vector, ε3 is the NN construction error, and |ε3 | ≤ ε3m . The elevator deflection is designed as gˆ 3 u = −k3 x˜3 + x˙3d − m 3 (X)u 3N − (1 − m 3 (X))u r3
(36)
where k3 > 0 is the gain parameter, gˆ 3 = Wˆ g3 θg3 and u 3N = fˆ3 (X) = ωˆ Tf 3 θ3 (X) x˜3 f 3U (X) r U u 3 = f 3 (X) tanh w3
(37) (38)
with ωˆ f 3 as the estimation of ω∗f 3 , Wˆ g3 as the estimation of Wg3 , and w3 as the positive parameter. Similar to the analysis in (27), the error dynamics of x˜3 is derived as x˜˙3 = g3 u + f 3 − x˙3d = (g˜ 3 + gˆ 3)u + f 3 − x˙3d = g˜ 3 u − k3 x˜3 + m3 (X)( f˜3+ ε3 ) + (1 − m 3 (X)) f 3 − u r3
(39)
where f˜3 = ω˜ Tf 3 θ3 , g˜ 3 = W˜ g3 θg3 , ω˜ f 3 = ω∗f 3 − ωˆ f 3 , W˜ g3 = Wg3 − Wˆ g3 . The adaptation laws of the estimated parameters are given as ω˙ˆ f 3 = f 3 (θ3 x˜3 m 3 (X) − δ f 3 ωˆ f 3 ) ˙ Wˆ g3 = Proj(g3 (θg3 x˜3 u − δg3 Wˆ g3 ))
Global design.
where f 3 is a symmetric positive definitive matrix, g3 , δ f 3 , and δg3 are positive design parameters. Remark 6: In (23) and (36), the item ki x˜i , i = 1, 3, provides the error feedback while with the switching function m i , neural function approximation item u iN and robust item u ri work together to make sure of the global tracking. Take (23) for example. The scheme is shown in Fig. 2. If |x 1 | ≤ r11 , x 1 falls into the compact 11 and u 1N is working. When |x 1 | ≥ r12 , x 1 goes out of the compact 12 , and then u r1 is working to pull the state back to 12 . If r11 < |x 1 | < r12 , then the signal m 1 (x 1 )u 1N + (1 − m 1 (x 1 ))u r1 tries to drag the state into the compact 11 . B. Stability Analysis Assumption 4: The FPA reference signal and its derivatives are smooth bounded functions. Assumption 5: There exists known constants g¯ i and g i such that g¯ i > |gi | > gi > 0, i = 1, 3. Theorem 1: Consider system (8) using virtual controls (23) and (31) and actual control (36) as well as adaptation laws (28), (29), (40), and (41) under Assumptions 1–5. Then, all the signals of (42) are uniformly ultimately bounded. Proof: Select a suitable Lyapunov function as below LG =
Li
(42)
i=1
with
1 2 −1 ˜ 2 ˜ f 1 + g1 Wg1 + y22 x˜1 + ω˜ Tf 1 −1 f1 ω 2 1 2 x˜2 + y32 L2 = 2 1 2 −1 ˜ 2 L3 = ˜ f 3 + g3 Wg3 . x˜3 + ω˜ Tf 3 −1 f3 ω 2 According to the definition of yi , i = 2, 3, the following equations can be obtained: L1 =
y˙i = x˙id − x˙ic yi = − + Bi (·) αi
(40) (41)
3
where Bi (·) = −x˙ic .
(43)
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According to the virtual controls (23) and (31) together with Assumption 4, we know there exist constants Mi > 0, i = 2, 3 with |Bi (·)| ≤ Mi .
