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Synergetic Synthesis Of Dc-Dc Boost Converter Controllers:

Theory And Experimental Analysis

A. Kolesnikov (+), G. Veselov (+), A. Kolesnikov (+), A. Monti (++), F. Ponci (++), E. Santi (++), and

R. Dougal (++

)

(+

)Department of Automatic Control SystemTaganrog State University of Radio-Engineering (TSURE)

44 Nekrasovsky St., Taganrog, 347928, Russia

(++

)Department of Electrical Engineering

University of South Carolina

Swearingen Center, Columbia, SC 29208 U.S.A.

Abstract- This paper describes a new approach to the

synthesis of controllers for power converters based on the

theory of synergetic control. The controller synthesis

procedure is completely analytical, and is based on fully

nonlinear models of the converter. Synergetic controllers

provide asymptotic stability with respect to the required

operating modes, invariance to load variations, and

robustness to variation of converter parameters. With respectto their dynamic characteristics, synergetic controllers are

superior to the existing types of PI controllers. We present

here the theory of the approach, a synthesis example for a

boost converter, simulation results, and experimental results.

I. INTRODUCTION

Design of controllers for power converter systems presents

interesting challenges. In the context of system theory,

power converters are non-linear time-varying systems; they

represent the worst condition for control design.

Much effort has been spent to define small-signal linear

approximations of power cells so that classical control

theory could be applied to the design. See for example

[1,2]. Those approaches guarantee the possibility to use a

simple linear controller, e.g. Proportional-Integral

controller, to stabilize the system.

The most critical disadvantage is that the so-determined

control is suited only for operation near a specific

operating point. Further analyses are then necessary to

determine the response characteristics under large signal

variations [3,4].

Other design approaches try to overcome the problem by

using the intrinsic non-linearity and time variation for the

control purpose. Significant examples of this approach

include the sliding mode control, used mostly for

continuous-time systems [5] and the deadbeat control, used

for digital systems [6]. Those two theories have beenapplied to power electronics mostly because of their

intrinsic capability to manage variably-structured systems.

In this paper we focus on a different approach, synergetic

control [7], that tries to overcome the previously described

problems by using the internal dynamic characteristics of

the system.

The synergetic approach is not limited by any non-

linearity; instead, it capitalizes on such non-linearities.

As will be discussed in the paper, this approach makes full

use of the intrinsic proprieties of the system. While this is a

strong point, it is also a weak point -- definition of the

system model plays a more strategic role than in any other

control approach. This introduces a great possibility for

sensitivity to system parameters. However, as we will

demonstrate with experimental results, this problem can besolved.

One obvious solution is the adoption of sophisticated

observers for parameter determination. This solution is

reasonable only if the cost of the control is not a significant

concern (e.g. high-power or high voltage applications). For

situations where the control costs are of concern, we will

show that suitable selection of the control macro-variables

can largely resolve any sensitivity to uncertainty in system

parameters.

In this paper we will describe the theory of synergetic

control, demonstrate its application in the case of a boost

converter, describe both simulation and experimental

results, and finally introduce some interesting practical

considerations.

II. THEORETICAL BACKGROUND

Synthesis of a synergetic controller begins by defining a

macro-variable, which is a function of the system state

variables:

),()( txt = (1)The control objective is to force the system to operate on

the manifold 0= . The designer can select thecharacteristics of this macro-variable according to the

control specifications (e.g. limitation in the control output,

and so on). In the trivial case is a simple linear

combination of the state variables. This process is thenrepeated, defining as many macro-variables as there are

control channels

Next, the dynamic evolution of the macro-variables is

fixed according to the equation:

( ) 0;0 >=+ TtT (2)where T is a design parameter describing the speed of

convergence to the manifold specified by the macro-

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variable. Finally, the control law (evolution in time of the

control output) is synthesized according to equation (2) and

the dynamic model of the system.

Briefly, any manifold introduces a new constraint on the

domain of the state space, and thereby reduces the order of

the system and forces it in the direction of global stability.

The procedure summarized above can be easily

implemented as a computer program for automatic

synthesis of the control law or it can be performed by hand

for simple systems, such as for the boost converter, that

have a small number of state variables.

By suitable selection of macro-variables the designer can

obtain interesting characteristics for the final system such

as:

Global stability Parameter insensitivity Noise suppression

These results are obtained while working on the full

nonlinear system and the designer does not need to

introduce simplifications in the modeling process to obtain

a linear description as is required for classical control

theory.

III. SYNTHESIS OF A SYNERGETIC CONTROLLER FOR A BOOST

CONVERTER

We now synthesize a controller for a DC-DC boost

converter (see Fig. 1). The classical time-averaged model

of the converter is:

( ) ( )

( ) ( ) ,1

;1

1

212

21

RC

xu

C

xtx

VL

uL

xtx g

=

+=

C

C

(3)

10 u (4)where 1x is the inductor current, 2x the capacitor voltage

and u the switch duty cycle. Our objective is to obtain the

control law ( )21,xxu as a function of state co-ordinates 1x ,

2x , which provides the required values of converter output

voltage sxx 22 = and, therefore, current Sxx 11 = forvarious operating modes, while satisfied limitation (4).

