1Inductor Current State-space based Estimation [614958]

1Inductor Current State-space based Estimation
within a Point-of-Load Buck Converter

Dorin O. Neac șu
Department of Applied Electronics and Intelligent S ystems,
Technical University of Ia și, Romania
Email: [anonimizat]

Abstract – The recent emergence of digital control for DC/DC
converters requires the deployment of modern contro l algorithms.
This paper explores further the usage of state-spac e based control
to buck converters with a discussion of the current estimation
methods. These are coded in assembly language for f aster calling
from a master C program within a microcontroller pl atform.
Keywords – DC-DC power converters, State-space meth ods,
Microcontrollers, Linear programming, estimation.
I. INTRODUCTION
Digital DC/DC power converters are seeing most cont rol laws
derived after modeling in state-space [1, 2, 3, 4, 5, 6], followed
with a conversion into a Laplace transfer function. Isolated
examples of control with state-space are reported i n research
literature [7, 8, 9, 10, 11, 12, 13, 14]. Therein, a feedback control
law derived with state-space equations assumes the change of
the pole locations in order for the dynamic perform ance to
improve or change. The feedback is defined now afte r the state
variables rather than the output(s). The deployment of state-
space based control is facing difficulty of impleme ntation on
low-cost platforms of entire analytical form of the control law.
This paper
o Builds upon an equivalent form of the state-space c ontroller
with input voltage feed-forward component to the cl assical
PI controller plus a compensation component [4,5,6, 15,16];
o Consider direct calculation and observer methods.
o Introduces a simplified form for implementation of the
estimation on low-cost microcontroller platforms.
II. STATE -SPACE MODELING
A buck converter is considered with 225 kHz switchi ng
frequency, 20 A nominal load current, 5 V nominal i nput
voltage, an output voltage in the range 0.5 .. 3.5 V, with a
nominal at 2.5 V, buck inductance 1.6 mH, with loss of 1 m Ω,
composite output capacitor bank with equivalent cap acitance of
1600 µF and equivalent loss resistance of 50 m Ω.

Fig.1 DC/DC Buck Converter The simplified state-space model yields [1,2,3,4].
(1)
III. STATE -SPACE BASED CONTROL
The state-space control law aims at system pole dis placement
into another location, helping better transient res ponse and
stability. Such control system would depend upon th e parameter
variation and would generate steady-state error. Th e solution
consists in adding a new state variable as the erro r integral.
An empirical approach is used herein to selection o f the novel
pole location: switching frequency is 225 kHz (4.44 µs) and the
effect of digitization allows a proper signal recon struction up to
half the sampling period, that is a delay of 2.22 µs. Since any
delay reduces phase reserve and pushes the system t owards
instability, consider the phase shift introduced at 16 kHz, that is
~10.4 o. The pole’ location is therefore chosen around 16 kHz:
Pd = [-90000+43000i -90000-43000i -100000 ] (2)
The control gains are calculated in MATLAB [4,5,6].
Adding an integral term to the converter poles has a negative
effect on the dynamic behavior as it tends to slow down the
transient response. To compensate for this, a feed- forward term
[N] from reference to the plant input is added. This g ain [N] is
chosen to produce a zero at the same frequency with the location
of the pole resulted after moving the integral pole at 0 Hz.
The control system can be re-arranged like in Fig. 2 in order
to show an equivalence to the more conventional PI control
[4,5,6,15,16]. This is very advantageous for implem entation
since it uses library code only. The first term, th e PID control
was herein implemented in library (assembly languag e) and
called from a system-level C-based control program. The
general form of the PID controller is coded after
[ ] − ⋅ ⋅ + ⋅+ ⋅ =
kd i p k ek eTKi eKk eKk y ] [ ] [1] [ ] [ ] [ (3)
] [ ] 1 [ ] [1k eK T kACC kACC ⋅⋅ + − = (4)
] [ ] [ ] 1 [ kACC k e NDk d + ⋅+=+ (5)
where e[k] is the last calculated error for the output voltage .
The second term needed for state-space control repr esents a
pure linear relationship between various measured a nd reference
Perturbation
=0, when Vin=ct input
⎩⎪⎨⎪⎧
=0−1

