SongDSC - INT J CONTROL 2002 VOL 75 NO 12 870 881 Robust...

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Robust stabilization and ultimate boundedness of dynamic surface control systems via convex optimization BONGSOB SONG { *, J. KARL HEDRICK { and ADAM HOWELL { In this paper, a new method of analysing the controller gains and ®lter time constants for dynamic surface control (DSC) is presented. First, since DSC provides linear error dynamics with perturbation terms for a class of non-linear systems, the design method can be used to assign the system matrix eigenvalues of the closed loop error dynamics. Then a procedure for testing the stability and performance of the ®xed controller in the face of uncertainties is presented. Finally, an ellipsoidal approximation of the tracking error bounds for a tracking problem is obtained via convex optimization. 1. Introduction Lyapunov-based control design techniques, such as integrator backstepping and sliding mode control, have had signi®cant success in the area of robust non-linear control (Slotine and Li 1991, Kristic et al. 1995, Khalil 1996). However, the integrator backstepping control is problematic because of the explosion of terms in the control law, while sliding mode control cannot generally be used for a system with mismatched uncertainties (Won and Hedrick 1996). An alternative control design method called multiple sliding surface control (MSS) was proposed to achieve robustness without this large number of terms, and has been successfully used for real-time engine control (Green and Hedrick 1990, Won and Hedrick 1996). Despite this improvement, MSS has its own limitations because the control law requires ®nding numerical derivatives of the reference trajectories. Therefore, dynamic surface control (DSC) was developed to overcome this problem through the use of dynamic ®lters (Gerdes 1996). A more detailed description of the design methodology and stability ana- lysis for DSC as applied to both Lipschitz and non- Lipschitz non-linear systems can be found in Swaroop et al. (2000). In order to illustrate the DSC design procedure, as well as a choice of surface gains and ®lter time con- stants, consider the following example introduced in Swaroop et al. (2000) _ x 1 ˆ x 2 ¢ f 1 x 1 _ x 2 ˆ x 3 _ x 3 ˆ u 9 > > = > > ; 1 where ¢ f 1 x 1 is the non-Lipschitz uncertainty bounded by a known C 1 function such that j ¢ f 1 x 1 †j μ Ƈ 1 x 1 . The goal is to make x 1 to track the desired value, x 1 d t † ˆ sin t , in presence of the uncertainty. Let S 1 : ˆ x 1 ¡ x 1 d t † ) _ S 1 ˆ x 2 ¢ f 1 ¡ _ x 1 d t Using the idea of non-linear damping and a low-pass ®lter as in the DSC design procedure (Swaroop et al. 2000), set · x 2 ˆ _ x 1 d t † ¡ L 1 S 1 ¡ S 1 Ƈ 2 1 2 ° ) ½ 2 _ x 2 d x 2 d ˆ · x 2 ; x 2 d 0 † ˆ · x 2 0 Similarly, de®ne the second sliding surface as S 2 : ˆ x 2 ¡ x 2 d ) _ S 2 ˆ x 3 ¡ _ x 2 d · x 3 ˆ _ x 2 d ¡ L 2 S 2 ) ½ 3 _ x 3 d x 3 d ˆ · x 3 ; x 3 d 0 † ˆ · x 3 0 S 3 : ˆ x 3 ¡ x 3 d ) _ S 3 ˆ u ¡ _ x 3 d u ˆ _ x 3 d ¡ L 3 S 3 ˆ · x 3 ¡ x 3 d ½ 3 ¡ L 3 S 3 Generally, the standard DSC design procedure
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