An almost optimal control design method for nonlinear time-delay systems

References

• Araya , Y . 2008 . Ekeland's Variational Principle and its Equivalent Theorems in Vector Optimization . Journal of Mathematical Analysis and Applications , 346 : 9 – 16 .
   Web of Science ®Google Scholar
• Anderson , EJ and Nash , P . 1987 . Linear Programming Infinite Dimensional Space: Theory and Applications , New York : Wiley & Sons .
   Google Scholar
• Bazara , MS , Jarvis , JJ and Sherali , HD . 2009 . Linear Programming and Network Flows, , 4th , New York : Wiley & Sons .
   Google Scholar
• Boukas , E-K and Liu , ZK . 2002 . Deterministic and Stochastic Time-Delay Systems , Boston , MA : Birkhäuser .
   Google Scholar
• Busenberg , S and Cooke , K . 1992 . Vertically Transmitted Diseases: Models and Dynamics , Berlin : Springer-Verlag .
   Google Scholar
• Burl , JB . 1999 . Linear Optimal control , Menlo Park , CA : Addison-Wesley .
   Google Scholar
• Delfour , MC . 1986 . The Linear Quadratic Control Problem with Delays in Space and Control Variables: A State Space Approach . SIAM Journal on Control and Optimization , 24 : 835 – 883 .
   Web of Science ®Google Scholar
• Dion , JM . 2001 . Linear Time-delay Systems , London : Pergamon .
   Google Scholar
• Ekeland , I . 1974 . On the Variational Principle . Journal of Mathematical Analysis and Applications , 47 : 324 – 354 .
   Web of Science ®Google Scholar
• Eller , DH , Aggarwal , JK and Banks , HT . 1969 . Optimal Control of Linear Time-delay Systems . IEEE Transactions on Automatic Control , 14 : 678 – 687 .
   Web of Science ®Google Scholar
• Farahi , MH , Rubio , JE and Wilson , DA . 1996 . The Optimal Control of Linear Wave Equation . International Journal of Control , 63 : 833 – 848 .
   Web of Science ®Google Scholar
• Fattorini , HO . 2005 . Infinite Dimensional Linear System: The Time Optimal and Norm Optimal , Amsterdam : Elsevier Sciences .
   Google Scholar
• Fleming , WH and Rishel , RW . 1975 . Deterministic and Stochastic Optimal Control , New York : Springer .
   Google Scholar
• Fridman , E . 2001 . New Lyapunov–Krasovskii Functionals for Stability of Linear Retarded and Neutral Type Systems . Systems and Control Letters , 43 : 309 – 319 .
   Web of Science ®Google Scholar
• Fridman , E , Shaked , U and Suplin , V . 2005 . Input/Output Delay Approach to Robust Sampled-data H 1 Control . Systems and Control Letters , 54 : 271 – 282 .
   Web of Science ®Google Scholar
• Ge , SS , Hong , F and Lee , TH . 2003 . Adaptive Neural Network Control of Nonlinear Systems with Unknown Time-delays . IEEE Transactions on Automatic Control , 48 : 2004 – 2010 .
   Web of Science ®Google Scholar
• Germani , A , Manes , C and Pepe , P . 2003 . Input-output Linearization with Delay Cancellation for Nonlinear Delay Systems: The Problem of the Internal Stability . International Journal of Robust and Nonlinear Control , 13 : 909 – 937 .
   Web of Science ®Google Scholar
• Göllmann , L , Kern , D and Maurer , H . 2008 . Optimal Control Problems with Delays in State and Control Variables Subject to Mixed Control-state Constraints . Optimal Control Applications and Methods , 30 : 341 – 365 .
   Google Scholar
• Gu , K , Kharitonov , VL and Chen , J . 2003 . Stability of Time-Delay Systems , Boston , MA : Birkhäuser .
   Google Scholar
• Hua , C , Guan , X and Shi , P . 2005 . Robust Backstepping Control for a Class of Time-delayed Systems . IEEE Transactions on Automatic Control , 50 : 894 – 899 .
