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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