Stability of Nonlinear Systems

The equilibria of a system can tell us a great deal about the stability of the system. For nonlinear systems, stability is a property of the equilibrium points, and to be stable is to converge to or stay equilibrium.

Definition 36

Lyapunov Stability essentially says that a finite deviation in the initial condition from equilibrium means the resulting trajectory of the system stay close to equilibrium. Notice that this definition is nearly identical to Theorem 10. That means stability of an equilibrium point is the same as saying the function which returns the solution to a system given its initial condition is continuous at the equilibrium point.

Definition 37

Definition 38

Attractive equilibria guarantee that trajectories beginning from initial conditions inside of a ball will converge to the equilibrium. However, attractivity does not imply stability since the trajectory could go arbitarily far from the equilibrium so long as it eventually returns.

Definition 39

Asymptotic stability fixes the problem of attractivity where trajectories could go far from the equilibrium, and it fixes the problem with stability where the trajectory may not converge to equilibrium. It means that trajectories starting in a ball around equilibrium will converge to equilibrium without leaving that ball. Because the constant for attractivity may depend on time, defining uniform asymptotic stability requires some modifications to the idea of attractivity.

Definition 40

Definition 41

Just as we can define stability, we can also define instability.

Definition 42

Lyapunov Functions

In order to prove different types of stability, we will construct functions which have particular properties around equilibrium points of the system. The properties of these functions will help determine what type of stable the equilibrium point is.

Definition 43

Definition 44

Definition 45

LPDF functions are locally “energy-like” in the sense that the equilibrium point is assigned the lowest “energy” value, and the larger the deviation from the equilibrium, the higher the value of the “energy”.

Definition 46

Definition 47

Descresence means that for a ball around the equilibrium, we can upper bound the the energy.

Quadratic Lyapunov Functions

Definition 48

A Quadratic Lypunov function is of the form

Theorem 14

Sum-of-Squares Lyapunov Functions

Definition 49

Theorem 15

A polynomial is SOS if and only if it can be written as

is SOS.

Proving Stability

Theorem 16

Theorem 17

Theorem 18

Theorem 19

Theorem 20

The results of Theorem 16, Theorem 17, Theorem 18, Theorem 19, Theorem 20 are summarized in Table 1.

Theorem 21 (LaSalle's Invariance Principle)

Indirect Method of Lyapunov

Definition 50

The state transition matrix is useful in determining properties of the system.

Theorem 22 (Lyapunov Lemma)

Theorem 23 (Tausskey Lemma)

The Lyapunov Lemma has extensions to the time-varying case.

Theorem 24 (Time-Varying Lyapunov Lemma)

It turns out that uniform asymptotic stability of the linearization of a system corresponds to uniform, asymptotic stability of the nonlinear system.

Theorem 25 (Indirect Theorem of Lyapunov)

Proving Instability

Theorem 26

Region of Attraction

For asymptotically stable and exponential stable equilibria, it makes sense to wonder which initial conditions will cause trajectories to converge to the equilibrium.

Definition 51

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