(44)
Take the derivative of L 1 = −k1 x˜12 + g1 x˜2 x˜1 + g1 y2 x˜ 1 + m 1 (x 1 )ε1 x˜1 + (1 − m 1 (x 1 )) f 1 − u r1 x˜ 1 − ω˜ T −1 ω˙ˆ f 1 − m 1 (x 1 )θ1 x˜1 f1
−1 ˙ y2 Wˆ g1 − θg1 x˜1 x 2c − 2 + B2 y2 . (45) − W˜ g1 g1 α2 Substituting the adaptation laws (28) and (29) into (45), we have L˙ 1 = −k1 x˜12 + g1 x˜2 x˜ 1 + g1 y2 x˜1 + m 1 (x 1 )ε1 x˜ 1 + (1 − m 1 (x 1 )) f 1 − u r1 x˜ 1 y2 + δg1 W˜ g1 Wˆ g1 − 2 + B2 y2 . α2 The following inequalities exist: + δ f 1 ω˜ Tf 1 ωˆ f 1
ω˜ Tf 1 ωˆ f 1 = ω˜ Tf 1 (ω∗f 1 − ω˜ f 1 ) 1 1 ≤ − ω˜ f 1 2 + ω∗f 1 2 2 2 W˜ g1 Wˆ g1 = W˜ g1 (Wg1 − W˜ g1 ) 1 2 1 2 ≤ − W˜ g1 + Wg1 2 2 1 2 1 2 m 1 (x 1 )ε1 x˜1 ≤ ε1m + x˜1 2 2 and
f 1 − u r1
x˜1 ≤ f 1U x˜1 − f 1U x˜ 1 tanh
x˜1 f1U w1
(46)
(47)
(48)
≤ κw1 . Using inequality ab ≤ (a 2 /2) + (b2 /2), we have L˙ 1 ≤ −k1 x˜12 + g¯ 1 |x˜2 x˜1 | + g¯ 1 |y2 x˜1 | −
L˙ 2 = x˜2 x˙˜ 2 + y3 y˙3 y32 + y3 B 3 α3 x˜ 2 x˜ 2 y2 M2 x˜ 2 y2 y2 ≤ 2 + 3 + 2 + 3 − k2 x˜22 − 3 + 3 + 3 2 2 2 2 α3 2 2 2 2 M x ˜ g ¯ 1 = −k20 x˜22 − k21 y32 + 3 + 3 − x˜22 2 2 2 x˜32 g¯ 1 2 − x˜2 + k23 ≤ −η2 L 2 + (50) 2 2
= x˜2 (x˜3 + y3 − k2 x˜2 ) −
˙ˆ f 1 − −1 W˜ g1 W˙ˆ g1 + y2 y˙2 L˙ 1 = x˜1 x˙˜ 1 − ω˜ Tf 1 −1 g1 f1 ω
f1
2 + and k13 = (M22 /2) + (1/2)δ f 1 ω∗f 1 2 + (1/2)δg1 Wg1 2 (1/2)ε1m + κw1 . Take the derivative of L 2
y22 + |B2 y2 | α2
1 1 − δ f 1 ω˜ f 1 2 + δ f 1 ω∗f 1 2 2 2 1 1 1 2 1 2 ∗2 − δg1 W˜ g1 + δg1 Wg1 + ε1m + x˜12 + κw1 2 2 2 2 1 g¯ 1 1 ≤ − k1 − g¯ 1 − x˜ 2 + x˜22 − δ f 1 ω˜ f 1 2 2 1 2 2 1 1 1 g¯ 1 2 − − δg1 W˜ g1 − − y 2 + k13 2 α2 2 2 2 g¯ 1 = −k10 x˜12 + x˜22 − k11 y22 2 1 1 2 − δ f 1 ω˜ f 1 2 − δg1 W˜ g1 + k13 2 2 g¯ 1 ≤ −η1 L 1 + x˜22 + k13 (49) 2 −1 where η1 = min[2k10 ,2k11 ,δ f 1/(λmax ( −1 f 1 )), δg1 /(λmax (g1 ))], k10 = k1 − g¯ 1 −(1/2) > 0, k11 = (1/α2 )−(g¯ 1 /2)−(1/2) > 0,
where η2 = min[2k20, 2k21 ], k20 = k2 − 1 − (g¯ 1 /2) > 0, k21 = (1/α3 ) − 1 > 0, and k23 = (M32 /2). Similar to the analysis of L˙ 1 , by taking the derivative of L 3 , we have ˙ˆ f 3 − −1 W˜ g3 W˙ˆ g3 L˙ 3 = x˜3 x˙˜ 3 − ω˜ Tf 3 −1 g3 f3 ω
= −k3 x˜32 + m 3 (X)ε3 x˜3 + (1 − m 3 (X)) f 3 − u r3 x˜3 + δ f 3 ω˜ Tf 3 ωˆ f 3 + δg3 W˜ g3 Wˆ g3 1 1 1 2 ≤ −(k3 − 1)x˜32 − x˜32 − δ f 3 ω˜ f 3 2 − δg3 W˜ g3 + k33 2 2 2 1 = −η3 L 3 − x˜32 + k33 (51) 2
−1 where η3 = min[2k30, δ f 3 /(λmax ( −1 f 3 )), δg3 /(λmax (g3 ))], ∗ 2 + 2 k30 = k3 −1 > 0, and k33 = (1/2)δ f 3 ω f 3 +(1/2)δg3 Wg3 2 (1/2)ε3m + κw3 . Finally, we have
L˙ G =
3
L˙ i
i=1
≤−
3
ηi L i + K
i=1
≤ −ηL G + K
(52)
