Digital Control of a Power Inverter Fabio Celani† , Michele Macellari‡ , Alfiero Schiaratura‡ and Luigi Schirone‡ † Department of Computer and Systems Science Antonio Ruberti

Sapienza University of Rome Italy

‡ Department of Aerospace and Astronautical Engineering Sapienza University of Rome Italy

The Eleventh IASTED Conference on Control and Applications Cambridge, UK July 14th , 2009

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Outline

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background on digital control of power inverters

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Outline

I

background on digital control of power inverters

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novel approach based on robust linear control

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Outline

I

background on digital control of power inverters

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novel approach based on robust linear control

I

experimental results

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Power Inverters

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(static) power inverter DC power supply → AC power supply

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strict specifications on total harmonic distorsion I I

over wide range of DC input voltages in presence of severe load variations

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Pulse Witdth Modulation (PWM) Power Inverter




TON + TOFF = TS constant

duty cycle d =

TON TOFF

Vr (t) biased sinusoidal reference voltage Controller enforces Vo (t) ' Vr (t)

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Pulse Witdth Modulation (PWM) Power Inverter




TON + TOFF = TS constant

duty cycle d =

TON TOFF

Vr (t) biased sinusoidal reference voltage Controller enforces Vo (t) ' Vr (t) Vo (s) = G (s)Vin (s)

G (s) =

1 LCs 2 + RL s+1

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Pulse Witdth Modulation (PWM) Power Inverter




TON + TOFF = TS constant

duty cycle d =

TON TOFF

Vr (t) biased sinusoidal reference voltage Controller enforces Vo (t) ' Vr (t) Vo (s) = G (s)Vin (s)

G (s) =

1 LCs 2 + RL s+1

Vin (t) = Vcc d(t)

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Pulse Witdth Modulation (PWM) Power Inverter




TON + TOFF = TS constant

duty cycle d =

TON TOFF

Vr (t) biased sinusoidal reference voltage Controller enforces Vo (t) ' Vr (t) Vo (s) = G (s)Vin (s)

Vr (t) +

−

G (s) =

Controller

1 LCs 2 + RL s+1

d(t)

Vin (t) = Vcc d(t)

VccG(s)

Vo(t)

circuit

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Digital Control

Vr∗(k)

Vr (t) Ts

+

e∗(k) digital d(t) d∗(k) VccG(s) ZOH controller −

Vo(t)

Vo∗(k) Ts

digital controller enforces Vo (t) ' Vr (t)

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Digital Control

Vr∗(k)

Vr (t) Ts

+

e∗(k) digital d(t) d∗(k) VccG(s) ZOH controller −

Vo(t)

Vo∗(k) Ts

digital controller enforces Vo (t) ' Vr (t) previous designs 1. sliding-mode control (Jezernik et al., Biel et al., Ramos et al.) I I

pros: robustness w.r.t. parametric uncertainties and external disturbances cons: chattering

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Digital Control

Vr∗(k)

Vr (t) Ts

+

e∗(k) digital d(t) d∗(k) VccG(s) ZOH controller −

Vo(t)

Vo∗(k) Ts

digital controller enforces Vo (t) ' Vr (t) previous designs 1. sliding-mode control (Jezernik et al., Biel et al., Ramos et al.) I I

pros: robustness w.r.t. parametric uncertainties and external disturbances cons: chattering

2. dead-beat control (Kawabata et el, Kojima et al.) I I

pros: fast transient response cons: sensitivity to parametric uncertainties and external disturbances

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Digital Control

Vr∗(k)

Vr (t) Ts

+

e∗(k) digital d(t) d∗(k) VccG(s) ZOH controller −

Vo(t)

Vo∗(k) Ts

digital controller enforces Vo (t) ' Vr (t) previous designs 1. sliding-mode control (Jezernik et al., Biel et al., Ramos et al.) I I

pros: robustness w.r.t. parametric uncertainties and external disturbances cons: chattering

2. dead-beat control (Kawabata et el, Kojima et al.) I I

pros: fast transient response cons: sensitivity to parametric uncertainties and external disturbances

novel design: robust linear control

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Robust Linear Control w(t) Vr (t) Ts

Vr∗(k)

∗

e (k) +

−

R∗(z)

∗

d (k)

ZOH

d(t)

VccG(s)

+ V (t) o +

Vo∗(k) Ts

Vr (t) = Vdc + Vac cos(2πf0 t)

w (t) = step(t)

uncertain Vcc

objective Vo (t) − Vr (t) → 0

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Robust Linear Control w(t) Vr∗(k)

Vr (t) Ts

∗

e (k) +

−

R∗(z)

∗

d (k)

ZOH

d(t)

+ V (t) o

VccG(s)

+

Vo∗(k) Ts

Vr (t) = Vdc + Vac cos(2πf0 t)

w (t) = step(t)

uncertain Vcc

objective Vo (t) − Vr (t) → 0

Discrete-time Equivalent Vr∗(k)

e∗(k) +

−

R∗(z)

Vr∗ (k) = Vdc + Vac cos(θk) G ∗ (z)

d∗(k)

VccG∗(z)

θ = 2πf0 TS

w∗(k) + +

Vo∗(k)

w ∗ (k) = step∗ (k)

zero-order hold equivalent of G (s)

objective Vo∗ (k) − Vr∗ (k) → 0 6 / 14

Robust Linear Control (cont’d) w∗(k) Vr∗(k)

e∗(k) +

−

R∗(z)

d∗(k)

VccG∗(z)

