Robust Adaptive Control of Nonlinear Systems with Bounded Disturbances

3b-05 3

Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA

ROBUST ADAPTIVE CONTROL OF NONLINEAR SYSTEMS WITH BOUNDED DISTURBANCES

Zhengtao Ding

Department of Mechanical Engineering, Ngee Ann Polytechnic 535 Clementi Road, Singapore 599489, Repuhlic of Singapore [email protected]

Abstract: This paper presents a dead-zone modification to the algorithm presented in (Marino and Tomei, 1993) for adaptive output feedback control of nonlinear singleinput, single-output, minimum phase systems of relative degree one which are under bounded disturbanceB. Keywords: Adaptive Control, Bounded Disturbances, Robustness, Linearizable Systems, Nonlinear Control Systems, Observers.

1. INTRODUCTION

Recently, some progresses in adaptive control of nonlinear systems have been achieved using transformations based on differential geometry (see Kanellakopoulos et al., 1991; Marino and Tomei, 1992; Marino and Tomei, 1993; Sastry and Isidori, 1989). It is shown by Marino and Tomei (1993) that under certain geometric conditions, a nonlinear system can be transformed into a special observer form. Furthermore, output feedback adaptive control, with the knowledge of the sign of the high frequency gain, can be achieved using the filtered transformation technique which removes the requirement of matching conditions. An algorithm's robustness to disturbances is an important factor to consider before applying it to industrial processes where disturbances are inevitable. A number of modifications have been developed for adapti ve control algorithms of linear systems (see Kreisselmeier and Narendra, 1982;

Narendra and Annaswamy, 1987; Narendra and Annaswamy, 1989; Ortega and Lozano-leal, 1987; Peterson and Nerendra, 1982; Samson. 1983) to tackle bounded disturbances. Due to the difference in the control structure used in the algorithm proposed by Marino and Tomei (1993), existing modifications cannot be directly applied. In this paper, a modification 1,0 the algorithm presented in (Marino and Tomei, 199:i) is proposed to accommodate bounded disturbances. Inter-relations are explored between the outputs and the state variables of the systems with the special formats due to the reference model and the observer model introduced by Marino and Tomei (1993). This leads to simplification of the dynamics of the output errors between the plant and the reference model, and between the plant and the observer. First an extra term driven by the control output error is introduced in the observer based on the simpli-

5096

5097

5098

5099

5100

According to the adaptive laws (60), if er + 1]Zr ::; ~I, the adaptation stops, otherwise using (36), (48) and (60), it is obtained that

2( -Acei -1]AoZ; + elDe + 1]ZlDz) 62 62 -A e 2 - 1]A £2 + ....£ + 1]2... c 1 0 I Ac Ao

\'

< :;

<

(e1- )2

6z Ao -Acer - 1]AoZ; + min(Ac, Ao)~l - min(Ac, /\0)[ + 1]Z; - ~11 O. -A c

6e

Ac

-1] Ao (Zl-

Marino, R. and P. Tomei (1992). Global adaptive observers for nonlinear systems via filtered transformations. IEEE trons. Automat. Contr., AC-37, 1239-1245. Marino, R. and P. Tomei (1993). Global adaptive output feedback control of nonlinear systems, Part I: Linear parameterization. IEEE trons. Automat. Contr., AC-38, 17-32.

)2

er

(62)

Narendra, K. S. and A. M. Annaswamy (1987). A new adaptive law for robust adaptive control without persistent excitation. IEEE trons. Automat. Contr., AC-32, 134-14[,.

Therefore V is bounded, which implies that el, Zl, ih, and k are bounded. The boundedness of J.l1 can be shown using the minimum phase property of the system and the boundedness of el, in the same way as shown in (Marino and Tomei, 1993). I; is also bounded. Hence, all the variables are bounded.

Ortega, R. and R. Lozano-Ieal (1987). A note on direct adaptive control of systems with bounded disturbanc-es. Automatica, 23, 253-254.

4. CONCLUSION

Peters on , B. B. and K. S. Narendra (1982). Bounded error adaptive control. IEEE trons. Automat. Contr., AC-27, 1161-1168.

B1 , &1

Due to the special formats of the reference model and the observer model, the relations between the state variables and the outputs have been established, which enable us to determine up-bounds of the disturbances to the derivatives of the measurable control output error and the observer error. A term related to the output error has been introduced in the observer structure to eliminate an interactive term between the output error and the observer error which could appear in the stability analysis. Based on the above, an algorithm has been proposed for adaptive control of a class of nonlinear systems under bounded disturbances. The algorithm ensures that all the variables remain bounded.

Narendra, K. S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, New Jersey.

Samson, C. (1983). Stability analysis of adaptive controlled systems subject to bounded disturbances. Automatica, 19, 81-86. Sastry, S. S. and A. Isidori (1989). Adaptive control of linearizable systems. IEEE trons. Automat. Contr., AC-34, 1123-1l31. Slotine, J. E. and W. Li (I991). Applied Nonlinear Control. Prentice-Hall, ~ew Jersey.

REFERENCES

Isidori, A. (1989). Nonlinear Control Systems. SpringerVeriag, Berlin. Kanellakopoulos, I., P. V. Kokotovic and A. S. Morse (1991). Systematic design of adaptive controllers for feedback linearizable systems. IEEE trons. Automat. Contr., AC-36, 1241-1253. Kreisselmeier, G. and K. S. Narendra (1982). Stable model reference control in the presence of bounded disturbances. IEEE trons. Automat. Contr., AC27,1169-1175

5101