controller design in the pc epoch: cad-tools and auto-tuning

DEPARTMENT OF CONTROL ENGINEERING AND AUTOMATIONGHENT UNIVERSITY

- FREQUENCY RESPONSE DESIGN TOOL - Robin DE KEYSER & Cristian VLASIN - 1

Controller Design in the PC Epoch:

CAD-TOOLS and AUTO-TUNING

Prof.Dr.ir. Robin DE KEYSER

[email protected]



SUMMARY



• simple dynamical compensators (of type PID, Phase-Lead-Lag,…) have done in the past a remarkably good and efficient job in real-life control applications• some powerful and well-developed theories are usually considered as basic knowledge for every control engineer: Root Locus and Frequency Response techniques• in the past, these theories have mainly been used as analysis tools, e.g. to analyse the stability of a given control system: closed-loop poles (RL) and Nyquist (FR)• they have never extensively been used as tools in order to design a good and useful dynamical compensator for a practical control engineering problem• nowadays, thanks to the computational and graphical power of modern PC, the RL and FR methods can be implemented as interactive, graphical design tools• in this way, control engineering finally moves away from being an abstract and mathematical-oriented discipline and it evolves gradually towards a mature engineering discipline, with design approaches similar to those used in other engineering fields, such as mechanical engineering, electrical engineering,…• recently, a ‘RLtool’ has been introduced in Matlab®; here, a ‘FRtool’ is presented• as a by-product, it is shown that modern ‘Auto-Tuners’ can lead to remarkably good performance, although they are based on very limited process knowledge



1 100.10.01

50

0

-50

-100

0-50

-100-150-200

M(dB)

20*log10(M)

ϕϕ

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 100 200 300

0

1

2

-2

-1

Ai

Ao

τ

TTπ2πf2ω ==

i

o

AA)M( =ω ωτ=ωϕ )(

Bode Diagrams Nyquist Plot

-1 M ϕ

ω

0.5

0

-0.5

-1

-1.50.50-0.5-1-1.5

M(dB)

ϕ

ω50

0

-50

-100-90-135-180-225-270

20*log10(M)

ϕ

Nichols Chart

Frequency Response (FR)

Linear System

Frequency ω (log)



CAD-TOOL



WHY FRtool?

• FR can deal with time-delay systems• The interesting part of the FR can be easily obtained experimentally• No such design tool available in Matlab®

FEATURES

• User friendly graphical interface (drag&drop and zoom included)• Possibility to display design specs as graphical restrictions on the Nichols plot• Real-time update of Nichols plot while dragging controller’s poles/zeros• Import/export process and controller from/to the Matlab® workspace• Option to print Nichols, Nyquist or Bode curves and closed-loop responses



- GUI - (1)



- GUI - (2)



PRACTICAL design SPECS Additional (traditional) design specs

1. Phase margin (PM)2. Gain margin (GM)

1. Robustness (Ro)2. Performance: settling time (Ts)3. Overshoot (OS)

1 2p

1 2

1 2

1 ( )( )PID: K (1 )

( )( )...General: ( )( )...

di

s z s zT s KT s ss z s zKs p s p

+ ++ + =

+ ++ +

C(s) P(s)-+ +

+ ++

nd

uew ySP ER MV CV



Robustness( ) ( ) ( )G s C s P s=

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

Y s G sT sW s G s

= = ≈+

( ) 1 ( ) ( 0)( ) 1 ( )Y s P sD s G s

= ≈+

( ) 1 ( 0)( ) 1 ( )

Y sN s G s

= ≈+

( ) / ( ) ( ) ( ) 1 ( 0)( ) / ( ) ( ) ( ) 1 ( )T s T s dT s G sG s G s dG s T s G sδδ

= = ≈+

Im

Re( )G jω1 ( )G jω+

-1

CL-sensitivity:

11 ( )G jω+

sensitivity-function: as small as possible (for all frequencies)

)G jω+1 (robustness-function: as large as possible (for all frequencies)

0

1log 01 ( )

dG j

ωω

∞

=+∫The Balloon Theorem (Bode):

Im

Re-1

Roα 2 220log( Re Im )dBM = +

degIm180 arctan( )Re

Φ = − +

Re 1 *cos( )Ro α= − +

Im *sin( )Ro α=



Overshoot and Settling-Time2

2 22n

n ns sωζω ω+ +Dominant 2nd-order CL:

Time-domain:

%OS ζ⇒21% 100OS e ζπ ζ− −=

4s

n

Tζω

≅ (2% settling time) s nT ω⇒

Frequency-domain:

2

12 1

pMζ ζ

=−

(smaller than!)pMζ ⇒

2 4 21 2 4 4 2BW nω ω ζ ζ ζ= − + − + n BWω ω⇒ (larger than!)



Design specs – Graphical equivalents

OS

PM

Ro

GM

Ts



- Design example – Process & Specs (1)

0.22500( ) *( 25)

sP s es s

−=+

Given the plant:

Design a phase-lead controller in the frequency domain so that the feedback system will satisfy the specs:• overshoot %OS< 5%• settling time Ts < 0.8s




Sys = tf(2500,[1 25 0],’InputDelay’, 0.2);

Defining the system in the Matlabcommand window, importing it in the P block of FRtool and defining the given constraints,

the Nichols curve will be the oneshown beside.

Phase-lead controller has:1( ) , 1

1TsC sTs

αα+

= <+

Pole Zero>




Chosen:

Pole = -25Zero = -2



- Design example – Design steps (1)

Zero = -5 Zero = -1




Pole = -60 Pole = -10




Gain = 3 Gain = 0.3



- Design example – Results

Gain = 0.14631Pole = -61.3185Zero = -11.8188

OS = 2% < 5%Ts = 0.574 < 0.8



BASIC control problem

Flight Control System: (angular) velocity control(servo, robot, antenna, disk drive, DVD,…)

32( 4)( 16)s s+ +

Nominal system:

Specs:• robustness > 0.7• overshoot < 10%• settling time < 1



Ro > 0.7OS < 10%Ts < 1

Zero: -6PI: Kp=2.2 Ti=0.17Gain:2.2*6=13.2

Ro > 0.7OS < 10%Ts < 1

Zero: -3PI: Kp=7.7 Ti=0.33Gain:7.7*3=23.1



Time (s ec.)

Am

plitu

de

S tep Res pons e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

To: Y

(1)

z=-3

z=-6

Time (s ec.)A

mpl

itude

S tep Res pons e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

To: Y

(1) nominal

robustness! z=-3 (both)

48( 5)( 12)s s+ +

32( 4)( 16)s s+ +

nominal: robustness!:

controller design in the pc epoch: cad-tools and auto-tuning

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