# American Institute of Mathematical Sciences

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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
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