Fig. 1: Boost Converter scheme

According to this method, we introduce the following

macro variable

0; 12111 >= xx (5)

Substitution of 1 (3) into the functional equation:

( ) 0;0 1111 >=+ TtT (6)yields:

( ) ( ) 01

1

1

211 =+ T

txtx (7)

Now substituting the derivatives ( )tx1 and ( )tx2 from (3)and (4), the control law is obtained:

+++

== 121

112

1

111

TV

LRC

x

LxCx

LCuU g (8)

The expression for u1 is the control action for the converter

controller. Substituting macro variable and RCT = into (8), we obtain the control law as:

( )

++

+= gV

Lx

RCRC

x

LxCx

LCu

111 2

11

112

1

(9)

When 1= , i.e. RCT =1 , we get:

+

+= gV

LRC

x

LxCx

LCu

11 1

112

1

(10)

Control laws (8), (9), or (10), according to (6), inevitably

move the representing point (RP) of object (1) firstly to in-

variant manifold 01 = (3), and then along this manifoldto the converters steady state: sxx 11 = ; sxx 22 = . Let usstudy the behavior of the closed loop system:

( )

( )RC

x

TV

LRC

x

CxLx

Lxtx

VLTVLRC

x

CxLx

Cxtx

g

gg

21

1

21

211

12

1

1

21

211

21

11

;111

++

+=

+

+++=

D

D (11)

on the manifold 0= (3). For this purpose, we substituterelation 21 xx = into (11). This results in:

( )( )

( )( )

.1

;

21

1

221

2

2

1

21

2

1

1

1

CL

Vx

CLRtx

CL

V

CLR

xtx

g

g

++

+=

++

+=

D

D

(12)

Each separate equation of (12) describes the behavior ofthe corresponding converter coordinate 1x or 2x on the

manifold 01 = . Evidently, equation (12) isasymptotically stable with respect to the converters steady

state:

gsgs RVxRVx 122

11 ; == (13)From relation (13) we see that converters steady state

operating point depends on the power source voltage

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gV and on the load resistance R. After we set the required

reference value of the converters output voltage sx2 , (13)

gives us a possibility to find 1 , present in macro variable

1 , i.e.

g

s

g

s

RVx

RVx 121 == . (14)

The steady state value of control su , which provides the

converters steady state (13), will be determined by the

following equation:

111 =Su . (15)

Knowing 1 in (14) and Su1 in (15), we can find the

steady state parameters of the converter.

So, the synthesized control law 1u after a time interval

approximately equal to T3 moves the RP of the plant to

the manifold 01 = , and then, according to equation (12),provides asymptotically stable movement along 01 = tothe converters steady state (13). According to (12), the

time to move RP along 01 = is determined by the

expression ( )CLR +21 .

Control law 1u provides converter motion from an

arbitrary initial state 010 x , 020 >x to the steady state

sx1 , sx2 . In other words, the synthesized control law 1u

with 01 >T and 01 > guarantees asymptotic stability (inthe whole) of the closed loop system with respect to the

converters steady state.

IV. OTHER POSSIBLE CONTROL SYNTHESIS FOR THE BOOST CASE

The previous case illustrated a very simple case of control

synthesis that transformed the boost circuit into a first

order system always working in the manifold described by

the macro-variable. This case does not cover all the

possible situations we could face in reality, where more

complex macro-variables must be introduced.

One classical problem is accounting for limitation of one

of the state variables, for example, limiting the maximum

input current. This problem can be simply solved by

defining a new macro-variable:)tanh( 2212 xAx = (16)

where max1xA = . This defines a new manifold where thecurrent is naturally limited. In the rest of paper other

assumptions will bring us to the definition of other possible

macro-variables.

V. SYSTEM MODELING RESULTS

Extensive simulation analysis has been conducted to verify

the control performance. The simulations have been

performed using both Matlab and the VTB simulator [8].

Fig. 2 and Fig. 3 show the transients created by changes in

the load and in the power source amplitude, as predicted by

Matlab models. Fig. 4 shows the phase portrait of thesystem and the stability characteristics of the control

system as demonstrated by convergence to the manifold.

Other simulation results, obtained in the VTB

environment, are shown in Fig. 5 and Fig. 6. This second

step was useful insofar as guiding construction of the real

converter because of the possibility to use more detailed

models of the power cell. For example, the capacitance

model in VTB contains also the equivalent series

resistance, giving the opportunity to explore more realistic

problems.

Fig. 2: The voltage transients

Fig. 3: The load changing

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Fig. 4: System phase portrait

VI. LABORATORY EXPERIENCE

Following theoretical analysis, a laboratory prototype was

designed and built. Since synergetic control is well suited

for digital implementation a DSP-based platform wasselected for migration from the VTB environment to the

real world.

