Chaos, Solitons and Fractals 35 (2008) 383–389 www.elsevier.com/locate/chaos
Composite tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities Yeong-Jeu Sun
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Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan, ROC Accepted 17 May 2006
Communicated by Prof. L. Marek-Crnjac
Abstract In this paper, the tracking problem for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities is considered. A composite tracking control is proposed such that the states of the feedback-controlled system track the desired trajectories with any pre-specified convergence rate. Meanwhile, an estimate of the tracking time is derived for such systems. A numerical example is also given to illustrate our main result. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The subject of robust tracking control is a very active research area. For decades, various approaches in robust tracking control have been investigated, such as adaptive control approach [1], variable structure approach [2–4], composite tracking control approach [5], and others [6,7]. In particular, the composite tracking control approach is a substantial methodology for the tracking control design of uncertain control systems with nonlinearities. Nonlinearities frequently appear in automatic control systems. In the past, researchers have been concerned with nonsmooth nonlinearities common in physical systems, such as saturation, hysteresis, relays, deadzones, and others; see, for example, [1,4,7–17]. These nonlinearities are frequently due to friction, which may vary with temperature and wear. Meanwhile, these nonlinearities may appear in mass produced components, such as valves and gears, which may vary from one component to the other [15]. In particular, the deadzone nonlinearity is important in itself, but also other nonsmooth nonlinearities, such as hysteresis, can be modeled using deadzone [15]. In this paper, the tracking problem for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities is investigated. Our objective is to propose a composite tracking control such that the states of the feedbackcontrolled system track the desired trajectories with any pre-specified convergence rate. In addition, an estimate of the tracking time is also derived. Finally, a numerical example is given to illustrate our main result.
*
Tel.: +886 7 6577711x6626; fax: +886 7 6577205. E-mail address:
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0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.034
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Nomenclature Rn Rþ I Q>0 kmax(Q) AT kAk m
the n-dimensional real space the set of all non-negative reals the unit matrix the matrix Q is a symmetric positive definite matrix (res. kmin(Q)) the maximum (res. minimum) eigenvalue of the symmetric matrix Q the transpose of the matrix A the induced Euclidean norm of the matrix A {1, 2, . . . , m}
