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Article

Research on Linear Active Disturbance RejectionControl in DC/DC Boost Converter

Hui Li 1, Xinxiu Liu 1 and Junwei Lu 2,*1 College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China;

[email protected] (H.L.); [email protected] (X.L.)2 School of Engineering and Built Environment, Griffith University, Gold Coast, QLD 4222, Australia* Correspondence: [email protected]; Tel.: +61-07-5552-7596

Received: 24 September 2019; Accepted: 28 October 2019; Published: 31 October 2019

Abstract: This paper proposes a cascade control strategy based on linear active disturbance rejectioncontrol (LADRC) for a boost DC/DC converter. It solves the problem that the output voltage of boostconverter is unstable due to non-minimum phase characteristics, input voltage and load variation.Firstly, the average state space model of boost converter is established. Secondly, a new output variableis selected, and a cascade control is adopted to solve the problems of narrow bandwidth and poordynamic performance caused by non-minimum phase. LADRC is used to estimate and compensatethe fluctuations of input voltage and loads in time. Linear state error feedback (LSEF) is used toachieve smaller errors than traditional control method, which ensures the stability and robustness ofthe system under internal uncertainty and external disturbance. Subsequently, the stability of thesystem is determined by frequency domain analysis. Finally, the feasibility and superiority of theproposed strategy is verified by simulation and hardware experiment.

Keywords: DC/DC boost converter; LADRC; LESO; cascade control

1. Introduction

With the vigorous promotion of renewable energy such as photovoltaics and wind turbines, the DCmicrogrid has received extensive attention [1–3]. Compared with the AC microgrid, the DC microgridhas the advantages of high efficiency and simple control structure [4]. In addition, the DC microgridavoids some of the problems in the AC microgrid. For instance, reactive power flow, harmonic current,and synchronization [5]. Therefore, the DC microgrid will become the main power architecture forbuildings, parks, and power electronic loads. The instability of the output voltage will cause someproblems, including protection device malfunction and damage to the electrical equipment, so it isimportant to ensure the stability of the DC output voltage.

In a DC microgrid, renewable energy units, loads, and other units are connected by inverters,such as DC/DC converters, DC/AC converters, AC/DC converters, etc. Moreover, it can be connected tothe AC microgrid or the large grid through bidirectional DC/AC. Not only can the AC side disturbanceand fault be effectively isolated, but also the DC side load can be reliably powered [6]. The DC/DC boostconverter is an important part of the connection between each unit and the bus. It achieves voltageboost by a specific circuit structure and adjusting the on-off time of the switching device [7]. Open loopcontrol is a simple method of controlling a step-up DC/DC converter that calculates the PWM pulseduty cycle from the input voltage and the output voltage. However, this method is extremely sensitiveto external disturbances and changes from parameters. Moreover, it is easy to cause problems such aslarge overshoot of the output voltage and high amplitude of the inductor current. Therefore, most ofthe DC/DC converter control uses a closed-loop control method.

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Electronics 2019, 8, 1249 2 of 19

Scholars and researchers have done a lot of work on the control of DC/DC converters. In [8,9],the isolated converter is used to increase the output voltage by adjusting the turns ratio. However,the excessive number of turns leads to a large leakage inductor, which makes it difficult to improve theefficiency. In [10,11], a cascade converter is used, which can control the voltage by adjusting the dutyratio of the two parts of the switch tube, but the converter is inefficient and difficult to control. In [12],a high boost converter with a switched capacitor is proposed. High voltage gain is achieved at theappropriate duty cycle by introducing a switched capacitor into the switched mode DC-DC converter.However, the switched capacitor introduces a pulsating input current that can cause poor line andload regulation issues. In [13], state feedback control is adopted, which has strong suppression of loaddisturbance, but does not consider input voltage disturbance. In [14], the high-order synovial observeris used to estimate the load change in advance and cancel it in the feedforward channel, and combine itwith the feedback control loop. It not only improves the transient characteristics of the system, but alsoimproves the stability margin of the system. However, the calculation method of this method is toolarge and sensitive to noise.

By analyzing the transfer function of the step-up DC/DC converter, it can be seen that the systemhas a zero point in the right half plane. Capacitance voltage is directly controlled as an output, whichcauses the bandwidth to be narrowed and the dynamic performance of the system to deteriorate, whichincreases the control difficulty [15,16]. In [17], an input–output feedback linearization technique isproposed and applied to an AC/DC converter. In [18], an internal model control with a conditionalintegrator is proposed for the robust output regulation of a DC/DC buck converter. Due to thenonlinearity and unstable zero dynamics of the boost converter, a novel nonlinear control strategybased on input–output feedback linearization is proposed in [19]. In [20], a wide series of controltechniques are presented for well-known power electronics converters, including the conventionalDC-DC boost converter. In response to this problem, two main solutions can be obtained from theperspective of control and system: (1) changing the topology of DC/DC booster converter to eliminatethe unstable zero dynamic; and (2) indirect control of the output voltage. By selecting a new outputvariable, the unstable zero dynamics of the converter are eliminated, and the relationship betweenthe new output variable and the output voltage is used for control. The former directly controls theoutput voltage and easily satisfies the control effect, but increasing the power electronic switchingdevice leads to an increase in cost. The latter does not change the topology of the converter, and theoutput voltage is controlled by indirect control, but the output voltage is susceptible to input voltageand resistance changes. The classical PI control relies on the error between the control target and theactual output to determine the control strategy to eliminate this error [21], so the design is simple andthe applicability is good. However, this method designs the output feedback control according to thetarget error, so that the dynamic response characteristic is slow, and the system is adjusted only whenthe output has changed.

Based on the in-depth study of PID, active disturbance rejection control (ADRC) attributes allthe uncertain factors acting on the controlled object to “unknown disturbances”, which are estimatedand compensated by the extended state observer. This method is based on process error to reducethe error [22]. The active disturbance rejection control consists of three parts: tracking differentiator(TD), extended state observer (ESO), and nonlinear state error feedback (NLSEF) control law. To makethe design of the controller simpler and easier to promote the application, Gao et.al. simplified theself-disturbance control to linear active disturbance rejection control (LADRC) [23].

The parameters’ setting of controller is simplified by using proportional-derivative (PD) control andestablishing the proportional coefficient and differential time constant to the controller bandwidth [24].Linearization of the ESO, conversion to a linear extended state observer (LESO), and the establishmentof a link between the observer bandwidth and parameters, simplifies the design of the LESO. Since theactive disturbance rejection controller does not depend on the system model and the algorithm issimple, it is applied in many fields. In [25], ADRC is applied to solve industrial problems. Industrialautomation equipment was tested. In the worst case, performance improvement from just under 20% to

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290% in all mechanical configuration was found. Coupled with the simplicity in the control algorithmitself, the ease of adjustment and operation, this improvement makes ADRC a viable alternative toexisting industrial controller in manufacturing industry. In [26], from the perspective of uncertainsystem control, the main methods in modern control theory are sorted out, and the theoretical analysisof the main part of ADRC and the flexible application of ADRC method in specific problems arediscussed. In [27], a fixed-time ADRC control method is proposed to solve the attitude control problemof the four-rotor unmanned aerial vehicle in the presence of dynamic wind, mass eccentricity andactuator fault. In [28], the LADRC method is applied in inverter control, which expands the range ofvirtual impedance and improves the immunity of the system. In [29], to improve the anti-interferenceability of the speed control system, ADRC controller using phase-locking loop observer is proposed.In [30], ADRC is applied to the gyro control of micro-electro-mechanical systems, which not only drivesthe drive shaft to vibrate along a predetermined trajectory, but also compensates for manufacturingdefects in a robust manner, making the performance of the gyro insensitive to parameter variations andnoise. In [31], LADRC is proposed to be applied to a double-switch buck-boost converter. To ensuredynamic control performance and smooth switching in different operating modes, the model deviationin different working modes is regarded as a generalized disturbance, and a unified current controldevice can be derived for current controller design.