where η = min[η1 , η2 , η3 ] and K = k13 + k23 + k33 . According to the Lasalle–Yoshizawa theorem, the closed-system stability is uniformly ultimately bounded. This completes the proof. Remark 7: If the initial value L¯ G |t =t0 = p0 , then L˙ G ≤ −ηp0 + K. If p0 > (K /η), then L˙ G < 0. System is stable in the following compact set: K ¯ ϒ = x˜i , yi | L G ≤ . (53) η The radius can be made arbitrary small by increasing the control gains (k1 , k2 , k3 ) and decreasing the filter parameters (α2 , α3 ). Remark 8: It can be easily deduced from (52) that K −ηt K 1 2 x˜i ≤ L G (t) ≤ L¯ G − (54) + . 2 η η
XU et al.: GLOBAL NEURAL DYNAMIC SURFACE TRACKING CONTROL OF STRICT-FEEDBACK SYSTEMS
V. G LOBAL N EURAL D IRECT C ONTROL OF ATTITUDE S UBSYSTEM
Design the updating laws as below
Different from the indirect design where the control gain function gi is required to be estimated, and the projection function is employed to avoid the singularity, the global DSC direct design will be analyzed in this section. Assumption 6: From [20], the sign of control gain function gi , i = 1, 3 is known where g1 > 0 and g3 < 0. A. Controller Design Step 1: The FPA surface error is defined as x˜1 = x 1 − x 1d . Taking the derivative of x˜1 and using (8), we have x˙˜1 = x˙1 − x˙ 1d = g1 x 2 + f 1 − x˙ 1d = g1 (x 2 + g1−1 ( f 1 − x˙1d )).
(55)
Define z 1 = [x 1 , x˙1d ]T , and on the compact set z1 , we have h 1 (z 1 ) = g1−1 ( f 1 − x˙1d ) = ω1∗T θ1 (z 1 ) + ε1
(56)
ω1∗
where is the optimal NN weights vector, θ1 (z 1 ) is the basis vector, and ε1 is the NN approximation error with |ε1 | ≤ ε1m . Obviously, the following inequality exists: U |h 1 (z 1 )| ≤ g −1 (57) f 1 + |x˙1d | 1 such that we can rewrite (57) into the following form: |h 1 (z 1 )| ≤ ϑ1m H1(z 1 )
(58)
is unknown positive constant and where ϑ1m H1(z 1 ) = f 1U + |x˙1d |. Take x 2 as the virtual control of (55) and design the signal x 2c as x 2c = −k1 x˜1 − m 1 (z 1 )u 1N − (1 − m 1 (z 1 ))u r1
(59)
with k1 > 0 as the gain parameter, and u 1N = ωˆ 1T θ1 (z 1 ) u r1 = ϑˆ 1 H1(z 1 ) tanh
χ1 w1
(60) (61)
where χ1 = H1 (z 1 )x˜1 , ωˆ 1 is the estimation of ω1∗ , w1 is the positive parameter, and ϑˆ 1 is the estimation of ϑ1m . Let us introduce a new state variable x 2d , which can be obtained by the following first-order filter: α2 x˙2d + x 2d = x 2c , x 2d (0) = x 2c (0)
(62)
where α2 > 0 is the filter parameter. Define y2 = x 2d − x 2c and x˜2 = x 2 − x 2d . Then, the error dynamics of x˜1 becomes x˙˜ 1 = g1 (x˜2 + y2 + x 2c + h 1 ) x˜2 + y2 − k1 x˜1 − m 1 (z 1 )u 1N + h 1 = g1 −(1 − m 1 (z 1 ))u r1 = g1 x˜2 + y2 − k1 x˜1 + m 1 (z 1 ) ω˜ 1T θ1 + ε1 χ1 + g1 (1 − m 1 (z 1 )) h 1 − ϑ1m H1(z 1 ) tanh w 1 χ1 + g1 (1 − m 1 (z 1 ))ϑ˜ 1 H1(z 1 ) tanh (63) w1 where ω˜ 1 = ω1∗ − ωˆ 1 and ϑ˜ 1 = ϑ1m − ϑˆ 1 .
