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Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 M.G. Safonov University of Southern California Zames Falb Multipliers.

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Presentation on theme: "Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 M.G. Safonov University of Southern California Zames Falb Multipliers."— Presentation transcript:

1 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 M.G. Safonov University of Southern California [msafonov@usc.edu] Zames Falb Multipliers for MIMO Nonlinearities

2 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Background: Robustness Analysis Role of Zames-Falb Multiplier M(s) xy Stable if topological separation monotone Theorem (Conic-Sector/IQC) Zames-Falb M(s)

3 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Typical Applications: relay (& hysteresis) deadzone actuator/sensor saturation anti-windup compensator analysis

4 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Historical Highlights Zames-Falb Discovery (1967-1968): –includes small-gain, positivity, Popov, off-axis circle, RL/RC,… –ad hoc graphical criteria, no attempt to optimize No better multipliers exist ( Willems 1969 ) …long hiatus… Multiple SISO   (x)’s ( Safonov 1984 ) Optimal ZF multipliers SISO   (x)’s : –Safonov-Wyetzner, 1987; Gapski-Geromel 1994 –Chen-Wen 1996; Kothare-Morari 1999 Repeated SISO   (x)’s: D’Amato-Rotea-Jonsson-Megretski 2001 TODAY’s TALK: Kulkarni-Safonov –Generalizing Zames-Falb multipliers for MIMO nonlinearities

5 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Overview Zames-falb multipliers (1968) Multiplier Theory, IQCs Zames-Falb Claim (1968) A Counter-Example Valid MIMO Extensions Repeated Nonlinearity Results Concluding Remarks

6 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Monotone, TI, memoryless and norm-bounded Stable and LTI Investigated System

7 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Multipliers: causal and stable with finite gain Strongly positive MH & finite normed N  Stability Zames and Falb (1968) Transformed System

8 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Zames-Falb Multipliers

9 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Re Im Zames-Falb Multipliers (contd.)

10 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 < Stable if graphs are topologically separated Background: IQC Stability

11 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 IQC for   block-diagonal u y IQCs for diagonal 

12 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Some IQCs : Known Results

13 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Obtain component IQCs  i (j  Stack them together Optimize multipliers M i (j  ) Each  i (j  ) depends linearly on a multiplier M i (j  ) > 0 Convex LMI problem Multipliers - A Broad Overview

14 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Fact: It Does the Opposite ! The Zames-Falb Claim MIMO Generalization

15 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 A Counter-example

16 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 The Source of Trouble

17 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 The Correct MIMO Extension

18 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Zames-Falb Claim (1968) A Counter-Example Valid MIMO Extensions Main Results Concluding Remarks Overview

19 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Three Problems for MIMO Nonlinearities Problem 1: Find the subclass of ZF multipliers that does preserve positivity Problem 2: Find the subclass of MIMO nonlinearities for which ZF multipliers do preserve positivity Problem 3: Find the greatest superclass of ZF multipliers the preserves positivity of repeated SISO nonlinearities.

20 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Main Result 1

21 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Main Result 2

22 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 and Main Result 3 Problem 3 Find the greatest superclass of ZF multipliers the preserves positivity of repeated SISO nonlinearities. Solution Multiplier matrices with impulse response That is, multipliers for repeated nonlinearities are L 1 -norm diagonally dominant matrices

23 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 SISO Artificial neural networks … Haykin (1997) Communication networks Electric circuits, e.g. Chua’s circuit … Gibbens & Kelly (1999) MIMO Flexible structures: spacecraft, aircraft, drives Static electric fields: semiconductors, motors MEMS: microbots Instances of Monotone Nonlinearities

24 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 Zames-Falb multipliers M are positivity preserving for incrementally positive MIMO nonlinearities N if and only if either or For repeated SISO nonlinearities, we now have L 1 -norm diagonally dominant matrices –Ideal for repeated saturation & deadzone –Best anti-windup stability test More reliable robustness analysis for aircraft, spacecraft, networks, missiles,… Conclusions

25 Workshop in honor of J. Boyd Pearson Jr Rice University 9-10 March 2001 ftp://routh.usc.edu/pub/safonov/safo99f.pdf ftp://routh.usc.edu/pub/safonov/safo01e.pdf ftp://routh.usc.edu/pub/safonov/safo01b.pdf Q U E S T I O N S ?? routh.usc.edu

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