According to the above analysis, to solve the problem of output voltage instability caused byinput voltage and load fluctuation and non-minimum phase of the DC/DC boost converter, this paperproposes a cascade control based on LADRC. Aiming at the instability of output voltage caused by thefluctuation of input voltage and load, and the change of system parameters, the LADRC is adoptedto observe the fluctuation of voltage and load through the observer and eliminate it through thedisturbance suppression channel. For the non-minimum phase problem of the system, a new outputvariable is selected to eliminate the unstable zero dynamics. The stability of the system is determinedby frequency domain analysis. This control method can effectively eliminate the change of the outputvoltage caused by the renewable energy power generation unit and the load disturbance, and does notdepend on the model of the system, the algorithm is simple, and is beneficial to engineering applications.

Through MATLAB/Simulink simulation analysis and experimental comparison with thecontrol effect of PI controller, the results verify the effectiveness and superiority of the proposedcontrol algorithm.

2. Modeling of DC/DC Boost Converter

Figure 1 shows a circuit diagram of a DC/DC boost converter. In the figure, Ui is the input voltage,IL is the inductive current, Uo is the output voltage, L is the inductance, C is the capacitance, and R isthe resistive load.

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in the control algorithm itself, the ease of adjustment and operation, this improvement makes ADRC

a viable alternative to existing industrial controller in manufacturing industry. In [26], from the

perspective of uncertain system control, the main methods in modern control theory are sorted out,

and the theoretical analysis of the main part of ADRC and the flexible application of ADRC method

in specific problems are discussed. In [27], a fixed-time ADRC control method is proposed to solve

the attitude control problem of the four-rotor unmanned aerial vehicle in the presence of dynamic

wind, mass eccentricity and actuator fault. In [28], the LADRC method is applied in inverter control,

which expands the range of virtual impedance and improves the immunity of the system. In [29], to

improve the anti-interference ability of the speed control system, ADRC controller using phase-

locking loop observer is proposed. In [30], ADRC is applied to the gyro control of micro-electro-

mechanical systems, which not only drives the drive shaft to vibrate along a predetermined

trajectory, but also compensates for manufacturing defects in a robust manner, making the

performance of the gyro insensitive to parameter variations and noise. In [31], LADRC is proposed

to be applied to a double-switch buck-boost converter. To ensure dynamic control performance and

smooth switching in different operating modes, the model deviation in different working modes is

regarded as a generalized disturbance, and a unified current control device can be derived for current

controller design.

According to the above analysis, to solve the problem of output voltage instability caused by

input voltage and load fluctuation and non-minimum phase of the DC/DC boost converter, this paper

proposes a cascade control based on LADRC. Aiming at the instability of output voltage caused by

the fluctuation of input voltage and load, and the change of system parameters, the LADRC is

adopted to observe the fluctuation of voltage and load through the observer and eliminate it through

the disturbance suppression channel. For the non-minimum phase problem of the system, a new

output variable is selected to eliminate the unstable zero dynamics. The stability of the system is

determined by frequency domain analysis. This control method can effectively eliminate the change

of the output voltage caused by the renewable energy power generation unit and the load

disturbance, and does not depend on the model of the system, the algorithm is simple, and is

beneficial to engineering applications.

Through MATLAB/Simulink simulation analysis and experimental comparison with the control

effect of PI controller, the results verify the effectiveness and superiority of the proposed control

algorithm.

2. Modeling of DC/DC Boost Converter

Figure 1 shows a circuit diagram of a DC/DC boost converter. In the figure, Ui is the input

voltage, IL is the inductive current, Uo is the output voltage, L is the inductance, C is the capacitance,

and R is the resistive load.

Figure 1. DC/DC boost converter circuit. Figure 1. DC/DC boost converter circuit.

The DC/DC boost converter uses PWM technology, which obtains the required output voltage bychanging the duty cycle of the pulse. This paper introduces the pulse model integration method to

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establish the average model of the DC/DC boost converter. The waveform of switch function µ(t) isshown in Figure 2.



The DC/DC boost converter uses PWM technology, which obtains the required output voltage

by changing the duty cycle of the pulse. This paper introduces the pulse model integration method

to establish the average model of the DC/DC boost converter. The waveform of switch function μ(t)

is shown in Figure 2.

The pulse function μ(t) can be expressed as:

1, ( )( )

0, ( ) ( 1)nT t n d T

tn d T t n T

(1)

Switching frequency fsw is 10 kHz in this paper, where T =1/fsw is the switching period, d is the

duty cycle, 0 < d < 1 and n = 0, 1, 2…

Figure 2. Impulse waveform.

As can be seen from the above pulse function, the DC/DC boost converter has two operating

states:

(1) Switch conduction mode, μ(t)=1: When the switch is turned on, the input voltage Ui charges

the inductor L, and the charging current is kept substantially constant. At the same time, the voltage

of capacitor C supplies power to load R.

(2) Switch cutoff mode, μ(t) = 0: When the switch is turned off, the input voltage Ui and the

inductor L together charge the capacitor C and supply power to the load resistor.

Switch conduction model equation:

1

Li

o

o

dIL U

dt

dUC U

dt R

(2)

Switch cutoff model equation:

o

o 1

Li

L o

dIL U U

dt

dUC I U

dt R

(3)

The average mathematical model equations of the DC/DC boost converter can be obtained by

combining Equations (2) and (3):

(1 )

(1 ) 1

o iL

o Lo

d U UdI

dt L L

dU d IU

dt C RC

(4)

3. Control Method Design

Figure 2. Impulse waveform.

The pulse function µ(t) can be expressed as:

µ(t) =

1, nT < t ≤ (n + d)T0, (n + d)T < t ≤ (n + 1)T

(1)

Switching frequency fsw is 10 kHz in this paper, where T =1/fsw is the switching period, d is theduty cycle, 0 < d < 1 and n = 0, 1, 2 . . .

As can be seen from the above pulse function, the DC/DC boost converter has two operating states:(1) Switch conduction mode, µ(t) = 1: When the switch is turned on, the input voltage Ui charges

the inductor L, and the charging current is kept substantially constant. At the same time, the voltage ofcapacitor C supplies power to load R.

(2) Switch cutoff mode, µ(t) = 0: When the switch is turned off, the input voltage Ui and theinductor L together charge the capacitor C and supply power to the load resistor.

Switch conduction model equation: L dILdt = Ui

C dUodt = − 1

R Uo(2)

Switch cutoff model equation: L dILdt = Ui −Uo

C dUodt = IL −

1R Uo

(3)

The average mathematical model equations of the DC/DC boost converter can be obtained bycombining Equations (2) and (3): dIL

dt = −(1−d)Uo

L + UiL

dUodt =

(1−d)ILC −

1RC Uo

(4)

3. Control Method Design

3.1. Cascade Control

According to the analysis of Equation (4), when the output voltage is directly controlled, the systemis a non-minimum phase system. This will lead to narrow bandwidth and slow dynamic responseto the system, which increases the difficulty of control. The output redefinition method is one of themethods to solve the non-minimum phase problem. By selecting a new output variable, a new outputfunction is established, so that the zero dynamic subsystem is stable.

