repetitive control : power electronics applications repetitive

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Jornadas de Ingeniería de Control Zaragoza Mayo 2005

Ramon Costa CastellóAdvanced Control of Energy Systems (ACES)

Instituto de Organización y Control (IOC)Universitat Politècnica de Catalunya (UPC)

Barcelona, Spain

Repetitive control : Power Electronics

Applications

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Contents• Repetitive Control Basics

– Introduction– Periodic Signals – Performance– Discrete Time– The Odd-Harmonic case– Control Scheme

• Cascade Approach• Plug-in Approach

• The active filter application– Introduction– Basic Concept– Architecture– Control Problem– Experimental Setup– Experimental Results

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Introduction• A key topic in classical control theory is the

Internal Model Principle (IMP). – B. Francis and W. Wonham, “Internal Model Principle

in control theory,” Automatica, vol. 12, pp. 457–465, 1976.

• This principle states that if a certain signal must be tracked or rejected without steady-state error, the generator must be inside the control loop, in the controller, or in the plant itself.

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Introduction : Type Concept

• Standard classical control subjects include the IMP concept implicitly when they introduce the system-type concept.

• The type concept can only be applied to polynomial signals (step, ramp, and parabola) whose generator has the form in the Laplace domain.

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Introduction : Type Concept (II)

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Introduction : Systems with periodical disturbances or

references• In practice, many real systems have to

handle tracking and rejecting periodic signals.

Magnet power supply for a protonsynchrotron (Nakano and others)

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Introduction : Systems with periodical disturbances or

references (II)

Demonstration of the Internal Model Principle by Digital Repetitive Control of an Educational Laboratory Plant. Ramon Costa-Castelló and Jordi Nebot and Robert Griñó.IEEETransactions on Education. Vol. 48, No.1, Pages 73- 80 (February 2005). ISSN : 0018-9359.

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Introduction : Power Electronics

• Inverter : Generating a 50/60 Hz signal from dc one (Tracking a reference signal)

• Active filter : Compensation of harmonic signals (Rejecting periodic signals)

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Periodical Signals

• Any periodical signal can be written as:

• The control loop should include:

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Periodical Signals : Generator

Yamamoto, Y. (1993). Learning control and relatedproblems in infinite-dimensional systems. In: Proceedings of the 1993 European Control Conference. pp. 191-222.

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Periodical Signals : Generator I

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Periodical Signals : Generator II

+pT

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Periodical Signals : Generator III

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Performance

C(s) P(s)

Open Loop Transfer Function

Sensitivity Function

Complementary Sensitivity Function

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Digital Case

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Digital Case II

Nz−+

p sT N T= ⋅

( ) 11NR z

z=

−

2 2j iN

i ip

z e iN T

π πω⋅

= → =⋅

pT

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Odd-Harmonic Case

Digital repetitive plug-in controller for odd-harmonic periodic references and disturbances Robert Griñó and Ramon Costa-Castelló. Automatica. Volume 41, Issue 1,Pages 153-157(January 2005)

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Odd-Harmonic Case II

Pole-Zero Map

Real Axis

Imag

inar

y Ax

is

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

N=3 odd harmonic

N=3 traditional

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Control Scheme

• Cascade form

• Plug-in Form

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Control Scheme : Cascade form

P(z)

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Control Scheme : Plug-in Approach

( )pG z( )cG z

( )xG z( )F zNz−

Repetitive Controller

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Control Scheme : Plug-in Approach II

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Control Scheme : Plug-in Approach III

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Plug-in Approach : Stability Conditions

1. First stability Condition : The System without the Repetitive Controller must be stable.

2. Second stability Condition

3. Third stability Condition :

( )cG z( ) 1F z

∞<

( )xG z

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Plug-in Approach : Filter• F(z) should fulfill the second stability condition.• Usually, a low-pass null-phase FIR filter is used.

• To assure unitary gain a DC frequency the parameters must fulfill :

• No causality problems exist because that the filter is in cascade with a N periods delay.

• The filter reduces the open-loop gain at those frequencies at which uncertainty exists (robustness). Unfortunately it slightly moves the open-loop pole positions in z-plane (precision loose).

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Plug-in Approach : Gx(z)• A common approach to design Gx(z) is

• Unfortunately, this approach cannot be applied to non-minimum-phase plants. Another approach is to cancel minimum-phase zeros and compensate the phase for the non minimum-phase ones:

• kr is fixed by a trade-off between robustness and transient response.