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ω˙ˆ 1 = 1 [m 1 (z 1 )θ1 x˜1 − δ f 1 ωˆ 1 ] χ1 ϑ˙ˆ 1 = ρ1 (1 − m 1 (z 1 ))χ1 tanh − δ1 ϑˆ 1 w1
(64) (65)
where 1 is a symmetric positive definitive matrix, ρ1 , δ f 1 , and δ1 are positive design parameters. Consider the following Lyapunov function: L1 =
1 2 1 T −1 1 1 ˜2 x˜ + ω˜ ω˜ 1 + y22 + ϑ . 2g1 1 2 1 1 2 2ρ1 1
(66)
With (63) and Remark 2, the derivative of L 1 is calculated as L˙ 1 = x˜1 (x˜2 + y2 − k1 x˜1 ) + m 1 (z 1 )x˜1 ω˜ 1T θ1 + ε1 + (1 − m 1 (z 1 ))x˜1 χ1 × h 1 − ϑ1m H1(z 1 ) tanh w1 χ1 + (1 − m 1 (z 1 ))ϑ˜ 1 χ1 tanh w1 1 −1 − ω˜ 1T 1 ω˙ˆ 1 + y2 y˙2 − ϑ˜ 1 ϑ˙ˆ 1 ρ1 = x˜1 (x˜2 + y2 − k1 x˜1 ) + m 1 (z 1 )x˜1 ε1 y2 + δ f 1 ωˆ 1T ω˜ 1 − 2 + B2 y2 + δ1 ϑ˜ 1 ϑˆ 1 α2 + (1 − m 1 (z 1 ))x˜1 χ1 × h 1 − ϑ1m H1(z 1 ) tanh w1
(67)
where B2 = −x˙2c and |B2 | ≤ M2 . Step 2: Taking the derivative of the pitch angle error x˜2 , we have x˙˜2 = x˙2 − x˙2d = x 3 − x˙2d .
(68)
Take x 3 as the virtual control of (68) and design the signal x 3c as x 3c = −k2 x˜2 + x˙2d − x˜ 1
(69)
where k2 > 0 is the gain parameter. Introduce a new state variable x 3d , which can be obtained by the following first-order filter: α3 x˙3d + x 3d = x 3c , x 3d (0) = x 3c (0)
(70)
where α3 > 0 is the filter parameter. Define y3 = x 3d − x 3c , x˜3 = x 3 − x 3d . Then x˙˜ 2 = x 3 − x˙2d = x˜3 + y3 − k2 x˜2 − x˜ 1 .
(71)
Select L 2 = (1/2)(x˜22 + y32 ) and its derivative is calculated as L˙ 2 = x˜2 x˙˜ 2 + y3 y˙3 = x˜2 (x˜3 + y3 − k2 x˜2 − x˜1 ) − where B3 = −x˙3c and |B3 | ≤ M3 .
y32 + y3 B 3 α3
(72)
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Step 3: Taking the derivative of the pitch rate error x˜3 , we have x˙˜3 = x˙3 − x˙3d = g3 u + f3 − x˙3d = g3 u + g3−1 ( f 3 − x˙3d ) .
there exist ki , ρi , i , δ f i , i = 1, 3, such that all the signals in (82) are bounded. Proof: From the procedure of controller design, it is already known L˙ 1 = x˜1 (x˜2 + y2 − k1 x˜1 ) + m 1 (z 1 )x˜1 ε1 y2 + δ f 1 ωˆ 1T ω˜ 1 − 2 + B2 y2 + ϑ˜ 1 δ1 ϑˆ 1 α2 + (1 − m 1 (z 1 ))x˜1 χ1 × h 1 − ϑ1m H1 (z 1 ) tanh w1 y2 L˙ 2 = x˜2 (x˜3 + y3 − k2 x˜2 − x˜1 ) − 3 + y3 B3 . α3
(73)
Define z 3 = [X T, x˙3d ]T , and on the compact set z3 , we have h 3 (z 3 ) = g3−1 ( f 3 − x˙3d )
= ω3∗T θ3 (z 3 ) + ε3
(74)
where ω3∗ is the optimal NN weights vector, θ3 (z 3 ) is the basis vector, and ε3 is the NN approximation error with |ε3 | ≤ ε3m . Similarly the following inequality exists: U f 3 + |x˙3d | . |h 3 (z 3 )| ≤ g −1 (75) 3
Taking the derivative of L 3 leads to
Then, we can rewrite (75) into the following form: |h 3 (z 3 )| ≤ ϑ3m H3(z 3 )
1 ˙˜ 3 + 1 ϑ˜ 3 ϑ˙˜ 3 L˙ 3 = − x˜3 x˙˜ 3 + ω˜ 3T −1 f3 ω g3 ρ3 ˙ˆ 3 − 1 ϑ˜ 3 ϑ˙ˆ 3 = −x˜3 [u + h 3 ] − ω˜ 3T −1 f3 ω ρ 3 = −x˜3 (k3 x˜3 + x˜2 ) − m 3 (z 3 )x˜3 ω˜ 3T θ3 + ε3 − (1 − m 3 (z 3 ))x˜3 χ3 × h 3 + ϑ3m H3(z 3 ) tanh w 3 χ 3 + (1 − m 3 (z 3 ))χ3 ϑ˜ 3 tanh w3 1 − ω˜ 3T 3−1 ω˙ˆ 3 − ϑ˜ 3 ϑ˙ˆ 3 . ρ3
