Proceedings of ICISIP - 2005

DESIGN OF FUZZY SLIDING MODE CONTROL FOR DC-DC CONVERTER

S.Arulselvi I.C.Ramesh Kumar�. G.Uma3 and M.Chidambaram4 U&3Dept. a/Electrical Engg.,A.nna University Chennai, India, arulselvi_lk3@yahoo.co.in1

'Dept. a/Chemical Engg., Indian Institute a/Technology Chennai, India.

Abstract

This paper presents a variable structure type Sliding Mode Control (S�\fC) under current mode control/or dc-dc boost ,-"pe converter haVing non-minimum phase chamferistic. The robustness 0/ the proposed conh'oller is verified/or load and line variations. To reduce the chaffering and to improve stability, a Fuzzy Sliding Mode Control (FS�\JC) is designed. .i detailed derivation of cOn/rot law is presented by using Flippov's method and LyapllrlO\." method/or SMC and FS,il1C respectively. Simulations we carried alit !Ising AIA.TLABI SIAlfJLINf.,: scjiware. The simulation restllts are presented

Key Words: Sliding Mode Control (S,HC); Fuzzy Sliding Mode Control (FS�\fC) Switch Mode Power :)'Upp(v r5i\1PSJ;

Pulse Widfh Modulation (P JrU): Flippov 's Jllethod; Lyapllnov Method.

1. INTRODlJCTION Switch-Mode Po\\,\.:r Supplics (SMPS) are non-linear and time-varying systems, and thus the design of high performance control is usuall\' a challenging is sue. The control should emure s�'stelll stability in any operating condition with good, stalIc <lnd dynamic pcrfonnances in tenns of rejection of input voltage disturbances and load

changes. Thesc charactenstics should be maintained even Wlder large input voltage <lnd load current changes under parameter variation (robustness). Two methods of control impkmented for dc-de converters are Voltage mode control and CUTTcnt mode control [1). Voltage mode control is designed hased on classical control approach and uses state space-averaging mcthod. This method derives an equivalent model by circuit-av\.:mging all the system variables in a switching period [2]. From the average model, a suitable sl1wll signa l model 15 deri\'ed by perturbation and linearizatlon around a precis\.: operating point. Control tec1mique dcsigned based lln this model, does not produce satisfactory rcsults for widc variation of system parameters. To over com\.: this, 111ultiloop control teclmiques, such as current mode control has been implemented to improve powcr converter behaviour [21-[ 3 J. Current mode control consists of tlVO loops. The outer voltage loop is designed , based on dassical control technique and the inner current control is designed based on PWM or SMC. The sliding

mode approach for variable structure s�'stel11 [3]-[5] offers an alternative way to implement a control action that exploits the inherent variable structure nature of SMPS. [n particular. the converter switches are driven as a function of the instantaneous values of the state yariahles to force the svstel11 trajectory to stay on a suitable selected surfac\.: on the phase space. This control technique offers converter stability even for large supply and load \'arialions, robustness, good dynamic response and implementation is simple.

In this paper an attempt is made to design a (i) sliding mode control boundary layer and Oi) fuzzy sliding mode control a non-minimum phase boost type dc - dc converter. For a detailed derivation of sliding mode and FSMC are presented. The outer loop controlkr is designed based on Ziegler Nichol's technique [7]. The dosed loop current mode control block diagram is shown 1I1 Fig.l .

Sl\IC! FS� ... [C

u

Fig. 1: Block diagram of the do.cd loop current mode contral for boost

converter

Simulation results for SMC ar\.: presented for load and line variations. Comparative study is l1lmle betwecn SMC and FSMC. It is observed that chattering. is reduced and stability is improved with a FSMC.

2. CIRCUIT DESCRIPTION"

The boost-type dc-dc s\\itching converter operating. in the continuous conduction mode is sho\\'n in Fig.2. The dynamics of this converter is analyscd using state variable technique.

0-7803-8840-2105/$20.00 ©2005 IEEE 217

Proceedings of IeISIP - 2005

Here iL (t) and v c (t; are chosen as the state variables [8]. When the switch is on, the dynamics of the inductor current

lie '.1. Rial . � c

+1

Fig 2. Boost -type de-de switching converter

iL (1) and the capacitor voltage \'c (t) are

diL It) I . --=-(V -I It)R ) . dt L III L . L

dvc(t) = -VO dt CRO

v.

