# American Institute of Mathematical Sciences

## An optimal pid tuning method for a single-link manipulator based on the control parametrization technique

 1 College of Electrical Engineering and Information Technology, Sichuan University, Chengdu 610065, China 2 Sichuan Institute of Aerospace Electronic Equipment, Chengdu 610100, China

*Corresponding author: Xiaodong Zeng

Received  September 2018 Revised  November 2018 Published  September 2019

Fund Project: This work was partially supported by a grant from National Natural Science Foundation of China under number 61701124, a grant from Science and Technology on Space Intelligent Control Laboratory, No. KGJZDSYS-2018-03, a grant from Sichuan Science and Technology Program under number 2019YJ0105, and a grant from Fundamental Research Funds for the Central Universities (China)

A control parametrization based optimal PID tuning scheme for a single-link manipulator is developed in this paper. The performance specifications of the control system are formulated as continuous state inequality constraints. Then, the PID optimal tuning problem of the single-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 a canonical form, which can be solved efficiently by control parametrization method. Since approach is using the gradient-based method, the corresponding gradient formulas for the cost function and the constraints are derived, respectively. The effectiveness of the proposed method is demonstrated by numerical simulations.

Citation: Bin Li, Xiaolong Guo, Xiaodong Zeng, Songyi Dian, Minhua Guo. An optimal pid tuning method for a single-link manipulator based on the control parametrization technique. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020107
##### References:
 [1] K. H. Ang, G. Chong and Y. Li, PID control system analysis, design and technology, IEEE Transactions on Control Systems Technology, 13 (2005), 559-576. doi: 10.1109/TCST.2005.847331. Google Scholar [2] K. J. Astrom, Toward intelligent control, IEEE Control Systems Magazine, 9 (1989), 60-64. doi: 10.1109/37.24813. Google Scholar [3] K. J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning, 2nd Edition, Research Triangle Park. North Carolina: Instrument Society of America, 1995.Google Scholar [4] K. J. Astrom and T. Hagglund, Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatica, 20 (1984), 645-651. doi: 10.1016/0005-1098(84)90014-1. Google Scholar [5] K. J. Astrom and T. Hagglund, Automatic Tuning of PID Controllers, Research Triangle Park, North Carolina: Instrument Society of America, 1988.Google Scholar [6] E. R. Daniel and S. J. Kyoung, An integrated identification and control design methodology for multivariable process system application, IEEE: Control Systems Society, 20 (2000), 25-37. doi: 10.1109/37.845036. Google Scholar [7] E. H. Bristol, Pattern recognition: An alternative to parameter identification in adaptive control, Automatica, 13 (1977), 197-202. doi: 10.1016/0005-1098(77)90046-2. Google Scholar [8] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198. doi: 10.3934/jimo.2017042. Google Scholar [9] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica J IFAC, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z. Google Scholar [10] L. Jennings, K. L. Teo, M. Fisher, C. J. Goh and L. S. Jennings., MISER3 Version 2, Optimal Control Software, Theory and User Manual, Departmentof Mathematics, The University of Western Australia, Australia, 1997. MISER3 version 2, Optimal Control Software, Theory and User Manual, Department of Mathematics, The University of Western Australia, Australia, 1997.Google Scholar [11] T. W. Kraus and T. J. Myron, Self-tuning PID controller uses pattern recognition approach, Control Engineering, (1984), 106–111.Google Scholar [12] B. Li and K. L Teo, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1101-1117. doi: 10.3934/dcdsb.2011.16.1101. Google Scholar [13] B. Li, C Xu and K. L. Teo, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092. Google Scholar [14] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. Google Scholar [15] J. K. Liu, Design of Robot Control System and MATLAB Simulation, Tsinghua University press, Beijing, Springer, Singapore, 2018. doi: 10.1007/978-981-10-5263-7. Google Scholar [16] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [17] C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264. doi: 10.1007/s11590-016-1105-6. Google Scholar [18] C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523. doi: 10.1137/16M1070530. Google Scholar [19] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182. doi: 10.1016/j.jprocont.2018.10.001. Google Scholar [20] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [21] C. Y. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems and Control Letters, 72 (2014), 53-60. doi: 10.1016/j.sysconle.2014.07.001. Google Scholar [22] C. Y. Liu, Z. H. Gong and E. M. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pacific Journal of Optimization, 9 (2013), 595-612. Google Scholar [23] C. Y. Liu, Z. H. Gong, B. Y. Shen and E. M. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706. doi: 10.1016/j.apm.2012.02.044. Google Scholar [24] C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer Berlin Heidelberg, 2014. doi: 10.1007/978-3-662-43793-3. Google Scholar [25] C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368. doi: 10.1016/j.apm.2017.09.007. Google Scholar [26] P. Mu, L. Wang and C. Y. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, (2017), 1–14. doi: 10.1007/s10957-017-1163-7. Google Scholar [27] Y. Nishikawa, N. Sannomiya and T. Ohta, A method for auto-tuning of PID control parameters, Automatica, 20 (1984), 321-332. doi: 10.1016/0005-1098(84)90047-5. Google Scholar [28] H. L. Shu, PID neural network multivariable control system analysis, Acta Automatica Sinica, 25 (1999), 105-111. Google Scholar [29] Y. H. Tao, Y. X. Yin and L. S. Ge, New PID Control and Application, Machinery Industry Press, Beijing, 1998.Google Scholar [30] K. L. Teo, L. S. Zhang and Y. Q, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequalit constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485. Google Scholar [31] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, copublished in the united states with John Wiley and Sons, Inc, NewYork, 1991. Google Scholar [32] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, J. Austral. Math. Soc. Ser. B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. Google Scholar [33] K. L. Teo, L. S. Zhang and Y. Q, Bai: A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar [34] W. Wang, J. T. Zhang and T. Y. Chai, Summary of advanced tuning methods for PID parameters, Acta Automatica Sinica, 26 (2000), 347-355. Google Scholar [35] Y. G. Wang and H. H. Shao, A self stabilization optimization based on sensitivity PID controller, Acta Automatica Sinica, 27 (2001), 140-143. Google Scholar [36] C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. Google Scholar [37] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. Google Scholar [38] C. J. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7. Google Scholar [39] J. L. Yuan, C. Y. Liu, X. Zhang, J. Xie, E. Feng, H. C. Yin and Z. L. Xiu, Optimal control of a batch fermentation process with nonlinear time- delay and free terminal time and cost sensitivity constraint, Journal of Process Control, 44 (2016), 41-52. doi: 10.1016/j.jprocont.2016.05.001. Google Scholar [40] J. L. Yuan, Y. D. Zhang, J. X. Ye, J. Xie, K. L. Teo, X. Zhu, E. Feng, H. Yin and Z. L. Xiu, Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time delayed process presenting metabolic discontinuities, Applied Mathematical Modeling, 46 (2017), 554-571. doi: 10.1016/j.apm.2017.01.079. Google Scholar [41] J. L. Yuan, X. Zhang, C. Y. liu, L. Chang, J. Xie, E. Feng, H. C. Yin and Z. L. Xiu, Robust optimization for nonlinear time delay dynamical system of dha regulon with cost sensitivity constraint in batch culture, Communications in Nonlinear Science and Numerical Simulation, 38 (2016), 140-171. doi: 10.1016/j.cnsns.2016.02.008. Google Scholar [42] J. L. Yuan, X. Zhu, X. Zhang, H. C. Yin, E. Feng and Z. L. Xiu, Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture, Applied Mathematics and Computation, textbf232 (2014), 150–163. doi: 10.1016/j.amc.2014.01.027. Google Scholar [43] Z. Q. Zhang and H. H. Shao, An optimal tuning method for PID parameters based on phase margin, Journal of Shanghai Jiaotong University, 34 (2000), 623-625. Google Scholar [44] M. Zhuang and D. P. Atherton, PID controller design for a TITO system, IEEE Proceedings: Control Theory Application, 141 (1994), 111-120. doi: 10.23919/ACC.1993.4793493. Google Scholar [45] J. G. Ziegler, Optimum seting for automatic controllers, Trans Asme, 64 (1942), 759-768. Google Scholar