2. Problem formulation and main results Definition 1 ([1,7,11,15,17]). The deadzone nonlinearities Dðu; b; m1 Þ, with b P 0 and m1 > 0, is defined to be the collection of all functions T m m D/ðu; b; m1 Þ :¼ D/1 ðu1 ; b; m1 Þ D/2 ðu2 ; b; m1 Þ D/m ðum ; b; m1 Þ : R R R ! R satisfying, for every i 2 m, 8 > < m1 ðui ri Þ; D/i ðui ; b; m1 Þ :¼ 0; > : m1 ðui þ ri Þ;
if ri 6 ui ; if ri 6 ui 6 ri ; if ui 6 ri
for any ri and ri with 0 6 ri, ri 6 b. In this paper, we consider the following uncertain nonlinear control system: x_ ðtÞ ¼ AxðtÞ þ Df ðt; xðtÞÞ þ BD/ðuðtÞ; b; m1 Þ; xð0Þ ¼ x0 ;
t P 0;
ð1aÞ ð1bÞ
where xðtÞ 2 Rn is the state vector, u ¼ ½ u1 u2 um T 2 Rm is the control, A 2 Rnn , B 2 Rnm , Df and D/ðu; b; m1 Þ 2 Dðu; b; m1 Þ, the deadzone nonlinearity, are uncertainties with appropriate dimensions, (A, B) is a completely controllable pair, and x0 is the initial state. The purpose of this paper is to design the tracking control u(t) such that the state x(t) of the system (1) can track the desired constant signal xd. Definition 2 [5]. For any positive constants e1 and e2, the tracking time ts(e1, e2), if it exists, of a control system is defined by keðtÞk 6 e1 keð0Þk þ e2
8t P ts ðe1 ; e2 Þ P 0;
where eðtÞ :¼ xðtÞ xd ðtÞ
ð2Þ
representing the error between the system state x and desired state xd. In real applications, e1 = 0.1 and e2 = 0.05 give reasonable choices. Clearly, a tracking control system having a small tracking time has a better transient behavior. Now, we make the following assumption. A 1 ([3,5,18]). The uncertain function Df is continuous and there exist continuous functions Dg, g, and constant vector h 2 Rm such that the following matching conditions are satisfied: Axd ¼ Bh; Df ðt; xÞ ¼ BDgðt; xÞ 8t 2 R; x 2 Rn ; kDgðt; xÞk 6 gðt; xÞ 8t 2 R; x 2 Rn :
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By (1), (2), and (A1), we have the following error dynamic system: e_ ðtÞ ¼ x_ ðtÞ x_ d ðtÞ ¼ AeðtÞ þ Axd þ Df ðt; eðtÞ þ xd Þ þ B½m1 uðtÞ þ DCðuðtÞÞ ¼ AeðtÞ þ Bh þ BDgðt; eðtÞ þ xd Þ þ B½m1 uðtÞ þ DCðuðtÞÞ ¼ AeðtÞ þ B½h þ Dgðt; eðtÞ þ xd Þ þ DCðuðtÞÞ þ m1 BuðtÞ ¼ AeðtÞ þ BDhðt; eðtÞ þ xd ; uðtÞÞ þ m1 BuðtÞ 8t P 0;
ð3Þ
where DCðuðtÞÞ :¼ D/ðuðtÞ; b; m1 Þ m1 uðtÞ; Dhðt; eðtÞ þ xd ; uðtÞÞ :¼ h þ Dgðt; eðtÞ þ xd Þ þ DCðuðtÞÞ: Clearly, by (A1), it follows that b: kDhðt; eðtÞ þ xd ; uðtÞÞk 6 Kðt; eðtÞ þ xd Þ :¼ khk þ gðt; eðtÞ þ xd Þ þ mm1
ð4Þ
First consider the case of the system (3) without any uncertainties and reference input, i.e., Dh = 0 and D/ðuðtÞ; b; m1 Þ ¼ m1 uðtÞ. Thus the nominal system of (3) can represented as e_ ðtÞ ¼ AeðtÞ þ m1 BuðtÞ:
ð5Þ
It can be shown by [19] that given any positive constant a and matrix Q > 0, the system (5) subjected to the control law uðtÞ ¼
1 T B PeðtÞ 2m1
ð6Þ
is globally exponentially stable with convergence rate a, where P > 0 is the unique solution of ðA þ aIÞT P þ P ðA þ aIÞ PBBT P þ Q ¼ 0:
ð7Þ
