Modeling

Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing Engineering

University of Toledo

Outline of Today’s Lecture

• Review• Feedback

• Open Loop Systems• Closed Loop System

• Positive Feedback• Negative Feedback• Basic Control actions

• Models• Dynamics• States• Phase Plots• Example: Predator Prey Model

Open Loop Control

• Usually “set point” systems• Advantages

• Simple• Insensitive to environment• Set and forget

• Disadvantages• Non correcting• Sensitive to disturbances• Insensitive to environment

• Examples• Irrigation systems• Washing machines

Sensing Compute Actuate

Closed Loop Control

• Adds a feedback loop to the control system

• For computational purposes, it is shown as

Sense

Compute

Actuate

Controller Plant

Sensor

Input Output

Disturbance

+ or -

+ or -+ or -

+ or -

Positive Feedback

Positive Feedback Clip

Controller Plant

Sensor

++

+ +

VibratingGuitar String

MagneticPickup

AmplifierSpeaker

Plucked String String Vibrations

Controller Plant

Sensor

++

+ + Sound

Guitar String w/ pickup

Amplifier Speaker

Ambient Sound

2 possible models

Background sound

Previous Vibrations

Positive Feedback

• Positive feedback is used to increase the actuation in the loop.• Advantages

• Increased results• Faster results• Finds extremes

• (maxima and minima)• Disadvantages

• Consumes energy• Subject to local extremes (introns)• May become unstable• May destroy system

• Examples:• Metal finders• Searches• Stock market programs• Genetic Algorithm

build population

createmutations

test

performancemeasure

Results

Culled fromPopulation

BestWorst

Negative Feedback

Input Controller Plant

Sensor

Output

Disturbance

+-

+ +

Error Signal

homeostasis

DesiredHeart Beat

Controller Plant

Sensor

Heart Beat

Salt

+-

+ +

Heart

Nerves

Parasympathetic/Sympathetic System

Negative Feedback

• Negative Feedback is used to reduce error• Advantages

• Controls to a set point• Robustness to disturbances (uncertainty)• Rejection of distortion

• Disadvantages• Prone to oscillation• Instability• Complexity• Coupling

• Examples• Set point control• Tracking• Chang the system dynamics

Basic Control Actions

• Bang-Bang (Off-On)• Fixed two state or multistate control actions• Control question: how to chose?

• Proportional• Control in proportion to error

• Integral• Control based on size and duration of error

• Derivative• Control based on size and change of error

• Combined (PID)• All three: Proportional, Integral and Derivative• Most used

Models

• A model is a representation of something• The something can be an idea, a concrete object or an abstract

object• It is NOT the real thing: they are simplifications

• it is a fiction of our imagination• Models can take many forms

• Solid• Blocks• Equations• Computer programs• Word descriptions• Symbols

( )u u

u u u t u t r

mz bz kz bz kzm z bz k k z k z bz kz

Models

• It is NOT the real thing: they are simplifications• it is a fiction of our imagination

• Models are used for• visualization• understanding• explaining to others• analysis• predict• improve

• The value of amodel is howwell it servesthe purposeused for ( )

u u

u u u t u t r

mz bz kz bz kzm z bz k k z k z bz kz

Models

• Different models answer different questions• As a model developer, you need to chose the

right model for your problemHow will costs change with the strengthof the landing gear?

How much vertical force is the landing gear putting on the nose?

How high should the tire pressure be?

At what point can the pilot rotate the aircraft to take weight off the nose wheel?

How much heat is built up in the tire on a takeoff roll?

Dynamics

• defn (TheFreeDictionary): The branch of mechanics that is concerned with the effects of forces on the motion of a body or system of bodies, especially of forces that do not originate within the system itself. Also called kinetics.

