Supervisory intelligent control system design for forward DC-DC converters

Supervisory intelligent control system design for

forward DC–DC converters

C.-F. Hsu, C.-M. Lin and K.-H. Cheng

Abstract: A supervisory intelligent control system is developed. The supervisory intelligent control system is comprised of a neural controller and a supervisory controller. The neural controller is investigated to mimic an ideal controller and the supervisory controller is designed to compensate for the approximation error between the neural controller and the ideal controller. In the proposed control scheme, an online parameter training methodology is developed based on the gradient descent method and the Lyapunov stability theorem, so that the control system can guarantee system stability. Finally, to investigate the effectiveness of the proposed control scheme, it is applied to control a forward DC–DC converter. A comparison between a PI controller, a fuzzy controller, a fuzzy neural network controller and the supervisory intelligent controller is made. Experimental results show that the proposed control system can achieve favourable regulation performances even for different input voltages and under load resistance variations.

1 Introduction

DC–DC converters are power electronic systems that convert one level of electrical voltage into another level by switching action [1, 2]. They can be used extensively in personal computers, computer peripherals and adapters of consumer electronic devices to provide DC voltages. From the control viewpoint, the controller design of the DC–DC converter is an intriguing issue owing to its intrinsic nonlinearity, which must cope with a wide input voltage and load resistance variations to ensure stability in any operating condition while providing fast transient response. For many years, the controller design was limited to PI control[3–5]. The selection of the controller parameters is a tradeoff between robustness and fast transient response. In general, it induces an overshoot in output voltage as the rise time of response is reduced. Recently, several approaches have been addressed by using sliding-mode control techniques[6–8]and fuzzy control techniques[7, 9, 10]for DC–DC converters. However, most of these approaches require time-consuming trial-and-error tuning procedures to achieve satisfactory performance; some of them cannot achieve satisfactory performance under change of operating point; and some of them do not give the stability analysis. The neural-network-based control technique has repre-sented an alternative design method for identification and control of these systems[11–15]. The successful key element is the approximation ability, where the parameterised neural network can approximate the unknown system dynamics of the ideal controller after learning. Recently, the concept of incorporating fuzzy logic into a neural network has grown

into a popular research topic[11]. The fuzzy neural network possesses the advantages of both fuzzy systems and neural networks since it combines fuzzy reasoning capability and neural network online learning capability. The fuzzy neural network has been widely adopted for control of complex dynamical systems owing to its fast learning property and good generalisation capability compared with the neural network[16–20]. These online learning algorithms are based on the gradient descent method [16, 18], the Lyapunov stability theorem[17, 19], and the genetic algorithm[20]. So the stability, convergence and robustness of the neural-network-based control system can be improved. For real-time applications, the basic issue of neural-network-based control techniques is to provide an online learning algorithm that does not require preliminary offline tuning.

The motivation of this paper is to design a supervisory intelligent control system using the fuzzy neural network approach and the Lyapunov stability technique for the DC–DC converter. In the supervisory intelligent control system, a neural controller is utilised as a main controller, in which the interconnection weights of the fuzzy neural network are tuned online in the sense of the gradient descent method; and a supervisory controller is designed to guarantee the system stability in the sense of the Lyapunov stability theorem. Finally, to investigate the effectiveness of the proposed supervisory intelligent control scheme, it is applied to control a forward DC–DC converter. A comparison between a PI controller, a fuzzy controller, a fuzzy neural network controller and the proposed super-visory intelligent controller is made. Experimental results show that the proposed control algorithm can achieve favourable responses, including fast rise time and settling time, and small overshoot even for different input voltages and under load resistance variations. Thus, the supervisory intelligent control is more suitable to control forward DC–DC converters since a self-learning scheme is applied.

