MEE10:112

Load Identification of

DC-DC converter

Shahruk Osman

Surkhru Osman

This thesis is presented as part of Degree of

Master of Science in Electrical Engineering

Blekinge Institute of Technology

February 2011

Blekinge Institute of Technology

School of Engineering

Department of Applied Signal Processing

Supervisor: Anders Hultgren

Abstract

This thesis is part of an ongoing project of Ericsson together with Blekinge Insti-tute of Technology, Växjö University and Linneaus University to develop SignalProcessing methods to increase the functionality, performance and reliability ofdc-dc converters using signal processing methods.

The aim of this work is to model the buck converter system of EricssonsBMR450 and develop a time-domain based system identi�cation method toidentify the capacitive load. The model is implemented in Matlab Simulinkand is composed of simple circuits. The system is analyzed through excitationfrom the command input using binary signal, which include all frequencies of it.Two types of identi�cation method, Black-box and Grey-box, used to analyzethe system and hence identify the load parameter. The sampled data are ana-lyzed and preprocessed to estimate a mathematical model and validate it. Theproposed method is able to identify the load parameter with good accuracy.

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Acknowledgement

We want to express our deepest gratitude to our supervisor Anders Hultgren forhis guidance and comments during the course of this project. We are grateful toAnders Hultgren for all the time he spent with us giving us constant guidanceand for his suggestions in carrying out the technical part of the project.

Furthermore, we would like to thank sta� at the department of ElectricalEngineering, Blekinge Institute of Technology who has helped us in many ways.

Finally, we would like to thank our father, mother, sister and brother fortheir unconditional support and love through our life.

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Contents

1 Introduction 1

1.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . 11.2 Summary of Contribution . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Buck Converter 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 DC-DC converters . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Why Buck Converter? . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Buck Converter Circuit topology . . . . . . . . . . . . . . . . . . 6

2.4.1 Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Two States of operation . . . . . . . . . . . . . . . . . . . . . . . 82.5.1 On State . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 OFF State . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Modes of operation . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Ericsons BMR 450 features . . . . . . . . . . . . . . . . . . . . . 92.8 State space model of Buck converter . . . . . . . . . . . . . . . . 10

3 System Identi�cation 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 What is System Identi�cation? . . . . . . . . . . . . . . . . . . . 153.3 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Model data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.7 Linear and non-linear models . . . . . . . . . . . . . . . . . . . . 183.8 Parametric and Non-parametric model . . . . . . . . . . . . . . . 183.9 Black Box Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.10 General Linear Model . . . . . . . . . . . . . . . . . . . . . . . . 193.11 AR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.12 ARX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.13 ARMAX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.14 Box-Jenkins Model . . . . . . . . . . . . . . . . . . . . . . . . . . 213.15 Output Error Model . . . . . . . . . . . . . . . . . . . . . . . . . 213.16 Transfer function Model . . . . . . . . . . . . . . . . . . . . . . . 223.17 State space model . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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3.18 Grey Box Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Simulink Implementation and simulation of the buck converter

model 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Simulink model for Buck Converter . . . . . . . . . . . . . . . . . 274.3 Preparing input and output data . . . . . . . . . . . . . . . . . . 314.4 Identi�cation Procedure . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.1 Black box modeling & results . . . . . . . . . . . . . . . . 334.4.2 Grey box modeling & results . . . . . . . . . . . . . . . . 40

4.4.2.1 Ideti�cation results of 3rd order state space model 424.4.2.2 Ideti�cation results of 2nd order state space model 47

4.4.3 Summary and Concluding remarks . . . . . . . . . . . . . 56

5 Conclusion and suggestions for future work 59

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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List of Figures

2.1 Circuit diagram of a linear regulator . . . . . . . . . . . . . . . . 62.2 Buck Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Buck Converter in ON state . . . . . . . . . . . . . . . . . . . . . 82.4 Buck Converter in OFF state . . . . . . . . . . . . . . . . . . . . 82.5 Current across di�erent components during the two states of op-

eration of Buck Converter . . . . . . . . . . . . . . . . . . . . . . 92.6 Ericsons BMR 450 features . . . . . . . . . . . . . . . . . . . . . 102.7 Buck Converter with capacitive load . . . . . . . . . . . . . . . . 112.8 Circuit diagram for buck converter when inductor current is known 13

3.1 System Identi�cation �owchart . . . . . . . . . . . . . . . . . . . 163.2 System Identi�cation . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 General Linear Model structure . . . . . . . . . . . . . . . . . . . 193.4 AR model structure . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 ARX Model structure . . . . . . . . . . . . . . . . . . . . . . . . 203.6 ARMAX Model structure . . . . . . . . . . . . . . . . . . . . . . 213.7 Box-Jenkins Model structure . . . . . . . . . . . . . . . . . . . . 213.8 Output Error Model . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Simulink model of buck converter . . . . . . . . . . . . . . . . . . 284.2 Generation of PWM signal . . . . . . . . . . . . . . . . . . . . . 284.3 Duty cycle generated from the saw tooth . . . . . . . . . . . . . . 294.4 : Buck converter system . . . . . . . . . . . . . . . . . . . . . . . 304.5 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6 Simulation of the output voltage . . . . . . . . . . . . . . . . . . 314.7 Estimation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.8 Input output signal . . . . . . . . . . . . . . . . . . . . . . . . . . 334.9 Spectra of the Input and output voltage . . . . . . . . . . . . . . 344.10 Frequency function of the data . . . . . . . . . . . . . . . . . . . 344.11 System identi�cation Tool GUI showing di�erent types of Quick

Start Linear models . . . . . . . . . . . . . . . . . . . . . . . . . 354.12 Step response of di�erent model . . . . . . . . . . . . . . . . . . 354.13 Bode plot from input voltage to output voltage . . . . . . . . . . 364.14 Impulse response from input voltage to output Voltage . . . . . . 374.15 Measured and simulated model output . . . . . . . . . . . . . . . 374.16 Input and output signals . . . . . . . . . . . . . . . . . . . . . . . 394.17 Measured and simulated model output . . . . . . . . . . . . . . . 404.18 Step response from input voltage to output voltage . . . . . . . . 424.19 Bode plot from input voltage to output voltage . . . . . . . . . . 42

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4.20 Online information about the minimization . . . . . . . . . . . . 434.21 Model output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.22 Prediction error for output voltage . . . . . . . . . . . . . . . . . 464.23 Comparison between estimated and actual capacitance values . . 464.24 Simulink model of a simple second order model . . . . . . . . . . 474.25 Noisy PRBS inductor input signal and load voltage . . . . . . . . 484.26 Identi�cation of second order model using third order model data 494.27 Comparison between estimated and original capacitance value for

reduced model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.28 Validation of the model with the data . . . . . . . . . . . . . . . 514.29 Comparison between the bode plots of the actual model and es-

timated model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.30 Comparison between estimated and original capacitance for re-

duced model using di�erent prbs signal . . . . . . . . . . . . . . . 534.31 Output voltage and inductor current of Controlled buck con-

verter(Courtesy Anders Hultgren) . . . . . . . . . . . . . . . . . 544.32 Comparison between estimated and actual capacitance value of

feedbacked buck converter . . . . . . . . . . . . . . . . . . . . . . 55

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List of Tables

3.1 System Identi�cation toolbox xupported data . . . . . . . . . . . 17

4.1 N4s2 (State space) model ouotput . . . . . . . . . . . . . . . . . 384.2 State space model output . . . . . . . . . . . . . . . . . . . . . . 454.3 Original Capacitance value, estimated value and error percentage 474.4 Original capacitance value, estimated capacitance value & error

percentage using a prbs signal as an inductor current input . . . 484.5 Original capacitance value, estimated capacitance value & error

percentage using inductor current . . . . . . . . . . . . . . . . . . 524.6 Original capacitance value, estimated capacitance value & error

percentage for inductor current using di�erent prbs signal . . . . 534.7 Original capacitance value, estimated capacitance value & error

percentage of a simple BMR 450 controller. . . . . . . . . . . . . 55

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Chapter 1

Introduction

In this chapter the motivations and background for this work are presented. Itprovides an analysis of the modeling method used for system identi�cation ofDC/DC converters.

1.1 Motivation and Background

DC/DC converters provide stable output voltage and variable output current. Itis used to maintain a steady, stable voltage as a power supply for semiconductordevices. DC/DC converters are a necessity for almost all electronic devices andare used to maintain a constant output voltage regardless of minor variations ininput voltage or load current. DC/DC converters are conjoined with the elec-tronic circuits to provide the desired voltage of wide ranges such as computers,home appliance, portable electronic devices etc.

On the other hand, the evolving applications for industrial, telecom, medical,military and consumer products are giving rise to the development of �DigitalPower�, which refers to computerizing the DC/DC converters and it is the mainfocus of the DC/DC converter area today[1]. Modeling and simulation is the in-strument used to shape the system level analysis and obtain better performanceof the converters. Normally, the models are based on the internal structure ofthe converter. However, the system dynamics are in�uenced by the load of thesystem. The load is an essential part of the buck converter system to obtain agood dynamic model. Moreover, insu�cient information about the system pa-rameters may cause error in designing the controller. A better controller couldbe obtained if the experimental data is used to determine the load information[2].

