This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS
1
Composite Neural Dynamic Surface Control of a Class of Uncertain Nonlinear Systems in Strict-Feedback Form Bin Xu, Zhongke Shi, Chenguang Yang, Member, IEEE, and Fuchun Sun
Abstract—This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strictfeedback form. Dynamic surface control technique is incorporated into radial-basis-function neural networks (NNs)-based control framework to eliminate the problem of explosion of complexity. To avoid the analytic computation, the command filter is employed to produce the command signals and their derivatives. Different from directly toward the asymptotic tracking, the accuracy of the identified neural models is taken into consideration. The prediction error between system state and serial–parallel estimation model is combined with compensated tracking error to construct the composite laws for NN weights updating. The uniformly ultimate boundedness stability is established using Lyapunov method. Simulation results are presented to demonstrate that the proposed method achieves smoother parameter adaption, better accuracy, and improved performance. Index Terms—Composite control, dynamic surface control, neural network, serial–parallel estimation model, strict-feedback.
I. Introduction OR NONLINEAR control systems, numerous approaches on adaptive and robust control have been proposed. Among these, back-stepping has been widely studied because it provides a systematic framework for the design of tracking and regulation strategies, suitable for a large class of state feedback linearizable nonlinear systems [1], [2]. For systems with high uncertainty, the universal approximation ability of fuzzy logic system (FLS)/neural network (NN) enhances efficiency of the intelligent control [3]–[10] and FLS/NN is widely employed for back-stepping based designs [11]–[14].
F
Manuscript received August 4, 2013; revised November 13, 2013 and February 26, 2014; accepted February 28, 2014. This work was supported in part by the National Science Foundation of China under Grant 61304098, Grant 61134004, and Grant 61210013, in part by the NWPU Basic Research Funding under Grant JC20120236, in part by the Royal Society Research Grant (RG130244), and in part by the Foundation of Key Laboratory of Autonomous Systems and Networked Control (Chinese Ministry of Education) under Grant 2012A04. This paper was recommended by Associate Editor T. H. Lee. B. Xu and Z. Shi are with the School of Automation, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]; [email protected]). C. Yang is with the School of Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, U.K. (e-mail: [email protected]). F. Sun is with the Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China (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/TCYB.2014.2311824
In [15], the back-stepping neural control approach was presented and higher tracking accuracy can be guaranteed by adding one robust item. In [16], an adaptive back-stepping NN control approach was extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections. In [17], variable structure control in combination with back-stepping was proposed for adaptive NN control design with guaranteed stability for uncertain multiple-input–multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities. In the conventional back-stepping design, the virtual controller designed at each step needs repeated differentiation. The explosion of complexity of back-stepping design can be eliminated by dynamic surface control (DSC) technique in which the virtual control is passed through a first-order filter to obtain the derivative [18]. In [19], the DSC approach in [18] was extended to an NN-based adaptive tracking control of a class of single-input–single-output (SISO) system and the pure feedback SISO system was studied in [20]. In [21], fuzzy DSC design was studied where FLS was combined with a Fourier series expansion to model unknown periodically disturbed system functions. In [22], to deal with the difficulty of unknown time-delay functions, the DSC was integrated with the function separation technique, the Lyapunov–Krasovskii functionals, and the desirable property of hyperbolic tangent functions. The problem of adaptive output-feedback control for a class of uncertain stochastic nonlinear strict-feedback systems with time-varying delays was studied in [23]. The observer-based adaptive fuzzy back-stepping DSC approach was developed for MIMO nonlinear systems, stochastic nonlinear strict-feedback systems, and uncertain nonlinear largescale systems with immeasurable states in [24]–[27], respectively, while the corresponding stability of the closed-loop systems was proved. In [28], command filter design was developed by generating compensating signals to remove the c d known effect of xi+1 . Due to simplicity, the DSC − xi+1 method is widely applied on flexible-joint robot control [29], ship control [30], power system control [31], marine vessels control [32], and flight control [33], [34]. Although the FLS/NN-based DSC has made much progress, the original intention employing FLS/NN for approximating the system uncertainty is missing. Intuitively, the more precise approximation of the nonlinear function is achieved, the better
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
IEEE TRANSACTIONS ON CYBERNETICS
performance is expected. However, most efforts have been directed toward achieving asymptotic stability and tracking. Little attention has been paid to the accuracy of the identified models as well as the transparency and interpretability [37]. Motivated by the modeling error, the hybrid adaptive fuzzy identification and control was proposed in [35], which achieves faster and improved tracking. However, the nth derivative of the plant output is required to be known, which is quite impractical. Due to the uncertainty, in [36], the signal e(n) was approximated by error-changing rate as e(n) (t) ≈
(n−1)
e
(t) − e t
(n−1)
(t − t)
.
II. Problem Formulation The class of SISO nonlinear plant [19] considered in this paper is described as follows:
T
In this paper, the RBF NNs [41] are employed to approximate the unknown nonlinearity f with the following form: fˆ (Xin ) = ωˆ T θ (Xin )
(3)
where Xin ∈ R is the input vector of the RBF NNs, fˆ ∈ R is the NN output, ωˆ ∈ RLN is the adjustable parameter vector, θ (·) : RM → RLN is a nonlinear vector function of the inputs, and LN is the number of NN nodes. The components of θ are the basis functions denoted by ρi . A commonly used basis function is the so-called Gaussian function of the following form: Xin − ξi 2 1 ρi (Xin ) = √ (4) exp − 2σi2 2πσi M
where ξi is an M-dimensional vector representing the center of the ith basis function and σi is the variance representing the spread of the basis function. For any given real continuous function f on a compact set Xin ∈ RM and an arbitrary εM > 0, there exist RBF NNs in the form of (5) and an optimal parameter vector ω∗ such that f (Xin ) = ω∗ T θ (Xin ) + ε sup | ε |< εM
Xin ∈ Xin
(5) (6)
where εM > 0 denotes the supremum value of the reconstruction error ε that is inevitably generated.
IV. Composite DSC Design With Prediction Error In this section, the prediction error derived from the difference between system state and serial–parallel estimation model will be incorporated into the DSC technique for the nthorder system described by (2). The recursive design procedure contains n steps. From Step 1 to Step n − 1, virtual control d xi+1 , i = 1, . . . , n − 1 is designed at each step and an overall control law u is constructed in the last step n. Step 1: Considering the first equation in (2) and using NN to approximate f1 (¯x1 ), we have x˙ 1 = f1 (¯x1 ) + x2 = ω1∗T θ1 (¯x1 ) + ε1 + x2
(7)
where ω1∗ is the optimal NN weights vector and ε1 is the NN approximation error with | ε1 |≤ εM . Define the tracking error (2)
where x(t) ∈ R is the state, x¯ i = [x1 , , . . . , xi ] , u ∈ R is the input, y ∈ R is the output, and fi , i = 1, . . . , n are unknown smooth functions of x¯ i . n
III. RBF Neural Networks
(1)
Since the information employed to construct the approximation error is not rigorous, during the controller design and stability analysis, it is difficult to analyze the influence of the estimation error. In [37], the composite design was studied using prediction error with a slightly different serial–parallel estimation model [38]. In [39], the method was applied on controlling two uncertain generalized Lorenz systems. In [40], to deal with the problem that the H ∞ control terms may degrade the fuzzy approximation ability, the composite design was employed and the fuzzy approximation ability was improved. It is noticed that the model studied in [35], [37], and [40] was restricted to the canonical form. The convenience provided by this model is that the derivative information can be directly derived from the states, but this information is not available for strict-feedback system with uncertainty. In this paper, a class of SISO strict-feedback system is studied. Different from [36], the NN modeling-related prediction error is defined between the state and the serial–parallel estimation model. Furthermore, the compensating signal is incorporated to eliminate the effect of the known error caused by command filter. The composite DSC approach is presented employing both compensated tracking error and prediction error. With the new composite DSC method, we show that better tracking performance is achieved. This paper is organized as follows. In Section II, the class of SISO nonlinear systems is characterized. The brief introduction of radial-basis-function neural networks (RBF NNs) is given in Section III. In Section IV, the composite DSC control is designed, and the stability analysis is presented in Section V. The effectiveness of the proposed approach is verified by simulation in Section VI. The conclusion is drawn in Section VII.
x˙ i = fi (¯xi ) + xi+1 , 1 ≤ i ≤ n − 1 x˙ n = fn (¯xn ) + u y = x1
For a given reference yr (t), t ≥ 0, which is a sufficiently smooth function of t with its time derivative y˙ r bounded for t ≥ 0, the control objective is to find an adaptive neural control law such that all signals and states defined in the closedloop system are bounded and the output tracks the bounded reference trajectory yr .
e1 = x1 − yr .
(8)
Choose virtual control x2d as x2d = −ωˆ 1T θ1 (¯x1 ) − k1 e1 + y˙ r
(9)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
where ωˆ 1 is the estimation of ω1∗ , k1 > 0 is the design constant. Introduce a new state variable x2c and let x2d pass through a first-order filter with time constant α2 > 0 to obtain x2c α2 x˙ 2c + x2c = x2d , x2c (0) = x2d (0).