2 PWM DC–DC converter

The switch-mode DC–DC converter can convert one level of electrical voltage into another level by switching action. Nowadays, it is very popular because of its high efficiency and small size. In switch-mode DC–DC converters, power

E-mail: fei@cn.nctu.edu.tw

C.-F. Hsu is with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, 300, Taiwan, Republic of China C.-M. Lin is with the Department of Electrical Engineering, Yuan-Ze University, Chung-Li, Tao-Yuan, 320, Taiwan, Republic of China

K.-H. Cheng is with the Department of Electrical Engineering, Chang Gung University, Tao-Yuan, 333, Taiwan, Republic of China

rThe Institution of Engineering and Technology 2006 IEE Proceedings online no. 20050376

doi:10.1049/ip-epa:20050376

switches cut off the load current within the turn-on and turn-off times under switching conditions. The output voltage is controlled by adjusting the on time of the power switch, which in turn adjusts the width of a voltage pulse at the output. This is known as pulse-width modulator ( PWM) control, where the switch frequency is constant and the duty cycle, d(N), varies with load resistance variations at the Nth sampling time. The output of the designed controller, dd(N), is the change of the duty cycle. Then, the duty cycle is determined by adding the change of duty cycle dd(N) to the previous duty cycle d(N
1), i.e.

dðN Þ ¼ dðN
1Þ þ ddðN Þ ð1Þ

This duty cycle signal is then sent to a PWM output stage that generates the appropriate switching pattern for the switch in the DC–DC converter. In this paper, a widely used forward DC–DC converter is discussed and is shown in Fig. 1, where Viand Voare the input and output voltages

of the converter, respectively, and assume Viis assumed to

be a constant voltage; D1and D2are the diodes; L is the

inductor, C is the output capacitor; and Q is the transistor which controls the converter circuit operating in different modes. When the transistor is on, Vi appears across the

primary and then generates Vx¼

Nm NP

ðVi
VlostÞ ð2Þ

where Vlostis the voltage drop occurring by transistor and

diodes, and represents the unmodelled dynamics in practical applications. The diode D1on the secondary ensures that

only positive voltages are applied to the output circuit while diode D2provides a circulating path for inductor current if

the transformer voltage is zero or negative. By the averaging method, the output voltage can be expressed as[2]

VoðN Þ ¼ Nm NP

ðVi
VlostÞdðN Þ ð3Þ

where NPis the turns of the primary power winding and Nm

is the turns of the slave power winding. The control problem of the forward DC–DC converter is to control the duty cycle so that the output voltage Vo(N) can provide a

fixed voltage under the occurrence of the uncertainties such as different input voltages and load resistance variations. The output error voltage is defined as

eðN Þ ¼ VoðN Þ
Vref ð4Þ

where Vrefis the reference output voltage. The control law

of the duty cycle is determined by the error voltage signal to provide fast transient response and small overshoot in the output voltage.

3 Supervisory intelligent controller design

The block diagram of the supervisory intelligent control for the power electronic system is shown in Fig. 2, in which the control law is taken as

ddsi¼ ddncþ ddsc ð5Þ

where the neural controller ddncis investigated to mimic an

ideal controller and the supervisory controller ddsc is

designed to compensate for the approximation error. The inputs of the neural controller are the output error voltage e and its derivative, and the input of the supervisory controller is the tracking index, which is defined as

s¼ e þ l Z t

0

edt ð6Þ

where l is a positive constant. Fig. 1 Forward DC–DC converter

nc d δ sc d δ + + + si d δ − + dt d i

V

V

V

e

V

supervisory intelligent control DC-DC dt d ) ( ) 1 ( ) (N d N d N d = − +δ si update laws (26), (27), (28) (14) controller (34) bound law (43) tracking index (6) + s nc d δ sc d δ + + + si d δ − + dt d i

V

V

V

e

V

supervisory intelligent control DC-DC dt d ) ( ) 1 ( ) (N d N d N d = − +δ si update laws (26), (27), (28) (14) control (34) bound estimation law (43) tracking index (6) + s neural controller supervisory converter system

