Lyapunov Theory and Design

TEQIP Workshop on

Control Techniques and Applications IIT Kanpur, 19-23 September 2016

Dr. Shubhendu Bhasin

Department of Electrical Engineering IIT Delhi

Feedback Control 101

Syste

m Controller

Sensors

desired

behavior

disturbance

Actuator

s

actual

behavior

noise

Control = Sensing + Computation + Actuation Control Design Process

Control System Objectives

• Modeling – ODE, PDE.

• Analysis – stability, robustness, performance

• Synthesis – Feedback Design Tools

• Regulation

• Tracking Stability

• Modeling Uncertainties

• Disturbances

• Sensor Noise

Robustness

• Transient

• Steady State

• Minimizing cost function

Performance

Inverted pendulum regulation Satellite attitude tracking

disturbance rejection

u + -

Why Study Nonlinear Systems? Real world is inherently nonlinear !

Mass-Spring-Coulomb Damper

Pendulum

Saturation Deadzone u u

Quantization u

Inherently nonlinear physical

laws

Actuator

nonlinearities

50

-50

e.g. on-off control, adaptive control laws Intentional nonlinearities

Why Study Linear Systems?

• Linear approximation about operating point

Steady level flight

• Superposition: Impulse response characterizes LTI system behavior

• Closed-form solution

• Universal controllers: Pole-placement, LQR etc.

Limitations of Linearization

• Linearization of and produce the same linear

system!

• Linearization captures local behavior around the operating point

• Linearization cannot capture rich nonlinear behavior

Limit Cycle

Multiple Equilibria

Bifurcation

Chaos

• Linearization not possible for “hard” nonlinearities e.g. backlash, saturation etc.

Nonlinear System Analysis

• No general method to solve nonlinear differential equations

• Superposition does not hold

• No general method to design controllers

Lyapunov (1857-1918)

Challenges

Lyapunov Theory (1892)

• Select a scalar positive function

• Choose u such that V(x, t) decreases i.e.

|e(t)|

t 0

|e(t)|

t 0

Bounded or

Ultimately

Bounded Exponential |e(t)|

t 0

Asymptotic

Nonlinear Systems

Autonomous System:

Non-Autonomous System:

Existence and Uniqueness of Solutions

Equilibrium Point (s)

Solution of

Example: Pendulum System

Stability of Equilibrium Points

Van der Pol Oscillator

Stable or unstable ?

Asymptotic Stability

Asymptotic Stability = Stability + Convergence Convergence Stability ?

Exponential Stability (Rate of Convergence)

Exponential Asymptotic ?

Asymptotic Exponential ?

Local Vs Global Stability

Lyapunov’s Indirect Method (Linearization)

Lyapunov’s Direct Method (Motivating Example)

Motivating Example (contd..)

Key Observations:

• Zero Energy corresponds to equilibrium

• Asymptotic stability convergence of mechanical energy to zero

• Stability properties are related to variation of mechanical energy

Lyapunov’s Direct Method (Basic Idea)

Lyapunov’s Stability Theorem (Local)

V(x) is positive definite

negative-definite

negative semi-definite

Exercise

Asymptotically stable?

Lyapunov’s Stability Theorem (Global)

V(x) is radially unbounded

Radial Unboundedness is Necessary

Divergence of states while

moving to lower “energy” curves

Exercise

Remarks:

• Lyapunov theorems give sufficient conditions for stability

• Failure of a Lyapunov function candidate to satisfy the theorem does not

mean that the eq. point is unstable.

Example (Pendulum with Friction)

Stability Analysis

Using a different V(x)

LaSalle’s Invariance Set Theorem

• Useful for proving asymptotic stability when

derivative of V(x) is only negative semi-definite

Pendulum with friction (Revisit)

Non-Autonomous Systems

Non-Autonomous Systems (Stability Definitions)

Lyapunov Theorems for Non-Autonomous Systems

Barbalat’s Lemma

Asymptotic Properties of functions and its derivatives

Lyapunov-Like Lemma

Example continued…

Boundedness of Solutions

Uniformly Ultimately Bounded Stability

Adaptive Control

TEQIP Workshop on

Control Techniques and Applications IIT Kanpur, 19-23 September 2016

Dr. Shubhendu Bhasin

Department of Electrical Engineering IIT Delhi

Introduction

Historical Perspective

X-15

Basic Idea

Adaptive Control: A Parametric Framework

Indirect and Direct Adaptive Control

Indirect and Direct Adaptive Control

Indirect Adaptive Control

Direct Adaptive Control

Gain Scheduling Adaptive Control

No parameter estimation

Model Reference Adaptive Control (MRAC)

Adaptive Control Topics

• Direct MRAC (SISO)

• Indirect MRAC (SISO)

• Lyapunov-Based Nonlinear Adaptive Control

• Parameter Convergence

• Parameter Drift

• Adaptive Backstepping