Peaking phenomenon
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Petar V. Kokotovic and Hector J. Sussmann are renowned control theorists who have made significant contributions to nonlinear control systems. Kokotovic founded the Center for Control Engineering and Computation at UC Santa Barbara and developed constructive nonlinear control methods. Sussmann taught at Rutgers University and authored over 140 research papers in nonlinear and optimal control. The document discusses the concept of "peaking" in nonlinear control systems, where the state or output variables can grow unboundedly fast as the system evolves over time. Peaking poses a structural obstacle to achieving global or semiglobal stabilization of nonlinear systems using feedback control. The paper presents theoretical results showing that peaking can cause the region of attraction
Peaking phenomenon
- 1. H. J. Sussmann, Senior Member, IEEE, and P. V. Kokotovic, Fellow, IEEE Citations: 195 IEEE Transactions on Automatic Control, Vol. 36, No. 4, April, 1991
- 2. Two Giants in the Kingdom of Nonlinear Control Dr. Kokotovic is the founder and director of the University of California-Santa Barbara’s Center for Control Engineering and Computation. This center has become a role model of cross disciplinary research and education. One of the Center’s achievements is a fully integrated cross- disciplinary graduate program for electrical and computer, mechanical and environmental, and chemical engineering fields. While at the University of Illinois at Urbana, he pioneered singular perturbation techniques, used today in power systems and adaptive controllers. At the University of California his group developed constructive nonlinear control methods and applied them, with colleagues from MIT, Caltech and United Technologies, to new jet engine designs. As a long-term industrial consultant, he has contributed to computer controls at Ford and to Petar V. Kokotovic power system stability at General Electric. With his 30 Ph.D. students and 20 postdoctoral researchers, Dr. Kokotovic has co-authored numerous papers (1934 – ) and 10 books. Residence – US Publications: 306, Citations: 9589 Citizenship – American Fields – Control theory Professor – Dep. Of Eng. at the Award Recipient UC, Santa Barbara, USA 2002 - Richard E. Bellman Control Heritage Award B.S. (1958), M.S. (1963) 1995 - IEEE Control Systems Award U. of Belgrade 1993 - George S. Axelby Outstanding Paper Award Faculty of Electrical Eng. 1991 - Hendrik W. Bode Lecture Prize Ph.D. (1965) 1984 - George S. Axelby Outstanding Paper Award USSR Academy of Sciences (Institute of Automation Hassan K. Khalil and Control), Moscow Michigan State University
- 3. The Richard E. Bellman Control Heritage Award is an annual award (since 1979) given by the American Automatic Control Council (AACC) for (Cont’) achievements in control theory, named after the 1991: John G. Truxal applied mathematician Richard E. Bellman. The 1992: Rutherford Aris award is given for "distinguished career 1993: Eliahu I. Jury contributions to the theory or applications 1994: Jose B. Cruz, Jr. of automatic control", and it is the "highest 1995: Michael Athans recognition of professional achievement for 1996: Elmer G. Gilbert U.S. control systems engineers and scientists". 1997: Rudolf E. Kalman The following people received 1998: Lotfi Asker Zadeh the AACC Richard E. Bellman Control Heritage Award: 1999: Yu-Chi Ho 2000: W. Harmon Ray 1979: Hendrik Wade Bode 2001: A.V. Balakrishnan 1980: Nathaniel B. Nichols 2002: Petar V. Kokotovic 1981: Charles Stark Draper 2003: Kumpati S. Narendra 1982: Irving Lefkowitz 2004: Harold J. Kushner 1983: John V. Breakwell 2005: Gene F. Franklin 1984: Richard E. Bellman 2006: Tamer Basar 1985: Harold Chestnut 2007: Sanjoy K. Mitter 1986: John Zaborszky 2008: Pravin Varaiya 1987: John C. Lozier 2009: George Leitmann 1988: Walter R. Evans 2010: Dragoslav D. Šiljak 1989: Roger W. Brockett 2011: Manfred Morari 1990: Arthur E. Bryson, Jr. 2012: Art Krener
- 4. The IEEE Control Systems Award is a technical field award given to an individual by the Institute of Electrical and Electronics Engineers(IEEE) "for outstanding contributions to control systems engineering, The following people received science or technology". It is an IEEE-level award, created the IEEE Control Systems Award: in 1980 by the Board of Directors of the IEEE, but sponsored by the IEEE Control Systems Society. 1992: Harold J. Kushner Originally the name was IEEE Control Systems 1993: Moshe M. Zakai Science and Engineering Award, but after 1991 the IEEE 1994: Elmer G. Gilbert changed it to IEEE Control Systems Award. 1995: Petar V. Kokotovic Recipients of this award receive a bronze $10,000, 1996: Vladimir A. Yakubovich Bronze Medal, Certificate in IEEE Folder, and travel 1997: Brian D. O. Anderson expenses to the CDC (round trip restricted coach air fare, 1998: Jan C. Willems conference registration, and four conference-rate hotel 1999: A. Stephen Morse nights), and a honorarium. 2000: Sanjoy K. Mitter The following people received 2001: Keith Glover the IEEE Control Systems Science and Engineering Award: 2002: Pravin Varaiya 2003: N. N. Krasovski 1982: Howard H. Rosenbrock 2004: John C. Doyle 1983: No award 2005: Manfred Morari 1984: Arthur E. Bryson, Jr. 2006: P. R. Kumar 1985: George Zames 2007: Lennart Ljung 1986: Charles A. Desoer 2008: Mathukumalli Vidyasagar 1987: Walter Murray Wonham 2009: David Q. Mayne 1988: Dante C. Youla 2010: Graham Clifford Goodwin 1989: Yu-Chi Ho 2011: Eduardo D. Sontag 1990: Karl Johan Åström 2012: Alberto Isidori 1991: Roger W. Brockett 2013: Stephen P. Boyd