1

−1
∙
∙+

0∙+

0∙
=01∙

2values, and it is named " compensation ". The analytical form of
the "compensation" term yields:
] [ ] [ ) 3 ( ] [ ) 2 ( ] 1 [ Δ kv NkvKk iKk dout out L ⋅+ ⋅ − ⋅ − = + (6)
Fig. 2 PI equivalent form of the state-space contro ller.
IV. CURRENT ESTIMATION
4.1. Direct calculation
This method assumes direct use of the state equatio ns (1), with
replacement of the capacitor voltages (state variab le) samples
with the output voltage measurements after the Tust in
approximation:
(7)
After some arrangement, the inductor current can be
calculated with a generic sum-of-products form
(8)
where
(9)
It is worthwhile to mention that the estimation bas ed on plant
model does not account for a change in duty cycle. Hence, a
possible estimation steady-state error. Such steady -state error is
eventually rejected or attenuated within the closed -loop control
of the output voltage since estimated current i L in “compensation
term” becomes balanced within control’s digital acc umulator.
Figs. 3 shows some results with application of this method.

4.2. Reduced order estimator
It is possible to simplify the theoretical estimato r equations
since the output voltage (“system output”) is also a “state
variable”. This means to implement an estimator for a single
state variable, that is the inductor current. Gener ally, this is the
theory for a reduced-order estimator . The second state equation
is used along with the actual value of the output v oltage for the
state variable vc. A true estimator is able to adjust operation
based on error of estimated state equation (Fig. 4) .
The general-form state-space equations for estimato r yield
(10)
and it can be applied to the estimated inductor cur rent
(11) (a)
(b)
Fig. 3. (a) Change of input voltage from 5 V to 6 V (20%) with the same output
voltage of 2.5 V; (b) Step change of load current f rom 20 A to 8 A.

Fig. 4 Reduced order estimator within the entire sy stem.

The estimator error is next defined as . (12)
The dynamic equation for the estimator error signal yields by
subtraction from (1) (13)
as
(14)
with the characteristic equation s + (E/C)=0 . (15)
There is a single estimator pole for the estimation of a single
state variable (inductor current herein). For a res ponse as fast as
possible, a pole with a frequency as high as possib le would
work. Therefore, the estimator pole needs to be at least 2 ..8
times higher than any other system poles so that it would not
influence the system behavior. That is any estimato r dynamics
is done before the main control law can react. On t he other hand,
a very large pole frequency would allow noise inter vene.
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Buck converter
Compensation
Δ 
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F3 ⋅  IJK  ⋅  IJK -KI 1/s
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Fig. 5 Results for a 20 A to 8 A load change with a n estimator pole at 70 kHz.
Fig. 6 Results for a 20 A to 8 A load change with a n estimator pole at 150 kHz.

