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Application of Minimum-time Optimal Control 
System in Buck-Boost Bi-linear Converters 

 

Seyed Mohammad Shariatmadar 
Electrical Engineering Department 

Naragh Branch, Islamic Azad University 
Naragh, Iran 

Shariatmadar@iau-naragh.ac.ir 

Seyed Mohammad Javad Jafarian 
Electrical Engineering Department 

Shahed University 
Tehran, Iran  

mohammad.jafarian@gmail.com 
 

 

Abstract—In this study, the theory of minimum-time optimal 
control system in buck-boost bi-linear converters is described, so 
that output voltage regulation is carried out within minimum 
time.  For this purpose, the Pontryagin's Minimum Principle is 
applied to find optimal switching level applying minimum-time 
optimal control rules. The results revealed that by utilizing an 
optimal switching level instead of classical switching patterns, 
output voltage regulation will be carried out within minimum 
time. However, transient energy index of increased overvoltage 
significantly reduces in order to attain minimum time optimal 
control in reduced output load. The laboratory results were used 
in order to verify numerical simulations. 

Keywords-Buck-boost converter; sliding mode control; 
minimum-time control; bi-linear converter 

I. INTRODUCTION  
Several methods have been introduced to control DC-to-DC 

switching converters [1-7]. Common controllers employed in 
these converters include the PWM traditional controller based 
on current control or voltage control [1-2]. Design based on the 
averaging method from the state space based on the small 
signal method in these controllers will not bring about a proper 
dynamic performance in large-signal disturbances. Due to the 
non-linear nature of these converters, nonlinear control 
methods have been recommended to improve the stability and 
resistance properties of controller in these systems [4-6]. The 
sliding mode control technique is one of the modern non-linear 
methods utilized in variable structure systems such as DC-to-
DC switching converters [7-8]. Ease of implementation, proper 
dynamic performance and stability are achieved by the 
application of this type of nonlinear control technique. 
Nevertheless, non-fixed and high switching frequency, 
regulation error of steady state and complicated design are 
among the difficulties of implementing the sliding mode 
controller in DC-to-DC switching converters [9].  

Non-minimum phase systems encounter certain limitations. 
This feature in voltage control method leads to complex design 
and inadequate dynamic response to identify frequency 
traversing via resonance frequency [1]. Despite the failure to 
maintain system stability and large signal disturbances, the 
current control method has faster transient response when 

compared to voltage control in a similar system. Several 
methods have been proposed to control boost converter in 
sliding mode, and especially in the latter years in an effort to 
enhance their performance [9-16], including PWM constant 
frequency based on voltage or current control, adaptive sliding 
mode controller and dual integral sliding level [12-16]. 
Looking back on these methods, we can conclude that in the 
majority of these systems, system stability and performance 
have targeted steady state while static behavior of system have 
been considered by researchers. Nevertheless, several studies 
have selected non-linear switching level to enhance system 
transient performance under sliding mode controllers [17]. In 
these systems, despite an improvement in system behavior, in 
contrast to linear switching level, transient response of system 
in large-signal disturbances do not exhibit proper behavior.  

In this study, the Pontryagin's Minimum Principle in buck-
boost bilinear converters is investigated and regulations of 
minimum-time optimal control are derived. The optimized 
closed-loop controller works on output voltage regulation in 
each output disturbance. Simulation results reveal that the 
performance of buck-boost time optimal converters exhibited a 
better transient performance when compared to the sliding 
mode controller. 

II. INTRODUCTION TO SLIDING MODE CONTROLLER 
Variable nonlinear dynamic system with the timeline below 

is assumed as follows: 

1 2( ) ( ) ( ) ( ) ( )inx t A x t B x t u t B V u t    (1) 
where x(t),v u(t), Vin are mode vector, disturbance input and 

input voltage of system respectively. A, B1, B2 are also fixed 
matrices with appropriate dimensions. The initial conditions 
and balance points in system in tf

 
are assumed as follows: 

0(0)

( )f f

x x

x t x




 

(2) 

In addition, restrictions of control input are as follows: 

( )N Pu u t u   (3) 



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where uP>0 and uN≤0 are upper and lower limits of control 
inputs. In classical systems design of switching level, Ssm(x) is a 
linear combination of system status: 

( ) ( ( ) ( ))Tsm fS x x t x t   (4) 
Where β is the sliding coefficient vector of system. The 

provisions to reach switching level and sliding on switching 
level are two sufficient conditions for the implementation of 
sliding mode controller in one of the customized systems. The 
first provision is to examine the reaching of system trajectory 
to switching level which is achieved by one of the following 
conditions: 

  0   0

  0   0
P sm N sm

P sm N sm

If x S Then x S

If x S Then x S

   

     
(5) 

Where  xP and xN are steady state values of state vector with 
uP and uN 

control inputs. The second condition (sliding) will be 
provided using the following relationship: 

1 2 1 2
0

lim  ( , ) ( , )  0
sm

sm sm
S

S x x S x x



 (6) 

 Where
1 2( ( , ))smS x x

 are time function of switching level 
function. The sliding coefficients of β is selected in (4) in such 
a way that the condition of (6) will be established. According to 
(5) and (6), control rule of infinite switching frequency in 
sliding mode regime is as follows: 

    ( ) 0
( )

   ( ) 0     

p sm

N sm

u S x
u t

u S x


 

  
(7) 

 The stability of the ystem in second grade systems is 
provided by trajectory guidance of system towards a stable 
equilibrium point in sliding mode regime [7]. 

