Neural adaptive control for a class of nonlinear systems with unknown ...
Neural Comput & Applic (2008) 17:339–345 DOI 10.1007/s00521-007-0124-8
ORIGINAL ARTICLE
Neural adaptive control for a class of nonlinear systems with unknown deadzone Zhonghua Wang Æ Yong Zhang Æ Hui Fang
Received: 28 January 2007 / Accepted: 24 April 2007 / Published online: 12 June 2007 Springer-Verlag London Limited 2007
Abstract This paper focuses on the adaptive control of a class of nonlinear systems with unknown deadzone using neural networks. By constructing a deadzone precompensator, a neural adaptive control scheme is developed using backstepping design techniques. Transient performance is guaranteed and semi-globally uniformly ultimately bounded stability is obtained. Another feature of this scheme is that the neural networks reconstruction error bound is assumed to be unknown and can be estimated online. Simulation results are given to demonstrate the effectiveness of the proposed controller. Keywords Deadzone Adaptive control Neural networks Nonlinear systems Backstepping Robust control
1 Introduction In practical control systems, deadzone characteristics are quite commonly encountered in actuators, such as hydraulic servo valves, electric servomotors and electronic circuits. The deadzone can severely limit system performance, such as excessive steady state error and poor transient system response. Proportional-derivative (PD) controllers have been observed to result in limit cycles if Z. Wang (&) Y. Zhang H. Fang School of Control Science and Engineering, University of Jinan, Jinan 250022, People’s Republic of China e-mail: [email protected] Y. Zhang e-mail: [email protected] H. Fang e-mail: [email protected]
the actuators have deadzone. Due to the nonanalytic nature of the deadzone in actuators and the fact that deadzone parameters are often unknown and time-variant, such systems present a challenge for the control design engineers. The most straightforward way to cope with deadzone nonlinearities is to cancel them by employing their inverses. However, this can be done only when the deadzone nonlinearities are exactly known. Adaptive control is considered a good solution to the control of plants with unknown parameters. The study of constructing adaptive deadzone inverse was initiated by Recker et al. [1] where an adaptive scheme was proposed for the case of full state measurement. Adaptive deadzone inverse control designs with output feedback have been developed for plants with an unknown dead-zone at the input of a linear part [2, 3]. Cho and Bai [4] continued the above research and a perfect asymptotical adaptive cancellation of an unknown deadzone was achieved analytically to systems in which the output of a dead-zone is measurable. Adaptive output feedback deadzone compensation scheme for systems with an unknown dead-zone at the input of an nth-order smooth nonlinear dynamics in the output-feedback canonical form is built by Tian and Tao [5]. Much of other efforts for deadzone compensation have been obtained, such as variable structure control [6–8], Neural networks[9], the fusion of relay feedback control with robust nominal-model following control [10] and fuzzy logic deadzone compensation [11–14]. Such approaches promise to improve the tracking performance of motion system in presence of unknown dead-zones. Most recently, Wang et al. [15] proposed a robust adaptive control scheme by exploring the properties of its deadzone model without using adaptive deadzone inverse, the fact that deals with deadzone is similar to the general method that deals with bounded disturbance. Considering
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the same class of systems in [15] with bounded external disturbance, Zhou et al. [16] proposed two robust adaptive backstepping algorithms by estimating the bound of disturbance-like term that is composed of deadzone and external disturbance. Since neural networks have a capability of approximating nonlinear functions, it is naturally that the neural networks have been used for approximation of control system dynamics [17]. The backstepping design techniques [21] have been combined with the neural networks control for nonlinear systems recently [18–20]. In this paper, a neural adaptive control scheme for a class of nonlinear systems with unknown deadzone is proposed by using adaptive backstepping. The deadzone pre-compensator is constructed to deal with non-symmetrical deadzone parameters. We use the radial basis function (RBF) neural networks as function approximators and assumed that the approximation error bound is not known. The tuning algorithm is rigorously shown to guarantee small tracking errors by using Lyapunov stability theory. Compared with [15] and [16], the contribution of this paper focus on the following two aspects. First, the system considered here are more general in that it includes unknown nonlinear function uncertainty, while the system treated in [15] and [16] only included unknown linearly parameters uncertainty. Second, the unknown deadzone parameters can be estimated online to compensate their defection while the deadzone was dealt with in a similar to disturbance in [15]. Another feature is that the neural networks reconstruction error bound is assumed to be unknown and is estimated online. This paper is organized as follows. in Sect. 2, a description of the nonlinear system with deadzone is presented and RBF neural networks are briefly explained. In Sect. 3, the deadzone precompensation design, adaptive backstepping design and the stability analysis are described in detail. To show the effectiveness of the proposed controller, a simulation example is given in Sect. 4. Conclusions are given in the last section.