3.1

Description of neural controller

A four-layer fuzzy neural network is shown in Fig. 3, which comprises the input (the i layer), membership (the j layer), rule (the k layer), and output (the o layer) layers. Layer 1 accepts the input variables. Layer 2 is used to calculate the Gaussian membership values. The nodes of layer 3 represent the fuzzy rules. The links before layer 3 represent the preconditions of the rules, and the links after layer 3 represent the consequences of the rule nodes. Layer 4 is the output layer. The node in this layer is the output of the fuzzy neural network. The interactions for the layers are given as follows[11, 16]:

Layer 1, Input layer: For every node i in this layer, the net input and the net output are represented as

net1_i ¼ x1 i ð7Þ y_i1¼ f1 iðnet 1 iÞ ¼ net 1 i; i¼ 1; 2 ð8Þ where x1

i represents the i-th input to the node of layer 1.

Layer 2, Membership layer: In this layer, each node performs a membership function and acts as an element for membership degree calculation, where the Gaussian function is adopted as the membership function. For the jth node net2_j ¼
ðx 2 i
m2ijÞ 2 ðs2 ijÞ 2 ð9Þ y_j2¼ fj2ðnet 2 jÞ ¼ expðnet 2 jÞ; j¼ 1; 2; . . . ; l ð10Þ where m2

ij and s2ij are the mean and standard deviation of

the Gaussian function in the jth term of the ith input linguistic variable x2

i to the node of layer 2, respectively.

Layer 3, Rule layer: Each node k in this layer is denoted by P, which multiplies the incoming signals and outputs the result of the product. For the kth rule node

net3_k¼Y j w3_jkx3_j ð11Þ y_k3¼ fk3ðnet 3 kÞ ¼ net 3 k; k¼ 1; 2; . . . ; n ð12Þ where x3

j represents the jth input to the node of layer 3, and w3

jk are the weights between the membership layer and the

rule layer, which are assumed to be unity.

Layer 4, Output layer: The single node o in this layer is labelled as S, which computes the overall output as the summation of all incoming signals:

net_o4¼X k w4_kx4_k ð13Þ y_o4¼ fo4ðnet 4 oÞ ¼ net 4 o ð14Þ

where the link weight w4

k is the output action strength

associated with the kth rule, x4k represents the kth input to

the node of layer 4, and y4

ois the output of the fuzzy neural

network.

3.2

Online learning algorithm

The central part of the learning algorithm for the neural controller concerns how to recursively obtain a gradient vector, which is defined as the derivative of an energy function with respect to a parameter of the neural network using the chain rule [10]. To describe the online learning algorithm of the fuzzy neural network, first the energy functionEis defined as:

E¼1 2eðN Þ

2 _ð15Þ

Then, the learning algorithm based on the gradient descent method is described below[11, 16].

Layer 4: d4o¼
@E @net4 o ¼
@E @y4 o @y_o4 @net4 o ð16Þ The Jacobian term of the plant, @E=@y4

o, can be expressed as @E @y4 o ¼ @1 2ðV0ðN Þ
VrefÞ 2 @ddncðN Þ ¼ eðN Þ @VoðN Þ @ddncðN Þ ¼Nm NP ðVi
VlostÞeðN Þ @dðN Þ @ddncðN Þ ¼Nm NP ðVi
VlostÞeðN Þ @ðdðN
1Þ þ @dsiðN ÞÞ @ddncðN Þ ¼Nm NP ðVi
VlostÞeðN Þ ð17Þ Layer 1