In general, the �eld of identi�cation can be classi�ed into parametric iden-ti�cation and non-parametric identi�cation[3]. In parametric identi�cation, themodel structure and the experimental data is used to estimate non-measurableparameters. The goal is to optimize the function that �ts the data to the systemdynamics. Maximum likelihood, Instrumental variable and subspace methodsare some of the approaches used to describe the system and some of the basicmethods like prediction error and instrumental variable method are importantconcepts in parametric identi�cation [4].

Non-parametric identi�cation can perform systems frequency response anal-

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ysis, transient response analysis; spectral analysis and correlation analysis arethe tools [6] and [7].

Usually, parametric model structures are classi�ed in two categories, black-box model and grey-box model. A black box model does not require any physicalmodeling. The parameters of the model can be adjusted without any consent ofthe physics of the system. The model depends on the external behavior of thesystem known as a behavioral model. Whereas, a grey-box model has a betterinsight about the model structure that partly de�nes the system dynamics withfew known parameters. The unknown parameters in the grey model can beestimated by means of system identi�cation.

System identi�cation toolbox in Matlab is a software to build mathematicalmodels using the measured input-output data [8]. It allows creating both typeof model structures, black-box and grey box to �t data into the model.

Works have been done previously to generate model and identify Dc-Dcconverters. In[9] and [10], a parameter model for the buck converter is presented.The parameters are identi�ed by analyzing the frequency response using a linearampli�er and a network analyzer. In [11] and [12] the transient response of theinput and output data is used to represent a model for the dc-dc converter.Wienner-Hammerstein structure is used to model the system non-linearity. Asimilar method is also proposed in [13] and [14]. The transient response isanalyzed to built a black box model of the dc-dc converters. Simple step testswere applied to identify the model and the step response was adjusted usingdi�erent �tting algorithms in [14]. In [15], non-parametric methods were used tocalculate the systems frequency response by means of correlation analysis. Butthis type of identi�cation techniques requires long processing of data sequenceand further exploitation of the data by means of cross correlation and fouriertransform. It may generate inaccurate model due to truncation and quantizationerror and the estimation while transforming from s to z[16]. To avoid this sortof inadequacies, a sensible way is to obtain a short data sequence.

In[17] a parametric method using a least square technique is used to identifythe parameters of a DC/DC converter. By means of polynomial interpolation,the derivative equations representing the voltage and current signal are solved.

1.2 Summary of Contribution

In this paper a method to identify the load capacitance of the DC-DC converteris presented. A simpli�ed model of ericssons BMR 450 was used and the externalload is considered to be a capacitor. The system equations and the systems inputand output data sequence were used to build the model for system identi�cation.Two approaches of system identi�cation methods are presented in this paper.The main contribution of this thesis are:-

1. Model the Buck Converter system in Matlab Simulink.

2. Compute di�erent black box models using systems input and output datasequence.

3. Compute grey box model using systems input and output data sequence.

4. Compare the two methods and identify the load parameter.

5. Identify the load parameter adding noise to the systems input and output.

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1.3 Outline of the thesis

The structure if the paper is organized as follows: Chapter 2 presents the basictopologies of the dc-dc buck converter and its operation. It provides an analysisof the converter model and its application. A brief review of the state spaceanalysis of the buck converter is also presented. In chapter 3, a brief review ofthe system identi�cation procedure. Di�erent modeling methods are discussed.It explains how to use the experimental data to build the model through systemidenti�cation techniques. It also explains the model validation procedure. Inchapter 4, Matlab Simulink implementation of the buck converter is presented.Two types of system identi�cation procedure is used to analyze the buck con-verter system and identify the load parameter. The simulation and experimentalresults are discussed. Chapters 5 present the discussion and suggest future work.

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Chapter 2

Buck Converter

2.1 Introduction

Over the decades, the usage of electrical components is growing at a very rapidpace, so is the competition to make the devices as portable and durable aspossible. Power consumption is one of the prime factors for any electrical com-ponents to run e�ciently. As the components are getting compact, the powerconversion techniques are changing too as linear regulators are replaced withswitching regulators.

Switching converter, such as switched-mode power supply (SMPS), is animportant element in processing the electrical power in the �eld of power elec-tronics [18]. Like typical power converters it converts power from a source toa load. SMPS are nonlinear and discontinuous in nature but can achieve veryhigh e�ciency and high voltage accuracy. This is due to high switching ratebetween the full-on and full-o� state.

Buck converter, a switched-mode power supply, is a DC-DC voltage stepdown converter. It is one of the most widely used dc-dc converter topology inpower management due to its high e�ciency over a wide load current range.Load current usually changes abruptly in battery operated devices and is animportant factor for the battery durability [19]. For example, in a laptop,the voltage needs to be stepped down for the microprocessors to run, and thelower voltages need to be maintained in a narrow range. For these reasons,Buck converters can be used for fast load and line transient response and highe�ciency over a large load current range [20].

2.2 DC-DC converters

DC-DC power converters are employed in majority of the digital systems todayfor its �exibility to step-up, step-down and invert. For example, in a laptopthe load voltage is regulated to 5V or 3.3V required by the integrated circuit(IC) or its sub-circuits. The voltage level is constant here unlike the voltagesupplied by a battery where it declines after a certain period. Di�erent voltagelevel can be obtained using the same circuit with the use to switched mode [21],where the switching is controlled by Pulse width modulated (PWM) signals. It

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maintains the voltage range regardless the variation of the load current and thebattery voltage drop [22].

Isolated and Non-isolated are two types of DC-DC converter. In IsolatedDC-DC converter there is isolation between the input and the output voltage tomaintain safety condition, typically used when the supply voltage is high [22].

2.3 Why Buck Converter?

Consider a conventional linear regulator, where a voltage Vs needs to be steppeddown to Vl across R1, and suppose voltage across Rl is V(Rl), as shown in �gure2.1. This means that V(Rl) volts must be dropped across Rl . If the current tothe circuit is i, then the power that must be dissipated across the regulator Rl

is given by :

P = i ∗ V(Rl)

This results the circuit to dissipate must of the energy as heat making thesystem less e�cient, also it requires cooling system making it bulky. In contrast,Buck converter, a basic step-down DC-DC converter uses switches (a transistorand a diode) to regulate the system between on and o� state at a �xed ratemaking it more e�cient and compact.

Figure 2.1: Circuit diagram of a linear regulator

2.4 Buck Converter Circuit topology

A basic buck converter operates in two modes is illustrated in �g 2.2. Whenthe switch is closed, the supply voltage is connected to the inductor and theload voltage. Current through the inductor increases, so does the energy storedin it. When the switch is open, the inductor transfers it energy to the loadresistor, decreasing the current through the system gradually. In this case, theinductor works as a source to the load keeping the �ow of current in the circuitcontinuous.

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Figure 2.2: Buck Converter

The Function of each component is described below:

2.4.1 Switch

The switch used in buck converters is basically a power switch, transistor. PulseWidth Modulated (PWM) signal is introduced at the gate of the transistor tocontrol the ON and OFF state of the transistor. When the switch is ON thesupply voltage equals sum of the voltages across the inductor and the loadresistor, and when it is OFF the voltages across the inductor equals the voltageacross the load resistor. Varying the PWM signal between the two states theaverage output voltage can be controlled.

If the power device is switched at a frequency f = 1/T and conduction dutycycle d = (ton)/T , then stepping a 50V supply to 10V needs a duty cycle of20% ideally.

2.4.2 Inductor

The main function of the inductor is to supply constant power to the loadresistor. When the switch is at OFF state, no power is supplied by the sourcevoltage. At this stage inductor acts as a source transferring current to the load.This also reduces the abrupt change in current through the power switch whenthe switch in ON, as shown in �gure 2.5.

2.4.3 Capacitor

The capacitor reduces ripples and overshoots in the output voltage by �lteringaway harmonic currents across the load. The capacitance acts as a low pass�lter and must be su�cient enough to avoid voltage ripples and overshoots.

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2.5 Two States of operation

2.5.1 On State

Figure 2.3: Buck Converter in ON state

In the ON state, the switch is closed transferring energy from the source voltage,Vs, to the inductor, L. At this stage, current through the inductor increases ata steady rate charging the inductor, as shown in �gure 2.5.

2.5.2 OFF State

Figure 2.4: Buck Converter in OFF state

In the OFF state, the switch is open and the inductor acts as a source main-taining constant transfer of energy to the load resistor. In this state, the diodeconducts and current through the circuit decreases linearly as the inductor dis-charges, as shown in �gure 2.5.

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Figure 2.5: Current across di�erent components during the two states of oper-ation of Buck Converter

Figure 2.5 shows the current across the switch, diode and the load during the twostates of operation of Buck Converter, where i0 represents the average currentacross the load.

2.6 Modes of operation

A buck converter can operate in two modes, Continuous and Discontinuousmode. In continuous mode, the current through the inductor never falls to zero.But in discontinuous mode, the current through the inductor falls to zero duringpart of the switching period. This occurs due to the time and energy requiredby the load small enough compare to the whole communication period. In thisthesis, the modelling is assumed to run on continuous mode.