Nevertheless, the nominal control law (6) may not suffice to render the uncertain error system (3) to be globally exponentially stable. Hence a corrective control term must be added to overcome the uncertain part of (3). In the following, a composite control, consisting of a nominal control and a corrective control, is proposed such that the uncertain error system (3) is globally exponentially stable. Moreover, an estimate of the tracking time for the feedback-controlled system is also provided. Theorem 1. If (A1) is satisfied for the uncertain nonlinear system (1), then the state x(t) of the system (1) tracks, with the pre-specified exponential convergence rate a, the desired constant signal xd under the composite tracking control law as uðtÞ ¼ u1 ðt; eðtÞÞ þ u2 ðt; eðtÞÞ; 1 T B PeðtÞ; u1 ðt; eðtÞÞ ¼ 2m1 u2 ðt; eðtÞÞ ¼ k 1 ðt; eðtÞÞBT PeðtÞ
ð8Þ ð9Þ ð10Þ
with k 1 ðt; eðtÞÞ ¼
1 2K2 ðt; eðtÞ þ xd Þ ; m1 2Kðt; eðtÞ þ xd Þ kBT PeðtÞk þ e2bt
P > 0 satisfying (7), and b > a. Moreover, the tracking time is estimated as 8 9 kmax ðP Þ 1 > > > 2 ln e 2 ln e ln ln 1 2> < = kmin ðP Þ 2kmin ðP Þ ðb aÞ ; ts ðe1 ; e2 Þ ¼ max 0; : > > 2a 2a > > : ;
ð11Þ
ð12Þ
Proof 1. Let V ðeðtÞÞ ¼ eT ðtÞPeðtÞ:
ð13Þ
The time derivative of V(e(t)) along the trajectories of system (3) with u defined by (8)–(11) is given by V_ ðeðtÞÞ ¼ eT ðtÞðAT P þ PAÞeðtÞ þ 2eT ðtÞPBDh þ 2m1 eT ðtÞPBuðtÞ 8t P 0:
ð14Þ
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It can be deduced, from (4), (7), (11), and (14), that V_ ðeðtÞÞ 6 eT ðtÞðAT P þ PAÞeðtÞ þ 2eT ðtÞPBDh eT ðtÞPBBT PeðtÞ 2m1 k 1 eT ðtÞPBBT PeðtÞ ¼ eT ðtÞ½ðA þ aIÞT P þ P ðA þ aIÞ PBBT P þ QeðtÞ 2aeT ðtÞPeðtÞ eT ðtÞQeðtÞ þ 2eT ðtÞPBDh 2m1 k 1 eT ðtÞPBBT PeðtÞ ¼ 2aeT ðtÞPeðtÞ eT ðtÞQeðtÞ þ 2eT ðtÞPBDh 2m1 k 1 eT ðtÞPBBT PeðtÞ 6 2aV ðeðtÞÞ þ 2kBT PeðtÞk kDhk 2m1 k 1 kBT PeðtÞk2 6 2aV ðeðtÞÞ þ 2kBT PeðtÞk K ¼ 2aV ðeðtÞÞ þ
4K2 kBT PeðtÞk2 2K kBT PeðtÞk þ e2bt
ð2K kBT PeðtÞkÞ ðe2bt Þ 2K kBT PeðtÞk þ e2bt
8t P 0:
ð15Þ
Using the inequality ab 6 b 8a; b P 0 with a þ b > 0 aþb in (15) yields V_ ðeðtÞÞ 6 2aV ðeðtÞÞ þ e2bt
8t P 0:
It follows that e2at V_ ðeðtÞÞ 6 2ae2at V ðeðtÞÞ þ e2at e2bt 8t P 0 ) ½e2at V_ ðeðtÞÞ þ 2ae2at V ðeðtÞÞ 6 e2ðbaÞt 8t P 0 d ) ½e2at V ðeðtÞÞ 6 e2ðbaÞt 8t P 0 dt Z t 1 1 ½1 e2ðbaÞt 6 e2ðbaÞs ds ¼ ) e2at V ðeðtÞÞ V ðeð0ÞÞ 6 2ðb aÞ 2ðb aÞ 0 1 ) V ðeðtÞÞ 6 V ðeð0ÞÞ e2at þ e2at 8t P 0: 2ðb aÞ
8t P 0 ð16Þ
By (13) and (16), one has kmin ðP ÞkeðtÞk2 6 V ðeðtÞÞ 6 V ðeð0ÞÞ e2at þ
1 1 e2at 6 kmax ðP Þkeð0Þk2 e2at þ e2at 2ðb aÞ 2ðb aÞ
8t P 0
which implies keðtÞk 6
kmax ðP Þ 1 keð0Þk2 þ kmin ðP Þ 2kmin ðP Þ ðb aÞ
1=2
eat
8t P 0:
ð17Þ
This completes the first part of our proof. By (12), one clearly has h i ðP Þ 2 ln e1 ln kkmax ðP Þ min ts ðe1 ; e2 Þ P 2a and ln ts ðe1 ; e2 Þ P
h
1 2kmin ðP ÞðbaÞ
i
2 ln e2
2a
which imply that 1=2 kmax ðP Þ 6 e1 eat kmin ðP Þ
8t P ts ðe1 ; e2 Þ P 0
ð18Þ
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and
1 2kmin ðP Þ ðb aÞ
1=2
6 e2 eat
8t P ts ðe1 ; e2 Þ P 0:
ð19Þ
By (18) and (19), and using the inequality pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi a þ b 6 a þ b; a; b P 0 in (17), we have keðtÞk 6
kmax ðP Þkeð0Þk2 kmin ðP Þ
!1=2 at
e
1 þ 2kmin ðP Þ ðb aÞ
1=2
eat 6 e1 keð0Þk þ e2
8t P ts ðe1 ; e2 Þ P 0:
This completes the proof. h Remark 1. It is noted that ts(e1, e2), with e1 > 0 and e2 > 0, is a decreasing function of e1 and e2 in view of (12), which coincides with our intuition.