is known. Without losing generality, we assume b > 0. w(t) is the output of an unknown deadzone graphically described in Fig. 1 and mathematically defined as follows: 8 < muðtÞ mbr wðtÞ ¼ 0 : muðtÞ mbl
for uðtÞ br for bl \uðtÞ\br for uðtÞ bl
where br ‡ 0, bl £ 0 and m > 0 are constants but unknown. Further, w(t) is not available for measurement as in [1]. The control objective is to design a control laws for u(t) in Eq. (2) such that the plant state vector x tracks the ðn1Þ specified desired trajectory xd ¼ ½xd ; x_ d ; . . . ; xd T ; and the tracking error x–xd converges to a small neighborhood around zero by appropriately selecting the design parameters. 2.2 Function approximation using RBF neural networks The RBF network can be considered for its ability to uniformly approximate smooth functions over compact sets to online approximate the function [17]. Using linearly parameterized approximation model of RBF networks with fixed centers and widths, system (1) may be expressed as xðnÞ ðtÞ þ hT nð xÞ þ Dð xÞ ¼ bwðtÞ
h Rp
2.1 System description
x2X
The systems to be controlled under consideration are described by xðnÞ ðtÞ þ f ð xÞ ¼ bwðtÞ
ð1Þ
where f(•) is an unknown continuous linear or nonlinear _ . . . ; xðn1Þ T is the plant state vector, b is function, x ¼ ½x; x; unknown, nonzero, but constant. Furthermore, the sign of b
123
ð3Þ
where h*2Rp is an unknown vector of adjustable weights, n (•) is a vector of RBF and D (•) represents the network reconstruction error, defined as D (•) = f(•)–h* Tn (•). In general, increasing the number of the adjustable weights reduces the network reconstruction error. Universal approximation results for neural networks indicate that if the number of weights and network nodes are sufficiently large, and then D can be made arbitrarily small on a compact set. The optimal weight vector h* defined in (3) is a quantity only for analytical weight purposes. Typically h* is chosen as the value of h that minimizes D(x) for all x2W, where X Rn is a compact subset, i.e., T h ¼ arg min sup j f ð x Þ h nð x Þ j 2
2 Problem formulation
ð2Þ
Fig. 1 The system model with deadzone
ð4Þ
Neural Comput & Applic (2008) 17:339–345
341
d~ ¼ d d^
It is assumed on a compact subset X Rn that jDð xÞj e 8 x2X
ð5Þ
where e > 0 is an unknown bound. It should be noted that the unknown bound e is not unique, here we define e to be the smallest constant such that (5) is satisfied.
3 Controller design In this section, we will propose an adaptive deadzone precompensation controller for plants preceded by a deadzone described in (1), which will guarantee global system stability and yields the system output tracking to a desired trajectory within a desired accuracy. 3.1 Deadzone precompensation
ð11Þ
Through the simply deriving, the throughput of the compensator plus deadzone can be written as bwðtÞ ¼ bmud d~T l þ d~T d
ð12Þ
¼ ½ l 1 l T ; d is the modeling mismatch term where l and satisfies the bound kdk 1 [13]. Remark 1 If d~ approximately equal to zero, i.e., the relation of w(t) and ud is linear according equation (12). Therefore the deadzone nonlinearity compensated effectively.
d^ ffi d; to the can be
3.2 Backstepping design The combination of Eqs. (3) and (12) can be rewritten in the following form
The deadzone model (2) may be written as bwðtÞ ¼ bmu satd ðuÞ
ð6Þ
where the saturation function satd(u) is defined as 8 < dr satd ðuÞ ¼ bmu : dl
for uðtÞ br for bl \uðtÞ\br for uðtÞ bl
bmu ¼ bmud þ ld^r þ ð1 lÞd^l
ð7Þ
ð8Þ
where l is defined as l¼
1 0
.. .
ð13Þ
x_ n1 ¼ xn
where dr and dl are defined as dr = bmbr and dl = bmbl respectively. To compensate the deleterious effects of the deadzone, one can place a precompensator plus desired control ud to counteract the deadzone. But the deadzone parameters are not known. The learning or adaptation method is considered intuitively. An estimate d^ ¼ ½d^r ; d^l T of the deadzone width parameter vector d = [dr,dl]T is as the precompensator to offset the effects of deadzone. The composite u can be expressed as
x_ 1 ¼ x2
if ud 0 if ud \0
xÞ Dð xÞ þ xud d~T l þ d~T d x_ n ¼ hT nð _ ; xn ¼ xðn1Þ and x = bm. where x1 ¼ x; x2 ¼ x; Following the design procedure of backstepping [21], one can define the following error variables z1 ¼ x1 xd ði1Þ
zi ¼ x i x d
ð14Þ ai1
i ¼ 2; 3; ; n
ð15Þ
where ai-1 is the virtual control at the ith step of backstepping design and will be determined in the later design. Step 1. From Eqs. (13) to (15), we can derive z_1 ¼ x2 x_ d
ð16Þ
Viewing x2 as the virtual control, the stabilizing function a1 is defined as ð9Þ
a1 ¼ c1 z1
ð17Þ
The composite throughput from ud to w(t) of the adaptive compensator plus the deadzone is
where c1 > 0 is a design prameter. By substituting (16) and (17) into (15), the z_1 equation becomes
bwðtÞ ¼ bmud þ ld^r þ ð1 lÞd^l satd ðbmud þ ld^r þ ð1 lÞd^l Þ
z_1 ¼ c1 z1 þ z2
If the deadzone width estimation error is defined as
ð10Þ
ð18Þ
Step 2. Considering the second equation x_ 2 ¼ x3 of (13) and (15), we have
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ð19Þ
~h ¼ h h
ð32Þ
Viewing x3 as the virtual
~ ¼ x ^ x x
ð33Þ
~e ¼ e ^e
ð34Þ
z_2 ¼ x3 x€d a_ 1 oa1 ox1 x2
oa1 oxd x_ d :
þ where a_ 1 ¼ control, we design a2 as a2 ¼ c2 z2 z1 þ a_ 1
ð20Þ
where c2 > 0 is a design parameter. Through the simply deriving, (19) can be rewritten as
^ ¼ 1 xx; ~ from the Eqs. (25), Noting the fact that xx (26) and (27), we can obtain the closed-loop error equation ~ n xÞ xxa z_n ¼ cn zn zn1 Dð
z_2 ¼ z1 c2 z2 þ z3
ð21Þ
þ d~T d ~hT nð d~T l xÞ ^e tanh
Step i. Repeating step 2, we have ðiÞ
z_i ¼ xiþ1 x€d a_ i1 Pi1 oai1 where a_ i1 ¼ k¼1 oxk xkþ1 þ
oai1 ðkÞ : Choosing ðk1Þ xd
ð22Þ
oxd
ai ¼ ci zi zi1 þ a_ i1
ð23Þ
where ci > 0, i = 2,3,...,n–1, are the design parameters. The error equation can be rewritten as z_i ¼ zi1 ci zi þ ziþ1
ð24Þ
Step n. Differentiating (15) and substituting the last equation of (13), we can obtain ðnÞ þ d~T d xd z_n ¼ hT nð xÞ Dð xÞ þ xud d~T l a_ n1
ð25Þ
Finally, let the control law and neural adaptive law be as follows ^ n ud ¼ xa
ð26Þ ðnÞ
zn /
3.3 Stability analysis The closed-loop system resulted from our control law can retain the uniformly ultimate bounded. To demonstrate this property, we define the Lyapunov function candidate as n 1 X 1 2 ~ xx ~ ~ 2 þ d~T C1 V¼ z2 þ ~hT C1 e h hþ d d þ ke ~ 2 i¼1 i c
!
