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Robust adaptive control scheme for optical tracking telescopes with unknown disturbances
Optik 126 (2015) 1185–1190
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Optik journal homepage: www.elsevier.de/ijleo
Robust adaptive control scheme for optical tracking telescopes with unknown disturbances Rong Mei a , Mou Chen b,∗ , William W. Guo c a b c
Criminal Investigation Department, Nanjing Forest Police College, Nanjing 210023, China College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China School of Engineering and Technology, Central Queensland University, North Rockhampton, QLD 4702, Australia
a r t i c l e
i n f o
Article history: Received 14 February 2014 Accepted 27 February 2015 Keywords: Optical tracking telescope Neural network Nonlinear disturbance observer Robust adaptive control Tracking control
a b s t r a c t In this paper, a robust adaptive control scheme is proposed for optical tracking telescopes with parametric uncertainty, unknown external disturbance and input saturation. To improve tracking performance of this robust adaptive control scheme, a nonlinear disturbance observer (NDO) is employed to tackle the integrated effect amalgamated from unknown parameters, unknown external disturbance and input saturation. At the same time, the radial basis function neural network (RBFNN) is introduced to approximate the input of an unknown function. Utilizing the estimated outputs of NDO and RBFNN, the robust adaptive control scheme is developed for optical tracking telescopes. Stability of the closed-loop system is rigourously proved via Lyapunov analysis and the convergent tracking error is guaranteed for optical tracking telescopes. Numerical simulation results are presented to illustrate the effectiveness of the proposed robust adaptive control scheme based on RBFNN and NDO for the uncertain dynamic of optical tracking telescopes. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction To obtain a good observation performance from a telescope on the Earth surface, adaptive optics systems are usually used [1]. Optical tracking telescope is one of the most important pieces of machinery in adaptive optics systems widely used in many practical observations. Thus, the highly precise tracking control of optical tracking telescopes remains a hot research topic in the control area. However, the load inertia, the torque and the friction coefficient may change when the optical tracking telescope is moving on Earth surface. Meanwhile, the nonlinear characteristic and input constraints will further increase the design difficulty for the robust adaptive control of optical tracking telescopes. To obtain a good position tracking control performance for uncertain optical tracking telescopes, robust control design is a challenging problem. In [2], a robust adaptive control was developed for optical tracking telescopes with unmodeled dynamics. On the other hand, robust adaptive tracking control can be developed using the neural networks or fuzzy logical systems to tackle the unknown continuous function term for uncertain nonlinear systems in [3–8]. In [9], a decentralized output-feedback neural control was proposed
∗ Corresponding author. Tel.: +86 2584893084. E-mail address: [email protected] (M. Chen). http://dx.doi.org/10.1016/j.ijleo.2015.02.088 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
for systems with unknown interconnections. Robust backstepping control was developed for a class of time delayed systems in [10]. In [11], an adaptive fuzzy Control was presented for synchronization of nonlinear teleoperators with stochastic time-varying communication delays. Composite neural dynamic surface control was proposed for a class of uncertain nonlinear systems in strictfeedback form in [12]. However, the control input saturation and time-varying unknown disturbance should be further considered in the robust tracking control design. Since actuator output of an optical tracking telescope is limited due to actuator’s physical constraints, it is very important to develop effective robust control techniques for uncertain dynamic of the optical tracking telescope with input saturation. In recent years, there have been extensive studies of various systems with input saturation [13–18]. In [19], a robust adaptive control was proposed for uncertain nonlinear systems with input saturation and external disturbance. A robust adaptive neural network (NN) control was proposed for a class of uncertain multi-input and multioutput (MIMO) nonlinear systems with input nonlinearities [4]. Backstepping control was studied for hovering unmanned aerial vehicle (UAV) including input saturations in [20]. In [21], an adaptive tracking control was developed for uncertain MIMO nonlinear systems with input saturation. Adaptive control was presented for minimum phase single-input and single-output (SISO) plants with input saturation [22]. In [23], an adaptive output feedback
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sliding-mode maneuvring and vibration control was proposed for the flexible spacecraft with input saturation. However, there is almost no research outcome in the robust control of optical tracking telescopes with parametric uncertainty, unknown external disturbance and input saturation. To improve the performance of robust control, a disturbance observer based robust control (DOBC) scheme can be developed for optical tracking telescopes. Recently, disturbance observer design and application have attracted considerable interest for robust control design of uncertain systems. Up to now, various disturbance observers have been extensively developed and corresponding robust control schemes were designed by using the disturbance observer output [24–31]. A general framework was developed for uncertain nonlinear systems using DOBC method in [32]. In [33], composite DOBC and terminal sliding mode control were studied for uncertain structural systems. The disturbance attenuation and rejection problem was proposed for MIMO nonlinear systems using DOBC technique in [34]. In [35], composite DOBC and H∞ control were proposed for complex continuous models. Adding robustness to nominal output feedback controller was presented for uncertain nonlinear systems using disturbance observer in [36]. In [37], a nonlinear disturbance observer-based robust control was proposed for systems with mismatched disturbances/uncertainties. Although significant progress has been made for the disturbance observer design, the nonlinear disturbance observer needs to be developed to estimate the compounded disturbance combining the parametric uncertainty and the unknown external disturbance with the effect of control input saturation. Furthermore, the robust adaptive control should be extended to covering uncertain optical tracking telescopes. This work is motivated by the disturbance observer-based robust adaptive control for optical tracking telescopes with parametric uncertainty, unknown disturbance and input saturation. The main contributions of this paper include: (i) A nonlinear disturbance observer (NDO) is developed to estimate the compounded disturbance amalgamated from the unknown parameters, unknown external disturbance and the input saturation; (ii) To fully utilize the dynamic information of the compounded disturbances, a robust adaptive control for optical tracking telescopes is developed using outputs of the developed NDO and RBFNN; (iii) Stability of the proposed robust adaptive tracking control scheme is rigorously established using Lyapunov theory, which shows that the semiglobal uniform boundedness of all closedloop signals is achieved. The organization of the paper is as follows. The problem description of robust adaptive control for optical tracking telescopes is given in Section 2. Section 3 describes the development of NDO and the robust adaptive control scheme is proposed using NDO and RBFNN. Simulation studies are provided in Section 4 to demonstrate the effectiveness of the proposed nonlinear disturbance observer based robust adaptive control approach, followed by some concluding remarks in Section 5. 2. Problem description
