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 laws

-­‐‑  Lack  of  global  stability  and  performance  results

+  Good  engineering  intuition  for  linear  designs  (local  stability  and  performance)

-­‐‑  Poor  physical  intuition

Nonlinear control laws

+  Powerful  robust  stability  analysis  tools

+  Possible  deep  physical  insight

-­‐‑  Need  for  stronger    theoretical  background

-­‐‑  Limited  tools  for  performance  analysis

Nonlinear Control: Key Ingredients   T

T −βv − fv 2 =mTdvdt;mT =m +ma

vAUV speed control

Dynamics

Nonlinear  Plant

T v

)(tvrObjective: generate T(t) so that )(tv tracks the reference speed

Tracking error vve r −=

mTdedt

=mTdvrdt

−mTdvdtError Dynamics

Nonlinear Control: Key Ingredients  

mTdedt

=mTdvrdt

−mTdvdt

Error Dynamics

2)( fvvTdtdvm

dttdem r

TT ++−= β

22)( fvvfvvkedtdvm

dtdvm

dtdem r

Tr

TT ++⎥⎦

⎤⎢⎣

⎡ +++−= ββ

0=+ kedtde 00 ≥−= tktete );exp()()(

Nonlinear Control Law

2)( fvvKedtdvmT r

T +++= β

Nonlinear Control: Key Ingredients  

00 ≥−= tktete );exp()()(

Tracking error tends to

zero exponentially fast.

Simple and elegant!

Catch: the nonlinear dynamics are known EXACTLY.

Key idea: i) use “simple” concepts, ii) deal with robustness against parameter uncertainty.

2)( fvvKedtdvmT r

T +++= β

New tools are needed: LYAPUNOV theory

Lyapunov theory of stability: a soft Intro  

0=+ fvdtdvm

(free mass, subjected to a simple motion resisting force)

v fv

vmf

dtdv

−=

)()( 0

)0(tvetv

ttmf

−−=

v

m/f

0 v

t

v=0 is an equilibrium point; dv/dt=0 when v=0!

v=0 is attractive

(trajectories converge to 0)

SIMPLE EXAMPLE

Lyapunov theory of stability: a soft Intro   vfv

0 v

How can one prove that the trajectories go to the equilibrium point

WITHOUT SOLVING the differential equation?

2

21)( mvvV =

(energy function)

0,0;0,0)(

==

≠

vVvvV

0

)(.))((

2

)(|

fvdtdvmv

dtdV

dttdv

vV

dttvdV

tv

−==

→∂

∂=

V positive and bounded below by zero;

dV/dt negative implies convergence

of V to 0!

Lyapunov theory of stability: a soft Intro  

What are the BENEFITS of this seemingly strange approach to investigate

convergence of the trajectories to an equilibrium point?

V positive and bounded below by zero;

dV/dt negative implies convergence of V to 0!

0)( =+ vfdtdvm

v f(v)

f a general dissipative force

v 0

Q-I

Q-III

e.g. v|v|

2

21)( mvvV =

0)( vvfdtdvmv

dtdV

−==

Very general form of nonlinear equation!

vfv

Lyapunov theory of stability: a soft Intro  

)(

);(

2212

1121

xkxdtdx

xkxdtdx

−−=

−=

)(21)( 2

221 xxxV +=

⎥⎦

⎤⎢⎣

⎡=

2

1

xx

x

State vector

0;21)( IQQxxxV T ==

Q-positive definite

)(xfdtdx

=

2-D case

0,0;0,0)(

==

≠

vVvvV

Lyapunov theory of stability: a soft Intro   2-D case

)(

);(

2212

1121

xkxdtdx

xkxdtdx

−−=

−=⎥⎦

⎤⎢⎣

⎡=

2

1

xx

x )(xfdtdx

=

ttxtxV ⇐⇐ )())((

)(21)( 2

221 xxxV +=

RtRtxRtxV ∈⇐∈⇐∈ 2)())((

dtdx

xV

dtxdV T

∂

∂=)(

1x2 2x1 1x1

[ ] ⎥⎦

⎤⎢⎣

⎡

−−

−=

)()(

,)(

221

11221 xkx

xkxxx

dtxdV

0)()()(2221211121 xkxxxxkxxx

dtxdV

−−−=

[ ] ⎥⎦

⎤⎢⎣

⎡

−−

−=

)()(

,)(

221

11221 xkx

xkxxx

dtxdV

V positive and bounded below by zero;

dV/dt negative implies convergence of V to 0! x tends do 0!

Lyapunov theory of stability: a soft Intro   Shifting

Is the origin always the TRUE origin?

2

2

)()(dtydmmg

dtdyfyk =+−−

mg

)(yk

y

)(dtdyf

y-measured from spring at rest

Examine if yeq is “attractive”!

