Nonlinear Systems

A nonlinear dynamic system can usually be represented by a set of nonlinear differential equations in the form

It is directly applicable to feedback control systems. The reason is that the equation can represent the close-loop dynamics of a feedback control system, with the control input being a function of state and time , and therefore disappearing in the closed-loop dynamics.

If the plant dynamics is

and some control law has been selected

then the closed-loop dynamics is

A special class of nonlinear systems are linear systems. The dynamics of linear systems are of the form

where is an matrix.

Autonomous and Non-autonomous

Definition: The nonlinear system is said to be automomous if does depend explicitly on time, i.e., if the system’s state equation can be written

Otherwise, the system is called non-autonomous.

Equilibrium Points

Definition: A state is an equilibrium state(or equilibrium point) of the system if once is equal to , it remains equal to for all future time.

Stability and Instability

Definition: The equilibrium state is said to be stable if, for any , there exists , such that if , then for all . Otherwise, the equilibrium point is unstable.

Asymptotic Stability and Exponential Stability

Definition: An equilibrium point is asymptotically stable if it is stable, and if in addtion there exists some such that implies that as .

Definition: An equilibrium point is exponentially stable if there exist two strictly positive numbers and such that

in some ball around the origin.

In words, the equation means that the state vector of an exponentially stable system converge to the origin faster than an exponential function. The positive number is often called the rate of exponential convergence.

Local and Global Stability

Definition: If asymptotic(or exponential) stability holds for any initial states, the equilibrium point is said to be asymptotically(or exponentially) stable in the large. It is also called globally asymptoticly(or exponentially) stable.

Positive Definite Functions and Lyapunov Functions

Definition: A scalar continuous function is said to be locally positive definite if and, in a ball

If and the above property holds over the whole state space, then is said to be globally positive definite.

Definition: If, in a ball , the function is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system is negative semi-definite, i.e.,

then is said to be a Lyapunov function for the system .

Lyapunov Theorem for Local Stability

Theorem:(Local Stability) If, in a ball , there exists a scalar function with continuous first partial derivatives such that

  • is positive definite(locally in )
  • is negative semi-definite(locally in )

then the equilibrium point is stable. If, actually, the derivative is locally negative definite in , then the stability is asymptotic.

Lyapunov Theorem for Global Stability

Theorem:(Global Stability) Assume that there exists a scalar function of the state , with continuous first order derivatives such that

  • is positive definite
  • is negative definite
  • as .

Then the equilibrium at the origin is globally asympotically stable.

Invariant Set Theorems

Definite: A set is an invariant set for a dynamic system if every system trajectory which starts from a point in remains in for all future time.

Theorem:(Local Invariant Set Theorem) Consider an autonomous system of the form , with continuous, and let be a scalar function with continous first partial derivatives. Assume that

  • for some , the region defined by is bounded.
  • for all in .

Let be the set of all points within where , and be the largest invariant set in . Then, every solution originating in tends to as .

Theorem:(Global Invarient Set Theorem) Consider an autonomous system of the form , with continuous, and let be a scalar function with continous first partial derivatives. Assume that

  • as .
  • over the whole state space.

Let be the set of all points where , and be the largest invariant set in . Then, all solution globally asymptotically converge to as .

Krasovskii’s Method

Theorem:(Krasovskii) Consider the autonomous system defined by , with the equilibrium point of interest being the origin. Let denote the Jacobian matrix of the system, i.e.,

If the matrix is negative definite in a neighborhood , then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is

If is the entire space and, in addition, as , then the equilibrium point is globally asymptotically stable.

Barbalet’s Lemma

Lemma:(Barbalat) If the differential function has a finite limit as , and if is uniformly continuous, then as .

Lemma:(“Lyapunov-Like Lemma”) If a scalar function satisfies the following conditions

  • is lower bounded
  • is negative semi-definite
  • is uniformly continuous in time

then as .