function. (y, t) denotes the unknown system uncertainty and d(t) ∈ R is the external unknown disturbance. r is the desired bounded and continuous reference signal and t is a time variable. ˛p , kp , ˛ and kz are system parameters of the optical tracking telescope. In our study, all system parameters ˛p , kp , ˛ and kz are assumed as unknown, which will increase the design complexity of robust adaptive control. v(t) ∈ R is the control input command and u(.) denotes the actual control input which is subjected to saturation type nonlinearity described by [19]:
u(v(t)) = sat(v(t)) =
sign(v(t))umax ,
|v(t)| ≥ umax
v(t),
|v(t)| < umax
(2)
where umax is a bound of u(t). To facilitate the design of robust adaptive control scheme for the optical tracking telescope, we define F(y, r) = − ˛p y + 140kp z + f(y) + (y, t) and g = 140kp . Considering (1), we have y˙ = F(y, r) + gu(v(t)) + d(t)
(3)
To efficiently handle the input saturation u(v(t)) in the robust adaptive control design, it is approximated using the following smooth tanh function [19,39]: p(v) = umax tanh(
v umax
) = umax
ev/umax − e−v/umax ev/umax + e−v/umax
(4)
The difference ı(v) between sat(v(t)) and p(v) can be written as ı(v) = sat(v(t)) − p(v)
(5)
Invoking the bounded property of the tanh function and the sat function yields |ı(v)| = |sat(v(t)) − p(v)| ≤ umax (1 − tanh(1)) = d
(6)
where d is an unknown positive constant. Considering the saturation characteristic and the corresponding approximation error, the uncertain nonlinear system (3) can be written as the following general uncertain nonlinear system: y˙ = F(y, r) + gp(v) + gı(v) + d(t)
(7)
From the general uncertain nonlinear system (7), the unknown function F(y, r) and the uncertain control gain g will make the design of the robust tracking scheme more difficult. To design the robust adaptive control scheme for the uncertain nonlinear system (7), invoking the mean-value theorem, p(v(t)) can be written as [39] p(v(t)) = p(v0 ) +
∂p(v) | (v − v0 ) ∂v v=v
(8)
where v = v + (1 − )v0 with 0 < < 1. By choosing v0 = 0, we have p(v(t)) = p(0) +
∂p(v) | v ∂v v=v
(9)
Due to p(0) = 0, we obtain p(v(t)) =
∂p(v) | v ∂v v=v
(10)
Then, the uncertain nonlinear system (7) can be rewritten as
Consider an optical tracking telescope with unknown parameters, unknown disturbance and input saturation in the form of [38]:
y˙ = F(y, r) + g
∂p(v) | v + gı(v) + d(t) = F(y, r) + g0 v + D0 (t) ∂v v=v (11)
y˙ + ˛p y = 140kp z + f (y) + 140kp u(v(t)) + (y, t) + d(t)
(1)
where y ∈ R is the system output and u ∈ R is the control input, respectively. f(y) = − ˛sign(y), z = kz (r − y) and sign(.) is the sign
where D0 (t) = g ∂p(v) |v=v v − g0 v + gı(v) + d(t) and g0 is a design ∂v parameter which needs to be properly chosen for the control scheme design.
R. Mei et al. / Optik 126 (2015) 1185–1190
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Defining e = y − r and considering (11), we have e˙ = y˙ − r˙ = F(y, r) + g0 v + D0 (t) − r˙
d r
(12)
Designned robust adaptive control
To tackle the unknown function F(y, r), the RBFNNs can be employed as approximation models to approximate F(y, r) due to their inherent approximation capabilities [40]. The RBFNN can smoothly approximate any continuous function f(Z) over the compact set Z ∈ Rq to any arbitrary accuracy as f (Z) = W ∗T S(Z) + ε(Z), r]T
∀Z ∈ Z ⊂ Rq
∈ R2
(14)
where ci and bi are the center and width of the neural cell of the ith hidden layer. Here, the RBFNN is used to approximate k0 F(y, r) with k0 > 0. Considering (13), (12) can be rewritten as k0−1 (W ∗T S(Z) + ε(Z)) + g0 v + D0 (t) − r˙
Defining D(t) =
v
Uncertain optical tracking telescope
Input saturation
Wˆ
Nonlinear Disturbance observer
y
Neural network
(13)
si (Z) = exp[−(Z − ci )T (Z − ci )/b2i ], i = 1, 2, . . ., p
k0−1 ε(Z) + D0 (t)
Dˆ
W*
is the input vector of NN, is the optimal where Z = [y, weight value, S(Z) = [s1 (Z), s2 (Z), . . ., sp (Z)]T ∈ Rp is the basis function, ε is the approximation error and q is a positive integer. The RBFNN is a particular network architecture which uses Gaussian functions in the form of
e˙ =
e
the uncertain tracking error dynamics of optical tracking telescopes (16). The block diagram of the proposed robust adaptive control based on NDO and RBFNN is shown in Fig. 1. In this paper, the NDO is designed as ˆ = k0 (e − s) D
(17)
ˆ T S(Z) + g0 v − r˙ + D ˆ s˙ = k0−1 W
(15)
ˆ is the estimate of where k0 > 0 is a design parameter of the NDO; D ˆ is the estimate of the optimal compounded disturbance D(t) and W weight value W* . Considering (16) and (17), we obtain
(16)
ˆ˙ = k0 (e˙ − s) ˆ ˆ T S(Z)) ˙ = k0 (D(t) − D(t)) D + (W ∗T S(Z) − W
and considering (15) yields
e˙ = k0−1 W ∗T S(Z) + g0 v + D(t) − r˙
Fig. 1. Structure of robust adaptive control using NDO and RBFNN.
To design the nonlinear disturbance-observer-based tracking control for uncertain nonlinear systems (7), the following lemma and assumption are required: Assumption 1. There exists an unknown positive constant such that the time derivative of unknown compounded disturbance D(t) ˙ ≤ with > 0. is bounded, i.e., |D| Assumption 2. For a practical optical tracking telescope, there exist an actual control u which can track the given desired output signal r of optical tracking telescopes in the presence of the unknown external disturbance and the input saturation. Namely, the controllability of the studied dynamic of the optical tracking telescope (1) should be satisfied under the control input saturation. Lemma 1. [41] For bounded initial conditions, if there exists a C1 continuous and positive definite Lyapunov function V(x) satisfying 1 ( x ) ≤ V(x) ≤ 2 ( x ), such that V˙ (x) ≤ − V (x) + c, where 1 , 2 : Rn → R are class K functions and c is a positive constant, then the solution x(t) is uniformly bounded. For the continuous desired trajectory r, the control objective is to use the robust adaptive control input v to render the tracking error convergent for the optical tracking telescope (1) in the presence of uncertain system parameter, input saturation and time-varying unknown external disturbances. Remark 1. For the unknown compounded disturbance D(t) in (16), it contains ∂p(v) |v=v , g0 , v, g, ı(v), d(t) and the approxima∂v tion error ε(Z) of RBFNN. All time derivatives of these parameters are always bounded for the practical optical tracking telescope. Thus, the Assumption 1 is reasonable. On the other hand, the design parameter g0 should be properly chosen to obtain the good tacking control performance. In general, we can select an arbitrary nonzero v and use 140 ∂p(v) |v=v to obtain g0 in the tracking control design. ∂v
3. Design of robust adaptive control using nonlinear disturbance observer To improve the robust tracking control performance, a NDO is adopted to monitor the unknown compounded disturbance D(t) in
(18)
˜ =D−D ˆ and W ˜ = W∗ − W ˆ . Considering (18) yields Define D ˜˙ = D˙ − D ˆ˙ = D˙ − k0 (D(t) − D(t)) ˆ ˆ T S(Z)) D − (W ∗T S(Z) − W
(19)
˜ −W ˜ T S(Z) = D˙ − k0 D Chose the Lyapunov function candidate as Vo =
1 2 ˜ D 2
(20)
Invoking (19), we obtain ˜2 − D ˜D ˜˙ = D ˜ D˙ − k0 D ˜W ˜ T S(Z) V˙ o = D
(21)
In accordance with Assumption 1 and considering the following fact ˜W ˜ T S(Z) ≤ 2|D||| ˜ W ˜ ||||S(Z)|| ≤ 2 |D| ˜ 2+ −2D
1 ˜ ||2 ||W
(22)
we have ˜ 2 + 0.5|D| ˙ 2 − k0 D ˜ 2+ ˜ 2 + 2 |D| V˙ o ≤ 0.5|D| ˜2 + ≤ −(k0 − (0.5 + 2 ))D
1 ˜ ||2 ||W
1 ˜ ||2 + 0.5 2 ||W
(23)
where ||S(Z)|| ≤ and > 0 is a design parameter. So far, the NDO design has been completed for the unknown compounded disturbance D(t) in the uncertain tracking error dynamics of optical tracking telescopes (16). The robust tracking control scheme will be proposed next. Based on outputs of the designed NDO and RBFNN, the robust adaptive control law is designed as
v=−
1 ˆ ˆ T S(Z) + D(t) ˙ (ke + k0−1 W − r) g0
(24)
where k > 0 is a design constant. ˆ and W ˜ = W∗ − W ˆ and substituting (24) ˜ =D−D Considering D into (16) yields ˆ T S(Z) + D(t) − D(t) ˆ e˙ = −ke + k0−1 W ∗T S(Z) − k0−1 W ˜ T S(Z) + D(t) ˜ = −ke + k0−1 W
(25)
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ˆ as Choose the adaptive law for W
10
ˆ) ˆ˙ = (k−1 S(Z)e + W W 0
Theorem 1. Considering the uncertain optical tracking telescope dynamic (7) with the parametric uncertainty, the unknown disturbance and the input saturation, the NDO is designed as (17) and the parameter updated law is chosen as (26). Then, using estimated outputs of the developed NDO and RBFNN, the robust tracking control law is given by (24). Under the developed robust tracking control scheme, the tracking error of uncertain optical tracking telescope dynamic (7) is convergent. Proof 1.
