8/24/2004 ISC 2004 1

-Based Controller Design

for Switching Regulators

with Input Filters

8/24/2004 ISC 2004 2

Authors

Mike Elmore

E&M Power

Johnson City, NY USA

elmore@EandMpower.com

Victor Skormin

Electrical and Computer Engineering Department

Watson School, Binghamton University

Binghamton, NY USA

8/24/2004 ISC 2004 3

Presentation Outline

• Problem Definition

• Background

Previous Work

Switching Power Converter and

Input Filter Interactions

Robust Control with -synthesis

• Analysis and Simulation Results

Stability Analysis

Frequency and Transient Responses

• Summary and Conclusions

8/24/2004 ISC 2004 4

Problem Definition

• Parametric uncertainty in switching power supplies and their loads

can result in unstable operation or failure to meet performance

specifications.

These uncertainties include: power supply component initial

tolerance and variations with age, temperature, the manufacturing

process, and electrical loading.

• Interactions between input filters and switching power supplies can

also lead to unstable operation.

• Taken together, parametric uncertainty and input filter interactions,

present a challenging design problem.

This study proposes a controller, designed with an -synthesis,

to mitigate the effects of parametric uncertainty

in switching power supplies with input filters.

8/24/2004 ISC 2004 5

Previous Work

• R. D. Middlebrook, “Input filter considerations in design and application of switching regulators,” Proc. IEEE Ind. Applicat. Soc. Annual Meeting, pp. 366 – 382, Chicago, IL, Oct. 11 – 14, 1976.

• R. W. Erickson, “Optimal single resistor damping of input filters,” Proc. of 14 th

Annual IEEE Applied Power Electron. Conf., pp. 1073 – 1079, Dallas, TX, Mar. 14 – 18, 1999.

• X. Feng et al, “Impedance specifications for stable DC distributed powersystems,” IEEE Trans. on Power Electron., vol. 17, no. 5, pp. 157 – 162, Mar. 2002.

• R. Tymerski, “Worst-case stability analysis of switching regulators using thestructured singular value,” IEEE Trans. on Power Electron., vol. 11, no. 5, pp. 723 - 730, Sept. 1996.

• S. Buso, “Design of a robust voltage controller for a buck-boost converter using -synthesis,” IEEE Trans. on Control System Technology, vol. 7, no. 2, pp. 222

– 229, Mar. 1999.

8/24/2004 ISC 2004 6

RsL

Cs

Ls

vs

Converter and Input Filter Interactions

Input Filter

Zs

Zi

Buck Switching Power Supply

L

C

Rl

Modulator A

DCReference

vc

d

vovg

R

,1

1

11

1

1 where

,011

22

1

DZTDRT

T

Z

TZ

Z

eii

i

s

Zei

T

T is the main loop gain, T1 is a minor loop gain, and

Zei is the output filter input impedance.

Condition for stability:

Rc

8/24/2004 ISC 2004 7

is a sufficient, but

more-than-necessary condition for

system stability.

Stability Analyses

Most design strategies to mitigate interactions have focused on reducing peaking in the

input filter output impedance.

Less attention has been given to reducing

peaking in the input impedance of the

switching converter.

11 is ZZT

T1 must satisfy the Nyquist criterion.

8/24/2004 ISC 2004 8

Buck Parametric Uncertainty

L = 1.80H 50%

C = 1750F 50%

R = 1.0 50%

Rc = 9m 50%

Rl = 30m 50%

Vg = 12V (constant)

5.21kHz < fc < 15.0kHz

31.5 < m < 99.9

8/24/2004 ISC 2004 9

Linear Fractional Transformation

u

w

MM

MM

y

v

2221

1211

12

1

112122,

where,,

MΔMIΔMMΔM

uΔMy

u

u

F

F

.y uncertaint the todue from deviation theis

. to from gain loop-closed nominal therepresents

.invertible is if defined,- wellis ,

2212

1

1121

22

11

ΔMMΔMIΔM

yuM

ΔMIΔM

uF

My u

wv We can partition M, so that the input-output

relationships may be given as:

8/24/2004 ISC 2004 10

Averaged Descriptor State-Space Affine LFR

cc

c

cl

RRRR

R

RR

RRRR

1A

GRR cl

10

00

00

01

00

01

01

10

Assume Rc << R

Averaged, Descriptor

State-Space Form

Affine Parameter Dependent Form

(G = 1/R )

In similar fashion the B, C, D, and E matrices are transformed.

LFR

R

RR cl

11

1

v

8/24/2004 ISC 2004 11

The Structured Singular Value and Stability

Question: What is the smallest uncertainty in the sense of

such that det(I - M) = 0 ?

0detmin

1

ΔMIΔ

M

Δ

Δ

Δ

Structured Singular Value:

bound'upper least ' theis sup where

,sup:

jMMH Norm:

.sup iff 111

jMΔ

If the nominal system M11(s) is stable, then the perturbed system

(I -M11)-1 is stable for all stable with

This is a generalization of the small-gain theorem.