For the control of the DC/DC boost converter, a new output variable can be selected to eliminatethe unstable zero dynamics and achieve the effect of change the non-minimum phase characteristicsof the system. According to the model in Equation (4), if the inductor current is the system output,the new subsystem is the minimum phase system. Therefore, the output voltage control problem of

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the system can be converted into an inductor current control problem. Further consideration is givento how to achieve the desired output voltage value by controlling the inductor current. There is arelationship between the output voltage and the inductor current, can be written as:

IL(∞) = U2o (∞)/RUi (5)

where IL(∞) and Uo(∞) are the steady state values of IL and Uo, respectively.It can be seen from Equation (5) that the control of the output voltage can be achieved by controlling

the inductor current. If the desired voltage is known, the reference value of the inductor current can beobtained according to Equation (6).

ILre f = U2ore f /RUi (6)

ILref and Uoref are reference values of the inductor current and the output voltage, respectively.According to Equation (6), ILref is calculated from Uoref, and there are two problems: (1) The

relationship between the inductor current and the output voltage is based on the stability of the system.Therefore, it is difficult to ensure the dynamic performance of the output voltage. (2) Since the inputvoltage Ui and the load resistance R of the DC/DC boost converter are uncertain, the Uoref cannot beobtained directly from the ILref. To solve the above problem, a cascade control system is formed bydesigning an output voltage controller in the front stage of the current controller. The output voltagecontroller produces an ILref by the error between the desired value of the output voltage and outputvoltage observation.

3.2. Linear Active Disturbance Rejection Control

LADRC consists of a linear extended state observer (LESO), and a linear state error feedback(LSEF) control law. LADRC is composed of disturbance suppression loop and feedback control loop.The structure is shown in Figure 3. The uncertain factors of the system are estimated in advance bythe LESO and eliminated by the disturbance suppression loop, which is combined with the feedbackcontrol loop to improve the transient and stability performance of the system.



3.1. Cascade Control

According to the analysis of Equation (4), when the output voltage is directly controlled, the

system is a non-minimum phase system. This will lead to narrow bandwidth and slow dynamic

response to the system, which increases the difficulty of control. The output redefinition method is

one of the methods to solve the non-minimum phase problem. By selecting a new output variable, a

new output function is established, so that the zero dynamic subsystem is stable.

For the control of the DC/DC boost converter, a new output variable can be selected to eliminate

the unstable zero dynamics and achieve the effect of change the non-minimum phase characteristics

of the system. According to the model in Equation (4), if the inductor current is the system output,

the new subsystem is the minimum phase system. Therefore, the output voltage control problem of

the system can be converted into an inductor current control problem. Further consideration is given

to how to achieve the desired output voltage value by controlling the inductor current. There is a

relationship between the output voltage and the inductor current, can be written as:

2( )= ( ) /L o iI U RU (5)

where IL(∞) and Uo(∞) are the steady state values of IL and Uo, respectively.

It can be seen from Equation (5) that the control of the output voltage can be achieved by

controlling the inductor current. If the desired voltage is known, the reference value of the inductor

current can be obtained according to Equation (6).

2= /Lref oref iI U RU (6)

ILref and Uoref are reference values of the inductor current and the output voltage, respectively.

According to Equation (6), ILref is calculated from Uoref, and there are two problems: (1) The

relationship between the inductor current and the output voltage is based on the stability of the

system. Therefore, it is difficult to ensure the dynamic performance of the output voltage. (2) Since

the input voltage Ui and the load resistance R of the DC/DC boost converter are uncertain, the Uoref

cannot be obtained directly from the ILref. To solve the above problem, a cascade control system is

formed by designing an output voltage controller in the front stage of the current controller. The

output voltage controller produces an ILref by the error between the desired value of the output voltage

and output voltage observation.

3.2. Linear Active Disturbance Rejection Control

LADRC consists of a linear extended state observer (LESO), and a linear state error feedback

(LSEF) control law. LADRC is composed of disturbance suppression loop and feedback control loop.

The structure is shown in Figure 3. The uncertain factors of the system are estimated in advance by

the LESO and eliminated by the disturbance suppression loop, which is combined with the feedback

control loop to improve the transient and stability performance of the system.

Figure 3. Structure of first-order LADRC. Figure 3. Structure of first-order LADRC.

r is the input reference; u0 is the controlled quantity; ω is external disturbance of the system; Gp

is the controlled object; y is the system output; y is the estimated value of system output; f is theestimated value of the total disturbance; kp is the controller parameter; and b0 is the system gain.

The first-order single-input single-output system is analyzed as follows:

.y = b0u0 + f (y, u0) (7)

where f is the total disturbance of the system.Let x1 = y, x2 = f, then .

x = Ax + Bu0 + E.f

y1 = Cx(8)

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where A =

[0 10 0

], B =

[b0

0

], E =

[01

], C =

[1 0

], x =

[x1

x2

].

The corresponding continuous LESO is .x = Ax + Bu0 + L0(y− y)y = Cx

(9)

where Lo =

[β1

β2

](10)

For first-order system, proportional control is adopted:

u = kp(r− x1) (11)

The total disturbance action is observed by the LESO as the expansion state of the system.To achieve automatic compensation of the total disturbance, the final control law is designed as

u0 =u− x2

b0=

kp(r− x1) − x2

b0= Kp(R− x) (12)

where R =

[r0

]is the input reference and Kp =

[kp 1

]/b0 is gain of controller.

Substitute Equation (12) into Equation (6) to get

.y = −x2 + u + f ≈ u (13)

In summary, LADRC can be described as the form of state space as follows: .x = (A− LoC)x + Bu0 + Loyu0 = Kp(R− x)

(14)

The gain Lo of the LESO and Kp of the LSEF are designed. By introducing the concept of bandwidth,the setting of Lo and Kp is converted into the observer’s bandwidth ωo and the controller’s bandwidthωc, which simplifies the parameter tuning process.

The characteristic equation of LESO can be obtained from Equation (14)

∣∣∣sI − (A− L0C)∣∣∣ = [

s + β1 −1β2 s

]= s(s + β1) + β2 = s2 + β1s + β2 (15)

All poles of the observer are assigned to −ωo, then

s2 + β1s + β2 = (s +ωo)2 (16)

where ωo is the bandwidth of observer.The closed-loop characteristic equation of the feedback control system is as follows:

∣∣∣sI − (A− BKp)∣∣∣ = [

s + kp 00 s

]= s(s + kp) (17)

All poles of the controller are assigned to −ωc, where ωc is the controller bandwidth.Combining Equations (15) and (16), the equivalent equation are obtained as

β1 = 2ωo, β2 = ω2o , kp = ωc (18)

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4. Controller Design

For the average model of the DC/DC boost converter established by Equation (3), this paperdesigns inner-loop current controller and outer-loop voltage controller based on LADRC. It has thecharacteristics of two-channel control with interference suppression and control feedback. LESO canbe used to estimate the fluctuations of input voltage and loads in advance, which can be eliminated bydisturbance suppression loop to improve the rapidity of the system. To avoid the problem of controlledquantity delay caused by error feedback from PI control, capacitive voltage signal and inductive currentsignal are, respectively, obtained by the LESO, which are used as the feedback quantity of voltage outerloop and current inner loop. Moreover, LESO has a filter function, which can suppress the noise in thesystem and reduce the low-pass filtering in the traditional cascade control.

4.1. Design of Current Controller

Design the current controller according to the first formula in Equation (4); the current controlleris designed as follows:

dIL

dt= −

(1− d)UL

+Ui

L(19)

Let IL = y1, d = u1, then

.y1 =

Uo

Lu1 +

UiL−

Uo

L= b1u1 + f1(y1, u1) (20)

where b1 = UoL is the object gain, y1 is the system output, u1 is the system input, and f1 = Ui

L −UoL = Ui−Uo

Lis the total disturbance of the system.

