LMI BASED Anti-Windup Controller Designing for Ball and ... - Ibcast

LMI BASED Anti-Windup Controller Designing 

for Ball and Beam Control System 

Zeeshan Shareef, Abrar Ahmed 

Department of Electrical Engineering, PIEAS, Nilore, Islamabad 

E-mail: zeeshan_iiee@yahoo.com 

Abstract- This paper describes the mathematical modeling of 

the multi-loop Ball & Beam Control System; the PID 

Controller designing as well as the Linear Matrix 

Inequalities (LMI) based Anti-Windup Controller (AWC) 

design to remove the winding effect present due to the 

integral actions present in the transfer function and in the 

controller. Balancing a ball on a motor driven beam is one of 

the most difficult and classical control problems. The ball 

and beam system is widely used because it is very simple to 

understand as a system and yet control techniques that can 

be studied. The classical PID Controller to stabilize the 

system is designed using the ITAE equation. Due to the 

integral terms present in the transfer function and 

Controller produce the winding effect in the actuator’s 

output due to saturation. From a Control Engineering 

perspective, one of the main attractions of LMI is that they 

can be used to solve problems which involve several matrix 

variables. The main advantage of LMI based AWC is that is 

gives the optimal result with the dynamic gains and it only 

dependent on the plant parameters, mean if the controller is 

changed then LMI based AWC will remain same for that 

Controller. The LMI based AWC removes the winding effect 

and doesn’t affect the transient response. In the paper the 

robust AWC for the cascade control is applied to the ball on 

beam control system. 

Keywords: LMI, AWC, optimal result, Ball and beam, winding, 

ITAE Equations, Cascade Control, Full Order. 

I. INTRODUCTION 

In process industry most of the systems are unstable by 

nature and essentially require feedback to for performance 

and stability. The main problem is to bring these unstable 

systems into the laboratory for analyses. Due to its 

simplistic design and dynamic characteristics, ball 

balancing beam seems to be an ideal model for complex, 

nonlinear control methods. 

The ball and beam control system is widely used 

system because of its simplicity to understand as a system 

and provides the opportunity to analyze the control 

techniques. The control task of this control system is to 

adjust the position of the ball on the beam [11]. 

This is a complex task because the ball does not stay 

in one place on the beam but moves all the way with an 

acceleration that is proportional to the tilt of the beam. 

This system is an open loop unstable [12]. Therefore, a 

feedback control must be employed to maintain the ball in a 

desired position on the beam [13]. 

There are many techniques for designing the PID 

controller like Ziegler Nichols methods [16]. But the ITAE 

method is simplest all of the methods because it doesn’t 

require any graph or chart. The Integral of Time multiply 

by Absolute Error (ITAE) index is a popular performance 

criterion used for control system design. The index was 

proposed by Graham and Lathrop (1953), who derived a set 

of normalized transfer function coefficients from 2nd to 

8th-order to minimize the ITAE criterion for a step input 

[17]. It based on the comparison of standard ITAE equation 

with the closed loop equations of the system. The ITAE 

provides the best selectivity of the performance indices. 

There are many other techniques to control the system. 

The Fuzzy control is one of the most rising control 

techniques nowadays. The fuzzy control technique has been 

applied to the ball and beam system with different 

methodologies [14,15]. 

The uncertainty in the plant and the actuator saturation are 

the two irritating problems in control theory. The antiwindup 

compensator (AWC) is an additional controller 

used to improve the overall performance of the closed loop 

system when a control signal undergoes saturation [4]. The 

AWC scheme has been developed for the stable and 

unstable plants. Nowadays, researchers are working on the 

robustness of the AWC [1-3]. 

The LMI based AWC for tackling the actuator 

saturation has become the trend in the control applications 

[10]. The procedure is to first design the linear controller 

for the unconstrained system in the absence of saturation. 

Then the AWC is applied for the improvement of 

performance. The existence of an AWC for general stable 

linear plant is shown in [8]. Having gone through literature, 

it has been found that robustness has been incorporated for 

the synthesis of anti-windup compensators [2-4]. 

There are number of cascade plant in the process 

industry with actuator saturation. The ball and beam control 

system can be used to demonstrate these cascaded control 

systems. In cascade systems the secondary loop must be 

faster than the primary loop. The cascade control has been 

described in the [5, 9]. 

In this paper, the robust AWC for cascade control is 

applied to the ball on the beam system. The AWC is taken 

as part of co-prime factorization of each plant. In the design 

of AWC, the global stability of closed loop is ensured. 

Proceedings of International Bhurban Conference on Applied Sciences & Technology, 

Islamabad, Pakistan, 10 - 13 January, 2011 154

Thus it also is referred to as global anti-windup 

compensator. 

loop, (2) outer ball and beam control loop. The design 

strategy is to first stabilize the inner loop followed by the 

outer loop. 

