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Material Type: Notes; Class: SYSTEMS ANALYSIS TECHNIQUES; Subject: Electrical & Comp. Sys. Engr.; University: Rensselaer Polytechnic Institute; Term: Fall 2004;
Typology: Study notes
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. (^) find equilibrium points.
define and give an example of asymptotic stability.
define and give an example of Lyapunov
.
.
. (^) define and give an example of global
definite. '
determine stability using Lyapunov's 2nd Method.
.
.
. (^) derive and use Lyapunov's Equation.
Ref: Sections 9.1-9.3, 9.6. Fall 2004
Equilibrium Points ~ f
A ball can be made to rest at A, E, F, G and anywhere between Band D:
Classify the Equilibrium Points
A andF are
E and G are
Cis
Time varying systems:
Asymptotic Stability
Def. An equilibrium point, xe, is asymptotically
stable if 1
. 1m II. el - t-> 00 II <1>(t,to' x(to), 0) - x 11 - 0
for any to and for all t > to'
If a trajectory reaches -xe,and no input is
ie(t) = (^0) for all t > to.
Find the equilibrium points for a pendulum with damping. (^) ... Set) + Set) + ksinS(t) = 0 Solution
Stability in the Sense of
Lyapunov
sense oJ Lyapunov (i.s.L.) iff for any E> 0 , ~ 8(E) 3
IIxO - xell < 8(E) ~ II (t;to' xO, 0) - xell < E
for any to and for all t > to.
Loosely speaking, if an initial state, xo, is near xe then the response to Xowill also stay near xe.
Lyapunov's 2ndMethod
The system i(t) = f(x(t)) is asymptotically stable if:3 V(x(t)) 3.
Note: V(x(t)) satisfies the definition of a positive definite function.
V(x (1)) satisfies the definition of a negative definite function.
If x(t) is 2 xl, explain asymptotic stability when
2 2 V(X (t» = Xl (t) + X2(t)
Explanation
Investigate the stability of the following
X2 (t) = -Xl (t) - ax2 (t) [X12 (t) + X22 (t) ]
Solution
Classify 2 2 a.) Xl +X2 + 1
b.) Xl +X
c.) sin2XI +sin2X 2 2 d.) -Xl -X 2 2 2 e.) Xl +2X2 +X3 +2xlx
x is 2x 1 x is 2x 1
x is 2x 1
x is 3x 1
x is 3xl
Answers
a.)
b.)
c.)
d.)
Table 9.8-1, page 478
IAI Rank andIndex CharacteristicValues A Form^ PrincipalLeading Minors Positive Definite Real quadraticform ~1'~2'... , ~fI
Hermitian form IAI>O r=p=fr^ Ai^ >^0
A =C7'C ICI>O A = C*7'C Iq> 0
Q = Yl1 + y,: +... + Yfl
Q (^) .-- z *z 1 1 + z *zIt. + ... +^ z *z" fI all positive
Positive. Semideflnite Real quadraticform. IAI = () Hermitian fortn
n - r roots r = p < n equalothers zero; positive
A = C7'C Iq = 0 A = CU'C ICI=O
Q = Yll+ YII + ... + yrl
all positive;
. Ar+1>'" , AfI-l non-negative Q - -11^ z *z^ +^ z *zII + ... +^ z *zrr
Negative Definite Real.quadratic form Hermitian form (-1)"^ IAI^ > () J'
r=1I p=O Ai<^0
A = -C7'C ICI>O A = -C*7'C ICI>O
Q = -Yll - Ytl-... - Ynl -A1'~I"'" (-:-l)"AfI Q - -^ -z *z 1 1 -^ z *%II -'"^ -^ z *zn n all positive
Negative Semidefinite Real quadraticform
Hermitian form IAI^ =^ ()
r<n p=O
n ,. r roots othersequal zero; negative
A = -CTC ICI=O A = -C*TC ICI=O
Q = -Yll- YII- ... - Yrl -AhAI".. , (-l)rAr all positive; Q = -zl*ZI - ZIzl - ... - zrzr (-l)r~r+1"'" (-l)"A"- non positive
i(t) + O.li(t) + x(t) = 0 or
0 1 i(t) = I Ix(t) -1 -0.
Lyapunov' s Equation
Let V(x(t)) = X(t)T Qx(t) where Q> 0, then,
V(x(t)) = d V(x(t)) - dt
matrix N :3Q 3
