一类欠驱动系统输出跟踪控制问题的理论和方法研究
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摘要
本文综合运用非线性控制的基本方法,讨论了一类欠驱动系统的输出跟踪控制问题。
在反步法框架下,选择联系函数及其导数为对应直接驱动自由度的待镇定系统动态,研究了一类欠驱动系统的非线性输出跟踪控制问题。采用过参数化的递归设计,讨论了一类具有结构化不确定性的欠驱动系统非线性自适应输出跟踪控制问题。再针对可能存在的非结构化不确定性,研究了一类欠驱动系统的非线性鲁棒输出跟踪控制问题。
建立了板球系统各个环节的模型,分析了其运动行为和结构特性。引入了非线性系统“主要动态”的概念。确定了板球系统中的“主要动态”。
研究了板球系统的镇定控制问题。构造了非线性摩擦力观测器,结合精确反馈线性化和最优控制理论提出了考虑摩擦补偿的镇定控制方法。又在系统初始值大范围变化及控制量受饱和约束的条件下,提出了自调整待设计参数的反步镇定控制方法。实验结果表明两种镇定控制方法都提高了板球系统的控制精度。
结合板球系统实验平台BPVS-JLU-Ⅱ,研究了板球系统输出跟踪控制问题。在球参数已知的条件下,以滑模变结构控制、反馈线性化结合极点配置两种方法分别构造了位置控制器。当球半径、质量和转动惯量未知时,采用Lyapunov直接方法与自适应反步法分别构造了非线性自适应位置控制器。在跟踪速度为6mm/s、18 mm/s、30 mm/s三种条件下,采用反步法、自适应反步法、滑模变结构、反馈线性化、模糊控制完成了轨迹跟踪控制实验。实验结果表明,与其它的一些控制方法相比,应用本文提出的非线性控制或者非线性自适应控制方法,较好地完成了跟踪控制任务,提高了板球系统的控制精度。
研究得到教育部高等学校博士学科点专项科研基金资助(项目名称:高速运动条件下板球系统镇定与高精度轨迹控制研究,项目编号:20060183006),并受到吉林大学“985工程”研究生创新基金的资助(项目名称:复杂环境下欠驱动机电系统的自主运动规划与控制,项目编号20080212)。
Underactuated systems have several irreplaceable advantages in saving energy, reducing costs, reducing cost and improving system reliability compared with fully actuated systems. Underactuated systems are used in fields of aviation, aerospace, marine exploration and etc. As underactuated systems have fewer actuators than degrees of freedoms, control problem of underactuated systems is challenging. Control problem of underactuated systems has been studied intensively by many researchers.
Output tracking control problem of a class of underactuated systems is investigated in this paper with nonlinear control methods. The plant for control is ball and plate system. This research is supported by College Doctor Special Scientific Research Fund of China under Grant 20060183006. Name of the project is 'stabilization and trajectory tracking control of the ball and plate system under high speed'. Main topics of this paper are summarized as below.
1. Nonlinear control problem is studied for a class of underactuated systems. The underactuated system has one underactuated degree and one fully actuated degree. The underactuated system can not be written into the upper triangular system. If system dynamics to be stabilized is chosen to be state variables as the common backstepping design, it is very difficulty to obtain a stable closed loop system. Concept of contacting function is introduced to solve the problem. Contacting function and its first derivative are selected as the dynamics to be stabilized in the backstepping design. Control Lyapunov functions are constructed step by step, and nonlinear output tracking controller is built recursively.
2. As some parameters of the underactuated system may be unknown, uncertain system parameters are added into mathematical model of the underactuated systems above. Nonlinear adaptive control problem of a class of underactuated systems is discussed. And backstepping design procedures are used to stabilize the dynamics of underactuated degree of freedom and fully actuated degree of freedom. Adaptive update laws for the uncertain parameters are also constructed. Uncertain parameters are estimated by overparametrization method.
3. As unknown disturbances are inevitable in engineering systems, unstructured uncertainty of the underactuated system is considered. And nonlinear robust output tracking control problem for a class of underactuated systems is investigated. A Priori bounded functions are introduced to reduce the influence of unknown disturbances. Nonlinear robust output tracking controller is constructed recursively for the underactuated system.
4. Motion and structure characteristics of the ball and plate system are investigated. Model of the ball and plate system are built including dynamics of ball moving on the plate, stepper motor, transmission sets and friction between the ball and plate. Controllability and observability of the ball and plate system is investigated with geometric theory of nonlinear system. Dynamics discussed in the analysis of controllability and observability is the dynamics of ball moving on the plate.
