Dynamic Nonlinear Control Systems Lecture 1: Introduction Dr. Hatem Elaydi Islamic University of Gaza Electrical Engineering Department Fall 2015
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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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