Linear Programming for Optimum PID Controller Tuning


This work presents a new methodology based on Linear Programming (LP) to tune Proportional-Integral-Derivative (PID) control parameters. From a specification of a desired output time domain of the plant, a linear optimization system is proposed to adjust the PID controller leading the output signal to stable operation condition with minimum oscillations. The constraint set used in the optimization process is defined by using numerical integration approach. The generated optimization problem is convex and easily solved using an interior point algorithm. Results obtained using familiar plants from literature have shown that the proposed linear programming problem is very effective for tuning PID controllers.

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Oliveira, E. , Honorio, L. , Anzai, A. and Soares, T. (2014) Linear Programming for Optimum PID Controller Tuning. Applied Mathematics, 5, 886-897. doi: 10.4236/am.2014.56084.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Dantzig, G.B. (1963) Linear Programming and Extensions. Princeton University, Princeton.
[2] Karmarkar, N. (1984) A New Polynomial Time Algorithm for Linear Programming. Combinatorica, 4, 373-395.
[3] Delson, J.K. and Shahidehpour, S.M. (1992) Linear Programming Applications to Power System Economics, Planning and Operations. IEEE Transactions on Power Systems, 7, 1155-1163.
[4] Chattopadhyay, B., Sachdev, M.S. and Sidhu, T.S. (1996) An On-Line Relay Coordination Algorithm for Adaptive Protection Using Linear Programming Technique. IEEE Transactions on Power Delivery, 11, 165-173.
[5] Boyd, S.P., Balakrishnan, V., Barratt, C.H., Khraishi, N.M., Li, X., Meyer, D.G. and Norman, S.A. (1988) A New CAD Method and Associated Architectures for Linear Controllers. IEEE Transactions on Automatic Control, 33, 268283.
[6] Padhy, N.P. (2004) Unit Commitment—A Bibliographical Survey. IEEE Transactions on Power Delivery, 19, 11961205.
[7] Astrom, K.J. and Hagglund, T. (2001) The Future of PID Control. Control Engineering Practice, 9, 1163-1175.
[8] Cominos, P. and Munro, N. (2002) PID Controllers: Recent Tuning Methods and Design to Specification. IEE Proceedings on Control Theory and Applications, 149, 46-53.
[9] Toscano, R. and Lyonnet, P. (2010) A New Heuristic Approach for Non-Convex Optimization Problems. Information Sciences, 180, 1955-1966.
[10] Ntogramatzidis, L. and Ferrante, A. (2011) Exact Tuning of PID Controllers in Control Feedback Design. Control Theory and Applications, 5, 565-578.
[11] Alfaro-Cid, E., McGookin, E. and Murray-Smith, D. (2006) GA-Optimised PID and Pole Placement Real and Simulated Performance When Controlling the Dynamics of a Supply Ship. IEEE Proceedings on Control Theory and Applications, 153, 228-236.
[12] Zhao, S., Qu, B., Suganthan, P., Iruthayarajan, M. and Baskar, S. (2010) Multi-Objective Robust PID Controller Tuning Using Multi-Objective Differential Evolution. IEEE 11th International Conference on Control Automation Robotics & Vision (ICARCV), 2398-2403.
[13] Ren, Z., San, Y. and Chen, J. (2006) Improved Particle Swarm Optimization and Its Application Research in Tuning of PID Parameters. Journal of System Simulation, 18, 2870-2873.
[14] Honorio, L., da Silva, A., Barbosa, D. and Delboni, L. (2010) Solving Optimal Power Flow Problems Using a Probabilistic α-Constrained Evolutionary Approach. Generation, Transmission & Distribution, 4, 674-682.
[15] Leite da Silva, A., Rezende, L., Honório, L. and Manso, L. (2011) Performance Comparison of Metaheuristics to Solve the Multi-Stage Transmission Expansion Planning Problem. Generation, Transmission Distribution, 5, 360-367.
[16] de Oliveira, L., Carneiro Jr, S., de Oliveira, E., Pereira, J., Silva Jr., I. and Costa, J. (2010) Optimal Reconfiguration and Capacitor Allocation in Radial Distribution Systems for Energy Losses Minimization. International Journal of Electrical Power & Energy Systems, 32, 840-848.
[17] de Souza, A., Honorio, L., Torres, G. and Lambert-Torres, G. (2004) Increasing the Loadability of Power Systems through Optimal-Local-Control Actions. IEEE Transactions on Power Systems, 19, 188-194.
[18] Zhang, Y. (1996) Solving Large-Scale Linear Programs by Interior-Point Methods under the Matlab Environment, Tech. rep., Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore.
[19] Mehrotra, S. (1992) On the Implementation of a Primal-Dual Interior Point Method. SIAM Journal on Optimization, 2, 575-601.
[20] Ogata, K. (1997) Modern Control Engineering Systems. 3rd Edition, Prentice Hall.
[21] Munoz, L.E. and Castillo, P. and Garcia, P. (2013) Observer-Control Scheme for Autonomous Navigation: Flight Tests Validation in a Quadrotor Vehicle. Unmanned Aircraft Systems (ICUAS), International Conference, IEEE Organization.

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