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November  2011, 16(4): 1101-1117. doi: 10.3934/dcdsb.2011.16.1101

## An optimal PID controller design for nonlinear constrained optimal control problems

 1 Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Building 2F, Technology Park of Harbin Institute of Technology, Harbin, 150001, China 2 Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, W.A. 6845 3 School of Electrical and Electronic Engineering, The University of Adelaide, SA 5005 4 Center for Control Theory and Guidance Technology, Harbin Institute of Technology 210, Building 2F, Technology Park of Harbin Institute of Technology, Harbin, 150001, China

Received  September 2010 Revised  March 2011 Published  August 2011

In this paper, we consider a class of optimal PID control problems subject to continuous inequality constraints and terminal equality constraint. By applying the constraint transcription method and a local smoothing technique to these continuous inequality constraint functions, we construct the corresponding smooth approximate functions. We use the concept of the penalty function to append these smooth approximate functions to the cost function, forming a new cost function. Then, the constrained optimal PID control problem is approximated by a sequence of optimal parameter selection problems subject to only terminal equality constraint. Each of these optimal parameter selection problems can be viewed and hence solved as a nonlinear optimization problem. The gradient formulas of the new appended cost function and the terminal equality constraint function are derived, and a reliable computation algorithm is given. The method proposed is used to solve a ship steering control problem.
Citation: Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101
##### References:
 [1] J. Van Amerongen, Adaptive steering of ships-a model reference approach,, Automatica, 20 (1984), 3.  doi: 10.1016/0005-1098(84)90060-8.  Google Scholar [2] M. I. Bech and L. Wangner Smitt, "Analogue Simulation of Ship Manoeuvres,", Hydro and Aerodynamics Lab. Report No. Hy-14, (1969).   Google Scholar [3] D. L. Brooke, "The Design of a New Automaticpilot for the Comerical Ship,", First IFAC/IFIP Symosium on Ship Operation Automation, (1973).   Google Scholar [4] J. Du, C. Guo, S. Yu and Y. Zhao, Adaptive autopilot design of time-varying uncertain ships with completely unknown control coefficient,, IEEE Journal of Oceanic Engineering, 32 (2007), 346.   Google Scholar [5] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, "MISER3, Version 3: Optimal Control Software, Theory and User Manual,", 1991., ().   Google Scholar [6] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica J. IFAC, 26 (1990), 371.  doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar [7] C. C. Lim and W. Forsythe, Autopilot for ship control,, IEE Procedings, 130 (1983), 281.   Google Scholar [8] C. Y. Liu, Z. H. Gong and E. M. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,, Journal of Industrial and Management Optimization, 5 (2009), 835.  doi: 10.3934/jimo.2009.5.835.  Google Scholar [9] R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of swtiched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455.  doi: 10.1109/TAC.2009.2029310.  Google Scholar [10] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar [11] V. Rehbock, C. C. Lim and K. L. Teo, A stable constrained optimal model following controller for discrete-time nonlinear systems affine in control,, Control Theory and Advanced Technology, 10 (1994), 793.   Google Scholar [12] 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, 55 (1991).   Google Scholar [13] 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.  doi: 10.1017/S0334270000010936.  Google Scholar [14] K. L. Teo and C. C. Lim, Time optimal control computation with application to ship steering,, Journal of Optimization Theory and Applicaitons, 56 (1988), 145.  doi: 10.1007/BF00938530.  Google Scholar [15] K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems,, Automatica J. IFAC, 29 (1993), 789.  doi: 10.1016/0005-1098(93)90076-6.  Google Scholar [16] C. Z. Wu and K. L. Teo, Global impulsive optimal control computation,, Journal of Industrial and Management Optimization, 2 (2006), 435.  doi: 10.3934/jimo.2006.2.435.  Google Scholar

show all references

##### References:
 [1] J. Van Amerongen, Adaptive steering of ships-a model reference approach,, Automatica, 20 (1984), 3.  doi: 10.1016/0005-1098(84)90060-8.  Google Scholar [2] M. I. Bech and L. Wangner Smitt, "Analogue Simulation of Ship Manoeuvres,", Hydro and Aerodynamics Lab. Report No. Hy-14, (1969).   Google Scholar [3] D. L. Brooke, "The Design of a New Automaticpilot for the Comerical Ship,", First IFAC/IFIP Symosium on Ship Operation Automation, (1973).   Google Scholar [4] J. Du, C. Guo, S. Yu and Y. Zhao, Adaptive autopilot design of time-varying uncertain ships with completely unknown control coefficient,, IEEE Journal of Oceanic Engineering, 32 (2007), 346.   Google Scholar [5] L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, "MISER3, Version 3: Optimal Control Software, Theory and User Manual,", 1991., ().   Google Scholar [6] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica J. IFAC, 26 (1990), 371.  doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar [7] C. C. Lim and W. Forsythe, Autopilot for ship control,, IEE Procedings, 130 (1983), 281.   Google Scholar [8] C. Y. Liu, Z. H. Gong and E. M. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture,, Journal of Industrial and Management Optimization, 5 (2009), 835.  doi: 10.3934/jimo.2009.5.835.  Google Scholar [9] R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of swtiched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455.  doi: 10.1109/TAC.2009.2029310.  Google Scholar [10] R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar [11] V. Rehbock, C. C. Lim and K. L. Teo, A stable constrained optimal model following controller for discrete-time nonlinear systems affine in control,, Control Theory and Advanced Technology, 10 (1994), 793.   Google Scholar [12] 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, 55 (1991).   Google Scholar [13] 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.  doi: 10.1017/S0334270000010936.  Google Scholar [14] K. L. Teo and C. C. Lim, Time optimal control computation with application to ship steering,, Journal of Optimization Theory and Applicaitons, 56 (1988), 145.  doi: 10.1007/BF00938530.  Google Scholar [15] K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems,, Automatica J. IFAC, 29 (1993), 789.  doi: 10.1016/0005-1098(93)90076-6.  Google Scholar [16] C. Z. Wu and K. L. Teo, Global impulsive optimal control computation,, Journal of Industrial and Management Optimization, 2 (2006), 435.  doi: 10.3934/jimo.2006.2.435.  Google Scholar
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