ð36Þ The derivative of Lyapunov function can be found as follows n X
_~ 1 _ ~_ x x ~ x ~_ þ d~T C1 zi z_i þ h~T C1 e~e h hþ d d þ ke ~ c i¼1 n X 1_ 2 ~ ^ an zn x c i zi þ x x ¼ c i¼1 h i _ h xÞzn C1 þ ~hT nð h h i _ þ d~T lzn þ dzn C1 d^
V_ ¼
xÞ þ xd þ a_ n1 an ¼ cn zn zn1 þ hT nð zn ^e tanhð Þ /
ð27Þ
^ ^ x 0 Þ _ ¼ c½an zn þ rx ðx x
ð28Þ
h_ ¼ Ch ½nð xÞzn þ rh ðh h0 Þ
ð29Þ
xÞ ^e tanhðzn =/Þ ke1~e^e_ H ¼ zn ½Dð
_ d^ ¼ Cd ½ lzn þ kd ðd^ d0 Þ _^e ¼ ke zn tanh zn þ re ð^e e0 Þ /
ð30Þ
we have
d
ð31Þ
where cn, c, rx ; rh, kd, ke, re and u are positive design ^ h parameters. Gh and Gd are positive definite matrices. x; and ^e are estimates of x1 ; h* and e respectively. tanh(•) denotes the hyperbolic tangent function. We define three estimate error variables as
123
ð35Þ
n
o þ zn ½Dð xÞ ^e tanhðzn =/Þ ke1~e^e_
ð37Þ
By define the contents of the curly brackets as Q, i.e.,
H e½j zn j zn tanhðzn =/Þ h i þ ~e zn tanhðzn =/Þ ke1^e_
ð38Þ
ð39Þ
Since the inequality jzn j zn tanhðzn =sÞ js holds for any s > 0, where j is a constant that satisfies j = e–(j+1), i.e. j = 0.2785 [22]. By applying this inequality to the equation (39), we have
Neural Comput & Applic (2008) 17:339–345
1 H je/ þ ~e zn tanhðzn =/Þ ^e_ ke
343
ð40Þ
By substituting the update law (28)–(31) into (37) and (40), we obtain V_
n X
ci z2i
~ ^ x x 0 Þ þ rh ~ hT ðh h0 Þ þ je/ þ rx xxð
We conclude that zi and parameter estimates are uniformly ultimately bounded. Furthermore, since 12 z2i V; we have z2i 2 exp ðctÞVð0Þ þ
2k ð1 exp ðctÞÞ: c
8t 0 ð52Þ
i¼1
~ nj þ kd d~T ðd^ d0 Þ þ re~eð^e e0 Þ þ jdjjz ð41Þ Noting that the facts i 1h 2 ~ ^ ~ ^ x x 0Þ ¼ x þ ðx x 0 Þ2 ðx x 0 Þ2 xð 2 i 1 h T~ ~ h h þ ðh h0 Þ2 ðh h0 Þ2 hT ðh h0 Þ ¼ ~ 2 i 1h d~T ðd^ d0 Þ ¼ d~T d~ þ ðd^ d0 Þ2 ðd d0 Þ2 2 i 1h ~eT ð^e e0 Þ ¼ ~e2 þ ð^e e0 Þ2 ðe e0 Þ2 2 ~ nj jdjjz
1 ~T ~ 2 ðd d þ zn Þ 2
( N¼ ð42Þ ð43Þ ð44Þ ð45Þ ð46Þ
we have
V_
n1 X i¼1
1 1 1 ~ 2 rh ~ hT ~ h ci z2i ðcn Þz2n rx xx 2 2 2
1 1 ðkd 1Þd~T d~ re~e2 þ k 2 2
ð47Þ
where 1 x 0 Þ2 þ rh ðh h0 Þ2 k ¼ ½rx ðx 2 þ kd ðd d0 Þ2 þ re ðe e0 Þ2 þ je/
rffiffiffiffiffi)
2k
zi 2 R jzi j
c
ð53Þ
Remark 2 According the Eq. (48), the size of k can be adjusted by design parameters rx ; rh, kd, re and u. Therefore, the radius of the tracking error convergence region can be freely adjusted in a known form. In summary, we obtain the following results. Theorem 1 Given the desired trajectory x1d,the control law (26) and adaptive laws (28)–(31) applied to the system (1) with the deadzone (2) ensures that all the closed-loop signals are uniformly ultimately bounded. Furthermore, the tracking error zi exponentially converges to a ball, whose radius is given by the Eq. (53) and can freely adjusted in known form by design parameters. Remark 3 From a practical perspective, the more prior parameter knowledge we know, the more higher tracking accuracy we can obtain. If we have roughly estimated the h*, d and e, we can chose the parameters x 0; bound of x; h*, d and e respectively. The h0, d0 and e0 close to x; parameter k will become smaller so that tracking precision is more accurate.