(10)
Define e2 = x2 − x2c . Then the derivative of e1 is obtained as e˙ 1 = x˙ 1 − y˙ r = ω1∗T θ1 (¯x1 ) + ε1 + x2 − y˙ r =
ω˜ 1T θ1 (¯x1 )
(x2c
−
x2d )
− ωˆ1 . where ω˜ 1 = To remove the effect of the known error (x2c − x2d ), the compensating signal z1 is designed as z˙ 1 = −k1 z1 + z2 +
−
c d z˙ i = −ki zi − zi−1 + zi+1 + (xi+1 − xi+1 ), zi (0) = 0.
z1 (0) = 0
(12)
where z2 will be defined in the next step. Now we obtain the compensated tracking error signals ν1 = e1 − z1 , ν2 = e2 − z2 .
(13)
The prediction error is defined as z1NN = x1 − xˆ 1
(14)
where the derivative of NN modeling information is defined with the serial–parallel estimation model [42], [43] xˆ˙ 1 = ωˆ 1T θ1 (¯x1 ) + x2 + β1 z1NN , xˆ 1 (0) = x1 (0)
(15)
with β1 > 0 as the user-defined positive constant. For the NN updating law, the signal z1NN is employed to construct the learning design ωˆ˙ 1 = γ1 (ν1 + γz1 z1NN ) θ1 (¯x1 ) − δ1 ωˆ 1 (16) where γ1 , γz1 , and δ1 are positive design constants. Step i: i = 2, . . . , n − 1. Considering the ith equation in (2) and using NN to approximate fi (¯xi ), we know x˙ i = fi (¯xi ) + xi+1 = ωi∗T θi (¯xi ) + εi + xi+1
(17)
where ωi∗ is the optimal NN weights vector and εi is the NN approximation error with | εi |≤ εM . Define the ith error surface ei to be ei = xi − xic
=
−ωˆ iT θi (¯xi )
νi = ei − zi
(23)
ziNN = xi − xˆ i
(24)
(18)
where the derivative of NN modeling information is defined with the serial–parallel estimation model xˆ˙ i = ωˆ iT θi (¯xi ) + xi+1 + βi ziNN , xˆ i (0) = xi (0) with βi > 0 as the user-defined positive constant. The update law of ωˆ i is designed to be ωˆ˙ i = γi (νi + γzi ziNN ) θi (¯xi ) − δi ωˆ i
− ki ei − ei−1 +
x˙ ic
(19)
ωi∗ ,
ki > 0 is the design constant. where ωˆ i is the estimation of d c and let xi+1 Introduce a new state variable xi+1 pass through a first-order filter with the time constant αi+1 > 0 to obtain c xi+1 c c d d c αi+1 x˙ i+1 + xi+1 = xi+1 , xi+1 (0) = xi+1 (0).
(20)
c . Then the derivative of ei is obtained Define ei+1 = xi+1 − xi+1 as
e˙ i = x˙ i − x˙ ic c d = ω˜ iT θi (¯xi ) + εi − ki ei − ei−1 + ei+1 + (xi+1 − xi+1 ) (21)
(25)
(26)
where γi , γzi , and δi are positive design constants. Step n: Considering the nth equation in (2) and using NN to approximate fn (¯xn ), we know x˙ n = fn (¯xn ) + u = ωn∗T θn (¯xn ) + εn + u
(27)
where ωn∗ is the optimal NN weights vector and εn is the NN approximation error with | εn |≤ εM . Define the nth error surface en to be en = xn − xnc .
(28)
The final control u is designed as u = −ωˆ nT θn (¯xn ) − kn en − en−1 + x˙ nc
(29)
ωn∗ ,
where ωˆ n is the estimation of kn > 0 is the design constant. Then the derivative of en is obtained as e˙ n = x˙ n − x˙ nc = ω˜ nT θn (¯xn ) + εn − kn en − en−1
(30)
ωn∗
where ω˜ n = − ωˆ n . The compensating signal is defined as z˙ n = −kn zn − zn−1 , zn (0) = 0.
(31)
Define the compensated tracking error signal
d is designed as The virtual control xi+1 d xi+1
(22)
Define the compensated tracking error signal
(11)
ω1∗
x2d ),
where ω˜ i = ωi∗ − ωˆ i . d c To remove the effect of the known error (xi+1 − xi+1 ), the compensating signal zi is defined as
and the prediction error
+ ε 1 − k 1 e1 + e 2 +
(x2c
3
νn = en − zn
(32)
znNN = xn − xˆ n
(33)
and the prediction error
where the derivative of NN modeling information is defined with the serial–parallel estimation model xˆ˙ n = ωˆ nT θn (¯xn ) + u + βn znNN , xˆ n (0) = xn (0) with βn > 0 as the user-defined positive constant. The update law of ωˆ n is designed to be ωˆ˙ n = γn (νn + γzn znNN ) θn (¯xn ) − δn ωˆ n where γn , γzn , and δn are positive design constants.
(34)
(35)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4
IEEE TRANSACTIONS ON CYBERNETICS
Remark 1: The NN approximation in (25) is used to obtain xˆ˙ i and the prediction error is employed for parameter adjustment in (26). There is no need to calculate the derivatives of the states and thus there is no such approximation as in (1). Remark 2: The δ-modification [15] is introduced to improve the robustness and to avoid the weight parameters to drift to very large values. Remark 3: For back-stepping design, the tracking error vector is defined as e¯ BS = [e1,BS , · · · , en,BS ] with ei,BS = xi − x¯ id for i = 1, · · · , n, where x¯ 1d = yr . The vector of function x¯ d = [¯x2d , · · · , x¯ nd ], referred to as virtual control signals, is defined as
For the NN prediction error, using (9), (19), and (29) with (15), (25), and (34), we have z˙ iNN = x˙ i − xˆ˙ i = ω˜ iT θi (¯xi ) + εi − βi ziNN .
Define mi = ω˜ iT θi (¯xi ), the following equation can be obtained: z˙ iNN ziNN = ziNN (mi + εi ) − βi z2iNN .
d
V˙ = (36)
n
n νi ν˙ i − ω˜ iT γi−1 ωˆ˙ i + γzi ziNN z˙ iNN
i=1
=
n
i=1
−ki νi2 + νi mi + νi εi
i=1
The updating of NN weights vector ωˆ i,BS refers to the design [19] using tracking error ei,BS . As the order n increases, d analytic computation of x¯˙ i becomes increasingly complicated. Remark 4: Compared with back-stepping design (36), the composite neural DSC controller proposed in this paper has the following features. 1) The method is based on the command filter that produces certain command signals and their derivatives. In this way, there is no load of the aforementioned analytic computation. 2) Only the information of yr (t) and y˙ r (t) is required in this paper while back-stepping design needs the knowledge of yr(i) (t). 3) The prediction error of NN modeling together with the compensated tracking error is included in the composite NN updating law to provide fast adaption ability.
−
Theorem 1: Let yr (t) and y˙ r (t) be continuous, bounded for t ≥ 0. For system (2) with the DSC laws (9), (19), and (29), the NN updating law (16), (26), and (35), and the compensated error signals νi defined in (23), it is guaranteed that the signals νi , ω˜ i , and ziNN are uniformly ultimately bounded. Proof: The Lyapunov function is chosen as n n 1 1 2 νi + ω˜ iT γi−1 ω˜ i + V = γzi z2iNN . (37) 2 i=1 2 i=1 For error dynamics of the compensated tracking error, we have ν˙ 1 = e˙ 1 − z˙ 1 = ω˜ 1T θ1 (¯x1 ) + ε1 − k1 (e1 − z1 ) + (e2 − z2 ) = ω˜ 1T θ1 (¯x1 ) + ε1 − k1 ν1 + ν2
(38) ν˙ i = e˙ i − z˙ i = ω˜ iT θi (¯xi ) + εi − ki (ei − zi ) − (ei−1 − zi−1 ) + (ei+1 − zi+1 ) = ω˜ iT θi (¯xi ) + εi − ki νi − νi−1 + νi+1 , i = 2, · · · , n − 1 (39)
ν˙ n = e˙ n − z˙ n = ω˜ nT θn (¯xn ) + εn − kn (en − zn ) − (en−1 − zn−1 ) (40)
n
mi (νi + γzi ziNN ) +
i=1
+
n
=
n
n
δi ω˜ iT ωˆ i
i=1
γzi ziNN (mi + εi ) −
i=1
n
γzi βi z2iNN
i=1
−ki νi2 + νi εi + γzi ziNN εi − γzi βi z2iNN
i=1 −δi ω˜ iT ω˜ i + δi ω˜ iT ωi∗ .
(43)
Consider the following facts:
εi 2 1 2 νi εi − ki νi2 = −ki νi − + ε 2ki 4ki i
εi 2 1 2 2 ziNN εi − βi ziNN = −βi ziNN − + εi 2βi 4βi 2 ωi∗ + 1 ω∗ 2 . ω˜ iT ωi∗ − ω˜ iT ω˜ i = − ω ˜ − i 2 4 i
V. Stability Analysis
= ω˜ nT θn (¯xn ) + εn − kn νn − νn−1 .
(42)
The derivative of V is derived as
T θ1 (x1 ) − k1 e1,BS + y˙ r x¯ 2d = −ωˆ 1,BS d T x¯ i+1 = −ωˆ i,BS θi (¯xi ) − ki ei,BS + x¯˙ i − ei−1,BS i = 2, · · · , n − 1 d T u = −ωˆ n,BS θn (¯xn ) − kn en,BS + x¯˙ n − en−1,BS .
