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Controlling DC-DC converters is to ensure the converter closed loop stability and to enhance its efficiency and dynamics. Many control approaches, dealing with this subject, can be found in the literature -. However, control performances are related to the converter model accuracy.
In this chapter the control design for DC-DC converters using linear control methods is presented. An accurate model is essential to design linear controllers. Small signal models for boost converters were obtained using the standard state-space averaging techniques.
Frequency response and root locus methods  may be utilized to design linear controllers. In the frequency response method, analog PID and PI controllers were designed based on the converters' small signal models . The system was compensated to achieve high loop gain, wide bandwidth and sufficient phase margin.
Linear controllers for DC-DC converters are often designed based on mathematical models. To obtain a certain performance objective, an accurate model is essential. A number of AC equivalent circuit modeling techniques like circuit averaging, averaged switch modeling, the current injected approach, and the state-space averaging method are there. Among these methods, the state-space averaged modeling is most widely used to model DC-DC converters.
In general the dynamic and output equations for any converter may be put in the form shown below for the ON and OFF duration.
ON Duration (during the time kTS t (k+d) TS)
OFF Duration (during the time (k+d) TS t (k+1) TS)
The above equations represents that the converter alternates between the two switched states at high frequency. So to represent the converter through a single equivalent dynamic representation, valid for both the ON and OFF durations is as given in the equation below.
Where is the average rate of change of dynamic variables over a full switching period.
Then for the averaged dynamic variables
Where A = A1d+A2(1-d); b = b1d+b2(1-d); e = e1d+e2(1-d);
n = n1d+n2(1-d); q = q1d+q2(1-d)
Equations (4.6) and (4.7) represent the equivalent dynamic and output equations of the converter.
Steady State Solution:
The steady state solution is obtained by equating the rate of change of dynamic variables to zero.
The ON and OFF state equivalent circuits of the boost converter are shown in figure 4.1 and figure 4.2 respectively.
Fig.4.1 ON-state Equivalent Circuit
The OFF state dynamic equations are as follows:
Fig.4.2 OFF-state Equivalent Circuit
The averaged state space equations are obtained by averaging the switching action and are as follows:
Where x is the averaged state variable vector, vg is the input voltage and A and b are:
By introducing a small perturbation in the input and state variables of the above equations, the system equations are given by:
For the linear small signal analysis the first order terms are only considered and therefore the new equations are:
From the above linear small signal model of the converter the desired control to transfer functions can be obtained as
Therefore the output-to-control small signal transfer function of the boost converter is given by the equation (4.22)
For a boost converter with the following parameters,
C=100F, L=3mH, R=150, D=0.74, Vg=50V, and fS=25KHz
The above transfer functions are evaluated and it can be seen that the above transfer functions has a Right Half Plane Zero (RHPZ) which is a special characteristic of boost converters. Before discussing the design of the controller the RHPZ is to be eliminated. The zero in the right half plane is caused by switching action, and it introduces a 90 degree phase delay. The RHPZ is eliminated by varying the value of rC. In general it can be shown that after simplifying the numerator of, for eliminating the RHPZ the value of rC must satisfy the following inequality.
The bode plot of the transfer functions are as shown in the figures Fig.4.3 and Fig.4.4.
Fig.4.3. Bode plot for the equation (4.23).
Fig.4.3. Bode plot for the equation (4.24).
Digital Controller Design for DC-DC Converters:
In DC-DC converters, the output voltage is a function of the input line voltage, the duty cycle and the load current. It is desirable to have a constant output voltage in the event of disturbances such as a sudden change of input voltage or load current. Negative feedback control is applied to DC-DC converters to automatically adjust the duty cycle to obtain the desired output voltage with high accuracy in spite of disturbance.
In this section, frequency response techniques are used to design digital controllers for DC-DC converters. The compensated system is expected to have the following characteristics.
Firstly, the loop gain should be high at lower frequencies to minimize steady-state error and increase rejection to disturbances of input voltage and load current variations.
Secondly, the crossover frequency should be as high as possible, but about an order of magnitude below the switching frequency to allow the DC-DC converter to respond quickly to the transients.
Thirdly, the phase margin should be sufficient to ensure the system's stability. When the phase margin of the loop gain is positive, the system is stable.
Phase margin determines the transient response of the DC-DC converter. An increase of the phase margin makes the system more stable with less ringing and oscillation. There is a qualitative relationship between the phase margin and the closed loop damping factor Q. The damping factor Q determines the shape of the transient response. When Q is equal to 0.5, the closed-loop system has two real poles at the same frequency, and the system is critically damped. The transient response will be fast without overshoot. When Q is larger than 0.5, there are two complex conjugate poles, and the system is under damped. The transient response will have an oscillatory-type waveform with decaying magnitude. The higher Q, the higher overshoot the transient response will have. When Q is less than 0.5, the closed-loop system has two real poles at two different frequencies, and the system is over damped. The transient response is a decaying exponential function of time with the time constant determined by the pole at the lower frequency.