- 5. Hector J. Sussmann was born in 1946 in Buenos Aires, Argentina. He was a research fellow in Decision and Control at Harvard University (1970-71), and then taught at the University of Chicago (1971-72). Since 1972 he has been with Rutgers University, in New Brunswick, NJ, where he is presently a Professor of Mathematics. He has given invited addresses at several national and international conferences, such as the 1978 International Congress of Mathematics (Helsinki), the 1989 MTNS (Amsterdam), the 1989 NOLCOS (Capri, Italy), the 1991 winter meeting of the American Mathematical Society (San Francisco), and the 1992 25th anniversary meeting of the INRIA (Paris). He is the author of more than 140 research papers in nonlinear and optimal control, and mathematical problems of robotics and neural Hector J. Sussmann networks. He is a Fellow of the IEEE, was a member of the Council of the American Mathematical Society (1981-1984), (1946 – ) and has served in the editorial boards of several mathematics journals. He chaired the 1992-1993 Control Residence – US Theory Year at the Institute for Mathematics and its Fields – Control theory Applications at the University of Minnesota. Professor – Dep. of Mathematic Hill Center-Busch Campus Rutger, the State U. of New Jersey, USA Publications: 173, Citations: 3763 M.S. (1966) – Mathematics U. of University of Buenos Aires Ph.D. (1965) – Mathematics New York University
- 9. “ Peaking phenomenon (state space) Normal Form”
- 10. Consider the system in Isidori’s Normal Form (NF) The explicit solution is internal dynamics (nonlinear) external dynamics (linear) *** Notice that the nonlinearity is dangerous even when multiplied by *** Alberto Isidori (1942- ) - An Italian control theorist - A Professor of Automatic The Stable Region can be extended by using Control at the U. of Rome instead of . - An Affiliate Professor of Electrical & Systems Engineering at Washingto U. in St Louis First edition in 1985 ***This high-gain idea achieves Semiglobal***
- 11. Now let’s we consider a cascade of a double integrator Note: linear subsystem and a scalar nonlinear subsystem: where partial-state feedback law It easy to see that For when increase a, and the escape of x(t) the response And the “peak” of to +happens becomes unbounded is at sooner. *** is peaking linearly with a*** 2
- 12. Note: It can be semiglobal stability semiglobally stabilized can’t be achieved What is the effect of this linear control on the nonlinearities? Will the dangerous term be negligible? Can global stabilization also be achieved?
- 13. The BB-Syndrome This model disregards a “jumping ball” Two nonlinear terms A chain of four integrators If you notice that the ball can be stabilized Fear of Peaking only through . If we wanna place two poles at For , our “control” is We‘ll see in later that for some initial conditions weaker than the centrifugal force . on the unit sphere, the necessarily reaches The term represents a strong positive peak values of order . feedback, which combined with the peaking of , will lead to instability and make What is the effect of this linear control the ball fly off the beam. on the nonlinearities? Will the dangerous term be negligible? Can global stabilization also be achieved? Reference: Kokotovic, P.V., The Joy of Feedback: Nonlinear and Adaptive, 1991 Bode prize lecture, IEEE Control Magazine, June 1992.
- 14. A general cascade system A general cascade system with a disturbance term “The interconnection term acts as disturbance which must be driven to zero without destroying the GAS property of the z-subsystem” Full-state feedback law - Backstepping We called the - Forwarding “system ” Partial-state feedback law - Pole placement - Two-time scale design
- 15. Assumption 1 (Subsystem stability/stabilizability) minimumphase Assumption 2 (Local exponential stabilizability of the -subsystem) Assumption 3 (Interconnection growth restriction) (linear growth in z) Theorem 4 (Global stabilization with partial-state feedback) 2 Proposition 5 (Polynomial W(z)) Positive semidefinite and radially unbounded polynomial W(z) satisfies the growth condition (v) of Theorem 4
- 16. Definition (Nonpeaking systems) If for each and there exists a bounded input such that and the output satisfies where is called the peaking exponent. (####) The peaking phenomenon occurs if the growth where the constants and do not of as a function of is polynomial. depend on . In all other cases, is a peaking system. If : Nonpeaking If : Peaking “We will design stabilizing control laws which satisfy that nonpeaking output condition”
- 17. Proposition (Peaking of output derivative) (****) (****) (####). (****)
- 18. Peaking Nonpeaking design Linear partial state feedback: Linear partial state feedback: (Two-time-scale design) The explicit solution of the z-subsystem: The solution is Hence, is nonpeaking system. The solution is A transient can be made as short by increasing , and its peak is at . is reduced to . Hence The state remains bounded for arbitrary large For any and large enough, this implies which means that does not exist!! For given a set of i.c., we can always select large that is, escapes to before . enough to make exist. Shrinking of region of attraction:
- 19. ($$$$) Theorem (Nonpeaking cascade) 1 ($$$$) ($$$$) Theorem (Lack of semiglobal stabilizability) 1 ($$$$)
- 20. The simplest cascades: the linear -subsystem is controllable and the z-subsystem is GAS - Even in these cascades, the peaking phenomenon in the -subsystem can destabilize the z –subsystem Every nonminimum phase system is peaking - Its output cannot be rapidly regulated to zero without first reaching a high peak which is determined by the unstable modes of the zero dynamics. Peaking is a structural obstacle to global and semiglobal stabilization in both partial and full-state feedback designs - It is not an obstacle to local stabilization, peaking causes the region of attraction to shrink as the feedback gain increases Global stabilization can be achieved with partial-state feedback if the stability properties of the z-subsystem are guaranteed by either polynomial