(41)
Then
εi 2 1 2 V˙ = − k i νi − − ε 2ki 4ki i i=1
n εi 2 1 2 γzi βi ziNN − − ε − 2βi 4βi i i=1
n ∗ 2 ∗ 2 ω 1 i − δi ω˜ i − 2 − 4 ωi i=1
n εi 2 εi 2 ≤− k 0 νi − + γz min β0 ziNN − 2ki 2βi i=1
2 ωi∗ +P ω ˜ +δ0 (44) − i 2 n
where k0 = min [ki ], β0 = min [βi ], γz min = min γzi , n 2 nγz max 2 nδmax 2 δ0 = min [δi ], P = ωmax , γz max = εM + εM + 4k 4β 4 0 0 ∗ max γzi , ωmax = max ωi , and δmax = max [δi ].
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
5
P εi P εi If νi − or ziNN − or ≥ ≥ 2k k 2β γ 0 i z min β0 i ∗ ω˜ i − ωi ≥ P , then V˙ ≤ 0. Then we know that νi , ziNN , 2 δ0 and ω˜ i are invariant to the sets defined as follows: P εM + νi = νi |νi | ≤ k0 2k0 εM P ziNN = ziNN |ziNN | ≤ + γz min β0 2β0 P ωmax ω˜ i = ω˜ i ω˜ i ≤ . (45) + δ0 2 So all the signals are uniformly ultimately bounded. This completes the proof. Remark 5: By increasing the values of ki and βi , the P εM P εM quantity , , , and can be made arbitrarily k0 2k0 γz min β0 2β0 small. Thus, the error νi and ziNN may be made arbitrarily small. Remark 6: The singular perturbation analysis can be applied with the similar way as in [28] that by decreasing αi , the solution of the design can be made arbitrarily close to the back-stepping solution based on analytic derivatives. Remark 7: In [44, Lemma 3], it is presented that the compensating signals zi are bounded. With the conclusion of Theorem 1, we know that signals νi are bounded. Then from the definition of compensated signal in (23), the boundedness of ei can be deduced. Remark 8: It is easy to combine the robust design [15], minimal parameter technique [45] with the composite DSC approach to enhance the performance. The purpose here is to clearly demonstrate how the composite design is constructed; in this paper, we do not put much emphasis on it. Remark 9: It is easy to extend the result in this paper to the strict-feedback system with known gi (¯xi ) = 0. The dynamics are in the following formulation:
Fig. 1.
System tracking. (a) Tracking Performance. (b) Tracking Error.
Fig. 2.
NN weights adaptation.
x˙ i = fi (¯xi ) + gi (¯xi )xi+1 , 1 ≤ i ≤ n − 1 x˙ n = fn (¯xn ) + gn (¯xn )u y = x1 .
(46)
Accordingly, the controller is designed as −ωˆ 1T θ1 (¯x1 ) − k1 e1 + y˙ r g1 T −ωˆ i θi (¯xi ) − ki ei − gi−1 ei−1 + x˙ ic d xi+1 = ,2 ≤ i ≤ n − 1 gi −ωˆ nT θn (¯xn ) − kn en − gn−1 en−1 + x˙ nc u= (47) gn x2d =
and
z˙ n = −kn zn − gn−1 zn−1 , zn (0) = 0
(50)
xˆ˙ n = ωˆ nT θn (¯xn ) + gn u + βn znNN , xˆ n (0) = xn (0).
(51)
while the compensating signals and the estimation are modified as c d − xi+1 ), zi (0)=0(48) z˙ i = −ki zi −gi−1 zi−1 +gi zi+1 + gi (xi+1 T (49) xˆ˙ i = ωˆ i θi (¯xi ) + gi xi+1 + βi ziNN , xˆ i (0) = xi (0) 1 ≤ i≤n−1
It is easy to construct the same Lyapunov function as (37). Following the same procedure, the boundedness of closedloop signals can be proved and thus the analysis is omitted here.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6
IEEE TRANSACTIONS ON CYBERNETICS
Fig. 3.
Estimation of fi .
Fig. 4.
System tracking. (a) Tracking Performance. (b) Tracking Error.
VI. Simulation
Fig. 5.
In this section, the third-order uncertain nonlinear systems in strict-feedback form are considered x˙ 1 = f1 (¯x1 ) + g1 (¯x1 )x2
To show the effectiveness of the method proposed in this paper, the comparison is conducted with the DSC method in [19], where the NN updating law has the following formulation: ωˆ˙ i = γi ei θi (¯xi ) − γi δi ωˆ i .
x˙ 2 = f2 (¯x2 ) + g2 (¯x2 )x3 x˙ 3 = f3 (¯x3 ) + g3 (¯x3 )u y = x1
NN weights adaptation.
(52)
where fi are unknown smooth functions while gi , i = 1, 2, 3 are known. The objective is to design a control law u such that the output of the closed-loop system can approximately track a reference input yr = sin(t) asymptotically. To simulate the overall control system, the differential equations of the system and the adaptive NN updating law are integrated with the MATLAB function ODE45.
(53)
The initial state of the system is 0. The control parameters chosen for simulation are: ki = 20, γi = 10, δi = 0.001, i = 1, 2, 3, and α2 = α3 = 0.005. For the NN design, the centers for x1 , x2 , and x3 are evenly spaced in [−1, 1]×[−3, 3]×[−5, 5]. For f1 , the NN nodes number is N1 = 21. For f2 , the numbers of the RBFs for x1 and x2 are seven and seven, respectively, and N2 = 49. For f3 , the numbers of the RBFs for x1 , x2 , and x3 are seven, seven, and three, respectively, and N3 = 147. To show the different tracking result, we denote the DSC method in [19] as DSC-CLASSIC and the method proposed with prediction error in Section IV as DSC-PE. The related parameters are selected as βi = 5, γzi = 1 for DSC-PE, i =
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
Fig. 6.
Estimation of fi .
Fig. 7.
System tracking. (a) Tracking Performance. (b) Tracking Error.
1, 2, 3. The systems studied in the first two examples are the same as [19] while the third example will consider the case with gi = 1. Example 1: Definitions of fi , gi , i = 1, 2, 3 are as follows: f1 (x1 ) = x13 f2 (¯x2 ) = x12 + x22 f3 (¯x3 ) = 0 gi = 1, i = 1, 2, 3.
(54)
The tracking performance is presented in Figs. 1–3. Fig. 1(a) shows the tracking performance. From the tracking error depicted in Fig. 1(b), it is clearly demonstrated that the DSC-PE achieves faster adaption and improved accuracy. The conclusion can be further confirmed from the NN response in Fig. 2 and the nonlinear tracking of fi shown in Fig. 3. It should be noted that the fi , i = 1, 2 are the functions of the states so that the values with DSC-PE are different from the ones with DSC-CLASSIC.
Fig. 8.
7
NN weights adaptation.
Example 2: The definitions of fi , gi , i = 1, 2, 3 are as follows: f1 (x1 ) = 2x12 sin(x1 ) f2 (¯x2 ) = x12 + x1 x2 + x2 cos x1 f3 (¯x3 ) = x1 x3 + x22 + x3 sin x2
(55)
gi = 1, i = 1, 2, 3. The tracking performance is presented in Figs. 4–6. Similar conclusion could be obtained that in comparison with the DSC-CLASSIC, the proposed DSC-PE method can achieve faster tracking and the NN modeling error decreases faster. Example 3: The control gains gi , i = 1, 2, 3 are given as g1 (x1 ) = 1 + x12 g2 (¯x2 ) = 2 g3 (¯x3 ) = 1 + x12 + x22 .
(56)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8
IEEE TRANSACTIONS ON CYBERNETICS
Fig. 9.
Estimation of fi .
and the definitions of fi are the same as in Example 2. The controller and NN updating laws described in Remark 9 are employed. The tracking performance is presented in Figs. 7–9. From the system response depicted in Fig. 7, with the DSC-PE method, the improved tracking performance is obtained. From Fig. 9, it is known that the proposed method still achieves better approximation with faster tracking. It is obvious that the method could be easily extended to more general case with known control gain function. VII. Conclusion In this paper, the composite neural DSC design has been investigated for a class of strict-feedback systems. The control scheme adjusts both adaptive neural controller and adaptive neural identification model parameters. The uniformly ultimate boundedness stability is established using Lyapunov method. The simulation shows that the proposed method can improve the uncertain functional approximation and make the adaption faster and more reliable. Acknowledgment The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions that improved the presentation of this paper. References [1] P. Kokotovic, “The joy of feedback: Nonlinear and adaptive: 1991 Bode prize lecture,” IEEE Contr. Syst. Mag., vol. 12, no. 3, pp.7–17, Jun.1992. [2] M. Chen, S. S. Ge, and B. Ren, “Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints,” Automatica, vol. 47, no. 3, pp. 452–465, 2011. [3] M. Wang, B. Chen, K. Liu, X. Liu, and S. Zhang, “Adaptive fuzzy tracking control of nonlinear time-delay systems with unknown virtual control coefficients,” Inform. Sci., vol. 178, no. 22, pp. 4326–4340, 2008. [4] Y. Pan, M. J. Er, D. Huang, and Q. Wang, “Adaptive fuzzy control with guaranteed convergence of optimal approximation error,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 807–818, Oct. 2011. [5] C. Yang, Z. Li, R. Cui, and B. Xu, “Neural network based motion control of an underactuated wheeled inverted pendulum model,” IEEE Trans. Neural Netw. Learn. Syst. doi: 10.1109/TNNLS.2014.2302475. [6] M. Chen and S. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug. 2013. [7] B. Xu, C. Yang, and Z. Shi, “Reinforcement learning output feedback NN control using deterministic learning technique,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 3, pp. 635–641, Mar. 2014. [8] S. Ge, C. Yang, and T. Lee, “Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time,” IEEE Trans. Neural Netw., vol. 19, no. 9, pp. 1599–1614, Sep. 2008.
