review of introductory control theory - phoneoximeter · review of introductory control theory ......

EECE 460 Winter 2007 Guy Dumont 1

Review of Introductory Control Theory

Guy Dumont


Review

Properties of Feedback From open loop to closed loop Important transfer functions Robustness and performance

Stability Routh Hurwitz Root locus Nyquist and Bode diagrams

Basic controllers Lead/lag compensators PID control

State space methods State space representation Controllability/observability State feedback, pole placement


High Gain Feedback and Inversion

+

-

R(s) Y(s)U(s)H(s) G(s)

G(s)

Open-loopcontroller

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) 1 ( ) ( )

Note that for ( ) 1, then

( ) ( ) 1

( ) ( ) ( ) ( )

U s H s R s H s G s U s

U s H s

R s H s G s

H s

U s H s

R s H s G s G s

= !

=+

�

� �

High gain feedback implicitly generates the inverse of G(s)without having to actually carry the inversion!


From Open Loop to Closed Loop

+

-

R(s) Y(s)U(s)H(s) G(s)

G(s)

+

-

R(s) U(s)H(s) G(s)

Y(s)

For the relationship betweenR(s) and Y(s), both areequivalent


Trade-offs

Although it seems that all is needed is highgain feedback, there is a cost attached to theuse of high-gain feedback It will result in very large control actions It increases the risk of instability It increase the sensitivity to measurement noise

When choosing the feedback gain, one mustbe conscious of the trade off between thosevarious issues

This is the essence of controldesign!


The Feedback Loop

Setpoint Output

Output disturbance

Measurement noise


The Feedback LoopTracking error

1 is called the sensitivity function

1

is called the complementary sensitivity function1

1Note that 1

1 1

SGC

GCT

GC

GCS T

GC GC

=+

=+

+ = + =+ +


The Feedback Loop


Four Important Transfer Functions

Input disturbance Output disturbance

Reference signal

Simple Feedback Control Loop


Four Important Transfer Functions

The following four transfer functionsplay an important role in control design

1( ) ( ) ( ) ( )

1 1 1

( ) ( ) ( ) ( )1 1 1

GC GY s R s W s D s

GC GC GC

C C GCU s R s W s D s

GC GC GC

= + ++ + +

= ! !+ + +

Control sensitivity

Input sensitivity Sensitivity Complementary sensitivity


RobustnessSensitivity of the closed-loop transfer function T to variationsin the process parameters:

2

2

2

2

2

We want to relate dT/T to dG/G

With we can write1

1 (1 )

(1 )

(1 )

(1 )

GCT

GC

dT C GC

dG GC GC

C GC GC

GC

C

GC

=+

= !+ +

+ !=

+

=+


Robustness

2

2

(1 )

(1 )

1

1 1

1

1

dT C

dG GC

CdT dG

GC

GC dG

GC GC G

dT dG

T GC G

dT dGS

T G

=+

=+

=+ +

=+

=


The Sensitivity Function

1

1

Typically is large at low frequencies and small at

high frequencies, hence

(0) 0 while ( ) 1

This implies

(0) 1 and ( ) small

SGC

GC

S S

T T

=+

! " =

= "


Specifications

Reduce sensitivity and disturbances at lowfrequencies

Close to perfect setpoint tracking at lowfrequencies

At high frequencies, when S=1 the systemhas the same sensitivity and disturbancerejection properties as the open-loop plant

Typically S can be decreased in a frequencyrange at the cost of an increase in anotherfrequency range


The Bode Integral

For a stable system Bode showed that

0log | ( ) | 0S j d! !

"

=#

“The waterbed effect ”


Steady-State Error

T

T

0

0

The tracking error is

1E (s)= ( )

1+GC

For a step input

1 1E (s)=

1+GC

Using the final value theorem

lim ( ) lim ( )

1 1lim

1+GC

1( )

1 (0) (0)

t s

s

R s

s

e t sE s

ss

eG C

!" !

!

=

=

" =+

We want the steady-state error to go tozero.

Hence, if G(0) is finite,we need C(0) to beinfinite, i.e. C mustcontain one integrator1/s


Test Input Signals

2

2 3

( ) ( ) ( )

( ) ( ) ( )

r t A r t At r t At

A A AR s R s R s

s s s

= = =

= = =


Performance of a second-order system

2

2

2 2

2

2 2

( )( ) ( ) ( )

1 ( )

( )2

1 for a step input

2

n

n n

n

n n

G s KY s R s R s

G s s ps K

R ss s

s s s

!

