Dynamic Nonlinear Control Systems

Lecture 1: Introduction Dr. Hatem Elaydi

Islamic University of Gaza Electrical Engineering Department

Fall 2015

Analysis & Design Philosophy

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Linear vs. Nonlinear

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Linear systems vs nonlinear systems

Linear systems

Nonlinear systems

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Linear systems vs nonlinear systems

Linear systems

Nonlinear systems

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Linear systems

• Linear systems are systems that have a certain set of properties.

• Linear systems are very nice objects to study because of their regularity. Why? We need structure.

System

ic

output input

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What is tricky about nonlinear systems?

LACK OF STRUCTURE! Cannot take everything for granted.

• Existence and uniqueness of solution to diff. eqns.

• Finite escape time

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Nonlinear from linear

• A lot of techniques that are used for nonlinear systems come from linear systems, because:

– Nonlinear systems can (sometime) be approximated by linear systems.

– Nonlinear systems can (sometime) be “transformed” into linear systems.

– The tools are generalized and extended.

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Why study nonlinear systems?

• Linearity is idealization. E.g. a simple pendulum.

• A lot of phenomena are only present in nonlinear systems. – Multiple (countable) equilibria. Why?

– Robust oscillations: where?

– Bifurcations

– Complex dynamics

• Why simulation is not always enough

• Why simulation is not always necessary

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Nonlinear Phenomena

• Finite Escape Time State goes to infinity in finite time

• Multiple Isolated Equilibria Nonlinear systems has more than one isolated equilibrium

points. The state convergence depends on the initial conditions.

• Limit Cycles Go into an oscillation of fixed magnitude and frequency,

irrespective of the initial state.

• Sub-harmonic, harmonic, almost periodic oscillations Oscillation frequencies are submultiples or multiples of

the input frequency.

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Nonlinear Phenomena…

• Chaos • Complicated steady-state behavior • Sometime random

• Multiple modes of behavior – Unforced systems may have more than one limit cycle – Forced systems with periodic excitation may exhibit

harmonic, sub-harmonic, or complicated steady-state behavior, depending upon the amplitude and frequency of the input.

– Exhibit discontinuity jump even though under smoothly changed input.

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Examples

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Examples

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Examples of Nonlinear Systems

Common Nonlinearities Examples:

Relay Pendulum Equation

Saturation Tunnel-Diode Circuit

Dead zone Mass-Spring System

Quantization Negative-Resistance Oscillator

Artificial Neural Network

Adaptive Control

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