2.7 Ericsons BMR 450 features

Research has been going on to develop methods to identify the converter loadproperties. Using the properties the functionality, reliability and performanceof the converters could be enhanced. The load model can be comprised ofcapacitance, inductance, negative resistance and current source. BMR450 isone of the �rst products within �digital power� developed by Ericssson. �Digitalpower� refers to computerizing the DC/DC converters, and it is the main focusof the DC/DC converter area today [23]

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Digital PWM with adaptive dead-time control, precision delay and ramp-up arethe main features of BMR450 [20]. Other features are listed below.Input: 4.5-14 VOutput: 20 A� DiPOL connect� Max height 8.2 mm (0.323 in)� 96.8 percent e�ciency at 3.3 V (typical value at half load)� PMBus read and write compliant OTHER FEATURES� Voltage/current/temperature monitoring� Precision delay and ramp-up� Non-linear transient response� Wide output voltage adjust function� Start up into a pre-biased output� Output short-circuit protection� On/O� remote control� Output voltage sense� Start up into pre-biased output� On/O�, OCP/OTP/OVP and voltage adjust

Figure 2.6: Ericsons BMR 450 features

2.8 State space model of Buck converter

Generally, models based on di�erential equations are composed of transfer func-tions that are interconnected to describe a system, where each transfer functiondescribes a subsystem. Such method could provide better understanding of thesystems response and stability. However, the method became more complicatedas the complexity ( higher order, multiple-input multiple-output, non-linear,time varying) of the system grew further [32]. On the other hand, state spacemodel is more versatile in dealing with complex nature of the system. It candescribe large systems in matrix form, also able to deal with time varying andnon-linearity of the system. Moreover, it can represent multiple-input multiple-output using B and C matrices. Since the system is described using matrix, itis easily manipulated by computers[33].

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The Buck converter can be modeled as state space parameters. A state spacemodel is the mathematical representation of a physical system as a �rst orderdi�erential equation. The goal is to model the system in the form of

x(t) = Ax(t) + Bu(t) + Ke(t)y(t) = Cs(t) + Du(t) + e(t)

Where y(t) is the output equation, x(t) is the state vector, u(t) is the inputvector and e(t)stochastic error.

� A is an n x n matrix, where n is the number of states.� B is an n x m matrix, where m is the number of inputs.� C is an p x n matrix, where p is number of outputs.� D is an p x m matrix.� K is the kalman gain matrix.

Figure 2.7: Buck Converter with capacitive load

The general structure of the dc-dc converter to be analyzed is shown in Figure2.7.

It consists of an inductor resistance RL and converter capacitor equivalentseries resistance RC

Selection of an appropriate model is an important aspect in control design.The choice of model structure is based an understanding of the system dynamicsgoing identi�cation. For a DC-DC converter, state-space models is the mostgeneral and e�cient method for dynamic modeling. The model of the buckconverter circuit is obtained by applying the Kirchho� laws that result in a setof equations describing the power circuit. The state-space model of the circuitcould be obtained by properly selecting the state variables, inductor current,iL , Capacitor voltage, vC and Output Capacitor voltage, vCl. The product ofsource voltage U and the duty cycle d is the input (dU) to the system. TheDC-DC converter was considered as an ideal electric circuit. The di�erentialequation related to state variables are

dU = ilRl + Ldildt + Vo

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By re-earranging we get, Ldildt = dU − ilRl − Vo

Also C dVc

dt = Vo−Vc

Rc.

Also il = CldVo

dt −Vc−Vo

Rc+ io

By re-arranging we get, CldVo

dt = il − io + Vc−Vo

Rc

By further rearranging the above equations, the desired state space modelis achieved

dildt = 1

L [dU − ilRl − Vo]

dVc

dt = 1C [Vo−Vc

Rc]

dVCl

dt = 1Cl

[il − io + Vc−Vo

Rc]

Vo = Vcl

And in matrix notation

ddt

ilvcvcl

=

−Rl

L 0 − 1L

0 − 1C∗RC

1C∗RC

1Cl

1Cl∗RC

− 1Cl∗RC

ilvcvcl

+

1L 00 00 − 1

C1

[ dUi0

]

v0 =[

0 0 1] il

vcvcl

+[

0 0] [ dU

i0

]

Where il is the inductor current and vc(t) is the capacitor voltage and Cl isthe capacitive load and d is the duty cycle.

Converter parameters are given byL = 3.3μH,RL = 100mΩ, C = 100μF,Rc = 100mΩ, fs = 330kHz,U = 12V

In this model the input voltage and the constant load capacitor current istaken as the inputs to the state space model and the output voltage is taken asthe output to the state space model.

However, if the three states which represents the model are reduced to twothe estimation time and the memory requirement can be reduced to some extent.If the inductor current is measured, a second order system will describe theconverter. Hence, the state space model derived earlier could be reduced to a2nd order system and the input voltage is replaced by the inductor current inone of the inputs to the model. The circuit diagram of the buck converter wheninductor current is used is shown in �g 2.8.

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Figure 2.8: Circuit diagram for buck converter when inductor current is known

The equations derived earlier

dVc

dt = 1C [Vo−Vc

Rc]

&

dVCl

dt = 1Cl

[il − io + Vc−Vo

Rc]

could be utilized to make the 2nd order state space model for the circuit.Assuming measurement signal vCL the state space model in matrix form can

be written as

ddt

[vcvcl

]=

[− 1

C∗RC

1C∗RC

1Cl∗RC

− 1Cl∗RC

][vcvcl

] [0 01Cl− 1

Cl

] [iLi0

]

vc=[

0 1][ vc

vcl

]+[

0 0][ iL

i0

]The reduced model has an order of two, which is also the minimum number

of states to represent this system using the available data. The inductor current,iL replaces the input voltage vin as the new input to the system.

13

14

Chapter 3

System Identi�cation

3.1 Introduction

Building a model from a given data is an important aspect in Science. Severaltechniques have been developed in di�erent application areas, and in the controlsystem the term is known as System Identi�cation [24]. System Identi�cationis an important topic nowadays in the control system application. In adaptivecontrol and in power converters its usage is absolute. It gathers the systemrealistic information directly from the system experimental data making theperformance of the control system at premium.

3.2 What is System Identi�cation?

System Identi�cation is the construction of a dynamic model for a dynamicsystem using a set of experimental input and output data. Which means, for agiven system, random inputs will be fed and the corresponding outputs will bemeasured, then a model will be constructed. The main objective of the processis to identify the system generating the data [25]. The process may include thefollowing stages [26]:

� Acquire data using proper experimental design. Proper experimental datawill represent the true dynamics of a system. Data acquisition must beachieved minimizing the external factors such as noise to ensure construc-tion of an e�cient model.

� Analysis of data and preprocessing. After the data acquisition, it ischecked whether is ready for modeling. If not, then it is �ltered or re-sampled to achieve the best possible sequence.

� Model estimation and validation. There is will be a group of model thatwill �t close to the dynamics of the system. Each model is estimated andvalidated until a model which resembles best with the system is chosen.

15

Figure 3.1: System Identi�cation �owchart

A �owchart of the system identi�cation procedure is shown in �gure 3.1. Theexperiment is designed �rst and based on that data is collected. The data ispre-�ltered before choosing the proper model structure. The model is thenevaluated and validated by estimating the parameters. The process ends if themodel is ok; otherwise, the procedure is carried again out with the samecollected data from the previous stages.

3.3 System

A system may be de�ned having a structure of components which processesinput to produce observable output signals. External factor, such as noise, mayinterfere the system and can be described in �gure 3.2.

Figure 3.2: System Identi�cation

Where, xt, nt and ytare the input, noise , and output of the system at acertain time t. A system is said to be a Dynamic System if the output isin�uenced by the past values of the input. It is very important that the output

16

at a certain time is dependent on the immediate past values of the input. Anexample could be street lights; the time it turns on or o� is dependent on theprevious values of the light sensitive photocells.

3.4 Model

A model can be described as a physical description of a system. The reasonwhy the term 'model' is referred instead of using the system is that the modelis not the system itself, rather a description of it. The system can be complexor inaccessible whereas the model can be much simpler and accessible havingcharacteristics close to the system. It may be expensive or time consuming torun experiments on a system; such factors can be overcome by using a model ofthe system.

3.5 Model data

The data for the system identi�cation procedure can have single or multipleinput or output. The input data must be discrete and uniformly spaced in timeshown by the equation below:

x ={x(t), x(2t), x(3t), x(4t), x(5t).................x(Nt)

}which produces and output sequence:y =

{y(t), y(2t), y(3t), y(4t), y(5t).................y(Nt)

}where, xt and yt are the values of the input and output signals at time t.A chart below shows the types of data system identi�cation toolbox can

support

SupportedData

Description

TimeDomain

One or more input or output data variablesampled at a function of time.

Can be real or complex.

Time SeriesOne or more outputs but no measured input.Can be time domain or frequency domain.

