CMW-IJBC00

CMW-IJBC00 - Tutorials and Reviews International Journal of...

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Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 10, No. 3 (2000) 511–548 c ± World Scientific Publishing Company BIFURCATION CONTROL: THEORIES, METHODS, AND APPLICATIONS GUANRONG CHEN * Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4793, USA JORGE L. MOIOLA Departamento de Ingenier´ ıa El´ ectrica, Universidad Nacional del Sur, Avda. Alem 1253, ( 8000 ) Bah´ ıa Blanca, Argentina HUA O. WANG Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708-0291, USA Received May 10, 1999; Revised August 1, 1999 Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation, stabilizing a bifurcated solution or branch, chang- ing the parameter value of an existing bifurcation point, modifying the shape or type of a bifurcation chain, introducing a new bifurcation at a preferable parameter value, monitoring the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a combination of some of these objectives. This article offers an overview of this emerging, challenging, stimulating, and yet promising field of research, putting the main subject of bifurcation control into perspective. 1 .I n t r o d u c t i o n................................ 5 1 2 2 .B i f u r c a t i o nC o n t r o l—T w oE x am p l e s..................... 5 1 3 2 . 1 .T h e l o g i s t i cm a p............................ 5 1 3 2 . 2 .A ne l e c t r i cp o w e rm o d e l ........................ 5 1 4 3 i f u r c a t i o n s i o n t r o lS y s t em s ....................... 5 1 5 4 i f u r c a t i o nP r e l im i n a r i e s .......................... 5 1 6 4 . 1 i f u r c a t i o n so fo n e - d e n s i o n a lm a p s................... 5 1 7 4.2. Hopf bifurcation of higher-dimensional systems . . . . . . . . . . . . . . 518 5 a s i cS t a t e - F e e d b a c kB i f u r c a t i o o n t r o lM e t h o d s ............... 5 1 9 5.1. Controlling saddle-node, transcritical, and pitchfork bifurcations . . . . . . . 519 5 . 2 h ep e r i o d - d o u b l i n gb i f u r c a t i o na n d i t sc o n t r o l .............. 5 2 0 5.3. Controlling the Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 522 * Current address: Department of Electronic Engineering, City University of Hong Kong. E-mail: gchen@uh.edu E-mail: comoiola@criba.edu.ar E-mail: hua@ee.duke.edu 511
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512 G. Chen et al. 6 .V a r i o u sB i f u r c a t i o nC o n t r o lM e t h o d s ..................... 5 2 3 6.1. Bifurcation control via state feedback and washout filter-aided d y n am i c f e e d b a c kc o n t r o l l e r s ...................... 5 2 3 6.2. Bifurcation control via normal forms and invariants . . . . . . . . . . . . 526 6.3. Bifurcation control via harmonic balance approximations . . . . . . . . . . 527 7. Controlling Hopf Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 528 7 . 1 .G r a p h i c a lH o p fb i f u r c a t i o nt h e o r em . .................. 5 2 8 7.1.1. The single-input single-output case . . . . . . . . . . . . . . . . 529 7.1.2. An example of Hopf bifurcation control . . . . . . . . . . . . . . . 530 7.2. Controlling the birth of multiple limit cycles . . . . . . . . . . . . . . . . 531 7.2.1. Necessary conditions for multiple limit cycles . . . . . . . . . . . . 531 7.2.2. An example of multiple limit cycle control . . . . . . . . . . . . . 532 7.3. Controlling the amplitudes of limit cycles . . . . . . . . . . . . . . . . . 533 7.3.1. Degenerate Hopf bifurcations and control of oscillations
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This note was uploaded on 01/04/2012 for the course CDS 101 taught by Professor Murray,r during the Fall '08 term at Caltech.

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CMW-IJBC00 - Tutorials and Reviews International Journal of...

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