State Feedback Control based Genetic Algorithms of a Modified Cuk Convertor

Nashwan Saleh Sultan

Electrical Power Technology Engineering

Technical Engineering College / Mosul, Northern Technical University

Mosul, Iraq

Abstract- The present work that includes a genetic algorithm technique programed as intelligent controller. It designed to control the output voltage response of the modified DC-DC Cuk converter. The modified Cuk converter analyzed and studied. Modifications involve coupling the two coils around one mutual core. A mathematical model of converter is represented and derived using differential equations and state space representation, then a state feedback controller designed for the modified DC-DC Cuk converter using a decimal coded genetic algorithm with a modified selection method. The open and closed loop output voltage responses have been analyzed, and studied; it shows a great improvement in the output performance.

Introduction

Switching Mode Power Supplies (SMPSs) get more interest and importance with the technologic advance all over the world.1 They can be classified into a few basic types such as buck, boost, and Cuk convertors,2 in addition to that, many various types were derived from the basic types. These modifications produce more efficiency over the basic types, or suffice specific requirements of certain operational circumstances.3

Evolutionary algorithms ,on the other hand, have been occupied a considerable portion of the researches in many fields of science.4 Since 1975 when (John Holland) proposed the idea of simulating the operators of the Darwinian evolution computationally to reproduce a group of possible solutions for a certain problem from a random generated group. Then evaluates these solution throughout the consecutive generations in order to obtain the best (optimal) solution for the problem 5 . The improvement performance of modified Cuk converter based on practical swarm optimization 6 . The present work tries to design a feedback controller for the Cuk convertor to improve its performance using genetic algorithm.

System Description:

The modified dc-dc Cuk converter is one of the many succeeded modified converters, which have been done to the normal Cuk convertor. It has been made by winding the two coils of the convertor around a single core with the high mutual between these coils as shown in figure (1).

This modified has made to achieve many advantages over the common Cuk convertor, such as the ability to minimize the ripple current to zero, if proper turn ratio and coupling coefficient chosen. Also, the modified Cuk converter

produced a converter with a lower weight, lower cost, and smaller size.

The study of a system in the control process that produced three main problems which can be examined in: system dynamics, system identification or modeling, and system control. Since modified dc-dc Cuk convertor contains a four energy storage elements (C1, C2, L1, L2), therefore; the system dynamics are expressed by a fourth order state space representation.

Fig. 1. The Modified dc-dc Cuk Converter Circuit

The converter works with two modes to produce complete dc level output conversion .6:

Mode 1: When the switch is ON, as in figure (2.a):

The capacitor (C1) charged by the electrical source when the switch turned ON, the current through (L1) increases proportionally with the duty cycle of the switch. The diode will be reverse biased, and the capacitor (C1) will be lost its charge through two directions; through the closed switch to the load and (C2), and through (L2) then (L1) by the mutual core.

Mode 2: When the switch is OFF, as in figure (2.b): The diode will be forward biased and the capacitor (C1) will be recharged through the source, the induced voltage through the mutual core and (L1).

Fig. 2. a. Mode 1 (switch on) b. Mode 2 (switch off)

The system mathematical model and the current path of are the same into the both inductors, and then one can use the algebraic sum to express the relation of the inductances, as follows:

V_(L_1 )=L_1 ?di?_1/dt+M ?di?_2/dt (1)

V_(L_2 )=M ?di?_1/dt+L_2 ?di?_2/dt (2)

The Solving of these instantaneous equations, which can get:

?di?_1/dt=L_2/(L_1 L_2-M^2 ) V_(L_1 )+(-M)/(L_1 L_2-M^2 ) V_(L_2 ) (3)

?di?_2/dt=(-M)/(L_1 L_2-M^2 ) V_(L_1 )+L_1/(L_1 L_2-M^2 ) V_(L_2 ) (4)

The overall mathematical model of the representation system that analyzed in the differential equations for the two modes, as follows:

Mode 1: Applying Kirchhoff law for the equivalent circuit of this mode, shown in figure (2.a), and by neglecting the voltage across the switch, one can get:

V_(L_1 )=V_in-i_1 R_1 (5)

V_(L_2 )=V_1-V_2-i_2 R_2 (6)

By substituting, these tow equations (5) and (6) in (3) and (4):

?di?_1/dt=M/? V_2+(-M)/? V_1+(MR_2)/? i_2+(-L_2 R_1)/? i_1+L_2/? V_in (7)

?di?_2/dt=(-L_1)/? V_2+L_1/? V_1+(-L_1 R_2)/? i_2+(MR_1)/? i_1+(-M)/? V_in (8)

As: ?=L_1 L_2-M^2

While:

(9)

(10)

Then, by applying Kirchhoff current law at node (n) :

(11)

(12)

(13)

(14)

(15)

The output matrix represented by the relation

Vo=Vc2 (16)

Mode 2: Applying Kirchhoff voltage law for the equivalent circuit of this mode, shown in figure (2.b), and by neglecting the voltage across the switch, one can get:

