Comparative Evaluation of PID Voltage Mode, PI Current Mode, Fuzzy and PWM Based Sliding Mode

Control for DC-DC converters Omar Ellabban, and Joeri Van Mierlo

Vrije Universiteit Brussel, IR-ETEC, Pleinlaan 2, B-1050 Elsene, Belgium Omar.Ellabban@vub.ac.be

Abstract- This paper presents a comparison between the application of four control techniques in DC-DC converters. PID voltage mode (PID-VM), PI current mode (PI-CM), fuzzy, and PWM based sliding mode (SM) controllers are applied to buck converter. The design procedures of controllers are reviewed. The dynamic performance of these controllers under star-up, steady state, input voltage variation, load current disturbances and EMI spectrum are presented and compared.

I. INTRODUCTION

DC-DC converters are nonlinear systems due to their inherent switching operation, they represent a big challenge for control design. Many linear and nonlinear control techniques had been applied to control DC-DC converters. Linear PID voltage mode (PID-VM) and PI current mode (PI-CM) controllers are usually designed for DC-DC converters using standard frequency response techniques based on the small signal model of the converter [1-3]. A Bode plot is used in the design to obtain the desired loop gain, crossover frequency and phase margin. The stability of the system is guaranteed by an adequate phase margin. However, linear PID and PI controllers can only be designed for one nominal operating point. A buck converter’s small signal model changes when the operating point varies. Therefore, it is difficult for the PID-VM and PI-CM controllers to respond well to changes in operating point. Nonlinear controllers which are more robust and have faster dynamic response are applied to power converters to solve this problem. Many nonlinear control schemes have been proposed for DC-DC converters. Between these schemes, sliding mode (SM) and fuzzy control have advantages such as simple and model free implementation [5-14].

This paper comprises the application of PID-VM, PI-CM, Fuzzy and PWM based SM control methods for buck converter. Simulation results using Matlab for the four control techniques are evaluated and compared.

II. BUCK CONVERTER MODELING

The buck converter is one of the simple but most useful power converters; a chopper circuit that converts a dc inputs to a dc output at a lower voltage. The buck converter shown in Fig.1, which is operating with the switching period of and duty cycle is considered [4].

During continuous conduction mode of operation, the state

Fig. 1 Buck converter: (a) circuit; (b) switch on; (c) switch off

space equations when the switch is ON (Fig. 1-b) are given by,

, 0 (1-a)

and when the switch is OFF (Fig. 1-c) are presented by,

, (1-b)

Using the state space averaging method, these sets of equations can be written as,

(2)

where is the inductor current and is the capacitor voltage. The small signal modeling for the buck converter is done assuming perturbations in the supply voltage and the duty cycle. Equation (3) gives the output to control transfer function

of a voltage mode buck converter, it has two poles and can be modeled as a second-order equation, (3)

A current mode buck converter also has two poles. However, the second pole is located near the switching frequency away from the dominant pole. Therfore, the model can be approximated into a first-order equation with a single pole. Control to output transfer function G s of a current mode buck converter can be expressed as,

(4)

where is the control signal.

III. PID VOLTAGE MODE CONTROL (PID-VM)

The output voltage of a buck converter is regulated by closing a feedback loop between the output voltage and the duty ratio signal. The output voltage is compared with a constant reference signal to form the error, which is then passed through the controller transfer function to generate a control signal proportional to the duty ratio; finally the PWM modulator converts the control signal into the switch drive signal , as shown in Fig. 2 [1].

The magnitude and phase asymptotes of the uncontrolled buck converter are sketched in Fig. 3-a, based on Eq. (3) and the parameters of buck converter given in Table 1. The uncontrolled buck converter frequency characteristic has a crossover frequency of 6.5 KHz with a phase margin of less than five degrees. In order to compensate the low-frequency loop gain and improving the phase margin, a PID controller has been designed, whose transfer function is

(5)

To design the controller [1]: first, the new crossover frequency is chosen to be one twentieth of the switching frequency; then, choose the phase margin (somewhat arbitrary) to be 52 ; then, the pole and zero frequencies of the part are chosen as;

(6)

After that, the frequency of the inverted zero of part is chosen (somewhat arbitrary) to be one tenth of the crossover frequency; finally, the controller gain is, (7)

Fig. 3-b, shows the bode plot of buck converter with PID voltage mode control, the system has a crossover frequency of 10 with a phase margin of 48.4 .

