Lyapunov Theory and Design
Dr. Shubhendu Bhasin
Department of Electrical Engineering IIT Delhi
TEQIP Workshop on
Control Techniques and Applications
IIT Kanpur, 19-23 September 2016
Feedback Control 101
Control = Sensing + Computation + Actuation
disturbance
desired +
behavior
-
Controller
u
Actuator
s
Syste
m
actual
behavior
Control Design Process
• Modeling – ODE, PDE.
• Analysis – stability,
robustness, performance
• Synthesis – Feedback
Design Tools
Sensors
noise
Control System Objectives
Stability
Robustness
• Regulation
• Tracking
• Modeling
Uncertainties
• Disturbances
• Sensor Noise
Inverted pendulum regulation
Satellite attitude tracking
• Transient
Performance
• Steady State
• Minimizing cost
function
disturbance rejection
Why Study Nonlinear Systems?
Real world is inherently nonlinear !
Inherently nonlinear physical
laws
Mass-Spring-Coulomb Damper
Pendulum
Actuator
nonlinearities
50
-50
Saturation
u
Intentional nonlinearities
Deadzone
u
e.g. on-off control, adaptive control laws
u
Quantization
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 captures local behavior around the operating point
• Linearization of
system!
and
produce the same linear
• Linearization not possible for “hard” nonlinearities e.g. backlash, saturation etc.
• Linearization cannot capture rich nonlinear behavior
Limit Cycle
Multiple Equilibria
Bifurcation
Chaos
Nonlinear System Analysis
Challenges
• No general method to solve nonlinear differential equations
• Superposition does not hold
• No general method to design controllers
Lyapunov Theory (1892)
• Select a scalar positive function
• Choose u such that V(x, t) decreases i.e.
Lyapunov
(1857-1918)
|e(t)|
0
Bounded or
Ultimately
Bounded
t
|e(t)|
0
Asymptotic
t
|e(t)|
0
Exponential
t
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 semi-definite
negative-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
Dr. Shubhendu Bhasin
Department of Electrical Engineering IIT Delhi
TEQIP Workshop on
Control Techniques and Applications
IIT Kanpur, 19-23 September 2016
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