ð48Þ 4 Simulation results
From the Eqs. (36) and (47), we can obtain V_ cV þ k
or the tracking error zi exponentially converges to a ball, whose radius is given by the set
ð49Þ
where c > 0 is defined as ( ) 1 rh kd 1 c ¼ min 1; 2ci ; cn ; rx ; ; ; re 1 2 kmin ðC1 h Þ kmin ðCd Þ ð50Þ Therefore, V satisfies k VðtÞ exp ðctÞVð0Þ þ ð1 exp ðctÞÞ: 8t 0 c ð51Þ
In this section, a simulation is presented to demonstrate the effectiveness of the proposed above. We consider the nonlinear system described as [15] x€ ¼
1 ex ðx_2 þ 2xÞ sinx_ 0:5x sin3t þ bwðtÞ 1 þ ex
ð54Þ
x
2 where f ð xÞ ¼ 1e 1þex þ ðx_ þ 2xÞ sinx_ þ 0:5x sin3t are assumed to be unknown, which is approximated by RBF neural networks. The deadzone parameters are b = 1, br = 0.5, bl = –0.6 and m = 1. The control objective is to _ T follow the desired reference let the system state ½x x T signal ½xd x_ d :
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Neural Comput & Applic (2008) 17:339–345 500
3 x(t) xd(t)
400
2
300 200
1 100 0
0
−100
−1
−200 −300
−2 −400
−3
−500
0
5
10 Time(sec)
15
20
0
5
10 Time(sec)
15
20
Fig. 2 Tracking performance of the system with deadzone
Fig. 4 Control input u(t) acting as the input of deadzone
In the simulation, the basis functions of the RBF neural networks are chosen as Gaussian functions [17]
–0.6]T and e0 = 100 respectively. The simulation results ^ with initial values xð0Þ ¼ ½2:5 3:5T ; xð0Þ ¼ 1; ^ ¼ ½0:6; 0:5T and ^e ¼ 100 are h(0) = [1,1,1,1,1]T, dð0Þ shown in Figs. 2–4. Figure 2 shows the position tracking performance and Fig. 3 shows the corresponding tracking error. From Figs. 2 and 3, we can see that good tracking performance is obtained. Figure 4 shows the corresponding control input signal u(t) of the system with deadzone.
"
ð x ti ÞT ð x ti Þ nð xÞi ¼ exp r2i
# ð55Þ
where ti = [ti1 ti2] is the center of the receptive field and ri is the width of radial Gaussian function. The neural networks contain five nodes with centers ti evenly spaced in [–6,6] and width ri = 3(i = 1,2,3,4,5). The desired trajectory is chosen as xd(t) = 2.5sin(t). The design parameters are selected as c1 = c2 = ke = 10, / ¼ rx ¼ re ¼ 0:001; Ch ¼ diagð5; 5; 5; 5; 5Þ; rh ¼ diag ð0:001; 0:001; 0:001; 0:001; 0:001Þ; Cd ¼ diagð10; 10Þ; 0 ¼ 1; h0 = [1 1 1 1 1]T, d0 = [0.5 kd ¼ diagð0:001; 0:001Þ; x
3
2.5
2
1.5
A neural adaptive control technique for a class of nonlinear systems with unknown deadzone has been presented. The stability of the controlled system has been guaranteed by Lyapunov theory. The tracking error converges to a small residual set which is adjustable by tuning the design parameters. The effectiveness of the proposed control scheme has also been highlighted with a simulation example.
References
1
0.5
0
−0.5
5 Conclusion
0
5
10 Time(sec)
15
Fig. 3 Tracking error z1 of the system with deadzone
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20
1. Recker D, Kokotovic´ P, Rhode D, Winkelman J (1991) Adaptive nonlinear control of systems containing a deadzone. In: Proceedings of the 30th IEEE conference on decision and control, Brighton, England, pp 2111–2115 2. Gang T, Kokotovic´ PV (1994) Adaptive control of plants with unknown deadzones. IEEE Trans Autom Control 39:59–68 3. Gang T, Tian M (1995) Design of adaptive dead-zone inverse for nonminimum phase plants. In: Proceedings of the 1995 ACC, Seattle, pp 2059–2063 4. Cho H-Y, Bai E-W (1998) Convergence results for an adaptive deadzone inverse. Int J Adapt Control Signal Process 12:451–466
Neural Comput & Applic (2008) 17:339–345 5. Tian M, Gang T (1997) Adaptive dead-zone compensation for output-feedback canonical systems. Int J Contr 67:791–812 6. Utkin VI (1993) Sliding mode control design principles and applications to electric drives. IEEE Trans Ind Electron 40:23–36 7. Jie S, Sun JQ (2003) Sliding control accounting for hardware limitation of mechanical actuators with deadzone. J Sound Vib 266:905–911 8. Corradini ML, Orlando G (2002) Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator. IEEE Trans Control Systems Technol 10:158–166 9. Sˇelmic´ RR, Lewis FL (2000) Deadzone compensation in motion control systems using neural networks. IEEE Trans Autom Control 45:602–613 10. Tsang KM, Li G(2001) Robust nonlinear nominal-model following control to overcome deadzone nonlinearities. IEEE Trans Ind Electron 48:177–184 11. Kim JH, Park JH, Lee SW, Edwin KP Chong: a two-layered fuzzy logic controller for systems with deadzones. IEEE Trans Ind Electron 41:155–162 12. Campos J, Lewis FL (1999) Deadzone compensation in discrete time using adaptive fuzzy logic. IEEE Trans Fuzzy Systems 7:697–707 13. Lewis FL, Woo KT, Li XW, Li ZX (1999) Deadzone compensation in motion control systems using adaptive fuzzy logic control. IEEE Trans Control Systems Technol 7:731–742
345 14. Oh JJ (2001) A deadzone compensator of a DC motor system using fuzzy logic control. IEEE Trans Systems Man Cybern C Appl Rev 31:42–48 15. Wang XS, Su CY, Hong H (2004) Robust adaptive control of a class of nonlinear systems with unknown dead-zone. Automatica 40:407–413 16. Zhou J, Wen C, Zhang Y (2004) Adaptive backstepping control of a class of uncertain nonlinear systems with unknown deadzone. In: Proceedings of 2004 IEEE conference on robotics, automation and mechatronics, Singapore, pp 513–518 17. Sanner RM, Slotine JJE (1992) Gaussian networks for direct adaptive control. IEEE Trans Neural Netw 3:837–863 18. Zhang T, Ge SS, Hang CC (2000) Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica 36:1835–1846 19. Li Y, Sheng Q, Zhuang X, Okyay K (2004) Robust and adaptive backstepping control for nonlinear system using RBF neural networks. IEEE Trans Neural Netw 15:693–701 20. Hsu CF, Lin CM, Lee TT (2006) Wavelet adaptive backstepping control for a class of nonlinear systems. IEEE Trans Neural Netw 17:1175–1183 21. Krstic´ M, Kanellakopoulos I, Kokotovic´ P (1995) Nonlinear and adaptive control design. Wiley, London 22. Polycarpou MM (1996) Stable adaptive neural control scheme for nonlinear system. IEEE Trans Autom Control 41:447–451