(76)
where ϑ3m is unknown positive H3(z 3 ) = f 3U + |x˙3d |. The control input u is designed as
constant
u = k3 x˜3 + x˜2 − m 3 (z 3 )u 3N + (1 − m 3 (z 3 ))u r3
and
(77)
with k3 > 0 as the gain parameter, and u 3N = ωˆ 3T θ3 (z 3 ) u r3 = ϑˆ 3 H3(z 3 ) tanh
χ3 w3
(78) (79)
where χ3 = H3(z 3 )x˜3 , ωˆ 3 is the estimation of ω3∗ , and w3 is the positive parameter. Design the updating laws as ω˙ˆ 3 = −3 [m 3 (z 3 )θ3 x˜3 + δ f 3 ωˆ 3 ] χ3 ˙ ˆ ˆ ϑ3 = ρ3 (1 − m 3 (z 3 ))χ3 tanh − δ 3 ϑ3 w3
(80) (81)
(84)
Substituting (80) and (81) into (84), we have L˙ 3 = −x˜3 (k3 x˜3 + x˜2 ) − m 3 (z 3 )x˜3 ε3 + δ f 3 ωˆ 3T ω˜ 3 + δ3 ϑ˜ 3 ϑˆ 3 − (1 − m 3 (z 3 ))x˜3 χ3 × h 3 + ϑ3m H3(z 3 ) tanh . w3
where 3 is a symmetric positive definitive matrix, ρ3 , δ f 3 , and δ3 are positive design parameters.
(85)
For i = 1, 3 and j = 2, 3, the following inequalities hold:
B. Stability Analysis Theorem 2: Consider the following Lyapunov function: LG =
3
Li
ω˜ iT ωˆ i ≤
(82)
ϑ˜ i ϑˆ i ≤
i=1
where
x˜i εi ≤
1 2 1 T −1 1 1 2 x˜1 + ω˜ 1 1 ω˜ 1 + y22 + L1 = ϑ˜ 2g1 2 2 2ρ1 1 1 2 L2 = x˜2 + y32 2 1 2 1 T −1 1 2 L3 = − ϑ˜ x˜3 + ω˜ 3 3 ω˜ 3 + 2g3 2 2ρ3 3
Bj yj ≤ x˜ j −1 y j ≤ and
where ω˜ 3 = ω3∗ − ωˆ 3 and ϑ˜ 3 = ϑ3m − ϑˆ 3 . Given constant κ, for any initial conditions fulfill L(0) =
3 i=1
L i (0) ≤ κ
1 ∗ 2 1 ω − ω˜ i 2 2 i 2 1 2 1 2 ϑ − ϑ˜ 2 im 2 i 1 2 1 2 x˜ + εim 2 i 2 1 2 1 2 M + yj 2 j 2 1 2 1 x˜ j −1 + y 2j 2 2
(83)
χ1 x˜1 −ϑ1m H1(z 1 ) tanh + h 1 ≤ ϑ1m κw1 w 1 χ3 + h 3 ≤ ϑ3m κw3 . −x˜3 ϑ3m H3(z 3 ) tanh w3
XU et al.: GLOBAL NEURAL DYNAMIC SURFACE TRACKING CONTROL OF STRICT-FEEDBACK SYSTEMS
Then, the derivative of L 1 is calculated as L˙ 1 ≤ −k1 x˜12 + x˜12 + y22 +
y2 M22 − 2 2 α2
1 1 − δ f 1 ω˜ 1 2 − δ1 ϑ˜ 12 + x˜1 x˜2 2 2 1 2 1 1 2 + ε1m + δ f 1 ω1∗ 2 + δ1 ϑ1m + ϑ1m κw1 2 2 2 1 1 = −(k1 − 1)x˜12 − − 1 y22 − δ f 1 ω˜ 1 2 α2 2 1 ˜2 − δ1 ϑ1 + k13 + x˜1 x˜2 2 ≤ −η1 L 1 + k13 + x˜ 1 x˜2 (86) where η1 = min[2k11 g¯ 1 , δ f 1 /(λmax (1−1 )), δ1 ρ1 , 2k12 ], k11 = k1 − 1 > 0, k12 = (1/α2 ) − 1 > 0, and 2 + (1/2)δ ω∗ 2 + (1/2)δ ϑ 2 + ϑ κw + k13 = (1/2)ε1m f1 1 1m 1m 1 1 2 (M2 /2). The derivative of L 2 is calculated as y32 + y3 B 3 α3 y2 M2 1 ≤ −k2 x˜22 + x˜22 + y32 − 3 + 3 − x˜1 x˜2 + x˜2 x˜3 2 α3 2 ≤ −η2 L 2 − x˜1 x˜2 + x˜2 x˜3 + k23 (87)
L˙ 2 = x˜2 (x˜3 + y3 − k2 x˜2 − x˜1 ) −
where η2 = min [2k21 , 2k22 ], k21 = k2 − (1/2) > 0, and k22 = (1/α3 ) − 1 > 0, k23 = (M32 /2). The derivative of L 3 is calculated as 1 1 1 1 2 L˙ 3 ≤ −k3 x˜ 32 + x˜ 32 − δ f 3 ω˜ 3 2 − δ3 ϑ˜ 32 − x˜2 x˜3 + ε3m 2 2 2 2 1 1 ∗ 2 2 + δ f 3 ω3 + δ3 ϑ3m + ϑ3m κw3 2 2 1 1 1 2 = − k3 − x˜3 − δ f 3 ω˜ 3 2 − δ3 ϑ˜ 32 + k33 − x˜2 x˜3 2 2 2 ≤ −η3 L 3 + k33 − x˜2 x˜3 (88) = min[2k31 g¯ 3 , (δ f 3 /λmax (3−1 )), δ3 ρ3 ], where η3 2 + (1/2)δ ω∗ 2 + k31 = k3 − (1/2) > 0, and k33 = (1/2)ε3m f3 3 2 (1/2)δ3 ϑ3m + ϑ3m κw3 . Finally, we have L˙ G =