After trail-and-error efforts, a 70 kHz estimator p ole has been
adopted herein and the gain E is easily calculated from (11).
Results for both 70 kHz and 150 kHz are produced in Figs.5-6.
The digitization (3) is next applied to (7) and lea ds to:
(16)
where (17)
This reduced-order estimator design has considered a gain E
without any integral action and a steady-state erro r may occur at
large parameter variation. The steady-state error i s not important
in this case since the estimated current is used wi thin a closed-
loop control loop for output voltage, which will ta ke care of any
internal steady-state error.
If improvement is critical or control is conducted after
inductor current, one can seek a larger E gain – that is a higher
estimator pole frequency – to cut the steady-state error, and also improve a little the transient. Such selection shou ld be carefully
considered since a higher estimator pole frequency will make the
system more sensitive to noise. Alternatively, an i ntegral action
can be adopted near the gain E in Fig. 4.
IV. MICROCONTROLLER IMPLEMENTATION
The control system using optimized forms (4-5, 11-1 2) is
easy to implement on a low-cost microcontroller sin ce it uses
library dot-product code in assembly language.
These methods have been implemented within the Micr ochip
Explorer Controller board along with a PIC Tail Buc k/Boost
converter board. The core software has been written in C
language using assembly language library routines f or PI-
equivalent (from Microchip control library) and dot -product
(from Vector Functions of the Microchip DSP Library ) modules.
The ISR was implemented in assembly language and oc curs
with measurement sampling in middle of the ON state , to avoid
switching effect on waveforms.
The program code has a total program memory under 0 x7FF.
Most previous solutions using state-space control t o dc/dc
converters were based on analog control IC, or dedi cated digital
ICs. Rare software-based state-space control soluti ons are
reported to run the control loop under 100 kHz, hen ce the
advantage of the arrangement from Figs. 2-4 and equ ations (4-
5) or (11-12).
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P⋅IJK # P⋅#1 #
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4

Fig. 7 Execution times for various code within the system, tested at a lower
switching frequency (142 kHz), with resources for 2 25 kHz operation.

[1] U.R.Prasanna, A.K.Rathore, "Small-Signal Modeli ng of Active-Clamped
ZVS Current-Fed Full-Bridge Isolated DC/DC Converte r and Control System
Implementation Using PSoC", IEEE Trans. on IE, vol. 61,iss.3,2014, pp.1253 –
1261.
[2] D.O.Neac șu, W.Bonnice, E. Holmansky, "On the Small-Signal Mo deling of
Parallel/Interleaved Buck/Boost Converters", IEEE I nternational Symposium in
Industrial Electronics, Bari, Italy, July 2010, pp. 2708-2713.
[3] N.Jantharamin, L.Zhang, "Analysis of multiphase interleaved converter by
using state-space averaging technique", 6th Interna tional Conference on
Electrical Engineering/Electronics, Computer, Telec ommunications and
Information Technology, 2009. ECTI-CON 2009, vol.01 , 2009, pp.288 – 291.
[4] D.O.Neacsu, "A Simplified Approach to Implement ation of State-Space
Control of DC/DC Converters on Low-Cost Microcontro llers", IEEE IECON
2015, Yokohama, November 2016, pp.631-636.
[5] D.O.Neacsu, "A State-Space Design Approach to Digit al Feedback
Control of DC/DC Converters", Professional Educatio n Seminar, IEEE APEC
2016, Long Beach, CA, USA, March 21st, 2016.
[6] D.O. Neacsu, "Switched Linear State-Space Control o f DC/DC Converters
with Optimal Dwell-Time", IEEE IECON Conference, Fl orence, Italy, October
2016, pp. 1423 – 1428.
[7] R. Priewasser, "Modeling, Control, and Digital Implementation of DC/DC
Converters under Variable Switching Frequency Opera tion", PhD Dissertation,
Alpen-Adria-Universität Klagenfurt, 2012.
[8] M.B. Poodeh, S.Eshtehardiha, M.Namnabat, "Optim ized state controller on
DC-DC converter", 2007. ICPE '07, pp.153-158.
IV. CONCLUSION
This paper reviews the usage of state-space control for point-
of-load converter. This class of converters work in difficult
conditions, seeing large currents and very low volt ages. The
state-space based control can help with a series of improvements
and deem valuable under s digital platform.
The paper details two estimation methods, one based on
converter equation and the second following the the oretical path
for a reduced-order estimator. Both solutions are c alculated into
a simplified dot-product form that presents advanta ges in
implementation. Along with the previous arrangement of the
state-space control with two generic terms (a PI te rm and a dot-
product term), the overall software implementation becomes
very attractive using three assembly routines calle d from an
interrupt also coded in assembly. The main program runs on C
for system compatibility.

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