Peaking phenomenon

The Peaking Phenomenon and the GlobalStabilization of Nonlinear Systems

H. J. Sussmann, Senior Member, IEEE, and P. V. Kokotovic, Fellow, IEEE

IEEE Transactions on Automatic Control, Vol. 36, No. 4, April, 1991Cita

tions

: 195

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 power system stability at General Electric. With his 30 Ph.D. students and 20 postdoctoral researchers, Dr. Kokotovic has co-authored numerous papers and 10 books.

Publications: 306, Citations: 9589

Hassan K. KhalilMichigan State University

Two Giants in the Kingdom of Nonlinear Control

Award Recipient2002 - Richard E. Bellman Control Heritage Award1995 - IEEE Control Systems Award1993 - George S. Axelby Outstanding Paper Award1991 - Hendrik W. Bode Lecture Prize1984 - George S. Axelby Outstanding Paper Award

Petar V. Kokotovic(1934 – )

Residence – USCitizenship – AmericanFields – Control theoryProfessor – Dep. Of Eng. at the UC, Santa Barbara, USA B.S. (1958), M.S. (1963) U. of Belgrade Faculty of Electrical Eng.Ph.D. (1965) USSR Academy of Sciences (Institute of Automation and Control), Moscow

The Richard E. Bellman Control Heritage Award is an annual award (since 1979) given by the American Automatic Control Council (AACC) for achievements in control theory, named after the applied mathematician Richard E. Bellman. The award is given for "distinguished career contributions to the theory or applications of automatic control", and it is the "highest recognition of professional achievement for U.S. control systems engineers and scientists".

The following people received the AACC Richard E. Bellman Control Heritage Award:

1979: Hendrik Wade Bode1980: Nathaniel B. Nichols1981: Charles Stark Draper1982: Irving Lefkowitz1983: John V. Breakwell1984: Richard E. Bellman1985: Harold Chestnut1986: John Zaborszky1987: John C. Lozier1988: Walter R. Evans1989: Roger W. Brockett1990: Arthur E. Bryson, Jr.

(Cont’)1991: John G. Truxal1992: Rutherford Aris1993: Eliahu I. Jury1994: Jose B. Cruz, Jr.1995: Michael Athans1996: Elmer G. Gilbert1997: Rudolf E. Kalman1998: Lotfi Asker Zadeh1999: Yu-Chi Ho2000: W. Harmon Ray2001: A.V. Balakrishnan2002: Petar V. Kokotovic2003: Kumpati S. Narendra2004: Harold J. Kushner2005: Gene F. Franklin2006: Tamer Basar2007: Sanjoy K. Mitter2008: Pravin Varaiya2009: George Leitmann2010: Dragoslav D. Šiljak2011: Manfred Morari2012: Art Krener

The following people received the IEEE Control Systems Science and Engineering Award:

1982: Howard H. Rosenbrock1983: No award1984: Arthur E. Bryson, Jr.1985: George Zames1986: Charles A. Desoer1987: Walter Murray Wonham1988: Dante C. Youla1989: Yu-Chi Ho1990: Karl Johan Åström1991: Roger W. Brockett

The following people received the IEEE Control Systems Award:

1992: Harold J. Kushner1993: Moshe M. Zakai1994: Elmer G. Gilbert1995: Petar V. Kokotovic1996: Vladimir A. Yakubovich1997: Brian D. O. Anderson1998: Jan C. Willems1999: A. Stephen Morse2000: Sanjoy K. Mitter2001: Keith Glover2002: Pravin Varaiya2003: N. N. Krasovski2004: John C. Doyle2005: Manfred Morari2006: P. R. Kumar2007: Lennart Ljung2008: Mathukumalli Vidyasagar2009: David Q. Mayne2010: Graham Clifford Goodwin2011: Eduardo D. Sontag2012: Alberto Isidori2013: Stephen P. Boyd

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, science or technology". It is an IEEE-level award, created in 1980 by the Board of Directors of the IEEE, but sponsored by the IEEE Control Systems Society. Originally the name was IEEE Control Systems Science and Engineering Award, but after 1991 the IEEE changed it to IEEE Control Systems Award. Recipients of this award receive a bronze $10,000, Bronze Medal, Certificate in IEEE Folder, and travel expenses to the CDC (round trip restricted coach air fare, conference registration, and four conference-rate hotel nights), and a honorarium.

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 networks. He is a Fellow of the IEEE, was a member of the Council of the American Mathematical Society (1981-1984), and has served in the editorial boards of several mathematics journals. He chaired the 1992-1993 Control Theory Year at the Institute for Mathematics and its Applications at the University of Minnesota.