Invoking (23) and (25), the time derivative of V is given by
1 ˜ ||2 ||W
2 ˜ T S(Z) + eD(t) ˜ ˜ T −1 W ˜˙ + 0.5 2 − ke + k0−1 eW +W 1 ˜ ||2 + 0.5 2 ||W
˜ T S(Z) + W ˜ T −1 W ˜˙ + k0−1 eW ˜ = W∗ − W ˆ and the weight value adaptation law Considering W (26), we have 1 ˜ ||2 ||W
(29)
According to the following fact ˜ TW ˜ 2 + W ˆ 2 − W ∗ 2 ≥ W ˜ 2 − W ∗ 2 ˆ = W 2W
(30)
we have ˜ 2 − (k − 0.5)e2 ≤ −(k0 − (1.0 + 2 ))D
1 2 ˜ || + 0.5 2 + W ∗ 2 − ( − )||W 2
2
(31)
≤ − V + C where 1
− ) 2
: = min((k0 − (1.0 + 2 )), (k − 0.5), ) max ( −1 )
C : = 0.5 2 + W ∗ 2 2 2(
(32)
To ensure the closed-loop system stability, the corresponding design matrices k0 , k, and should be chosen to make k0 − (1.0 + 2 ) > 0, k − 0.5 > 0 and 2 − 1 > 0. Considering (31), it ˜ and W ˜ are semiglobal unimay directly show that the signals e, D formly ultimately bounded (SGUUB) using Lemma 1. Invoking (31), we have 0≤V ≤
C C + V (0) −
e− t
4 3
1 0
0
2
4
6
8
10
Time [s]
According to (33), we know that V is exponentially convergent, i.e., lim V = C . Thus, the tracking error e and the estimated errors t−→∞
˜2 + ≤ −(k0 − (0.5 + 2 ))D
˜ TW ˆ + 0.5 2 − W
5
˜ and W ˜ of the closed-loop system are bounded and converge to a D compact set. Analysis above demonstrates that the tracking error e is convergent and the control objective is achieved. This concludes the proof.
˜˙ ˜D ˜˙ + ee˙ + W ˜ T −1 W V˙ = D
˜ 2 − (k − 0.5)e2 + V˙ ≤ −(k0 − (1.0 + 2 ))D
6
Fig. 2. Tracking control result for the constant signal.
(27)
˜ 2 − (k − 0.5)e2 + ≤ −(k0 − (1.0 + 2 ))D
7
2
Consider the Lyapunov function candidate
1 ˜ T −1 W ˜ V = Vo + e2 + W 2
V˙
8
Tracking control result
where = T > 0 and > 0. The above robust tracking control design for the uncertain optical tracking telescope dynamic (7) using NDO and RBFNN can be included in the following theorem which is described as:
r y
9
(26)
(33)
Remark 2. In this paper, the RBFNN is employed to tackle the unknown function term F(y, r). To handle the approximation error of RBFNN, it combines the external time-varying disturbance with the effect of control input saturation as a compounded disturbance which is estimated via the developed NDO. For the proposed NDO, we know that the estimate error with suitable approximation performance can be achieved by choosing the appropriate disturbance observer gain parameter k0 . Since NDO and RBFNN are used, the disturbance and uncertainty rejection ability is improved for uncertain optical tracking telescopes under the proposed robust adaptive control scheme. 4. Simulation study In this section, the extensive numerical simulation results are presented to illustrate the effectiveness of the proposed robust adaptive control scheme based on RBFNN and NDO for the uncertain dynamic of optical tracking telescopes. In this simulation, all system parameters in (1) are give as ˛ = 5, ˛p = 2, kp = 0.05 and kz = 1. The system uncertainty is chosen as (y, t) = 0.2 sin(y)e−2y and the time-varying disturbance is d(t) = 0.1 sin(t) + 0.3e−t . The initial conˆ 0 = 0 and D ˆ = 0. ditions are arbitrarily chosen as y0 = 0.1, W To design the NDO and the robust adaptive controller for the uncertain optical tracking telescope, all design parameters are chosen as ko = 10, k = 4, g0 = 5, = diag{2}, = 2, umax = 30 and = 0.05. The NDO is designed as (17) and the parameter updated law is chosen as (26). The robust tracking control law is designed as (24) based on the estimated outputs of the developed NDO and RBFNN. Under the designed tracking control law (24), the extensive numerical simulation results and corresponding analysis are appended as follows. Firstly, the constant tracking signal is chosen to illustrate the effectiveness of the proposed robust adaptive control scheme. The desired constant tracking signal is chosen as r(t) = 7 for optical tracking telescopes. The tracking control results under the designed robust adaptive control scheme are shown in Figs. 2 and 3. Fig. 2 shows that a satisfactory tracking control result is obtained under the proposed robust adaptive control scheme. The tracking error is presented in Fig. 3 and it is convergent. The control input is shown in Fig. 4 and the bounded property is demonstrated. The plot of the
R. Mei et al. / Optik 126 (2015) 1185–1190
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1
yr
1 0
Tracking control result
−1
Tracking error e
y
0.8
−2 −3 −4 −5
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
−6
−1 −7
0
2
4
6
8
10
0
Time [s]
5
10
15
20
Time [s]
Fig. 3. Tracking control error for the constant signal.
Fig. 6. Tracking control result for the time-varying signal.