[9] C. Yang, S. S. Ge, C. Xiang, T. Chai, and T. H. Lee, “Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1873–1886, Nov. 2008. [10] C. Yang, Z. Li, and J. Li, “Trajectory planning and optimized adaptive control for a class of wheeled inverted pendulum vehicle models,” IEEE Trans. Cybern., vol. 43, no. 1, pp. 24–36, Feb. 2013. [11] M. Chen, S. Ge, and B. Ren, “Robust attitude control of helicopters with actuator dynamics using neural networks,” IET Contr. Theory Applicat., vol. 4, no. 12, pp. 2837–2854, 2010. [12] M. Wang, B. Chen, X. Liu, and P. Shi, “Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems,” Fuzzy Sets Syst., vol. 159, no. 8, pp. 949–967, 2008. [13] B. Xu, D. Wang, F. Sun, and Z. Shi, “Direct neural discrete control of hypersonic flight vehicle,” Nonlinear Dynam., vol. 70, no. 1, pp. 269–278, 2012. [14] B. Xu, Y. Pan, D. Wang, and F. Sun, “Discrete-time hypersonic flight control based on extreme learning machine,” Neurocomputing, vol. 128, pp. 232–241, Mar. 2014. [15] Y. Li, S. Qiang, and X. Zhuang, “Robust and adaptive backstepping control for nonlinear systems using RBF neural networks,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 693–701, May 2004. [16] W. Chen and J. Li, “Decentralized output-feedback neural control for systems with unknown interconnections,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 1, pp. 258–266, Feb. 2008. [17] M. Chen, S. S. Ge, and B. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [18] D. Swaroop, J. Hedrick, P. Yip, and J. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 45, no. 10, pp. 1893–1899, Oct. 2000. [19] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strictfeedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005. [20] T. Zhang and S. Ge, “Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form,” Automatica, vol. 44, no. 7, pp. 1895–1903, 2008. [21] W. Chen, L. Jiao, R. Li, and J. Li, “Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 674–685, Aug. 2010. [22] M. Wang, X. Liu, and P. Shi, “Adaptive neural control of purefeedback nonlinear time-delay systems via dynamic surface technique,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1681–1692, Dec. 2011. [23] W. Chen, L. Jiao, J. Li, and R. Li, “Adaptive NN backstepping outputfeedback control for stochastic nonlinear strict-feedback systems with time-varying delays,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 939–950, Jun. 2010. [24] S. Tong, Y. Li, G. Feng, and T. Li, “Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 4, pp. 1124–1135, Aug. 2011. [25] S. Tong, T. Wang, Y. Li, and B. Chen, “A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 314–327, Apr. 2013.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
[26] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [27] S. Tong, Y. Li, and H. Zhang, “Adaptive neural network decentralized backstepping output-feedback control for nonlinear large-scale systems with time delays,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1073–1086, Jul. 2011. [28] J. A. Farrell, M. Polycarpou, M. Sharma, and W. Dong, “Command filtered backstepping,” IEEE Trans. Automat. Contr., vol. 54, no. 6, pp. 1391–1395, Jun. 2009. [29] S. J. Yoo, J. B. Park, and Y. H. Choi, “Adaptive dynamic surface control of flexible-joint robots using self-recurrent wavelet neural networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 6, pp. 1342–1355, Dec. 2006. [30] D. Chwa, “Global tracking control of underactuated ships with input and velocity sonstraints using dynamic surface control method,” IEEE Trans. Contr. Syst. Technol., vol. 19, no. 6, pp. 1357–1370, Nov. 2011. [31] S. Mehraeen, S. Jagannathan, and M. Crow, “Power system stabilization using adaptive neural network-based dynamic surface control,” IEEE Trans. Power Syst., vol. 26, no. 2, pp. 669–680, May 2011. [32] M. Chen, S. S. Ge, B. V. E. How, and Y. S. Choo, “Robust adaptive position mooring control for marine vessels,” IEEE Trans. Contr. Syst. Technol., vol. 21, no. 2, pp. 395–409, Mar. 2013. [33] B. Xu, X. Huang, D. Wang, and F. Sun, “Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation,” Asian J. Contr., vol. 16, no. 1, pp. 162–174, 2014. [34] B. Xu, D. Wang, H. Wang, and S. Zhu, “Command filter based robust nonlinear control of hypersonic aircraft with magnitude constraints on states and actuators,” J. Intell. Robot. Syst., vol. 73, nos. 1–4, pp. 219–231, 2014. [35] M. Hojati and S. Gazor, “Hybrid adaptive fuzzy identification and control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 10, no. 2, pp. 198–210, Apr. 2002. [36] S. Labiod and T. Guerra, “Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems,” Fuzzy Sets Syst., vol. 158, no. 10, pp. 1126–1137, 2007. [37] D. Bellomo, D. Naso, B. Turchiano, and R. Babuˇska, “Composite adaptive fuzzy control,” in Proc. 16th IFAC World Congr., Prague, Czech Republic, 2005, pp. 97–102. [38] L. Wang, “Design and analysis of fuzzy identifiers of nonlinear dynamic systems,” IEEE Trans. Automat. Contr., vol. 40, no. 1, pp. 11–23, Jan. 1995. [39] Y. Pan, M. J. Er, and T. Sun, “Composite adaptive fuzzy control for synchronizing generalized Lorenz systems,” Chaos, vol. 22, no. 2, p. 023144, 2012. [40] Y. Pan, Y. Zhou, T. Sun, and M. J. Er, “Composite adaptive fuzzy H∞ tracking control of uncertain nonlinear systems,” Neurocomputing, vol. 99, no. 1, pp. 15–24, Jan. 2013. [41] R. Sanner and J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [42] L. Wang, “Design and analysis of fuzzy identifiers of nonlinear dynamic systems,” IEEE Trans. Automat. Contr., vol. 40, no. 1, pp. 11–23, Jan. 1995. [43] K. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 1, pp. 4–27, Mar. 1990. [44] W. Dong, J. A. Farrell, M. M. Polycarpou, V. Djapic, and M. Sharma, “Command filtered adaptive backstepping,” IEEE Trans. Contr. Syst. Technol., vol. 20, no. 3, pp. 566–580, May 2012. [45] T. Li, D. Wang, G. Feng, and S. Tong, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 915–927, Jun. 2010.
9
Bin Xu received the B.S. degree in measurement and control from Northwestern Polytechnical University, Xi’an, China, in 2006, and the Ph.D. degree in computer science from Tsinghua University, Beijing, China, in 2012. From 2010 to 2011, he was with ETH Zurich, Zurich, Switzerland, and, from 2012 to 2013, he was a Research Fellow with the Nanyang Technological University, Singapore. Since 2012, he has been a Lecturer with the School of Automation, Northwestern Polytechnical University. His current research interests include intelligent control and adaptive control with application on flight dynamics, multirobot formation, and transportation systems.
Zhongke Shi received the Ph.D. degree in control theory and its applications from Northwestern Polytechnical University, Xi’an, China, in 1994. He is currently a Professor and a Doctorial Advisor with the College of Automatic Control, Northwestern Polytechnical University. He was recognized as a Distinguished Young Scholar in 1999 by the Natural Science Foundation of China. His current research interests include robust control, nonlinear control, flight control, system identification, and parameter estimation.
Chenguang Yang (M’10) received the B.E. 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. From 2009 to 2010, he was a Research Associate with the Imperial College London, London, U.K., where he worked on human–robot interaction. Since 2010, he has been a Lecturer in robotics with Plymouth University, Plymouth, U.K. His current research interests include robotics, control, and human–robot interaction.
Fuchun Sun received the B.S. and M.S. degrees from the Naval Aeronautical Engineering Academy, Yantai, China, in 1986 and 1989, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1998. He was with the Department of Automatic Control, Naval Aeronautical Engineering Academy. From 1998 to 2000, he was a Post-Doctoral Fellow with the Department of Automation, Tsinghua University. He is currently a Professor with the Department of Computer Science and Technology, Tsinghua University. His current research interests include intelligent control, neural networks, fuzzy systems, variable structure control, nonlinear systems, and robotics. Dr. Sun was the recipient of the excellent Doctoral Dissertation Prize of China in 2000 and the Choon-Gang Academic Award by Korea in 2003. He was recognized as a Distinguished Young Scholar in 2006 by the Natural Science Foundation of China.