 

III. OPTIMAL CONTROL OF NON-OSCILLATORY DYNAMIC 
SYSTEMS. 

 
In minimum time problem, we focus on minimizing 

function as follows: 

0
( )  

ft
J u dt   (8) 

We suppose that in (1), the values of A, B1, B2 are as 
follows: 

1
1 2

2

0 0 0
, ,

0 0 1
A B B




     
       

 



    
(9)

 

Where a1 and a2 are real eigenvalues of system. The 
Hamiltonian function of systems (1) with constant values (9) in 
minimum-time optimal control system is as follows: 

      
       1 1 1 2 2 2 2

, ,

1 ( ) ( ) in

H x t t u t

t x t t x t V t u t



    



  
 

(10)
 

Where λ(t) is auxiliary variable of system status which is 
obtained as follows: 

 
 

 11
2 2

( )
0 0

0 1
int

t
u t V

t



 


    

    
    






 
(11) 

The optimal condition to apply Pontryagin's Minimum 
Principle as well as Hamiltonian’s Minimum function (10) 
when compared to u(t) limited control input could be obtained 
as follows: 

 
 
 

* max 2

min 2

0

0
u

u t
t

u t








  

(12)
 

Due to lack of freedom of primary values in status auxiliary 
variables, minimum-time optimal control command is 
impossible by this method. On the contrary, according to (1), 
(3) and (9), system status variables will be obtained as follows: 

    11 1 0 txx t e   (13) 
        2

*
*

2 2
2

0t x
u t

x t e u t


     (14) 

Where 
*

u  could be one and /or zero. Because of the nature 
of the bang-bang, u(t) command and constant control input, the 
fastest way to damped fluctuations of system is to place 
trajectory system on the trajectory leading to equilibrium point. 
So, the optimal switching level could be obtained by (12)-(13) 
as reverse time dynamic so that the dynamics of the system 
could be utilized to move towards the equilibrium point [17]. 
So, as a result of independent variables’ dynamics in the above 
equations, the optimal switching level could be obtained by 
eliminating time from (12)-(13) and expressing parameter 
changes in terms of each other as follows: 

   
 
      

12

1 1

* ln
*

2 2
2

fx t

x t
fx

u t
x t e u t t








   
 

(15) 

IV. OPTIMAL CONTROL OF FLUCTUATING DYNAMIC 
SYSTEMS 

It is assumed that A, B1, B2 values in (1) are as follows: 

1 2

0 0
, ,

0 1
A B B






     

             

  
(16) 

Where a and ω are real and imaginary eigenvalues of 
system. The Hamiltonian’s function in system (1) with constant 
values (15) in minimum-time optimal control system are as 
follows: 

      
   
       
1 1 1 2

2 2 2 1 2

, ,

1 ( ) ( )

( ) ( ) in

H x t t u t

t x t t x t

t x t t x t V t u t



   

    



  

  

 
(17) 

Where λ(t) is the auxiliary variable of system status which 
is obtained as follows: 

 
 

 1
2

( )
0

1
int

t
u t V

t






 


    

    
     

  
(18) 

 According to (16) and application of  Pontryagin's 
Minimum Principle as well as Hamiltonian’s Minimum 



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function (10) compared to  limited control input, (11)  will be 
repeated. Due to the nature of the bang-bang,  *u t command, 
constant control input and unlimited number of input control 
status changes, the minimum-time control trajectory for every 
control input status will be obtained separately: 


   

1

1 2

*
*

2

( )

(0) cos (0) sin( )

sin( )

t

n n

x t

e x t x t

u t
t u t

  


 

 





 


  



 

(19) 


   

2

2 1

*
*

2

( )

(0) cos( ) (0) sin( )

cos( ) ( )

t

n n

x t

e x t x t

u t
t u t

  


 

 





 


  



 

(20) 

While: 

2 2
n     (21) 

As earlier mentioned, minimum-time optimal control is 
obtained by positioning system variables on trajectory resulting 
from system dynamics and leading them to the point. In 
fluctuating dynamic systems, with regard to bang-bang 
property of u(t) control input, Hamiltonian’s Minimum 
function (16) for λ2(t) changes as well as dependence of λ2(t)  
status auxiliary function signal on ω frequency, in minimum-
time optimal function, u(t) control input for π/ω maximum time 
unit will be constant. So, minimum-time switching function to 
reach ultimate time or equilibrium point of system for lower 
π/ω  time unit (as reverse time) related to moving on trajectory 
will result from system dynamics.  