35
0.15
30 0.1
Tracking error e
Control input u
25
20
15
10
0.05
0
−0.05
5
0 −0.1
0
2
4
6
8
10
0
Fig. 4. Control input response for the constant signal.
5
10
15
20
Time [s]
Time [s]
Fig. 7. Tracking control error for the time-varying signal.
results of the optical tracking telescopes are shown in Figs. 6 and 7. Fig. 6 shows that the actual output of the uncertain optical tracking telescope can well track the desired trajectory. Although the time-varying tracking signal is used, a satisfactory tracking error is achieved as shown in Fig. 7. In accordance with above simulation results for the constant tracking signal and the time-varying tracking signal, the satisfactory performance of position tracking control is achieved under the developed robust adaptive tracking control scheme for uncertain optical tracking telescopes. Therefore the proposed robust adaptive control scheme using NDO and RBFNN is effective for uncertain optical tracking telescopes. 5. Conclusion Fig. 5. Norm of the neural network weight for the constant signal.
norm of weight value for the radial basis function neural network is given in Fig. 5 and convergent. These simulation results from the constant tracking signal demonstrate that the developed robust tracking control law for the uncertain optical tracking telescope is valid. The time-varying tracking signal is then chosen to illustrate the effectiveness of the proposed robust adaptive control scheme for optical tracking telescopes. Here, the desired time-varying tracking signal is chosen as r(t) = sin(t). Under the designed robust adaptive control scheme using NDO and RBFNN, the tracking control
A robust adaptive tracking control scheme has been developed for uncertain optical tracking telescopes in the presence of the unknown parameters, the external time-varying disturbance and control input saturation in this paper. To improve the robust tracking performance, the RBFNN has been introduced to approximate the unknown continuous system uncertainty. Furthermore, the unknown external disturbance, the unknown neural network approximation error with the effect of input saturation are integrated as a compounded disturbance and the NDO has been developed to estimate the effect of this compounded disturbance. The robust adaptive tracking control scheme for uncertain optical tracking telescopes has been developed based on the outputs of
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the RBFNN and the developed NDO. By using Lyapunov analysis, the convergence of all closed-loop signals has been guaranteed. Finally, the simulation results have demonstrated that the proposed robust adaptive tracking scheme is effective for uncertain optical tracking telescopes. Acknowledgements This work is partially supported by Jiangsu Natural Science Foundation of China (granted number: SBK20130033), Program for New Century Excellent Talents in University of China (granted number: NCET-11-0830), “Liu Da Ren Cai Gao Feng” Scheme of Jiangsu Province in China (granted number: 2012-XXRJ-010) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. References [1] S. Azeemuddin, M.R. Sayeh, Effect of various parameters on working of alloptical Schmitt trigger, Optik 122 (10) (2011) 1935–1938. [2] J. Chen, Y.S. Liu, L.Z. Pan, Robust adaptive control for optical tracking telescope with unmodeled dynamics, J. Sichuan Univ. (English Sci. Ed.) 36 (3) (2004) 89–92. [3] Y.J. Liu, W. Wang, S.C. Tong, Y.S. Liu, Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters, IEEE Trans. Syst. Man Cybern. Part A: Syst. Hum. 40 (1) (2010) 170–184. [4] M. Chen, S.S. Ge, B. How, Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities, IEEE Trans. Neural Netw. 21 (5) (2010) 796–812. [5] T.S. Li, D. Wang, N.X. Chen, A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40 (3) (2011) 915–927. [6] W.S. Chen, L.C. Jiao, R.H. Li, J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Trans. Fuzzy Syst. 18 (4) (2010) 674–685. [7] B. Xu, C.G. Yang, Z.K. Shi, Reinforcement learning output feedback NN control using deterministic learning technique, IEEE Trans. Neural Netw. Learn. Syst. 25 (3) (2014) 635–641. [8] M. Chen, G. Tao, B. Jiang, Dynamic surface control using neural networks for a class of uncertain nonlinear systems with input saturation, IEEE Trans. Neural Netw. Learn. Syst. (2014), http://dx.doi.org/10.1109/TNNLS.2014.2360933 [9] W.S. Chen, J.M. Li, Decentralized output-feedback neural control for systems with unknown interconnections, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38 (1) (2008) 258–266. [10] C.C. Hua, X.P. Guan, P. Shi, Robust backstepping control for a class of time delayed systems, IEEE Trans. Autom. Control 50 (6) (2005) 894–899. [11] Z.J. Li, X.Q. Cao, N. Ding, Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays, IEEE Trans. Fuzzy Syst. 19 (4) (2011) 745–757. [12] B. Xu, Z.K. Shi, C.G. Yang, F.C. Sun, Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Cybern. 44 (12) (2014) 2626–2634. [13] Y.Y. Cao, Z.L. Lin, Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation, IEEE Trans. Fuzzy Syst. 11 (1) (2003) 57–67. [14] Q. Hu, G.F. Ma, L.H. Xie, Robust and adaptive variable structure output feedback control of uncertain systems with input nonlinearity, Automatica 44 (4) (2008) 552–559. [15] H.N. Wu, H.X. Li, H∞ fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints, IEEE Trans. Fuzzy Syst. 16 (2) (2008) 502–516.
