1. An Introduction to Lyapunov Stability Theory
3-5 Julho Ciência 2017 , Lisbon
Nonlinear Systems:
An Introduction to Lyapunov Stability Theory
Antonio Pascoal
Modelação e Simulação,

2. Linear versus Non-Linear
Linear versus Nonlinear Control
Nonlinear Plant
Nonlinear control laws
+ Powerful robust stability analysis tools
+ Possible deep physical insight
- Need for stronger theoretical background
- Limited tools for performance analysis
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 Stability Analysis

3. Nonlinear control: key ideas
AUV speed control
Dynamics
Nonlinear Plant
Nonlinear Stability Analysis
Objective: generate T(t) so that
tracks the reference speed
Tracking error
Error Dynamics

4. Nonlinear control: key ideas
Error Dynamics
Nonlinear Stability Analysis
Nonlinear Control Law

5. Nonlinear control: key ideas
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.
New tools are needed: LYAPUNOV theory

6. Lyapunov theory of stability: a soft Intro
SIMPLE EXAMPLE
SIMPLE EXAMPLE
(free mass, subjected to a simple motion resisting force) v fv v
m/f
Nonlinear Stability Analysis t
v=0 is an equilibrium point; dv/dt=0 when v=0!
v=0 is attractive
(trajectories converge to 0) v

7. Lyapunov theory of stability: a soft Intro
How can one prove that the trajectories go to the equilibrium point
WITHOUT SOLVING the differential equation?
(energy function)
Nonlinear Stability Analysis v
V positive and bounded below by zero;
dV/dt negative implies convergence
of V to 0!

8. Lyapunov theory of stability: a soft Intro
2-D case
State vector
Nonlinear Stability Analysis
Q-positive definite

9. 2-D case Nonlinear Stability Analysis 1x2 2x1 1x1 x tends do 0!
V positive and bounded below by zero;
dV/dt negative implies convergence of V to 0!
x tends do 0!

10. Lyapunov theory of stability: a soft Intro
Shifting
Is the origin always the TRUE origin? mg y
y-measured from spring at rest
Nonlinear Stability Analysis
Equilibrium point yeq: dx/dt=0
Examine if yeq is “attractive”!
Examine the
ZERO eq. Point!

11. Lyapunov theory of stability: a soft Intro
Shifting
Is the origin always the TRUE origin?
Nonlinear Stability Analysis
xref(t) is a solution
Examine if xref(t) is “attractive”!
Examine the
ZERO eq. Point!

12. Lyapunov theory of stability: a soft Intro
Control Action u y
Nonlinear
plant
Nonlinear Stability Analysis
Static control
law
Investigate if 0
is attractive!

13. Lyapunov Stability Theory
Stability of the zero solution
The zero solution is STABLE if
Nonlinear Stability Analysis
x-space

14. Lyapunov Stability Theory
Attractiveness of the zero solution
The zero solution is locally ATTRACTIVE if
Nonlinear Stability Analysis
x-space

15. Lyapunov Stability Theory
The zero solution is locally
ASYMPTOTICALLY STABLE if
it is STABLE and ATTRACTIVE
Nonlinear Stability Analysis
One may have attractiveness but NOT
Stability!
(the two conditions are required for
Asymptotic Stability!)

16. Lyapunov Theory: a formal approach
Nonlinear Stability Analysis

17. Lyapunov Theory: a formal approach
Nonlinear Stability Analysis
(the two conditions are required for
Asymptotic Stability!)

18. Stability Analysis Nonlinear Stability Analysis
There are at least three ways of assessing the stability (of
an equilibrium point of a) system:
Nonlinear Stability Analysis
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)

19. Lyapunov Theory: a formal approach
Nonlinear Stability Analysis

20. Lyapunov Theory: a formal approach
Nonlinear Stability Analysis If
then the origin is globally asymptotically stable

21. Lyapunov Theory: a formal approach
What happens when
Is the situation hopeless?
No!
Nonlinear Stability Analysis
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

22. Example: Mass-Spring-Dashpot System
Nonlinear Stability: Examples

23. Example: Mass-Spring-Dashpot System
f(.), k(.) – 1st and 3rd quadrants
f(0)=k(0)=0 y
Nonlinear Stability: Examples
V(x)>0!

24. Example: Mass-Spring-Dashpot System
Nonlinear Stability: Examples
Trajectory leaves W unless x1=0!
M is the origin.
The origin is asymptotically stable! W
Examine dynamics here!

25. Example: AUV position control
AUV moving in the water with speed v
under the action of an applied force u. v
-v|v|
Nonlinear Stability: Examples x* x
Objective: drive the position x of the AUV to x* (by proper choice of u)

26. Example: AUV position control
Suggested control law
Nonlinear Stability: Examples
Control law exhibits Proporcional + Derivative actions
The plant itself has a pure integrator (to drive the static error to 0)

27. Example: AUV position control
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:
Nonlinear Stability: Examples

28. Example: AUV position control
is an equilibrium point of the system!
Nonlinear Stability: Examples
Step 2. Prove global asymptotic stability of the origin
Seek inspiration from the spring-mass-dashpot system y

29. Example: AUV position control
V(x)>0!
Nonlinear Stability: Examples
Using La Salle´s theorem it follows that the origin is globally asymptotically stable (GAS)

30. AUV Path Following Path to be followed AUV trajectoryψ
Path to be followed
Nonlinear Stability: Examples
AUV
Functional Specifications / Systems Theory

31. AUV Path Following Set the speed of the AUV equal to V>0 (constant)
AUV trajectory ψ
Set the speed of the AUV equal to V>0 (constant)
Nonlinear Stability: Examples
Recruit the heading angle
so that
the lateral distance to the
vertical path
will converge to 0!

32. AUV Path Following Plant Model I: Objective
AUV trajectory ψ
Plant Model I:
Nonlinear Stability: Examples
Objective

33. AUV Path Following Objective: reduce the error to o! Simplified
AUV trajectory ψ
Simplified
(linearized) model
Nonlinear Stability: Examples

34. AUV Path Following Objective: reduce the error to o!
Nonlinear Stability: Examples

35. AUV Path Following Objective: reduce the error to o!
1. Lyapunov function candidate
Nonlinear Stability: Examples
2. Check that V is positive definite
3. Check that dV(t)/dt is neg. def.
Because V(.) is radially unbounded, 0 is GAS!

36. An Introduction to Lyapunov Stability Theory
3-5 Julho Ciência 2017 , Lisbon
Nonlinear Systems:
An Introduction to Lyapunov Stability Theory
Antonio Pascoal
Modelação e Simulação,