Thereafter, using the generalized results, minimum-time 
optimal control will be proved in bilinear systems.  

V. OPTIMAL CONTROL OF BILINEAR SYSTEMS 
Hamiltonian’s function (1) with limited control input in 

minimum-time optimal control is as follows: 

      
  1 2

, ,

1 ( ) ( ) ( )T in

H x t t u t

t A x t B x t u t B V







  
 

(22) 

Where λ(t)  is the auxiliary variable of system status which 
is defined as follows: 

1

1

( , , , , )
( )

 ( ) ( ) ( )

H x u v t
t

x

A t B t u t




 


 




  



 
(23) 

The optimal condition to apply Pontryagin's Minimum 
Principle and Hamiltonian’s Minimum Function when 
compared to limited control input could be obtained (17). 

In bilinear systems with bang-bang input, depending on the 
type of control subsystems in each control input (dynamic, 
fluctuating or dynamic fluctuating), the switching optimum 
level will be determined. 

Figure 1 illustrates the schematic of a buck-boost multiple 
area converter in nonlinear control of sliding mode. R1, R2 and 
Rl are leakage resistance of the inductor, capacitor and output 
load respectively, vi represents the input voltage, T1 and T2 
which are defined as u(t) in equations which are open and close 
at zero value respectively. In this converter, the status variables 
are defined as follows: 

 

Rl

iL(t)

1

-1

+
vout(t)

-
vout(t)Vref

iref

R2

T2

C

Switching surface

K2K1

+

_

+

_

Classical SMC:
k1=β1
k2=β2

Optimal SMC:
    k1=Vout /Vp

k2=1
R3= R1/(R1-1)

Vi

R1

R3

L

R1

T2
+

-

 
Fig. 1.  Buck-boost multiple area converter schematic 

VI. MINIMUM-TIME OPTIMAL CONTROL IN BUCK-BOOST 
MULTIPLE AREA CONVERTER 

1

2

1
0

( )( )

( )( ) 1
0

o

li

i

L
i tx t V C

v tx t

V

 
 

            
 
 

 
(24) 

where v0(t) and il(t) are capacitor voltage and inductor 
current, Vi is fixed non- regulated input voltage. With 
normalization of state-space model of buck-boost converter 
(Figure 1), new equations could be expressed as follows: 

( )

( )

2 1

1
1 2

( ) 1 ( ) ( )

( )
( ) 1 ( ) (

(

)

)x u x u

x
x u

Q
x

t t t t
t

t t t

+= -

=- - -




 

(25) 

where lQ R C L  and t LCt =  are quality 
coefficient and normalized time variable in buck-boost 
converter respectively. By comparing  (1) and (34), normalized 
matrix of state space system is displayed as follows: 

1
1 0 1 0

1 0 1
1 0

Q
                   

1 1 2A B B
 

(26) 

 

In this model, u(τ) is controllable input with two 0 and 1 
variables. For u(τ)=1 value, system equations will be 
transformed into non-fluctuating dynamic form (9) and 

1 21 , 0Q    will be resulted. So, in this case, based on  
 (12) and (13),  x1(τ) and x2(τ) status variables could be defined 
as follows: 



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   1 1 0 Qxx e





  (27) 

   2 2. 0iV xx     (28) 
Therefore, minimum-time optimal switching level for 

( ) 1u    command will be equal to: 

   
 
 
1

1

ln

2 2
f

x
Q

x
f ixx V e


 



   (29) 

For u(τ)=0, the system equations (25) will be transformed 
into fluctuating dynamic form (15). Accordingly, a and ω, by 
calculating eigen values of A matrix will be obtained as 

21 41
,

2 2

Q

Q Q
 


  . In this condition, based on (18) and 

(19) , x1new(τ) and x2new(τ) status variables could be defined as 
follows: 

 
     

1

1 20 cos (0) sin ]

new

new new

x

e x x


 



 (30) 

 

 
   

2

2 1(0) cos (0) sin ]

new

new new

x

e x x


 



 (31) 

and: 

1 1

2 2

( ) ( )0

( ) ( )1
new

new

x t x t

x t x t




    
        

 (32) 

Figure 2 illustrates the minimum-time optimal switching 
level function in buck-boost system. 

