Abstract — Compensators and Proportional-Integral (PI) controllers have been designed and used for control of Zeta converters. However, frequent voltage variations in some applications such as maximum power point tracking of solar power require a high profile voltage tracking controller. In addition, load resistance and circuit parameter variations such as inductance, capacitance and their internal resistance influence the performance of conventional controllers. This paper illustrates the design and application of a complementary model reference adaptive controller for output voltage tracking control of zeta converters. The complementary controller structure will reduce the adaptive controller’s control effort to 2%-9% variation. The results demonstrate a close tracking profile with minimal control effort and elimination of the load resistance dependencies. Experimental results are provided to demonstrate the high performance of output voltage tracking profile.

I. INTRODUCTION Zeta converters are non-inverting buck-boost circuits with

applications in power quality improvement, power factor correction, and interfacing the renewable energy sources to the grid. Therefore, they have high potential for applications in microgrids and smart grids [1],[2]. These converters are also used in industrial applications such as: LED lamp drivers [3], electronic ballast (EB) for fluorescent lamps [2], power rating correction and power quality improvements (PFC) [1], DC/DC converter interfaces between photovoltaic systems and the grid [4], power electronic interface between storage devices (battery and ultra-capacitor) in hybrid electric vehicles [5], AC inverters [6], and DC converter used for permanent magnet synchronous machines (PMSM) to interface for applications such as air conditioning systems, refrigerators, washing machines and medical equipment [7-10].

To achieve non-inverting, low harmonics, and high power factor, multiple resonant elements are used in their structure. These elements make the modeling and control of these converters complicated. Various techniques use peak and average of current and voltage values in a Proportional Integrator (PI), linear compensator, and feed-forward in single- or double-loop configurations [11]. These techniques generate high sensitivity to noise, exhibit error in averaged values, and require slope compensation [12]. Average current control techniques [7], [8], [13] are becoming the dominant approach in controlling these converters. The average current control loop is usually used inside a voltage

A. Izadian is the founder and director of the Energy Systems and Power Electronics Laboratory at the Purdue School of Engineering and Technology, Indianapolis, IN. 46202. E-mail: aizadian@iupui.edu.

P. Khayyer is with the Electrical and Computer Engineering Department at The Ohio State University, Columbus, OH, 43210. khayyer.1@osu.edu

control loop where the error signal from the voltage loop is sent through a controller. The controller amplifies the current error and in comparison with a saw-tooth carrier waveform, it generates PWM pulses [13]. This controller has fixed gains for a voltage and current references, and needs tuning when circuit parameters or references change. Therefore, it has poor voltage tracking performance and generates overshoot and steady state error.

Feed-forward control technique is also used specifically for grid-connected applications of zeta converter [14], [15]. In either single-loop or double- loop configurations, two major controllers are used which are the PI controller [5], [16], [17] and compensator designed based on pole placement techniques [8], [18-19]. While results obtained from pole placement have low error, this technique may not be useful in situations where variable reference voltage or current are the targets and when load and system parameters shift over time. In our previous work, we have introduced adaptive control of zeta converter [29]. To overcome this issue and to have a more accurate control for zeta converter, this paper focuses on application of a Model Reference Adaptive Controller to regulate the output voltage of zeta converters. The system operation, model, and transfer function will be obtained in section II, III, and IV. Closed loop control design and simulation results are in section V, and experimental results are provided in section VI.

II. CIRCUIT OPERATION AND MODELING

Power electronic converters have numerous applications and have important role in overall system efficiency and performance. Accurate control of power converters often guarantees the voltage and frequency stability of the power system. Dynamic modeling of converters is required to design controller for power electronic converters. Many linear or nonlinear modeling techniques are used for mathematical expression of power converters. Nonlinear techniques such as component connection modeling and signal flow graph (SFG) are used for complicated circuits and are generally more accurate. State space averaging technique represents a linear technique in power electronic circuit modeling [4], [5], [19], [21-24].

Shown in Figure 1, a non-inverting buck-boost zeta converter has higher number of resonant elements. This imposes a higher order system and higher modes of operation than a conventional converter. The two main operation modes are continuous current mode (CCM) and discontinuous inductor current mode (DCM).

Afshin Izadian, Senior Member, IEEE, and Pardis Khayyer

Complementary Adaptive Control of Zeta Converters

978-1-4673-4974-1/13/$31.00 ©2013 IEEE 1338

Fig. 1. Schematic of zeta buck-boost converter

Modes of operation are generated when the status of

switches change. When the switch Q is on, the input voltage is applied across L1 and causes a linear increase of current to charge the inductor. The input voltage and the charged capacitor C1 will increase the load current through the inductor L2. When the switch Q is off, L1 charges C1 through the diode and L2 supplies the load. The inductor L2 and capacitor C2 in zeta converter create a filter, which lowers the output ripple. In CCM mode, the input and output voltages of the zeta converter are related as follows . . , (1)

where Dcycl. is the converter’s duty ratio. For values of D less than 0.5, the converter ideally operates in buck mode, and for values of Dcycl. larger than 0.5 the converter ideally operates in boost mode. The critical equivalent inductance of this circuit [24] is 1 . , (2) where, . (3)

More details on component selection and sizing for zeta converter are explained in [20].

