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intro to lyapunov
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Nonlinear Systems: an Introduction to Lyapunov Stability Theory
A. Pascoal , April 2013 (draft under revision)
Linear versus Nonlinear Control
Nonlinear Plant
u y
Linear based control lawsLinear based control laws
-- Lack of global Lack of global stability and stability and performance resultsperformance results
++ Good engineering Good engineering intuition for linear intuition for linear designs (local stability designs (local stability and performance)and performance)
-- Poor physical Poor physical intuitionintuition
Nonlinear control lawsNonlinear control laws
++ Powerful robust Powerful robust stability analysis toolsstability analysis tools
++ Possible deep Possible deep physical insightphysical insight
-- Need for stronger Need for stronger theoretical theoretical backgroundbackground-- Limited tools for Limited tools for performance analysisperformance analysis
Nonlinear Control: Key Ingredients
T
vAUV speed controlAUV speed control
DynamicsDynamics
Nonlinear Plant
T v
)(tvrObjective: Objective: generate T(t) so thatgenerate T(t) so that )(tv tracks the reference speed tracks the reference speed
Tracking errorTracking error vve r
Error DynamicsError Dynamics
Nonlinear Control: Key Ingredients
Error DynamicsError Dynamics
2)(fvvT
dt
dvm
dt
tdem r
TT
22)( fvvfvvkedt
dvm
dt
dvm
dt
dem r
Tr
TT
0 kedt
de 00 tktete );exp()()(
Nonlinear Control LawNonlinear Control Law
2)( fvvKedt
dvmT r
T
Nonlinear Control: Key Ingredients
00 tktete );exp()()(
Tracking error tends to Tracking error tends to
zero exponentially fast.zero exponentially fast.
Simple and elegant!Simple and elegant!
Catch: the nonlinear dynamics are known EXACTLY.Catch: the nonlinear dynamics are known EXACTLY.
Key idea: i) use “simple” concepts, ii) deal with Key idea: i) use “simple” concepts, ii) deal with robustness against parameter uncertainty.robustness against parameter uncertainty.
2)( fvvKedt
dvmT r
T
New tools are needed: LYAPUNOV theoryNew tools are needed: LYAPUNOV theory
Lyapunov theory of stability: a soft Intro
0 fvdt
dvm
(free mass, subjected to a simple motion resisting force)(free mass, subjected to a simple motion resisting force)
vvfvfv
vm
f
dt
dv
)()( 0
)0(tvetv
ttm
f
vv
m/fm/f
00 vv
tt
v=0 is an equilibrium point; dv/dt=0 when v=0!v=0 is an equilibrium point; dv/dt=0 when v=0!
v=0 is attractive v=0 is attractive
(trajectories (trajectories converge to 0)converge to 0)
SIMPLE EXAMPLESIMPLE EXAMPLE
Lyapunov theory of stability: a soft Intro
vvfvfv
00 vv
How can one prove that the trajectories go to the equilibrium point How can one prove that the trajectories go to the equilibrium point
WITHOUT SOLVING the differential equation?WITHOUT SOLVING the differential equation?
2
2
1)( mvvV
(energy function)(energy function)
0,0
;0,0)(
vV
vvV
0
)(.
))((
2
)(|
fvdt
dvmv
dt
dV
dt
tdv
v
V
dt
tvdV
tv
V positive and bounded below by zero;V positive and bounded below by zero;
dV/dt negative implies convergence dV/dt negative implies convergence
of V to 0!of V to 0!
Lyapunov theory of stability: a soft Intro
What are the BENEFITS of this seemingly strange approach to investigate What are the BENEFITS of this seemingly strange approach to investigate
convergence of the trajectories to an equilibrium point?convergence of the trajectories to an equilibrium point?
V positive and bounded below by zero;V positive and bounded below by zero;
dV/dt negative implies convergence of V to 0!dV/dt negative implies convergence of V to 0!