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ORIGINAL ARTICLE
Neural adaptive control for a class of nonlinear systems with unknown deadzone Zhonghua Wang Æ Yong Zhang Æ Hui Fang
Received: 28 January 2007 / Accepted: 24 April 2007 / Published online: 12 June 2007 Springer-Verlag London Limited 2007
Abstract This paper focuses on the adaptive control of a class of nonlinear systems with unknown deadzone using neural networks. By constructing a deadzone precompensator, a neural adaptive control scheme is developed using backstepping design techniques. Transient performance is guaranteed and semi-globally uniformly ultimately bounded stability is obtained. Another feature of this scheme is that the neural networks reconstruction error bound is assumed to be unknown and can be estimated online. Simulation results are given to demonstrate the effectiveness of the proposed controller. Keywords Deadzone Adaptive control Neural networks Nonlinear systems Backstepping Robust control
1 Introduction In practical control systems, deadzone characteristics are quite commonly encountered in actuators, such as hydraulic servo valves, electric servomotors and electronic circuits. The deadzone can severely limit system performance, such as excessive steady state error and poor transient system response. Proportional-derivative (PD) controllers have been observed to result in limit cycles if Z. Wang (&) Y. Zhang H. Fang School of Control Science and Engineering, University of Jinan, Jinan 250022, People’s Republic of China e-mail: [email protected] Y. Zhang e-mail: [email protected] H. Fang e-mail: [email protected]
the actuators have deadzone. Due to the nonanalytic nature of the deadzone in actuators and the fact that deadzone parameters are often unknown and time-variant, such systems present a challenge for the control design engineers. The most straightforward way to cope with deadzone nonlinearities is to cancel them by employing their inverses. However, this can be done only when the deadzone nonlinearities are exactly known. Adaptive control is considered a good solution to the control of plants with unknown parameters. The study of constructing adaptive deadzone inverse was initiated by Recker et al. [1] where an adaptive scheme was proposed for the case of full state measurement. Adaptive deadzone inverse control designs with output feedback have been developed for plants with an unknown dead-zone at the input of a linear part [2, 3]. Cho and Bai [4] continued the above research and a perfect asymptotical adaptive cancellation of an unknown deadzone was achieved analytically to systems in which the output of a dead-zone is measurable. Adaptive output feedback deadzone compensation scheme for systems with an unknown dead-zone at the input of an nth-order smooth nonlinear dynamics in the output-feedback canonical form is built by Tian and Tao [5]. Much of other efforts for deadzone compensation have been obtained, such as variable structure control [6–8], Neural networks[9], the fusion of relay feedback control with robust nominal-model following control [10] and fuzzy logic deadzone compensation [11–14]. Such approaches promise to improve the tracking performance of motion system in presence of unknown dead-zones. Most recently, Wang et al. [15] proposed a robust adaptive control scheme by exploring the properties of its deadzone model without using adaptive deadzone inverse, the fact that deals with deadzone is similar to the general method that deals with bounded disturbance. Considering
123
340
Neural Comput & Applic (2008) 17:339–345
the same class of systems in [15] with bounded external disturbance, Zhou et al. [16] proposed two robust adaptive backstepping algorithms by estimating the bound of disturbance-like term that is composed of deadzone and external disturbance. Since neural networks have a capability of approximating nonlinear functions, it is naturally that the neural networks have been used for approximation of control system dynamics [17]. The backstepping design techniques [21] have been combined with the neural networks control for nonlinear systems recently [18–20]. In this paper, a neural adaptive control scheme for a class of nonlinear systems with unknown deadzone is proposed by using adaptive backstepping. The deadzone pre-compensator is constructed to deal with non-symmetrical deadzone parameters. We use the radial basis function (RBF) neural networks as function approximators and assumed that the approximation error bound is not known. The tuning algorithm is rigorously shown to guarantee small tracking errors by using Lyapunov stability theory. Compared with [15] and [16], the contribution of this paper focus on the following two aspects. First, the system considered here are more general in that it includes unknown nonlinear function uncertainty, while the system treated in [15] and [16] only included unknown linearly parameters uncertainty. Second, the unknown deadzone parameters can be estimated online to compensate their defection while the deadzone was dealt with in a similar to disturbance in [15]. Another feature is that the neural networks reconstruction error bound is assumed to be unknown and is estimated online. This paper is organized as follows. in Sect. 2, a description of the nonlinear system with deadzone is presented and RBF neural networks are briefly explained. In Sect. 3, the deadzone precompensation design, adaptive backstepping design and the stability analysis are described in detail. To show the effectiveness of the proposed controller, a simulation example is given in Sect. 4. Conclusions are given in the last section.