 

 
Fig. 2.  Optimal switchinglevel function in buck-boost converter 

VII. IMPLEMENTATION AND COMPARISON OF MINIMUM-
TIME OPTIMAL CONTROLLER AND CLASSICAL SLIDING MODE IN 

BUCK-BOOST MULTIPLE AREA CONVERTER 

The switching level for sliding mode converter based on 
(4), is defined as follows: 

1 2

1 1 2 2

( , )

( - ) ( ) 0
sm

ref ref

S x x

x V x I 

  

 (33) 

while β1=0, β2=1. Closed loop under reduced output load 
of increased transient energy index is introduced to compare 
the transient operation of the two systems as follows: 

2 

0
( ) ( ( ) )

t

out refj t v t v dt   (34) 
The simulation results of output voltage and inductor 

current in buck-boost converter under reduction of load from 
Rl=5Ω to Rl=15Ω  in classical and minimum-time optimal 
systems are illustrated in Figures 3-4. The Buck-boost 
parameters are shown in Τable I. 

 

0 50 100 150 200

0

1

2

3

4

5

Time��s �
i l
�A�

Optimal SMC

Classical SMC

 

0 50 100 150 200

10

12

14

16

18

20

22

Tim e��s �

v o
u
t�V�

Classical SMC

Optimal SMC

 
Fig. 3.  Fig. 3. Buck-boost multi area converter to reduce output load 

from 5 to 15Ω. A) output voltage B) inductor current 

 

 
Fig. 4.  Increased energy index of classical multi area converter and 
optimal buck-boost minimum-time systems under reduction of load from 5 to 
15Ω. 

The results indicated that despite the equality of maximum 
voltage amplitude of current in minimum-time optimal and 
classical sliding mode systems, the disturbance in minimum-
time optimal systems was omitted within time less than similar 
classical systems. The increased energy index induced from 
output voltage (33) in minimum-time optimal systems is lower 

A 

B 



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than classical systems (Figure 4). Figure 5 shows the 
experimental setup that provided for the test. Figure 6 
illustrates the applied results of reduced load which confirms 
the findings of analytical simulation analysis. In second 
condition, system performance under increased load from 
Rl=15Ω to Rl=5Ω  in classical as well as minimum-time 
optimal systems has been compared and analyzed. The output 
voltage simulation results and inductor current with VP=7 V, 
VN=0 and Vref=10 under increased load from Rl=5Ω to Rl=15Ω  
are illustrated in Figure 7. According to this figure, we could 
say that inductor current needs to save more energy so as to 
achieve output voltage regulation within minimum time.  

 

  

Fig. 5.  Experimental Setup 

 

 

 
Fig. 6.  Buck-boost multi area converter in reduced load from 5 to 15Ω.  

A) Schematic, B) Optimal controller. 

Despite the access to minimum regulation of output voltage 
in increased output load, voltage drop amplitude in minimum-

time optimal systems is larger than similar classical sliding 
mode systems. So, application of minimum-time optimal 
control in buck-boost multi area converter enables us to remove 
disturbances in the least possible time and energy. In this 
situation, converter switching is a function of status variables 
(output voltage and inductor current) as though it will reach the 
maximum value simultaneously in output phase voltage phase 
and inductor current. So, the minimum-time optimal controller 
enhances the system dynamic performance and decreases the 
transient parameter of output overvoltage. VN 

TABLE I.  BUCK- BOOST PARAMETERS 

Nominal Value Parameter Description 
7 V VP  Positive voltage  

{0 or -7}
 

VN Negative voltage 
3.4 μF

  
C  Capacitance 

96 μΗ
 

L  Inductance 
{11 Ω, 15 Ω}

  
R  Load resistance 

10 V  Vref  Desired output voltage  

0.2 Ω  R1  
Leakage resistance 

inductor  

10-8 Ω R2 
Leakage resistance 

capacitor 

 

0 50 100 150 200
1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Time��s �

i l
�A�

Optimal

SMC

Classical

SMC

0 50 100 150 200

2

4

6

8

10

Time��s �

v o
ut
�V� ClassicalSMC

Optimal

SMC

 
Fig. 7.  Classical converter and buck-boost minimum-time optimal systems 
in increased output load from 15 to 5Ω.  A) Output voltage, B) Inductor 
current 

VIII. CONCLUSION 
The present study examined the modeling and designing of 

minimum-time optimal controller of sliding mode in buck-
boost multi area converter systems. Findings were also 
examined and analyzed. The results show that the application 
of switching level instead of classic model causes the output 
voltage regulation to be implemented within minimum time. 

B 

A 

A 

B 



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However, the transient energy index of increased voltage in 
minimum-time optimal controller significantly decreases. By 
substituting optimal instead of classical switching surface, the 
output voltage could be regulated in minimum time. In 
addition, transient over voltage energy index in optimal buck-
boost SMC for load step decreased was effectively reduced. 

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[2] R. Jadeja, A. Ved, S. Chauhan, “An Investigation on the performance of 
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