## 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 $x$ and time $t$, 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 $A(t)$ is an $n\times n$ matrix.

## Autonomous and Non-autonomous

Definition: The nonlinear system is said to be automomous if $f$ 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 $x^*$ is an equilibrium state(or equilibrium point) of the system if once $x(t)$ is equal to $x^*$, it remains equal to $x^*$ for all future time.

## Stability and Instability

Definition: The equilibrium state $x=0$ is said to be stable if, for any $R > 0$, there exists $r > 0$, such that if $\|x(0)\|< r$, then $\|x(t)< R\|$ for all $t \geq 0$. Otherwise, the equilibrium point is unstable.

## Asymptotic Stability and Exponential Stability

Definition: An equilibrium point $0$ is asymptotically stable if it is stable, and if in addtion there exists some $r > 0$ such that $\|x(0)\| < r$ implies that $x(t) \to 0$ as $t \to \infty$.

Definition: An equilibrium point $0$ is exponentially stable if there exist two strictly positive numbers $\alpha$ and $\lambda$ such that

in some ball $B_r$ 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 $\lambda$ 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 $V(x)$ is said to be locally positive definite if $V(0) = 0$ and, in a ball $B_{R_0}$

If $V(0) = 0$ and the above property holds over the whole state space, then $V(x)$ is said to be globally positive definite.

Definition: If, in a ball $B_{R_0}$, the function $V(x)$ is positive definite and has continuous partial derivatives, and if its time derivative along any state trajectory of system $\dot x = f(x)$ is negative semi-definite, i.e.,

then $V(x)$ is said to be a Lyapunov function for the system $\dot x = f(x)$.

## Lyapunov Theorem for Local Stability

Theorem:(Local Stability) If, in a ball $B_{R_0}$, there exists a scalar function $V(x)$ with continuous first partial derivatives such that

• $V(x)$ is positive definite(locally in $B_{R_0}$)
• $\dot V(x)$ is negative semi-definite(locally in $B_{R_0}$)

then the equilibrium point $0$ is stable. If, actually, the derivative $\dot V(x)$ is locally negative definite in $B_{R_0}$, then the stability is asymptotic.

## Lyapunov Theorem for Global Stability

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

• $V(x)$ is positive definite
• $\dot V(x)$ is negative definite
• $V(x) \to \infty\$ as $\|x\| \to \infty$.

Then the equilibrium at the origin is globally asympotically stable.

## Invariant Set Theorems

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

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

• for some $l>0$, the region $\Omega_l$ defined by $V(x) < l$ is bounded.
• $\dot V \leq 0$ for all $x$ in $\Omega_l$.

Let $R$ be the set of all points within $\Omega_l$ where $\dot V(x) = 0$, and $M$ be the largest invariant set in $R$. Then, every solution $x(t)$ originating in $\Omega_l$ tends to $M$ as $t \to \infty$.

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

• $V(x) \to \infty$ as $\|x\| \to \infty$.
• $\dot V(x) \leq 0$ over the whole state space.

Let $R$ be the set of all points where $\dot V(x) = 0$, and $M$ be the largest invariant set in $R$. Then, all solution globally asymptotically converge to $M$ as $t \to \infty$.

## Krasovskii’s Method

Theorem:(Krasovskii) Consider the autonomous system defined by $\dot x = f(x)$, with the equilibrium point of interest being the origin. Let $A(x)$ denote the Jacobian matrix of the system, i.e.,

If the matrix $F = A + A^T$ is negative definite in a neighborhood $\Omega$, then the equilibrium point at the origin is asymptotically stable. A Lyapunov function for this system is

If $\Omega$ is the entire space and, in addition, $V(x) \to \infty$ as $\|x\| \to \infty$, then the equilibrium point is globally asymptotically stable.

## Barbalet’s Lemma

Lemma:(Barbalat) If the differential function $f(t)$ has a finite limit as $t \to \infty$, and if $\dot f$ is uniformly continuous, then $\dot f(t) \to 0$ as $t \to \infty$.

Lemma:(“Lyapunov-Like Lemma”) If a scalar function $V(x, t)$ satisfies the following conditions

• $V(x,t)$ is lower bounded
• $\dot V(x,t)$ is negative semi-definite
• $\dot V(x,t)$ is uniformly continuous in time

then $\dot V(x,t)\to 0$ as $t \to \infty$.