1
Composite Neural Dynamic Surface Control of a Class of Uncertain Nonlinear Systems in Strict-Feedback Form Bin Xu, Zhongke Shi, Chenguang Yang, Member, IEEE, and Fuchun Sun
Abstract—This paper studies the composite adaptive tracking control for a class of uncertain nonlinear systems in strictfeedback form. Dynamic surface control technique is incorporated into radial-basis-function neural networks (NNs)-based control framework to eliminate the problem of explosion of complexity. To avoid the analytic computation, the command filter is employed to produce the command signals and their derivatives. Different from directly toward the asymptotic tracking, the accuracy of the identified neural models is taken into consideration. The prediction error between system state and serial–parallel estimation model is combined with compensated tracking error to construct the composite laws for NN weights updating. The uniformly ultimate boundedness stability is established using Lyapunov method. Simulation results are presented to demonstrate that the proposed method achieves smoother parameter adaption, better accuracy, and improved performance. Index Terms—Composite control, dynamic surface control, neural network, serial–parallel estimation model, strict-feedback.
I. Introduction OR NONLINEAR control systems, numerous approaches on adaptive and robust control have been proposed. Among these, back-stepping has been widely studied because it provides a systematic framework for the design of tracking and regulation strategies, suitable for a large class of state feedback linearizable nonlinear systems [1], [2]. For systems with high uncertainty, the universal approximation ability of fuzzy logic system (FLS)/neural network (NN) enhances efficiency of the intelligent control [3]–[10] and FLS/NN is widely employed for back-stepping based designs [11]–[14].
F
Manuscript received August 4, 2013; revised November 13, 2013 and February 26, 2014; accepted February 28, 2014. This work was supported in part by the National Science Foundation of China under Grant 61304098, Grant 61134004, and Grant 61210013, in part by the NWPU Basic Research Funding under Grant JC20120236, in part by the Royal Society Research Grant (RG130244), and in part by the Foundation of Key Laboratory of Autonomous Systems and Networked Control (Chinese Ministry of Education) under Grant 2012A04. This paper was recommended by Associate Editor T. H. Lee. B. Xu and Z. Shi are with the School of Automation, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]; [email protected]). C. Yang is with the School of Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, U.K. (e-mail: [email protected]). F. Sun is with the Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China (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/TCYB.2014.2311824
In [15], the back-stepping neural control approach was presented and higher tracking accuracy can be guaranteed by adding one robust item. In [16], an adaptive back-stepping NN control approach was extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections. In [17], variable structure control in combination with back-stepping was proposed for adaptive NN control design with guaranteed stability for uncertain multiple-input–multiple-output (MIMO) nonlinear systems with unknown control coefficient matrices and input nonlinearities. In the conventional back-stepping design, the virtual controller designed at each step needs repeated differentiation. The explosion of complexity of back-stepping design can be eliminated by dynamic surface control (DSC) technique in which the virtual control is passed through a first-order filter to obtain the derivative [18]. In [19], the DSC approach in [18] was extended to an NN-based adaptive tracking control of a class of single-input–single-output (SISO) system and the pure feedback SISO system was studied in [20]. In [21], fuzzy DSC design was studied where FLS was combined with a Fourier series expansion to model unknown periodically disturbed system functions. In [22], to deal with the difficulty of unknown time-delay functions, the DSC was integrated with the function separation technique, the Lyapunov–Krasovskii functionals, and the desirable property of hyperbolic tangent functions. The problem of adaptive output-feedback control for a class of uncertain stochastic nonlinear strict-feedback systems with time-varying delays was studied in [23]. The observer-based adaptive fuzzy back-stepping DSC approach was developed for MIMO nonlinear systems, stochastic nonlinear strict-feedback systems, and uncertain nonlinear largescale systems with immeasurable states in [24]–[27], respectively, while the corresponding stability of the closed-loop systems was proved. In [28], command filter design was developed by generating compensating signals to remove the c d known effect of xi+1 . Due to simplicity, the DSC − xi+1 method is widely applied on flexible-joint robot control [29], ship control [30], power system control [31], marine vessels control [32], and flight control [33], [34]. Although the FLS/NN-based DSC has made much progress, the original intention employing FLS/NN for approximating the system uncertainty is missing. Intuitively, the more precise approximation of the nonlinear function is achieved, the better
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
IEEE TRANSACTIONS ON CYBERNETICS
performance is expected. However, most efforts have been directed toward achieving asymptotic stability and tracking. Little attention has been paid to the accuracy of the identified models as well as the transparency and interpretability [37]. Motivated by the modeling error, the hybrid adaptive fuzzy identification and control was proposed in [35], which achieves faster and improved tracking. However, the nth derivative of the plant output is required to be known, which is quite impractical. Due to the uncertainty, in [36], the signal e(n) was approximated by error-changing rate as e(n) (t) ≈
(n−1)
e
(t) − e t
(n−1)
(t − t)
.
II. Problem Formulation The class of SISO nonlinear plant [19] considered in this paper is described as follows:
T
In this paper, the RBF NNs [41] are employed to approximate the unknown nonlinearity f with the following form: fˆ (Xin ) = ωˆ T θ (Xin )
(3)
where Xin ∈ R is the input vector of the RBF NNs, fˆ ∈ R is the NN output, ωˆ ∈ RLN is the adjustable parameter vector, θ (·) : RM → RLN is a nonlinear vector function of the inputs, and LN is the number of NN nodes. The components of θ are the basis functions denoted by ρi . A commonly used basis function is the so-called Gaussian function of the following form: Xin − ξi 2 1 ρi (Xin ) = √ (4) exp − 2σi2 2πσi M
where ξi is an M-dimensional vector representing the center of the ith basis function and σi is the variance representing the spread of the basis function. For any given real continuous function f on a compact set Xin ∈ RM and an arbitrary εM > 0, there exist RBF NNs in the form of (5) and an optimal parameter vector ω∗ such that f (Xin ) = ω∗ T θ (Xin ) + ε sup | ε |< εM
Xin ∈ Xin
(5) (6)
where εM > 0 denotes the supremum value of the reconstruction error ε that is inevitably generated.
IV. Composite DSC Design With Prediction Error In this section, the prediction error derived from the difference between system state and serial–parallel estimation model will be incorporated into the DSC technique for the nthorder system described by (2). The recursive design procedure contains n steps. From Step 1 to Step n − 1, virtual control d xi+1 , i = 1, . . . , n − 1 is designed at each step and an overall control law u is constructed in the last step n. Step 1: Considering the first equation in (2) and using NN to approximate f1 (¯x1 ), we have x˙ 1 = f1 (¯x1 ) + x2 = ω1∗T θ1 (¯x1 ) + ε1 + x2
(7)
where ω1∗ is the optimal NN weights vector and ε1 is the NN approximation error with | ε1 |≤ εM . Define the tracking error (2)
where x(t) ∈ R is the state, x¯ i = [x1 , , . . . , xi ] , u ∈ R is the input, y ∈ R is the output, and fi , i = 1, . . . , n are unknown smooth functions of x¯ i . n
III. RBF Neural Networks
(1)
Since the information employed to construct the approximation error is not rigorous, during the controller design and stability analysis, it is difficult to analyze the influence of the estimation error. In [37], the composite design was studied using prediction error with a slightly different serial–parallel estimation model [38]. In [39], the method was applied on controlling two uncertain generalized Lorenz systems. In [40], to deal with the problem that the H ∞ control terms may degrade the fuzzy approximation ability, the composite design was employed and the fuzzy approximation ability was improved. It is noticed that the model studied in [35], [37], and [40] was restricted to the canonical form. The convenience provided by this model is that the derivative information can be directly derived from the states, but this information is not available for strict-feedback system with uncertainty. In this paper, a class of SISO strict-feedback system is studied. Different from [36], the NN modeling-related prediction error is defined between the state and the serial–parallel estimation model. Furthermore, the compensating signal is incorporated to eliminate the effect of the known error caused by command filter. The composite DSC approach is presented employing both compensated tracking error and prediction error. With the new composite DSC method, we show that better tracking performance is achieved. This paper is organized as follows. In Section II, the class of SISO nonlinear systems is characterized. The brief introduction of radial-basis-function neural networks (RBF NNs) is given in Section III. In Section IV, the composite DSC control is designed, and the stability analysis is presented in Section V. The effectiveness of the proposed approach is verified by simulation in Section VI. The conclusion is drawn in Section VII.
x˙ i = fi (¯xi ) + xi+1 , 1 ≤ i ≤ n − 1 x˙ n = fn (¯xn ) + u y = x1
For a given reference yr (t), t ≥ 0, which is a sufficiently smooth function of t with its time derivative y˙ r bounded for t ≥ 0, the control objective is to find an adaptive neural control law such that all signals and states defined in the closedloop system are bounded and the output tracks the bounded reference trajectory yr .
e1 = x1 − yr .
(8)
Choose virtual control x2d as x2d = −ωˆ 1T θ1 (¯x1 ) − k1 e1 + y˙ r
(9)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
where ωˆ 1 is the estimation of ω1∗ , k1 > 0 is the design constant. Introduce a new state variable x2c and let x2d pass through a first-order filter with time constant α2 > 0 to obtain x2c α2 x˙ 2c + x2c = x2d , x2c (0) = x2d (0).
