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Aerial Robotics

Control Theory

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-491/691

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Signal Processing Fundamentals

  • Fourier Transform

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  • A mapping between representations in different domains:
    • Time Domain

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    • “Frequency” Domain

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  • Sinusoids are the only functions (Note: together with exponentials) that retain their “shape” when differentiated
    • Physical Laws work in Differential Equations
    • We can associate the transformation of a signal fed through a dynamical system as an operation/transformation on its Frequency Domain features (Amplitude & Phase)

CS491/691 C. Papachristos

 

 

Time (s)

Frequency - 2πω (rad/s)

 

Amplitude

Frequency - 2πω (rad/s)

Phase

Dynamical System

Differential Equations

Original�Amplitude �& Phase

Sinusoid

Input

Modified�Amplitude �& Phase

Sinusoid

Output

Note: Indicative images of different domains, but for the Fourier Series

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Signal Processing Fundamentals

 

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Signal Processing Fundamentals

 

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Signal Processing Fundamentals

 

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For

ω=3

 

 

For

ω=5

By Thenub314 - Own work, CC BY-SA 3.0

 

 

Fourier Transform Magnitude

 

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Linear Time Invariant Dynamical Systems

  • Linear Time Invariant (LTI) dynamical systems are characterized by:

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    • Homogeneity (Linearity):

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    • Superposition (Linearity):

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    • Time Invariance:

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  • Important class of systems because “We can solve them”, i.e. we have developed a set of mathematical tools to systematically solve them
  • Also, we can approximate many physical processes with LTI dynamical systems

CS491/691 C. Papachristos

 

L T I

Dynamical System

 

 

 

 

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Modeling of LTI Dynamical Systems

 

Time Domain based representation

Complex Domain based representation

 

 

CS491/691 C. Papachristos

 

By Zach Star, https://www.youtube.com/watch?v=n2y7n6jw5d0

 

 

 

 

 

 

 

 

 

 

 

 

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Modeling of LTI Dynamical Systems

 

Time Domain based representation

Complex Domain based representation

 

 

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E.g.,:

 

 

 

 

 

 

 

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Modeling of LTI Dynamical Systems

 

 

 

 

 

 

 

 

 

Note 2: Used in Systems Analysis

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  • Transforms linear Differential Equations into Algebraic ones
  • Transforms Convolution into Multiplication

 

CS491/691 C. Papachristos

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Modeling of LTI Dynamical Systems

 

Time Domain based representation

Complex Domain based representation

 

 

CS491/691 C. Papachristos

 

 

 

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Modeling of LTI Dynamical Systems

 

 

 

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L T I

Dynamical System

Note: The Transfer Function is the Laplace Transform of the LTI’s Impulse Response

 

 

 

 

 

 

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Modeling of LTI Dynamical Systems

 

 

 

 

 

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

C. Papachristos

 

Gain Origin z/p Real z/p Complex z/p Transp. Delay

On the whiteboard:

 

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Analysis of LTI Dynamical Systems

 

 

 

 

 

CS491/691 C. Papachristos

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Analysis of LTI Dynamical Systems

 

 

 

 

 

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Step Response

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Analysis of LTI Dynamical Systems

2nd Order Systems Response Analysis

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Second order system representation:

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In transfer function form:

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DC-gain :

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Damping ratio:

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Natural frequency:

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Poles:

 

 

 

 

 

 

 

 

CS491/691 C. Papachristos

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Analysis of LTI Dynamical Systems

 

 

 

 

 

 

CS491/691 C. Papachristos

Note: A critically damped system reaches the Steady-State Response in the minimum time possible

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Analysis of LTI Dynamical Systems

2nd Order Systems Response Analysis

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Pole Maps on the Complex Plane:

 

 

 

 

Poles are Complex numbers

Poles have only Real components

Poles have only Imaginary component

Complex Root-Pair

Locus:

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constant “damping ratio”

lines

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Analysis of LTI Dynamical Systems

2nd Order Systems Response Analysis

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Time-Domain Responses:

 

 

 

 

Some overshoot & oscillation exists

No overshoot (or oscillation)

Fixed-amplitude oscillation

Complex Root-Pair

Locus:

CS491/691 C. Papachristos

Step Response

Step Response

Step Response

constant “damping ratio”

lines

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

 

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

Critical

Stability

CS491/691 C. Papachristos

 

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Analysis of LTI Dynamical Systems

 

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No oscillations

oscillations

oscillations

Stable

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

C. Papachristos

Note: A right-half-plane Zero’s effect on a (step) response is an initial “dip” in the wrong direction (harder to control!)

A Non-Minimum-Phase system:

The Minimum Phase system:

Another Non-Minimum Phase system:

 

 

 

By: MATLAB: https://www.youtube.com/watch?v=jGEkmDRsq_M

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

 

 

 

 

C. Papachristos

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Analysis of LTI Dynamical Systems

 

C. Papachristos

 

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Analysis of LTI Dynamical Systems

 

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Note: Upper-half of Nyquist plot is a mirror of lower-half (corresponds to response for negative frequencies)

 

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Analysis of LTI Dynamical Systems

 

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“Unstable” Zeros

(or closed-loop Unstable Poles)

 

 

(only) Unstable open-loop Poles

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Analysis of LTI Dynamical Systems

Stability Margins

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  • Bode-based Gain & Phase Margins inform us of tolerance when only one of either the Gain/Phase changes

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  • Bode plot not useful for Unstable / Marginally Unstable systems

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  • Discuss how to find margins on Nyquist

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  • Discuss Disc Margins

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  • Low-pass / High-pass / Band-pass / Notch filters?

CS491/691 C. Papachristos

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Analysis of LTI Dynamical Systems

 

 

 

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Analysis of LTI Dynamical Systems

 

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Analysis of LTI Dynamical Systems

 

C. Papachristos

 

 

 

 

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State-Space Modeling of LTI Dynamical Systems

 

 

 

 

CS491/691 C. Papachristos

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State-Space Modeling of LTI Dynamical Systems

 

 

 

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State-Space Modeling of LTI Dynamical Systems

 

 

 

 

 

 

 

CS491/691 C. Papachristos

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State-Space Modeling of LTI Dynamical Systems

Example

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SISO system Ordinary Differential Equation (2nd order):

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In State-Space form:

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(In State-Space form: )

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CS491/691 C. Papachristos

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State-Space Modeling of LTI Dynamical Systems

 

 

C. Papachristos

Eigenvalues Equation

 

 

 

 

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State-Space Analysis of LTI Dynamical Systems

 

 

 

Continuous

Time:

C. Papachristos

 

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State-Space Analysis of LTI Dynamical Systems

 

 

 

 

 

Continuous

Time:

Discrete

Time:

 

CS491/691 C. Papachristos

 

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State-Space Analysis of LTI Dynamical Systems

 

 

Continuous

Time:

Discrete

Time:

 

 

CS491/691 C. Papachristos

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State-Space Analysis of LTI Dynamical Systems

 

 

 

 

 

Continuous

Time:

Discrete

Time:

 

CS491/691 C. Papachristos

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State-Space Analysis of LTI Dynamical Systems

 

 

 

 

Continuous

Time:

Discrete

Time:

 

CS491/691 C. Papachristos

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Time for Questions !

CS-491/691

CS491/691 C. Papachristos