Tutorial on Zames-Falb multipliers
J. Carrasco1 , M. C. Turner2 and W. P. Heath3
Multiplier theory is a well-established technique for studying input-output stability of so-called Lur’e
systems: feedback interconnections of linear time-invariant elements and static, possibly time-varying
nonlinearities. When that nonlinearity is slope-restricted, the most appropriate class of multiplier for
studying stability (in the sense that the least conservative estimates will be provided) is that of the
Zames-Falb multiplier [17, 7]. Zames-Falb multipliers have received steady attention in the literature,
not least due to their application to systems with saturation, including anti-windup [9]. They are
also the multiplier of choice used to introduce integral quadratic constraint theory [11]. Despite their
long history in control theory, their use in mainstream control has been rather limited due to the
difficulties in finding a suitable multiplier within the Zames-Falb class for a given problem. This has
generated considerable recent interest in search methods for Zames-Falb multipliers, both from the
current authors [14, 13, 2, 15, 3, 4] and from others [5, 16].
In this special session we will provide a tutorial on both the underlying theory of Zames-Falb
multipliers and their use in modern control analysis and synthesis. Our treatment will be sufficiently
introductory to be accessible to PhD students. In particular we only consider continuous single-input
single-output systems. Nevertheless we will include up-to-date research questions and the tutorial
should be of interest and use to all researchers in this and related fields.
Topics covered will include:
1. Multiplier theory
• Motivating anti-windup example.
• Passivity and absolute stability.
• Derivation of classic results (e.g. circle criterion) as L2 -stability results.
2. Development of Zames-Falb multipliers
• Early history (O’Shea’s contribution; the original paper of Zames and Falb; the classic
books).
• Integral quadratic constraints.
3. Fundamental properties and limitations.
• Positivity.
• The Kalman conjecture.
• Megretski’s result on L1 norm restrictions [10].
• Phase limitations of first order multipliers.
4. Searches for SISO continuous-time Zames-Falb multipliers.
• Chen and Wen’s method [6].
• The method of deltas [8, 5].
• Searches for causal multipliers [14].
• Searches for anti-causal multipliers [3].
1
Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Sackville St
Building, Manchester M13 9PL. Email: [email protected]
2
Control Systems Research Group, Department of Engineering, University of Leicester, Leicester LE1 7RH. Email:
[email protected]
3
Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Sackville St
Building, Manchester M13 9PL. Email: [email protected] (corresponding author)
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5. Phase equivalence theory. Recent analysis [4] has shown that other classes of multiplier in the
literature are formally equivalent to subclasses of Zames-Falb multiplier. We will discuss both
the consequences and the following classes:
• The classical Popov criterion.
• Park’s criterion [12].
• Various extensions to the Zames-Falb class [13, 1].
6. Applications and open questions. In particular the challenge of control synthesis for nonlinear
systems with L2 specifications.
References
[1] D. Altshuller. Delay-integral-quadratic constraints and stability multipliers for systems with MIMO nonlinearities. IEEE Transactions on Automatic Control, 56(4), 738–747, 2011.
[2] J. Carrasco, W. P. Heath, G. Li and A. Lanzon. Comments on ‘on the existence of stable, causal multipliers for systems with slope-restricted nonlinearities’. IEEE Transactions on Automatic Control, 57(9),
2422–2428, 2012.
[3] J. Carrasco, M. Maya-Gonzalez, A. Lanzon and W. P. Heath. LMI search for rational anticausal Zames–Falb
multipliers. IEEE Control and Decision Conference, Maui, 2012.
[4] J. Carrasco, W. P. Heath and A. Lanzon. Equivalence between classes of multipliers for slope-restricted
nonlinearities. Automatica, Vol 49, pp1732-1740, 2013.
[5] M. Chang, R. Mancera and M. Safonov. Computation of Zames–Falb multipliers revisited. IEEE Transactions on Automatic Control, 57(4), 1024–1029, 2011.
[6] X. Chen and J. T. Wen. Robustness analysis for linear time-invariant systems with structured incrementally
sector bounded feedback nonlinearities. Journal of Applied Mathematics and Computer Science, 6(4),
pp625-648, 1996
[7] C. A. Desoer and M. Vidyasagar. Feedback systems: input–output properties. Academic Press, Inc., 1975.
[8] P. Gapski and J. Geromel. A convex approach to the absolute stability problem. IEEE Transactions on
Automatic Control, vol. 39, no. 9, pp. 1929 –1932, 1994.
[9] M. V. Kothare and M. Morari. Multiplier theory for stability analysis of antiwindup control systems.
Automatica, 35(5), 917–928, 1999.
[10] A. Megretski. Combining L1 and L2 Methods in the Robust Stability and Performance Analysis of Nonlinear
Systems. IEEE CDC, New Orleans, 1995.
[11] A. Megretski and A. Rantzer. System analysis via integral quadratic constraints. IEEE Transactions on
Automatic Control, 42(6), 819–830, 1997.
[12] P. Park. Stability criteria of sector- and slope-restricted Lur’e systems. IEEE Transactions on Automatic
Control, 47(2), 308–313, 2002.
[13] M. C. Turner and M. L, Kerr. L2 gain bounds for systems with sector bounded and slope-restricted
nonlinearities. International Journal of Robust and Nonlinear Control, 22(13), 1505–1521, 2012.
[14] M.C Turner, M.L Kerr and I. Postlethwaite. On the existence of stable causal multipliers for systems with
slope restricted nonlinearities. IEEE Transactions on Automatic Control 54(11):2697-2702, 2009.
[15] M. C. Turner, M. L. Kerr and I. Postlethwaite. Authors’ reply to comments on ‘on the existence of
stable, causal multipliers for systems with slope-restricted nonlinearities’. IEEE Transactions on Automatic
Control, 57(9), 2428–2431, 2012.
[16] J. Veenman and C. W. Scherer. IQC-synthesis with general dynamic multipliers. IFAC world congress,
Milan, 2011.
[17] G. Zames and P. L. Falb. Stability conditions for systems with monotone and slope-restricted nonlinearities.
SIAM Journal on Control, 6(1), 89–108, 1968.
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