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##### References:
 [1] K. H. Ang, G. Chong and Y. Li, PID control system analysis, design and technology, IEEE Transactions on Control Systems Technology, 13 (2005), 559-576. doi: 10.1109/TCST.2005.847331. Google Scholar [2] K. J. Astrom, Toward intelligent control, IEEE Control Systems Magazine, 9 (1989), 60-64. doi: 10.1109/37.24813. Google Scholar [3] K. J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning, 2nd Edition, Research Triangle Park. North Carolina: Instrument Society of America, 1995.Google Scholar [4] K. J. Astrom and T. Hagglund, Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatica, 20 (1984), 645-651. doi: 10.1016/0005-1098(84)90014-1. Google Scholar [5] K. J. Astrom and T. Hagglund, Automatic Tuning of PID Controllers, Research Triangle Park, North Carolina: Instrument Society of America, 1988.Google Scholar [6] E. R. Daniel and S. J. Kyoung, An integrated identification and control design methodology for multivariable process system application, IEEE: Control Systems Society, 20 (2000), 25-37. doi: 10.1109/37.845036. Google Scholar [7] E. H. Bristol, Pattern recognition: An alternative to parameter identification in adaptive control, Automatica, 13 (1977), 197-202. doi: 10.1016/0005-1098(77)90046-2. Google Scholar [8] Z. H. Gong, C. Y. Liu and Y. J. Wang, Optimal control of switched systems with multiple time-delays and a cost on changing control, Journal of Industrial and Management Optimization, 14 (2018), 183-198. doi: 10.3934/jimo.2017042. Google Scholar [9] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica J IFAC, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z. Google Scholar [10] L. Jennings, K. L. Teo, M. Fisher, C. J. Goh and L. S. Jennings., MISER3 Version 2, Optimal Control Software, Theory and User Manual, Departmentof Mathematics, The University of Western Australia, Australia, 1997. MISER3 version 2, Optimal Control Software, Theory and User Manual, Department of Mathematics, The University of Western Australia, Australia, 1997.Google Scholar [11] T. W. Kraus and T. J. Myron, Self-tuning PID controller uses pattern recognition approach, Control Engineering, (1984), 106–111.Google Scholar [12] B. Li and K. L Teo, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1101-1117. doi: 10.3934/dcdsb.2011.16.1101. Google Scholar [13] B. Li, C Xu and K. L. Teo, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092. Google Scholar [14] B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5. Google Scholar [15] J. K. Liu, Design of Robot Control System and MATLAB Simulation, Tsinghua University press, Beijing, Springer, Singapore, 2018. doi: 10.1007/978-981-10-5263-7. Google Scholar [16] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [17] C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264. doi: 10.1007/s11590-016-1105-6. Google Scholar [18] C. Y. Liu, R. Loxton, Q. Lin and K. L. Teo, Dynamic optimization for switched time-delay systems with state-dependent switching conditions, SIAM Journal on Control and Optimization, 56 (2018), 3499-3523. doi: 10.1137/16M1070530. Google Scholar [19] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2019), 170-182. doi: 10.1016/j.jprocont.2018.10.001. Google Scholar [20] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20. doi: 10.1016/j.nahs.2017.01.006. Google Scholar [21] C. Y. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems and Control Letters, 72 (2014), 53-60. doi: 10.1016/j.sysconle.2014.07.001. Google Scholar [22] C. Y. Liu, Z. H. Gong and E. M. Feng, Optimal control for a nonlinear time-delay system in fed-batch fermentation, Pacific Journal of Optimization, 9 (2013), 595-612. Google Scholar [23] C. Y. Liu, Z. H. Gong, B. Y. Shen and E. M. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706. doi: 10.1016/j.apm.2012.02.044. Google Scholar [24] C. Y. Liu and Z. H. Gong, Optimal Control of Switched Systems Arising in Fermentation Processes, Springer Berlin Heidelberg, 2014. doi: 10.1007/978-3-662-43793-3. Google Scholar [25] C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368. doi: 10.1016/j.apm.2017.09.007. Google Scholar [26] P. Mu, L. Wang and C. Y. Liu, A control parameterization method to solve the fractional-order optimal control problem, Journal of Optimization Theory and Applications, (2017), 1–14. doi: 10.1007/s10957-017-1163-7. Google Scholar [27] Y. Nishikawa, N. Sannomiya and T. Ohta, A method for auto-tuning of PID control parameters, Automatica, 20 (1984), 321-332. doi: 10.1016/0005-1098(84)90047-5. Google Scholar [28] H. L. Shu, PID neural network multivariable control system analysis, Acta Automatica Sinica, 25 (1999), 105-111. Google Scholar [29] Y. H. Tao, Y. X. Yin and L. S. Ge, New PID Control and Application, Machinery Industry Press, Beijing, 1998.Google Scholar [30] K. L. Teo, L. S. Zhang and Y. Q, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequalit constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485. Google Scholar [31] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, copublished in the united states with John Wiley and Sons, Inc, NewYork, 1991. Google Scholar [32] K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, J. Austral. Math. Soc. Ser. B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936. Google Scholar [33] K. L. Teo, L. S. Zhang and Y. Q, Bai: A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial Management and Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895. Google Scholar [34] W. Wang, J. T. Zhang and T. Y. Chai, Summary of advanced tuning methods for PID parameters, Acta Automatica Sinica, 26 (2000), 347-355. Google Scholar [35] Y. G. Wang and H. H. Shao, A self stabilization optimization based on sensitivity PID controller, Acta Automatica Sinica, 27 (2001), 140-143. Google Scholar [36] C. Z. Wu and K. L. Teo, Global impulsive optimal control computation, Journal of Industrial and Management Optimization, 2 (2006), 435-450. doi: 10.3934/jimo.2006.2.435. Google Scholar [37] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810. doi: 10.3934/jimo.2016.12.781. Google Scholar [38] C. J. Yu, B. Li, R. Loxton and K. L. Teo, Optimal discrete-valued control computation, Journal of Global Optimization, 56 (2013), 503-518. doi: 10.1007/s10898-012-9858-7. Google Scholar [39] J. L. Yuan, C. Y. Liu, X. Zhang, J. Xie, E. Feng, H. C. Yin and Z. L. Xiu, Optimal control of a batch fermentation process with nonlinear time- delay and free terminal time and cost sensitivity constraint, Journal of Process Control, 44 (2016), 41-52. doi: 10.1016/j.jprocont.2016.05.001. Google Scholar [40] J. L. Yuan, Y. D. Zhang, J. X. Ye, J. Xie, K. L. Teo, X. Zhu, E. Feng, H. Yin and Z. L. Xiu, Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time delayed process presenting metabolic discontinuities, Applied Mathematical Modeling, 46 (2017), 554-571. doi: 10.1016/j.apm.2017.01.079. Google Scholar [41] J. L. Yuan, X. Zhang, C. Y. liu, L. Chang, J. Xie, E. Feng, H. C. Yin and Z. L. Xiu, Robust optimization for nonlinear time delay dynamical system of dha regulon with cost sensitivity constraint in batch culture, Communications in Nonlinear Science and Numerical Simulation, 38 (2016), 140-171. doi: 10.1016/j.cnsns.2016.02.008. Google Scholar [42] J. L. Yuan, X. Zhu, X. Zhang, H. C. Yin, E. Feng and Z. L. Xiu, Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture, Applied Mathematics and Computation, textbf232 (2014), 150–163. doi: 10.1016/j.amc.2014.01.027. Google Scholar [43] Z. Q. Zhang and H. H. Shao, An optimal tuning method for PID parameters based on phase margin, Journal of Shanghai Jiaotong University, 34 (2000), 623-625. Google Scholar [44] M. Zhuang and D. P. Atherton, PID controller design for a TITO system, IEEE Proceedings: Control Theory Application, 141 (1994), 111-120. doi: 10.23919/ACC.1993.4793493. Google Scholar [45] J. G. Ziegler, Optimum seting for automatic controllers, Trans Asme, 64 (1942), 759-768. Google Scholar
Performance Specifications
Manipulator angle $q(t)$
Angle velocity $\dot{q}(t)$
Torque $\tau(t)$
Manipulator angle $q(t)$ with disturbance
Angle velocity $\dot{q}(t)$ with disturbance
Torque $\tau(t)$ with disturbance
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