"! !

!

"! !

= =+ + +

=+ +

=+ +


Second-Order System

Percent Overshoot 100p v

v

M f

f

!= "


Second-Order System

2

2

/ 1

4With 0.02 then

1

1

v s

n

p

n

p

f T

T

M e!" !

#!$

"

$ !

% %

=

=%

= +

�

Swiftness of response: rise time and peak timeCloseness of response: overshoot and settling time


Second-Order System


Steady-State Error

Reduction or elimination of steady-state error is a fundamentalreason for using feedback

0

( )( ) ( )

1 ( ) ( )

with ( ) 1

1( ) ( ) ( )

1 ( )

The steady state error is then

( )lim ( ) lim

1 ( )

a

at s

G sY s R s

G s H s

H s

E s R s Y sG s

sR se t

G s!" !

=+

=

= # =+

=+


Steady-State ErrorThe number of integrators in G(s) defines its type number

0

2

0 0 0

( / )Step input: lim . For to be zero, ( ) must contain

1 ( )

at least one integrator, i.e. be of at least type 1

( / )Ramp input: lim lim lim

1 ( ) ( ) ( )

For to be

ss sss

sss s s

ss

s A se e G s

G s

s A s A Ae

G s s sG s sG s

e

!

! ! !

=+

= = =+ +

3

2 2 20 0 0

zero, ( ) must contain at least two integrators, i.e. be of at least type 2

( / )Acceleration input: lim lim lim

1 ( ) ( ) ( )

For to be zero, ( ) must contain at least th

sss s s

ss

G s

s A s A Ae

G s s s G s s G s

e G s

! ! != = =

+ +

ree integrators, i.e. be of at least type 3


Error Constants2

0 0 0lim ( ) lim ( ) lim ( )

Then for

Step input 1

Ramp input

Acceleration input

p v as s s

ss

p

ss

ss

K G s K sG s K s G s

Ae

K

Ae

Kv

Ae

Ka

! ! != = =

=+

=

=


Performance Indices

“A performance index is a quantitativemeasure of the performance of a system andis chosen so that emphasis is given to theimportant system specifications”

When the system parameters are chosen sothe performance index reaches an extremum(typically a minimum), then it is consideredan optimal control system

To be useful, a performance index should beeasy to optimize


Performance Indices

ISE

Convenient foranalytical andcomputationalreasons

2

0

Integrated square error

( )

T

ISE e t dt= !


Performance Indices

0

0

2

0

| ( ) |

| ( ) |

( )

T

T

T

IAE e t dt

ITAE t e t dt

ITSE te t dt

=

=

=

!

!

!

Integrated absolute error

Integrated time-multipliedabsolute error

Integrated time-multipliedsquared error


Definitions of Stability

BIBO stability: A system is said to beBIBO stable if for any bounded input, itsoutput is also bounded.

Absolute stability: Stable /Unstable Relative stability: Degree of stability

(i.e. how far from instability) A stable linear system described by a

transfer is such that all its poles havenegative real parts


Routh-Hurwitz Criterion

Consider the polynomial

Routh-Hurwitz stability criterion is a test toascertain without computing the roots,whether or not all roots of a polynomial havenegative real parts.

Today, with MATLAB, roots of a polynomialare so easily calculated that the Routh-Hurwitz criterion is hardly ever used.

1 1 1 0( )n n n n

Q s a s a s a s a! !

= + + + +L


Root Locus Concept

1 ( ) 0KG s+ =

The root locus is the path of the roots of the characteristicequation traced out in the s-plane as a system parameter ischanged

K


Root Locus Concept

2

2

2

2 2

* 2

* 2

11 ( ) 1 0

1

1 0

( 1) ( 1) 0

( 1) 4( 1) 2 3

1 1For 0, ( ) ( 1) 2 3

2 2

1 1For 0, ( ) ( 1) | 2 3 |

2 2

(this occurs for 1 3)

sKG s K

s s

s s Ks K

s K s K

K K K K

s K K K K

s K K j K K

K

++ = + =

+ +

+ + + + =

+ + + + =

! = + " + = " "

! > = " + ± " "

! < = " + ± " "

" < <


Root Locus Concept*

*

* 2

2

1 3For 0, (0)

2 2

1For 3, (3) (3 1) 2

2

For large ,

1 1( ) ( 1) 2 1 4

2 2

1 1 ( 1) ( 1) 4

2 2

1 1 ( 1) ( 1) 1,

2 2

K s j

K s

K

s K K K K

K K

K K K

= = ! ±

= = ! + = !