Let x1 = y1, x2 = f 1, then .x = A1x + B1u1 + E1

.f 1

y1 = C1x(21)

where A1 =

[0 10 0

], B1 =

[b1

0

], E1 =

[01

], C1 =

[1 0

], and x =

[x1

x2

].

Second-order LESO is .x = A1x + B1u1 + Lo1(y1 − y1)

y1 = C1x(22)

where Lo1 =

[β11

β12

].

According to Equation (19), the relative of the current is first-order, and proportional controlis used:

u = k1(r1 − x1) (23)

According to the expansion state x2 = f1 given by the LESO. To achieve automatic compensationof the total disturbance, the final control law is designed as

u1 =u− x2

b1=

k1(r− x1) − x2

b1= K1(R1 − x) (24)

where R1 =

[r1

0

]is the input reference and K1 =

[k1 1

]/b1 is gain of controller.

Substituting Equation (24) into Equation (20),

.y1 = −x2 + u1 + f1 ≈ u1 (25)

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The tuning of Lo1 and K1 is converted to the tuning of the observer bandwidth ωo1 and thecontroller bandwidth ωc1. All poles of the observer are configured to −ωo1, and all poles of thecontroller are configured to −ωc1.

β11 = 2ωo1, β12 = ω2o1, k1 = ωc1 (26)

4.2. Design of Voltage Controller

Design a voltage controller according to the second formula in Equation (4); the voltage controlleris designed as follows:

dUo

dt=

(1− d)IL

C−

1RC

Uo (27)

Let Uo = y2, IL = u2, then

.y2 =

1− dC

u2 −y1

RC= b2u2 + f2(y2, u2) (28)

where b2 = 1−dC is the object gain, y2 is the system output, u2 is the system input, and f2 = −

y2RC is the

total disturbance of the system.Let x1 = y2, x2 = f 2, then .

x = A2x + B2u2 + E2.f 2

y2 = C2x(29)

where A2 =

[0 10 0

], B2 =

[b2

0

], E2 =

[01

], C2 =

[1 0

], x =

[x1

x2

].

Second-order LESO is .x = A2x + B2u2 + Lo2(y2 − y2)

y2 = C2x(30)

where Lo2 =

[β21

β22

].

According to Equation (27), the dynamic characteristics of the voltage is first-order, so proportionalcontrol is used:

u = k2(r2 − x1) (31)

To achieve automatic compensation of the total disturbance, the final control law is designed as

u2 =u− x2

b2=

k2(r− x2) − x2

b2= K2(R2 − x) (32)

whereR2 =

[r2

0

]is the input reference and K2 =

[k2 1

]/b2 is the controller gain.


.y2 = −x2 + u2 + f2 ≈ u2 (33)

The Lo2 and K2 tuning is converted to the tuning of the observer bandwidth ωo2 and the controllerbandwidth ωc2. All poles of the observer are configured to −ωo2, and all poles of the controller areconfigured to −ωc2.

β21 = 2ωo2, β22 = ω2o2, k2 = ωc2 (34)

Based on the above controller, the structure of the proposed controller is shown in Figure 4.

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2 2 2 22 2

2 2

ˆ ˆ ˆ( )ˆ ˆ

ox A x B u L y y

y C x

(30)

where

212

22oL

.

According to Equation (27), the dynamic characteristics of the voltage is first-order, so

proportional control is used:

2 2 1( )u k r x (31)

To achieve automatic compensation of the total disturbance, the final control law is designed as

2 222 2

2

22

2

ˆ ˆ( )ˆ ˆ ˆ( )k r x xu K R x

b b

u x (32)

where 22

ˆ0r

R

is the input reference and 2 2 21 /K k b is the controller gain.


2 2 2 2 2ˆy x u f u (33)

The Lo2 and K2 tuning is converted to the tuning of the observer bandwidth ωo2 and the controller

bandwidth ωc2. All poles of the observer are configured to -ωo2, and all poles of the controller are

configured to -ωc2.

221 2 22 2 2 22 , ,o o ck (34)

Based on the above controller, the structure of the proposed controller is shown in Figure 4.

Figure 4. The structure of proposed controller.

4.3. System Stability Analysis

The transfer function of the DC/DC boost converter can be obtained from Equation (4)

Gp(s) =∆Uo(s)∆d(s)

=−sLRIL + DRUo

s2LCR + sL + D2R(35)

where D = 1 − d and R is the load. As is shown in Equation (35), there is a zero in the right half plane.The bode diagram of Gp(s) is shown in Figure 5.






2 2( )

( )( )o L o

p

U s sLRI DRUG s

d s s LCR sL D R

(35)

where D=1-d and R is the load. As is shown in Equation (35), there is a zero in the right half plane.

The bode diagram of Gp(s) is shown in Figure 5.

Figure 5. Bode diagram of Gp(s).

As can be seen in Figure 5, the phase margin of Gp(s) is -16.3 deg and the amplitude margin is -

33.6 dB. According to the stability criterion of closed-loop control system, the system is unstable.

Figure 6 shows the control structure for one converter with the implementation of the LADRC

method.

Figure 6. Control structure of boost DC/DC converter.

In Figure 6, Uref is the reference voltage of the voltage loop, Iref is the reference current of

the current loop, and Uo is the output voltage. The LADRC voltage outer loop is composed of H1(s)

and Gc1(s), and the LADRC current inner loop is composed of H2(s) and Gc2(s). Gp1(s) is the transfer

function of the controlled quantity to the output voltage, and Gp2(s) is the transfer function of the

controlled quantity to the inductor current.

According to Equations (22), (23), (31) and (32), the transfer function of the first-order LADRC

can be obtained as


Electronics 2019, 8, 1249 10 of 19

As can be seen in Figure 5, the phase margin of Gp(s) is −16.3 deg and the amplitude margin is−33.6 dB. According to the stability criterion of closed-loop control system, the system is unstable.

Figure 6 shows the control structure for one converter with the implementation of theLADRC method.






2 2( )

( )( )o L o

p

U s sLRI DRUG s

d s s LCR sL D R

(35)

where D=1-d and R is the load. As is shown in Equation (35), there is a zero in the right half plane.

The bode diagram of Gp(s) is shown in Figure 5.


As can be seen in Figure 5, the phase margin of Gp(s) is -16.3 deg and the amplitude margin is -

33.6 dB. According to the stability criterion of closed-loop control system, the system is unstable.

Figure 6 shows the control structure for one converter with the implementation of the LADRC

method.


In Figure 6, Uref is the reference voltage of the voltage loop, Iref is the reference current of

the current loop, and Uo is the output voltage. The LADRC voltage outer loop is composed of H1(s)

and Gc1(s), and the LADRC current inner loop is composed of H2(s) and Gc2(s). Gp1(s) is the transfer

function of the controlled quantity to the output voltage, and Gp2(s) is the transfer function of the

controlled quantity to the inductor current.

According to Equations (22), (23), (31) and (32), the transfer function of the first-order LADRC

can be obtained as


In Figure 6, ∆Uref is the reference voltage of the voltage loop, ∆Iref is the reference current of thecurrent loop, and ∆Uo is the output voltage. The LADRC voltage outer loop is composed of H1(s)and Gc1(s), and the LADRC current inner loop is composed of H2(s) and Gc2(s). Gp1(s) is the transferfunction of the controlled quantity to the output voltage, and Gp2(s) is the transfer function of thecontrolled quantity to the inductor current.

