introduction to control systems - umass...

1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.451 Dynamic Systems – Intro to Controls

Introduction to Control Systems

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell

Stochastic

Dynamic

Deterministic

Distributed ParameterLumped Parameter

NonlinearLinear

Variable CoefficientConstant Coefficient

Continuous TimeDiscrete Time

Static

Increasing Ease of Analysis Increasing Realism

First Order Second Order Higher Order

CLASSIFICATION OF MATHEMATICAL MODELS


Classification of Mathematical Models

Source: Dynamic Systems – Vu & Esfandiari


Introduction to Control SystemsDefinitions

Closed Loop System – there always exists some ‘feedback’ of one or more variables that influence the input excitation.

Disturbance Inputs – signal that exists but no control is possible.

Negative/Positive Feedback – signals that are added to or subtracted from an input signal.

Open Loop System – no feedback signals are available to the control system.Plant – device, system or component that is described by some transfer relation.Essential Components of a Control System – Plant, Sensor, Actuator, Controller



(Reference))s(R + )s(Gc

)s(H

−∑

)s(Ea )s(Ga)s(U +

+

)s(DDisturbance

Sensor

Controller Actuator

Plant )s(Gp

INPUT

)s(YCONTROLLED OUTPUTGeneral Block Diagram

of a Control System


Example – Control System

Sour

ce: D

ynam

ic S

yste

ms

–Vu

& Es

fand

iari(a)

(b)

(c)



Knowledge of the system Poles and Zeros is important information in the design and analysis of dynamic systems, vibrations, and controls.Assume a system transfer function such as:

poleszeros

rdenominatonumerator

)s(D)s(N.F.T ⇒==

Poles and Zeros

The numerator polynomial can be factored to identify the system transfer function ZEROS. The denominator polynomial can also be factored to identify the system transfer function POLES.

( )( )2s1s5.0s

2s3s5.0s.F.T 2 ++

+=

+++

=

Xby denotedOby denoted

2,1POLES5.0ZERO→→

−−→−→

Im

Re

POLEZERO

X XO2- 1-


Time Constant – First Order System

)t(fvv =+τ&

Differential Equation

01s =+τ

Characteristic Equation

Pole

τ−=

1s

Im

ReXτ1

−


Time Constant – First Order System

)t(fxcxm =+ &&&

Consider the mechanical system shown

then

OR

cm

=τ

m fcx,x &

)t(fcvvm =+&

ORc

)t(fvvcm

=+

&c

)t(fvv =+τ&

The time constant is


First Order System – RC Circuit

( )( )( )20s7s1s140)s(h

+++=

Consider the function of an RC circuit

Slowest Pole

Written in POLE-ZERO form, the block diagram is

( ) ( ) ( )20s20

7s7

1s1

FX

+++=

F1s

1+ 7s

7+ 20s

20+

Quicker Pole

Fastest Pole

X


First Order System – RC CircuitA SIMULINK Model can be used to quickly see the response of each pole.

STEP

1s1+

7s7+

20s20+

SUMX( )( )( )20s7s1s

140+++

1X

7X

20X

The pole ‘closest’ to the jω axis dominates the time response of the system.


SIMULINK Model


Time Constant – Second Order System

)t(fxx2x 2nn =ω+ζω+ &&&

Differential Equation

Pole - Overdamped

1s 2nn2,1 −ζω±ζω−=

0s2s 2nn

2 =ω+ζω+Characteristic Equation

Pole – Critically damped

n2,1s ω−= (repeated)Pole – Underdamped

2nn2,1 1js ζ−ω±ζω−=

djω±σ−=Pole – Undamped

n2,1 js ω±=

ImReX

ImRe

ImRe

ImRe

X

XX

X

XX


Time Constant – Second Order System

)t(fkxxcxm =++ &&&

Consider the mechanical system

with

OR

which is written in standard form as

m)t(fx

mkx

mcx =++ &&&

mk

n =ω

km2c

m2c

cc

nc=

ω==ζ

m)t(fxx2x 2

nn =ω+ζω+ &&&


Transient Response Specification TermsMaximum Percent Overshoot – Mp%

The maximum overshoot is the maximum amplitude expressed as a percentage of the steady state value.

Settling Time - ts

21p eM ζ−

πζ−

=

The time required for the response to reach a small percentage of steady state value.

σ=

ζω=τ=≈

444'ttn

%2s%2s

σ=

ζω=τ=≈

333'ttn

%5s%5s


Transient Response Specification Terms

Rise Time – tr

The time for an underdamped2nd order system to rise from 0 to 100% of the final steady state value.

ζζ−

ω−= −

21

nr

1tan1t

Peak Time – tp

The time required to reach maximum overshoot is called peak time. d

ptωπ

=


Transient Response Specification Terms


Transfer Functions Building Block MethodologyAll transfer functions can be broken down into pieces (called building blocks) for evaluation and assessment.

( )( )( )500s101s2ss

90s750)s(H 2 +++

+=

These can be categorized as•constant•first order pole (or zero) at origin•first order pole (or zero) not at origin•second order pole (or zero) with damping ratio less than 1.0

For Example:Constant 1st Order Zero

(not at origin)

1st Order Pole (not at origin)2nd Order Pole

Damping < 1.01st Order Pole (at origin)


Transfer Functions Building Block MethodologyFirst Order Pole At Origin

s1)s(G =

ω=ω

j1)j(G

First Order Pole

Second Order Pole

dB Mag

log f

20 dB/octave

dB Mag

log f

20 dB/octave

dB Mag

log f

40 dB/octave

aj1)j(G+ω

=ω

as1)s(G+

=

2nn

2

2n

s2s)s(G

ω+ζω+

ω=

( ) ( ) 2nn

2

2n

j2j)j(G

ω+ωζω+ω

ω=ω


Transfer Functions Building Block MethodologyFirst Order Zero At Origin

s)s(G =ω=ω j)j(G

First Order Zero

Second Order Zero

dB Mag

log f

20 dB/octave

dB Mag

log f

20 dB/octave

dB Mag

log f

40 dB/octave

aaj)j(G +ω

=ω

aas)s(G +

=

2n

2nn

2 s2s)s(Gω

ω+ζω+=

( ) ( )2

n

2nn

2 j2j)j(Gω

ω+ωζω+ω=ω


Controller Types•On/Off•Differential On/Off•Proportional Control (P control)•Derivative Control (D control)•Integral Control (I control)•PD Control •PI Control•PID Control

Others (Advanced Methods)•Lead Network Control•Lag Network Control•Lead-Lag Network Control•Feed Forward Control


Controller TypesOn/Off

)t(e 01

01M

2MOR

Differential

Differential

Example: Home heating system


Controller Types

P Control is a simple gain and should be always tried first.

pK)s(Ea )s(U

)s(EK)s(E)s(G)s(U apac ==

sKd)s(Ea )s(U

Proportional Control

D Control adds damping to the system and therefore provides stability to the system.

Derivative Control

)s(sEK)s(E)s(G)s(U adac ==


Controller Types

I Control reduces steady state error but can increase instability.

sK I)s(E a )s(U

)s(Es

K)s(E)s(G)s(U aI

ac ==

Integral Control


Controller Types

sKs

KK dI

p ++)s(E a )s(U

Desired Value of Control Variable

PID Control

)s(EsKs

KK)s(E)s(G)s(U adI

pac

++==

+

−∑ sK

sKK d

Ip ++

Plant

)s(M

introduction to control systems - umass...

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