�)

(I)

(2)

When the switch is off, the dvnamics equations are

diL It) _ I ----(V. -ILlORi -yO) dt L m ,

dvc(t) I .. Vo --=-(lL(1)--) dt C RO

(3)

(4)

In both cases, capacitor voltage Vc(t) and output voltage Vo are related via the following equation:

, dvc (0 . Vo = RcC -- + \'c (t) (5)

'dt The simulink model of the boost converter is developed using equatIOn (1 )-(5). The nominal converter parameters are given in Table 1.

TABLE 1: CONVERTER PARAMETER

Vm VA L RL C � Ro Fs Input Output l\lilli Voltage Voltage Henry Q ftF Q Q kHz

10 20 170 0.1 1000 0.25 100 Ino

Thl;: load line characteristics of boost converter are non linear [3]. Hence the voltage mode control fails to give satisfactory results for wide range of operation. In this paper design of current mode control is discussed.

3. DERN ATION OF SLIDING MODE CONTOL LAW The effectiveness of the sliding mode control for boost

converter is verified by considering the following transfer function.

G (s) =

VO(s) =

58823.5(1-.00068s) d(s) s

2 +1.7s+1471

(6)

The phase variable canonical foml derived from (6) is given by

[0 A­-1471

For boost converter XI = V" and X 2 = i L . are considered. Similarly the errors el and ez are given by

e =V f--VO I re

e 2 =iL,ref -iL

(7)

(8)

The sliding surface CJ is defined in the error space. It is a function of the tracking error between the plant state and the desired, bounded, reference signal. The sliding surface is given by

(j = G e (9)

During sliding mode, using the principle of sliding mode the following conditions are satisfied

(j = 0 and d o/dt = G (de/dO =0 (10)

Substituting for error in tenns of reference signal a�d t�e plant states into G(de/dt)= 0 ,the Filippov's average ot sWltch controlr9], u.q is given by

. -I [ ] ueq = (G8) G Yd -AYd +Ae (ll)

Substituting this control law into

(12)

will give the error dynamics as given in (13) and (14):

218

e=�'d

-AYd

+Ae-B(GBflG[�'d-Avd +Ae: (13)

e= [ l-B(GB)-I

G }�'d-AY'd +Ae) (14)

When sliding mode occurs on cr "'Ge =0, the variable stmctme system is insensitive to any disturbances. Therefore

with the assumption that Y d and AYdare distmbance to the

system, the invariance condition is derived from (14) as

[1-B(GBf1G }'d =0

and [1-13(GB)-IGJ AVd = 0 ( 15) It can be seen that the perfonnance characteristics of the

t:ITor tn�iectory in (14) can be mel bv suitably choosing the eigen values of Acq in (15)

(16)

The matrix G is derived such that the eigen values of Aeq have certain negative real \·alw:s. This would guarantee the asymptotic convergence of error to zero at the desired rate. The sliding mode function cr is obtained from G matrix.

The matrix A,q is chosen to satisfy (16) and it is given by

l-I

A -eq - 0

o ] -0.000001

From equation (16), thL; lllatrix G can be obtained as

G = [0.000001 I]

The control law is of the limn

u = M sgn (cr) y

where y = Xl

sgn (cr) = {. I for cr > 0.01 o for 0'< - 0.01

Hence the sliding surface cr is given by

where G I = (JOOOOO L 02 = I and M = 1.

(17)

(18)

(19)

The control law for SMC is obtained by referring (17) to (19) and it is given by

4. SIl\WLATION ANDRESVLTS

Proceedings of fCISIP - 2005

The Fig.3 shows s!mulink model for implementing-variable stru��ture Sli�'������-J-';�

I I'ill.'

. '."'-- . ...

. , .. l��_ .>'.' II

.. �

._. .... _:--1. .--

""-I Y� I i� .. -§., �J:J T �n. ·- �··II"-" , .. _.

Fig':\: Simulink model implementing Sliding mode control

The regulated' output voltage and inductor current wavcfoffil for a step change in reference voltage from 20 to 25 volts applied at 1.5 seconds and 25 to 20 volts applied at 3 seconds are shown in FigS The corresponding sliding function cr and switching pulse are shown in Fig.6 and Fig.7

respectively. It is observed thaC when a is within the hOUl1dary layer ± .01, switching pulse of constant 'ON' time is g<:nerateu. But when a exceeus the positive bounuury, the switch is fully ON until it rcaches -.0 I and vice versa for negative bOUl1dary. Also it is ohserved that the controller tracks the output with zero steady state error with a settling time of 1.5 second. The dynamic perftmnance is also studied for load variation and supply voltage disturbance. The results are shown in Figures 8 and 9 respective1�'. The state phase trajectory of inductor current and capacitor voltage is shown in Fig. 10. It is observed that controller is immune to supp ly , load variations and parameter variations by producing constant switching pulses but with chattering.

