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Simulation and experiment of event-triggered control for inverted pendulum system
YU Hao , OUYANG Li , HAO Fei
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-09-24; Accepted: 2015-12-18; Published online: 2016-01-20 16:54
Foundation item: National Natural Science Foundation of China (61573036, 61174057); Beijing Municipal Natural Science Foundation (4112034); the Fundamental Research Funds for the Central Universities
Corresponding author. Tel.:010-82314517, E-mail:fhao@buaa.edu.cn
Abstract: This paper investigates the event-triggered control for single linear inverted pendulum system from theoretical, simulation and experimental points of view to illustrate the feasibility and practicability of event-triggered control. Firstly, the parameter in event condition was designed based on linear matrix inequality, and the estimation of lower bound of inter-event times was provided. Secondly, the feasibility of theoretical results without quantization is illustrated by Simulink simulations. Finally, the influence of quantization was studied by experiments, which is quite negative on the system with model-based controller. According to the experiment results, the mixed event condition and zero-order-hold controller are more suitable for practical inverted pendulum system. They can guarantee the stability with efficient reduction of communication frequency, which reflects the potential practical value of event-triggered control.
Key words: inverted pendulum     state feedback     event-triggered control     model-based networked control     Simulink simulation

1 事件触发控制倒立摆系统

 图 1 一级直线倒立摆示意图 Fig. 1 Sketch map of single linear inverted pendulum

 (1)

 (2)

 (3)

 (4)
 (5)

 (6)

tsN≥0(N≥0表示全体非负整数)为事件触发时刻，由如下的事件条件得到。

 (7)

 (8)
2 理论结果

 (9)

 (10)

 (11)

 (12)

 (13)

 (14)

 (15)

n(ts)=0可得

 图 4 相对型事件条件及控制器(4)作用下的仿真结果 Fig. 4 Simulation results of system with relative event condition and controller (4)
 图 5 相对型事件条件及控制器(5)作用下的仿真结果 Fig. 5 Simulation results of system with relative event condition and controller (5)

 图 6 混合型事件条件及控制器(4)作用下的仿真结果 Fig. 6 Simulation results of system with mixed event condition and controller (4)
3.2 实验结果

 图 7 一级直线倒立摆实物设备 Fig. 7 Equipment of single linear inverted pendulum

 图 8 相对型事件条件及控制器(4)作用下的实验结果 Fig. 8 Experimental results of system with relative event condition and controller (4)
 图 9 混合型事件条件及控制器(4)作用下的实验结果 Fig. 9 Experimental results of system with mixed event condition and controller (4)

 图 10 相对型事件条件及控制器(5)作用下的实验结果 Fig. 10 Experimental results of system with relative event condition and controller (5)

 图 11 带有量化及控制器(4)作用下的仿真结果 Fig. 11 Simulation results of system with quantization and controller (4)
 图 12 带有量化及控制器(5)作用下的仿真结果 Fig. 12 Simulation results of system with quantization and controller (5)

 图 13 带有大初始状态及控制器(4)的系统的仿真结果 Fig. 13 Simulation results of system with large initial state and controller (4)
 图 14 带有大初始状态及控制器(5)的系统的仿真结果 Fig. 14 Simulation results of system with large initial state and controller (5)

4 结论

1) 本文研究了事件触发控制策略在倒立摆系统中的应用。考虑了两类事件触发控制器，即零阶保持控制器及基于模型控制器，在相对型和混合型事件条件作用下对于倒立摆系统稳定性及触发性能的影响。

2) 对于事件条件，只期望保证系统最终有界时，混合型事件条件可以获得更好的触发性能。对于控制器，当可以准确获得系统状态时，基于模型控制器可以得到更好的触发性能。

3) 当量化效应无法被忽略时，两者触发性能无明显差别，但考虑到基于模型控制需要消耗更多的计算资源，则零阶保持控制器更为优秀。因此，对于倒立摆系统，混合型事件条件及零阶保持控制器更为适用。其在保证闭环系统稳定性的前提下可以有效降低控制律的更新频率。

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#### 文章信息

YU Hao, OUYANG Li, HAO Fei

Simulation and experiment of event-triggered control for inverted pendulum system

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(10): 2107-2117
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0631