[16] Q. Hu, B. Xiao, M.I. Friswell, Robust fault-tolerant control for spacecraft attitude stabilisation subject to input saturation, IET Control Theory Appl. 5 (2) (2011) 271–282. [17] M. Chen, Q.X. Wu, C.S. Jiang, B. Jiang, Guaranteed transient performance based control for near space vehicles with input saturation, Sci. China Inform. Sci. 57 (5) (2014) 1–12. [18] M. Chen, B. Jiang, R.X. Cui, Actuator fault-tolerant control of ocean surface vessels with input saturation, Int. J. Robust Nonlinear Control (2015), http://dx. doi.org/10.1002/rnc.3324 [19] C.Y. Wen, J. Zhou, Z.T. Liu, H.Y. Su, Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance, IEEE Trans. Autom. Control 56 (7) (2011) 1672–1678. [20] J.R. Azinheira, A. Moutinho, Hover control of an UAV with backstepping design including input saturations, IEEE Trans. Control Syst. Technol. 16 (3) (2008) 517–526. [21] M. Chen, S.S. Ge, B. Ren, Adaptive tracking control of uncertain MIMO nonlinear systems with input saturation, Automatica 47 (3) (2011) 452–465. [22] Y.S. Zhong, Globally stable adaptive system design for minimum phase SISO plants with input saturation, Automatica 41 (11) (2005) 1539–1547. [23] Q. Hu, Adaptive output feedback sliding-mode manoeuvring and vibration control of flexible spacecraft with input saturation, IET Control Theory Appl. 2 (6) (2008) 467–478. [24] W.H. Chen, D.J. Ballance, P.J. Gawthrop, J. O’Reilly, Nonlinear PID predictive controller, IEE Proc. Control Theory Appl. 146 (6) (1999) 603–611. [25] F.J. Lin, P.H. Chou, Y.S. Kung, Robust fuzzy neural network controller with nonlinear disturbance observer for two-axis motion control system, IET Control Theory Appl. 2 (3) (2008) 151–167. [26] M. Chen, W.H. Chen, Disturbance observer based robust control for time delay uncertain systems, Int. J. Control Autom. Syst. 8 (2) (2010) 445–453. [27] M. Chen, W.H. Chen, Sliding mode controller design for a class of uncertain nonlinear system based disturbance observer, Int. J. Adapt. Control Signal Process. 24 (1) (2010) 51–64. [28] A. Mohammadi, M. Tavakoli, H.J. Marquez, Disturbance observer-based control of non-linear haptic teleoperation systems, IET Control Theory Appl. 5 (18) (2011) 2063–2074. [29] J. Yang, S.H. Li, X.H. Yu, Sliding-mode control for systems with mismatched uncertainties via a disturbance observer, IEEE Trans. Ind. Electron. 60 (1) (2013) 160–169. [30] S.H. Li, J. Yang, Robust autopilot design for bank-to-turn missiles using disturbance observers, IEEE Trans. Aerosp. Electron. Syst. 49 (1) (2013) 558–579. [31] J. Yang, W.H. Chen, S.H. Li, Robust autopilot design of uncertain bank-to-turn missiles using state-space disturbance observers, Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 226 (1) (2012) 97–107. [32] W.H. Chen, Disturbance bserver based control for nonlinear systems, IEEE Trans. Mechatron. 9 (4) (2004) 706–710. [33] X.J. Wei, H.F. Zhang, L. Guo, Composite disturbance-observer-based control and terminal sliding mode control for uncertain structural systems, Int. J. Syst. Sci. 40 (10) (2009) 1009–1017. [34] L. Guo, W.H. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Int. J. Robust Nonlinear Control 15 (3) (2005) 109–125. [35] X.J. Wei, L. Guo, Composite disturbance-observer-based control and H∞ control for complex continuous models, Int. J. Robust Nonlinear Control 20 (1) (2009) 106–118. [36] J. Back, H. Shim, Adding robustness to nominal output feedback controllers for uncertain nonlinear systems: a nonlinear version of disturbance observer, Automatica 44 (10) (2008) 2528–2537. [37] J. Yang, W.H. Chen, S.H. Li, Non-linear disturbance observer-based robust control for systems with mismatched disturbances/uncertainties, IET Control Theory Appl. 5 (18) (2011) 2053–2062. [38] Z.J. Han, Adaptive Control, Tsinghua University, Beijing, 1995. [39] M. Chen, B. Jiang, Robust bounded control for uncertain flight dynamics using disturbance observer, J. Syst. Eng. Electron. 25 (4) (2014) 640–647. [40] S.S. Ge, C.C. Hang, T.H. Lee, T. Zhang, Stable Adaptive Neural Network Control, Kluwer Academic, Norwell, USA, 2001. [41] S.S. Ge, C. Wang, Adaptive neural control of uncertain MIMO nonlinear systems, IEEE Trans. Neural Netw. 15 (3) (2004) 674–692.
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Optik journal homepage: www.elsevier.de/ijleo
Robust adaptive control scheme for optical tracking telescopes with unknown disturbances Rong Mei a , Mou Chen b,∗ , William W. Guo c a b c
Criminal Investigation Department, Nanjing Forest Police College, Nanjing 210023, China College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China School of Engineering and Technology, Central Queensland University, North Rockhampton, QLD 4702, Australia
a r t i c l e
i n f o
Article history: Received 14 February 2014 Accepted 27 February 2015 Keywords: Optical tracking telescope Neural network Nonlinear disturbance observer Robust adaptive control Tracking control
a b s t r a c t In this paper, a robust adaptive control scheme is proposed for optical tracking telescopes with parametric uncertainty, unknown external disturbance and input saturation. To improve tracking performance of this robust adaptive control scheme, a nonlinear disturbance observer (NDO) is employed to tackle the integrated effect amalgamated from unknown parameters, unknown external disturbance and input saturation. At the same time, the radial basis function neural network (RBFNN) is introduced to approximate the input of an unknown function. Utilizing the estimated outputs of NDO and RBFNN, the robust adaptive control scheme is developed for optical tracking telescopes. Stability of the closed-loop system is rigourously proved via Lyapunov analysis and the convergent tracking error is guaranteed for optical tracking telescopes. Numerical simulation results are presented to illustrate the effectiveness of the proposed robust adaptive control scheme based on RBFNN and NDO for the uncertain dynamic of optical tracking telescopes. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction To obtain a good observation performance from a telescope on the Earth surface, adaptive optics systems are usually used [1]. Optical tracking telescope is one of the most important pieces of machinery in adaptive optics systems widely used in many practical observations. Thus, the highly precise tracking control of optical tracking telescopes remains a hot research topic in the control area. However, the load inertia, the torque and the friction coefficient may change when the optical tracking telescope is moving on Earth surface. Meanwhile, the nonlinear characteristic and input constraints will further increase the design difficulty for the robust adaptive control of optical tracking telescopes. To obtain a good position tracking control performance for uncertain optical tracking telescopes, robust control design is a challenging problem. In [2], a robust adaptive control was developed for optical tracking telescopes with unmodeled dynamics. On the other hand, robust adaptive tracking control can be developed using the neural networks or fuzzy logical systems to tackle the unknown continuous function term for uncertain nonlinear systems in [3–8]. In [9], a decentralized output-feedback neural control was proposed
∗ Corresponding author. Tel.: +86 2584893084. E-mail address: [email protected] (M. Chen). http://dx.doi.org/10.1016/j.ijleo.2015.02.088 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
for systems with unknown interconnections. Robust backstepping control was developed for a class of time delayed systems in [10]. In [11], an adaptive fuzzy Control was presented for synchronization of nonlinear teleoperators with stochastic time-varying communication delays. Composite neural dynamic surface control was proposed for a class of uncertain nonlinear systems in strictfeedback form in [12]. However, the control input saturation and time-varying unknown disturbance should be further considered in the robust tracking control design. Since actuator output of an optical tracking telescope is limited due to actuator’s physical constraints, it is very important to develop effective robust control techniques for uncertain dynamic of the optical tracking telescope with input saturation. In recent years, there have been extensive studies of various systems with input saturation [13–18]. In [19], a robust adaptive control was proposed for uncertain nonlinear systems with input saturation and external disturbance. A robust adaptive neural network (NN) control was proposed for a class of uncertain multi-input and multioutput (MIMO) nonlinear systems with input nonlinearities [4]. Backstepping control was studied for hovering unmanned aerial vehicle (UAV) including input saturations in [20]. In [21], an adaptive tracking control was developed for uncertain MIMO nonlinear systems with input saturation. Adaptive control was presented for minimum phase single-input and single-output (SISO) plants with input saturation [22]. In [23], an adaptive output feedback