is known. Without losing generality, we assume b > 0. w(t) is the output of an unknown deadzone graphically described in Fig. 1 and mathematically defined as follows: 8 < muðtÞ mbr wðtÞ ¼ 0 : muðtÞ mbl
for uðtÞ br for bl \uðtÞ\br for uðtÞ bl
where br ‡ 0, bl £ 0 and m > 0 are constants but unknown. Further, w(t) is not available for measurement as in [1]. The control objective is to design a control laws for u(t) in Eq. (2) such that the plant state vector x tracks the ðn1Þ specified desired trajectory xd ¼ ½xd ; x_ d ; . . . ; xd T ; and the tracking error x–xd converges to a small neighborhood around zero by appropriately selecting the design parameters. 2.2 Function approximation using RBF neural networks The RBF network can be considered for its ability to uniformly approximate smooth functions over compact sets to online approximate the function [17]. Using linearly parameterized approximation model of RBF networks with fixed centers and widths, system (1) may be expressed as xðnÞ ðtÞ þ hT nð xÞ þ Dð xÞ ¼ bwðtÞ
h Rp
2.1 System description
x2X
The systems to be controlled under consideration are described by xðnÞ ðtÞ þ f ð xÞ ¼ bwðtÞ
ð1Þ
where f(•) is an unknown continuous linear or nonlinear _ . . . ; xðn1Þ T is the plant state vector, b is function, x ¼ ½x; x; unknown, nonzero, but constant. Furthermore, the sign of b
123
ð3Þ
where h*2Rp is an unknown vector of adjustable weights, n (•) is a vector of RBF and D (•) represents the network reconstruction error, defined as D (•) = f(•)–h* Tn (•). In general, increasing the number of the adjustable weights reduces the network reconstruction error. Universal approximation results for neural networks indicate that if the number of weights and network nodes are sufficiently large, and then D can be made arbitrarily small on a compact set. The optimal weight vector h* defined in (3) is a quantity only for analytical weight purposes. Typically h* is chosen as the value of h that minimizes D(x) for all x2W, where X Rn is a compact subset, i.e., T h ¼ arg min sup j f ð x Þ h nð x Þ j 2
2 Problem formulation
ð2Þ
Fig. 1 The system model with deadzone
ð4Þ
Neural Comput & Applic (2008) 17:339–345
341
d~ ¼ d d^
It is assumed on a compact subset X Rn that jDð xÞj e 8 x2X
ð5Þ
where e > 0 is an unknown bound. It should be noted that the unknown bound e is not unique, here we define e to be the smallest constant such that (5) is satisfied.
3 Controller design In this section, we will propose an adaptive deadzone precompensation controller for plants preceded by a deadzone described in (1), which will guarantee global system stability and yields the system output tracking to a desired trajectory within a desired accuracy. 3.1 Deadzone precompensation
ð11Þ
Through the simply deriving, the throughput of the compensator plus deadzone can be written as bwðtÞ ¼ bmud d~T l þ d~T d
ð12Þ
¼ ½ l 1 l T ; d is the modeling mismatch term where l and satisfies the bound kdk 1 [13]. Remark 1 If d~ approximately equal to zero, i.e., the relation of w(t) and ud is linear according equation (12). Therefore the deadzone nonlinearity compensated effectively.
d^ ffi d; to the can be
3.2 Backstepping design The combination of Eqs. (3) and (12) can be rewritten in the following form
The deadzone model (2) may be written as bwðtÞ ¼ bmu satd ðuÞ
ð6Þ
where the saturation function satd(u) is defined as 8 < dr satd ðuÞ ¼ bmu : dl
for uðtÞ br for bl \uðtÞ\br for uðtÞ bl
bmu ¼ bmud þ ld^r þ ð1 lÞd^l
ð7Þ
ð8Þ
where l is defined as l¼
1 0
.. .