According to Equations (22), (23), (31) and (32), the transfer function of the first-order LADRC canbe obtained as

H1(s) = k1s2 + β11s + β12

k1s2 + (k1β11 + β12)s + k1β12(36)

Gc1(s) =1b1

k1s2 + (k1β11 + β12)s + k1β12

s2 + β11s(37)

H2(s) = k2s2 + β21s + β22

k2s2 + (k2β21 + β22)s + k2β22(38)

Gc2(s) =1b2

k2s2 + (k2β21 + β22)s + k2β22

s2 + β21s(39)

The Gp1(s) and Gp2(s) transfer functions are obtained by the DC/DC boost circuit small signal model.The transfer function of the controlled quantity ∆d to the output voltage ∆Uo is

Gp1(s) =∆Uo(s)∆d(s)

=−sLRIL + DRUo


The transfer function of the controlled quantity ∆d(t) to the output current ∆IL is

Gp2(s) =∆IL(s)∆d(s)

=sUoRC + DILR + Uo


From Figure 6, the transfer function of the inner loop current loop can be obtained as follows:

G1(s) =Gc2(s)H2(s)

1 + Gc2(s)Gp2(s)(42)

The system open loop transfer function is

G0(s) =H1(s)Gc1(s)H2(s)Gc2(s)Gp2(s)Gp1(s)

1 + Gc2(s)Gp2(s)(43)

Electronics 2019, 8, 1249 11 of 19

To verify the effectiveness of the proposed method, bode diagram is used to analysis the stabilityof the system. The current loop bode diagram is shown in Figure 7a, and the voltage loop bode diagramis shown in Figure 7b.



211 12

1 1 21 1 11 12 1 12

( )( )s s

H s kk s k s k

(36)

21 1 11 12 1 12

1 21 11

( )1( )c

k s k s kG s

b s s

(37)

221 22

2 2 22 2 21 22 2 22

( )( )s s

H s kk s k s k

(38)

22 2 21 22 2 22

2 22 21

( )1( )c

k s k s kG s

b s s

(39)

The Gp1(s) and Gp2(s) transfer functions are obtained by the DC/DC boost circuit small signal

model.

The transfer function of the controlled quantity d to the output voltage Uo is

1 2 2( )

( )( )o L o

p

U s sLRI DRUG s

d s s LCR sL D R

(40)

The transfer function of the controlled quantity d(t) to the output current IL is

2 2 2( ()(

))

LL o o

p

sU RC DI R UIG s

d s

s

s LCR sL D R

(41)

From Figure 6, the transfer function of the inner loop current loop can be obtained as follows:

2 21

2 2

( ) ( )( )1 ( ) ( )

c

c p

G s H sG s

G s G s

(42)

The system open loop transfer function is

1 1 2 2 2 10

2 2

( ) ( ) ( ) ( ) ( ) ( )( )

1 ( ) ( )c c p p

c p

H s G s H s G s G s G sG s

G s G s

(43)

To verify the effectiveness of the proposed method, bode diagram is used to analysis the stability

of the system. The current loop bode diagram is shown in Figure 7a, and the voltage loop bode

diagram is shown in Figure 7b.

(a)



(b)

Figure 7. (a) Current loop open loop transfer function; and (b) open-loop bode diagram of systems

with Controller.

According to Figure 7a, the phase margin of current loop is 26 deg and the amplitude margin is

infinite. As shown in Figure 7b, the phase margin of Gp(s) is 53.7 deg and the amplitude margin is 38

dB. According to the stability criterion of the closed-loop control system, the closed-loop control

system satisfies the stability characteristic, and the stability margin of system is excellent.

5. Simulation Results

To verify the performances of the proposed strategy, simulation tests were designed in

Matlab/Simulink. Under the premise that PI cascade control and LADRC cascade control have no

overshoot, the classical dual-loop PI controller was compared with the proposed controller to

demonstrate the superiority of the proposed method. The circuit parameters and controller

parameters in the simulation are shown in Table 1. To verify the stability of the system under different

cases, three cases were designed, as shown in Table 2. For clarity, we will describe the variables and

acronyms in Table A1 of the Appendix.

Table 1. Simulation parameters.

Parameters Description Value

Uoref Reference value of output voltage 24 V

L inductance value 1 mH

C capacitance value 920 μF

fsw switching frequency 10 KHz

ωc1, ωo1, b1 LADRC parameters of current control loop 165, 270, 543.5

ωc2, ωo2, b2 LADRC parameters of voltage control loop 1600, 8800, 24,000

kp1, ki1 PI parameters of voltage control loop 0.3, 7

kp2, ki2 PI parameters of current control loop 0.25, 30

Table 2. Different cases.

Case Input Voltage Load

1 12 V→10 V 50 Ω

2 12 V→8 V 50 Ω

3 12 V 50 Ω→25 Ω

Figure 7. (a) Current loop open loop transfer function; and (b) open-loop bode diagram of systemswith Controller.

According to Figure 7a, the phase margin of current loop is 26 deg and the amplitude margin isinfinite. As shown in Figure 7b, the phase margin of Gp(s) is 53.7 deg and the amplitude margin is38 dB. According to the stability criterion of the closed-loop control system, the closed-loop controlsystem satisfies the stability characteristic, and the stability margin of system is excellent.

5. Simulation Results

To verify the performances of the proposed strategy, simulation tests were designed inMatlab/Simulink. Under the premise that PI cascade control and LADRC cascade control haveno overshoot, the classical dual-loop PI controller was compared with the proposed controller todemonstrate the superiority of the proposed method. The circuit parameters and controller parametersin the simulation are shown in Table 1. To verify the stability of the system under different cases,three cases were designed, as shown in Table 2. For clarity, we will describe the variables and acronymsin the Appendix A.

Electronics 2019, 8, 1249 12 of 19

Table 1. Simulation parameters.

Parameters Description Value

Uoref Reference value of output voltage 24 VL inductance value 1 mHC capacitance value 920 µF

fsw switching frequency 10 KHzωc1, ωo1,b1 LADRC parameters of current control loop 165, 270, 543.5ωc2, ωo2,b2 LADRC parameters of voltage control loop 1600, 8800, 24,000

kp1, ki1 PI parameters of voltage control loop 0.3, 7kp2, ki2 PI parameters of current control loop 0.25, 30

Table 2. Different cases.

Case Input Voltage Load

1 12 V→10 V 50 Ω2 12 V→8 V 50 Ω3 12 V 50 Ω→25 Ω

Case 1: The converter input voltage was changed to examine the stabilization performance. At thebeginning, the output voltage was regulated at 24 V with a 50 Ω resistance load. At 0.6 s, the inputvoltage of the converter was reduced from 12 V to 10 V. The voltage response is shown in Figure 8;the converter is controlled by LADRC cascade, the output voltage drops from 24 V at 23.6 V and reachesthe desired value after 0.05 s. In contrast, the converter is controlled by PI cascade, the output voltagedrops from 24 V to 23.3 V, and it takes 0.25 s to reach the desired value. The converter controlled by theproposed method can respond immediately and achieve the desired effect. At the same time, a longerresponse time and a large voltage deviation can be observed under the PI controller. The currentresponse is shown in Figure 9; by further observing the change of the inductor current, both the LADRCcontrol and the PI control can quickly reach the desired value, but the current shock of the PI control islarger than proposed method.Electronics 2019, 8, x FOR PEER REVIEW 13 of 19


Figure 8. Output voltage responses in Case 1.

Figure 9. Inductance current responses in Case 1.




Electronics 2019, 8, 1249 13 of 19








Case 2: To further validate the proposed control method, the changes of the input voltage weredoubled. The basic setting was identical with the Case 1. Firstly, the output voltage was regulated at24 V, and a 50 Ω resistance was connected to the DC bus. At 0.6 s, the input voltage of the converterwas reduced from 12 V to 8 V. The output voltage response is shown in Figure 10; the converter iscontrolled by LADRC cascade, the output voltage drops from 24 V at 23.2 V and reaches the desiredvalue after 0.07 s. In contrast, the converter is controlled by PI cascade, the output voltage drops from24 V to 22.6 V, and it takes 0.28 s to reach the desired value. The current response is shown in Figure 11;the verification results are consistent with the results of Case 1.