V_(L_1 )=V_in-i_1 R_1-V_1 (17)

? V?_(L_2 )=?-V?_2-i_2 R_2 (18)

By substituting these two equations in (3) and (4):

?di?_1/dt=M/? V_2+(-L_2)/? V_1+(MR_2)/? i_2+(-L_2 R_1)/? i_1+L_2/? V_in (19)

?di?_2/dt=(-L_1)/? V_2+M/? V_1+(-L_1 R_2)/? i_2+(MR_1)/? i_1+(-M)/? V_in (20)

Such as in the mode 1, the representation of the voltages across the each inductor can be produced by the following equations:

(21)

(22)

(23)

The output voltage is expressed by the same equation (16):

In order to write the general state space mathematical model for the overall modified Cuk converter system, let the state variable to be:

x1=i L1

x2= i L2

x3= vc1

X4=vc2

Applying the standard form of the state space representation: 7

? x?^’=Ax+Bu (24)

(25)

In order to find the system matrix (A) of the converter system, (Aon) is multiplied by the connection period (the switch is ON), and (Aoff) is multiplied by the disconnection period (the switch is Off), then summation the two matrices to find the overall modified Cuk converter system matrix,8 and as follows:

Doff = 1 – Don (26)

A = A1 * Don + A2 * Doff (27)

Genetic Algorithm

Genetic algorithm is searching technique that used as a general purpose, which used successfully for more types of computation application and optimal system problems. It proposed by John Holland, 1975 up to date. It follows by its operators the principles of development proposed by Darwin. Genetic algorithm applies selection, crossover, and mutation operators with the goal of finding the best solution to a problem. 9

1. Initial population: A random initial population of 50 individual, decimally encoded, has been used, each single individual with four genes, representing the produce four state feedback gains, (each individual gene represents a possible state feedback gain). The initial population forms a matrix of (50*4) dimensions. 10

A programming method for scaling the initial choice of the random individuals from the period (0 to 1) to any other period has been proposed. Most of the running cases were started with an initial population of values within the space

(-10) to (20).

2. Fitness function 1 (objective function): The principle of finding the integral absolute error (IAE) as a performance index has been used to leader the search of the genetic algorithm, and appearing to minimize this error during the selection and recombination of the coupled individuals during all the generations.

=

Where: N = number of samples.

Fig.3. the step response components of the fitness function

3. Selection: A selection technique used in this algorithm that called the modified roulette selection method, which mixed of the both selection methods of the weighted random selection 11 and the elitism selection methods, produced by Kenneth De Jong 1975. 12 The purpose of using a hybrid selection technique is approach to : 1) overcome the drawbacks of the roulette wheel 13 ( the wheel may stop at the divider of the worst possibility more than one time), 2) to overcome the drawback of using the selectiveness criteria with a very high selectiveness rate (steady state GA) which is the slow overshooting to notice and explore new search areas. So, the selection method which was used keeps the best 20% individuals as elite, and fee them forward to the next generation without any changing, and applied the roulette wheel selection on the remaining 80% of the population. 14

4. Crossover15-18: Because of the chosen individuals were decimally encoded, the “Arithmetic Crossover method” that used to recombine the individuals and form those of the next produced generation (offspring), and as follows:

Table 2. The elements of the modified dc-dc Cuk Converter.

Vs : Supply voltage 12 V R1 : Inductance resistor 0.01 ?

Vref : Reference Voltage 12V R2 : Inductance resistor 0.01 ?

L1 : First Inductance 18mH C1 : First Capacitance 200?F

L2 : Second Inductance 18 mH C2 : Second Capacitance 470?F

M : Mutual Inductance -1.6mH RL : Load resistance 2 ? 10 ? 45 ?

Offspring 1 = a* Parent 1 + (1 – a)* Parent 2 (29)

Offspring 2 = ( 1 – a )* Parent1 + a*Parent 2 (30)

Whereas the factor number selected randomly at each coupling. Also, the matrix establishing the initial population was divided vertically into two matrices, each of (50*2) dimensions, and made the crossover worker separately for each matrix, and then connecting the two matrices after the crossover operation. This technique can be used in order to simplify the ability of choosing the limits of the initial population values for the similar mechanisms separately. Because that the effect of the similar components is significantly similar, that we chose to started each two alike components from their proper search space. Many crossover rates were tried, and the main rate used was 100% of the recombined individuals, (there was an elite of 20%).

5. Mutation: The mathematics mutation used in this algorithm. Also because of the decimal encoding of the each individuals, and produced as shown:

Mutated Individual 1 = Mutated Individual 1 + a (31)

Mutated Individual 2 = Mutated Individual 1 – b (32)

Whereas (a) and (b) are the random generated values.

Many rates of mutation were tested in the algorithm, and finally the rate of (3.5 %) was used. It noticed that the many higher mutation rates produced not useful performances.

6. Stopping criteria: it is containing a multi-condition stop criteria, which has been used in the genetic algorithm, and presented as shown below.

a. End if at sufficing three of the following four limitations:

(1) Steady state error=0.

(2) Peak over shoot