FrequencyDomain

Fourier transform of the time domain input oroutput signals

FrequencyResponse

Complex frequency response values of a linearsystem also known as frequency function data

Table 3.1: System Identi�cation toolbox xupported data

3.6 Dynamic Model

A model is said to be dynamic if the inputs and outputs of the model has somerelationship with the future output. In mathematical term, if the output of themodel is a function of the previous value of the inputs and outputs then themodel is said to be dynamic, represented by the following equation:

17

y ={x(t−1), t(t−1), x(t−2), t(t−2), x(t−2), t(t−2).............

}where,xt and yt are the values of the input and output signals at time t.

3.7 Linear and non-linear models

A model is linear if the output is a function of the previous value of the inputsand outputs and is linear. If the function is nonlinear then the model is nonlin-ear as well. System Identi�cation can estimate linear and nonlinear models ofdynamic systems from measured data. It can also model additional inputs tothe system such as noise. In practice all systems are non-linear and the outputis a nonlinear function of the input signal. But the system can be representedby linear model having dynamic properties close to it.

When a system physical property corresponds a nonlinear system, the inputand output data can be manipulated such that it becomes linear. The responseof the system is plotted to a speci�c input and compared with the input level.If the response is found di�erent then nonlinear model is used. Linear modelof varying complexity can also be identi�ed. If the model doesn't reproducemeasured output, nonlinear model is used.

3.8 Parametric and Non-parametric model

Methods to identify linear models can be divided into two groups, Parameterestimation method and Non-Parametric methods [27, 28]. Parametric modeltries to estimate the parameters in a user speci�ed model such as transfer func-tion and state space matrices. Black box and grey box are the two types ofmodel structure. A model that has some unknown parameters and some knownparameters called a grey-model structure. A model without known paramtersare called a black-model structure. Whereas non-parametric model estimates ageneric models such as step response, impulse response and frequency response.Depending on the model type we need to choose di�erent ways to identify.Suppose for a parametric model we need to estimate the parameters and fornon-parametric model, such as frequency response, we need to �t the graph.

3.9 Black Box Model

Black Box Model is a systems model whose internal structure and implementa-tion knowledge is unknown. The parameters may not have a physical interpreta-tion and the system can be represented as having an input, output and transfercharacteristics only. The structure of a model can be di�erent in modeling ablack box, many models can be estimated and the best one which describes itcan be chosen. The system can be describes using di�erential equation or trans-fer function. Black Box model can either be linear or nonlinear, where linearmodel can be either a continuous time system or a discrete time signal. It isconstructed to describe a large class of system. If a black box model structureis used no physical modeling is required, such as the state space averaging[27].

18

3.10 General Linear Model

The choice of the model structure depends on the dynamic and characteristicsof the system [25]. Model having more freedom and described by higher or-der polynomial is vulnerable as it may lead to that the model parameters areadapted to the characteristics of the experimental noise and dynamic character-istic which does not exist. To provide model �exibility to the system dynamicsand the noise characteristics, system model structure can be describe as shownin the �gure below.

Figure 3.3: General Linear Model structure

Figure 3.3 shows the general linear model structure, the estimation of it iscomputed by nonlinear optimization method. The method involves highcomputational complexity with no certainty of global convergence. Simplemodel, such as AR, ARX, ARMAX, Box-Jenkins, and Output error structure,can be created by putting one or more polynomials equal to 1. Each of themodels has their distinctive characteristics and is widely used in modelingsystem.

3.11 AR Model

The Autoregressive (AR) model is basically the structure for modeling a signalrather than a system, since a system consists of an input as well as an output,where the current output is dependent on the previous outputs. The input isthe time-series signals with no system inputs or noise contribute in modeling,and the output is a sequence of multiple models. Common application of theAR model is the linear prediction coding. Figure 3.4 shows the structure of ARmodel where e represents the system disturbances and y is the system output.

Figure 3.4: AR model structure

19

3.12 ARX Model

If the value of C(q), D(q)and F(q) is 1 then the general linear model reduces toAutoregressive with exogenous terms (ARX) model. In polynomial estimationmethods, the ARX model is more e�cient since it is formed using least squaremethod. Where the solution is unique and always satis�es the global minimum ofthe loss function. Figure 3.5 shows the structure of an ARX where e representsthe system disturbances, x represents the system input and y is the systemoutput.

Figure 3.5: ARX Model structure

The model equation can be writte as

y[k] = B(q)A(q)u[k] + 1

A(q)e[k]

The model has some disadvantages and can be overcome if the signal-to-noiseratio is high. The transfer function of the deterministic part of the signal hassame set of poles as the transfer function of the stochastic part of the signal.And the coupling can bias the estimation of the ARX model if the disturbanceof the system is not white noise. However, the calculations are very simple.

3.13 ARMAX Model

If the value of D(q)and F(q) is 1 then the general linear model reduces to Autore-gressive moving average with exogenous terms (ARMAX) model. The modelstructure includes system dynamics unlike the ARX model. Where there areearly disturbances in the process, such as at the input, the ARMAX model inmore e�cient and �exible in modeling the disturbance than the ARX model.Fifure 3.6 shows the structure of an ARMAX where e represents the system dis-turbances, x represents the system input and y is the system output.The modelequation can be writte as

y[k] = B(q)A(q)u[k] + C(q)

A(q)e[k]

20

Figure 3.6: ARMAX Model structure

3.14 Box-Jenkins Model

If the value of A(q) is 1 then the general linear model reduces to Box-Jenkinsmodel. Here the disturbance characteristics are modeled separately from systemdynamics; as a result it provides a complete modeling of a system. Box-Jenkinsis good in modeling system that has late disturbances, such as measurementnoise at the output. Figure 3.7 shows the structure of Box-Jenkins where erepresents the system disturbances, x represents the system input and y is thesystem output.The model equation can be writte as

y[k] = B(q)F (q)u[k] + C(q)

D(q)e[k]

Figure 3.7: Box-Jenkins Model structure

3.15 Output Error Model

If the value of A(q), C(q)and D(q) is 1 then the general linear model reduces tooutput error model. Here the model describes the system dynamics separatelyand does not use any parameters to simulate the disturbance properties. Figure3.8 shows the structure of Output error model where e represents the systemdisturbances, x represents the system input and y is the system output.Themodel equation can be writte as

21

y[k] = B(q)F (q)u[k] + A(q)

C(q)e[k]

Figure 3.8: Output Error Model

3.16 Transfer function Model

Transfer function model are used to describe the deterministic part of a sys-tem. General linear models are commonly used to de�ne stochastic part ofa system as it can describe the stochastic and deterministic part of a systemseparately. In classical control engineering, deterministic part is more impor-tant than stochastic part of a system. Transfer function model can be use tode�ne either a continuous time system or a discrete time system, which canbe describes using the following equation, from which the transfer function isderived.

y(t) = G(s)x(t) + e(t)y(k) = G(z)x(k) + e(k)

Where,y(t)and y(k) are the system outputx(t)and x(k) are the system inpute(t)and e(k) are the system disturbanceG(s)and G(z) is the transfer function between the stimulus and the response

3.17 State space model

The state space model is the most e�cient method to describe a system hav-ing multiple inputs and multiple outputs (MIMO systems). For complex sys-tems having high order and a large number of calculations the above mentionedmethods faces complications, as a result they lack to converge in the globalminima[25]. Also the computational time for the iterative procedure neededto converge in the local minima maybe excessive. In contrast, the state spacerepresents the system more completely than the polynomial models since theidenti�cation procedure does not require nonlinear optimization. As a resultwithout the initial guess the estimation reaches the solution. In addition theparameter setting in state space model is simpler than the polynomial modelsas it requires only setting the order and the number of states of the model.

The state space model can be describes by the following equation. Bothcontinuous and discrete state space models can be estimated.

x(n+1) = Ax(n) + Bu(n) + Ke(n)

22

y(n+1) = Cx(n) + Du(n) + e(n)

Where,

x(n)= State vector

y(n)= System output

u(n)= System input

e(n)= Stochastic error A,B,C,D and K are the system matrices

3.18 Grey Box Model

Grey Box model is used to model a system, whose internal dynamics and physicalparameters are partially known, what is going inside the system is not entirelyknown. The remaining unknown parameters are then estimated with someestimation methods using a model which is constructed based on insight ofthe system and experimental data. The unknown parameters are the heart ofthe system identi�cation e�orts If a state space averaging produces a modelhaving some parameters unknown then it's a gray box model structure, wherethe parameters can be estimated with system identi�cation methods [27].

A grey box model allows modi�cations in model design rather than working ona �xed design. This kind of �exibility provides to determine the e�ects due tovariations in design. It also helps to analyze the design sensitivity for aparticular design feature. Grey box identi�cation procedure includes:

A) Designing experiment for the system, in our case the buck convertercircuit. To collect the input-output data, a model of the buck converter circuitis constructed on a simulation platform. This could be done in Matlab Simulinkwhich can perform multi-domain simulation and model based design for dyamicand embedded system [8]. The experiment is design in a manner to collect asmuch useful information about the system as possible, which would eventuallydescribe the buck converter system.