Figure 11. Inductance current responses in Case 2. Figure 11. Inductance current responses in Case 2.

Case 3: For load changes, the proposed control method was verified by Case 3. Initially, the outputvoltage was stable at 24 V and a 50 Ω resistance is connected. Then, the resistance decreased to25 Ω in 0.6 s. The output voltage response is shown in Figure 12. With the same load change,the voltage deviation of PI control is 2.6 V and the recovery process is 0.35 s. The recovery process ofthe proposed control algorithm is 0.1 s, which is shorter than the recovery process of PI control by0.25 s. Simulation results show that the proposed control algorithm has good dynamic performance.In addition, the output current response is shown in Figure 13; by observing changes of the inductor

Electronics 2019, 8, 1249 14 of 19

current, LADRC control can quickly reach the desired value, while PI control takes a long time to reachthe desired value.Electronics 2019, 8, x FOR PEER REVIEW 14 of 19




Case 1: The converter input voltage was changed to examine the stabilization performance. At

the beginning, the output voltage was regulated at 24 V with a 50 Ω resistance load. At 0.6 s, the input

voltage of the converter was reduced from 12 V to 10 V. The voltage response is shown in Figure 8;

the converter is controlled by LADRC cascade, the output voltage drops from 24 V at 23.6 V and

reaches the desired value after 0.05 s. In contrast, the converter is controlled by PI cascade, the output

voltage drops from 24 V to 23.3 V, and it takes 0.25 s to reach the desired value. The converter

controlled by the proposed method can respond immediately and achieve the desired effect. At the

same time, a longer response time and a large voltage deviation can be observed under the PI

controller. The current response is shown in Figure 9; by further observing the change of the inductor

current, both the LADRC control and the PI control can quickly reach the desired value, but the

current shock of the PI control is larger than proposed method.

Case 2: To further validate the proposed control method, the changes of the input voltage were

doubled. The basic setting was identical with the Case 1. Firstly, the output voltage was regulated at

24 V, and a 50 Ω resistance was connected to the DC bus. At 0.6 s, the input voltage of the converter

was reduced from 12 V to 8 V. The output voltage response is shown in Figure 10; the converter is

controlled by LADRC cascade, the output voltage drops from 24 V at 23.2 V and reaches the desired

value after 0.07 s. In contrast, the converter is controlled by PI cascade, the output voltage drops from

24 V to 22.6 V, and it takes 0.28 s to reach the desired value. The current response is shown in Figure

11; the verification results are consistent with the results of Case 1.

Case 3: For load changes, the proposed control method was verified by Case 3. Initially, the

output voltage was stable at 24 V and a 50 Ω resistance is connected. Then, the resistance decreased

to 25 Ω in 0.6 s. The output voltage response is shown in Figure 12. With the same load change, the

voltage deviation of PI control is 2.6 V and the recovery process is 0.35 s. The recovery process of the

proposed control algorithm is 0.1 s, which is shorter than the recovery process of PI control by 0.25 s.

Simulation results show that the proposed control algorithm has good dynamic performance. In

addition, the output current response is shown in Figure 13; by observing changes of the inductor






Case 1: The converter input voltage was changed to examine the stabilization performance. At

the beginning, the output voltage was regulated at 24 V with a 50 Ω resistance load. At 0.6 s, the input

voltage of the converter was reduced from 12 V to 10 V. The voltage response is shown in Figure 8;

the converter is controlled by LADRC cascade, the output voltage drops from 24 V at 23.6 V and

reaches the desired value after 0.05 s. In contrast, the converter is controlled by PI cascade, the output

voltage drops from 24 V to 23.3 V, and it takes 0.25 s to reach the desired value. The converter

controlled by the proposed method can respond immediately and achieve the desired effect. At the

same time, a longer response time and a large voltage deviation can be observed under the PI

controller. The current response is shown in Figure 9; by further observing the change of the inductor

current, both the LADRC control and the PI control can quickly reach the desired value, but the

current shock of the PI control is larger than proposed method.

Case 2: To further validate the proposed control method, the changes of the input voltage were

doubled. The basic setting was identical with the Case 1. Firstly, the output voltage was regulated at

24 V, and a 50 Ω resistance was connected to the DC bus. At 0.6 s, the input voltage of the converter

was reduced from 12 V to 8 V. The output voltage response is shown in Figure 10; the converter is

controlled by LADRC cascade, the output voltage drops from 24 V at 23.2 V and reaches the desired

value after 0.07 s. In contrast, the converter is controlled by PI cascade, the output voltage drops from

24 V to 22.6 V, and it takes 0.28 s to reach the desired value. The current response is shown in Figure

11; the verification results are consistent with the results of Case 1.

Case 3: For load changes, the proposed control method was verified by Case 3. Initially, the

output voltage was stable at 24 V and a 50 Ω resistance is connected. Then, the resistance decreased

to 25 Ω in 0.6 s. The output voltage response is shown in Figure 12. With the same load change, the

voltage deviation of PI control is 2.6 V and the recovery process is 0.35 s. The recovery process of the

proposed control algorithm is 0.1 s, which is shorter than the recovery process of PI control by 0.25 s.

Simulation results show that the proposed control algorithm has good dynamic performance. In

addition, the output current response is shown in Figure 13; by observing changes of the inductor


6. Hardware Experiment

To verify the proposed control method, a DC/DC boost converter experimental platform wasbuilt, as shown in Figure 14. The platform consists of a DC power source, a DC/DC boost converter,an electronic load, and a scope. Among them, the electronic load was set to the constant resistancemode. The circuit parameters are the same as those in the simulation, as shown in Table 1. To obtainthe optimal actual control performance, ωc1, ωo1, ωc2, ωo2, kp1, ki1, kp2 and ki2 were redesigned as 33,160, 300, 900, 0.03, 1, 0.05 and 2.



current, LADRC control can quickly reach the desired value, while PI control takes a long time to

reach the desired value.


To verify the proposed control method, a DC/DC boost converter experimental platform was

built, as shown in Figure 14. The platform consists of a DC power source, a DC/DC boost converter,

an electronic load, and a scope. Among them, the electronic load was set to the constant resistance

mode. The circuit parameters are the same as those in the simulation, as shown in Table 1. To obtain

the optimal actual control performance, ωc1, ωo1, ωc2, ωo2, kp1, ki1, kp2 and ki2 were redesigned as 33,

160, 300, 900, 0.03, 1, 0.05 and 2.

Figure 14. DC/DC boost converter experimental platform.

(a) (b)

Figure 15. (a) LADRC controller: experimental comparison responses with Case 1; and (b) PI

controller: experimental comparison responses with Case 1.


Electronics 2019, 8, 1249 15 of 19

First, the input voltage of the boost converter was changed to examine the stability of the system.The output voltage was controlled at 24 V and a 50 Ω resistive load was connected to the voltageoutput. As shown in Figure 15, the input voltage of the boost converter drops from 12 V to 10 V.To ensure a constant output voltage, the current amplitude rises from 0.8 A to 1 A in a short time.The recovery time of the output voltage controlled by LADRC is shorter than the recovery time of thePI control. To further verify the stability of the system, the comparison of similar conditions continued,and the input voltage was changed by twice as much. Consistent with the previous results, the LADRCcontroller can bring the output voltage to the desired value faster. Experimental results are shown inFigure 16.



current, LADRC control can quickly reach the desired value, while PI control takes a long time to

reach the desired value.


