Composite tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities

Composite tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities

Chaos, Solitons and Fractals 35 (2008) 383–389 www.elsevier.com/locate/chaos Composite tracking control for a class of uncertain nonlinear control sy...

153KB Sizes 2 Downloads 218 Views

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

*

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: [email protected]

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.034

384

Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389

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 :

Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389

385

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

386

Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389

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

Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389

387

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.

388

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.

References [1] Xu JX, Lee TH, Jia QW. Design and analysis of a new adaptive robust control scheme for a class of nonlinear uncertain systems. Int J Syst Sci 1999;30:239–45. [2] Park KB, Tsuji T. Sliding mode control of second-order nonlinear uncertain systems. Int J Robust Nonlinear Control 1999;9:769–80.

Y.-J. Sun / Chaos, Solitons and Fractals 35 (2008) 383–389

389

[3] Shyu KK, Yan JJ. Variable-structure model following adaptive control for systems with time-varying delay. Control-Theor Adv Technol 1994;10:513–21. [4] Yan JJ, Shyu KK, Lin JS. Adaptive variable structure control for uncertain chaotic systems containing dead-zone nonlinearity. Chaos, Solitons & Fractals 2005;25:347–55. [5] Sun YJ, Yu GJ, Chao YH, Hsieh JG. Exponential stability and guaranteed tracking time for a class of model reference control systems via composite feedback control. Int J Adapt Control Signal Process 1997;11:155–65. [6] Toivonen HT, Pensar J. A worst-case approach to optimal tracking control with robust performance. Int J Control 1996;65:17–32. [7] Sun YJ, Hsieh JG. Exponential tracking control for a class of uncertain systems with time-varying arguments and deadzone nonlinearities. ASME J Dyn Syst, Meas Control 1997;119:825–30. [8] Biswas A. Perturbation of solitons with non-Kerr law nonlinearity. Chaos, Solitons & Fractals 2002;13:815–23. [9] Brandon JA, Benoit E, Jezequel L. Subtle bifurcation in the vibration of a system with an interface nonlinearity. Chaos, Solitons & Fractals 1998;9:393–400. [10] Das CK, Roy CA. On a model for optical pulse propagation with saturation nonlinearity. Chaos, Solitons & Fractals 1999;11:947–52. [11] Franklin GF, Powell JD, Emani-Naeini A. Feedback control of dynamic system. Massachusetts: Addison-Wesley; 1994. [12] Konar S, Mishra M, Jana S. The effect of quintic nonlinearity on the propagation characteristics of dispersion managed optical solitons. Chaos, Solitons & Fractals 2006;29:823–8. [13] Ng L, Rand R. Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos, Solitons & Fractals 2002;14:173–81. [14] Puu T, Sushko I. A business cycle model with cubic nonlinearity. Chaos, Solitons & Fractals 2004;19:597–612. [15] Recker DA, Kokotovic PV, Rhode D, Winkelman J. Adaptive nonlinear control of systems containing a deadzone. In: Proceedings of the 30th conference on decision and control, 1991. p. 2110–5. [16] Remeo F, Rega G. Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach. Chaos, Solitons & Fractals 2006;27:606–17. [17] Thaler GJ, Pastel MP. Analysis and design of nonlinear feedback control systems. New Jersey: McGraw-Hill; 1962. [18] Park SK, Ahn HK. Robust controller design with novel sliding surface. IEE-Proc Control Theory Appl 1999;146:242–6. [19] Anderson BDO, Moore JB. Optimal control-linear quadratic methods. New Jersey: Prentice-Hall; 1990.