= ! + ± ! + !

= ! + ± ! !

" ! + ± ! = ! !

Poles of G(s) !

Zero of G(s) !

Tends to !"


Root Locus Concept1 ( ) 0

( )1 0

( )

( ) ( ) 0

When 0, this collapses to ( ) 0.

Since the roots of ( ) 0 are the poles of ( ), those are the

closed-loop poles for 0.

1 ( )When is large, ( ) ( ) 0 t

( )

KG s

N sKD s

D s KN s

K D s

D s G s

K

N sK D s KN s

K D s

+ =

+ =

+ =

= =

=

=

+ = + =( )

ends to 0( )

thus the closed-loop poles tend to the roots of ( ) 0, i.e. the

open-loop zeros, and also to infinity if is strictly proper.

N s

D s

N s

N

D

=

=


Fourth-Order System

4 3 21 0 1 0

12 64 128 ( 4)( 4 4 )( 4 4 )

K K

s s s s s s s j s j+ = ! + =

+ + + + + + + "


Fourth-Order System


Improving Transient Response

Cannot be obtainedby gain adjustment


Polar Plot or Nyquist Diagram

2

2

2 4 2 2 4 2

1

2 4 2 2

1

( )( 1)

( )

( )

1( ) tan

K KG j

j j j

K jK

KG

!! !" ! ! "

! " !

! ! " ! ! "

!

! ! "

# !!"

$

= =+ $

$= $

+ +

=

+

% &= $ ' (

$) *


Bode Diagram

1

1 s!+


The Nyquist Criterion

Open-loop stable plant ( ) ( 0)

A feedback system is stable if and only if the contour

in the ( )-plane does not encircle the (-1,0) point.

Open-loop unstable plant ( ) ( 0)

A feedback system is s

L

L s P

L s

L s P

=

!

"

table if and only if, for the contour

in the ( )-plane, the number of counterclockwise of

the (-1,0) point is equal to the number of poles of ( )

with positive real parts.

LL s

P L s

!


Relative Stability

1 2

1 2

1 2

1 2

1 2

( )( 1)( 1)

crosses the real axis at

System is marginally stable when

1, when

KGH j

j j j

Ku

u K

!! !" !"

" "

" "

" "

" "

=+ +

#=

+

+= # =

Ultimate point


Gain and Phase Margins on the Bode Diagram


Gain Margin

180

180

The factor by which the gain at ultimate point can be

increased to reach marginal stability

1 1

( )

Expressed in dB:

120log( ) 20log

20log ( ) dB

G

G

d L j

M dd

M L j

!

!

=

= = "

= "

Reasonable values aregain margins in the range[2 5], or in dB [6 14]dB.


Phase Margin

The phase margin is , the difference between and

the phase of the loop at the crossover frequency

arg ( )

c

cM L j!

! "

#

" #

$

= +

Reasonable values are phase margins In the range [30 60] degrees


Sensitivity and Nyquist Plot

( )L j!

s!

1 ( )s

L j!+

The radius r of the circle centered at -1 and tangent to L is the reciprocal of thenominal sensitivity peak

1|1 ( ) | | ( ) |s sr L j S j! ! "= + =

Critical point


Lead-Lag Compensators


Lead-lag Compensators

Phase Lead Phase Lag


Systems With a Prefilter

The lead filter zero will significantlyaffect the response of the closed-loopsystem

( )( )

( ) ( ) ( ) ( )( ) ( )( )

1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( )

p c

c c

z s zG s

G s G s G s zG ss z s pT s

G s G s G s G s s p s z G s

+

+ += = =

+ + + + +


Phase-Lead

Adds phase lead near crossover Increases bandwidth Increases high frequency gain Improves dynamic response Prefilter needed Requires additional amplifier gain Increases susceptibility to noise Applicable when fast response is desired Not applicable when phase decreases rapidly

near crossover frequency


Phase Lag

Adds phase lag to increase error constantwhile maintaining phase margin

Decreases system bandwidth Suppresses high-frequency noise Reduces steady-state error Slows down response Applicable when error constants are specified Not applicable when no low-frequency range

exists where desired phase margin exists


PID Controller

“Textbook” PID controller:

In practice:

( )

which corresponds to

( )( ) ( ) ( )

I

c P D

P I D

KG s K K s

s

de tu t K e t K e d K

dt! !

