New design methodology for adaptive switching gain based discrete-time sliding mode control

Figures & data

Figure 1. Possible conditions of adaptive sliding mode. (a) Case-1. (b) Case-2. (c) Case-3. (d) Case-4.

Figure 2. Free body diagram of mass–spring system.

Figure 3. Simulation results of rectilinear plant. (a) Switching gain. (b) Control input. (c) Sliding function. (d) State $x_1$.

Table 1. Parameter values in simulation 1.

Sl.no	Parameter	Value
1	φ	0.5
2	β	1
3	ξ	0.1
4	γ	0.2

Table 2. Comparison of performance in simulation 1.

Sl.no	Control algorithm	$\sum _k\left|ϵ\right(k\left)\right|$	$\sum _k\left|u\right(k\left)\right|$
1	Existing adaptive law (Monsees & Scherpen, 2002)	30.4	45.5
2	Existing adaptive law (Salhi et al., 2016)	19.2	38.7
3	Proposed adaptive law	4.3	33.6

Table 3. Parameter values in simulation 2.

Sl.no	Parameter	Value
1	φ	0.5
2	β	1
3	ξ	0.1
4	γ	0.2
5	${\gamma }_0$	0.8
6	${\gamma }^0$	0.6

Figure 4. Simulation results of CNC milling machine. (a) Switching gain. (b) Control input. (c) Sliding function. (d) State $x_1$.

Table 4. Comparison of performance in simulation 2.

Sl.no	Control algorithm	$\sum _k\left|ϵ\right(k\left)\right|$	$\sum _k\left|u\right(k\left)\right|$
1	Existing adaptive law (Monsees & Scherpen, 2002)	52.9	1032.7
2	Existing adaptive law (Salhi et al., 2016)	61.0	1031.8
3	Proposed adaptive law	21.5	1009.8

Monsees, G., & Scherpen, J. M. A. (2002). Adaptive switching gain for a discrete-time sliding mode controller. International Journal of Control, 75(4), 242–251. doi: 10.1080/00207170110101766

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Salhi, H., Kamoun, S., Essounbouli, N., & Hamzaoui, A. (2016). Adaptive discrete-time sliding-mode control of nonlinear systems described by Wiener models. International Journal of Control, 89(3), 611–622. doi: 10.1080/00207179.2015.1088964

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