II. 

MATHEMATICAL MODELING 

The Ball & Beam Control system is a multi-loop system 

which comprises of two parts. One is the mathematical 

model of the dc motor followed by mathematical model of 

the ball and beam system. The typical ball & beam control 

system is shown in the Fig. 1. 

Fig.2 Block Diagram of Ball and Beam Control System 

Ball 

Ball Position 

Motor 

Fig.1 Ball and Beam Control System 

Desired 

Position 

Beam 

There are number of techniques to design the PID 

controller which involves Ziegler and Nichols, SISO Tool, 

ITAE Equations, root locus, frequency response etc. In our 

design, we have used the ITAE equations for the PD and 

PID gains computation. The performance index ITAE 

provides the best selectivity of the performance indices; 

that is, the minimum value of the integral is readily 

discernible as the system parameters are varied. The 

general form of the performance integral is 

Fig.1 shows that there are three main components are 

involved in the system which includes moments and 

forces acting on them, the motor, the beam, and the ball. 

Using the equation for the sum of forces and torques the 

final mathematical model that we get is 

GT 

G 

Φ 

I 

() s 

() s 

in 

, 

g 

= 

2 

⎛ ⎞ 

2⎜ 

2 ⎛ a ⎞ 

s 1+ 

⎟ 

⎜ ⎟ 

⎝ 

5 ⎝ a′ 

⎠ ⎠ 

Based on the motor parameters, Km 

= 0.7/ rev/ sce/ 

volts, 

−6 2 

τ = 0.014 Sec , J = 1.4× 10 Kg− m , motor model can 

be written as 

θ ( s) 

0.7 

Gm 

( s) 

= = 

I ( s) 

0.014s 

+ 1 

in 

Based on the ball and beam specifications of 

a= 1.5cm(radius of the ball) and a' = 9.4227cm(for 2π) 

the transfer function of ball and beam can be written as 

1 

() s = 

1.713s 

BallBeam( Open) 2 

0.7 

() s = 

0.0234s + 1.713s + 1.1991s 

4 3 2 

(1) 

(2) 

(3) 

The overall transfer function of the ball and beam in open 

loop will become: 

(4) 

III. PID CONTROLLER DESIGNING METHODOLOGY 

I 

G 

T 

= ∫ 

0 

motor 

f ( e( 

t), 

r( 

t), 

y( 

t), 

t) 

dt 

The procedure for calculating the PID gains using the ITAE 

equations is as follow. 

Step-1: In this calculation method first we multiply the 

general form of the PID with the open loop transfer 

function. 

Step-2: The characteristics equation is compared with the 

standard ITAE equations. The value of damping ratio (ζ) is 

constant that is 0.6. 

Using standard equations of ITAE the PD and the PID 

controllers designed for the inner and outer loop can be 

written as respectively [14,15]: 

( Controller )( 

s) 

= 2s 

+ 11 

2 

5s 

+ 6s 

+ 0. 2 

G Ball ( Controller ) 

( s) 

= 

s 

The main advantage for using the ITAE equation method 

for the calculation of gain is that there is no requirement for 

any graph or chart like the root locus and the bode plot. 

These equations can easily be used for the gains calculation 

of up to 6th order characteristics equation. After calculating 

the gains from the ITAE equations we require the little bit 

tuning of the gains. 

(5) 

(6) 

(7) 

The block diagram of the whole system is shown in Fig.2. 

The system consists of two loops: (1) Inner motor control 



IV. SIMULATION RESULT OF UNCONSTRAINT SYSTEM 

1 

In this section the simulation results of the system with the 

controllers are presented. First we see the step response of 

the overall system without controller. The block diagram 

of the system without controller with unity feedback and 

its step response is shown in the Fig.3 and Fig. 4 

respectively. 

0.8 

0.6 

0.4 

0.2 

Reference Signal 

Inner loop Response with Controller 

0 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 

Fig.3: Block Diagram of the Overall Closed Loop system with 

unity feedback without controller 

Fig.6: Step response of inner closed loop with PD controller 

120 

100 

80 

Output of the closed loop system without controller 

1.4 

1.2 


System Output 

60 

1 

40 

0.8 

Output 

20 

0 

0.6 

-20 

0.4 

-40 

-60 

0.2 

-80 

0 5 10 15 20 25 30 

Time 

Fig.4: Step Response of the Overall Closed Loop system with unity 

feedback without controller 

0 

0 1 2 3 4 5 6 7 8 9 10 

Fig.7: Step response Overall closed loop system with controllers 

From this simulation result e can analyze that the system 

in close loop without controllers is unstable. To stabilize 

the system we start from the inner loop and apply the PD 

controller. The block diagram of the inner loop with PD 

controller and its response is shown in the Fig.5 and Fig.6 

respectively. 

