Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems
Abstract
:1. Introduction
- Different from the combined SMC and ADP frameworks of [39,40], this paper proposes a new dynamic event-triggered (DET) mechanism. By introducing an auxiliary variable, which is non-negative. This can increase the length of the time intervals between triggering events, further reducing the communication burden compared with [39,40].
- A novel Integral Sliding Mode Control (ISMC) scheme based on DET for uncertain nonlinear systems is proposed, consisting of two control laws. A first event-triggered controller is designed to tackle the matched uncertainties and force the trajectory of the system on the sliding mode surface. A second event-triggered controller is designed to tackle the unmatched uncertainties and guarantee optimal performance.
- To solve the resulting optimal control problem, a critic-only neural network (NN) based on ADP is proposed via the experience replay technique, which helps relaxing the excitation condition typically required for ADP methods to work. Stability of the closed-loop system is proven in the sense of uniformly ultimately boundedness, while guaranteeing Zeno-free behavior of the triggering mechanism.
2. System Formulation and Preliminaries
3. DET-Based ISMC Design
- (1)
- (2)
- The modulation gain associated with is minimized, which means that the amplitude of chattering can be reduced;
- (3)
- avoids amplifying the effect of the unmatched disturbance.
4. DET-Based Optimal Controller Design
4.1. Dynamically Triggering Rule for Optimal Input
4.2. Dynamically Triggered ADP with Single Critic NN
4.3. Stability Analysis
- (1)
- Event is not triggered:
- (2)
- On the triggering instant:
4.4. Algorithm Design of the Event-Triggered ISM Optimal Tracking Control
Algorithm 1 Event-Triggered ISM Optimal Tracking Control. |
Input: initial states of the sliding mode dynamics (11) |
1: Select an initial admissible policy and a proper small scalar |
2: To tackle the uncertainty affecting the nonlinear system (5), an integral type sliding surface is designed as
|
3: To extend the continuous-time control in the event-triggered paradigm, we define a virtual control input satisfying , where
|
4: Hence, the event-triggered ISMC becomes
|
5. Simulation
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Strategies | Samples | Average Interval | Minimal Interval |
---|---|---|---|
2000 | 0.05 | 0.05 | |
632 | 0.1582 | 0.1 | |
476 | 0.2101 | 0.1 | |
577 | 0.1733 | 0.1 | |
418 | 0.2392 | 0.15 |
Triggering Rate (%) | 0.01 | 0.1 | 0.5 | 1 | ||
---|---|---|---|---|---|---|
1 | 22.35 | 23.75 | 27.45 | 29.00 | ||
3 | 17.95 | 22.4 | 28.55 | 30.15 | ||
5 | 15.55 | 22.25 | 29.20 | 30.65 | ||
10 | 12.55 | 22.85 | 30.05 | 31.10 |
Triggering Rate (%) | 0.01 | 0.1 | 0.5 | 1 | ||
---|---|---|---|---|---|---|
0.1 | 24.90 | 25.10 | 25.45 | 25.65 | ||
0.5 | 22.10 | 22.90 | 25 | 25.80 | ||
1 | 17.10 | 20.80 | 25.25 | 26.60 | ||
3 | 1.75 | 3.6 | 26.30 | 27.50 |
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Tan, W.; Yu, W.; Wang, H. Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems. Symmetry 2022, 14, 1264. https://doi.org/10.3390/sym14061264
Tan W, Yu W, Wang H. Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems. Symmetry. 2022; 14(6):1264. https://doi.org/10.3390/sym14061264
Chicago/Turabian StyleTan, Wei, Wenwu Yu, and He Wang. 2022. "Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems" Symmetry 14, no. 6: 1264. https://doi.org/10.3390/sym14061264
APA StyleTan, W., Yu, W., & Wang, H. (2022). Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems. Symmetry, 14(6), 1264. https://doi.org/10.3390/sym14061264