(10)
Define e2 = x2 − x2c . Then the derivative of e1 is obtained as e˙ 1 = x˙ 1 − y˙ r = ω1∗T θ1 (¯x1 ) + ε1 + x2 − y˙ r =
ω˜ 1T θ1 (¯x1 )
(x2c
−
x2d )
− ωˆ1 . where ω˜ 1 = To remove the effect of the known error (x2c − x2d ), the compensating signal z1 is designed as z˙ 1 = −k1 z1 + z2 +
−
c d z˙ i = −ki zi − zi−1 + zi+1 + (xi+1 − xi+1 ), zi (0) = 0.
z1 (0) = 0
(12)
where z2 will be defined in the next step. Now we obtain the compensated tracking error signals ν1 = e1 − z1 , ν2 = e2 − z2 .
(13)
The prediction error is defined as z1NN = x1 − xˆ 1
(14)
where the derivative of NN modeling information is defined with the serial–parallel estimation model [42], [43] xˆ˙ 1 = ωˆ 1T θ1 (¯x1 ) + x2 + β1 z1NN , xˆ 1 (0) = x1 (0)
(15)
with β1 > 0 as the user-defined positive constant. For the NN updating law, the signal z1NN is employed to construct the learning design ωˆ˙ 1 = γ1 (ν1 + γz1 z1NN ) θ1 (¯x1 ) − δ1 ωˆ 1 (16) where γ1 , γz1 , and δ1 are positive design constants. Step i: i = 2, . . . , n − 1. Considering the ith equation in (2) and using NN to approximate fi (¯xi ), we know x˙ i = fi (¯xi ) + xi+1 = ωi∗T θi (¯xi ) + εi + xi+1
(17)
where ωi∗ is the optimal NN weights vector and εi is the NN approximation error with | εi |≤ εM . Define the ith error surface ei to be ei = xi − xic
=
−ωˆ iT θi (¯xi )
νi = ei − zi
(23)
ziNN = xi − xˆ i
(24)
(18)
where the derivative of NN modeling information is defined with the serial–parallel estimation model xˆ˙ i = ωˆ iT θi (¯xi ) + xi+1 + βi ziNN , xˆ i (0) = xi (0) with βi > 0 as the user-defined positive constant. The update law of ωˆ i is designed to be ωˆ˙ i = γi (νi + γzi ziNN ) θi (¯xi ) − δi ωˆ i
− ki ei − ei−1 +
x˙ ic
(19)
ωi∗ ,
ki > 0 is the design constant. where ωˆ i is the estimation of d c and let xi+1 Introduce a new state variable xi+1 pass through a first-order filter with the time constant αi+1 > 0 to obtain c xi+1 c c d d c αi+1 x˙ i+1 + xi+1 = xi+1 , xi+1 (0) = xi+1 (0).
(20)
c . Then the derivative of ei is obtained Define ei+1 = xi+1 − xi+1 as
e˙ i = x˙ i − x˙ ic c d = ω˜ iT θi (¯xi ) + εi − ki ei − ei−1 + ei+1 + (xi+1 − xi+1 ) (21)
(25)
(26)
where γi , γzi , and δi are positive design constants. Step n: Considering the nth equation in (2) and using NN to approximate fn (¯xn ), we know x˙ n = fn (¯xn ) + u = ωn∗T θn (¯xn ) + εn + u
(27)
where ωn∗ is the optimal NN weights vector and εn is the NN approximation error with | εn |≤ εM . Define the nth error surface en to be en = xn − xnc .
(28)
The final control u is designed as u = −ωˆ nT θn (¯xn ) − kn en − en−1 + x˙ nc
(29)
ωn∗ ,
where ωˆ n is the estimation of kn > 0 is the design constant. Then the derivative of en is obtained as e˙ n = x˙ n − x˙ nc = ω˜ nT θn (¯xn ) + εn − kn en − en−1
(30)
ωn∗
where ω˜ n = − ωˆ n . The compensating signal is defined as z˙ n = −kn zn − zn−1 , zn (0) = 0.
(31)
Define the compensated tracking error signal
d is designed as The virtual control xi+1 d xi+1
(22)
Define the compensated tracking error signal
(11)
ω1∗
x2d ),
where ω˜ i = ωi∗ − ωˆ i . d c To remove the effect of the known error (xi+1 − xi+1 ), the compensating signal zi is defined as
and the prediction error
+ ε 1 − k 1 e1 + e 2 +
(x2c
3
νn = en − zn
(32)
znNN = xn − xˆ n
(33)
and the prediction error
where the derivative of NN modeling information is defined with the serial–parallel estimation model xˆ˙ n = ωˆ nT θn (¯xn ) + u + βn znNN , xˆ n (0) = xn (0) with βn > 0 as the user-defined positive constant. The update law of ωˆ n is designed to be ωˆ˙ n = γn (νn + γzn znNN ) θn (¯xn ) − δn ωˆ n where γn , γzn , and δn are positive design constants.
(34)
(35)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4
IEEE TRANSACTIONS ON CYBERNETICS
Remark 1: The NN approximation in (25) is used to obtain xˆ˙ i and the prediction error is employed for parameter adjustment in (26). There is no need to calculate the derivatives of the states and thus there is no such approximation as in (1). Remark 2: The δ-modification [15] is introduced to improve the robustness and to avoid the weight parameters to drift to very large values. Remark 3: For back-stepping design, the tracking error vector is defined as e¯ BS = [e1,BS , · · · , en,BS ] with ei,BS = xi − x¯ id for i = 1, · · · , n, where x¯ 1d = yr . The vector of function x¯ d = [¯x2d , · · · , x¯ nd ], referred to as virtual control signals, is defined as
For the NN prediction error, using (9), (19), and (29) with (15), (25), and (34), we have z˙ iNN = x˙ i − xˆ˙ i = ω˜ iT θi (¯xi ) + εi − βi ziNN .
Define mi = ω˜ iT θi (¯xi ), the following equation can be obtained: z˙ iNN ziNN = ziNN (mi + εi ) − βi z2iNN .
d
V˙ = (36)
n
n νi ν˙ i − ω˜ iT γi−1 ωˆ˙ i + γzi ziNN z˙ iNN
i=1
=
n
i=1
−ki νi2 + νi mi + νi εi
i=1
The updating of NN weights vector ωˆ i,BS refers to the design [19] using tracking error ei,BS . As the order n increases, d analytic computation of x¯˙ i becomes increasingly complicated. Remark 4: Compared with back-stepping design (36), the composite neural DSC controller proposed in this paper has the following features. 1) The method is based on the command filter that produces certain command signals and their derivatives. In this way, there is no load of the aforementioned analytic computation. 2) Only the information of yr (t) and y˙ r (t) is required in this paper while back-stepping design needs the knowledge of yr(i) (t). 3) The prediction error of NN modeling together with the compensated tracking error is included in the composite NN updating law to provide fast adaption ability.
−
Theorem 1: Let yr (t) and y˙ r (t) be continuous, bounded for t ≥ 0. For system (2) with the DSC laws (9), (19), and (29), the NN updating law (16), (26), and (35), and the compensated error signals νi defined in (23), it is guaranteed that the signals νi , ω˜ i , and ziNN are uniformly ultimately bounded. Proof: The Lyapunov function is chosen as n n 1 1 2 νi + ω˜ iT γi−1 ω˜ i + V = γzi z2iNN . (37) 2 i=1 2 i=1 For error dynamics of the compensated tracking error, we have ν˙ 1 = e˙ 1 − z˙ 1 = ω˜ 1T θ1 (¯x1 ) + ε1 − k1 (e1 − z1 ) + (e2 − z2 ) = ω˜ 1T θ1 (¯x1 ) + ε1 − k1 ν1 + ν2
(38) ν˙ i = e˙ i − z˙ i = ω˜ iT θi (¯xi ) + εi − ki (ei − zi ) − (ei−1 − zi−1 ) + (ei+1 − zi+1 ) = ω˜ iT θi (¯xi ) + εi − ki νi − νi−1 + νi+1 , i = 2, · · · , n − 1 (39)
ν˙ n = e˙ n − z˙ n = ω˜ nT θn (¯xn ) + εn − kn (en − zn ) − (en−1 − zn−1 ) (40)
n
mi (νi + γzi ziNN ) +
i=1
+
n
=
n
n
δi ω˜ iT ωˆ i
i=1
γzi ziNN (mi + εi ) −
i=1
n
γzi βi z2iNN
i=1
−ki νi2 + νi εi + γzi ziNN εi − γzi βi z2iNN
i=1 −δi ω˜ iT ω˜ i + δi ω˜ iT ωi∗ .
(43)
Consider the following facts:
εi 2 1 2 νi εi − ki νi2 = −ki νi − + ε 2ki 4ki i
εi 2 1 2 2 ziNN εi − βi ziNN = −βi ziNN − + εi 2βi 4βi 2 ωi∗ + 1 ω∗ 2 . ω˜ iT ωi∗ − ω˜ iT ω˜ i = − ω ˜ − i 2 4 i
V. Stability Analysis
= ω˜ nT θn (¯xn ) + εn − kn νn − νn−1 .
(42)
The derivative of V is derived as
T θ1 (x1 ) − k1 e1,BS + y˙ r x¯ 2d = −ωˆ 1,BS d T x¯ i+1 = −ωˆ i,BS θi (¯xi ) − ki ei,BS + x¯˙ i − ei−1,BS i = 2, · · · , n − 1 d T u = −ωˆ n,BS θn (¯xn ) − kn en,BS + x¯˙ n − en−1,BS .