B) Data collection and processing- The experimental data is collected withproper selection of the sampling time. The choice of the sampling time hasa great in�uence on the identi�cation process. The sampled data from theexperiment has many imperfections and needs be pre-�ltered before using inidenti�cation algorithms. There might be many shortcomings within the datasuch as o�sets, trends, drift, periodic variations and low frequency disturbances[26]. These types of disturbances could be removed either by making a noisemodel or pretreatment of data. The later will be performed during the simula-tion. Generally the measurements are recorded in physical units. The levels inthe input and output level may have some deviation from normal levels. Thiswill allow the models to waste some parameters correcting the levels. Subtract-ing mean from the input and output data before the estimation procedure willcope with this deviation from equilibrium [26].

∆u[t] = u[t]− 1N

∑Nt=1u[t]

∆y[t] = y[t]− 1N

∑Nt=1y[t]

23

There are possibilities that few or many input-output data samples could bemisinterpreted. This kind of outliers and missing data could happen due tocertain measurement failures while acquiring the data. There could be partlymisinterpreted data. To deal with this problem the data could be divided in tosegments of informative part and later merging di�erent portions to make adata object. The function misdata can be used to reconstruct any missingdata from the data object [26]. The data could be �ltered before theestimation procedure to enhance certain frequency regions in order to attain abetter �t of the model.

C) Constructing Model Structure and validation- Choosing an appropriatemodel structure is an important aspect in the identi�cation procedure. Themodel structure will consist of known and unknown parameters that will bestdescribe the buck converter circuit. There are varieties of models available whichis discussed in later section. Based on the properly selected model structure wecan estimate the unknown parameters and perform tests using di�erent sets ofexperimental data in order to validate it.

The grey box model could be used to specify partially known physicalparameters to estimate unknown parameters using model estimation method.The physical system could be represented by state space model consisting ofknown and unknown parameters and can estimate linear/non-lineardiscrete/continuous time models for arbitrary di�erence or di�erentialequation. The physical system can be single-input single-output (SISO),multiple input-multiple output (MIMO) or multiple input-multiple output [8].The state space model of the the buck converter could be found by applyingKircho�s laws as described in section 2.8. It is used to create an arbitraryparameterized state space model function in M�le which contains user de�nedparameters and information about the model.

In continuous time the state space model could be written as

x(t) = Ax(t) + Bu(t) + Ke(t)

y(t) = Cs(t) + Du(t) + e(t)

In discrete time, the state space model could be written as

x(kT+T ) = Fx(kT ) + Gu(kT ) + Ke(kT )

y(kT ) = Hs(kT ) + Du(kT ) + w(kT )

There are varieties of parameter estimation methods to build mathematicalmodel of dynamical system using recorded input-output data. Prediction errormethods can be used to predict and estimate arbitrary parametric models. Ifwe consideru(t) and y(t) to be the input and output of the system at time tand ZN = [u(1, . . .N), y(1, . . .N)] accumulate previous data up to time N andwe want to predict the value y(t)based on the observations of the past sample.The predictor of the model could be de�ned as y(t t − 1) = f(Zt−1) Wherey(t t − 1) is the one step ahead predictor, which depends on the arbitraryfunction of previous observed data. A �nite dimensional parameter vector θ isintroduced to parameterize the predictor such that an estimate of θ could befound from the measured dataZN and the model parameterization in order to

24

minimize the distance between the predicted values y(t θ), . . . . . . ., y(t θ) andthe measured values y(1), . . . . . . y(N) in a suitable norm[34]

y(t θ) = f(Zt−1),θ)

Where y(t θ) is a general predictor model.

(θN ) = argminj

VN (θ)

VN (θ) =∑

Nt=1l(y(t)− f(Zt−1,θ))

Here, l de�nes a suitable distance measure such that l(ε) = ||ε||2. Detailedinformation about the minimization problem is discussed in [5, Chapter 10].

For a given data the unknown parameters could be obtained using iterativeprediction error minimization (PEM) method. Grey box identi�cation usesoptimization to minimize the di�erence between the measured real output andthe estimated output. PEM uses this optimization to minimize the cost function.

VN (G,H) =∑

Nt=1e

2(t)

Where e(t) is the di�erence between the measured real output and theestimated output, VN (G,H) is a scalar value and N indicated the number ofdata sample.

For linear model the error could be de�ned by

e(t) = H−1(q)[y(t)−G(q)u(t)]

For multiple output models the equation becomes more complex. PEM usesthe same algorithm as armax apart from some alteration in computation ofprediction errors and gradients [8].

In order to optimize the prediction error several estimation algorithm optionsare available which can a�ect the quality of the results[8].

Simulation Method- The simulation method could be selected using�SimulationOption� �eld. Various �xed step and variable step solvers areavailable. For continuous time the default solver is ode4[matlab]. Searchmethod- The search method could be selected using the �SearchMethod� �eldto generate the maximum likelihood estimated of the model parameters[8].

Gradient options- Method to calculate the gradients could be selected using�GradientOptions� �eld. Gradients are calculated using the derivatives oferrors with respect to unknown parameters and initial states.

The �rst step in the estimation procedure is to de�ne a model structure.The model of the dc-dc converter is constructed with known dynamics. Thestate space model with respect to the converter circuit can be set up

ddt

ilvcvcl

=

−Rl

L 0 − 1L

0 − 1C∗RC

1C∗RC

1Cl

1Cl∗RC − 1

Cl∗RC

ilvcvcl

+

1L 00 00 − 1

C1

[ dUi0

]

v0 =[

0 0 1] il

vcvcl

+[

0 0] [ dU

i0

]

25

The unknown parameter Cl is to be estimated with the help of the observeddata and the known parameters. The parameterized state space modelstructure could be represented using IDGREY objects and the estimationcould then be performed using the experimental data.

A state space model object consists of the all the information about themodel and the parameters. The parameters that are going to be estimatedusing the experimental data could be speci�ed. The model object is created inthe following way. Initially a nominal parameter model is set up using knownentries. Then a model object is made and the unknown parameter values arespeci�ed for parametric analysis. The unknown parameters which are to beestimated are speci�ed using par symbol and is initialized with best guesses orzero is put if unknown. Both continuous time and discrete time models can bestored as objects.

A=[-Rl/L 0 -1/L;0 -1/(C*Rc) 1/(C*Rc);par(1) par(1)/(Rc) -par(1)/(Rc)];B=[1/L 0;0 0;0 -par(1)];C=[0 0 1];D=[0 0];K = [par(3);par(4);par(2)];x0 = [par(5);par(6);par(7)];

For a given value of the parameter vector the model will predict the outputy(t) at time t− 1of the estimated model.

The quality of the predictor could be assess by calculating the prediction erroras discussed earlier in the chapter.

ε(t) = y(t)− y(t).

The estimated models could then be simulated using Matlab sim command.Same procedure was followed for reduced order model but with di�erent modelobjects.

26

Chapter 4

Simulink Implementation and

simulation of the buck

converter model

4.1 Introduction

Simulink is a basic way to model and analyze LTI systems with the ability todeal with complex systems in the �eld of Control Systems. It can perform multi-domain simulation and performs mathematical simulations of a wide range ofdynamic and embedded systems[8]. It is an extension to Matlab, which providesa graphical user interface (GUI), where models of the numerical equations couldbe entered as block diagrams. The system could be modi�ed at a higher levelusing click and drag mouse operation. New blocks could easily be created ormodi�ed using the extensive Simulink block library which provides useful blockssuch as sinks, sources, continuous, discrete, linear, non-linear and mathematicaloperations. Simulink models are classi�ed into various criteria in to successivelevels. As a result it provides better insight of the model and its connectingparts.

In order to simulate the buck converter model, the whole model is dividedinto blocks and its corresponding numerical equations are found. The equa-tions are represented using simulink library blocks in the simulink browser withspeci�c parameter values. In order to determine sets of linear and non-lineardi�erential equations MATLAB's ordinary di�erential equations(ODEs) solveris used. The type of solver and the simulation time are set according to thedata needed.

4.2 Simulink model for Buck Converter

The model of the buck converter circuit is obtained by applying Kircho�s voltageand current laws which leads to sets of equation describing the buck convertercircuit. The procedure of deriving the model equation and selecting the statevariables was shown in section 2.8.

dU = ilRl + Ldildt + Vo

27

C dVc

dt = Vo−Vc

Rc

il = CldVo

dt −Vc−Vo

Rc+ io

The above equations represent the exact behavior of the buck convertercircuit. The equations are implemented in Matlab Simulink.

Figure 4.1: Simulink model of buck converter

The simulation diagram is a time domain representation of the buck convertercircuit. The numbers of state variables are equal to the number of integratorsin the circuit. The simulink model comprises of sources, sinks and variousfunctional blocks. The model is represented using simple functional block sothat the state space equations could be easily derived from the circuit.Instead, matlab built in state space block could also be used. In that case thedata obtained does not match exactly but gives a similar behaviour. Themodel is con�gured with a number of parameters as shown in �gure 4.1. Theparameters are: the capacitance C, the load capacitance Cl, the inductance L,the internal resistance of the capacitor RC and the internal resistance of theinductor respectively. The system model is highlighted in to three majorsegments: Pulse width modulator, buck converter system and the load. Thesegments are brie�y discussed later in this chapter.