To design a controller using the frequency response method, phase-lead, phase lag or lead-lag compensation is usually used. A proportional-derivative (PD) controller is phase-lead compensation. PD controllers are used to increase the phase margin and improve the cross-over frequency. A zero is placed at frequency ωz far below the cross over frequency to improve the phase margin. The transfer function of a PD controller is shown in equation (4.25).
The pole at ωp is placed well below the switching frequency to avoid amplification of the switching noise. The maximum phase shift occurs at the geometric mean of the pole ωp and the zero ωz. To obtain maximum phase margin improvement, the maximum phase shift should be placed at the cross-over frequency.
A proportional-integral (PI) controller is a phase-lag controller. A PI controller is used to increase the low frequency loop gain, thus reducing steady-state error. The transfer function of a PI controller is shown in equation (4.26).
The PI controller has a pole at the origin. Both PD and PI controllers are first-order controllers.
By using a lead-lag compensator, the advantages of lead compensation and lag compensation can be combined to obtain sufficient phase margin, high loop gain and wide control bandwidth. A proportional-integral-derivative (PID) controller is a lead-lag compensator. It is the most widely used compensator in feedback control systems. The PID controller is defined by (4.27), where e(t) is the compensator input and m(t) is the compensator output.
The Laplace transform of equation (4.27) yields the transfer function in equation (4.29).
The integral term is phase-lag and the derivative term is phase-lead. The low frequency gain is improved by the integral term, and the low-frequency components of the output voltage are accurately regulated. At high frequency, the phase margin and cross-over frequency are improved by the derivative term, which improves the system's stability and the speed of the transient response. An increase in the proportional term will increase the speed of system response; however, too much proportional gain will make the system unstable.
A PI controller is designed for the proposed boost converter in the following sections.
PI Controller Design for Two Inductor Boost Converters
A PI controller is designed for the steady state to reduce oscillations of the duty cycle. A pole is placed at the origin and a zero was placed at 127 radians/s. The Bode plot of the PI-controller-compensated system is shown in Fig. 4.4. The phase margin of the PI compensated system is 93.4 degrees. The Root locus of the PI-controller-compensated system is shown in Fig. 4.5.
Fig.4.4 Bode plot of PI controller compensated Two Inductor Boost converter.
Fig.4.5 Bode plot of PI controller compensated Two Inductor Boost converter.
With the above analysis the Two Inductor Boost converter is simulated in closed loop system using the PI controller. The output of the converter is sensed and compared with a reference voltage. The error obtained is processed through a PI controller and fed back to the system. PI controller adjusts the pulse width to maintain output voltage constant. It is observed that the output voltage remains constant due to control action of the closed loop system.
The system is subjected to a step input signal of 50 V magnitude as shown in Fig.4.6. The boost converter improves (boosted) the signal to a magnitude of 200 V as shown in Fig.4.7. But we observe some peak overshoot in the response of the system.
Fig.4.6 Input of the Two Inductor Boost converter.
Fig.4.7 Output of the Two Inductor Boost converter.
FUZZY CONTROLLER DESIGN FOR TWO INDUCTOR BOOST CONVERTERS:
Linear controllers for DC-DC converters are usually designed based on mathematical models. To obtain a certain performance objective, an accurate model is essential. In the previous section, linear controllers were designed for Two Inductor Boost converters based on each converter's small signal model using frequency response and root locus design methods. The small signal model changes due to variations in operating point. For the boost converter's small signal model, the poles and a right half plane zero, as well as the magnitude of the frequency response, are all dependent on the duty cycle D. This makes the transfer function of the boost converter's small signal model a nonlinear function of the duty cycle. The right-half plane zero and the nonlinear nature of the boost converter's small signal model makes the control design for this converter more challenging from the point of view of stability and bandwidth .
To achieve a stable and fast response, two solutions are possible. One is to develop a more accurate model for the converter. However, the model may become too complex to use in controller development. A second solution is to use a nonlinear controller . Since fuzzy controllers don't require a precise mathematical model, they are well suited to nonlinear, time-variant systems. The design of fuzzy controllers is presented in this chapter. The first section introduces the concept of fuzzy control. The second section is mainly focused on the fuzzy controller based Two Inductor Boost converter.
Introduction to Fuzzy Control:
Fuzzy control is an artificial intelligence technique that is widely used in control systems. It provides a convenient method for constructing nonlinear controllers from heuristic information.
Conventional controllers are designed based on a mathematical model. Closed loop control specifications include disturbance rejection properties, insensitivity to plant parameter variations, stability, rise time, overshoot and settling time and steady-state error. Based on these specifications, conventional controllers are designed. Major conventional control methods include classical control methods (frequency response and root locus techniques), state-space methods, optimal control, robust control, adaptive control, sliding mode control and other nonlinear control methods such as feedback linearization and back stepping. These conventional control methods provide a variety of ways to utilize information from mathematical models on how to obtain good control.
