The Fourth International Conference on Control and Automation (ICCA’03), 10-12 June 2003, Montreal, Canada
ThA01-5
Adaptive Variable Structure Control Design Without a Priori Knowledge of Control Directions Jian-Xin Xu∗ , Ya-Jun Pan and Tong-Heng Lee Department of Electrical and Computer Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore 117576 ∗ E-mail:
[email protected] Abstract In this paper, a new adaptive variable structure control (VSC) scheme is proposed for nonlinear systems without a prior knowledge of control directions. By incorporating a Nussbaum-type function, the new adaptive VSC law can ensure the asymptotic convergence of the tracking error in the existence of non-parametric uncertainties. Simulation results are shown to validate the effectiveness of the proposed control scheme.
The paper is organized as follows. In Section 2, the problem formulation is presented. In Section 3, the new adaptive variable structure control scheme design and the convergence property are given. In Section 4, a numerical example is presented which demonstrates the effectiveness of the proposed scheme. Finally, Section 5 gives the conclusions. For simplicity, we focus on SISO nonlinear uncertain systems in this paper.
2 Problem Formulation Consider the following uncertain nonlinear system 1 Introduction x˙ = Variable Structure Control (VSC) is a well-known robust control design method which only requires the upper bounds of the parametric or non-parametric uncertainties. Owing to the superb robustness of VSC, over the past several decades, increasing attention has been drawn to VSC and hitherto, significant approaches have been developed [1]-[7]. Most of the existing VSC schemes are proposed to deal with the control task when the control directions are known a priori. In this paper, we will show the possibility of performing tracking control without prior knowledge of control directions, which is a challenging problem in general for any control methods. Up to now, there are mainly two approaches to address this problem. One approach is to incorporate the technique of Nussbaum-type gains into the control design. The first result was proposed by Nussbaum [8], and later extended to adaptive control systems [9], [10]. Note that those adaptive control schemes were developed to deal with parametric uncertainties. Another approach is to directly estimate unknown parameters involved in the control directions [11]-[14]. In this paper, the first approach is integrated with the typical variable structure control (VSC). By incorporating a Nussbaum-type function, the new adaptive VSC law can ensure the convergence of the tracking error in the existence of non-parametric uncertainties.
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883
η(x, t) + bu(x, t),
(1)
where x ∈ R is a physically measurable state, b = 0 is an unknown constant parameter and u ∈ R is the control input. η is a lumped disturbance. The sign of b, which determines the control direction, is assumed to be unknown. The system is required to track the trajectory xd ∈ C 1 [0, ∞). The switching surface is selected as σ = xd − x. Assumption: η(x, t) is norm bounded by a known non-negative bounding function, i.e., |η(x, t)| ≤ βη (x, t). ∀x in any closed set, ∃β, βη (x, t) < β, where β is a finite constant. When the parameter b is known, this tracking problem can be solved simply by the classical variable structure control. When b is unknown, we need to look for a new adaptive VSC approach. In this paper, the following Nussbaum-type function will be used in the controller design. Definition: [8] v( ) is an even smooth Nassbaum-type
function, if the function has the following properties 8 1 s lim sup v(k)dk = ∞, s→∞ s 8 1 s lim inf v(k)dk = −∞. s→∞ s
≤
V (0) +
=
V (0) +
8
t
˙ [bv(k) + 1] kdτ
0
8
k(t)
An example of such a continuous function is v(k) = k 2 cos(k). Regarding the Nussbaum-type function, the following property holds [10]. Property: Let V ( ) and k( ) be smooth functions defined on [0, T ) with V (t) ≥ 0, ∀t ∈ [0, T ), v( ) be an even smooth Nassbaum-type function, and b a nonzero constant. If the following inequality holds 8 k(t) V (t) ≤ [bv(ω) + 1] dω + c, ∀t ∈ [0, T ), where c is an arbitrary constant, then V (t), k(t) and 8 k(t) [bv(ω) + 1] dω must be bounded on [0, T ). 0
0
βk < ∞, βi < ∞, such that
V (t) ≤ βV , k(t) ≤ βk , 8 k(t) [bv(ω) + 1] dω ≤ βi . 0
Note that σ=2
0 0 V (t) ≤ 2 βV
implies σ ∈ L∞ , in the sequel x ∈ L∞ and βη ∈ L∞ . From (5), we have 8 t σ 2 dτ ≤
V (0) +
0
=
3 Adaptive Variable Structure Controller Design The adaptive variable structure controller is designed as (2)
z(x, t) = −σ − x˙ d − (βη + ε)sign(σ),
(3)
where v( ) is an even smooth Nussbaum-type function and ε is a positive constant. Theorem: For system (1) under the adaptive variable structure controller (2) and (3), σ → 0 as t → ∞. Proof: Selecting a positive definite scalar function 1 2 V = σ . Under the control law (2), the time deriva2 tive of V is V˙
=
σ σ˙ = σ e˙ = σ(x˙ d − η − bu)
= ≤
+σ [−σ − x˙ d − (βη + ε)sign(σ)] − ση −σ 2 + [bv(k) + 1] k˙ − σβη sign(σ) + βη |σ|
=
˙ −σ 2 + [bv(k) + 1] k.
