nonlinear systems: an introduction to lyapunov stability theory a. pascoal, jan. 2014

Nonlinear Systems: an Introduction to Lyapunov Stability Theory

A. Pascoal , Jan. 2014

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


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


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


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!


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?


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


)(

);(

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

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


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!


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 yEquilibrium point yeq: 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!


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!


Control ActionControl Action

0)0();( hyhu

Nonlinear Nonlinear

plantplant

yyuu

Static controlStatic control

lawlaw

0)0();( FxFdt

dxInvestigate if 0Investigate if 0

is attractive! is attractive!

0)0();();,( gxgyuxfdt

dx

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


1

0

22 )(

2

1)(

x

dkmxxV

)(1

)(1

21

2

2

1

xfm

xkm

x

dt

dxdt

dx

!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


)(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!

Lyapunov Theory Krazovskii-La Salle An application exampl (physically motivated control law)An application exampl (physically motivated control law)

vv-v|v|-v|v|

AUV moving in the water with speed AUV moving in the water with speed v v

under the action of the applied force u. under the action of the applied force u.

Objective: drive the position x of the AUV to x* (by proper Objective: drive the position x of the AUV to x* (by proper choice of u)choice of u)

x0 x*

Lyapunov Theory Krazovskii-La Salle An application exampleAn application example

Suggested control lawSuggested control law

Control law exhibits Proporcional + Derivative actionsControl law exhibits Proporcional + Derivative actions

The plant itself has a pure integrator (to drive the static error to 0)The plant itself has a pure integrator (to drive the static error to 0)

Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of theShow asymptotic stability of the

(equilibrium point of the) system (equilibrium point of the) system

Step 1. Start by re-writing the equations in terms of the variables that Step 1. Start by re-writing the equations in terms of the variables that must be driven to o.must be driven to o.

Objective: Objective:



Step 2. Prove global asymptotic stability of the originStep 2. Prove global asymptotic stability of the origin

is an equilibrium point of the system!is an equilibrium point of the system!

Seek inspiration from the spring-mass-dashpot systemSeek inspiration from the spring-mass-dashpot system


1

0

22 )(

2

1)(

x

dkmxxV )(yk

yy)(dt

dyf




V(x)>0!V(x)>0!

Using La Salle´s theorem it follows that the origin is Using La Salle´s theorem it follows that the origin is globally asymptotically stable (GAS) globally asymptotically stable (GAS)

nonlinear systems: an introduction to lyapunov stability theory a. pascoal, jan. 2014

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