5. Contacts between dynamics of the underactuated systems and motions of the underactuated systems are discussed. Concept of main dynamics is introduced. Relationships between main dynamics and motions of the underactuated systems are studied. And system main dynamics of the ball and plate system is illustrated.
6. After mathematical model of the ball and plate system is established, stabilization control problem of the ball and plate system is discussed. The stabilization control problem is investigated on the experimental platform BPVS-JLU-Ⅱ. Frictions between the ball and the plate are estimated by nonlinear state observer in augmented state space. And then stabilization controller with friction compensation is built with theory of feedback linearization and optimal control. Further, when initial conditions of the ball and plate system vary in a wide range and control inputs are limited by saturation, backstepping design with automatic tuning parameters is proposed for the stabilization control problem. Fuzzy logic is constructed to tune the design parameters automatically in the backstepping control. Parameters of the fuzzy logic are optimized by genetic algorithm under restricted conditions introduced by control saturations.
7. Nonlinear output tracking control problem is studied on the experimental platform BPVS-JLU-Ⅱ. When ball parameters are known, position controller is constructed with variable structure control method first. And then feedback linearization method is used to avoid chattering phenomenon. Adaptive control problem is further investigated for the ball and plate system with structured uncertainty. When radius, mass and inertia of the ball is not specific, nonlinear adaptive controller based on Lyapunov direct method is built first. As the transmission sets of ball and plate system is influenced by backlash nonlinearities, adaptive backstepping design is employed to avoid chattering phenomenon of the nonlinear adaptive controller using Lyapunov direct method. Circle tracking experiments were established on the experimental platform BPVS-JLU-Ⅱwith expected tracking velocity 6 mm/s, 18 mm/s, 30 mm/s respectively. Five control schemes are used to construct the position controller including backstepping, adaptive backstepping, variable structure control, feedback linearization and fuzzy control. Relationship between expected tracking velocity and control precision of the ball and plate system was discussed for different control schemes. Some experimental phenomenon of the ball and the plate in the tracking control was analyzed. Trajectory tracking experiments established on the ball and plate system experimental platform BPVS-JLU-Ⅱhave indicated that when nonlinear model of the ball and plate system is considered and position controller is constructed with backstepping or adaptive backstepping design, trajectory tracking control could be done with higher control precision compared with other control schemes.
在反步法框架下,选择联系函数及其导数为对应直接驱动自由度的待镇定系统动态,研究了一类欠驱动系统的非线性输出跟踪控制问题。采用过参数化的递归设计,讨论了一类具有结构化不确定性的欠驱动系统非线性自适应输出跟踪控制问题。再针对可能存在的非结构化不确定性,研究了一类欠驱动系统的非线性鲁棒输出跟踪控制问题。
建立了板球系统各个环节的模型,分析了其运动行为和结构特性。引入了非线性系统“主要动态”的概念。确定了板球系统中的“主要动态”。
研究了板球系统的镇定控制问题。构造了非线性摩擦力观测器,结合精确反馈线性化和最优控制理论提出了考虑摩擦补偿的镇定控制方法。又在系统初始值大范围变化及控制量受饱和约束的条件下,提出了自调整待设计参数的反步镇定控制方法。实验结果表明两种镇定控制方法都提高了板球系统的控制精度。
结合板球系统实验平台BPVS-JLU-Ⅱ,研究了板球系统输出跟踪控制问题。在球参数已知的条件下,以滑模变结构控制、反馈线性化结合极点配置两种方法分别构造了位置控制器。当球半径、质量和转动惯量未知时,采用Lyapunov直接方法与自适应反步法分别构造了非线性自适应位置控制器。在跟踪速度为6mm/s、18 mm/s、30 mm/s三种条件下,采用反步法、自适应反步法、滑模变结构、反馈线性化、模糊控制完成了轨迹跟踪控制实验。实验结果表明,与其它的一些控制方法相比,应用本文提出的非线性控制或者非线性自适应控制方法,较好地完成了跟踪控制任务,提高了板球系统的控制精度。
研究得到教育部高等学校博士学科点专项科研基金资助(项目名称:高速运动条件下板球系统镇定与高精度轨迹控制研究,项目编号:20060183006),并受到吉林大学“985工程”研究生创新基金的资助(项目名称:复杂环境下欠驱动机电系统的自主运动规划与控制,项目编号20080212)。
Underactuated systems have several irreplaceable advantages in saving energy, reducing costs, reducing cost and improving system reliability compared with fully actuated systems. Underactuated systems are used in fields of aviation, aerospace, marine exploration and etc. As underactuated systems have fewer actuators than degrees of freedoms, control problem of underactuated systems is challenging. Control problem of underactuated systems has been studied intensively by many researchers.