ð13Þ
x_ n1 ¼ xn
where dr and dl are defined as dr = bmbr and dl = bmbl respectively. To compensate the deleterious effects of the deadzone, one can place a precompensator plus desired control ud to counteract the deadzone. But the deadzone parameters are not known. The learning or adaptation method is considered intuitively. An estimate d^ ¼ ½d^r ; d^l T of the deadzone width parameter vector d = [dr,dl]T is as the precompensator to offset the effects of deadzone. The composite u can be expressed as
x_ 1 ¼ x2
if ud 0 if ud \0
xÞ Dð xÞ þ xud d~T l þ d~T d x_ n ¼ hT nð _ ; xn ¼ xðn1Þ and x = bm. where x1 ¼ x; x2 ¼ x; Following the design procedure of backstepping [21], one can define the following error variables z1 ¼ x1 xd ði1Þ
zi ¼ x i x d
ð14Þ ai1
i ¼ 2; 3; ; n
ð15Þ
where ai-1 is the virtual control at the ith step of backstepping design and will be determined in the later design. Step 1. From Eqs. (13) to (15), we can derive z_1 ¼ x2 x_ d
ð16Þ
Viewing x2 as the virtual control, the stabilizing function a1 is defined as ð9Þ
a1 ¼ c1 z1
ð17Þ
The composite throughput from ud to w(t) of the adaptive compensator plus the deadzone is
where c1 > 0 is a design prameter. By substituting (16) and (17) into (15), the z_1 equation becomes
bwðtÞ ¼ bmud þ ld^r þ ð1 lÞd^l satd ðbmud þ ld^r þ ð1 lÞd^l Þ
z_1 ¼ c1 z1 þ z2
If the deadzone width estimation error is defined as
ð10Þ
ð18Þ
Step 2. Considering the second equation x_ 2 ¼ x3 of (13) and (15), we have
123
342
Neural Comput & Applic (2008) 17:339–345
ð19Þ
~h ¼ h h
ð32Þ
Viewing x3 as the virtual
~ ¼ x ^ x x
ð33Þ
~e ¼ e ^e
ð34Þ
z_2 ¼ x3 x€d a_ 1 oa1 ox1 x2
oa1 oxd x_ d :
þ where a_ 1 ¼ control, we design a2 as a2 ¼ c2 z2 z1 þ a_ 1
ð20Þ
where c2 > 0 is a design parameter. Through the simply deriving, (19) can be rewritten as
^ ¼ 1 xx; ~ from the Eqs. (25), Noting the fact that xx (26) and (27), we can obtain the closed-loop error equation ~ n xÞ xxa z_n ¼ cn zn zn1 Dð
z_2 ¼ z1 c2 z2 þ z3
ð21Þ
þ d~T d ~hT nð d~T l xÞ ^e tanh
Step i. Repeating step 2, we have ðiÞ
z_i ¼ xiþ1 x€d a_ i1 Pi1 oai1 where a_ i1 ¼ k¼1 oxk xkþ1 þ
oai1 ðkÞ : Choosing ðk1Þ xd
ð22Þ
oxd
ai ¼ ci zi zi1 þ a_ i1
ð23Þ
where ci > 0, i = 2,3,...,n–1, are the design parameters. The error equation can be rewritten as z_i ¼ zi1 ci zi þ ziþ1
ð24Þ
Step n. Differentiating (15) and substituting the last equation of (13), we can obtain ðnÞ þ d~T d xd z_n ¼ hT nð xÞ Dð xÞ þ xud d~T l a_ n1
ð25Þ
Finally, let the control law and neural adaptive law be as follows ^ n ud ¼ xa
ð26Þ ðnÞ
zn /
3.3 Stability analysis The closed-loop system resulted from our control law can retain the uniformly ultimate bounded. To demonstrate this property, we define the Lyapunov function candidate as n 1 X 1 2 ~ xx ~ ~ 2 þ d~T C1 V¼ z2 þ ~hT C1 e h hþ d d þ ke ~ 2 i¼1 i c
!
ð36Þ The derivative of Lyapunov function can be found as follows n X
_~ 1 _ ~_ x x ~ x ~_ þ d~T C1 zi z_i þ h~T C1 e~e h hþ d d þ ke ~ c i¼1 n X 1_ 2 ~ ^ an zn x c i zi þ x x ¼ c i¼1 h i _ h xÞzn C1 þ ~hT nð h h i _ þ d~T lzn þ dzn C1 d^
V_ ¼
xÞ þ xd þ a_ n1 an ¼ cn zn zn1 þ hT nð zn ^e tanhð Þ /
ð27Þ
^ ^ x 0 Þ _ ¼ c½an zn þ rx ðx x
ð28Þ
h_ ¼ Ch ½nð xÞzn þ rh ðh h0 Þ
ð29Þ
xÞ ^e tanhðzn =/Þ ke1~e^e_ H ¼ zn ½Dð
_ d^ ¼ Cd ½ lzn þ kd ðd^ d0 Þ _^e ¼ ke zn tanh zn þ re ð^e e0 Þ /
ð30Þ
we have
d
ð31Þ
where cn, c, rx ; rh, kd, ke, re and u are positive design ^ h parameters. Gh and Gd are positive definite matrices. x; and ^e are estimates of x1 ; h* and e respectively. tanh(•) denotes the hyperbolic tangent function. We define three estimate error variables as
123
ð35Þ
n
o þ zn ½Dð xÞ ^e tanhðzn =/Þ ke1~e^e_
ð37Þ
By define the contents of the curly brackets as Q, i.e.,
H e½j zn j zn tanhðzn =/Þ h i þ ~e zn tanhðzn =/Þ ke1^e_
ð38Þ
ð39Þ
Since the inequality jzn j zn tanhðzn =sÞ js holds for any s > 0, where j is a constant that satisfies j = e–(j+1), i.e. j = 0.2785 [22]. By applying this inequality to the equation (39), we have
Neural Comput & Applic (2008) 17:339–345
1 H je/ þ ~e zn tanhðzn =/Þ ^e_ ke
343
ð40Þ
By substituting the update law (28)–(31) into (37) and (40), we obtain V_
n X
ci z2i
~ ^ x x 0 Þ þ rh ~ hT ðh h0 Þ þ je/ þ rx xxð
We conclude that zi and parameter estimates are uniformly ultimately bounded. Furthermore, since 12 z2i V; we have z2i 2 exp ðctÞVð0Þ þ
2k ð1 exp ðctÞÞ: c
8t 0 ð52Þ
i¼1