Π

Layer 2 Layer 3 Layer 4

1 x 2 x

Σ

Π

Π

Π

Π

Layer 2 Layer 3 Layer 4

1 x 2 x

Σ

Π

Π

Π

Π

and the weight is updated by an amount Dw4_k¼
Zw @E @w4 k ¼
Zw @E @y4 o @y4 o @net4 o
 _@net4 o @w4 k
 ¼
Zwd4ox 4 k ð18Þ

where Zwis the learning rate of the connecting weights of the

fuzzy neural network. The weights of the output layer are updated according to the following:

w4_kðN þ 1Þ ¼ w4

kðN Þ þ Dw4k ð19Þ

Layer 3: Since the weights in this layer are unities, only the error term needs to be calculated and propagated:

d3_k¼
@E @net3 k ¼
@E @y4 o @y4 o @net4 o

@net 4 o @y3 k @y3 k @net3 k
 ¼ d4ow 4 k ð20Þ

Layer 2: The multiplication operation is done in this layer. The error term is computed as follows:

d2j ¼
@E @net_j2¼
@E @y4 o @y_o4 @net4 o @net4_o @y3_k @y3_k @net3_k
  @net 3 k @y2 i @y2_j @net2 j " # ¼ d4o X k w4_ky_k3 ð21Þ and the update law of m2

ij is Dm2_ij ¼
Zm @E @m2 ij ¼
Zm @E @y4 o @y4 o @net4 o @net4 o @y3 k @y_k3 @net3 k @net3_k @y2 j @y3 j @net2 j @net2 j @m2 ij " # ¼
Zmd4o X k w4_ky_k32ðx 2 i
m2ijÞ ðs2 ijÞ 2 ð22Þ

where Zmis the learning rate of the mean. The update law of

s2 ij is Ds2ij¼
Zs @E @s2 ij ¼
Zs @E @y4 o @y_o4 @net4 o @net4_o @y3 k @y3 k @net3 k @net3 k @y2 j @y_j2 @net2 j @net2_j @s2 ij " # ¼
Zsd4o X k w4_ky3_k2ðx 2 i
m2ijÞ 2 ðs2 ijÞ 3 ð23Þ

where Zsis the learning rate of the standard deviation. The

mean and standard deviation of the hidden layer are updated as follows: m2_ijðN þ 1Þ ¼ m2 ijðN Þ þ Dm2ij ð24Þ s2_ijðN þ 1Þ ¼ s2ijðN Þ þ Ds 2 ij ð25Þ

Since NP, Nm, Vi and Vlost in (17) are unavailable, these

parameters in the learning algorithms can be reorganised as positive constants in practical applications. Therefore, the update laws (18), (22), and (23) can be reconstructed as follows: Dw4_k¼
Z0wex 4 k ð26Þ Dm4_ij¼
Z0me X k w4_ky3_k2ðx 2 i
m2ijÞ ðs2 ijÞ 2 ð27Þ Ds2ij¼
Z 0 se X k w4_ky_k32ðx 2 i
m2ijÞ 2 ðs2 ijÞ 3 ð28Þ where Z0 w¼ Zw Nm NPðVi
VlostÞ, Z 0 m¼ Zm Nm NPðVi
VlostÞ and Z0_s¼ ZsNNmPðVi
VlostÞ. Z 0

w, Z0m and Z0s can be taken as the

new learning rates.

3.3

Supervisory controller

Differentiating both sides of (3) with respect to time yields _

Vo¼ Nm NP

ðVi
VlostÞdd ð29Þ

If the parameters of the converter are well known and the external disturbance is measurable, an ideal controller can be obtained as[21]

dd ¼ NP

NmðVi
VlostÞ

ð _Vref
leÞ ð30Þ

Substituting (30) into (29), gives

_eþ le ¼ 0 ð31Þ

Since l is a positive constant, it implies that lim

t!1e¼ 0.