III. STATE SPACE AVERAGE MODEL State space averaging is a dynamic modeling technique

used for mathematical representation of converters. In this technique, the state space representation of each mode of operation is obtained, and the overall system is represented as an averaged system over a complete cycle of operation. In zeta converters, continuous current mode (CCM) has two modes of operation. Therefore, two sets of equations can be written for this circuit, which can include internal resistances of the capacitors and inductors.

In Mode 1 the switch is on and diode is off, and Mode 2 the switch is off and diode is on. Each set of state space equations consist of five equations, four of which represent the four state variable dynamics , , , , and one equation to represent the output voltage of the circuit . Equations representing mode 1 are as follows [18]

Mode 1: .

(4)

In Mode 2: Equations representing mode 2 are as follows

Mode 2: . (5)

Considering x=[iL1,iL2,vC1,vC2], the averaged state space model is

. (6)

The averaged model parameters can be obtained as 00 1 0 0 0 00

00 , 0 0 , 0 . (7)

IV. TRANSFER FUNCTION The output voltage-to-duty cycle transfer function of the

circuit can be obtained from (7) using the Laplace operator s [18], [23] by

(18)

(19) Where 1 1 , 1 111 , 1 1 21 , ,

(10)

, , and is the load current.

The system characteristic equation has the following coefficients [18],[23]: , 1 , 1 1

, (11)

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1 11 1 1

The transfer function has 4 poles and three zeros. The poles of

the circuit can be proven to have negative real values. Therefore, the transfer function is Hurwitz. The 1 term has a left-half-plane (LHP) zero as is positive value. The minimum phase condition is satisfied if the term exhibits all LHP zeros. That is guaranteed if the following conditions hold: 4 0 . (12)

If (12) holds, the transfer function demonstrates a 4th-order

system with 3 LHP zeros and relative degree 1. Considering the Hurwitz transfer function, the system becomes positive real.

V. CLOSED-LOOP CONTROL To maintain the output voltage at the desired value, an

adaptive system will be used as a complementary controller. Adaptive controllers can adjust the duty cycle to track the reference waveform. However, they need higher adaptation gains which increases the risk of instability in the system. A complementary system can decrease the gains of adaptive controller and enhance the voltage tracking performance.

The design of the controller, the output tracking performance, and the control effort are obtained for this converter. The controller and the converter will be implemented in simulation using Matlab/ SimPowerSystem toolbox, and prototyped to demonstrate the experimental results.

A. Complementary Controller The output voltage of a buck boost converter can be

ideally obtained from (1). Therefore, the ideal duty cycle from this equation can be obtained by: . . (13)

This duty cycle does not account for parameter shifts and

load variation. Therefore, steady state error will occur as the load or operating conditions of the system change. To complement the duty cycle and compensate for parameter variations, the adaptive control command is added to ideal duty cycle . making the ultimate control command as . . (14)

B. Design of Adaptive Controller This section introduces a model reference adaptive control

approach to control the output voltage of the zeta converter. Since the main application of these converters in renewable energies is in interfaces of photovoltaic systems, a high profile tracking performance of the output voltage is required. This needs continuous gain adjustments, and elimination of the control system dependencies on load resistance. In addition, frequent load changes in a power

system will change the current of the inductor and consequently the output voltage of the system. Adaptive controllers are designed to provide high tracking performance and to eliminate the effect of the load resistance.

The control law and gain adaptation techniques [27], [25-28] are represented for the averaged state space model of the zeta converter. For a first order positive real system, the control law is expressed as , (15) where and are the reference model and plant output signals, denotes the reference input and , are controller gains that are adjusted simultaneously according to a gain adaptation law to mitigate the tracking error.

Considering the estimated values of the controller gains, the equivalent control command is defined as , (16) where , are the estimations of the control gain and are computed according to the gain adaptation technique as , (17) , (18) where is the adaptation gain, e denotes the tracking error, and kb is the DC gain of the transfer function. The proof of stability of this controller is provided in several references including [27], [28].

Figure 2 illustrates the control system configuration with respect to zeta converter and the voltage reference. The model generates the reference signal that the controller has to track. The controller compensates for the load variation and unknown parameters such as internal resistance of inductors and capacitors that may shift over time and operating temperature. In this section, the controller is used to compensate for the load resistance that has the highest impact on the output voltage and control performance.

