We say that a belongs to class K, if

1. α(0)=0

2. α(x)>0, for all x>0

3. α is nondecreasing, i.e. α(x1)α(x2) for all x1>x2.

䊏 2.7 Lyapunov Stability Theorems

class K function because (1) fails. On the other hand, α(x)=-x K function because (2) and (3) fail.

䊏

DEFINITION 2.7–2 In the following, ℜ⁺=[0,∞).

1. Locally Positive Definite: A continuous function V:ℜ⁺×ℜⁿ→R is locally positive definite (l.p.d) if there exists a class K function a(.) and a neighborhood N of the origin of ℜⁿ such that

V(t, x)α(||x||) for all t0, and all x∈N.

2. Positive Definite: The function V is said to be positive definite (p.d) if N=ℜⁿ.

3. Negative and Local Negative Definite: We say that V is (locally) negative definite (n.d) if -V is (locally) positive definite.

䊏

EXAMPLE 2.7–2: Locally Positive Definite Functions

[Vidyasagar 1992] The function is l.p.d but not p.d, since V(t, x)=0 at x=(0, π/2). On the other hand,

1. Locally Decrescent: A continuous function is locally decrescent if There exists a class K function ß(.) and a neighborhood N of the origin of such that

V(t, x)ß(||x||) for all t0 and all x∈Ν.

69 2. Decrescent: We say that V is decrescent if N=ℜⁿ.

䊏

EXAMPLE 2.7–3: Decrescent Functions

[Vidyasagar 1992] The function is locally but

not globally decrescent. On the other hand, is globally decrescent.

䊏

DEFINITION 2.7–4 Given a continuously differentiate function V: ℜ⁺×ℜⁿ→ R together with a system of differential equations (2.7.1), the derivative of V along (2.7.1) is defined as a function V: ℜ⁺×ℜⁿ→R given by

䊏

EXAMPLE 2.7–4: Lyapunov Functions

Consider the function of Example 2.7.3 and assume given a system

Then, the derivative of V(t, x) along this system is

䊏

Lyapunov Theorems

We are now ready to state Lyapunov Theorems, which we group in Theorem 2.7.1. For the proof, see [Khalil 2001], [Vidyasagar 1992].

THEOREM 2.7–1: Lyapunov

Given the nonlinear system 2.7 Lyapunov Stability Theorems

Then

1. Stability: The origin is stable in the sense of Lyapunov, if for x∈Ν, there exists a scalar function V(t, x) with continuous partial derivative such that

(a) V(t, x) is positive definite (b) V is negative semi-definite

2. Uniform Stability: The origin is uniformly stable if in addition to (a) and (b) V(t, x) is decrescent for x∈Ν.

3. Asymptotic Stability: The origin is asymptotically stable if V(t, x) satisfies (a) and is negative definite for x∈Ν.

4. Global Asymptotic Stability: The origin is globally, asymptotically stable if V(t, x) verifies (a), and V(t, x) is negative definite for all x∈ℜⁿ i.e. if N=ℜⁿ.

5. Uniform Asymptotic Stability: The origin is UAS if V(t, x) satisfies (a), V(t, x) is decrescent, and V(t,x) is negative definite for x∈Ν.

6. Global Uniform Asymptotic Stability: The origin is GUAS if N=ℜⁿ,and if V(t, x) satisfies (a), V(t,x) is decrescent, V(t,x) is negative definite and V(t, x) is radially unbounded, i.e. if it goes to infinity uniformly in time as ||x||→∞.

7. Exponential Stability: The origin is exponentially stable if there exists positive constants α, ß, γ such that

8. Global Exponential Stability: The origin is globally exponential stable if it is exponentially stable for all x∈ℜⁿ.

䊏 The function V(t, x) in the theorem is called a Lyapunov function. Note that the theorem provides sufficient conditions for the stability of the origin and that the inability to provide a Lyapunov function candidate has no indication on the stability of the origin for a particular system.