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Contents• Repetitive Control Basics √

– Introduction– Periodic Signals – Performance– Discrete Time– The Odd-Harmonic case– Control Scheme

• Cascade Approach• Plug-in Approach

• The active filter application– Introduction– Basic Concept– Architecture– Control Problem– Experimental Setup– Experimental Results

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Introduction• Proliferation of nonlinear loads ->This

fact has deteriorated the power quality of electrical power systems.

• More stringent requirements proposals IEC-61000-3-{2,4} and IEEE-519.

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Basic Concepts

sv

si Linear Load

Nonlinear Load

Active Filter

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Architecture : Complete Picture

Full BridgeBoost Converter

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Control Problem: Control Goals

• Current in phase with the voltage waveform:

• Constant average value of the voltage at the DC bus capacitor:

*2 0 dx V=

( )* sins d ri I tω= ⋅ ⋅

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Architecture : Boost Converter

1 2 1

2 1

1 sL x x r x vu

C x x⋅ + + ⋅ =⎧

= → ⎨ ⋅ =⎩

&

&

1 2 1

2 1

1 sL x x r x vu

C x x⋅ − + ⋅ =⎧

= − → ⎨ ⋅ = −⎩

&

&

r L

CsV 2x

1x

r L

CsV 2x

1x

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Architecture : Boost Converter II

1 2 1

2 1

1 sL x x r x vu

C x x⋅ + + ⋅ =⎧

= → ⎨ ⋅ =⎩

&

&1 2 1

2 1

1 sL x x r x vu

C x x⋅ − + ⋅ =⎧

= − → ⎨ ⋅ = −⎩

&

&

1 2 1

2 1

sL x ux r x vC x ux⋅ = − − ⋅ +

⋅ =

&

&

1 2 1

2 1

sL x u x r x vC x u x⋅ = − ⋅ − ⋅ +⋅ = ⋅

&

& [ 1,1]u = −The averaged model

{ }1,1u = −

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Control Problem: Current Control loop

1 2 1 sL x u x r x v⋅ = ⋅ − ⋅ +&

2 dx V=

1 1d sL x u V r x V⋅ = ⋅ − ⋅ +&

( )( ) ( )1

1

d

p

Vx s r G s

Lu s sr

−= =⎛ ⎞ +⎜ ⎟⎝ ⎠

( )1 ( )( ) 1 p

p

G sG z z Z

s−=

⎧ ⎫− ⎨ ⎬⎩ ⎭

ZOH, T

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Control Problem: Voltage Loop

2 1C x u x⋅ = − ⋅& 1 2C y u x x⋅ = − ⋅ ⋅&

22

2xy =

( ) ( ) ( ) ( )11

sin cos sind r l r l rl

Iload

x t I t a l t b l tω ω ω∞

=

= − ⋅ + ⋅∑1444442444443

Current loop in steady state

( ) ( ) ( ) ( )21 2 21 12

k T k kd l l dkT

l

E TC y r I b a b I b+ ⋅⎡ ⎤= − − + + + −⎢ ⎥⎣ ⎦

∑

( ) ( )112

k T kdkT

E TC y I b+ ⋅= −

r=0

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Control Problem: Voltage Loop

21

E T

z

⋅

−

1b

PI

2

2dV

y

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Control Problem: Proposed Scheme

• Two control loops :– Current loop : Digital Repetitive Control– Voltage loop : Classical PI Control

( )sin r tω ⋅

Boost Converter

Repetitive Controller

PI Controller2x

dVdI

1x lI

*sI

Odd-Harmonic Digital Repetitive Control of a Single-Phase Current Active Filter . Ramon Costa-Castelló, Robert Griñó & Enric Fossas IEEE Transactions on Power Electronics. Volume: 19, Issue: 4, Year: July 2004. E.Page(s):1060- 1068.

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Experimental Setup• Active filter parameters:

– Capacitor: 6600 uF, 450 V DC – Inductor: 0.8 mH– parasitic resistance: 0.04 Ohm– IGBT: 1200 V, 100 A

• Feedback paths (sensors):– Network voltage: voltage

transformer (220V/15V) – Network current: Hall-effect

sensor (TECSA-HA-050053) (50A)

– DC bus voltage: AD-215BY isolation amplifier

• Control hardware:– ADSP-21161 floating-point

DSP – ADMC-200 coprocessor:

A/D channels and PWM generation

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Experimental Setup : General view

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Experimental setup : IGBT drivers

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Experimental setup : Control hardware

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Experimental Results: Nonlinear Load

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Experimental Results: No-Load

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Experimental Results: Full NL load

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Experimental Results: Full load to No-load

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Experimental Results: No-load to full load

repetitive control : power electronics applications repetitive

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