Figure 4.2: Generation of PWM signal

28

There are two inputs to the model. The �rst input to the buck convertersystem is product of the constant voltage source V g and the duty cycle d. Theduty cycle is the ration of the pulse width to the switching period and has avalue in the interval 0 to 1. Figure 4.2 shows the pwm system which consistsof a saw-tooth waveform generator, duty cycle command and a zero-crossingcomparator. The switching period is equal to TS i.e the period of the Signaland is determined by the control signal δ(t). The control signal δ(t) is obtainedusing the pulse width modulator. The simin command pass the duty cyclegenerated from the workspace. The output of the zero-crossing comparator isa pulse train of amplitude 1. When the signal from the sawtooth exceeds dutycycle the output is 0 and when the sawtooth signal less than the duty cyclethe output is 1 as shown in �gure 4.3. The width of each pulse is determinedby the duty cycle d. The duty cycle was arbitrarily changed between thevalues 0.4 and 0.6 in order to make the system react to changes.

Figure 4.3: Duty cycle generated from the saw tooth

29

Figure 4.4: : Buck converter system

Figure 4.4 shows the buck converter model of the system. The model isde�ned with the system equations using various functional blocks (integrator,adder, multiplier, gain etc). The product of the input voltage and the dutycycle provides the required pwm signal for the buck converter system. Theaverage output voltage can be controlled using the duty cycle. The systemequations, shown in blue, show the part of the system it belongs to.

Figure 4.5: Load

The second input is the constant current source, I, provided by the outputload. Figure 4.5 shows the representation of capacitive load through simulinkblocks. It consist of the constant current source, a di�erentiator and a gainelement. Fig 4.6 shows the output voltage with a constant duty cycle.

30

Figure 4.6: Simulation of the output voltage

4.3 Preparing input and output data

The choice of the input data is very decisive for the process of system identi�ca-tion. A good choice of data will lead to successful identi�cation of the process.The data must contain substantial information to identify the system well withgood results. To fetch the experimental data the following adjustment wereconsidered to be vital

� Choice of input signal � To get enough information the input signal has toexcite the system [29]. An input with a single frequency, ω will not givesu�cient information about the system, as a result it should contain ade-quate number of frequencies to make a unique input-output combinationsince there are many systems which behaves similarly for the same input.If the signal levels are arbitrarily changed between two levels it will gener-ate maximum number of frequencies for the system to react considerably.Such signal could be generated using pseudo random binary sequence.

� Sampling frequency- The sampling interval may a�ect model accuracy andis chosen precisely. High sampling rate compared to the system dynamicswill lead to redundant information and relatively small information valuein the new data points. While a lower sampling rate in respect to thesystem dynamics will hinder the ability to identify the correct parametervalues [29]. This is due to the fact that the shape of the original signalcannot be matched very closely after sampling it at a lower rate. Whiledetermining the sampling interval the Nyquist criterion is kept under con-sideration. The condition states that the reconstructed signal will matchthe original signal provided that the original signal contains no frequenciesat or above this limit (fs ≥ 2B).

31

� Pre-processing of data- There might be certain lacking in our sampleddata that may interfere in our identi�cation procedure. This could bethe drifting of signal levels or high frequency disturbances, these kind ofdrifting is undesirable and as a result any trends in the data should beremoved. There could be obvious errors (outliers) among the data or anytransients, that needs to be removed, that might have been accumulatedbefore the desired operating point is reached for which the modeling wasperformed[29].

The identi�cation of the converter was performed by collecting the data fromthe simulation experiment and was used to determine a model for the converter.An input prbs signal was generated and the corresponding output signal wascollected during the experiment. The input and output data for the systemidenti�cation procedure were retrieved as an array in the workspace by thevariable name parameter. The sampling frequency was chosen to be 330kHz.To remove the o�sets the mean values were deducted from the data. Thisprocess bene�ts to make a more accurate linear model due to the fact that thelinear models are less responsive to slight deviation between input and outputlevels [8]. The observed data was used to make two sets of data. The �rst setwas used as an estimation data or training data. The information of the datais used to select a model for the system. The second sets of data were used asthe validation data. This set of data was used to make certain that the model isnot only valid for the estimation data but for other data as well. Fig 5.4 showsthe input output data. The graph it can be observed that the output voltageis lower than zero. This is because a linear model of the system was simulatedin order to get any working point for the system. The real system consists ofa diode, which allows current to �ow in one direction, making the current tobe non-negative, when the switch is open and the system then goes in to thediscontinuous mode. In our analysis this is not a problem since the model islinear.

Figure 4.7: Estimation Data

32

4.4 Identi�cation Procedure

The whole system identi�cation procedure was carried out in two ways. The �rstapproach is to consider a black-box model using GUI. The second approach isto consider a grey-box model using command line. Both methods use the timedomain input-output data to identify the system parameters. Below are theidenti�cation procedures for the two mehods described.

4.4.1 Black box modeling & results

System identi�cation toolbox software is used to build the mathematical struc-ture using the sampled data from the simulink model of the buck converter.. Fig4.8 shows the two sets of input and output data used to estimate and validatethe four model structures (ARX, ARMAX,OE,BJ).

Figure 4.8: Input output signal

Figure 4.8 shows the input and output data collected by running the simulinkmodel. The model consists of the same parameter values mentioned in section2.8. The data contains the output voltage[V] at the load capacitor and theinput voltage [V ] of the pwm signal. It is sampled at a frequency of 330 kHz .The data is divided in two segments in �gure 4.8; the red part is theestimation data and the magenta part is the validation data.

33

Figure 4.9: Spectra of the Input and output voltage

Figure 4.9 shows the Periodogram of the input voltage and the output voltageof the selected data set. It is the absolute square of the Fourier transform ofthe data sequence.

Figure 4.10: Frequency function of the data

Figure4.10 shows the estimated frequency functions of the input and outputdata. It gives the amplitude and phase response of the corresponding frequencyresponse and is estimated using ETFE(Empirical Transfer Function Estimate)

34

Figure 4.11: System identi�cation Tool GUI showing di�erent types of QuickStart Linear models

Fig 4.11 shows the system identi�cation tool GUI showing di�erent types linearmodels that could be estimated. It can produce the �nal linear models andcould provide information to con�gure the estimation of accurate parametricmodels, such as time constants, input delays and resonant frequencies [8]. Themodel order for the polynomial is chosen before the estimation. The modelorder was increased until a satisfactory accuracy was obtained.

Figure 4.12: Step response of di�erent model

35

Step response can be used to estimate the order and relative degree of asystem [30]. Figure 4.12 shows the step response of di�erent estimated model.There are two ways to identify the order of a system using the step response.If the response of a system is a zero slope to a non zero input whent = 0 and ifthere are oscillations in step resonse then the system is a second order orhigher. It can be observed from the �gure that the step response has a zeroinitial slope when t = 0. Moreover the system oscillates which concludes thatthe system is a second order.

Figure 4.13: Bode plot from input voltage to output voltage

The order of the system can also be found by analyzing the bode plot. Asecond or higher order system is con�rmed when the phase drops below -90degrees. The number of multiples of -90 degrees obtained asymptotically atthe lowest point of the phase plot of the system de�nes the relative degree ofthe system [30]. It can be observed from �gure 4.13 of bode plot that thephase plot dips asymptotically to -180 degrees relative to input. Since -180degrees is two multiple of -90 degrees, it can be con�rmed that the system is asecond order or higher. But it is already known that the model is a third ordersystem.

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Figure 4.14: Impulse response from input voltage to output Voltage

Figure 4.15: Measured and simulated model output

Figure 4.14 shows the comparison of measured and simulated model output.Di�erent model structures are veri�ed using a di�erent set of data known asvalidation data and the �t values are listed on the top-right corner of thewindow in descending order of the best �t models. A �t value 100 indicatesthat the model output �ts perfectly to the measured output as the mean of themeasured output. While 0 indicates a poor �t to the validation data.

37

Figure 4.15 shows second order oe221( Output Error) model best �ts thesystem dynamics compared to other modelsS. There are two basic method tocompute the state space model. N4s2 is 15 second order state space model andis calculated using n4sid method. It uses a sub space based method, it usesprojection instead of using iterative based method. While PEM is a standardprediction error likelihood method, based on iterative minimization of a crite-rion. To start the iterations N4SID is used to calculate the parameter values.Bj2221 is a second order Box-Jenkins model. Armx2221 and arx221 are secondorder ARMAX and ARX models. The coe�cients of ARX, ARMAX, OE andBJ models are computed using prediction error/maximum likelihood method[8].

State-space model:x(t+Ts) = Ax(t) + Bu(t) + Ke(t)y(t) = Cx(t) + Du(t) + e(t)A =

x1 x2

x1 0.91155 −0.018059x2 0.55357 1.0796B =

InputV oltag ConstantLoax1 7.0446e + 005 9.107e + 007x2 −1.6781e + 008 −4.1886e + 008C =

x1 x2

OutputV oltage 8.8194e−010 −2.3098e−011

D =InputV oltag ConstantLoa

OutputV oltage 0 0K =

OutputV oltagex1 2.2345e + 008x2 −2.2412e + 009x(0) =

x1 1.1957e + 009x2 5.0031e + 010Estimated using N4SID from data set EstimationLoss function 1.01009e−007 and FPE 1.02365e−007

Sampling interval:3e−006 sec

Table 4.1: N4s2 (State space) model ouotput

Table 4.1 shows the state space model output obtained by utilizing Matlabssystem identi�cation toolbox. Although the black-box state space model givesa good �t value against the data obtained but it is not able to identify theparameters correctly. The matrices in the second row of the B matrix shouldhave been the same according to the derived mathematical model. Moreover, Aand C matrix do not match the mathematical model as well.