= + +

= + +"

( )1

I D

c P

D

K K sG s K

s s!= + +

+


PID Controller

PI controller: Used extensively inprocess control on a broad range ofapplications due to simplicity andrelatively good performance

PD controller: Used extensively incontrolling electromechanical systems

PI controller: ( ) I

c P

KG s K

s= +

PD controller: ( )c P DG s K K s= +


The State Differential Equation


Transfer Function of a State Space Model

Substituting

into the output equation

( ) ( ) ( )y s Cx s Du s= +

yields


Control Canonical Form1 2 1

1

1 2

0

1

1 0

...( )

...

n n

n

n n

n n

b s b s b s bG s

s a s a s a

! !

!

! !

!

+ + + +=

+ + + +

[ ]

0 1 2 1

0 1 2 1

0 1 0 ... 0 0

0 0 1 ... 0 0

0 0 0 ... 1 0

... 1

... 0

n

n n

A B

a a a a

C b b b b D

!

! !

" # " #$ % $ %$ % $ %$ % $ %= =$ % $ %$ % $ %$ % $ %! ! ! ! & '& '

= =

M M M O M M


Observer Canonical Form1 2 1

1

1 2

0

1

1 0

...( )

...

n n

n

n n

n n

b s b s b s bG s

s a s a s a

! !

!

! !

!

+ + + +=

+ + + +

[ ]

0 0

1 1

2 2

1 1

0 0 0

1 0 0

0 1 0

0 0 1

0 0 ... 0 1 0

n n

a b

a b

A a B b

a b

C D

! !

!" # " #$ % $ %!$ % $ %$ % $ %= ! =$ % $ %$ % $ %$ % $ %!& ' & '

= =

L

L

L

M M O M M M

L


The State Transition Matrix2 ( )

2 2

2 2

2 2

1 1( ) (0) (0) (0) (0)

2! !

1 1(0) (0) (0) (0)

2! !

1 1( ) (0)

2! !

1 1

2! !

k k

k k

k k

At k k

x t x x t x t x tk

x Ax t A x t A x tk

I At A t A t xk

e I At A t A tk

= + + + + +

= + + + + +

= + + + + +

= + + + + +

& && L L

L L

L L

L L

This series converges for all finite t. It is called the matrix exponential

( ) (0)Atx t e x=


The State Transition Equation

1

t tA(t- )

0 0

( ) ( ) ( )

( ) ( ) ( )

The inverse Laplace transform gives

x(t)= e ( ) ( ) ( )

x Ax bu

sX s AX s bU s

X s sI A bU s

bu d t bu d! ! ! " ! ! !

#

= +

= +

= #

= #$ $

&

(Assumes x(0)=0)

When initial conditions are non zero:

0( ) ( ) (0) ( ) ( )

t

x t t x t bu d! ! " " "= + #$


Characteristic Equation and Eigenvalues

For a transfer function G(s)=N(s)/D(s),the roots of the characteristic equation D(s)=0 are the poles of the system.

The denominator of the transfer function of a state-space representation is det(sI-A)

The characteristic equation is then det(sI-A)=0

The roots of this equation are also called the eigenvalues of the matrix A


Controllability

Theorem: This system is controllable ifand only if the following controllabilitymatrix has rank n:

x Ax Bu

y Cx Du

= +

= +

&

2 1nS B AB A B A B

!" #= $ %L


Observability

Theorem: This system is observable ifand only if the following observabilitymatrix has rank n:

x Ax Bu

y Cx Du

= +

= +

&

2 1T

nV C CA CA CA

!" #= $ %L


State Feedback

Consider the state-space system

Let r(t) denote the reference signal Then, provided the pair [A,B] is completely

controllable, there exists a feedback of the form

such that the eigenvalues of [A-BK] can be arbitrarilyassigned

( ) ( ) ( )

( ) ( ) ( )

x t Ax t Bu t

y t Cx t Du t

= +

= +

&

1 2 3

( ) ( ) ( )

[ ]

u t r t Kx t

K k k k

= !

� L


Ackermann’s Formula The state feedback gain matrix

that gives the desired characteristic equation

is given by

where

1 2 3[ ] where ( ) ( ) ( )K k k k u t r t Kx t= !� L

1

1( ) n n

nq s s s! !"

= + + +L

[ ] 10 0 1 ( )K S q A!= L

1 1

1[ ] and ( )n n n

nS B AB A B q A A A I! !" "= = + + +L L

review of introductory control theory - phoneoximeter · review of introductory control theory ......

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