71 EXAMPLE 2.7–5: Stability via Lyapunov Functions

1. Consider the system described by

and choose a Lyapunov function candidate

Then the origin may be shown to be a stable equilibrium point.

2. Consider the Mathieu equation described in Example 2.6.3. Let the Lyapunov function candidate be given by

The origin is then shown to be a US equilibrium point.

3. The system given in Example 2.6.3–5, has a GES equilibrium point at the origin. This may be shown by considering a Lyapunov function candidate

V(x)=x² which leads to

Then,

0.5x²V(x)2x² and

The above inequalities hold for any x∈ℜⁿ. 2.7 Lyapunov Stability Theorems

and pick a Lyapunov function candidate

so that

so that the origin is SL.

䊏 Lyapunov Theorems may be used to design controllers that will stabilize a nonlinear system such as a robot. In fact, if one chooses a Lyapunov function candidate V(t, x), then finding its total derivative V(t, x) will exhibit an explicit dependence on the control signal. By choosing the control signal to make V(t, x) negative definite, stability of the closed-loop system is guaranteed. Unfortunately, it is not always easy to guarantee the global asymptotic stability of an equilibrium point using Lyapunov Theorem. This is due to the fact, that V(t, x) may be shown to be negative but not necessarily negative-definite. If the open-loop system were autonomous, Lyapunov theory is greatly simplified as shown in the next section.

The Autonomous Case

Suppose the open-loop system is not autonomous, i.e. is not explicitly dependent on t, then a time-independent Lyapunov function candidate V(x) may be obtained and the positive definite conditions are greatly simplified as described next.

LEMMA 2.7–1: A time invariant continuous function V(x) is positive definite if V(0)=0 and V(x)>0 for x≠0. It is locally positive definite if the above holds

in a neighborhood of the origin 䊏

Note that the condition that V(0)=0 is not necessary and that as long as V(0) is bounded above the Lyapunov results hold without modification.

73 With the above simplification, the Lyapunov results hold except that no distinction is made between uniform and regular stability results.

EXAMPLE 2.7–6: Uniform Stability via Lyapunov Functions Consider again the damped pendulum described by

and obtain a state-space by choosing x1=θ and then a Lyapunov function candidate is

so that and the origin is SL and actually USL.

䊏

EXAMPLE 2.7–7: Uniform Stability via Lyapunov Functions

This example illustrates the local asymptotic, uniform stability of the origin for the system

(1)

Choose then

which is strictly less than zero for all

䊏 Sometimes, and although V(x) is only non-positive, LaSalle’s theorem [LaSale and Lefschetz 1961], [Khalil 2001] may be used to guarantee the global asymptotic stability of the equilibrium point as described in the next theorem.

2.7 Lyapunov Stability Theorems

and let the origin be an equilibrium point. Then,

1. Asymptotic Stability: Suppose a Lyapunov function V(x) has been found such that for , V(x)>0 and Then the origin is asymptotically stable if and only if only at x=0.

2. Global Asymptotic Stability: The origin is GAS if above and V(x) is radially unbounded.

䊏 Unfortunately, in many applications with time-varying trajectories, the open-loop systems are not autonomous, and more advanced results such as the ones described later will be called upon to show global asymptotic stability.

EXAMPLE 2.7–8: LaSalle Theorem Consider the autonomous system

The origin is an equilibrium point. Moreover, consider a Lyapunov function candidate

Leading to

Since for all x1=x2, we need to check whether the origin is the only point where It can be seen from the state equation that x1=x2 can only happen at the origin, therefore the origin is GAS.

䊏

75 Note that LaSalle’s theorem is actually more general than we have described. In fact, it can be used to ascertain the stability of sets rather than just an equilibrium point. The basic idea is that since V(x) is lower bounded V(x)>c, then the derivative V has to gradually vanish, and that the trajectory is eventually confined to the set where V=0. The following definitions is useful in explaining the more general LaSalle’s theorem.

DEFINITION 2.7–5 A set G is said to be an invariant set of a dynamical