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sliding-mode maneuvring and vibration control was proposed for the flexible spacecraft with input saturation. However, there is almost no research outcome in the robust control of optical tracking telescopes with parametric uncertainty, unknown external disturbance and input saturation. To improve the performance of robust control, a disturbance observer based robust control (DOBC) scheme can be developed for optical tracking telescopes. Recently, disturbance observer design and application have attracted considerable interest for robust control design of uncertain systems. Up to now, various disturbance observers have been extensively developed and corresponding robust control schemes were designed by using the disturbance observer output [24–31]. A general framework was developed for uncertain nonlinear systems using DOBC method in [32]. In [33], composite DOBC and terminal sliding mode control were studied for uncertain structural systems. The disturbance attenuation and rejection problem was proposed for MIMO nonlinear systems using DOBC technique in [34]. In [35], composite DOBC and H∞ control were proposed for complex continuous models. Adding robustness to nominal output feedback controller was presented for uncertain nonlinear systems using disturbance observer in [36]. In [37], a nonlinear disturbance observer-based robust control was proposed for systems with mismatched disturbances/uncertainties. Although significant progress has been made for the disturbance observer design, the nonlinear disturbance observer needs to be developed to estimate the compounded disturbance combining the parametric uncertainty and the unknown external disturbance with the effect of control input saturation. Furthermore, the robust adaptive control should be extended to covering uncertain optical tracking telescopes. This work is motivated by the disturbance observer-based robust adaptive control for optical tracking telescopes with parametric uncertainty, unknown disturbance and input saturation. The main contributions of this paper include: (i) A nonlinear disturbance observer (NDO) is developed to estimate the compounded disturbance amalgamated from the unknown parameters, unknown external disturbance and the input saturation; (ii) To fully utilize the dynamic information of the compounded disturbances, a robust adaptive control for optical tracking telescopes is developed using outputs of the developed NDO and RBFNN; (iii) Stability of the proposed robust adaptive tracking control scheme is rigorously established using Lyapunov theory, which shows that the semiglobal uniform boundedness of all closedloop signals is achieved. The organization of the paper is as follows. The problem description of robust adaptive control for optical tracking telescopes is given in Section 2. Section 3 describes the development of NDO and the robust adaptive control scheme is proposed using NDO and RBFNN. Simulation studies are provided in Section 4 to demonstrate the effectiveness of the proposed nonlinear disturbance observer based robust adaptive control approach, followed by some concluding remarks in Section 5. 2. Problem description
function. (y, t) denotes the unknown system uncertainty and d(t) ∈ R is the external unknown disturbance. r is the desired bounded and continuous reference signal and t is a time variable. ˛p , kp , ˛ and kz are system parameters of the optical tracking telescope. In our study, all system parameters ˛p , kp , ˛ and kz are assumed as unknown, which will increase the design complexity of robust adaptive control. v(t) ∈ R is the control input command and u(.) denotes the actual control input which is subjected to saturation type nonlinearity described by [19]:
u(v(t)) = sat(v(t)) =
sign(v(t))umax ,
|v(t)| ≥ umax
v(t),
|v(t)| < umax
(2)
where umax is a bound of u(t). To facilitate the design of robust adaptive control scheme for the optical tracking telescope, we define F(y, r) = − ˛p y + 140kp z + f(y) + (y, t) and g = 140kp . Considering (1), we have y˙ = F(y, r) + gu(v(t)) + d(t)
(3)
To efficiently handle the input saturation u(v(t)) in the robust adaptive control design, it is approximated using the following smooth tanh function [19,39]: p(v) = umax tanh(
v umax
) = umax
ev/umax − e−v/umax ev/umax + e−v/umax
(4)
The difference ı(v) between sat(v(t)) and p(v) can be written as ı(v) = sat(v(t)) − p(v)
(5)
Invoking the bounded property of the tanh function and the sat function yields |ı(v)| = |sat(v(t)) − p(v)| ≤ umax (1 − tanh(1)) = d
(6)
where d is an unknown positive constant. Considering the saturation characteristic and the corresponding approximation error, the uncertain nonlinear system (3) can be written as the following general uncertain nonlinear system: y˙ = F(y, r) + gp(v) + gı(v) + d(t)
(7)
From the general uncertain nonlinear system (7), the unknown function F(y, r) and the uncertain control gain g will make the design of the robust tracking scheme more difficult. To design the robust adaptive control scheme for the uncertain nonlinear system (7), invoking the mean-value theorem, p(v(t)) can be written as [39] p(v(t)) = p(v0 ) +
∂p(v) | (v − v0 ) ∂v v=v
(8)
where v = v + (1 − )v0 with 0 < < 1. By choosing v0 = 0, we have p(v(t)) = p(0) +
∂p(v) | v ∂v v=v
(9)
Due to p(0) = 0, we obtain p(v(t)) =
∂p(v) | v ∂v v=v
(10)
Then, the uncertain nonlinear system (7) can be rewritten as
Consider an optical tracking telescope with unknown parameters, unknown disturbance and input saturation in the form of [38]:
y˙ = F(y, r) + g
∂p(v) | v + gı(v) + d(t) = F(y, r) + g0 v + D0 (t) ∂v v=v (11)
y˙ + ˛p y = 140kp z + f (y) + 140kp u(v(t)) + (y, t) + d(t)
(1)
where y ∈ R is the system output and u ∈ R is the control input, respectively. f(y) = − ˛sign(y), z = kz (r − y) and sign(.) is the sign
where D0 (t) = g ∂p(v) |v=v v − g0 v + gı(v) + d(t) and g0 is a design ∂v parameter which needs to be properly chosen for the control scheme design.
R. Mei et al. / Optik 126 (2015) 1185–1190
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Defining e = y − r and considering (11), we have e˙ = y˙ − r˙ = F(y, r) + g0 v + D0 (t) − r˙
d r
(12)
Designned robust adaptive control
To tackle the unknown function F(y, r), the RBFNNs can be employed as approximation models to approximate F(y, r) due to their inherent approximation capabilities [40]. The RBFNN can smoothly approximate any continuous function f(Z) over the compact set Z ∈ Rq to any arbitrary accuracy as f (Z) = W ∗T S(Z) + ε(Z), r]T
∀Z ∈ Z ⊂ Rq
∈ R2
(14)
where ci and bi are the center and width of the neural cell of the ith hidden layer. Here, the RBFNN is used to approximate k0 F(y, r) with k0 > 0. Considering (13), (12) can be rewritten as k0−1 (W ∗T S(Z) + ε(Z)) + g0 v + D0 (t) − r˙
Defining D(t) =
v
Uncertain optical tracking telescope
Input saturation
Wˆ
Nonlinear Disturbance observer
y
Neural network
(13)
si (Z) = exp[−(Z − ci )T (Z − ci )/b2i ], i = 1, 2, . . ., p
k0−1 ε(Z) + D0 (t)
Dˆ
W*
is the input vector of NN, is the optimal where Z = [y, weight value, S(Z) = [s1 (Z), s2 (Z), . . ., sp (Z)]T ∈ Rp is the basis function, ε is the approximation error and q is a positive integer. The RBFNN is a particular network architecture which uses Gaussian functions in the form of
e˙ =
e
the uncertain tracking error dynamics of optical tracking telescopes (16). The block diagram of the proposed robust adaptive control based on NDO and RBFNN is shown in Fig. 1. In this paper, the NDO is designed as ˆ = k0 (e − s) D
(17)
ˆ T S(Z) + g0 v − r˙ + D ˆ s˙ = k0−1 W
(15)
ˆ is the estimate of where k0 > 0 is a design parameter of the NDO; D ˆ is the estimate of the optimal compounded disturbance D(t) and W weight value W* . Considering (16) and (17), we obtain
(16)
ˆ˙ = k0 (e˙ − s) ˆ ˆ T S(Z)) ˙ = k0 (D(t) − D(t)) D + (W ∗T S(Z) − W
and considering (15) yields
e˙ = k0−1 W ∗T S(Z) + g0 v + D(t) − r˙
Fig. 1. Structure of robust adaptive control using NDO and RBFNN.