(41)
Then
εi 2 1 2 V˙ = − k i νi − − ε 2ki 4ki i i=1
n εi 2 1 2 γzi βi ziNN − − ε − 2βi 4βi i i=1
n ∗ 2 ∗ 2 ω 1 i − δi ω˜ i − 2 − 4 ωi i=1
n εi 2 εi 2 ≤− k 0 νi − + γz min β0 ziNN − 2ki 2βi i=1
2 ωi∗ +P ω ˜ +δ0 (44) − i 2 n
where k0 = min [ki ], β0 = min [βi ], γz min = min γzi , n 2 nγz max 2 nδmax 2 δ0 = min [δi ], P = ωmax , γz max = εM + εM + 4k 4β 4 0 0 ∗ max γzi , ωmax = max ωi , and δmax = max [δi ].
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
5
P εi P εi If νi − or ziNN − or ≥ ≥ 2k k 2β γ 0 i z min β0 i ∗ ω˜ i − ωi ≥ P , then V˙ ≤ 0. Then we know that νi , ziNN , 2 δ0 and ω˜ i are invariant to the sets defined as follows: P εM + νi = νi |νi | ≤ k0 2k0 εM P ziNN = ziNN |ziNN | ≤ + γz min β0 2β0 P ωmax ω˜ i = ω˜ i ω˜ i ≤ . (45) + δ0 2 So all the signals are uniformly ultimately bounded. This completes the proof. Remark 5: By increasing the values of ki and βi , the P εM P εM quantity , , , and can be made arbitrarily k0 2k0 γz min β0 2β0 small. Thus, the error νi and ziNN may be made arbitrarily small. Remark 6: The singular perturbation analysis can be applied with the similar way as in [28] that by decreasing αi , the solution of the design can be made arbitrarily close to the back-stepping solution based on analytic derivatives. Remark 7: In [44, Lemma 3], it is presented that the compensating signals zi are bounded. With the conclusion of Theorem 1, we know that signals νi are bounded. Then from the definition of compensated signal in (23), the boundedness of ei can be deduced. Remark 8: It is easy to combine the robust design [15], minimal parameter technique [45] with the composite DSC approach to enhance the performance. The purpose here is to clearly demonstrate how the composite design is constructed; in this paper, we do not put much emphasis on it. Remark 9: It is easy to extend the result in this paper to the strict-feedback system with known gi (¯xi ) = 0. The dynamics are in the following formulation:
Fig. 1.
System tracking. (a) Tracking Performance. (b) Tracking Error.
Fig. 2.
NN weights adaptation.
x˙ i = fi (¯xi ) + gi (¯xi )xi+1 , 1 ≤ i ≤ n − 1 x˙ n = fn (¯xn ) + gn (¯xn )u y = x1 .
(46)
Accordingly, the controller is designed as −ωˆ 1T θ1 (¯x1 ) − k1 e1 + y˙ r g1 T −ωˆ i θi (¯xi ) − ki ei − gi−1 ei−1 + x˙ ic d xi+1 = ,2 ≤ i ≤ n − 1 gi −ωˆ nT θn (¯xn ) − kn en − gn−1 en−1 + x˙ nc u= (47) gn x2d =
and
z˙ n = −kn zn − gn−1 zn−1 , zn (0) = 0
(50)
xˆ˙ n = ωˆ nT θn (¯xn ) + gn u + βn znNN , xˆ n (0) = xn (0).
(51)
while the compensating signals and the estimation are modified as c d − xi+1 ), zi (0)=0(48) z˙ i = −ki zi −gi−1 zi−1 +gi zi+1 + gi (xi+1 T (49) xˆ˙ i = ωˆ i θi (¯xi ) + gi xi+1 + βi ziNN , xˆ i (0) = xi (0) 1 ≤ i≤n−1
It is easy to construct the same Lyapunov function as (37). Following the same procedure, the boundedness of closedloop signals can be proved and thus the analysis is omitted here.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6
IEEE TRANSACTIONS ON CYBERNETICS
Fig. 3.
Estimation of fi .
Fig. 4.
System tracking. (a) Tracking Performance. (b) Tracking Error.
VI. Simulation
Fig. 5.
In this section, the third-order uncertain nonlinear systems in strict-feedback form are considered x˙ 1 = f1 (¯x1 ) + g1 (¯x1 )x2
To show the effectiveness of the method proposed in this paper, the comparison is conducted with the DSC method in [19], where the NN updating law has the following formulation: ωˆ˙ i = γi ei θi (¯xi ) − γi δi ωˆ i .
x˙ 2 = f2 (¯x2 ) + g2 (¯x2 )x3 x˙ 3 = f3 (¯x3 ) + g3 (¯x3 )u y = x1
NN weights adaptation.
(52)
where fi are unknown smooth functions while gi , i = 1, 2, 3 are known. The objective is to design a control law u such that the output of the closed-loop system can approximately track a reference input yr = sin(t) asymptotically. To simulate the overall control system, the differential equations of the system and the adaptive NN updating law are integrated with the MATLAB function ODE45.
(53)
The initial state of the system is 0. The control parameters chosen for simulation are: ki = 20, γi = 10, δi = 0.001, i = 1, 2, 3, and α2 = α3 = 0.005. For the NN design, the centers for x1 , x2 , and x3 are evenly spaced in [−1, 1]×[−3, 3]×[−5, 5]. For f1 , the NN nodes number is N1 = 21. For f2 , the numbers of the RBFs for x1 and x2 are seven and seven, respectively, and N2 = 49. For f3 , the numbers of the RBFs for x1 , x2 , and x3 are seven, seven, and three, respectively, and N3 = 147. To show the different tracking result, we denote the DSC method in [19] as DSC-CLASSIC and the method proposed with prediction error in Section IV as DSC-PE. The related parameters are selected as βi = 5, γzi = 1 for DSC-PE, i =
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
Fig. 6.
Estimation of fi .
Fig. 7.
System tracking. (a) Tracking Performance. (b) Tracking Error.
1, 2, 3. The systems studied in the first two examples are the same as [19] while the third example will consider the case with gi = 1. Example 1: Definitions of fi , gi , i = 1, 2, 3 are as follows: f1 (x1 ) = x13 f2 (¯x2 ) = x12 + x22 f3 (¯x3 ) = 0 gi = 1, i = 1, 2, 3.
(54)
The tracking performance is presented in Figs. 1–3. Fig. 1(a) shows the tracking performance. From the tracking error depicted in Fig. 1(b), it is clearly demonstrated that the DSC-PE achieves faster adaption and improved accuracy. The conclusion can be further confirmed from the NN response in Fig. 2 and the nonlinear tracking of fi shown in Fig. 3. It should be noted that the fi , i = 1, 2 are the functions of the states so that the values with DSC-PE are different from the ones with DSC-CLASSIC.
Fig. 8.
7
NN weights adaptation.
Example 2: The definitions of fi , gi , i = 1, 2, 3 are as follows: f1 (x1 ) = 2x12 sin(x1 ) f2 (¯x2 ) = x12 + x1 x2 + x2 cos x1 f3 (¯x3 ) = x1 x3 + x22 + x3 sin x2
(55)
gi = 1, i = 1, 2, 3. The tracking performance is presented in Figs. 4–6. Similar conclusion could be obtained that in comparison with the DSC-CLASSIC, the proposed DSC-PE method can achieve faster tracking and the NN modeling error decreases faster. Example 3: The control gains gi , i = 1, 2, 3 are given as g1 (x1 ) = 1 + x12 g2 (¯x2 ) = 2 g3 (¯x3 ) = 1 + x12 + x22 .
(56)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8
IEEE TRANSACTIONS ON CYBERNETICS
Fig. 9.
Estimation of fi .
and the definitions of fi are the same as in Example 2. The controller and NN updating laws described in Remark 9 are employed. The tracking performance is presented in Figs. 7–9. From the system response depicted in Fig. 7, with the DSC-PE method, the improved tracking performance is obtained. From Fig. 9, it is known that the proposed method still achieves better approximation with faster tracking. It is obvious that the method could be easily extended to more general case with known control gain function. VII. Conclusion In this paper, the composite neural DSC design has been investigated for a class of strict-feedback systems. The control scheme adjusts both adaptive neural controller and adaptive neural identification model parameters. The uniformly ultimate boundedness stability is established using Lyapunov method. The simulation shows that the proposed method can improve the uncertain functional approximation and make the adaption faster and more reliable. Acknowledgment The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions that improved the presentation of this paper. References [1] P. Kokotovic, “The joy of feedback: Nonlinear and adaptive: 1991 Bode prize lecture,” IEEE Contr. Syst. Mag., vol. 12, no. 3, pp.7–17, Jun.1992. [2] M. Chen, S. S. Ge, and B. Ren, “Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints,” Automatica, vol. 47, no. 3, pp. 452–465, 2011. [3] M. Wang, B. Chen, K. Liu, X. Liu, and S. Zhang, “Adaptive fuzzy tracking control of nonlinear time-delay systems with unknown virtual control coefficients,” Inform. Sci., vol. 178, no. 22, pp. 4326–4340, 2008. [4] Y. Pan, M. J. Er, D. Huang, and Q. Wang, “Adaptive fuzzy control with guaranteed convergence of optimal approximation error,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 807–818, Oct. 2011. [5] C. Yang, Z. Li, R. Cui, and B. Xu, “Neural network based motion control of an underactuated wheeled inverted pendulum model,” IEEE Trans. Neural Netw. Learn. Syst. doi: 10.1109/TNNLS.2014.2302475. [6] M. Chen and S. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug. 2013. [7] B. Xu, C. Yang, and Z. Shi, “Reinforcement learning output feedback NN control using deterministic learning technique,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 3, pp. 635–641, Mar. 2014. [8] S. Ge, C. Yang, and T. Lee, “Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time,” IEEE Trans. Neural Netw., vol. 19, no. 9, pp. 1599–1614, Sep. 2008.