3 i=1
L˙ i ≤ −
≤ −ηL G + K
3 i=1
ηi L i +
3
ki3
i=1
(89)
where η = min[η1 , η2 , η3 ], K = 3i=1 ki3 . According to the Lasalle–Yoshizawa theorem, the closed-system stability is uniformly ultimately bounded. This completes the proof. Remark 9: Since g3 is negative, the signs of k3 x˜3 , x˜2 , and ϑˆ 3 H3(z 3 ) tanh(χ3 /w3 ) are set as positive. In case of unknown control directions, the Nussbaum function-based robust adaptive design [15] could be employed. Remark 10: Different from the indirect design in Section IV where the control gain function gi , i = 1, 3 should be estimated in each step, the direct design is on approximating the complex unknown functions h i .
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Remark 11: By introducing the first-order filter, the inputs for NN can be selected intuitively, and it facilitates the design in [41]. The upper bound of the nonlinearity h i is derived by analyzing the components, and it is formulated into the estimation of ϑim . This is much more convenient since the function ϕi (see [41, eq. (8)]) is difficult to construct. VI. S IMULATIONS In this section, the simulation study is performed to verify the effectiveness of the proposed global neural controller. The initial values of the states are set as V0 = 7850 ft/s, h 0 = 86 000 ft, α0 = 3.5°, γ0 = 0, and q0 = 0. The step command is Vc = 400 ft/s, h c = 1000 ft. The reference commands of h r and Vr are generated 2 /(s + ω )(s 2 + 2ε ω s + ω2 )) and by the filters (ωn1 ωn2 n1 c n2 n2 2 2 2 (ωv1 ωv2 /(s + ωv1 )(s + 2εc ωv2 s + ωv2 )), respectively, with ωn1 = 0.5, ωn2 = 0.2, ωv1 = 0.3, ωv2 = 0.2, εc = 0.7. The FPA command parameters are set as kh = 0.5, ki = 0.1. The filter parameters are selected as αi = 0.05, i = 2, 3 and the parameters for switching function are i = 1, b = 2, ri2 = 1, and ri1 = 0.1. The robust parameters are set as w1 = 2 × 10−6 , w3 = 10−3 . For the velocity system, the PID controller is employed and the parameters are selected as K pv = 0.5, K i v = 0.001, and K dv = 0.01. The normalization parameters for x id are selected as [−0.008; 0.008], [0.04; 0.09], and [−0.08; 0.08], separately. Example 1: Global Indirect Neural Control. The control gains for the DSC are selected as k1 = 2, k2 = 2, and k3 = 5. Parameters for adaptive laws are selected as f i = 2I , gi = 10−6 , δ f i = 0.01, δgi = 0.02, i = 1, 3. The number of NN nodes is set as N1 = 21 and N3 = 7×7×3, with their centers x 1 , x 2 , and x 3 being evenly spaced in the normalization region. The bound for Wg1 is selected as [0.6(1/m)C Lα , 1.4(1/m)C Lα ], and the bound for Wg3 