一种基于控制参数化的双连杆机械臂最优PID参数整定方法
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摘要
考虑一类双连杆机械臂的PID控制问题,提出一种基于控制参数化的最优PID参数整定方法.首先,把系统的性能指标建模为最优控制中的连续状态不等式约束.其次,将双连杆机械臂的最优PID参数整定问题转化为含连续状态不等式约束的最优参数选择问题.然后,应用约束转录法结合局部平滑法来处理连续状态不等式约束.得到一个标准的最优参数选择问题,且这个标准问题可以用最优控制软件MISER 3.2来求解.由于是基于梯度的方法来求解问题,所以在文中推导了代价函数和经过处理后约束的梯度公式.最后,通过数值仿真验证了提出的方法的有效性.
A control-parametrization-based optimal PID tuning scheme for a two-link manipulator is developed in this paper. The performance specifications of the control system are modelled as continuous state inequality constraints. Then,the PID optimal tuning of the two-link manipulator can be formulated as an optimal parameter selection problem subject to continuous inequality constraints. These continuous inequality constraints are handled by the constraint transcription method together with a local smoothing technique. In such a way,the transformed problem becomes an optimal parameter selection problem in canonical form,which can be solved via the optimal control software MISER 3. 2. Considering that the proposed approach is designed based on the gradient-based method,the corresponding gradient formulas for the cost function and constraints are derived. The effectiveness of the proposed method is illustrated by the numerical examples.
A control-parametrization-based optimal PID tuning scheme for a two-link manipulator is developed in this paper. The performance specifications of the control system are modelled as continuous state inequality constraints. Then,the PID optimal tuning of the two-link manipulator can be formulated as an optimal parameter selection problem subject to continuous inequality constraints. These continuous inequality constraints are handled by the constraint transcription method together with a local smoothing technique. In such a way,the transformed problem becomes an optimal parameter selection problem in canonical form,which can be solved via the optimal control software MISER 3. 2. Considering that the proposed approach is designed based on the gradient-based method,the corresponding gradient formulas for the cost function and constraints are derived. The effectiveness of the proposed method is illustrated by the numerical examples.
引文
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[23]LI B,YU C J,LOXTON R,et al.Optimal discrete-valued control computation[J].Journal of Global Optimization,2013,56(2):503-518.
[24]YANG F,TEO K L,LOXTON R,et al.Visual miser:an efficient user-friendly visual program for solving optimal control problems[J].Journal of Industrial and Management Optimization,2016,12(2):781-810.
[25]JENNINGS L,TEO K L,FISGER M,et al.Miser3 version 2,optimal control software,theory and user manual[D].Australia:The University of Western Australia,1997.
[2]王伟,张晶涛,柴天佑.PID参数先进整定方法综述[J].自动化学报,2000,26(3):347-355.WANG W,TAO J T,CHAI T Y,A survey of advanced PID parameter tuningmethods[J].Acta Automatica Sinica.2000,26(3):347-355.
[3]ZIEGLER J G.Optimum seting for automatic controllers[J].Trans Asme,1942,64(2B):759-768.
[4]ANG K H,CHONG G,LI Y.PID control system analysis,design,and technology[J].IEEE Transactions on Control Systems Technology,2005,13(4):559-576.
[5]NISHIKAWA Y,SANNOMIY N,OHTA T.A method for auto-tuning of PID control parameters[J].Automatica,1984,20(3):321-332.
[6]陶永华,尹怡欣,葛芦生.新型PID控制及其应用[M].北京:机械工业出版社,1998.
[7]王亚刚,邵惠鹤.一种基于灵敏度的自镇定最优PI控制器[J].自动化学报,2001,27(1):140-143.WANG Y G,SHAO HH.Automatic tuning of optimal PI controllers based on sensitivity specification[J].Acta Automatica Sinica,2001,27(1):140-143.
[8]张志强,邵惠鹤.一种新的基于相位裕度PID参数最优整定方法[J].上海交通大学学报,2000,34(5):623-625.ZHANG Z Q,SHAO HH.New optimization tuning method of PID parameters based on phase margin[J].Journal of Shanghai Jiaotong University,2000,34(5):623-625.
[9]ASTROM K J.Toward intelligent control[J].IEEE Control systems magazine,1989(April):60-64.
[10]ASTROM K J,HAGGLUND T.Automatic tuning of simple regulators with specifications on phase and amplitude margins[J].Automatica,1984,20:645-651.
[11]ASTROM K J,HAGGLUND T.Automatic Tuning of PID Controllers[M].Research Triangle Park,North Carolina:Instrument Society of America,1988.
[12]ZHUANG M,ATHERTON D P.PID controller design for TITO system[J].IEEE Proceedings:Control Theory Application,1994,141(2):111-120.
[13]舒怀林.PID神经元网络多变量控制系统分析[J].自动化学报,1999,25(1):105-111.SHU H L.Analysis of PID neural networkmultiv-Ariable control systems[J].Acta Automatica Sinica,1999,25(1):105-111.
[14]KRAUS T W,MYRON T J.Self-tuning PID controller uses pattern recognition approach[J].Control Engineering,1984(June):106-111.
[15]BRISTOL E H.Pattern recognition:an alternative to parameter identification in adaptive control[J].Automatica,1977,13:197-202.
[16]WU C Z,TEO K L.Global impulsive optimal control computation[J].Journal of Industrial and Management Optimization,2006(2):435-450.
[17]TEO K L,GOH C J,WONG K H.A unified computational approach for optimal control problems[M].New York:Pitman Monographs and Surveys in Pure and Applied Mathematics,1991.
[18]JENNINGS L S,TEO K L.A computational algorithm for functional inequality constrained optimization problems[J].Automatica J IFAC,26(1990),371-375.
[19]TEO K L,JENNINGS L S,LEE H W J,et al.The control parameterization enhancing transform for constrained optimal control Problem[J].Australian Mathematica,1999,40(3):314-335.
[20]LI B,TEO K L.An optimal PID controller design for nonlinear constrained optimal control problems[J].Discrete and Continuous Dynamical Systems-Series B,2011,16(4):1101-1117.
[21]LI B,XU C.Time optimal Zermelo’s navigation problem with moving and fixed obstacle[J].Applied Mathematics&Computation,2013,224(2-3):866-875.
[22]LI B,YU C J,TEO K L,et al.An exact penalty function method for continuous inequality constrained optimal control problem[J].Journal of Optimization Theory and Applications,2011,151(2):260-291.
[23]LI B,YU C J,LOXTON R,et al.Optimal discrete-valued control computation[J].Journal of Global Optimization,2013,56(2):503-518.
[24]YANG F,TEO K L,LOXTON R,et al.Visual miser:an efficient user-friendly visual program for solving optimal control problems[J].Journal of Industrial and Management Optimization,2016,12(2):781-810.
[25]JENNINGS L,TEO K L,FISGER M,et al.Miser3 version 2,optimal control software,theory and user manual[D].Australia:The University of Western Australia,1997.