To verify the proposed control method, a DC/DC boost converter experimental platform was

built, as shown in Figure 14. The platform consists of a DC power source, a DC/DC boost converter,

an electronic load, and a scope. Among them, the electronic load was set to the constant resistance

mode. The circuit parameters are the same as those in the simulation, as shown in Table 1. To obtain

the optimal actual control performance, ωc1, ωo1, ωc2, ωo2, kp1, ki1, kp2 and ki2 were redesigned as 33,

160, 300, 900, 0.03, 1, 0.05 and 2.


(a) (b)



Figure 15. (a) LADRC controller: experimental comparison responses with Case 1; and (b) PI controller:experimental comparison responses with Case 1.Electronics 2019, 8, x FOR PEER REVIEW 16 of 19


(a) (b)


controller: experimental comparison responses with case 2.

(a) (b)



First, the input voltage of the boost converter was changed to examine the stability of the system.

The output voltage was controlled at 24 V and a 50 Ω resistive load was connected to the voltage

output. As shown in Figure 15, the input voltage of the boost converter drops from 12 V to 10 V. To

ensure a constant output voltage, the current amplitude rises from 0.8 A to 1 A in a short time. The

recovery time of the output voltage controlled by LADRC is shorter than the recovery time of the PI

control. To further verify the stability of the system, the comparison of similar conditions continued,

and the input voltage was changed by twice as much. Consistent with the previous results, the

LADRC controller can bring the output voltage to the desired value faster. Experimental results are

shown in Figure 16.

Second, the input voltage was set to 12 V and the load resistance was changed from 50 Ω to 25

Ω to verify the stability of the system under different loads. To ensure a constant output voltage, the

current amplitude rises from 0.8 A to 2 A over a certain period. The experimental results are shown

in Figure 17. The output voltage and inductor current controlled by LADRC reach the desired value

more quickly than the PI control.

7. Discussion

To improve the control performance of DC/DC boost converter, this paper proposes a cascade

control based on LADRC. To achieve the effect of eliminating unstable zero dynamics, the inductance

current of the current loop is taken as a new output. Then, the indirect control is performed by the

relationship between the inductance current and the output voltage to solve the non-minimum phase

problem of DC/DC boost converter system. The instability of the output voltage caused by the

fluctuation of input voltage and that of loads is solved by designing the LADRC controller. The

simulation and experimental results show that the output voltage by using LADRC cascade control

Figure 16. (a) LADRC controller: experimental comparison responses with Case 2; and (b) PI controller:experimental comparison responses with case 2.

Second, the input voltage was set to 12 V and the load resistance was changed from 50 Ω to25 Ω to verify the stability of the system under different loads. To ensure a constant output voltage,the current amplitude rises from 0.8 A to 2 A over a certain period. The experimental results are shownin Figure 17. The output voltage and inductor current controlled by LADRC reach the desired valuemore quickly than the PI control.

Electronics 2019, 8, 1249 16 of 19



(a) (b)


controller: experimental comparison responses with case 2.

(a) (b)



First, the input voltage of the boost converter was changed to examine the stability of the system.

The output voltage was controlled at 24 V and a 50 Ω resistive load was connected to the voltage

output. As shown in Figure 15, the input voltage of the boost converter drops from 12 V to 10 V. To

ensure a constant output voltage, the current amplitude rises from 0.8 A to 1 A in a short time. The

recovery time of the output voltage controlled by LADRC is shorter than the recovery time of the PI

control. To further verify the stability of the system, the comparison of similar conditions continued,

and the input voltage was changed by twice as much. Consistent with the previous results, the

LADRC controller can bring the output voltage to the desired value faster. Experimental results are

shown in Figure 16.

Second, the input voltage was set to 12 V and the load resistance was changed from 50 Ω to 25

Ω to verify the stability of the system under different loads. To ensure a constant output voltage, the

current amplitude rises from 0.8 A to 2 A over a certain period. The experimental results are shown

in Figure 17. The output voltage and inductor current controlled by LADRC reach the desired value

more quickly than the PI control.

7. Discussion

To improve the control performance of DC/DC boost converter, this paper proposes a cascade

control based on LADRC. To achieve the effect of eliminating unstable zero dynamics, the inductance

current of the current loop is taken as a new output. Then, the indirect control is performed by the

relationship between the inductance current and the output voltage to solve the non-minimum phase

problem of DC/DC boost converter system. The instability of the output voltage caused by the

fluctuation of input voltage and that of loads is solved by designing the LADRC controller. The

simulation and experimental results show that the output voltage by using LADRC cascade control

Figure 17. (a) LADRC controller: experimental comparison responses with Case 3; and (b) PI controller:experimental comparison responses with Case 3.

7. Discussion

To improve the control performance of DC/DC boost converter, this paper proposes a cascadecontrol based on LADRC. To achieve the effect of eliminating unstable zero dynamics, the inductancecurrent of the current loop is taken as a new output. Then, the indirect control is performed bythe relationship between the inductance current and the output voltage to solve the non-minimumphase problem of DC/DC boost converter system. The instability of the output voltage caused bythe fluctuation of input voltage and that of loads is solved by designing the LADRC controller.The simulation and experimental results show that the output voltage by using LADRC cascade controlnot only has good dynamic performance and steady state performance, but also has strong robustnessunder circumstances of the variation of input voltage and loads.

Author Contributions: Methodology, H.L. and X.L.; validation, X.L. and H.L.; writing—original draft preparation,X.L.; writing—review and editing, H.L. and X.L.; supervision, J.L. and H.L.; project administration, H.L. and J.L.;and funding acquisition, H.L. and J.L.

Funding: This research was supported by Shanghai International Science and Technology Cooperation Program(No.15220710500), Shanghai Science and Technology Commission Key Program (No.18DZ1203200), and 2019Shanghai civil military integration development project (No. 2019-jmrh1-kj40).

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Symbols DescriptionDC direct currentAC alternating currentLADRC linear active disturbance rejection controlLSEF linear state error feedbackLESO linear extended state observerADRC active disturbance rejection controlNLSEF nonlinear state error feedbackESO extended state observerTD tracking differentiatorPD proportional-derivativePWM pulse width modulationUi input voltage of DCDC boost converterUo output voltage of DCDC boost converterIL inductor currentL inductance valueC capacitance valueR resistive loadfsw switching frequency

Electronics 2019, 8, 1249 17 of 19

T switching periodd duty cycleµ(t) pulse functionUo(∞) the steady state values of Uo

IL(∞) the steady state values of ILUoref reference value of output voltageILref reference value of inductor currentr first-order LADRC input referenceu0 controlled quantityω external disturbance of the systemGp controlled objecty system outputy estimated value of system outputf total disturbance of the systemf estimated value of the total disturbanceb0 system gainKp gain of LSEFLo gain of LESOωo observer’s bandwidthωc controller’s bandwidthD 1-dGp(s) transfer function of the DC/DC boost converter∆Uref reference voltage of the voltage loop∆Iref reference current of the current loop∆Uo output voltageH1(s), Gc1(s) transfer function of LADRC voltage outer loopH2(s), Gc2(s) transfer function of LADRC current inner loopGp1(s) transfer function of the ∆d to ∆Uo

Gp2(s) transfer function of the ∆d(t) to the ∆ILG1(s) transfer function of the inner loop current loopG0(s) system open loop transfer functionωc1, ωo1, b1 LADRC parameters of current control loopωc2, ωo2, b2 LADRC parameters of voltage control loopkp1, ki1 PI parameters of voltage control loopkp2, ki2 PI parameters of current control loop

References

1. Dragicevic, T.; Lu, X.; Vasquez, J.; Guerrero, J. DC Microgrids–Part I: A Review of Control Strategies andStabilization Techniques. IEEE Trans. Power Electron. 2016, 31, 4876–4891. [CrossRef]

2. El-Shahat, A.; Sumaiya, S. DC-Microgrid System Design, Control, and Analysis. Electronics 2019, 8, 124.[CrossRef]

3. Baranwal, M.; Askarian, A.; Salapaka, S.; Salapaka, M. Distributed Architecture for Robust and OptimalControl of DC Microgrids. IEEE Trans. Ind. Electron. 2018. [CrossRef]

4. Kakigano, H.; Nishino, A.; Ise, T. Distribution Voltage Control for DC Microgrids Using Fuzzy Control andGain-Scheduling Technique. IEEE Trans. Power Electron. 2013, 28, 2246–2258. [CrossRef]

5. Shenai, K.; Shah, K. Smart DC micro-grid for efficient utilization of distributed renewable energy.In Proceedings of the Energytech, Cleveland, OH, USA, 25–26 May 2011.