However, since the system parameters and the voltage drop may be unknown or perturbed, the ideal controller cannot be implemented. To tackle this problem, a neural network is utilised to approximate this ideal controller. By the universal approximation theorem, there exists an optimal fuzzy neural network such that[22]

ddncðwÞ
dd¼ e ð32Þ where w _{¼ ½w}4 k m2  ij s2  ij T

is the ideal weight vector of the neural controller, and e denotes the approximation error and is assumed to be bounded by 0 jej  E, where E is a positive constant. The error bound is assumed to be a constant during the observation; however, it is difficult to measure it in practical applications. Therefore, a bound estimation is developed to observe the bound of the approximation error. Define the estimation error of the bound

~

E¼ E
^E ð33Þ

where ^E is the estimated error bound. The supervisory controller is designed to compensate for the effect of approximation error and is chosen as

ddsc¼
^EsgnðsÞ ð34Þ

in which sgn(.) is a sign function. By substituting (5) into (29), it is can be shown that

_ Vo¼

Nm NP

ðVi
VlostÞðddncþ ddscÞ ð35Þ

After some straightforward manipulations, the error equation governing the system can be obtained through (6), (30) and (35) as follows:

_eþ le ¼Nm NP

ðVi
VlostÞðddncþ ddsc
ddÞ ¼ _s ð36Þ

Define a Lyapunov function as Vðs; ~EÞ ¼s 2 2 þ ~ E2 2ZE ð37Þ where the positive constant ZE is a learning rate.

(34) and (36), we obtain _ Vðs; ~EÞ ¼ sNm NP ðVi
VlostÞðe þ ddscÞ þ ~ EE_~ ZE ¼Nm NP ðVi
VlostÞðes þ ^EjsjÞ þ ~ EE_~ ZE ð38Þ If the bound estimation law is chosen as

_~E ¼
_^E ¼
ZE Nm NP ðVi
VlostÞjsj ð39Þ then (38) becomes _ Vðs; ~EÞ ¼Nm NP

ðVi
VlostÞðes
EjsjÞ

Nm NP

ðVi
VlostÞðjejjsj
EjsjÞ

¼
Nm NP

ðVi
VlostÞðE
jejÞjsj  0 ð40Þ

Since V_ðs; ~EÞ is negative semi-definite, that is VðsðtÞ; ~EðtÞÞ  V ðsð0Þ; ~Eð0ÞÞ, it implies that s and ~E are bounded. Let function ONm

NPðVi
VlostÞ ðE
jejÞs 

Nm

NPðVi
VlostÞ ðE
jejÞjsj 
_V ðs; ~EÞ, and integrate O with respect to time, then we obtain

Z t 0

OðtÞdt  V ðsð0Þ; ~Eð0ÞÞ
V ðsðtÞ; ~EðtÞÞ ð41Þ Because Vðsð0Þ; ~Eð0ÞÞ is bounded, and V ðsðtÞ; ~EðtÞÞ is nonincreasing and bounded, the following result can be obtained: lim t!1 Z t 0 OðtÞdto1 ð42Þ

Also, _OðtÞ is bounded, so by Barbalat’s Lemma [22], lim

t!1O¼ 0. That is, s-0 as t-N. Hence, the supervisory

intelligent control of power electronic systems is asympto-tically stable. Similarly, since NP, Nm, Viand Vlostin (39) are

unavailable in practical applications, the bound estimation law (39) can be reconstructed as follows:

_^E ¼ Z0

Ejsj ð43Þ

where Z0_E ¼ ZE Nm

NPðVi
VlostÞ. The design algorithms of the supervisory intelligent control are summarised as follows: Step 1: The output error voltage e and the tracking index s are given in (4) and (6), respectively.

Step 2: The value of neural controller ddncis the output of

the fuzzy neural network as given in (14).

Step 3: The supervisory controller ddscis given in (34) with

the parameter ^Eadapted by (43).

Step 4: The duty cycle is determined by adding the change of duty cycle to the previous duty cycle as shown in (1). Step 5: The duty cycle signal is then sent to a PWM output stage. Then, go back to Step 1.