[9] C. Yang, S. S. Ge, C. Xiang, T. Chai, and T. H. Lee, “Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1873–1886, Nov. 2008. [10] C. Yang, Z. Li, and J. Li, “Trajectory planning and optimized adaptive control for a class of wheeled inverted pendulum vehicle models,” IEEE Trans. Cybern., vol. 43, no. 1, pp. 24–36, Feb. 2013. [11] M. Chen, S. Ge, and B. Ren, “Robust attitude control of helicopters with actuator dynamics using neural networks,” IET Contr. Theory Applicat., vol. 4, no. 12, pp. 2837–2854, 2010. [12] M. Wang, B. Chen, X. Liu, and P. Shi, “Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems,” Fuzzy Sets Syst., vol. 159, no. 8, pp. 949–967, 2008. [13] B. Xu, D. Wang, F. Sun, and Z. Shi, “Direct neural discrete control of hypersonic flight vehicle,” Nonlinear Dynam., vol. 70, no. 1, pp. 269–278, 2012. [14] B. Xu, Y. Pan, D. Wang, and F. Sun, “Discrete-time hypersonic flight control based on extreme learning machine,” Neurocomputing, vol. 128, pp. 232–241, Mar. 2014. [15] Y. Li, S. Qiang, and X. Zhuang, “Robust and adaptive backstepping control for nonlinear systems using RBF neural networks,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 693–701, May 2004. [16] W. Chen and J. Li, “Decentralized output-feedback neural control for systems with unknown interconnections,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 1, pp. 258–266, Feb. 2008. [17] M. Chen, S. S. Ge, and B. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010. [18] D. Swaroop, J. Hedrick, P. Yip, and J. Gerdes, “Dynamic surface control for a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 45, no. 10, pp. 1893–1899, Oct. 2000. [19] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strictfeedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005. [20] T. Zhang and S. Ge, “Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form,” Automatica, vol. 44, no. 7, pp. 1895–1903, 2008. [21] W. Chen, L. Jiao, R. Li, and J. Li, “Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 674–685, Aug. 2010. [22] M. Wang, X. Liu, and P. Shi, “Adaptive neural control of purefeedback nonlinear time-delay systems via dynamic surface technique,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1681–1692, Dec. 2011. [23] W. Chen, L. Jiao, J. Li, and R. Li, “Adaptive NN backstepping outputfeedback control for stochastic nonlinear strict-feedback systems with time-varying delays,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 939–950, Jun. 2010. [24] S. Tong, Y. Li, G. Feng, and T. Li, “Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 4, pp. 1124–1135, Aug. 2011. [25] S. Tong, T. Wang, Y. Li, and B. Chen, “A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 314–327, Apr. 2013.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. XU et al.: COMPOSITE NEURAL DYNAMIC SURFACE CONTROL OF A CLASS OF UNCERTAIN NONLINEAR SYSTEMS
[26] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [27] S. Tong, Y. Li, and H. Zhang, “Adaptive neural network decentralized backstepping output-feedback control for nonlinear large-scale systems with time delays,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1073–1086, Jul. 2011. [28] J. A. Farrell, M. Polycarpou, M. Sharma, and W. Dong, “Command filtered backstepping,” IEEE Trans. Automat. Contr., vol. 54, no. 6, pp. 1391–1395, Jun. 2009. [29] S. J. Yoo, J. B. Park, and Y. H. Choi, “Adaptive dynamic surface control of flexible-joint robots using self-recurrent wavelet neural networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 36, no. 6, pp. 1342–1355, Dec. 2006. [30] D. Chwa, “Global tracking control of underactuated ships with input and velocity sonstraints using dynamic surface control method,” IEEE Trans. Contr. Syst. Technol., vol. 19, no. 6, pp. 1357–1370, Nov. 2011. [31] S. Mehraeen, S. Jagannathan, and M. Crow, “Power system stabilization using adaptive neural network-based dynamic surface control,” IEEE Trans. Power Syst., vol. 26, no. 2, pp. 669–680, May 2011. [32] M. Chen, S. S. Ge, B. V. E. How, and Y. S. Choo, “Robust adaptive position mooring control for marine vessels,” IEEE Trans. Contr. Syst. Technol., vol. 21, no. 2, pp. 395–409, Mar. 2013. [33] B. Xu, X. Huang, D. Wang, and F. Sun, “Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation,” Asian J. Contr., vol. 16, no. 1, pp. 162–174, 2014. [34] B. Xu, D. Wang, H. Wang, and S. Zhu, “Command filter based robust nonlinear control of hypersonic aircraft with magnitude constraints on states and actuators,” J. Intell. Robot. Syst., vol. 73, nos. 1–4, pp. 219–231, 2014. [35] M. Hojati and S. Gazor, “Hybrid adaptive fuzzy identification and control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 10, no. 2, pp. 198–210, Apr. 2002. [36] S. Labiod and T. Guerra, “Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems,” Fuzzy Sets Syst., vol. 158, no. 10, pp. 1126–1137, 2007. [37] D. Bellomo, D. Naso, B. Turchiano, and R. Babuˇska, “Composite adaptive fuzzy control,” in Proc. 16th IFAC World Congr., Prague, Czech Republic, 2005, pp. 97–102. [38] L. Wang, “Design and analysis of fuzzy identifiers of nonlinear dynamic systems,” IEEE Trans. Automat. Contr., vol. 40, no. 1, pp. 11–23, Jan. 1995. [39] Y. Pan, M. J. Er, and T. Sun, “Composite adaptive fuzzy control for synchronizing generalized Lorenz systems,” Chaos, vol. 22, no. 2, p. 023144, 2012. [40] Y. Pan, Y. Zhou, T. Sun, and M. J. Er, “Composite adaptive fuzzy H∞ tracking control of uncertain nonlinear systems,” Neurocomputing, vol. 99, no. 1, pp. 15–24, Jan. 2013. [41] R. Sanner and J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [42] L. Wang, “Design and analysis of fuzzy identifiers of nonlinear dynamic systems,” IEEE Trans. Automat. Contr., vol. 40, no. 1, pp. 11–23, Jan. 1995. [43] K. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 1, pp. 4–27, Mar. 1990. [44] W. Dong, J. A. Farrell, M. M. Polycarpou, V. Djapic, and M. Sharma, “Command filtered adaptive backstepping,” IEEE Trans. Contr. Syst. Technol., vol. 20, no. 3, pp. 566–580, May 2012. [45] T. Li, D. Wang, G. Feng, and S. Tong, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 915–927, Jun. 2010.
9
Bin Xu received the B.S. degree in measurement and control from Northwestern Polytechnical University, Xi’an, China, in 2006, and the Ph.D. degree in computer science from Tsinghua University, Beijing, China, in 2012. From 2010 to 2011, he was with ETH Zurich, Zurich, Switzerland, and, from 2012 to 2013, he was a Research Fellow with the Nanyang Technological University, Singapore. Since 2012, he has been a Lecturer with the School of Automation, Northwestern Polytechnical University. His current research interests include intelligent control and adaptive control with application on flight dynamics, multirobot formation, and transportation systems.
Zhongke Shi received the Ph.D. degree in control theory and its applications from Northwestern Polytechnical University, Xi’an, China, in 1994. He is currently a Professor and a Doctorial Advisor with the College of Automatic Control, Northwestern Polytechnical University. He was recognized as a Distinguished Young Scholar in 1999 by the Natural Science Foundation of China. His current research interests include robust control, nonlinear control, flight control, system identification, and parameter estimation.
Chenguang Yang (M’10) received the B.E. 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. From 2009 to 2010, he was a Research Associate with the Imperial College London, London, U.K., where he worked on human–robot interaction. Since 2010, he has been a Lecturer in robotics with Plymouth University, Plymouth, U.K. His current research interests include robotics, control, and human–robot interaction.
Fuchun Sun received the B.S. and M.S. degrees from the Naval Aeronautical Engineering Academy, Yantai, China, in 1986 and 1989, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1998. He was with the Department of Automatic Control, Naval Aeronautical Engineering Academy. From 1998 to 2000, he was a Post-Doctoral Fellow with the Department of Automation, Tsinghua University. He is currently a Professor with the Department of Computer Science and Technology, Tsinghua University. His current research interests include intelligent control, neural networks, fuzzy systems, variable structure control, nonlinear systems, and robotics. Dr. Sun was the recipient of the excellent Doctoral Dissertation Prize of China in 2000 and the Choon-Gang Academic Award by Korea in 2003. He was recognized as a Distinguished Young Scholar in 2006 by the Natural Science Foundation of China.