is selected δe δe as [1.4(1/I yy )cC ¯ M , 0.6(1/I yy )cC ¯ M ]. The model parameters can be referred to [20]. The simulation results are presented in Figs. 3–5. From the tracking performance shown in Fig. 3, the altitude and velocity are controlled to follow the command references very well. From the response of system states shown in Fig. 3, it is obvious to know that the states converge to new trim values. Fig. 4 shows the control inputs of elevator deflection and fuel equivalence ratio. The trajectory of switching signal is demonstrated in Fig. 5, and it is observed that initially m 3 is zero since the pitch rate is out of the neural approximation region. With the robust controller, in about 15 s, the signals m 1 and m 3 achieve one, which means the states are pulled back to the transient. Initially, the NN weights adaption in Fig. 5 is changing fast due to the system tracking error, and then the value converges to some constant. It can be concluded that with the proposed global indirect design, the controller achieves good tracking results in case of unknown dynamics. Example 2: Global Direct Neural Control. The control gains for the DSC are selected as k1 = 20, k2 = 1, and k3 = 5. Parameters for adaptive laws are selected as i = 0.01I , ρi = 0.2, δ f i = 20, δi = 20,
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Fig. 3.
System tracking. Left: altitude tracking. Middle: velocity tracking. Right: system states response.
Fig. 4.
Control input. Left: elevator deflection. Right: fuel equivalence ratio.
Fig. 5.
Signal response. Left: switching signal. Right: NN weights adaption.
i = 1, 3. The initial values of ϑˆ i (0), i = 1, 3 are selected as 0. The normalization parameters for x˙id , i = 1, 3 are selected as [−0.01; 0.01].
The number of NN nodes is set as N1 = 7 × 3 and N3 = 7 × 7 × 3 × 3. The simulation results are presented in Figs. 6–8. Fig. 6 demonstrates that satisfied tracking
XU et al.: GLOBAL NEURAL DYNAMIC SURFACE TRACKING CONTROL OF STRICT-FEEDBACK SYSTEMS
Fig. 6.
System tracking. Left: altitude tracking. Middle: velocity tracking. Right: system states response.
Fig. 7.
Control input. Left: elevator deflection. Right: fuel equivalence ratio.
Fig. 8.
Signal response. Left: switching signal. Middle: NN weights adaption. Right: robust item.
performance is obtained with the direct design without approximating the control gain function. It is observed that the response of the control inputs in Fig. 7 is different from the indirect design shown in Fig. 4. The switching signals
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shown in Fig. 8 respond according to the designed approximation region. It is obvious that the NN weights adaption and the estimation of the robust item in Fig. 8 are bounded.