   Web of Science ®Google Scholar
• Jankovic , M . 2009 . Forwarding, Backstepping, and Finite Spectrum Assignment for Time-delay Systems . Automatica , 45 : 2 – 9 .
   Web of Science ®Google Scholar
• Kamayad , A , Keyanpour , M and Farahi , MH . 2007 . A New Approach for Solving of Optimal Nonlinear Control Problems . Applied Mathematics and Computation , 187 : 1461 – 1471 .
   Google Scholar
• Kamayad , A , Rubio , JE and Wilson , DA . 1991 . The Optimal Control of the Multidimensional Diffusion Equation . Journal of Optimization Theory and Applications , 70 : 191 – 209 .
   Google Scholar
• Kharatashvili , GL . 1967 . “ A Maximum Principle in External Problems with Delays ” . In Mathematical Theory of Control , Edited by: Balakrishnan , AV and Neustadt , LW . 26 – 34 . New York : Academic Press .
   Google Scholar
• Kolmanovskii , VB and Myshkis , AD . 1999 . Introduction to the Theory and Applications of Functional Differential Equations , The Netherlands : Kluwer Academic .
   Google Scholar
• Kolmanovskii , VB and Shaikhet , LE . 1996 . Control of Systems with Aftereffect, Translation of Mathematical Monographs , Vol. 157 , Providence , RI : American Mathematical Society .
   Google Scholar
• Koshkouei , AJ and Burnham , KJ . 2009 . Discontinuous Observers for Non-linear Time-delay Systems . International Journal of Systems Science , 40 : 383 – 392 .
   Web of Science ®Google Scholar
• Koshkouei , AJ . and Zinober, A.S.I. (1996), ‘Sliding Mode Time-delay Systems’, in Proceedings of the IEEE International Workshop on Variable Structure Control (VSS’96), Tokyo, pp. 97–101
   Google Scholar
• Krasovskii , NN . 1956 . On the Application of the Second Method of Lyapunov for Equations with Time Delays . Prikladnaya Matematika i Mekhanika , 20 : 315 – 327 (in Russian) .
   Google Scholar
• Krasovskii , NN . 1963 . Stability of Motion , Stanford , CA : Stanford University Press .
   Google Scholar
• Kwakernaak , H and Sivan , R . 1972 . Linear Optimal Control Systems , New York : Wiley-InterScience .
   Google Scholar
• Levaggi , L and Punta , E . 2006 . Analysis of a Second-order Sliding-mode Algorithm in Presence of Input Delays . IEEE Transactions on Automatic Control , 51 : 1325 – 1332 .
   Google Scholar
• Luo , N , De La Sen , M and Rodellar , J . 1997 . Robust Stabilization of a Class of Uncertain Time-delay Systems in Sliding Mode . International Journal of Robust and Nonlinear Control , 7 : 59 – 74 .
   Web of Science ®Google Scholar
• Malakhovski , E and Mirkin , L . 2006 . On Stability of Second-order Quasi-polynomials with a Single Delay . Automatica , 42 : 1041 – 1047 .
   Google Scholar
• Malek-Zavarei , M and Jamshidi , M . 1987 . Time-Delay Systems: Analysis, Optimization and Applications , Amsterdam : North-Holland .
   Google Scholar
• Márquez-Martínez , L and Moog , C . 2004 . Input-Output Feedback Linearization of Time-delay Systems . IEEE Transactions on Automatic Control , 49 : 781 – 785 .
   Web of Science ®Google Scholar
• Mazenc , F and Bliman , PA . 2006 . Backstepping Design for Time-delay Nonlinear Systems . IEEE Transactions on Automatic Control , 51 : 149 – 154 .
   Web of Science ®Google Scholar
• Mounier , H and Rudolph , J . 2008 . Flatness and Quasi-static State Feedback in Non-linear Delay Systems . International Journal of Control , 81 : 445 – 456 .
   Web of Science ®Google Scholar
• Niculescu , SI . 2001 . Delay Effects on Stability: A Robust Control Approach , London : Springer-Verlag .
