Nonlinear Control Design – EE5730
Title : Nonlinear Control Design
Course No : EE5730
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
NPTEL lecture notes on Nonlinear control design