Different from conventional control, fuzzy control is based on the expert knowledge of the system. Fuzzy control provides a formal methodology to represent and implement a human's heuristic knowledge about how to control the system. A block diagram of a fuzzy control system is shown in Figure. 4.8. A fuzzy controller contains four main components:
The fuzzification interface that converts its inputs into information that the inference mechanism can use to activate and apply rules.
The rule base which contains the expert's linguistic description of how to achieve good control.
The inference mechanism that evaluates which control rules are relevant in the current situation, and
The defuzzification interface which converts the conclusion from the inference mechanism into the control input to the plant.
Input u (t)
Output y (t)
Fig.4.8 Block diagram of fuzzy control system
The performance objectives and design constraints are the same as those for conventional control. Design of fuzzy controllers involves the following procedures:
Choose the fuzzy controller's inputs and outputs
Choose the preprocessing for the controller inputs and post processing for the controller outputs, and
Design each of the four components of the fuzzy controller shown in Figure 4.8.
Design of Fuzzy Logic Controller:
A fuzzy controller for a Two Inductor Boost converter has an input as the error in the output voltage and Ref is the digital value corresponding to the desired output voltage. The output of the fuzzy controller is the change in duty cycle.
First, the linguistic values are quantified using membership functions. Each universe of discourse is divided into fuzzy subsets. There were 3 fuzzy subsets in the fuzzy controller for the Two Inductor Boost converter. The membership functions for the input variable and the output variable are as shown in the Fig.4.9 and Fig.4.10. The variables are the membership degrees assigned to each fuzzy subset to quantify the certainty that the input can be classified linguistically into the corresponding fuzzy subsets. A triangle-shaped membership function was used for this controller design for the ease of implementation. Of the 3 subsets, there is one subset for the positive part and one subset for the negative part of the universe of discourse, respectively. The input and output membership functions are indicated with I1, I2, I3 and O1, O2 and O3 respectively.
Fig.4.9. Membership functions of the inputs for the Two Inductor Boost converterC:\Users\Navneeth\AppData\Local\Temp\msohtmlclip1\01\clip_image001.jpg
Fig.4.9. Membership functions of the outputs for the Two Inductor Boost converterC:\Users\Navneeth\AppData\Local\Temp\msohtmlclip1\01\clip_image001.jpg
The rule base is derived from general knowledge of DC-DC converters and adjusted based on required results. There is a tradeoff between the size of the rule base and the performance of the controller. A 3X3 rule base is designed for the Two Inductor Boost converter as described in Table-1 that can build by crossing the fuzzy sets considered for each input.
Table 1. Fuzzy rule base matrix.
In the fuzzy inference system, the fuzzified input variables are processed with fuzzy operators, and the IF-THEN rule implementation.
The linguistic output data is converted back into crisp output data by defuzzification process. This is given by Equation 4.30.
Where N is the number of rules that are effective at any one time
Wi is the weighing factor, Ci is the change in duty cycle.
With the above analysis the Two Inductor Boost converter is simulated in closed loop system using the Fuzzy controller. The output of the converter is sensed and compared with a reference voltage. The error obtained is processed through a Fuzzy controller and fed back to the system. Fuzzy controller adjusts the pulse width to maintain output voltage constant. It is observed that the output voltage remains constant due to control action of the closed loop system.
The Two Inductor Boost converter system in a closed loop using the Fuzzy controller is subjected to a step input signal of 50 V magnitude as shown in Fig.4.10. The error voltage sensed and which is given as input to the Fuzzy controller is as shown in Fig.4.11. The Fuzzy controller controls the control signals given to the switches such that the magnitude of the output voltage from the boost converter is of 200 V as shown in Fig.4.12.
Fig.4.10. Input to the Two Inductor Boost Converter with Fuzzy Controller
Fig.4.11. Error Input to the Fuzzy Controller
Fig.4.11. Output Voltage of the Two Inductor Boost Converter with Fuzzy Controller
This chapter presents the two-inductor, double-switch boost converter topology and its variations that can regulate the output voltage in a wide range of load current and input voltage in a closed loop method. With the analysis given for this two inductor boost converter system, the inductor values are significantly reduced. The two-inductor boost converter closed loop controls were studied with both PI controller and Fuzzy Logic controller. Results were analyzed and discussed using MATLAB Simulink. We observed that the output voltage of the system using Fuzzy logic controller is boosted and also produce a signal free from transient noise that makes reduced effect of Electromagnetic interference (EMI). Thus the utilization of smaller inductors and the fuzzy controller improves the efficiency of the proposed two inductor boost converter system.