~ nj þ kd d~T ðd^ d0 Þ þ re~eð^e e0 Þ þ jdjjz ð41Þ Noting that the facts i 1h 2 ~ ^ ~ ^ x x 0Þ ¼ x þ ðx x 0 Þ2 ðx x 0 Þ2 xð 2 i 1 h T~ ~ h h þ ðh h0 Þ2 ðh h0 Þ2 hT ðh h0 Þ ¼ ~ 2 i 1h d~T ðd^ d0 Þ ¼ d~T d~ þ ðd^ d0 Þ2 ðd d0 Þ2 2 i 1h ~eT ð^e e0 Þ ¼ ~e2 þ ð^e e0 Þ2 ðe e0 Þ2 2 ~ nj jdjjz
1 ~T ~ 2 ðd d þ zn Þ 2
( N¼ ð42Þ ð43Þ ð44Þ ð45Þ ð46Þ
we have
V_
n1 X i¼1
1 1 1 ~ 2 rh ~ hT ~ h ci z2i ðcn Þz2n rx xx 2 2 2
1 1 ðkd 1Þd~T d~ re~e2 þ k 2 2
ð47Þ
where 1 x 0 Þ2 þ rh ðh h0 Þ2 k ¼ ½rx ðx 2 þ kd ðd d0 Þ2 þ re ðe e0 Þ2 þ je/
rffiffiffiffiffi)
2k
zi 2 R jzi j
c
ð53Þ
Remark 2 According the Eq. (48), the size of k can be adjusted by design parameters rx ; rh, kd, re and u. Therefore, the radius of the tracking error convergence region can be freely adjusted in a known form. In summary, we obtain the following results. Theorem 1 Given the desired trajectory x1d,the control law (26) and adaptive laws (28)–(31) applied to the system (1) with the deadzone (2) ensures that all the closed-loop signals are uniformly ultimately bounded. Furthermore, the tracking error zi exponentially converges to a ball, whose radius is given by the Eq. (53) and can freely adjusted in known form by design parameters. Remark 3 From a practical perspective, the more prior parameter knowledge we know, the more higher tracking accuracy we can obtain. If we have roughly estimated the h*, d and e, we can chose the parameters x 0; bound of x; h*, d and e respectively. The h0, d0 and e0 close to x; parameter k will become smaller so that tracking precision is more accurate.
ð48Þ 4 Simulation results
From the Eqs. (36) and (47), we can obtain V_ cV þ k
or the tracking error zi exponentially converges to a ball, whose radius is given by the set
ð49Þ
where c > 0 is defined as ( ) 1 rh kd 1 c ¼ min 1; 2ci ; cn ; rx ; ; ; re 1 2 kmin ðC1 h Þ kmin ðCd Þ ð50Þ Therefore, V satisfies k VðtÞ exp ðctÞVð0Þ þ ð1 exp ðctÞÞ: 8t 0 c ð51Þ
In this section, a simulation is presented to demonstrate the effectiveness of the proposed above. We consider the nonlinear system described as [15] x€ ¼
1 ex ðx_2 þ 2xÞ sinx_ 0:5x sin3t þ bwðtÞ 1 þ ex
ð54Þ
x
2 where f ð xÞ ¼ 1e 1þex þ ðx_ þ 2xÞ sinx_ þ 0:5x sin3t are assumed to be unknown, which is approximated by RBF neural networks. The deadzone parameters are b = 1, br = 0.5, bl = –0.6 and m = 1. The control objective is to _ T follow the desired reference let the system state ½x x T signal ½xd x_ d :
123
344
Neural Comput & Applic (2008) 17:339–345 500
3 x(t) xd(t)
400
2
300 200
1 100 0
0
−100
−1
−200 −300
−2 −400
−3
−500
0
5
10 Time(sec)
15
20
0
5
10 Time(sec)
15
20
Fig. 2 Tracking performance of the system with deadzone
Fig. 4 Control input u(t) acting as the input of deadzone
In the simulation, the basis functions of the RBF neural networks are chosen as Gaussian functions [17]
–0.6]T and e0 = 100 respectively. The simulation results ^ with initial values xð0Þ ¼ ½2:5 3:5T ; xð0Þ ¼ 1; ^ ¼ ½0:6; 0:5T and ^e ¼ 100 are h(0) = [1,1,1,1,1]T, dð0Þ shown in Figs. 2–4. Figure 2 shows the position tracking performance and Fig. 3 shows the corresponding tracking error. From Figs. 2 and 3, we can see that good tracking performance is obtained. Figure 4 shows the corresponding control input signal u(t) of the system with deadzone.
"
ð x ti ÞT ð x ti Þ nð xÞi ¼ exp r2i
# ð55Þ
where ti = [ti1 ti2] is the center of the receptive field and ri is the width of radial Gaussian function. The neural networks contain five nodes with centers ti evenly spaced in [–6,6] and width ri = 3(i = 1,2,3,4,5). The desired trajectory is chosen as xd(t) = 2.5sin(t). The design parameters are selected as c1 = c2 = ke = 10, / ¼ rx ¼ re ¼ 0:001; Ch ¼ diagð5; 5; 5; 5; 5Þ; rh ¼ diag ð0:001; 0:001; 0:001; 0:001; 0:001Þ; Cd ¼ diagð10; 10Þ; 0 ¼ 1; h0 = [1 1 1 1 1]T, d0 = [0.5 kd ¼ diagð0:001; 0:001Þ; x
3
2.5
2
1.5
A neural adaptive control technique for a class of nonlinear systems with unknown deadzone has been presented. The stability of the controlled system has been guaranteed by Lyapunov theory. The tracking error converges to a small residual set which is adjustable by tuning the design parameters. The effectiveness of the proposed control scheme has also been highlighted with a simulation example.
References
1
0.5
0
−0.5
5 Conclusion
0
5
10 Time(sec)
15
Fig. 3 Tracking error z1 of the system with deadzone
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20
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