Page 105 -

86 Introduction to Control Theory

where φ is continuous in both arguments. Lure then stated the following
Absolute Stability problem which became known as Lure’s Problem: Suppose
the system described by the above equations is given where:
1. All eigenvalues of A have negative real parts or that A has one eigen-
value at the origin while the rest of them is in the open left-half plane
(OLHP), and
2. (A, b) is controllable, and
3. (c, A) is observable, and
4. The nonlinearity φ(.,.) satisfies

(a) φ(t,0)=0; t 0 and
(b)

Then, find conditions on the linear system (A, B, C, D) such that x=0 is a
GAS equilibrium point of the closed-loop system.
Note that sometimes when we know more about the nonlinearity φ, the
last condition above is replaced by the following:

where k 2 k 1 0 are constants. We then say that φ belongs to the sector
[k 1 , k 2 ].
The first attempt to solve this problem was made by Aizerman in what is
now known as Aizerman’s conjecture [Vidyasagar 1992] followed by the
efforts of Kalman [Vidyasagar 1992]. The correct solution however, was not
available until the
In order to present the correct results, we need to present the KY and
MKY lemmas.

The MKY Lemma

The following lemmas are versions of the Meyer-Kalman-Yakubovich
(MKY) lemma which appears in [Narendra and Taylor 1973], [Khalil 2001]
amongst other places, and will be useful in designing adaptive controllers
for robots.

LEMMA 2.9–1: Meyer-Kalman-Yakubovitch Let the system (2.9.3) with
-1
D=0 be controllable. Then the transfer function c(sI-A) b is SPR if and
only if

Copyright © 2004 by Marcel Dekker, Inc.