   Google Scholar
• Niu , Y , Ho , D and Lam , J . 2005 . Robust Integral Sliding Mode Control for Uncertain Stochastic Systems with Time-varying Delay . Automatica , 41 : 873 – 880 .
   Web of Science ®Google Scholar
• Oguchi , T , Watanabe , A and Nakamizo , T . 2002 . Input-output Linearization of Retarded Non-linear Systems by Using an Extension of Lie Derivative . International Journal of Control , 75 : 582 – 590 .
   Web of Science ®Google Scholar
• Oguztoreli , MN . 1963 . A Time Optimal Control Problem for Systems Described by Differential Difference Equations . SIAM Journal on Control and Optimization , 1 : 290 – 310 .
   Google Scholar
• Oguztoreli , MN . 1966 . Time-Lag Control Systems , New York : Academic Press .
   Google Scholar
• Pepe , P . 2009 . Input-to-state Stabilization of Stabilizable, Time-Delay, Control-affine, Nonlinear Systems . IEEE Transactions on Automatic Control , 54 : 1688 – 1693 .
   Google Scholar
• Rosenbloom , PC . 1952 . Quelques classes de problèmes éxtremaux . Bulletin de la Societé Mathématique de France , 80 : 183 – 216 .
   Google Scholar
• Rubio , JE . 1986 . Control and Optimization: The Linear Treatment of Nonlinear Problems , London : John Wiley Press .
   Google Scholar
• Rubio , JE . 1995 . The Global Control of Diffusion Equations . SIAM Journal on Control and Optimization , 33 : 308 – 322 .
   Google Scholar
• Rubio , JE . 2000 . The Optimal Control of Nonlinear Diffusion Equations with Rough Initial Data . Journal of the Franklin Institute , 337 : 673 – 690 .
   Google Scholar
• Royden , HL . 1970 . Real Analysis , London : The Macmillan Company .
   Google Scholar
• Shin , H , Choi , H and Lim , J . 2006 . Feedback Linearisation of Uncertain Nonlinear Systems with Time-delay . IEE Proceedings Part D: Control Theory and Applications , 153 : 732 – 736 .
   Google Scholar
• Shtessel , YB , Zinober , ASI and Shkolnikov , IA . 2003 . Sliding Mode Control for Nonlinear Systems with Output Delay via Method of Stable System Center . ASME Journal Dynamics Systems, Measurement and Control , 125 : 253 – 257 .
   Web of Science ®Google Scholar
• Soliman , MA and Ray , WH . 1971 . Optimal Control of Multivariable Systems with Pure Time Delays . Automatica , 7 : 681 – 689 .
   Web of Science ®Google Scholar
• Taksar , MI . 1997 . Infinite Dimensional Linear Programming Approach to Singular Stochastic Control . SIAM Journal on Control and Optimization , 35 : 604 – 625 .
   Google Scholar
• Uchida , K , Shimemura , E , Kubo , T and Abe , N . 1988 . The Linear-quadratic Optimal Control Approach to Feedback Control Design for Systems with Delay . Automatica , 24 : 773 – 780 .
   Web of Science ®Google Scholar
• Vershik , AM . 1970 . Some Remarks on the Infinite Dimensional Problems of Linear Programming . Russian Mathematical Surveys , 25 : 117 – 124 .
   Google Scholar
• Wu , L and Zheng , WX . 2009 . Weighted H ∞ Model Reduction for Linear Switched Systems with Time-varying Delay . Automatica , 45 : 186 – 193 .
   Web of Science ®Google Scholar
• Xia , X , Márquez-Martínez , L , Zagalak , P and Moog , C . 2002 . Analysis of Nonlinear Time-delay Systems Using Modules over Noncommutative Rings . Automatica , 38 : 1549 – 1555 .
   Web of Science ®Google Scholar
• Zamirian , M , Farahi , MH and Nazemi , AR . 2007 . An Applicable Method for Solving the Shortest Path Problems . Applied Mathematics and Computation , 190 : 1479 – 1486 .
   Web of Science ®Google Scholar