To design the nonlinear disturbance-observer-based tracking control for uncertain nonlinear systems (7), the following lemma and assumption are required: Assumption 1. There exists an unknown positive constant such that the time derivative of unknown compounded disturbance D(t) ˙ ≤ with > 0. is bounded, i.e., |D| Assumption 2. For a practical optical tracking telescope, there exist an actual control u which can track the given desired output signal r of optical tracking telescopes in the presence of the unknown external disturbance and the input saturation. Namely, the controllability of the studied dynamic of the optical tracking telescope (1) should be satisfied under the control input saturation. Lemma 1. [41] For bounded initial conditions, if there exists a C1 continuous and positive definite Lyapunov function V(x) satisfying 1 ( x ) ≤ V(x) ≤ 2 ( x ), such that V˙ (x) ≤ − V (x) + c, where 1 , 2 : Rn → R are class K functions and c is a positive constant, then the solution x(t) is uniformly bounded. For the continuous desired trajectory r, the control objective is to use the robust adaptive control input v to render the tracking error convergent for the optical tracking telescope (1) in the presence of uncertain system parameter, input saturation and time-varying unknown external disturbances. Remark 1. For the unknown compounded disturbance D(t) in (16), it contains ∂p(v) |v=v , g0 , v, g, ı(v), d(t) and the approxima∂v tion error ε(Z) of RBFNN. All time derivatives of these parameters are always bounded for the practical optical tracking telescope. Thus, the Assumption 1 is reasonable. On the other hand, the design parameter g0 should be properly chosen to obtain the good tacking control performance. In general, we can select an arbitrary nonzero v and use 140 ∂p(v) |v=v to obtain g0 in the tracking control design. ∂v
3. Design of robust adaptive control using nonlinear disturbance observer To improve the robust tracking control performance, a NDO is adopted to monitor the unknown compounded disturbance D(t) in
(18)
˜ =D−D ˆ and W ˜ = W∗ − W ˆ . Considering (18) yields Define D ˜˙ = D˙ − D ˆ˙ = D˙ − k0 (D(t) − D(t)) ˆ ˆ T S(Z)) D − (W ∗T S(Z) − W
(19)
˜ −W ˜ T S(Z) = D˙ − k0 D Chose the Lyapunov function candidate as Vo =
1 2 ˜ D 2
(20)
Invoking (19), we obtain ˜2 − D ˜D ˜˙ = D ˜ D˙ − k0 D ˜W ˜ T S(Z) V˙ o = D
(21)
In accordance with Assumption 1 and considering the following fact ˜W ˜ T S(Z) ≤ 2|D||| ˜ W ˜ ||||S(Z)|| ≤ 2 |D| ˜ 2+ −2D
1 ˜ ||2 ||W
(22)
we have ˜ 2 + 0.5|D| ˙ 2 − k0 D ˜ 2+ ˜ 2 + 2 |D| V˙ o ≤ 0.5|D| ˜2 + ≤ −(k0 − (0.5 + 2 ))D
1 ˜ ||2 ||W
1 ˜ ||2 + 0.5 2 ||W
(23)
where ||S(Z)|| ≤ and > 0 is a design parameter. So far, the NDO design has been completed for the unknown compounded disturbance D(t) in the uncertain tracking error dynamics of optical tracking telescopes (16). The robust tracking control scheme will be proposed next. Based on outputs of the designed NDO and RBFNN, the robust adaptive control law is designed as
v=−
1 ˆ ˆ T S(Z) + D(t) ˙ (ke + k0−1 W − r) g0
(24)
where k > 0 is a design constant. ˆ and W ˜ = W∗ − W ˆ and substituting (24) ˜ =D−D Considering D into (16) yields ˆ T S(Z) + D(t) − D(t) ˆ e˙ = −ke + k0−1 W ∗T S(Z) − k0−1 W ˜ T S(Z) + D(t) ˜ = −ke + k0−1 W
(25)
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ˆ as Choose the adaptive law for W
10
ˆ) ˆ˙ = (k−1 S(Z)e + W W 0
Theorem 1. Considering the uncertain optical tracking telescope dynamic (7) with the parametric uncertainty, the unknown disturbance and the input saturation, the NDO is designed as (17) and the parameter updated law is chosen as (26). Then, using estimated outputs of the developed NDO and RBFNN, the robust tracking control law is given by (24). Under the developed robust tracking control scheme, the tracking error of uncertain optical tracking telescope dynamic (7) is convergent. Proof 1.
Invoking (23) and (25), the time derivative of V is given by
1 ˜ ||2 ||W
2 ˜ T S(Z) + eD(t) ˜ ˜ T −1 W ˜˙ + 0.5 2 − ke + k0−1 eW +W 1 ˜ ||2 + 0.5 2 ||W
˜ T S(Z) + W ˜ T −1 W ˜˙ + k0−1 eW ˜ = W∗ − W ˆ and the weight value adaptation law Considering W (26), we have 1 ˜ ||2 ||W
(29)
According to the following fact ˜ TW ˜ 2 + W ˆ 2 − W ∗ 2 ≥ W ˜ 2 − W ∗ 2 ˆ = W 2W
(30)
we have ˜ 2 − (k − 0.5)e2 ≤ −(k0 − (1.0 + 2 ))D
1 2 ˜ || + 0.5 2 + W ∗ 2 − ( − )||W 2
2
(31)
≤ − V + C where 1
− ) 2
: = min((k0 − (1.0 + 2 )), (k − 0.5), ) max ( −1 )
C : = 0.5 2 + W ∗ 2 2 2(
(32)
To ensure the closed-loop system stability, the corresponding design matrices k0 , k, and should be chosen to make k0 − (1.0 + 2 ) > 0, k − 0.5 > 0 and 2 − 1 > 0. Considering (31), it ˜ and W ˜ are semiglobal unimay directly show that the signals e, D formly ultimately bounded (SGUUB) using Lemma 1. Invoking (31), we have 0≤V ≤
C C + V (0) −
e− t
4 3
1 0
0
2
4
6
8
10
Time [s]
According to (33), we know that V is exponentially convergent, i.e., lim V = C . Thus, the tracking error e and the estimated errors t−→∞
˜2 + ≤ −(k0 − (0.5 + 2 ))D
˜ TW ˆ + 0.5 2 − W
5
˜ and W ˜ of the closed-loop system are bounded and converge to a D compact set. Analysis above demonstrates that the tracking error e is convergent and the control objective is achieved. This concludes the proof.
˜˙ ˜D ˜˙ + ee˙ + W ˜ T −1 W V˙ = D
˜ 2 − (k − 0.5)e2 + V˙ ≤ −(k0 − (1.0 + 2 ))D
6
Fig. 2. Tracking control result for the constant signal.
