A complete and convex search for
discrete-time noncausal FIR ZamesFalb multipliers
Student: Shuai Wang
Supervisor: William P. Heath
Co-supervisor: Joaquin Carrasco
The University of Manchester
UKACC PhD Presentation Showcase
Discrete-time Lur’e system
G
If an LTI plant G is in negative feedback with
an S[0, k] slope-restricted nonlinearity, then
stability is guaranteed if there is a multiplier
M such that



Re M  e j  1+kG  e j  > 0,   [0,2 ]
G
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Slide 2
Overview of results
Types of methods
Continuous methods
Discrete methods
Finite Impulse
Response
Safonov, 1987; Gapski, 1994;
Chang, 2012
FIR Zames-Falb
Basis functions
Chen & Wen,1995; Vee,2013
Restricted structure
rational multipliers
Turner,2009; Carrasco,2012;
Turner,2011
Ahmad,2013
Jury-Lee criteria &
Lyapunov approach
Khalil2002; Kothare,1999;
Carrasco,2013
Ahmad,2012
M ( z) 
nb

i  n f
mi z i
 FIR Zames-Falb, noncausal, convex search, covers both
slope restricted and odd slope restricted
 Remarkably efficient and improvement on existing literature
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Slide 3
Phase equivalence
Lemma
Amplitude
Given P  RH  , if there exists a Zames-Falb multiplier M such that
Re M ( z ) P( z )  0,  z  1
then there exists an FIR Zames-Falb multiplier MFIR such that
Re M FIR ( z ) P( z )  0,  z  1
x(0)
4
x(1)
Definition
(Willems,1968)
i
M ( z )   mi z
x(2)
i 4
M ( z) 

mz
i 
i
i
Amplitude
n
n
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Slide 4
Numerical results
z 4  1.5 z 3  0.5 z 2  0.5 z  0.5
G( z) 
4.4 z 5  8.957 z 4  9.893 z 3  5.671z 2  2.207 z  0.5
SLOPE RESTRICTION RESULTS BY USING
VARIOUS STABILITY CRITERIA
Circle Criterion
Tsypkin Criterion
Park & Kim (1998)
Haddad & Bernstein (1994)
Ahmad et. al. (2013)
Ahmad et. al. (2012)
Gonzaga et. al. (2012)
FIR Zames-Falb
1.53
1.69
1.69
1.56
1.83
2.59
1.53
3.82
0
1
2
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3
4
5
Slide 5
Computation time
G( z) 
6000
0.1z
z 2  1.8 z  0.81
T  1.7333e0.1 seconds
computation time
evaluated time
5000
time/second
4000
3000
2000
1000
0
0
10
20
30
40
50
60
n
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70
80
Slide 6
Conclusion and future work
 Phase-equivalence
 Discrete-time FIR Zames-Falb multipliers are phase-equivalent to the
class of discrete-time rational Zames-Falb multipliers
 A convex search for discrete-time Zames-Falb
multipliers with FIR structure
 KYP lemma derived for discrete-time noncausal transfer functions
 No source of conservatism
 Complete search and it is expected to be the best for slope-restricted
nonlinearities
 Future work
 MIMO extension
 Anti-windup synthesis
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Slide 7