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Figure 4.16: Input and output signals

Figure 4.16 shows the input and output data collected by running the simulinkmodel. The data contains the output voltage[V ] at the load capacitor and theinductor current[A]. It can be seen from the �gure that a negative voltage andnegative current appears, this is due to the fact that the means were removedfrom the signal as we are interested in the dynamics but not the signal level.After identi�cation we can add a constant to the model to bring the output tothe correct level.

39

Figure 4.17: Measured and simulated model output

Figure 4.17 shows the comparison of measured and simulated model outputusing the inductor current as the input.

4.4.2 Grey box modeling & results

In greybox modeling, the system is represented using the unknown parameterCl

and known mathematical structure with observed data. Since most of the pa-rameters of the system and information about the model are known, a statespace model structure would better �t the model and will represent the systemcompetely.

The state space model could be represented by the following equation

x(t) = Ax(t) + Bu(t) + Ke(t)

y(t) = Cs(t) + Du(t) + e(t)

Where y(t) is the output equation, x(t) is the state vector, u(t) is the inputvector and e(t)stochastic error.

� A is an n x n matrix, where n is the number of states.

� B is an n x m matrix, where m is the number of inputs.

� C is an p x n matrix, where p is number of outputs.

� D is an p x m matrix.

40

� K is the kalman gain matrix.

The state equations for the model could be created by determining state vari-ables from the system model or from the nth order di�erential equation, whichis discussed in section 2.8. The state space model for the buck converter systemwas found found in the the following steps

After identifying the grey-box model, pem was used to estimate the values ofthe unknown parameters.

-Apply KVL to recognize energy storage elements.

-Identify the relation between each energy storage elements.

-Re arrange to the equation to relate the derivative of the state vector withthe state vector and the input.

-Apply physics and algebra to �nd other state equations.

-Relate the output with the state vector and the input.

-Perform an experiment on the real system, in order to get data.

The time domain data is stored of the input and output vector are stored in acolumn vector in iddata object

data = iddata(y, u, Ts)

Where, y is the time domain output signa with added noise sequence, u isthe time domain input signal with added noise sequence and Ts is the samplingtime of the experimental data. For continuous time Ts is set to be 0. Thedata stored is �ltered using a moving average �lter before being processed forestimation.

The parametrized state space model described in the section is representedusing idgrey objects. Idgrey is class for using ODE models that uses experimen-tal data to estimate the parameters.

A,B,C,D,K, xo = idgrey(par, T, aux)

Where, A,B,C,D andK are the system matrices described earlier and x0 isthe initial state values, pars contain the parameters, T is the sampling intervaland aux is any variable of auxiliary quantities.

When the system is described by the model object, the model parametersare estimated using iterative prediction-error minimization method

m = pem(data,m)

Where, data is the iddata object and m is the idgrey object.

41

4.4.2.1 Ideti�cation results of 3rd order state space model

Figure 4.18: Step response from input voltage to output voltage

Figure 4.18 shows the step response from input voltage to output voltage ofthe estimated model and the real state space model of the buck convertersystem. It can be observed from the �gure that the estimated model seem tobe quite good at matching the response of the actual model.

Figure 4.19: Bode plot from input voltage to output voltage

Figure 4.19 shows the bode plot from input voltage to output voltage of theestimated model and the real state space model of the buck converter system.It gives a further con�rmation of the identi�cation of the identi�cationmethod. The original and the estimated model seem to be in good agreement.

42

The response of the estimated model closely matches the response of theactual converter.

Figure 4.20: Online information about the minimization

43

Figure 4.21: Model output

Figure 4.21 gives an illustration of the model compared to the original outputusing the original input and hence evaluates the models quality in capturingthe system dynamics. The simulated output, m, gives a �t value of 72.92% , itis relatively good in capturing the system dynamics. While n is the real statespace model with known output capacitance. This is to make sure that thestate space model constructed according to the simlink model from which thedata was collected. Due to added noise to the input and output sequence,original model gives a �t value of 96%.

44

Time domain data set with 3001 samples.Sampling interval: 3e−006 sec

Outputs Unit(ifspecified)OutputV oltage V

Inputs Unit(ifspecified)InputV oltage V

ConstantLoadCurrent AState-space model:x(t+Ts) = Ax(t) + Bu(t) + Ke(t)y(t) = Cx(t) + Du(t) + e(t)A =

x1 x2 x3

x1 0.99794 −0.052378 0.015972x2 0.055868 0.99397 0.0034352x3 −0.046228 0.092054 0.04072B =

InputV oltag ConstantLoax1 −0.00010793 0.0012839x2 −0.00035296 0.00019392x3 0.011371 −0.079782C =

x1 x2 x3

OutputV olta 162.13 28.984 2.8011D =

InputV oltag ConstantLoaOutputV olta 0 0K =

OutputV oltagex1 0.0241x2 −0.023088x3 −0.72581x(0) =

x1 −0.00058571x2 −0.00091205x3 0.0076389Estimated using PEM using SearchMethod = Auto from data set Load Identi-�cationLoss function 3.67193e−025 and FPE 3.70864e−025

Sampling interval: 3e−006sec

Table 4.2: State space model output

45

Figure 4.22: Prediction error for output voltage

Figure 4.22 shows the prediction error obtained with the estimated DC/DCconverter model. It can be observed from the �gure that most of theprediction error obtained are small and they are centered at zero(non-bised).

Figure 4.23: Comparison between estimated and actual capacitance values

Figure 4.23 shows the comparison between estimated and actual capacitancevalues. The x-axis of the graph shows the actual value that was used for the

46

load capacitor and the y-axis shows the estimated values obtained. Theestimated values are obtained through the simulation with added noise toinput and output signals. The estimated value is not good compared to thereal output. But the linear behavior of the output compared to the estimatedvalue is utilized to re-calculate the output value using the gradient andintercept of the straight line graph. The re-calculated value gives a very goodresult and is able to estimate the capacitance values with very good accuracy.Table 4.3 shows the estimated value and error percentage using variouscapacitance values..

Real Cap Value Estimated with noise Error Percentage

2.00E-07 1.49177E-07 25.45.00E-07 3.72064E-07 25.51.00E-06 7.66854E-07 23.12.00E-06 1.73841E-06 13.13.00E-06 2.9591E-06 1.365.00E-06 4.9243E-06 1.517.00E-06 6.96726E-06 0.46

Table 4.3: Original Capacitance value, estimated value and error percentage

4.4.2.2 Ideti�cation results of 2nd order state space model

To implement the second order system with inductor current as an inputwhich was described in section 2.8, the identi�cation procedure was carried outwith a simple case. A prbs signal with noise was generated, which is then fedinto the second order model and the load capacitor value was obtained. This isto verify the valitdity of the model whether it wokrs with simple case or not.Figure 4.24 shows the simple second order model implemented on Simulink.

Figure 4.24: Simulink model of a simple second order model

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Figure 4.25: Noisy PRBS inductor input signal and load voltage

Figure 4.25 shows the prbs signal added to noise, which is used as an inductorcurrent signal to the second order model. From the graph it can be observedthat the output voltage is lower than zero. As discussed in section 4.3, the realsystem consists of a diode, which allows current to �ow in one direction,making the current to be non-negative, when the switch is open, it then goesinto discontinuous mode.The prbs signal consists of two levels (0.6 and 0.9)with added noise (+/-0.04). Table below shows the estimated values obtainedfor various load capacitance values and the corrosponding error percentage.

Real Cap Value Estimated with noise Error Percentage

2,00E-06 1.9842e-006 0.79173,00E-06 3.0383e-006 1.27605,00E-06 4.9928e-006 0.14347,00E-06 7.3284e-006 4.69101,00E-05 1.0066e-005 0.65685,00E-05 5.0443e-005 0.88548,00E-05 8.0106e-005 0.13281,00E-04 1.0018e-004 0.1797

Table 4.4: Original capacitance value, estimated capacitance value & error per-centage using a prbs signal as an inductor current input

The reduced state space model seems to work very good with this simple caseas the estimated values are very close to the original load capacitor values. Inthis case the control sequence was generated using prbs signal, alternatively asaw tooth signal can also be used. In the next step the control sequence isgenerated for the input voltage for the third order system. Then the

48

identi�cation procedure for the second order was carried out using theinductor and the load voltage values obtained through the third order modeldescribed in section 4.2. Figure 4.26 shows the third order buck convertermodel. To get the inductor values for the second order model, the third ordermodel shown in �gure 4.26 is simulated. The inductor and the output voltagedata were obtained from this model and were used as an identi�cation data forthe second order inductor model.