0)( vfdt
dvm
vvf(v)f(v)
f a general dissipative forcef a general dissipative force
vv00
Q-IQ-I
Q-IIIQ-III
e.g. v|v|e.g. v|v|
2
2
1)( mvvV
0)( vvfdt
dvmv
dt
dV
Very general form of nonlinear equation!Very general form of nonlinear equation!
vvfvfv
Lyapunov theory of stability: a soft Intro
)(
);(
2212
1121
xkxdt
dx
xkxdt
dx
)(2
1)( 2
22
1 xxxV
2
1
x
xx
State State vectorvector
0;2
1)( IQQxxxV T
Q-positive definiteQ-positive definite
)(xfdt
dx
2-D case2-D case
0,0
;0,0)(
vV
vvV
Lyapunov theory of stability: a soft Intro
2-D case2-D case
)(
);(
2212
1121
xkxdt
dx
xkxdt
dx
2
1
x
xx )(xf
dt
dx
ttxtxV )())((
)(2
1)( 2
22
1 xxxV
RtRtxRtxV 2)())((
dt
dx
x
V
dt
xdV T
)(
1x21x2 2x12x11x11x1
)(
)(,
)(
221
11221 xkx
xkxxx
dt
xdV
0)()()(
2221211121 xkxxxxkxxxdt
xdV
)(
)(,
)(
221
11221 xkx
xkxxx
dt
xdV
V positive and bounded below by zero;V positive and bounded below by zero;
dV/dt negative implies convergence of V to 0!dV/dt negative implies convergence of V to 0!x tends do 0!x tends do 0!
Lyapunov theory of stability: a soft Intro
ShiftingShifting
Is the origin always the TRUE origin?Is the origin always the TRUE origin?
2
2
)()(dt
ydmmg
dt
dyfyk
mgmg
)(yk
yy
)(dt
dyf
y-measured from spring at resty-measured from spring at rest
Examine if yeq is “attractive”!Examine if yeq is “attractive”!
eqxx
dt
d
dt
d
dt
dx
dt
dx eq
)()( GxF
dt
dx
dt
deq
Equilibrium point xEquilibrium point xeq: dx/dt=0: dx/dt=0 mgyk eq )(
0;
eqeq
yx
dt
dy
yx 0)();( eqxFxF
dt
dx
0)0()0( eqxFGExamine the Examine the
ZERO eq. Point!ZERO eq. Point!
Lyapunov theory of stability: a soft Intro
ShiftingShifting
Is the origin always the TRUE origin?Is the origin always the TRUE origin?
Examine if xExamine if xrefref(t) is “attractive”!(t) is “attractive”!
refxx
dt
dxF
dt
d
dt
dx
dt
dx
ref
ref
)(
),()()()( tGxFxFxFdt
dx
dt
drefrefref
0))(()0)((),0( txFtxFtG refref
))(()(
));(()(
txFdt
tdxtxF
dt
tdxref
ref
xxrefref(t) is a solution(t) is a solution
Examine the Examine the
ZERO eq. Point!ZERO eq. Point!
Lyapunov theory of stability: a soft Intro
Control ActionControl Action 0)0();();,( gxgyuxfdt
dx
0)0();( hyhu
Nonlinear Nonlinear
plantplant
yyuu
Static controlStatic control
lawlaw
0)0()));((,( fxghxfdt
dx
0)0();( FxFdt
dxInvestigate if 0Investigate if 0
is attractive! is attractive!
Lyapunov Theory
Stability of the zero solutionStability of the zero solution
0)0(;)( fxfdt
dx
00x-spacex-space
The zero solution is STABLE if The zero solution is STABLE if
0);0()()0()(:0)(,0 ttBtxBtx o
Lyapunov Theory
0)0(;)( fxfdt
dx
00x-spacex-space
The zero solution is locally ATTRACTIVE ifThe zero solution is locally ATTRACTIVE if
0)(lim)0()(:0 to txBtx
Attractiveness of the zero solutionAttractiveness of the zero solution
Lyapunov Theory 0)0(;)( fxfdt
dx
The zero solution is locally The zero solution is locally
ASYMPTOTICALLY STABLE ifASYMPTOTICALLY STABLE if
it is STABLE and ATTRACTIVEit is STABLE and ATTRACTIVE
(the two conditions are required for(the two conditions are required for
Asymptotic Stability!)Asymptotic Stability!)