4 Experimental results

The computer control experimental system for the forward DC–DC converter is shown in Fig. 4. A servo control card is installed in the control computer, which includes multi-channels of D/A, A/D, PIO and encoder interface circuits. The control problem is to control the duty cycle so that the output voltage can provide a fixed voltage (Vref¼ 10 V)

under the occurrence of uncertainties such as different input voltages and load resistance variations. The proposed control algorithm is realised on a Pentium processor using the ‘Turbo C’ language. Two experimental cases are addressed: (a) Case 1 (the input voltage is set as Vi¼ 20 V);

(b) Case 2 (the input voltage is set as Vi¼ 25 V). In both

cases, some load resistance variations with step changes are tested: (i) from 20 to 4 O at 300 ms, (ii) from 4 to 20 O at 500 ms, and (iii) from 20 to 4 O at 700 ms. The circuit parameter values of the forward DC–DC converter are chosen as NP: Nm¼ 4 : 3, R ¼ 20 O, L ¼ 500 mH and

C¼ 2200 mF. The converter runs at a switching frequency of 20 kHz and the controller runs at a sampling frequency of 1 kHz. The duty cycle is generated by a PWM IC SG1825. The generated duty cycle is directly proportional to the analogue output of the controller. To illustrate the effectiveness of the proposed design method, a comparison between a PI controller, a fuzzy controller, a fuzzy neural network controller and the proposed supervisory intelligent controller is made.

4.1

Comparison of different control method

ddpi¼
0:01e
0:2_e ð44Þ

It is a PD type for the change of duty cycle ddpi, therefore it

is a PI type for the duty cycle dpi. The experimental results

for the PI controller are shown in Fig. 5. For Case 1 and Case 2, the converter responses are shown in Figs. 5a and c; and the associated control efforts are shown in Figs. 5b and d, respectively. From the experimental results, the PI controller can achieve fast tracking performances; however, there exists 5% overshoot and the PI gains are determined through a lot of trials. Next, a fuzzy controller proposed in [9]is applied to the forward DC–DC converter. The fuzzy control rules are given in the following form:

Rule i: IF e is F_ei and _e is F_{_e}i; THEN ddfc is ri ð45Þ

where ri, i¼ 1, 2, y, n are the singleton control actions and

Fi

eand F_eiare the labels of the fuzzy sets. The defuzzification

of the controller output is accomplished by the method of centre-of-gravity: ddfcðN Þ ¼ Pn i¼1 vi ri Pn i¼1 vi ð46Þ Fig. 4 Experimental setup

where viis the firing weight of the ith rule. The fuzzy rules in

(45) can be constructed by the sense that e and _e will approach zero with fast rise time and without large overshoot. Generally, the determination of these rules comes from human knowledge and via some trial-and-error processes. In this sense, a 25 fuzzy rule system is summarised in Table 1, where the fuzzy labels are negative big (NB), negative small (NS), zero (ZO), positive small ( PS), and positive big ( PB). The experimental results of the

fuzzy controller for Case 1 and Case 2 are shown in Fig. 6. From the experimental results, the fuzzy controller can achieve fast tracking performance; however, the fuzzy rules base is constructed through much trial-and-error to ensure proper behaviour in the operating conditions. In the following, a fuzzy neural network controller proposed in [16] is applied to the forward DC–DC converter. The parameters of the fuzzy neural network controller are selected as Z0_w¼ Z0

m¼ Z 0

s¼ 0:001. These parameters are Fig. 5 Experimental results of PI controller

a, b Case 1 c, d Case 2

Fig. 6 Experimental results of fuzzy controller

a, b Case 1 c, d Case 2

Table 1: Fuzzy rules of fuzzy controller for power electronic system _ e e NB NS ZO PS PB NB 1.000 1.000 1.000 0.400 0.000 NS 1.000 1.000 0.400 0.000
0.400 ZO 1.000 0.400 0.000
0.400
1.000 PS 0.400 0.000
0.400
1.000
1.000 PB 0.000
0.400
1.000
1.000
1.000

chosen to achieve good transient control performance considering the requirement of stability. The experimental results of the fuzzy neural network controller for Case 1 and Case 2 are shown in Fig. 7. From the experimental results, the robust tracking performance of the fuzzy neural network controller is obvious under the occurrence of the load resistance variations after training. However, the initial transient response is not good.