(27)
˜ 2 − (k − 0.5)e2 + ≤ −(k0 − (1.0 + 2 ))D
7
2
Consider the Lyapunov function candidate
1 ˜ T −1 W ˜ V = Vo + e2 + W 2
V˙
8
Tracking control result
where = T > 0 and > 0. The above robust tracking control design for the uncertain optical tracking telescope dynamic (7) using NDO and RBFNN can be included in the following theorem which is described as:
r y
9
(26)
(33)
Remark 2. In this paper, the RBFNN is employed to tackle the unknown function term F(y, r). To handle the approximation error of RBFNN, it combines the external time-varying disturbance with the effect of control input saturation as a compounded disturbance which is estimated via the developed NDO. For the proposed NDO, we know that the estimate error with suitable approximation performance can be achieved by choosing the appropriate disturbance observer gain parameter k0 . Since NDO and RBFNN are used, the disturbance and uncertainty rejection ability is improved for uncertain optical tracking telescopes under the proposed robust adaptive control scheme. 4. Simulation study In this section, the extensive numerical simulation results are presented to illustrate the effectiveness of the proposed robust adaptive control scheme based on RBFNN and NDO for the uncertain dynamic of optical tracking telescopes. In this simulation, all system parameters in (1) are give as ˛ = 5, ˛p = 2, kp = 0.05 and kz = 1. The system uncertainty is chosen as (y, t) = 0.2 sin(y)e−2y and the time-varying disturbance is d(t) = 0.1 sin(t) + 0.3e−t . The initial conˆ 0 = 0 and D ˆ = 0. ditions are arbitrarily chosen as y0 = 0.1, W To design the NDO and the robust adaptive controller for the uncertain optical tracking telescope, all design parameters are chosen as ko = 10, k = 4, g0 = 5, = diag{2}, = 2, umax = 30 and = 0.05. The NDO is designed as (17) and the parameter updated law is chosen as (26). The robust tracking control law is designed as (24) based on the estimated outputs of the developed NDO and RBFNN. Under the designed tracking control law (24), the extensive numerical simulation results and corresponding analysis are appended as follows. Firstly, the constant tracking signal is chosen to illustrate the effectiveness of the proposed robust adaptive control scheme. The desired constant tracking signal is chosen as r(t) = 7 for optical tracking telescopes. The tracking control results under the designed robust adaptive control scheme are shown in Figs. 2 and 3. Fig. 2 shows that a satisfactory tracking control result is obtained under the proposed robust adaptive control scheme. The tracking error is presented in Fig. 3 and it is convergent. The control input is shown in Fig. 4 and the bounded property is demonstrated. The plot of the
R. Mei et al. / Optik 126 (2015) 1185–1190
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results of the optical tracking telescopes are shown in Figs. 6 and 7. Fig. 6 shows that the actual output of the uncertain optical tracking telescope can well track the desired trajectory. Although the time-varying tracking signal is used, a satisfactory tracking error is achieved as shown in Fig. 7. In accordance with above simulation results for the constant tracking signal and the time-varying tracking signal, the satisfactory performance of position tracking control is achieved under the developed robust adaptive tracking control scheme for uncertain optical tracking telescopes. Therefore the proposed robust adaptive control scheme using NDO and RBFNN is effective for uncertain optical tracking telescopes. 5. Conclusion Fig. 5. Norm of the neural network weight for the constant signal.
norm of weight value for the radial basis function neural network is given in Fig. 5 and convergent. These simulation results from the constant tracking signal demonstrate that the developed robust tracking control law for the uncertain optical tracking telescope is valid. The time-varying tracking signal is then chosen to illustrate the effectiveness of the proposed robust adaptive control scheme for optical tracking telescopes. Here, the desired time-varying tracking signal is chosen as r(t) = sin(t). Under the designed robust adaptive control scheme using NDO and RBFNN, the tracking control
A robust adaptive tracking control scheme has been developed for uncertain optical tracking telescopes in the presence of the unknown parameters, the external time-varying disturbance and control input saturation in this paper. To improve the robust tracking performance, the RBFNN has been introduced to approximate the unknown continuous system uncertainty. Furthermore, the unknown external disturbance, the unknown neural network approximation error with the effect of input saturation are integrated as a compounded disturbance and the NDO has been developed to estimate the effect of this compounded disturbance. The robust adaptive tracking control scheme for uncertain optical tracking telescopes has been developed based on the outputs of
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the RBFNN and the developed NDO. By using Lyapunov analysis, the convergence of all closed-loop signals has been guaranteed. Finally, the simulation results have demonstrated that the proposed robust adaptive tracking scheme is effective for uncertain optical tracking telescopes. Acknowledgements This work is partially supported by Jiangsu Natural Science Foundation of China (granted number: SBK20130033), Program for New Century Excellent Talents in University of China (granted number: NCET-11-0830), “Liu Da Ren Cai Gao Feng” Scheme of Jiangsu Province in China (granted number: 2012-XXRJ-010) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. References [1] S. Azeemuddin, M.R. Sayeh, Effect of various parameters on working of alloptical Schmitt trigger, Optik 122 (10) (2011) 1935–1938. [2] J. Chen, Y.S. Liu, L.Z. Pan, Robust adaptive control for optical tracking telescope with unmodeled dynamics, J. Sichuan Univ. (English Sci. Ed.) 36 (3) (2004) 89–92. [3] Y.J. Liu, W. Wang, S.C. Tong, Y.S. Liu, Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters, IEEE Trans. Syst. Man Cybern. Part A: Syst. Hum. 40 (1) (2010) 170–184. [4] M. Chen, S.S. Ge, B. How, Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities, IEEE Trans. Neural Netw. 21 (5) (2010) 796–812. [5] T.S. Li, D. Wang, N.X. Chen, A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40 (3) (2011) 915–927. [6] W.S. Chen, L.C. Jiao, R.H. Li, J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Trans. Fuzzy Syst. 18 (4) (2010) 674–685. [7] B. Xu, C.G. Yang, Z.K. Shi, Reinforcement learning output feedback NN control using deterministic learning technique, IEEE Trans. Neural Netw. Learn. Syst. 25 (3) (2014) 635–641. [8] M. Chen, G. Tao, B. Jiang, Dynamic surface control using neural networks for a class of uncertain nonlinear systems with input saturation, IEEE Trans. Neural Netw. Learn. Syst. (2014), http://dx.doi.org/10.1109/TNNLS.2014.2360933 [9] W.S. Chen, J.M. Li, Decentralized output-feedback neural control for systems with unknown interconnections, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 38 (1) (2008) 258–266. [10] C.C. Hua, X.P. Guan, P. Shi, Robust backstepping control for a class of time delayed systems, IEEE Trans. Autom. Control 50 (6) (2005) 894–899. [11] Z.J. Li, X.Q. Cao, N. Ding, Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays, IEEE Trans. Fuzzy Syst. 19 (4) (2011) 745–757. [12] B. Xu, Z.K. Shi, C.G. Yang, F.C. Sun, Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form, IEEE Trans. Cybern. 44 (12) (2014) 2626–2634. [13] Y.Y. Cao, Z.L. Lin, Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation, IEEE Trans. Fuzzy Syst. 11 (1) (2003) 57–67. [14] Q. Hu, G.F. Ma, L.H. Xie, Robust and adaptive variable structure output feedback control of uncertain systems with input nonlinearity, Automatica 44 (4) (2008) 552–559. [15] H.N. Wu, H.X. Li, H∞ fuzzy observer-based control for a class of nonlinear distributed parameter systems with control constraints, IEEE Trans. Fuzzy Syst. 16 (2) (2008) 502–516.
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