ζ+= eqxx

dtd

dtd

dtdx

dtdx eq ζζ

=+=

)()( ζζζ GxF

dtdx

dtd

eq =+==

Equilibrium point xeq: dx/dt=0 mgyk eq =)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= = 0

; eqeq

yx

dtdyy

x 0)();( == eqxFxFdtdx

0)0()0( =+= eqxFGExamine the

ZERO eq. Point!

Lyapunov theory of stability: a soft Intro   Shifting

Is the origin always the TRUE origin?

Examine if xref(t) is “attractive”!

ζ+= refxx

dtdxF

dtd

dtdx

dtdx

ref

ref

ζ

ζ

+

=+=

)(

),()()()( tGxFxFxFdtdx

dtd

refrefref ζζζ

=−+=−=

0))(()0)((),0( =−+= txFtxFtG refref

))(()(

));(()( txFdttdx

txFdttdx

refref ==

xref(t) is a solution

Examine the

ZERO eq. Point!

Lyapunov theory of stability: a soft Intro  

Control Action 0)0();();,( === gxgyuxfdtdx

0)0();( == hyhu

Nonlinear

plant

y u

Static control

law

0)0()));((,( == fxghxfdtdx

0)0();( == FxFdtdx

Investigate if 0

is attractive!

Lyapunov Theory  

ε

δ

Stability of the zero solution

0)0(;)( == fxfdtdx

0 x-space

The zero solution is STABLE if

0);0()()0()(:0)(,0 ttBtxBtx o ≥∈⇒∈>=∃>∀ εδεδδε

Lyapunov Theory  

α

0)0(;)( == fxfdtdx

0 x-space

The zero solution is locally ATTRACTIVE if

0)(lim)0()(:0 =⇒∈>∃ →∞to txBtx αα

Attractiveness of the zero solution

Lyapunov Theory   0)0(;)( == fxfdtdx

The zero solution is locally

ASYMPTOTICALLY STABLE if

it is STABLE and ATTRACTIVE

(the two conditions are required for

Asymptotic Stability!)

ε

δ

One may have attractiveness but NOT

Stability!

Key Ingredients for Nonlinear Control   Lyapunov Theory (a formal approach)

)1()(xfdtdx

=

Lyapunov Theory  

(the two conditions are required for

Asymptotic Stability!) ε

δ

Lyapunov Theory  

There are at least three ways of assessing the stability (of

an equilibrium point of a) system:

• Solve the differential equation (brute-force)

• Linearize the dynamics and examine the behaviour

of the resulting linear system (local results for hyperbolic

eq. points only)

• Use Lypaunov´s direct method (elegant and powerful,

may yield global results)

Lyapunov Theory  

Lyapunov Theory  

If

∞→∞→< x as )(;0)( xVdtxdV

then the origin is globally asymptotically

stable

Lyapunov Theory  

What happens when ?0)(≤

dtxdV

Is the situation hopeless? No!

⎭⎬⎫

⎩⎨⎧

==Ω

≤

0)(::

;0)(

dtxdVx

definedtxdVLet

Suppose the only trajectory of the system entirely contained in Ω is the null trajectory. Then, the origin is asymptotically stable

(Let M be the largest invariant set contained in Ω. Then all solutions converge to M. If M is the origin, the results follows)

Krazovskii-LaSalle

2

2

)()(dtydm

dtdyfyk =−−

)(yk

y )(dtdyf

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= = 0

0; eqx

dtdyy

x

0)0();( == FxFdtdx

Lyapunov Theory   Krazovskii-La Salle  

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

EnergyPotentialEnergyKineticxV +=)(

∫+=1

0

22 )(

21)(

x

dkmxxV ςς

)(yk

y )(dtdyf

Lyapunov Theory   Krazovskii-La Salle  

∫+=1

0

22 )(

21)(

x

dkmxxV ςς⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

=+=dtdxxk

dtdxmx

dtxdV 1

12

2)()(

!0)()())(1)(1( 2221212 ≤−=+−− xxfxxkxfm

xkm

mx

f(.), k(.) – 1st and 3rd quadrants

f(0)=k(0)=0

V(x)>0!

)(yk

y )(dtdyf

Lyapunov Theory   Krazovskii-La Salle  

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(1)(121

2

2

1

xfm

xkm

x

dtdxdtdx

!0)()(22 ≤− xxf

dtxdV

2x

1x

!00 2 == xfordtdV

Examine dynamics here!

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

)(10

12

1

xkmdt

dxdtdx

Trajectory leaves Ω

unless x1=0!

Ω M is the origin.

The origin is asymptotically

stable!

Lyapunov theory of stability: a soft Intro   What can we say about GLOBAL stability of ?

V positive and bounded below by zero;

dV/dt negative implies convergence of V to 0!

0)( =+ vfdtdvm

v f(v)

f a general dissipative force

v 0

Q-I

Q-III

e.g. v|v|

2

21)( mvvV =

0)( vvfdtdvmv

dtdV

−==

Very general form of nonlinear equation!

vfv

Nonlinear Systems: an Introduction to Lyapunov Stability Theory