Fig. 2. Model reference adaptive controller configuration

C. Simulation and Circuit Parameters The circuit element parameters are listed in Table 1. The

control performance is analyzed in several load resistance values of 12Ω, 13Ω, and 14Ω in order to evaluate the performance of the controller. As the input voltage from a battery is 12V, the reference voltage was selected to be 9V and 14V to excite the buck and boost modes of operation. The controller was initialized at 0.05 adaptation gain and 10kHz carrier frequency.

TABLE I. CIRCUIT ELEMENTS AND PARAMETERS L1,2 (mH) rL1,L2, rc1,c2 C1,2 (µF) R (Ω) Converter Parameters 0.5 Unknown 470 12, 13,14

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The simulation results are obtained fovoltage buck and boost commands. The along with the controlled output voltage Figure 3. As Figure 3 demonstrates, the outpthe reference very closely. The adaeliminates the effect of load variation andimperfect circuit elements such as inductointernal resistances. These values are hardmay change over time and operating condimplementation of complementary adaguarantees accurate tracking of referenvarious operating conditions. The operationbuck (9V reference) and boost (14V referenFigure 3. The adaptation gain has a direct ripple and overshoot generation. As showhigher gain values reduced the ripple and inshoot.

Fig. 3. Output voltage tracking performance of adapload resistance variation.

As the load resistance decreased to 12Ω

current naturally results in more voltage ripvoltage. However, the complementary adwith higher adaptation gain 0.05 could totavoltage ripples. The buck and boost moresulted in similar voltage ripples and thsensitivity to load resistance variation wcomplementary model reference adaptive cin a high performance voltage tracking profi

VI. EXPERIMENTAL RESULT

The controller was implemented on a dSprototyping device and connected to prototype. The experimental setup is showncarrier frequency of 10kHz the voltagegenerated to run the power buck a 12V sboost to 14V repeatedly. The voltage profileis shown in Figure 5 for a series of load refrom 12Ω-14Ω. As the figure demonstracontrol performance both in buck and boosexactly the reference voltage. The output v5% overshoot and no steady state error. The

0 2 4 66

8

10

12

14

16

18

Time/Second

Out

put

Vol

tage

(V

)

Adaptive Voltage Tracking Pro

for two cases of reference signal

are illustrated in put voltage tracks aptive controller d the influence of rs and capacitors

d to measure and dition. Therefore, aptive controller nce voltages in n of the circuit in nce) are shown in effect on voltage wn in Figure 3, ncreased the over

ptive controller under

Ω, the higher load pples in the output daptive controller ally eliminate the des of operation

he output voltage was reduced. The controller resulted file.

TS SPACE 1104 fast the zeta board

n in Figure 4. At e reference was source to 9V and e of the converter

esistances ranging ates, the adaptive st modes generate voltage generates e ideal duty cycle

generates a rough estimate of the doutput voltages from the board. Buvoltages as the load and board considered on the duty cycle complementary adaptive controller wthe duty cycle was adjusted to matthat of the desired reference. The ccan achieve voltages in both bucoperation.

Fig. 4. Experimental setup and zeta board.

As the load resistance changed,compensated for the current and vomatch the average output voltage wAs Figure 5 illustrates, there is no sor boost operating conditions whchanged

Fig. 5. Experimental results of adaptive voltaunder load resistance variation.

The control effort is shown in illustrates, the duty cycle varies changes in the circuit and load corequired less control effort and tharound 0.46 or 46% to generate 9obtained from (13) is 42%. A totalgenerated from adaptive controllercircuit voltage drops and load increased as the load current increasloads. In the boost mode of operatiofrom 0.55 (55%) to 0.62 (62%) to mwith that of the reference. The idea

8 10

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13 Ohm

14 Ohm

ofile

0 2 4 68

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10

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15

16

Time / Secon

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)

Adaptive Voltage Control Profile

duty cycle and produces ut these are not accurate

voltage drops are not calculations. As the

was added to the system, tch the output voltage to complemented controller ck and boost modes of

, the adaptive controller oltage drop variations to

with that of the reference. steady state error in buck hen the load resistance

age profile control performance

Figure 6. As the figure to compensate for the

ndition. The buck mode he duty cycle remained V. The ideal duty cycle l of ~4% duty cycle was r to compensate for the

variations. This value sed due to low resistance on, the duty cycle varied match the output voltage al duty cycle to generate

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under Load Variation

12 Ohm

13 Ohm

14 Ohm

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14V is obtained from (13) is 53%. A complement of 2% to 9% duty cycle was added by the adaptive controller to perfectly track the reference voltage.

Fig. 6. Experimental results of adaptive control effort (duty cycle variation) under load resistance variation.

VII. CONCLUSION A complementary adaptive controller was designed for

Zeta converters to directly track the reference output voltage. The main advantage of the controller was to eliminate the output voltage controls from load variation, and provide an independent control approach from the inductor current and parameter variations. Minimal control effort was required to track the voltage reference. High performance voltage tracking profile was achieved.

ACKNOWLEDGMENT Authors would like to thank Mr. Heng Yang for his effort

in conducting experiments.

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