# Nonlinear Control Design – EE5730

**Title :** Nonlinear Control Design

**Course No :** EE5730

**Credits :**

**Prerequisite :**

**Syllabus :**

**The course deals with nonlinear analysis for the most part and the remaining**

**is devoted to control design techniques.**

1. Mathematical preliminaries: Open and closed sets, compact set, dense

set, Continuity of functions, Lipschitz condition, smooth functions, Vector

space, norm of a vector, normed linear space, inner product space.

2. We will begin with an introduction to simple mechanical systems wherein

the notion of degree-of-freedom, configuration space, configuration vari-

ables will be brought out. The state-space models of a few benchmark

examples in nonlinear control will be derived using Euler-Lagrange for-

mulation. The notion of equilibrium points and operating points will help

us to extract linearized models based on Jacobian linearization.

3. Second-order nonlinear systems occupy a special place in the study of non-

linear systems since they are easy to interpret geometrically in the plane.

Here, we will touch upon the concept of a vector field, trajectories, vector

field plot, phase-plane portrait and positively invariant sets. The classi-

fication of equilibrium points based on the eigenvalues of the linearized

system will also be introduced and we will see why the analysis based on

linearization fails in some cases. Periodic solutions and the notion of limit

cycles will lead us to the Bendixson’s theorem and Poincar´e-Bendixson

criteria that provide sufficient conditions to rule-out and rule-in the exis-

tence of limit cycles respectively for a second-order system. We will end

this discussion with two methods for obtaining an approximate solutions

of periodic solutions.

4. Stability notions: Stability is central to control system design and here we

will study various notions of stability such as Lagrange stability, Lyapunov

stability, asymptotic stability, global asymptotic stability, exponential sta-

bility, relative stability and instability. The tools that we will use to infer

the stability properties include Lyapunov’s direct and indirect method, La

Salle’s invariance property and singular perturbations.

5. Design methods: Finally, we will see the design of control laws based on

Lyapunov function and Sliding mode control and illustrate the methodol-

ogy on a few benchmark examples.

**Text Books :**

1. Nonlinear System Analysis: M. Vidyasagar

2. Nonlinear Systems: H. K. Khalil

**References :**

NPTEL lecture notes on Nonlinear control design