Output tracking control problem of a class of underactuated systems is investigated in this paper with nonlinear control methods. The plant for control is ball and plate system. This research is supported by College Doctor Special Scientific Research Fund of China under Grant 20060183006. Name of the project is 'stabilization and trajectory tracking control of the ball and plate system under high speed'. Main topics of this paper are summarized as below.
1. Nonlinear control problem is studied for a class of underactuated systems. The underactuated system has one underactuated degree and one fully actuated degree. The underactuated system can not be written into the upper triangular system. If system dynamics to be stabilized is chosen to be state variables as the common backstepping design, it is very difficulty to obtain a stable closed loop system. Concept of contacting function is introduced to solve the problem. Contacting function and its first derivative are selected as the dynamics to be stabilized in the backstepping design. Control Lyapunov functions are constructed step by step, and nonlinear output tracking controller is built recursively.
2. As some parameters of the underactuated system may be unknown, uncertain system parameters are added into mathematical model of the underactuated systems above. Nonlinear adaptive control problem of a class of underactuated systems is discussed. And backstepping design procedures are used to stabilize the dynamics of underactuated degree of freedom and fully actuated degree of freedom. Adaptive update laws for the uncertain parameters are also constructed. Uncertain parameters are estimated by overparametrization method.
3. As unknown disturbances are inevitable in engineering systems, unstructured uncertainty of the underactuated system is considered. And nonlinear robust output tracking control problem for a class of underactuated systems is investigated. A Priori bounded functions are introduced to reduce the influence of unknown disturbances. Nonlinear robust output tracking controller is constructed recursively for the underactuated system.
4. Motion and structure characteristics of the ball and plate system are investigated. Model of the ball and plate system are built including dynamics of ball moving on the plate, stepper motor, transmission sets and friction between the ball and plate. Controllability and observability of the ball and plate system is investigated with geometric theory of nonlinear system. Dynamics discussed in the analysis of controllability and observability is the dynamics of ball moving on the plate.
5. Contacts between dynamics of the underactuated systems and motions of the underactuated systems are discussed. Concept of main dynamics is introduced. Relationships between main dynamics and motions of the underactuated systems are studied. And system main dynamics of the ball and plate system is illustrated.
6. After mathematical model of the ball and plate system is established, stabilization control problem of the ball and plate system is discussed. The stabilization control problem is investigated on the experimental platform BPVS-JLU-Ⅱ. Frictions between the ball and the plate are estimated by nonlinear state observer in augmented state space. And then stabilization controller with friction compensation is built with theory of feedback linearization and optimal control. Further, when initial conditions of the ball and plate system vary in a wide range and control inputs are limited by saturation, backstepping design with automatic tuning parameters is proposed for the stabilization control problem. Fuzzy logic is constructed to tune the design parameters automatically in the backstepping control. Parameters of the fuzzy logic are optimized by genetic algorithm under restricted conditions introduced by control saturations.
7. Nonlinear output tracking control problem is studied on the experimental platform BPVS-JLU-Ⅱ. When ball parameters are known, position controller is constructed with variable structure control method first. And then feedback linearization method is used to avoid chattering phenomenon. Adaptive control problem is further investigated for the ball and plate system with structured uncertainty. When radius, mass and inertia of the ball is not specific, nonlinear adaptive controller based on Lyapunov direct method is built first. As the transmission sets of ball and plate system is influenced by backlash nonlinearities, adaptive backstepping design is employed to avoid chattering phenomenon of the nonlinear adaptive controller using Lyapunov direct method. Circle tracking experiments were established on the experimental platform BPVS-JLU-Ⅱwith expected tracking velocity 6 mm/s, 18 mm/s, 30 mm/s respectively. Five control schemes are used to construct the position controller including backstepping, adaptive backstepping, variable structure control, feedback linearization and fuzzy control. Relationship between expected tracking velocity and control precision of the ball and plate system was discussed for different control schemes. Some experimental phenomenon of the ball and the plate in the tracking control was analyzed. Trajectory tracking experiments established on the ball and plate system experimental platform BPVS-JLU-Ⅱhave indicated that when nonlinear model of the ball and plate system is considered and position controller is constructed with backstepping or adaptive backstepping design, trajectory tracking control could be done with higher control precision compared with other control schemes.
引文
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