Fig. 2 Simulink model of a PID voltage mode control for buck converter:

(a) complete model; (b) controller model

TABLE 1 SPECIFICATION OF BUCK CONVERTER Parameter name Symbol Value Input voltage 24Output voltage 12Capacitance 150Inductance 100Minimum load resistance 3 ΩSwitching frequency 200

Fig. 3 Bode plot for : (a) uncontrolled; (b) with PID voltage mode

control

IV. PI CURRENT MODE CONTROL (PI-CM)

In this control method, the PWM modulator is replaced by the switch current feedback loop. The switch is switched on at the start of each cycle by a clock pulse which sets the output of the S-R latch. The switch current rises linearly while it is conducting. The current is fed back as and is compared by the reference signal . When is equal to the reference, the comparator output switches low, resetting the S-R latch and turning the switch off as shown in Fig. 4 [2, 3].

Fig. 5-a displays the converter frequency characteristics of . As the graph indicates, while the phase margin of the

converter is sufficient, the gain for low frequency is very low. In turn a PI controller is designed to compensate for low frequency gain whose transfer function is

1 (8)

Fig. 4 Simulink model of a PI current mode control for buck converter:

(a) complete model; (b) controller model

To design the PI controller: first, the new crossover frequency f is chosen to be one twentieth of the switching frequency; then the frequency of the inverted zero is chosen (somewhat arbitrary) to be one tenth of the cross over frequency; finally, the controller gain G is,

GG GPI

(9)

Fig. 5-b, shows the bode plot of buck converter with PI current mode control; the system has a sufficient phase margin and high low frequency gain.

V. FUZZY LOGIC CONTROL

The conventional control theory uses a mathematical model of a process to be controlled and some specifications of the expected behavior in close loop to design a controller. However obtaining a mathematical model could be difficult in many nonlinear or unknown systems, or if the system does not have constant parameters, what might limit the linear control strategies. Fuzzy controllers don’t require an exact mathematical model. Instead, they are designed based on general knowledge of the plant. A reason of this is that the human knowledge adds several types of information and it can mix different control strategies that cannot be added in an analytical control law. Then the knowledge based fuzzy control uses the experience and the knowledge of an expert. A kind of knowledge based fuzzy control is the rule based fuzzy control where the human knowledge is approximated by means of linguistic fuzzy rules in the form if-then, which describe the control action in a particular condition of the system. Due to the nonlinear behavior showed by the converter and to the inevitable variations in its parameters in real implementations, a nonlinear fuzzy control might be desirable to control the converter [5-9].

There are two inputs for the fuzzy controller for a buck converter. The first input is the error in the output voltage

, where is the converter output voltage at the sampling instant and is the desired output voltage. The second input, 1 , is change of error at the sample. Each input is composed by seven fuzzy sets as shown in Fig. 6. The two inputs are multiplied by the scaling factors and respectively, and

Fig. 5 Bode plot for : (a) uncontrolled; (b) with PI current mode control

then fed into the fuzzy controller. The output of the fuzzy controller is the change in duty cycle , which also has seven fuzzy sets as the inputs and it is scaled by a linear gain

. The scaling factors , and can be tuned to obtain a satisfactory response. The duty cycle , at the sampling time, is determined by adding the previous duty cycle 1 , to the calculated change in duty cycle: 1 · (10) The calculated duty cycle signal is then sent to a PWM output stage that generates the appropriate switching pattern for the buck converter, as shown in Fig. 7. The rule base used to implement this controller is composed by 49 rules and it is shown in the Table 2. All the fuzzy inference system was implemented in the fuzzy inference system (FIS) of Matlab. The control surface is shown in the Fig. 8.

Fig. 6 Membership functions for , and

Fig. 7 Simulink model of fuzzy control for buck converter: (a) complete

model; (b) controller model

Fig. 8 Control surface of fuzzy controller

TABLE 2 RULE BASE FOR FUZZY CONTROLLER

/ NB NM NS ZE PS PM PB NB NB NB NB NB NM NS ZE NM NB NB NB NM NS ZE PS NS NB NB NM NS ZE PS PM ZE NB NM NS ZE PS PM PB PS NM NS ZE PS PM PB PB PM NS ZE PS PM PB PB PB PB ZE PS PM PB PB PB PB

VI. PWM BASED SLIDING MODE CONTROL (PWM BASED SM)

Sliding mode controllers are well known for their

robustness and stability. The nature of the controller is to ideally operate at an infinite switching frequency such that the controlled variables can track a certain reference path to achieve the desired dynamic response and steady state operation. The main problem with using of this controller in control of DC-DC converters is variable and high switching frequency which increases switching losses, inductor and transformer core losses, and electromagnetic interference (EMI) noise issue [9-14].

