An Algorithm to Design Pareto Optimal Controllers for Linear Systems

In this paper, an algorithm is proposed for the multi-objective optimal design of controllers with fixed structure, but tunable parameters, for linear time-invariant systems. Differently from other tools available in the literature, the proposed method allows one to determine an exact solution to a multi-objective minimization problem without eliminating variables. Several applications of the proposed method, which span from the computation of a Pareto optimal solution for a differential game to the fast stabilization of a closed-loop system, are reported to corroborate the theoretical results.

[1]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[2]  Puduru Viswanadha Reddy,et al.  Necessary and Sufficient Conditions for Pareto Optimality in Infinite Horizon Cooperative Differential Games , 2014, IEEE Transactions on Automatic Control.

[3]  J. Lin.  Maximal vectors and multi-objective optimization , 1976 .

[4]  Jean Charles Faugère,et al.  A new efficient algorithm for computing Gröbner bases without reduction to zero (F5) , 2002, ISSAC '02.

[5]  Antonio Tornambè,et al.  On polynomial vector fields having a given affine variety as attractive and invariant set: application to robotics , 2015, Int. J. Control.

[6]  George Leitmann,et al.  A simple derivation of necessary conditions for Pareto optimality , 1974 .

[7]  Corrado Possieri,et al.  An algebraic geometry approach for the computation of all linear feedback Nash equilibria in LQ differential games , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[8]  B. Sturmfels.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[9]  William S. Levine,et al.  The Control Handbook , 2005 .

[10]  J. Faugère.  A new efficient algorithm for computing Gröbner bases (F4) , 1999 .

[11]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[12]  Kemin Zhou,et al.  Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. II. Optimal control , 1994 .

[13]  Magali Bardet,et al.  On the Complexity of a Grobner Basis Algorithm , 2005 .

[14]  Ramon Vilanova,et al.  PID Control in the Third Millennium , 2012 .

[15]  Yun Li,et al.  PID control system analysis, design, and technology , 2005, IEEE Transactions on Control Systems Technology.

[16]  P. Khargonekar,et al.  Multiple objective optimal control of linear systems: the quadratic norm case , 1991 .

[17]  Kalyanmoy Deb,et al.  Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms , 1994, Evolutionary Computation.

[18]  P. Siarry,et al.  Multiobjective Optimization: Principles and Case Studies , 2004 .

[19]  Manuel Laguna,et al.  Tabu Search , 1997 .

[20]  D. Bernstein,et al.  LQG control with an H/sup infinity / performance bound: a Riccati equation approach , 1989 .

[21]  Carsten W. Scherer,et al.  From mixed to multi-objective control , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[22]  Tore Hägglund,et al.  Robust PID Design Based on QFT and Convex–Concave Optimization , 2017, IEEE Transactions on Control Systems Technology.

[23]  Adrian Gambier,et al.  Multi-objective optimal control: An introduction , 2011, 2011 8th Asian Control Conference (ASCC).

[24]  B. Anderson,et al.  Output feedback stabilization—Solution by algebraic geometry methods , 1977, Proceedings of the IEEE.

[25]  W. Haddad,et al.  LQG control with an H∞ performance bound: a riccati equation approach , 1988, 1988 American Control Conference.

[26]  Stephen P. Boyd,et al.  Method of centers for minimizing generalized eigenvalues , 1993, Linear Algebra and its Applications.

[27]  L. Menini,et al.  Dead‐beat regulation of mechanical juggling systems , 2018 .

[28]  Guo-Ping Liu,et al.  Multiobjective Optimisation And Control , 2008 .

[29]  E. L. Ulungu,et al.  MOSA method: a tool for solving multiobjective combinatorial optimization problems , 1999 .

[30]  Kaisa Miettinen,et al.  Proper Pareto Optimality In Nonconvex Problems - Characterization With Tangent And Normal Cones , 1998 .

[31]  Laura Menini,et al.  Application of algebraic geometry techniques in permanent-magnet DC motor fault detection and identification , 2015, Eur. J. Control.

[32]  B. Francis.  The linear multivariable regulator problem , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[33]  Laura Menini,et al.  Switching Signal Estimator Design for a Class of Elementary Systems , 2016, IEEE Transactions on Automatic Control.

[34]  Laura Menini,et al.  Boolean network representation of a continuous-time system and finite-horizon optimal control: application to the single-gene regulatory system for the lac operon , 2017, Int. J. Control.

[35]  Tore Hägglund,et al.  The future of PID control , 2000 .

[36]  M Guerrisi,et al.  Advanced lung ventilation system (ALVS) with linear respiratory mechanics assumption for waveform optimization of dual-controlled ventilation. , 2007, Medical engineering & physics.

[37]  Laura Menini,et al.  Algebraic Methods for Multiobjective Optimal Design of Control Feedbacks for Linear Systems , 2018, IEEE Transactions on Automatic Control.

[38]  Corrado Possieri,et al.  On polynomial feedback Nash equilibria for two-player scalar differential games , 2016, Autom..

[39]  Carlos A. Coello Coello,et al.  An updated survey of GA-based multiobjective optimization techniques , 2000, CSUR.

[40]  Antonio Tornambè,et al.  On f-invariant and attractive affine varieties for continuous-time polynomial systems: The case of robot motion planning , 2014, 53rd IEEE Conference on Decision and Control.

[41]  Wei Yang,et al.  Robust Multi-Objective Feedback Design by Quantifier Elimination , 1997, J. Symb. Comput..