Comparacion

Control engineering

Dr. Alejandro Rodríguez Angeles

Modeling

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Mathematical models

- Mechanical systems

El i- Electric systems

- Transfer functions and typical responses

- Block diagram

- Linearization of non linear systems

- Euler-Lagrange formalism

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Mathematical modelMathematical model

- Helps to understand the system in order to design thep y gcontrol

- Tipically the system's knowledge is incorporated in a- Tipically, the system s knowledge is incorporated in amathematical model

- A mathematical model is a set of ecuations thatrepresent a system behavior with certain precision.

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Parameters & variables of a model

Perturbations

Input OutputModel

Internal Variables

Input Variables

OutputVariables

Parameters (constants, variants & design)

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M d l l ifi tiModel classification

• Deterministic & stochastic

• Linear and non linear• Linear and non linear

• Continuous and discrete

• Variants and invariants

• Static and dynamicStatic and dynamic

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Mechanical systemsMechanical systems

Translational mechanical systems (linear)

Rotational mechanical systems (linear)

Inverted pendulum (non linear)

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Newton's and Euler´s mechanics

Translational motion: (Newton equation)

( )d ( )d mvF madt

Rotational motion: (Euler equation)

( )d I( )d I I Idt

Translational mechanical systems

Most common variables in translational mechanical systems:x displacement (m)v velocity (m/s)a accelerationf force (N)

A th i bl f i t tAnother variable of interest:w energy (J)p power (w)

Power applied to a mobile robot at constant velocity v ,(1)

Corresponds to the velocity with which energy is applied or dissipated.(2)

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Translational mechanical systems

Mass

Second Newton’s law establishes that for a constant mass, the totalforce acting on a body is equal to the variation in movement, i.e.

(3)

Meanwhile, the energy can be stored in kinetic energy in case themass is in motion, and in potential energy in case of verticaldisplacement relative to the reference position.

Kinetic energy, (4)

Potential energy for a uniform gravitate field,(5)

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Translational mechanical systems

Damper (friction)Damper (friction)

Forces that are algebraic functions of relative velocities betweenbodies are modeled by friction phenomena the most common beingbodies are modeled by friction phenomena, the most common beingviscous friction

(6)

C [N·s/m] and v: relative velocity.

The sense of the resultant force is oposed to the relative motion

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Translational mechanical systems

Spring (flexibility, elasticity)

Mechanical element that deforms itself when expose to force, andp ,that represents an algebraic relation between the applied force andthe produced elongation.

In a spring the above mentioned relation corresponds to thecharacteristic curve. For a linear spring

(7)xK spring constant (N/m).

The potential energy stored at the spring is

x

The potential energy stored at the spring is,

(8) 2x

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Newton's mechanics

• Based on force balances along the x, y, and z coordinates, i.e.

0 0 0F F F 0, 0, 0, x y zF F F • Forces for most typical mechanical components

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mass2

2mdv d xF ma m mdt dt

x

springx1 x2

2 1( )cF c x x

ddamper

x1 x2

2 1 2 1( ) ( )ddF d v v d x xdt

Translational mechanical systems

Example

Obtain a model for the mechanical system assuming linear behavior ofy gall the elements.

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Translational mechanical systems

Solution

Considering the forces in the free body diagram and form Newton’s Considering the forces in the free body diagram and form Newton s law

The resulting sum of forces must be zero:

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Rotational mechanical systems

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Rotational mechanical systems

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Rotational mechanical systems

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Rotational mechanical systems

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Rotational mechanical systems

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Rotational mechanical systems

II

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Inverted pendulum

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Inverted pendulum

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Inverted pendulum

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Inverted pendulum

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Inverted pendulum

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Inverted pendulum

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Inverted Pendulum on Car

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