One may have attractiveness but NOT One may have attractiveness but NOT
Stability!Stability!
Key Ingredients for Nonlinear Control
Lyapunov Theory (a formal approach)Lyapunov Theory (a formal approach)
)1()(xfdt
dx
Lyapunov Theory
(the two conditions are required for(the two conditions are required for
Asymptotic Stability!)Asymptotic Stability!)
Lyapunov Theory
There are at least three ways of assessing the stability (of There are at least three ways of assessing the stability (of
an equilibrium point of a) system:an equilibrium point of a) system:
•Solve the differential equation (brute-force)Solve the differential equation (brute-force)
•Linearize the dynamics and examine the behaviourLinearize the dynamics and examine the behaviour
of the resulting linear system (local results for hyperbolic of the resulting linear system (local results for hyperbolic
eq. points only)eq. points only)
•Use Lypaunov´s direct method (elegant and powerful, Use Lypaunov´s direct method (elegant and powerful,
may yield global results)may yield global results)
Lyapunov Theory
Lyapunov Theory
If If
x as )(;0)(
xVdt
xdV
then the origin is then the origin is globally asymptotically globally asymptotically
stablestable
Lyapunov Theory
What happens whenWhat happens when ?0)(
dt
xdV
Is the situation hopeless? Is the situation hopeless? No!No!
0)(
::
;0)(
dt
xdVx
definedt
xdVLet
Suppose the only trajectory of the system Suppose the only trajectory of the system entirely contained in entirely contained in is the null trajectory. is the null trajectory. Then, the origin is asymptotically stable Then, the origin is asymptotically stable
(Let M be the largest invariant set (Let M be the largest invariant set contained in contained in . Then all solutions . Then all solutions converge to M. If M is the origin, the converge to M. If M is the origin, the results follows)results follows)
Krazovskii-LaSalleKrazovskii-LaSalle
2
2
)()(dt
ydm
dt
dyfyk
)(yk
yy)(dt
dyf
0
0; eqx
dt
dy
yx
0)0();( FxFdt
dx
Lyapunov Theory Krazovskii-La Salle
)(1
)(1
21
2
2
1
xfm
xkm
x
dt
dxdt
dx
EnergyPotentialEnergyKineticxV )(
1
0
22 )(
2
1)(
x
dkmxxV
)(yk
yy)(dt
dyf
Lyapunov Theory Krazovskii-La Salle
1
0
22 )(
2
1)(
x
dkmxxV
)(1
)(1
21
2
2
1
xfm
xkm
x
dt
dxdt
dx
dt
dxxk
dt
dxmx
dt
xdV 11
2
2
)()(
!0)()())(1
)(1
( 2221212 xxfxxkxfm
xkm
mx
f(.), k(.) – 1st and 3rd quadrantsf(.), k(.) – 1st and 3rd quadrants
f(0)=k(0)=0f(0)=k(0)=0
V(x)>0!V(x)>0!
)(yk
yy)(dt
dyf
Lyapunov Theory Krazovskii-La Salle
)(1
)(1
21
2
2
1
xfm
xkm
x
dt
dxdt
dx
!0)()(
22 xxfdt
xdV
2x
1x
!00 2 xfordt
dV
Examine dynamics here!Examine dynamics here!
)(1
0
12
1
xkm
dt
dxdt
dx
Trajectory Trajectory leaves leaves
unless xunless x11=0!=0!
M is the origin.M is the origin.
The origin is The origin is asymptotically asymptotically
stable!stable!
Nonlinear Systems: an Introduction to Lyapunov Stability Theory
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