Figure 4.26: Identi�cation of second order model using third order model data

49

Figure 4.27: Comparison between estimated and original capacitance value forreduced model

Fig 4.27 shows the comparison between estimated and actual capacitance values.The x-axis of the graph shows the actual value that was used for the loadcapacitor and the y-axis shows the estimated values obtained. The estimationwere performed using a prbs signal in which the duty cycle for the input voltagewere arbitarily changed betweem 0.4 and 0.6. The estimated values obtainedthrough simulation of the reduced model with added noise, also follow a linearrelationship with the actual capacitance values. The properties of the graph isutilized to obtain a re-calculated value through linear mapping.

50

Figure 4.28: Validation of the model with the data

Figure 4.28 shows the comparison between the model output and measuredoutput. It can be seen that a �t value of 92.92 is obtained which is quite good.The percentage of the output variation is explained by the mode

fit = 100 ∗ (1− norm(yh− y)/norm(y −mean(y)))

here yh is the estimated output and y is the measured output.

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Figure 4.29: Comparison between the bode plots of the actual model and esti-mated model

Figure 4.29 shows the comparison between the frequency responses of theactual model and estimated model. The responses of both the system matcheseach other.

Real Cap Value Estimated with noise Error Percentage

2.00E-07 1.66E-07 17.00745.00E-07 4.76E-07 4.72531.00E-06 1.07E-06 6.83612.00E-06 2.03E-06 1.60083.00E-06 2.97E-06 1.05995.00E-06 5.04E-06 0.7277.00E-06 6.94E-06 0.88051,00E-05 1.00E-05 0.0106

Table 4.5: Original capacitance value, estimated capacitance value & error per-centage using inductor current

Table 4.5 shows the values obtained using inductor current. It can be seenthat a good error percentage is obtained (within 10% of the original value formost of the estimated value) for di�erent capacitor values. Simulation usingthis structure produced similar results.

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Figure 4.30: Comparison between estimated and original capacitance for re-duced model using di�erent prbs signal

Figure 4.30 shows the comparison between estimated and actual capacitancevalues for reduced model using di�erent prbs signal. It also shows a linearrelationship and to re-calculate the values using linear mapping a slightadjustment was done intercept and the gradient of the straight line graph inorder to obtain more accurate result.

Real Cap Value Estimated with noise Error Percentage

2.00E-07 2.0336e-007 1.68155.00E-07 5.0625e-007 1.24981.00E-06 1.0033e-006 0.33232.00E-06 2.0017e-006 0.08593.00E-06 3.0006e-006 0.01885.00E-06 4.9995e-006 0.00927.00E-06 7.0004e-006 0.00531,00E-05 1.0002e-005 0.0181

Table 4.6: Original capacitance value, estimated capacitance value & error per-centage for inductor current using di�erent prbs signal

Finally, the identi�cation is performed with input-output data obtainedthrough feedbacked BMR 450 buck converter with a control sequence. Figure4.31 shows the data obtained through the simulation of a controlled BMR 450buck converter with feedback.

53

Figure 4.31: Output voltage and inductor current of Controlled buck con-verter(Courtesy Anders Hultgren)

54

Figure 4.32: Comparison between estimated and actual capacitance value offeedbacked buck converter

As with previous case, the estimated value in this case is not good but it alsoshows a linear reltionship with the original values. Thus a linear mapping isdone and the values are re-estimated. The new estimated values and the errorpercentage is shown for di�erent capacitor values in table 4.7.

Real Cap Value Estimated with noise Error Percentage

2,00E-06 2.0788e-006 3.94033,00E-06 3.0829e-006 2.76215,00E-06 5.0677e-006 1.35487,00E-06 7.0706e-006 1.00841,00E-05 1.0059e-005 0.59015,00E-05 4.9995e-005 0.01088,00E-05 8.0044e-005 0.05521,00E-04 1.0008e-004 0.0793

Table 4.7: Original capacitance value, estimated capacitance value & error per-centage of a simple BMR 450 controller.

The identi�cation result shown in table seems to match very well with the

55

original capacitor value. It can be seen that a good error percentage is obtained(within 5% of the original value) for di�erent capacitor values.The capacitorvalues for the reduced state space model was identi�ed successfully and alsoreduces the estimation time and the memory requirement.

4.4.3 Summary and Concluding remarks

The ARX model is the simplest model which solves linear regression equationsin analytic form and most e�ective of all polynomial models. The ARX modelis a good choice when the model order is high. In contrast, the system distur-bances contribute to the part of the system dynamics and the coupling can beimpossible. Whereas, ARMAX model structure consists of disturbance dynam-ics. ARMAX models are more e�ective in dealing with disturbances particularlyin early stage of the system. The Box-Jenkins (BJ) structure implements dis-turbance properties separately from system dynamics. It is the most e�ectivewhen there are disturbances later in the system. The output error (OE) modelde�nes system dynamics separately without any parameters describing the dis-turbances characteristics. These models minimize a performance function basedon sum of squared errors. Although these methods can be used in variety ofsituation but when dealing with systems of higher order (that consists of MIMOsystem with many parameters and need quite a bit of calculations); they canface di�culties. The performance functions may have many local minima whichresults in a de�ciency in global minima. Parameterization of system order anddelays can be di�cult. There might also be instability in numerical calculationsand to calculate the iterative numerical minimization higher computation timewill be required [25]. The state space identi�cation method is the ideal choiceunder the circumstances. It is the most convenient and compact way to modeland analyze a MIMO system. It can focus on particular unknown characteristicsof the system. It can represent MIMO system with more details than the poly-nomial models due to fact that it is similar to �rst principle models. The orderof the model could be selected and the state space model identi�cation arrivesto the solution despite the initial guess. In addition, it is much easier to set theparameters for the state-space model and the calculations are quite easier dueto the fact that the model is represented by matrices. With the reduced modelorder the estimation is fast and the memory requirement is also reduced.

The output of the black box model is quite good at capturing the systemdynamics. The input voltage and the inductor current were used separately tocompute di�erent models (ARMAX, ARX, BJ, OE and State-space). Althoughthe output of the models were quite similar and provided good result in terms of�t values, it could not identify the parameter values of the converter with goodaccuracy and hence the load parameter could not be successfully identi�ed. Onthe other hand the grey-box identi�cation procedure was started o� by puttingthe correct parameter values for most of the parameters in the system apartfrom the capacitance value which is to be found. A parameterized state spacemodel structure is created and the load capacitor value is estimed by minimizingthe cost function using prediction error method. To reduce the estimation timeand memory requirement, a second order model struture is derived and thewhole estimation procedure is repeated using measured inductor current. At�rst a simple inductor current is used as an input to verify the model strucure,then with control sequence at the input voltage and in the end with simple

56

BMR 450 controller. The response of the estimated model closely matches theresponse of the actual converter. But the parameter values obtained was notsatisfactory. Using prbs current for the second order model the accuracy washigh but using inductor current from the third order model the result was notgood. This problem was solved by linear mapping from the identi�cation andthe capacitor value. The re-calculated capacitor value matches quite well withthe actual value.

57

58

Chapter 5

Conclusion and suggestions

for future work

In this part, the work presented in this thesis is summarized. Furthermore,suggestions for future work are proposed.

5.1 Conclusion

In this paper, a method for estimating the load parameter of Ericssons BMR450(Dc-Dc Buck Converter) is presented. A simulink model of the buck converter wasdeveloped to generate the data needed for system identi�cation. To get su�-cient information about the system, a prbs signal was used, which consists ofwide range of frequencies. This excites the system and useful data is recorded.The data is used to build a black-box model and a grey-box model to analyzethe system and identify the load parameter. The method was implemented inMatlab and validated through simulation results. The responses of both theestimated model were compared with the actual output. They have matchedsuccessfully in most of the cases. At �rst, the load parameter is identi�ed, bycreating an arbitrary parameterized third order state space model of the systemand using iterative prediction error minimization method to calculate it. Theestimated load parameter value did not provide good result but it showed alinear relationship with the actual load parameter value. The estimated valueis re-calculated by linear mapping and a very good result was obtained. Inorder to reduce the computational complexity and the memory requirment thethird order model was reduced to a second order model with inductor currentas the new input to the model. To test the precision of the second order model,prbs input were used as an inductor current in to the second order model andit delivered a good result with high accuracy without linear mapping. Thenthe actual inductor values form the third order model were used to identify theload parameter of the second order model. The result was not good but it alsoshowed a linear relationship with the actual load capacitor value. Using linearmapping, the model re-estimated the load parameter values with good accu-racy. Moreover, to test the robustness of the model di�erent prbs signal wereused at the input voltage and the end result was also good with linear mapping.Finally, the identi�cation using the second order model was performed on the

59

feedbacked BMR 450 controller. Like the previous cases, it did not give goodresults initially but using linear mapping the re-estimated values were very closeto the original values with an error percentage less than 5.

5.2 Future work

Suggestions for future work are summarized below.� The model should be modi�ed in order to compute the load parameter

directly� The proposed method uses o�ine identi�cation; an online identi�cation

method should be �gure out.� In this model a capacitive load is considered, more complex load should

be investigated.

60

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