In recent years, fixed frequency SM control has been investigated as a better control then conventional SM control method for DC-DC converters. One of the most recent methods to obtain a constant switching frequency in SM is to change the modulation method of the SM controllers from hysteresis-modulation (HM) to pulse width modulation (PWM). The technique of PWM modulation is to compare a desired analogue control signal with a ramp signal, of which a pulse like output switching signal having the same frequency as the ramp signal, will be generated. The advantage is that the frequency of the output switching signal will be constant, regardless of how the duty cycle varies with the variation of the control signal. To achieve such a controller, a relationship between SM control and duty cycle control is required. This method in [12] offers mapping the equivalent control onto the duty cycle function of the pulse width modulator. It suggests sliding surface with three coefficients , and and results second order response for output variable response as, (11) where is the voltage error, is the rate of change of voltage error and is the integral of the voltage error. Comparing the equivalent control and the duty ratio control, the following relationships can be established: (12) The constant gain parameters, , and are depended on values of , and sliding coefficients . The dynamic performance of controller can be changed depended on values of . Fig. 9 shows the block diagram of the PWM based SM control for buck converter.

Fig. 9 Simulink model of a PWM-based SM control for buck converter:

(a) complete model; (b) controller model

VII. SIMULATION RESULTS

Performance analysis of four controllers is evaluated with Matlab simulation. PID-VM, PI-CM, Fuzzy and PWM based SM are developed based on last descriptions. Performance analysis of converters is evaluated under star-up, steady state, input voltage variation, load current disturbances and EMI spectrum. Fig. 10 shows the start up behavior of the controllers. PI-CM has the largest overshoot; PID-VM, Fuzzy and SM have a same overshoot for output voltage. SM has the shortest settling time but Fuzzy has the longest settling time. Therefore, SM has the best start up performance. Also Fig. 10 shows the steady state behavior of the controllers. PID-VM, PI-CM and Fuzzy have the same output voltage ripples, but Fuzzy has some out voltage oscillations. SM has the smallest out voltage ripples. SM is the most efficient control for steady state. Fig. 11 shows controllers behavior during 20% step change in the input voltage. SM control is not affected by input voltage step, PID-VM has the largest output spike, PI-CM has almost negligible effect and Fuzzy has larger output ripples. Therefore, SM has a strong immunity against input voltage variation. Fig. 12 and 13 show controllers behavior during 50% load step increases and decreases respectively. SM has the smallest overshoot and the smallest settling time, Fuzzy has some output voltage oscillations during step up load change and a longer time oscillation during load step down change, both PID-VM and PI-CM have a little transient. So, SM has the best performance during load change. Since the selected control strategy also effects on the generated electromagnetic interference (EMI) spectrum [15], Fig. 14 shows the EMI converter input current spectrum. SM has the smallest EMI spectrum.

VIII. CONCLUSION

Linear PID-VM and PI-CM controllers and Fuzzy, PWM based SM controllers were designed and implemented for the buck converter. Linear PID and PI controller were designed based on the frequency response of the buck converter using frequency response techniques. Fuzzy controllers were designed based on the expert knowledge of the buck converter and tuned using a trial and error method. The SM controller does not need accurate mathematical models but requires the knowledge of parameter variation range to ensure stability and satisfy reaching conditions. PWM based SM has a fixed switch frequency like the others and is well known for its robustness and stability. PWM based SM has the best performance during start-up, steady state, input voltage and load change, also it has the lowest input current spectrum. From the present study, PWM based SM seems to be a viable controller for application in power DC-DC converters.

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Fig. 10 Response of the controllers during start up

Fig. 11 Response of the controllers under 20% input voltage step

Fig. 12 Response of the controllers under 50% step load increase

Fig. 13 Response of the controllers under 50% step load decrease

Fig. 14 Input current spectrum for different control technique