Repetitive control : Power Electronics Applications Ramon Costa Castelló Advanced Control of Energy Systems (ACES) Instituto de Organización y Control (IOC) Universitat Politècnica de Catalunya (UPC) Barcelona, Spain
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
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.
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.
Introduction : Type Concept (II)
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 proton synchrotron (Nakano and others)
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.IEEE Transactions on Education. Vol. 48, No.1, Pages 73-80 (February 2005). ISSN : 0018-9359.
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)
Periodical Signals Any periodical signal can be written as: The control loop should include:
Periodical Signals : Generator Yamamoto, Y. (1993). Learning control and related problems in infinitedimensional systems. In: Proceedings of the 1993 European Control Conference. pp. 191-222.
Periodical Signals : Generator I
Periodical Signals : Generator II T p +
Periodical Signals : Generator III
Performance C(s) P(s) Open Loop Transfer Function Sensitivity Function Complementary Sensitivity Function
Digital Case
Digital Case II ( ) R z T p = z + N 1 T p 1 z N = N T s 2π j i N 2π z = e ω = i i i N T p
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)
Odd-Harmonic Case II N=3 traditional N=3 odd harmonic Imaginary Axis Pole-Zero Map 1 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1 Real Axis
Control Scheme Cascade form Plug-in Form
Control Scheme : Cascade form P(z)
Control Scheme : Plug-in Approach Repetitive Controller z N F( z) Gx ( z) Gc ( z) Gp ( z)
Control Scheme : Plug-in Approach II
Control Scheme : Plug-in Approach III
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 : Gx ( z) Gc F( z) < 1 ( z)
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).
Plug-in Approach : G x (z) A common approach to design G x (z) is Unfortunately, this approach cannot be applied to nonminimum-phase plants. Another approach is to cancel minimum-phase zeros and compensate the phase for the non minimum-phase ones: k r is fixed by a trade-off between robustness and transient response.
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
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.
Basic Concepts v s i s Linear Load Active Filter Nonlinear Load
Architecture : Complete Picture Full Bridge Boost Converter
Control Problem: Control Goals Current in phase with the voltage waveform: ( ) * sin s d r i = I ω t Constant average value of the voltage at the DC bus capacitor: * x = V 2 0 d
Architecture : Boost Converter u r L r L & = 1 C x& = x 1 x 1 Vs x2 L x1+ x2 + r x1 = v s x C 2 1 & = 1 C x& = x L x1 x2 + r x1 = v s 2 1 u Vs x2 C
Architecture : Boost u & + + = = 1 C x& 2 = x1 Converter II L x1 x2 r x1 v s & L x1 = ux2 r x1+ v s C x& = ux & 2 1 L x1 = u x2 r x1+ v s C x& = u x 2 1 & u = 1 C x& = x L x1 x2 + r x1 = v s 2 1 u = { 1,1} The averaged model u = [ 1,1]
Control Problem: Current L x& = u x r x + v Control loop 1 2 1 s x = V 2 d L x& = u V r x + V 1 d 1 V x ( ) 1 s = r = G ( ) ( ) p s u s L d s + 1 r s G () ( 1 ( ) 1 ) p s Gp z = z Z s ZOH, T
Control Problem: Voltage Loop C x y = x 2 C y& = u x x & = u x 2 1 1 2 ( ) 2 2 x1 () t = Id sin ( ωrt) al cos( l ωrt) + blsin ( l ωrt) l= 1 1444442444443 Iload Current loop in steady state k+ 1 T k 2 2 ET k 2 ( d 1) ( l l ) ( d 1) C y = r kt I b + a + b + I b l 2 r=0 ( k+ 1) T ET k C y = I b kt 2 ( ) d 1
Control Problem: Voltage Loop 2 V d 2 PI E T z 2 1 y b 1
Control Problem: Proposed Two control loops : Scheme Current loop : Digital Repetitive Control Voltage loop : Classical PI Control I d sin ( ωr t) Repetitive Controller PI Controller Boost Converter 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. x 2 x 1 V d * I s I l
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
Experimental Setup : General view
Experimental setup : IGBT drivers
Experimental setup : Control hardware
Experimental Results: Nonlinear Load
Experimental Results: No-Load
Experimental Results: Full NL load
Experimental Results: Full NL load
Experimental Results: Full load to No-load
Experimental Results: No-load to full load