The small-scale power converter system has the following

nominal characteristics:

- Rated Input Voltage: 12 V

- Rated Output Voltage: 40 V

- Maximum Load: 100 W

- Input Inductance: L = 46 mH

- Output Filter Capacitance: 1.360 mF

- Main Switch: IRF540N

The main targets of the experimental analysis were:

Verification of the control theory Analysis of problems related to the model

parameter sensitivity

By defining the controller in Simulink, we were able to

easily export the control algorithms to both a dSpace

platform for control of the real hardware, and to the VTB

environment for system simulation. The ease of inserting

the Simulink controller into both hardware model allowed

unique opportunities to rapidly experiment with a wide

variety of macro-variable definitions in order to identify

and resolve significant early problems.One interesting observation common to both experimental

environments (simulation and hardware) was the

possibility to introduce any kind of transient in the output

voltage reference without requiring any soft-start option.

The system easily remained stable under large non-linear

transients.

Fig. 5: VTB schematic

Fig. 6: VTB results

On the other hand, adoption of the simplest macro-variabledefinitions revealed significant problems with respect to

parameter sensitivity. This sensitivity mostly affected the

steady-state value of the output voltage -- which resulted to

be different from the reference value.

For this reason, after the first set of experiments, a new

macro-variable was defined:

)x-(xk)x-(x 1ref12ref2 += (17)

This new macro-variable significantly reduced the problem

of parameter sensitivity and allowed for the steady state to

be set more accurately.

Using this approach two main parameters had to be tunedfor control performance:

The value T involved in the main synergeticequation (2)

The value of K involved in the macro-variabledefinition.

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The role of T is extremely interesting. As far as equation

(2) is concerned, T defines the speed with which we reach

the manifold. On the other hand, this parameter also plays

an interesting role in noise reduction.

In the case of the boost control, the state vector is easily

accessible and so we can assume that the error introduced

in evaluation of the macro-variable is quite limited. On the

other hand, its derivative is obtained by means of the state

equations so then the system parameters play a significant

role.

Let us suppose that we have a systematic constant error in

the evaluation of the derivative. If we check for the steady-

state condition of this equation we will have:

( ) 0)( =++ etT D

and then in steady state:

Te=

This means that by decreasing T, we decrease the time

with which the manifold is reached. But also we reduce the

steady-state error that is introduced by wrong estimation of

the system parameters. During the experiments we found

that a reduction of T from 1ms to 0.1 ms yielded a

significant increase in accuracy of the steady-state.

K plays a significant role after reaching the manifold; it

determines the way that errors in the main state variable

are canceled by using the error on the current.

Decreasing K increases the control performance but also

calls for a higher current peak during any transient.

This situation, as pointed out in the introduction, could be

solved by definition of a more sophisticated macro-

variable which can account for current limitation.

The synergetic approach also gives an opportunity to solve

the steady-state problem by introducing a new state

variable that represents the integral of the reference-

feedback error. This is analogous to an integral term in a

standard linear controller.

We decided to avoid this option in order to keep the system

simpler and to better exploit the possibilities offered by

parameter tuning and macro-variable definition.

However, the introduction of the integral term is always

possible to force the error to go zero at steady state.

According to the laboratory experience, we also figured

out that this option should be considered eventually as a

refining option working in the direction of keeping theintegral charge as small as possible.

VII. COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL

RESULTS

All the results shown in the following have been obtained

using the macro-variable definition reported in (17).

We want now to show some comparisons between

simulated and experimental results that confirm the

theoretical discussion presented in the previous paragraph.

These results show the transient that follows step change of

the reference voltage from 20 to 40 V.

Fig. 7: Output Voltage (Simulation)

Fig. 8: Output Voltage (experiment)

Fig. 7 and Fig. 8 show the output voltage from experiment

and simulation. One can clearly see that the synergetic

control transformed the second-order system into a first-

order system. This can be easily justified by consideringthat when we are on the manifold we have a linear relation

between two state variables. Introducing the constraint, the

order is reduced. This is always true for synergetic

applications and it constitutes a similarity with the sliding

mode approach.

Fig. 9 and Fig. 10 focus on the evolution in time of the

macro-variable. The two transients looks very similar in

the first part moving in the direction of the manifold with

the same speed. In the experimental results, anyway, a

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second transient starts when we are close to steady state:

this can be considered another side effect of the imperfect

system modeling.

Fig. 9: macro-variable as function of time (simulated data)

Fig. 10: macro-variable as function of time real data)

Finally in Fig. 11 and in Fig. 12 the results for the input

current are presented. Also in this case the simulation

results and the experimental data match perfectly.

VIII. CONCLUSIONS

This paper introduced a new and interesting control

approach called Synergetic Control. The main feature ofthis approach is to manage with the same level of

simplicity both linear and non-linear systems.

The main aspect of control design is definition of a macro-

variable that specifies a manifold for the space variables.

We have discussed several different definitions of the

macro-variable and described the practical consequences of

the different selections. The theoretical aspects have been

discussed and then confirmed through experiment and

simulation.

Fig. 11: Input Current (averaged-simulated value)

Fig. 12: Input current (filtered-experimental data)

ACKNOWLEDGEMENT

This work was supported by the US Office of Naval

Research (ONR) under grant N00014-00-1-0131

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