## Reduced-order design of high-order sliding mode control system.(English)Zbl 1237.93037

Summary: To design an $$r$$-th $$(r>2)$$ order sliding mode control system, a sliding variable and its derivatives of up to ($$r - 1$$) are in general required for the control implementation. This paper proposes a reduced-order design algorithm using only the sliding variable and its derivatives of up to ($$r - 2$$) as the extension of the second-order asymptotic sliding mode control. For a linear time-invariant continuous-time system with disturbances, it is found that a high-order sliding mode can be reached locally and asymptotically by a reduced-order sliding mode control law if the sum of the system poles is less than the sum of the system zeros. The robust stability of the reduced-order high-order sliding mode control system, including the convergence to the high-order sliding mode and the convergence to the origin is proved by two Lyapunov functions. Simulation results show the effectiveness of the proposed control algorithm.

### MSC:

 93B12 Variable structure systems 93B11 System structure simplification 93C15 Control/observation systems governed by ordinary differential equations
Full Text:

### References:

 [1] Utkin, Sliding Modes in Control and Optimization (1992) [2] Edwards, Sliding Mode Control: Theory and Applications (1998) [3] Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control 76 pp 924– (2003) · Zbl 1049.93014 [4] Bartolini, A survey of applications of second-order sliding mode control to mechanical systems, International Journal of Control 76 (9-10) pp 875– (2003) · Zbl 1070.93011 [5] Levant, Principles of 2-sliding mode design, Automatica 43 (4) pp 576– (2007) · Zbl 1261.93027 [6] Levant, Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (6) pp 1247– (1993) · Zbl 0789.93063 [7] Sira-Ramirez, On the dynamical sliding mode control of nonlinear systems, International Journal of Control 57 (5) pp 1039– (1993) · Zbl 0772.93040 [8] Fridman, Higher order sliding modes as a natural phenomenon in control theory, Lecture Notes in Control and Information Sciences: Robust Control Via Variable Structure and Lyapunov Techniques 217 pp 107– (1996) · Zbl 0854.93021 [9] Lu, Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes, International Journal of Control 70 (5) pp 735– (1998) · Zbl 0931.93058 [10] Bartolini, On the convergence properties of a 2-sliding control algorithm for nonlinear uncertain systems, International Journal of Control 74 (7) pp 718– (2001) · Zbl 1010.93021 [11] Boiko, Analysis of second order sliding mode algorithms in the frequency domain, IEEE Transactions on Automatic Control 49 (6) pp 946– (2004) · Zbl 1365.93067 [12] Damiano, Second-order sliding-mode control of DC drives, IEEE Transactions on Industrial Electronics 51 (2) pp 364– (2004) [13] Levant, Homogeneous quasi-continuous sliding mode control, Lecture Notes in Control and Information Sciences: Advances in Variable Structure and Sliding Mode Control 334 pp 143– (2006) · Zbl 1140.93341 [14] Anosov, On stability of equilibrium points of relay systems, Automation and Remote Control 2 (1) pp 135– (1959) · Zbl 0093.09204 [15] Shtessel YB Krupp DR Shkolnikov IA 2-sliding-mode control for nonlinear plants with parametric andd dynamic uncertainties 1 9 [16] Chung, A general class of sliding surface for sliding mode control, IEEE Transactions on Automatic Control 43 (1) pp 115– (1998) · Zbl 0907.93014 [17] Johansson, Limit cycles with chattering in relay feedback systems, IEEE Transactions on Automatic Control 47 (9) pp 1414– (2002) · Zbl 1364.93341 [18] Shustin, Robust semiglobal stabilization of the second order system by relay feedback with an uncertain variable time delay, SIAM Journal on Control and Optimization 47 (1) pp 196– (2008) · Zbl 1172.34053 [19] Boiko, Oscillations and transfer properties of relay servo systems with integrating plants, IEEE Transactions on Automatic Control 53 (11) pp 2686– (2008) · Zbl 1367.93557 [20] Aguilar, Generating self-excited oscillations via two-relay controller, IEEE Transactions on Automatic Control 54 (2) pp 416– (2009) · Zbl 1367.93384 [21] Boiko I On convergence rate of second-order sliding mode control algorithms 491 497 [22] Boiko, Discontinuous Control Systems: Frequency-Domain Analysis and Design (2009) · Zbl 1165.93002 [23] Pan Y Liu G Kumar KD Robust stability analysis of asymptotic second-order sliding mode control system using lyapunov function [24] Fridman LM Stability of motions in singularly perturbed discontinuous control systems 367 370 [25] Fridman, Chattering analysis in sliding mode systems with inertial sensors, International Journal of Control 76 (9-10) pp 906– (2003) · Zbl 1062.93011 [26] Isidori, Nonlinear Control Systems (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.