4.2

Supervisory intelligent control

The proposed supervisory intelligent controller is applied to the forward DC–DC converter. The parameters of the proposed controller are selected as l¼ 1000, Z0

w¼ Z0m¼

Z0_s¼ 0:001 and Z0

e¼ 0:00001; the choice of these

para-meters is through some trials. If the learning rates are chosen too small, then the parameter convergence of the supervisory intelligent controller will be easily achieved;

Fig. 7 Experimental results of fuzzy neural network controller

a, b Case 1 c, d Case 2

however, this will result in slow learning speed. On the other hand, if the learning rates are chosen too large, then the learning speed will be fast; however, the super-visory intelligent controller system may become more unstable for the parameter convergence. The experimental results of the supervisory intelligent controller for Case 1 and Case 2 are shown in Fig. 8. Since the controller parameters are initialised from zero, the supervisory

intelligent controller has the drawback of large overshoot responses and control efforts at the initial learning phase. After training, the trained supervisory intelligent controller is applied to control the forward DC–DC converter system again. The experimental results of the trained supervisory intelligent controller for Case 1 and Case 2 are shown in Fig. 9. It is seen that the regulation performance of the trained supervisory intelligent controller

Fig. 8 Experimental results of supervisory intelligent controller

a, b Case 1 c, d Case 2

is further improved when the initial values of the controller parameters are trained. The comparisons of control performance and control characteristics for PI control, fuzzy control, fuzzy neural network control and supervisory intelligent control are summarised in Tables 2 and 3, respectively. It is seen that the supervisory intelligent control system has robust characteristics and fast transient response even for different input voltages and

under load resistance variations since the online learning scheme is applied. From the view point of computation time, the supervisory intelligent control and fuzzy neural network control will pay the price of more computation time than the fuzzy control and PI control for achieving better control performance. However, the increase of computation time is acceptable for real applications.

Fig. 9 Experimental results of trained supervisory intelligent controller

a, b Case 1 c, d Case 2

5 Conclusions

A PI control, a fuzzy control, a fuzzy neural network control and a supervisory intelligent control have been adopted to control a forward type DC–DC converter. The proposed supervisory intelligent control system comprises a neural controller and a supervisory controller, in which the controller parameters can be tuned online based on the gradient descent method and the Lyapunov stability theorem to achieve system stability and satisfactory performance. To illustrate the effectiveness of the proposed design method, several experiments have been performed. The experimental results demonstrate the efficiency of the proposed control method. Therefore, the proposed super-visory intelligent controller is suitable for DC–DC converter control.

6 Acknowledgment

The authors appreciate the financial support in part from the National Science Council of Republic of China under grant NSC 93-2213-E-155-038. The authors would like to express their gratitude to the reviewers for their valuable comments and suggestions.

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Table 2: Performance comparison

Controller Case 1 Case 2

Over-shoot (%) Settling time (ms) Over-shoot (%) Settling time (ms) PI controller 5 24 5 19 Fuzzy controller 0 38 0 34 Fuzzy neural network controller 25 54 30 56 Supervisory intelligent controller 20 48 22 44 Trained supervisory intelligent controller 0 21 0 19

Table 3: Characteristics comparison

Controller Controller parameters Load variation regulation ability Stability proof Computa-tion time (ms)

PI controller trial and error middle yes 0.180 Fuzzy controller trial and error good no 0.260 Fuzzy neural network controller online learning excellent no 0.295 Supervisory intelligent controller online learning excellent yes 0.298 Trained supervisory intelligent controller online learning excellent yes 0.298