3. Illustrative example Consider the uncertain nonlinear control system as (1) with 2 3 2 3 2 3 0 3 1 0 5; 5; B ¼ 4 5; Df ¼ 4 A¼4 DaðtÞ x21 ðtÞ þ DbðtÞ x1 ðtÞx2 ðtÞ þ DcðtÞx22 ðtÞ 1 1 1 D/ðu; b; m1 Þ ¼ D/ðu; 1; 1Þ 2 Dðu; 1; 1Þ; xðtÞ :¼ ½ x1 ðtÞ x2 ðtÞ T 2 R2 ; u 2 R; 1 6 DaðtÞ 6 1;
1 6 DbðtÞ 6 1;
1 6 DcðtÞ 6 1:
Obviously, m ¼ m1 ¼ b ¼ 1. The desired tracking signal is given by xd ¼ ½ 1 3 T . Thus, by (2), one has eðtÞ :¼ ½ e1 ðtÞ e2 ðtÞ T ¼ ½ x1 ðtÞ 1 x2 ðtÞ 3 T : It can be verified from (A1) and (4) that gðt; eðtÞ þ xd Þ ¼ ðe1 ðtÞ þ 1Þ2 þ jðe1 ðtÞ þ 1Þðe2 ðtÞ þ 3Þj þ ðe2 ðtÞ þ 3Þ2 ; Kðt; eðtÞ þ xd Þ ¼ ðe1 ðtÞ þ 1Þ2 þ jðe1 ðtÞ þ 1Þðe2 ðtÞ þ 3Þj þ ðe2 ðtÞ þ 3Þ2 þ 5: By (7), we have 0:8667 P¼ 1:1894
1:1894 4:5453
> 0;
if we select a = 1, b = 1.05, and 0:1 0 Q¼ > 0: 0 0:1 It follows from (11) that k 1 ðt; eðtÞÞ ¼
2K2 ðt; eðtÞ þ xd Þ : 2Kðt; eðtÞ þ xd Þ j1:1894e1 ðtÞ þ 4:5453e2 ðtÞj þ e2:1t
Finally, owing to (8)–(11), we obtain the design controller uðtÞ ¼ ½0:5 k 1 ðt; eðtÞÞ ½1:1894e1 ðtÞ þ 4:5453e2 ðtÞ: Consequently, by Theorem 1, we conclude that the state of the feedback-controlled system exponentially tracks the desired constant signal xd with the guaranteed convergence rate a = 1 and the tracking time is estimated as ts(0.1, 0.05) = 4.4783.
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Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389
e1(t), e2(t)
2 1.5 1
e 1(t)
0.5 0 e 2(t)
-0.5 -1 -1.5 -2 0
0.5
1
1.5
2
2.5 t
3
3.5
4
4.
5
Fig. 1. e1(t) and e2(t) of the tracking error signal.
Selecting the parameters values Da(t) = Db(t) = Dc(t) = 1 and 8 > < u 0:5; D/ðu; 1; 1Þ :¼ 0; > : u þ 0:7;
if 0:5 6 u; if 0:7 6 u 6 0:5; if u 6 0:7;
the error signal is depicted in Fig. 1.
4. Conclusions The tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities has been investigated in this paper. A composite tracking control has been proposed such that the states of the feedbackcontrolled system track the desired constant signal with any pre-specified convergence rate. Meanwhile, an estimate of the tracking time has been derived for such systems. A numerical example is also been provided to illustrate the use of our main result. However, the tracking problem for more general nonlinear systems with relay with deadzone still remains unanswered. This constitutes an interesting future research problem.
Acknowledgements The author thanks the National Science Council of Republic of China for supporting this work under grant NSC94-2213-E-214-020. The author wishes also to register a note of the financial support from I-Shou University under grant ISU-93-01-01.
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