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 10, OCTOBER 2015
VII. C ONCLUSION In this paper, both indirect and direct global neural controllers with the dynamic surface design are developed for the strict-feedback systems. The novel switching controller with the adaptive neural controller and the robust controller is presented. Different from the previous results on SGUUB, the GUUB stability is rigorously established by the Lyapunov approach. In the indirect design, the dynamic inversion control is designed by estimating functions f i and gi , separately, while the robust controller is constructed using the upper bound of | f i |. Differently, in the direct design, all the unknown dynamics h i is approximated by NN, and the upper bound |h i | is estimated to construct the robust controller. The simulation results of hypersonic flight dynamics are presented to demonstrate the feasibility of the proposed global neural DSC design. In the future work, there are still some interesting problems to be addressed. For global tracking, the system in this paper is in the strict-feedback form. Furthermore, the idea could be extended to more general systems, e.g., pure-feedback system, large-scale system. The time-varying disturbance might be taken into consideration for more effective control. For hypersonic flight control, this paper presents the general global neural design to eliminate the drawback in the previous intelligent design. In the next step, the important features [1] of a large flight envelope, aerothermoelastic effect, flexible effect, and nonminimum phase characteristic will be considered for hypersonic flight control. R EFERENCES [1] B. Xu and Z. Shi, “An overview on flight dynamics and control approaches for hypersonic vehicles,” Sci. China Inf. Sci., vol. 58, no. 7, pp. 1–19, Jul. 2015. [2] D. K. Schmidt, “Optimum mission performance and multivariable flight guidance for airbreathing launch vehicles,” J. Guid., Control, Dyn., vol. 20, no. 6, pp. 1157–1164, 1997. [3] F. R. Chavez and D. K. Schmidt, “Analytical aeropropulsive-aeroelastic hypersonic-vehicle model with dynamic analysis,” J. Guid., Control, Dyn., vol. 17, no. 6, pp. 1308–1319, 1994. [4] D. Sigthorsson, P. Jankovsky, A. Serrani, S. Yurkovich, M. Bolender, and D. B. Doman, “Robust linear output feedback control of an airbreathing hypersonic vehicle,” J. Guid., Control, Dyn., vol. 31, no. 4, pp. 1052–1066, 2008. [5] Z. Wilcox, W. MacKunis, S. Bhat, R. Lind, and W. Dixon, “Lyapunov-based exponential tracking control of a hypersonic aircraft with aerothermoelastic effects,” J. Guid., Control, Dyn., vol. 33, no. 4, pp. 1213–1224, 2010. [6] G.-X. Wen, Y.-J. Liu, and C. P. Chen, “Direct adaptive robust NN control for a class of discrete-time nonlinear strict-feedback SISO systems,” Neural Comput. Appl., vol. 21, no. 6, pp. 1423–1431, Sep. 2012. [7] S. Mehraeen, S. Jagannathan, and M. L. Crow, “Decentralized dynamic surface control of large-scale interconnected systems in strict-feedback form using neural networks with asymptotic stabilization,” IEEE Trans. Neural Netw., vol. 22, no. 11, pp. 1709–1722, Nov. 2011. [8] H. Liu, “A fuzzy qualitative framework for connecting robot qualitative and quantitative representations,” IEEE Trans. Fuzzy Syst., vol. 16, no. 6, pp. 1522–1530, Dec. 2008. [9] H. Li, L. Wu, H. Gao, X. Hu, and Y. Si, “Reference output tracking control for a flexible air-breathing hypersonic vehicle via output feedback,” Optim. Control Appl. Methods, vol. 33, no. 4, pp. 461–487, Jul./Aug. 2012. [10] N. Wang, H.-N. Wu, and L. Guo, “Coupling-observer-based nonlinear control for flexible air-breathing hypersonic vehicles,” Nonlinear Dyn., vol. 78, no. 3, pp. 2141–2159, Nov. 2014. [11] Q. Wang and R. F. Stengel, “Robust nonlinear control of a hypersonic aircraft,” J. Guid., Control, Dyn., vol. 23, no. 4, pp. 577–585, 2000.
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XU et al.: GLOBAL NEURAL DYNAMIC SURFACE TRACKING CONTROL OF STRICT-FEEDBACK SYSTEMS
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Chenguang Yang (M’10) received the B.Eng. degree in measurement and control from Northwestern Polytechnical University, Xi’an, China, in 2005, and the Ph.D. degree in control engineering from the National University of Singapore, Singapore, in 2010. He was with Imperial College London, London, U.K., as a Research Associate, where he was involved in human–robot interaction from 2009 to 2010. He has been with Plymouth University, Devon, U.K., as a Lecturer in Robotics, since 2010. His current research interests include robotics, control, and human–robot interaction.
Yongping Pan (M’14) received the B.Eng. degree in automation and the M.Eng. degree in control theory and control engineering from the Guangdong University of Technology (GDUT), Guangzhou, China, in 2004 and 2007, respectively, and the Ph.D. degree in control theory and control engineering from the South China University of Technology (SCUT), Guangzhou, in 2011. He was a Control Engineer with the Eaton Group, Santak Electronic Company, Ltd., Shenzhen, China, and Light Engineering Company, Ltd., Guangzhou, from 2007 to 2008. From 2011 to 2013, he was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is currently a Research Fellow with the Department of Biomedical Engineering, National University of Singapore, Singapore. He has authored or co-authored over 50 peer-reviewed research papers in journals and conferences. His current research interests include automatic control, computational intelligence, robotics, and automation. Dr. Pan was a recipient of the Rockwell Automation Master Scholarship and the GDUT Post-Graduate Academic Award in 2006, and the SCUT Innovation Fund of Excellent Doctoral Dissertations and the SCUT Excellent Graduate Student Award in 2010. He is an Associate Editor of the International Journal of Fuzzy Systems, and serves as a Reviewer for a number of flagship journals.