8/24/2004 ISC 2004 12

Robust Stability Versus Robust Performance

M(s)y u

wvp

M(s)

y u

wv

F

P

Δ

Δ

Δ

0

0

:The uncertainty matrix is augmented

to include performance criteria:

0detmin

1

11

11

ΔMIΔ

M

Δ

Δ

0detmin

1

PPP

P

ΔMIΔ

M

Δ

Δ

8/24/2004 ISC 2004 13

-Synthesis

sss

sss

sss

s

333231

232221

131211

)(

MMM

MMM

MMM

M

M(s)y u

wv

K

The goal of -synthesis is to minimize over all stabilizing controllers K

the peak values of () of the closed-loop transfer function FL(G, K).

This is done via an software process know as D-K iteration.

G is defined as a sub-matrix of M for nominal

performance (G = M22), robust stability

(G = [M11 M12 ; M21 M22 ]), or robust

performance (G = M ).

uKΔM

uΔKMy

,,

,,

u

u

FF

FF

8/24/2004 ISC 2004 14

Controller Design Software

• Define averaged, descriptor state-space equations

• Transform averaged, descriptor state-space equations to affine form

• Transform affine formto LFR form

Normalize uncertainty block to [-1, 1]

• Define the system matrix M

• Define weighting functions for performance uncertainties

• Create the interconnected system

Buck_all_dk.m

Buck_Plant_all.m buck_dk_defin.m dkit.m buck_all_dk_control.m

• Find optimum controller withD-K iteration

• Define D-K iterationparameters

• Import optimum controllerfrom D-K iteration result

• Define non-optimumcontroller

• Perform mu analyses withoptimum and non-optimumcontrollers

• Make plots of step responses,impedances, open-loop gain,and mu for stability andperformance

All m-files use commands found in analysis and

synthesis, LMI control, and LFR toolboxes.

Red boxes are user defined m-files.

(Uses Buck_Plant_all.m

andbuck_dk_defin.m)

8/24/2004 ISC 2004 15

Mitigation Design Approaches

The problem of interactions between a buck converter and its input filter is addressed

with

• a robust controller, designed with -synthesis (D-K iteration) to minimize

converter input impedance peaking

Conventional design methods mostly address main loop stability and load and input

voltage transient responses.

Design often proceeds by trial-and-error, either with a converter simulation model or

in the laboratory. The goal is to maximize bandwidth, while achieving acceptable

phase and gain margins.

The design objective rarely focuses on optimization of converter input and/or output

impedances or audiosusceptibility, explicitly. Interactions between the converter and

its input filter frequently occur.

• a 2-stage, phase-lead controller with an integrator is designed with PSpice

and a genetic algorithm to minimize converter input impedance peaking

For comparison purposes

8/24/2004 ISC 2004 16

Robust Controller Synthesis with D-K Iteration

11086.5

11016.55.0

15610

11012.2

4697

54

2

7

sss

ss

KOPT

Weighting function for tracking error Weighting function for input impedance

Reduced optimum controller

Controller comparison

D-K iteration

result

8/24/2004 ISC 2004 17

Input Impedance and Main Loop Gain T with Robust Controller

Note the low phase marginNote the sharp peak

Red results use a non-optimum, 2-stage, phase-lead controller K with an integrator.

Blue results use the optimum robust controller Kopt designed with D-K iteration.

8/24/2004 ISC 2004 18

Nyquist Plot with Robust Controller

This shows that the system

with worse case parameters

and a 2-stage, phase-leadcontroller with an integrator is unstable, since the Nyquistplot encircles (-1, j 0).

The system with the optimum controller has stable T1.

(Gm 90.1, m- = 115,

and m+ = 138)

m- = 115

m+ = 138

Gm = 90.1

8/24/2004 ISC 2004 19

Controller Optimization with GenSpice

11091.1

11039.4

11024.4

15291

19088

74

4

, sss

ss

K GAOPT

Input impedance

measurement

Main Loop Stability

measurements

Optimization

Setup Menu

Controller comparison

GenSpice result

8/24/2004 ISC 2004 20

Nyquist Plot with GenSpice Optimized Controller

This shows that the system

with worse case parameters

and a 2-stage, phase-leadcontroller with an integrator is unstable, since the Nyquistplot encircles (-1, j 0).

The system with the optimum controller has stable T1.

(Gm 7.6, m1- = 54,

and m2- = 143)

m2- = 143m1

- = 54

Gm = 7.6

8/24/2004 ISC 2004 21

Summary and Conclusions

• Interactions between switching regulators and their input filters can result in

unstable power converter systems.

Usually, input filters are designed (or redesigned) to mitigate interactions.

However, this often results in larger and/or more expensive implementations.

• -based, robust controllers designed with D-K iteration can prevent input

filter instabilities resulting from converter output filter peaking.

This has been demonstrated with an optimum controller designed with several

Matlab toolboxes.

This design approach accounts for uncertainty in power train components and

can result in excellent stability phase and gain margins.

• Conventional phase-lead controllers with an integrator can be optimized to

prevent input filter instabilities resulting from converter output filter peaking.

This has been demonstrated with an optimum controller designed with PSpice

and genetic algorithms (GenSpice).

However, this approach does not account for uncertainty in power train components.