Publications: 173, Citations: 3763

Hector J. Sussmann(1946 – )

Residence – USFields – Control theoryProfessor – Dep. of Mathematic Hill Center-Busch Campus Rutger, the State U. of New Jersey, USAM.S. (1966) – Mathematics U. of University of Buenos AiresPh.D. (1965) – Mathematics New York University

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ของระบบทางกายภาพท��สินใจ Peaking phenomenon มุ่าจากการศึ�กษาระบบ

บนปร�ภ)มุ่�สิเติท (state space) ในร)ป Normal Form”

I. Introduction: What is Peaking Phenomenon?

Consider the system in Isidori’s Normal Form (NF) The explicit solution is

external dynamics (linear)

internal dynamics (nonlinear)

*** Notice thatthe nonlinearity is dangerouseven when multiplied by ***

***This high-gain idea achieves Semiglobal***

The Stable Region can be extended by using instead of .

First edition in 1985

Alberto Isidori (1942- )- An Italian control theorist- A Professor of Automatic Control at the U. of Rome - An Affiliate Professor of

Electrical & Systems Engineering at Washingto U. in St Louis

Now let’s we consider a cascade of a double integrator linear subsystem and a scalar nonlinear subsystem:

It easy to see that when increase a,the escape of x(t) to + happens sooner.

And the “peak” of is at

***2 is peaking linearly with a***

I. Introduction: What is Peaking Phenomenon?

Note:

For andthe response becomes unbounded

where

partial-state feedback law

I. Introduction: What is Peaking Phenomenon?

Note:

It can be semiglobally stabilized

semiglobal stability 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?

I. Introduction: A Physical Interpretation Point View

The BB-Syndrome

Reference:Kokotovic, P.V., The Joy of Feedback: Nonlinear and Adaptive, 1991 Bode prize lecture, IEEE Control Magazine, June 1992.

This model disregards a “jumping ball”

A chain of four integrators

Two nonlinear

terms

If you notice that the ball can be stabilized only through .

For , our “control” is weaker than the centrifugal force .

The term represents a strong positive feedback, which combined with the peaking of , will lead to instability and make the ball fly off the beam.

Fear of PeakingIf we wanna place two poles at We‘ll see in later that for some initial conditions

on the unit sphere, the necessarily reaches peak values of order .

What is the effect of this linear control on the nonlinearities?

Will the dangerous term be negligible?Can global stabilization also be achieved?

II. Cascade Systems

“The interconnection term acts as disturbance which must be driven to zero without destroying the GAS property of the z-subsystem”

A general cascade system

A general cascade system with a disturbance term

We called the “system ”

Full-state feedback law - Backstepping - Forwarding

Partial-state feedback law - Pole placement - Two-time scale design

III. Growth Restriction for Global Stabilization

Assumption 1 (Subsystem stability/stabilizability)

Assumption 3 (Interconnection growth restriction) (linear growth in z)

Assumption 2 (Local exponential stabilizability of the -subsystem)

Theorem 4 (Global stabilization with partial-state feedback)

Proposition 5 (Polynomial W(z)) Positive semidefinite and radially unbounded polynomial W(z) satisfies the growth condition (v) of Theorem 4

2

minimumphase

IV. Nonpeaking Systems

The peaking phenomenon occurs if the growthof as a function of is polynomial.

If : NonpeakingIf : Peaking

where is called the peaking exponent.

“We will design stabilizing control laws which satisfy that nonpeaking output condition”

Definition (Nonpeaking systems)

If for each and there exists a bounded input such thatand the output satisfies

where the constants and do not depend on . In all other cases, is a peaking system.

(####)

Proposition (Peaking of output derivative)

(****)

(****)

(****)

(####).

V. An Example

Peaking Nonpeaking design

Linear partial state feedback: Linear partial state feedback:

(Two-time-scale design)The explicit solution of the z-subsystem:

Hence, is nonpeaking system.A transient can be made as short by increasing , is reduced to .

The state remains bounded for arbitrary large

The solution is

and its peak is at .

Hence

For any and large enough, this implies which means that does not exist!!that is, escapes to before .

Shrinking of region of attraction:

The solution is

For given a set of i.c., we can always select large enough to make exist.

Theorem (Lack of semiglobal stabilizability)

($$$$)1

Theorem (Nonpeaking cascade)

($$$$)

1

($$$$)

($$$$)

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

VI. Conclusion