• Things move!• If we are to control movement we need to know

how they move

Dynamics

• Models of dynamics used in this course:• Based on functions of time

• Differential equations• time is considered continuous

• Difference equations• time is considered discrete

( )u u

u u u t u t r

mz bz kz bz kzm z bz k k z k z bz kz

( 2 ) ( 2) value of z at time t + 2 time steps( 2) ( 1) ( ) ( 1) ( )

( 2) ( 1) ( ) ( ) ( ) ( 1) ( )u u

u u u t u t r

z t t z tmz t bz t kz t bz t kz tm z t bz t k k z t k z t bz t kz t

State

• A state is a set of variables whose values when known completely define the dynamics (motion)

• State variables for nose wheel example:

• The parameters of the example are

• So, what about ?

( , , , , )u r uz z z z z

( , , , , )u tm m b k k

( )u u

u u u t u t r

mz bz kz bz kzm z bz k k z k z bz kz

( , )uz z

State

• So, what about ?• These are completely determined by the state

variables!• We can rewrite the equations as

( , )uz z

( )

u u

u t uu

u

b z z k z kz

mb z z k k z kz

zm

( )u u

u u u t u t r

mz bz kz bz kzm z bz k k z k z bz kz

Phase Plot

• A phase plot is a plot of a state variable vs. another state variable• Useful in understanding how the dynamics

change with changes in state• To see how the dynamics are represented by the

phase plot, consider the predator prey problem• we will first use differential equations

then difference equations

Predator Prey Model

• Volterra – Lotke model• Vito Volterra and Alfred J. Lotke independently developed this useful model • Explains the growth of a thing that depends on the growth of another thing• Lynx and hares or whales and krill are typically used to demonstrate the

model but it could be two infantry units fighting each other or two stock firms trying to acquire the same limited commodity

• Let x(t) represent the prey (hares, krill, xth Inf, etc.) and y(t) represent the predator (lynx, whales, nth Inf, etc.)

• Prey population growth rate without predators is assumed proportional to population size

dx axdt

But….

Predator Prey Model

• But there are predators! • The predators [eat, destroy, acquire…] the prey in proportion to

the number of prey and predators.

• The predator population without sufficient prey dies out at a rate of

• But there are predators that are [eaten, destroyed, acquired …] that sustain the predators in proportion to both populations sizes:

dx ax bxydt

dy my nxydt

dy mydt

Predator Prey Model

• The model is

• subject to• x(t) = prey population size• y(t) = predator population size• a = growth rate of prey• b = rate of prey predation• m = death rate of predators• n = rate of predator sustenance

• Model Solution

• State Variables: x, y• Note the controls in

this nonlinear, coupled, model

dx ax bxydtdy my nxydt

/ ( )/ ( )

ln( ) ln( )m a

bx ny

dy dy dt m nx ydx dx dt a by x

a mb dy n dxy xa y by nx m x c

x y Ce

Predator Prey Model

• Lynx – Hare ( data from Leigh, 1968)

• Continuous Model

Discrete Predator Prey Model

• To discretize,• choose time step, h• replace

• The difference equations are

• If h = 1 year, the model is

• Evaluation:• with x(0)=80, y(0)=30• a = 0.7, b = 0.03,• m=0.99, n = 0.03• Initially h = 1 year

( ) ( )

( ) ( )

dx x t h x tdt hdy y t h y tdt h

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

x t h x t h ax t bx t y t

y t h y t h my t nx t y t

( 1) ( ) ( ) ( ) ( )( 1) ( ) ( ) ( ) ( )x t x t ax t bx t y ty t y t my t nx t y t

1 80 .7*80 .03*80*30 64

(1) 30 .99*30 .03*80*30 57.92 64 .7*64 .03*64*57.9 2.368

(2) 30 .99*30 .03*64*57.9 3.53

x

yx

y

Discrete Predator Prey Model

• Unstable Model!• Problem: the time step

is too big!• with h = 0.1 years

Choice of time step is critical to discrete modeling

The smaller the time step, the more accurate the model but more computations

Summary

• We discussed modeling• Models are simplifications• Dynamics: things move!

• differential equations• difference equations

• States• a set of variables whose values when known completely

define the dynamics (motion)• Phase Plots

• a plot of a state variable vs. another state variable• Example: Predator Prey Model

• Next: State Space Models