6. Dragicevic, T.; Vasquez, J.C.; Guerrero, J.M.; Skrlec, D. Advanced LVDC Electrical Power Architectures andMicrogrids: A step toward a new generation of power distribution networks. IEEE Elect. Mag. 2014, 2, 54–65.[CrossRef]

7. Boujelben, N.; Masmoudi, F.; Djemel, M.; Derbel, N. Design and comparison of quadratic boost and doublecascade boost converters with boost converter. In Proceedings of the 14th International Multi-Conference onSystems, Signals Devices (SSD), Marrakech, Morocco, 28–31 March 2017; pp. 245–252.

http://dx.doi.org/10.1109/TPEL.2015.2478859


http://dx.doi.org/10.1109/TIE.2018.2840506


http://dx.doi.org/10.1109/MELE.2013.2297033

Electronics 2019, 8, 1249 18 of 19

8. Jou, H.; Huang, J.; Wu, J.; Wu, K. Novel Isolated Multilevel DC–DC Power Converter. IEEE Trans. PowerElectron. 2016, 31, 2690–2694. [CrossRef]

9. Zhang, C.; Gao, Z.; Chen, T.; Yang, J. Isolated DC/DC converter with three-level high-frequency link andbidirectional power flow ability for electric vehicles. IET Power Electron. 2019, 12, 1742–1751. [CrossRef]

10. Leyva-Ramos, J.; Ortiz-Lopez, M.G.; Diaz-Saldierna, L.H.; Morales-Saldana, J.A. Switching regulator using aquadratic boost converter for wide DC conversion ratios. Power Electron. IET 2009, 2, 605–613. [CrossRef]

11. Wang, Y.; Qiu, Y.; Bian, Q.; Guan, Y.; Xu, D. A Single Switch Quadratic Boost High Step up DC-DC Converter.IEEE Trans. Ind. Electron. 2018, 66, 4387–4397. [CrossRef]

12. Wu, G.; Ruan, X.; Ye, Z. Non-isolated High Step-up DC-DC Converters Adopting Switched-capacitor Cell.Ind. Electron. IEEE Trans. 2015, 62, 383–393. [CrossRef]

13. Chen, F.; Cai, X.S. Design of feedback control laws for switching regulators based on the bilinear large signalmodel. Power Electron. IEEE Trans. 1989, 5, 236–240. [CrossRef]

14. Zhang, C.; Wang, X.; Lin, P.; Liu, P.X.; Yan, Y.; Yang, J. Finite-Time Feedforward Decoupling and PreciseDecentralized Control for DC Microgrids Towards Large-Signal Stability. IEEE Trans. Smart Grid 2019.[CrossRef]

15. Lee, T.S. Input-output linearization and zero-dynamics control of three-phase AC/DC voltage-sourceconverters. Power Electron. IEEE Trans. 2003, 18, 11–22.

16. Viswanathan, K.; Oruganti, R.; Srinivasan, D. Dual mode control of tri-state boost converter for improvedperformance. In Proceedings of the IEEE Power Electronics Specialist Conference, Acapulco, Mexico,15–19 June 2003.

17. Sastry, J.; Ojo, O.; Wu, Z. High performance control of a boost AC-DC PWM rectifier-induction generatorsystem. In Proceedings of the Industry Applications Conference, Hong Kong, China, 2–6 October 2005.

18. Wei, X.; Tsang, K.M.; Chan, W.L. DC/DC Buck Converter Using Internal Model Control. Electr. Mach.Power Syst. 2009, 37, 320–330. [CrossRef]

19. Shuai, D.; Xie, Y.; Wang, X. The Research of Input-Output Linearization and Stabilization Analysis ofInternal Dynamics on the CCM Boost Converter. In Proceedings of the International Conference on ElectricalMachines Systems, Wuhan, China, 17–20 October 2008.

20. Sira-Ramírez, H.; Silva-Ortigoza, R. Control Design Techniques in Power Electronics Devices; Springer: London,UK, 2006; pp. 235–355.

21. Han, J. From PID to Active Disturbance Rejection Control. IEEE Trans. Ind. Electron. 2009, 56, 900–906.22.[CrossRef]

22. Han, J. The auto-disturbance-rejection controller (ADRC) and its application. Control Decis. 1998, 1, 19–23.23. Gao, Z. Scaling and bandwidth-parameterization based controller tuning. In Proceedings of the American

Control Conference, Denver, CO, USA, 4–6 June 2003; pp. 4989–4996.24. Gao, Z. Active Disturbance Rejection Control: A paradigm shift in feedback control system design.

In Proceedings of the American Control Conference, Minneapolis, MN, USA, 14–16 June 2006.25. Gang, T.; Gao, Z. Benchmark tests of Active Disturbance Rejection Control on an industrial motion control

platform. In Proceedings of the 2009 American Control Conference, St. Louis, MO, USA, 10–12 June 2009;pp. 5552–5557.

26. Huang, Y.; Xue, W.; Gao, Z.; Sira-Ramirez, H.; Sun, M. Active Disturbance Rejection Control: Methodology,Practice and Analysis. In Proceedings of the Control Conference, Hilton, Portland, 4–6 June 2014.

27. Song, C.; Wei, C.; Yang, F.; Cui, N. High-Order Sliding Mode-Based Fixed-Time Active Disturbance RejectionControl for Quadrotor Attitude System. Electronics 2018, 7, 357. [CrossRef]

28. Li, H.; Qu, Y. A Composite Strategy for Harmonic Compensation in Standalone Inverter Based on LinearActive Disturbance Rejection Control. Energies 2019, 12, 2618. [CrossRef]

29. Zuo, Y.; Zhu, X.; Li, Q.; Chao, Z.; Yi, D.; Xiang, Z. Active Disturbance Rejection Controller for Speed Controlof Electrical Drives Using Phase-locking Loop Observer. IEEE Trans. Ind. Electron. 2019, 66, 1748–1759.[CrossRef]


http://dx.doi.org/10.1049/iet-pel.2018.6268

http://dx.doi.org/10.1049/iet-pel.2008.0169



http://dx.doi.org/10.1109/63.53161

http://dx.doi.org/10.1109/TSG.2019.2923536

http://dx.doi.org/10.1080/15325000802454500



http://dx.doi.org/10.3390/en12132618


Electronics 2019, 8, 1249 19 of 19

30. Zheng, Q.; Dong, L.; Lee, D.H.; Gao, Z. Active disturbance rejection control for MEMS gyroscopes.In Proceedings of the American Control Conference, St. Louis, MO, USA, 10–12 June 2009.

31. You, J.; Fan, W.; Yu, L.; Fu, B.; Liao, M. Disturbance Rejection Control Method of Double-Switch Buck-BoostConverter. Energies 2019, 12, 278. [CrossRef]

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