k(t)
[bv(ω) + 1] dω
0
V (0) + βi ,
(7)
Furthermore, in the controller (2), the smooth function v(k) is bounded because k(t) is bounded. In (3), z( )∈L∞ because σ ∈ L∞ . Thus u( ) ∈ L∞ which means that σ˙ = x˙ d − η − bu is bounded. According to Barbalat’s lemma [15], σ ∈ L∞ ∩L2 and the boundedness of σ˙ warrants lim σ = 0. t→∞
Remark: The above result can be easily extended to higher order systems F i = 1, 2, · · · , n − 1, x˙ i (t) = xi+1 (t), x˙ n (t) = η(x, t) + bu(x, t), where x = [x1 , x2 , · · · , xn ]T . Denote xd T [x1d , x2d , · · · , xnd ] the target trajectory, and ei = e˙ i−1 ,
=
i = 2, 3, · · · , n
the tracking errors. Choose the switching surface σ = σ(e1 , e2 , · · · , en ),
(4)
Integrating both sides of the inequality (4) from 0 to t, we have 8 t 8 t 2 ˙ (5) σ dτ + [bv(k) + 1] kdτ V (t) ≤ V (0) − 0
8
1 where V (0) = σ(0)2 is a finite positive constant. 2 Hence, σ is square integrable which means σ ∈ L2 .
e1 = x1 − x1d ,
σ [x˙ d − η − bv(k)z] σ [x˙ d − η − bv(k)z − z + z] σ x˙ d + [bv(k) + 1] k˙
= =
(6)
Hence, according to P roperty, V (t), k(t) and 8 k(t) [bv(ω) + 1] dω are bounded , i.e., ∃βV < ∞,
0
u(x, t) = v(k(t))z(x, t) ˙ k(t) = −z(x, t)σ
[bv(ω) + 1] dω.
0
0
884
∂σ = 0. ∂en
Then the adaptive variable structure control law can be designed analogous to (2) and (3), as below, which achieves the asymptotic tracking convergence u(x, t) = ˙ k(t) = z(x, t) =
v(k(t))z(x, t) −αz(x, t)σ −σ − g(x, t) − x˙ nd − (βη + ε)sign(ασ),
where g=
w
∂σ ∂en
W−1 n−1 3 i=1
2.5
∂σ ei+1 ∂en
2 Adapting parameter k
∂σ and α = . ∂en
4 Illustrative Examples Consider the uncertain nonlinear system (1) with x(0) = 0.5, b = 1 and η(x, t) = 2x2 sin(πt). The control direction, the sign of b, is assumed unknown. The known bound of η is βη = 2x2 . The system is required to track the target trajectory xd (t) = 0.2 + 0.2sin2 (πt).
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5 Time(sec)
3
3.5
4
4.5
5
Figure 2: The evolution of the adapting parameter k(t).
The switching surface is chosen to be σ = xd − x. The simulation is carried out by applying the proposed adaptive variable structure control law (2) and (3) with ε = 0.5. It is clearly shown in F ig.1 that σ = 0 is achieved. The boundedness of the adapting parameter k(t) is shown in F ig.2. 0.2
−0.2
[4] K. D. Young and U. Ozguner, Lecture notes in control and information sciences 247, Variable structure systems, sliding mode and nonlinear control, vol. 51, Springer-Verlag, London, 1999.
−0.4 Switching surface σ
[2] A. S. I. Zinober, Lecture Notes in Control and Information sciences, variable structure and Lyapunov control, vol. 64, Springer-Verlag, London, 1994. [3] C. Edwards and S. K. Spurgeon, Sliding mode control: theory and applications, vol. 7, Taylor and Francis, London, 1998.
0
−0.6
−0.8
[5] X. H. Yu and J. X. Xu, Advances in Variable Structure Systems — Analysis, Integration and Applications, World Scientific, Singapore, 2000, ISBN 981— 02—4464—9, (Hardcover).
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−1.2
−1.4
−1.6
−1.8
References [1] V. I. Utkin, Sliding Modes in Control and Optimization, vol. 34, Springer-Verlag, Berlin, 1992.
0
0.5
1
1.5
2
2.5 Time(sec)
3
3.5
4
4.5
5
Figure 1: The evolution of the switching surface σ(x, t).
5 Conclusions In order to deal with the tracking problem without any prior knowledge with regards to the control directions, a Nussbaum function is incorporated into the adaptive variable structure controller. Based on the Lyapunov’s direct method, the system with zero tracking error is ensured under the proposed control scheme. The effectiveness of the adaptive VSC design is demonstrated through a numerical example.
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[6] X. H. Yu and J. X. Xu, Variable Structure Systems: Towards the 21st Century, vol. 274 of Lecture Notes in Control and Information Sciences, SpringerVerlag, 2002, ISBN 3—540—42965—4. [7] W. Perruquetti and J. P. Barbot, Sliding mode control in engineering, Mark Dekker Inc., New York, 2002. [8] R.D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Systems and Control Letters, vol. 3, pp. 243—246, 1983. [9] E. P. Ryan, “A universal adaptive stabilizer for a class of nonlinear systems,” Systems and Control Letters, vol. 16, pp. 209—218, 1991. [10] X. D. Ye and J.P. Jiang, “Adaptive nonlinear design without a priori knowledge of control directions,” IEEE Transaction on Automatic Control, vol. 43, no. 11, pp. 1617—1621, 1998.
[11] D. R. Mudgett and A. S. Morse, “Adaptive stabilizing of systems with unknown high frequence gain,” International Journal of Control, vol. 30, pp. 549—554, 1985. [12] B. Brogliato and R. Lozano, “Adaptive control of a simple nonlinear system without a priori information on the plant parameters,” IEEE Transactions on Automatic Control, vol. 37, pp. 30—37, 1992. [13] B. Brogliato and R. Lozano, “Adaptive control of first-order nonlinear systems with reduced knowledge of the plant parameters,” IEEE Transactions Automatic Control, vol. 39, pp. 1764—1768, 1994. [14] J. Kaloust and Z. Qu, “Continuous robust control design for nonlinear uncertain systems without a priori knowledge of control direction,” IEEE Transactions on Automatic Control, vol. 40, no. 2, pp. 276—282, 1995. [15] K. S. Narendra and A. M. Annaswamy, Stable adaptive systems, vol. 3, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.
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