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Nonlinear Systems: an Introduction to Lyapunov Stability Theory
Lisbon, IST, A. Pascoal , April 2016
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 ≻
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 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!
x1 k (x1 ) > 0, x1 > 0; x2 k (x2 ) > 0, x2 > 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 GxFdtdx
dtd
eq =+==
Equilibrium point yeq: 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
f (0,0) = 0
0)0();( == hyhu
Nonlinear
plant
y u
Static control
law
dxdt
= f (x ,h (g (x ))); f (0,0) = 0
0)0();( == FxFdtdx
Investigate if 0
is attractive!
0)0();();,( === gxgyuxfdtdx
Lyapunov Theory
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
δ
ε
Lyapunov Theory 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
α
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!)
One may have attractiveness but NOT
Stability! ε
δ
Key Ingredients for Nonlinear Control Lyapunov Theory (a formal approach)
)1()(xfdtdx =
Lyapunov Theory
(the two conditions are required for
Asymptotic Stability!) εδ
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 W is the null trajectory. Then, the origin is asymptotically stable
(Let M be the largest invariant set contained in W. 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 +=)(
V (x) = 12mx2
2 + k(ς )dς0
x1
∫
)(yk
y )(dtdyf
Lyapunov Theory Krazovskii-La Salle
V (x) = 12mx2
2 + k(ς )dς0
x1
∫⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)(1)(121
2
2
1
xfm
xkm
x
dtdxdtdx
dV (x )dt
=mx2
dx2dt
+ k (x1)dx1dt
=
!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 W unless
x1=0!
WM is the origin.
The origin is asymptotically
stable!
Lyapunov Theory Krazovskii-La Salle An application example (physically motivated control law)
dvdt
= −v |v |+u
dxdt
=v
v -v|v|
AUV moving in the water with speed v
under the action of the applied force u.
Objective: drive the position x of the AUV to x* (by proper choice of u)
x 0 x*
Lyapunov Theory Krazovskii-La Salle An application example
u = −k1(x − x*)− k2v ; k1, k2 > 0
Suggested control law
Control law exhibits Proporcional + Derivative actions
The plant itself has a pure integrator (to drive the static error to 0)
dvdt
= −v |v |+u
dxdt
=v
dvdt
= −v |v |+u dxdt
=vk1
k2
x * xv
Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the
(equilibrium point of the) system
Step 1. Start by re-writing the equations in terms of the variables that must be driven to o.
Objective:
v (t )→ 0e (t ) = x (t )− x *(t )→ 0
dvdt
= −v |v |+u
dedt
=dxdt
−dx *dt
=dxdt
=v
dvdt
= −v |v |−k1e − k2v
dedt
=v
u = −k1(x − x*)− k2v ; k1, k2 > 0
Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the
(equilibrium point of the) system
Step 2. Prove global asymptotic stability of the origin
[v ,e ]T = [0,0]T
is an equilibrium point of the system!
Seek inspiration from the spring-mass-dashpot system
dvdt
= −v |v |−k1e − k2v
dedt
=v
EnergyPotentialEnergyKineticxV +=)(
V (x) = 12mx2
2 + k(ς )dς0
x1
∫)(yk
y )(dtdyf
Lyapunov Theory Krazovskii-La Salle Show asymptotic stability of the
(equilibrium point of the) system
EnergyPotentialEnergyKineticxV +=)( V (x ) = 12v 2 + k1ς d ς
0
e
∫
V(x)>0!
dVdt
=∂V∂vdvdt
+∂V∂ededt
=v dvdt
+ k1ededt
=v[−v |v |−k1e − k2v ]+ k1ev
=v 2 |v |−k2v2 ≤ o!
Using La Salle´s theorem it follows that the origin is globally asymptotically stable (GAS)
Nonlinear Systems: an Introduction to Lyapunov Stability Theory
Lisbon, IST, A. Pascoal , April 2016