5, FUZZY SLIDING MOD�: CONTROL

The draw back of sliding mode control is produces it chattering in the manipulated variable. This can be reduced by using bOUl1dary la�'ers In this case, the hitting phase control forces the states into the region bounded with the bounding layers and not onto the sliding surface. To overcome ·the difficulties and to speed up the response a fuzzy sliding mode control is designed for the boost converter and the results are compared with the SMC.

The input of the fuzzy logic sliding mode controller arc a(k) and 6cr(k) and the output of the fuzz�' sliding mode controller is change in duty ratio Ll.U(k) The range of cr(k),

L\O'(k) and 6u(k) are taken as ±O.5, ±.05 and 0 to I respectively. The cr(k) and Ll.cr(k) are fuzzified with gaussian member with 5 labels as shown in Fig. 4.a. The membership fUl1ction for 6u(k) is divided into 5 labels using triangular membership function as shown in Fig. 4.b.[ lOJ-[lI]. The fuzzy sliding control is implemented using lllamdani type inference"

(20) 219

Proteedings of ICISIP - 2005

t fl

05 IJ ��(k) OS

-->C!.a (k)

Fig 4.a : Membership function for ark) and 6.a(k)

J.l

Fig .tb: to.!embership function for ,1u(kl

The rule table can be established using Lyapunov function as given bv

Equation (21) implies that the initial state of the control 3ystem is nol (m the 3liding surface. The rate of increase of

V[cr (K)] is defined as follows:

t.vrcr(K)]=Vlcr(k+1 )]- Vlcr(k)] (22)

= cr(k) t.cr(k) + 'is t.a(ki

where c,cr (K) '" cr(k+ I ) - a (k) (23) If the condition t.. V[ cr(k)] < U is satisfied and cr(k) = 0 is a stable sliding surface, any state trajectory will be driven towards the origin of the state space, along the sliding surface.

Hence, a fuzzy logic contr ol law is developed in an attemp t to accomplish the above objective. The rule for ith iteration is given by

if (J(k) is PB and t..a(k) is PB then c,u(k) = PB (24) T . .lJlLE 2: THE FUZZY RULE TABLE

'icr NB NS Z PS PB u NB NB NB NS Z 'PS NS NB NS Z PS PS Z NS Z PS PS PB PS Z PS PS PB PB PB PS PS PB PB PB

The final control output is given by

u(k)= uCk-l) + c,u(k)

wh�re M=u(k)

(25)

(26)

This output is used to control the duty ratio of the switching pulse. Simulation is carried out using Fig.3 i mplementing the FSMC. It will be proved that better result is produced using FSMC in tenns or reduced chattering and settling time.

The regulated output voltage and inductor current waveform for a step change in r eference voltage from 20 to 25 volts applied at 1.5 seconds and 25 to 20 volts applied at 3 seconds are shQ\.\TI in Fig.II. The corresponding sliding

function (J, switching pulse and FSMC output sho",m in Fig. 12 and Fig.13 respectively. It IS ohsClycd that, wht:n cr IS within the boundary layer ± .0 I, switch pulse of constant

'ON' time is generated. But when (J e:...:ceeds the positive boundary, the switch is fully ON until it reaches -.01 and vice versa for negative boundary. The value of gain M is varied from 0 to I depends on a and L\cr. From Fig. 12 and 13. It is observed that, if cr is less than 0.1 the gain value M is . 3 and when cr is greater tlk'ln 0.1 M is arounu 0.65 when cr is with the ± 0.1 M is maintained at 0.7. Also it is observed that the controller tracks the output with zcro steady stale error with a settling time of 1.5 second. The dynamic perfonllance is also studied for load variation and supply voltage dis turbance. The results are shovill in Figures 14 and 15 respcctivel�'. It is observed that controller is imnllme to supply_ load and parameter variations bv producing constant switching pulses at steady state with reduced chattering. compared to SMC.

6. CONCLUSION

In this p aper a Sliding Mode Controller and FSMC are developed and verified in simulation for DC - DC Boost converter, It is observed that SMC is robust to supply, load and parameter variation. It produces zero steudy state with more chattering. But FSMC produces the smne results with less chattering. It is concluded that FSMC is better choice for voltage regulation of boost converter with parameter variation, load and supply disturbances.

220

o

REFERENCES [1] Ned Mohan and M.Underland, Power Electronics

Converters, Application and Design, John Wiley and Sons, 1995.