Vr∗ (k) = Vdc + Vac cos(θk) θ = 2πf0 TS

+ +

Vo∗(k)

w ∗ (k) = step∗ (k)

objective Vo∗ (k) − Vr∗ (k) → 0

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Robust Linear Control (cont’d) w∗(k) Vr∗(k)

e∗(k) +

−

R∗(z)

d∗(k)

VccG∗(z)

Vr∗ (k) = Vdc + Vac cos(θk) θ = 2πf0 TS

+ +

Vo∗(k)

w ∗ (k) = step∗ (k)

objective Vo∗ (k) − Vr∗ (k) → 0 ˜ R ∗ (z) = R(z)

z2 (z − 1)(z − e jθ )(z − e −jθ )

poles in z = 1, z = e ±jθ → asymptotic tracking and disturbance rejection

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Robust Linear Control (cont’d) w∗(k) Vr∗(k)

e∗(k) +

−

R∗(z)

d∗(k)

VccG∗(z)

Vr∗ (k) = Vdc + Vac cos(θk) θ = 2πf0 TS

+ +

Vo∗(k)

w ∗ (k) = step∗ (k)

objective Vo∗ (k) − Vr∗ (k) → 0 ˜ R ∗ (z) = R(z)

z2 (z − 1)(z − e jθ )(z − e −jθ )

poles in z = 1, z = e ±jθ → asymptotic tracking and disturbance rejection term z 2 at the numerator

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latency time reduced

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strictly proper R ∗ (z)

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Robust Linear Control (cont’d) w∗(k) Vr∗(k)

e∗(k) +

−

R∗(z)

d∗(k)

VccG∗(z)

Vr∗ (k) = Vdc + Vac cos(θk) θ = 2πf0 TS

+ +

Vo∗(k)

w ∗ (k) = step∗ (k)

objective Vo∗ (k) − Vr∗ (k) → 0 ˜ R ∗ (z) = R(z)

z2 (z − 1)(z − e jθ )(z − e −jθ )

poles in z = 1, z = e ±jθ → asymptotic tracking and disturbance rejection term z 2 at the numerator

−

˜ R(z)

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latency time reduced

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strictly proper R ∗ (z) z2 V G∗(z) (z−1)(z−ejθ )(z−e−jθ ) cc

˜ R(z) stabilizer robust over a wide range of Vcc ’s 7 / 14

Robust Stabilization −

f0 =400 Hz

Ts =20 µs

˜ R(z)

z2 V G∗(z) (z−1)(z−ejθ )(z−e−jθ ) cc

0 =50 V V 0 L=20µH C =50 µF R=4 Ω Vcc cc = Vcc ±30%

0 G ∗ (z) z 2 9.36(z + 0.97) z 2 Vcc = jθ −jθ j0.05 (z − 1)(z − e )(z − e ) (z − 1)(z − e )(z − e −j0.05 )(z 2 − 1.54z + 0.90)

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Robust Stabilization −

f0 =400 Hz

Ts =20 µs

˜ R(z)

z2 V G∗(z) (z−1)(z−ejθ )(z−e−jθ ) cc

0 =50 V V 0 L=20µH C =50 µF R=4 Ω Vcc cc = Vcc ±30%

0 G ∗ (z) z 2 9.36(z + 0.97) z 2 Vcc = jθ −jθ j0.05 (z − 1)(z − e )(z − e ) (z − 1)(z − e )(z − e −j0.05 )(z 2 − 1.54z + 0.90)

Root Locus

1 0.8 0.6

Imaginary Axis

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

0 Real Axis

0.5

1

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Robust Stabilization ˜ R(z)

−

z2 V G∗(z) (z−1)(z−ejθ )(z−e−jθ ) cc

35 ≤ Vcc ≤ 65

3.84 10−10 (z + 1)2 ˜ R(z) = (z − 0.9608)2 Root Locus

Root Locus 0.06

1.5

0.05

Imaginary Axis

0.04 0.03

1

0.02 0.01

Imaginary Axis

0

0.5

−0.01 −0.02 0.94

0.96

0.98 Real Axis1

1.02

1.04

0

−0.5

−1

−1.5

−1.5

−1

−0.5

0 Real Axis

0.5

1

1.5

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Digital Regulator

w(t) Vr (t) Ts

Vr∗(k)

e∗(k) +

−

R∗(z)

d∗(k)

ZOH

d(t)

VccG(s)

+ V (t) o +

Vo∗(k) Ts

˜ R ∗ (z) = R(z)

=

z2 (z − 1)(z − e jθ )(z − e −jθ )

3.845 10−10 z 4 + 7.689 10−10 z 3 + 3.845 10−10 z 2 z 5 − 4.919z 4 + 9.68z 3 − 9.527z 2 + 4.689z − 0.9231

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Experimental Results - Nominal Conditions

0 = 50 V Vcc = Vcc

Output voltage in steady-state

no load variations

FFT of output voltage in steady-state

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Experimental Results - Nonlinear Load 0 = 50 V Vcc = Vcc


 390 Ω R = 4 Ω C = 30 µF R= L

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Experimental Results - Nonlinear Load 0 = 50 V Vcc = Vcc


 390 Ω R = 4 Ω C = 30 µF R= L

Output voltage and current

FFT of output voltage in steady-state

transient ends within hundreds of µs’s

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Experimental Results - Variations in Vcc

0 = 50 V Vcc

Output voltage corresponding to a +20% step variation of Vcc

Output voltage corresponding to a -20% step variation of Vcc

transient ends within tens of ms’s

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Conclusions

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digital control system for a power inverter effective in presence of fast load variations and over a wide range of DC input voltages

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Conclusions

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digital control system for a power inverter effective in presence of fast load variations and over a wide range of DC input voltages

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the control algorithm is computationally complex

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