[2] Timoth�' LSKVaranina, Power Electronics Hand Book, CRC Press LLC, 2000.

[3] Muhammad H.Rashid, Powcr Elcctronics Handbook, Acadamic Press, 200 I.

[4] M.Aluned, M.Kuislul1a, Implementing sliding mode Control for Buck Converter, IEEE, 2003.

[5] Pradeep K.Nandam, p.c.Sen, "Industrial Applications of Sliding Mode Control", IEEE, ] 995.

[6] P.Mattavelli, L.Rossetto, "General-Purpose-Sliding­Mode controller for DCIDC converter Application 's", IEEE, 1993.

[7] S.Arulselvi, G.Uma and M.Chidambaram, "Design of PID controller based on IMC, Z-N, Luyben and Synthesis methods for Boost Converter with RHS z"ero",3,d International Conference on Upcoming Engineers, Toronto, Canada, May 13-14,2004.

[81 Juing-huieSu, Jian Chen, "Learning. Feedback Controller Design of SWItching Converter via MATLAB/SlMULINK " , IEEE, 2000.

[9J A.Davari, ADonthi, "Stable Robust Tracking of Non­Minimum phase systems". IEEE, 1998.

[101 CL.H. Wang, F.Y Sung, "Improved Fuzzy Sliding­Mode Control for a linear Servo motor System", 1. Control Engineering Practice. Vol. No.2, pp 219-227, 1997.

[I I] S.Arulselvi, A.Biju and G.Uma;'Design ilnd Simulation of Fuzzy Logic Controller for a Constant Frequency Quasi-Resonant DC-DC Converter", IEEE ICISIP, Chennai, India, pp.472-476, 2004.

Results for SMC

:�rr;rL; � "-__ -__ ':=�;��:_I_L.i-L I I : I : I I " ""-r . j ; I f· 1" 1 :+--..--1--.--1 , I, ! .1 cIo

Time( seconds)---?-Fig. 5. Wavefonns of aJ Regulated Output Voltage bJ inductor current For hoost converter with reference output voltage change 20 to 25 volts effected at 1.5 second and 25 to 20 volts effected M:' seconds.

221

-I" I ." ... - . ..;.

�'-.Io'�+--.Io----.-I�.������ Time( seconds >-7

Proceedings of ICISIP - 2005

Fig. 6. Wavefonns of a) Switching pulse and b) Sliding function (a) for reference voltage change 25 to 20 Volts effected at 3 seconds.

Time(seconds) �

Fig. 7. WavefolUls of oj SW.itching pulse and b) Sliding Function (a) for reference volt.lge lI'om 20 to 25Vnlts applied 1.5 seconds

Time (Seconds) ---?-Fig. 8. Regulated output voltage for lO%for [Dad variation (From 1000

to 900 applied at 2 seconds and settles within 1 second.

" 00 £l o •• ;. '3 c. g

Time (seconds) ---?-

Fig. 9. Regulated output voltage for 10" of or supply " oltage disturbance (from \0 to 11 Volts applied at 2 seconds and settles within 1 second.

Proceedings of ICISIP - 2005

'" .:l 0.. e � .,� C I .. ' � u� " v '''1 ..

c

;:; . "[ " ,� '"" .5 .I

Capacitor voltage (Volts) �

Fig. 10. Phase plane trajectory of boost converter for Operating Condition Xd (for20v. 4ampsj.

Results for FSMC

-r I !

rrrl I I--I

Time (�econds) ----7

::rl 'r 1'-I.,

'-, :. ;, ' . .

Time (seconds)

Fig.ll. Wavefonns of a) Regulated Output Voltage b) inductor current for boost converto:r with reference output "oltage change 20 to 2S volts effected

.1 1.5 second and 25 to 20 volts effected at 3 seconds.

Fig.l2. Waveforms of a) Fuzzy output and b) Sliding function (0) 0) Switching pulse for reference voltage change

2S tl} 20 Volts effected at 3 seconds.

222

o

Fig.B, Waveforms of a) Fuzzy output and b) Sliding function (a) 0) Switching pulse for reference voltage change 2S to 20 Volts effected at

1.5 secl}nds.

�

"If u or> .f! 0 > ,·1 "

"11 c:>.. " 0 "V

I:.

Time (seconds)

Fig. 14. Regulated output voltage for lO%for supply variation (From lOon to 900 applied at 2 second. and s�ttle. within 1 second.

Time (seconds)--7

Fig. IS. Regulated output voltage for lO%for load voltage disturbance (from 10 to 11 Volts applied at 2 seconds and seltles within 1 second.