From the response we can see that the overall system has 

been stabilized with a settling time of 2.3 Sec. Although the 

2.3sec is much high settling time but it is the requirement 

because at the low settling time the motor starts to take 

jerks and through the ball outside the beam. The dire time 

of the system with PID controller is 0.444 Sec and the 

percentage overshot is 28.9%. 

Fig.5: Block Diagram of inner closed loop with PD controller 

Form this result it is clear that the inner loop is stable now 

with a settling time of 0.08 Sec. In the next step the PID 

controller will be applied in the outer loop to stabilize the 

overall system. 

The response of the overall system with PID controller is 

shown in Fig.7 in right column: 

IV 

SIMULATION RESULT OF CONSTRAINT 

SYSTEM 

All real-world applications of feedback control involve 

control actuator with amplitude and rate limitations. The 

control design techniques that ignore these actuator limits 

may cause undesirable transient response, degrade the close 

loop performance and may cause even closed-loop 

instability. Thus actuator saturation constitutes fundamental 

limitations of many linear control design techniques and 

has attracted the attention of numerous researchers, 

especially in the last decade. 



The response of the ball on the beam system with 

saturation [0 0.5] applied is shown in Fig.8 below: 

20 

15 

10 


Unconstraint System 

Constraint System 

In the Fig.9 G in represent the motor transfer function and 

~ 

out 

~ 

G represent the transfer function of the ball and beam. 

This configuration can be decoupled into linear and 

nonlinear parts for the stability and performance analysis as 

shown in Fig.10. 

5 

Output 

0 

-5 

-10 

-15 

-20 

0 5 10 15 20 25 30 

Time 

Fig.8: Step response Overall closed loop Constraint system with 

controllers 

In the simulations shown in Fig.8 the pulses are applied to 

the system and the output is observed. From the above 

results we can see that the after applying the actuator 

saturation to the system, the system become unstable. The 

black dotted line is the response of the constraint closed 

loop system. 

V. ANTI-WINDUP CONTROLLER DESIGN 

Generally two basic types of AWC structures are used. 

One in which the AWC output is directly added into the 

output of system as in [1,3,4] and other in which AWC 

output is directly fed into controller like [8]. The first one 

is applicable to most of the engineering problems as 

shown in [10]. So the architecture we use is of first type in 

which AWC can be incorporated for existing 

unconstrained control scheme. 

In case of cascade control system, the plant G in (s) has 

input saturation which also disturbs the performance of 

outer loop. The AWC parameters M1 and M2 are chosen 

as part of co-prime factorization of plants G p1 (s) and G p2 (s) 

respectively as shown in Fig.9 [9]. 

Fig.10: Equivalent of Fig.9. Decoupling AWC into nonlinear loop and 

disturbance filters 

The role of ( M1− I −M2 

−I) 

in nonlinear loop is pivotal in 

extracting the system from saturation back to linear 

behavior. 

The coprime factorization of G p1 (s) and G p2 (s) can be 

written as: 

G () s = N () s M () s 

(8) 

−1 

p1 1 1 

G () s = N () s M () s 

−1 

p2 2 2 

Full-order right co-prime factorization of both plants can be 

described as follows: 

(9) 

(10) 

(11) 

Here F 1 and F 2 are used to determine co-prime 

factorizations of G p1 and G p2 respectively and chosen such 

that A p1 + B p1 F 1 and Ap 2 + B p2 F 2 are Hurwitz matrix. In this 

case the mapping Γ 

p 

: ulin → yd , y is given as 

1 d 2 

(12) 

Fig.9: Cascade control system with conditioning using M 1 and M 2 



Then in order to minimize the L 2 

norm of the system to a 

minimum value γ we have the following objective 

function [9]: 

d 

~ ~ 

1 

0 

T 

⎡P 

⎤ 

J = xp ⎢ xp + Wr ulin − ( Fx 

1 p1+ u) − ( F2xp2 

+ u) 

dt 0 P 

⎥ 

⎣ 2 ⎦ 

~ 

2 

~ 

2 

2 

p1 1 p1 p1 p2 2 p2 p2 

γ 

lin 

+ W C x + D u + W C x + D u − u < 0 

2 

(13) 

Output 

1 

0.8 

0.6 

0.4 


Constraint + Full Order Robust AWC 

By adding the dead-zone nonlinearity and after applying 

the Schur Complement, S-procedure and Congruence 

transformation, we finally get the LMI (14) for robust full 

order anti-windup compensator. 

0.2 

0 

0 5 10 15 20 25 30 

Time 

Fig.11: Step Response of the constraint system with robust full order 

dynamic Anti-windup controller 

(14) 

Here Q 1 > 0, Q 2 > 0, U > 0, and γ > 0. Here Q 1 = P 1 -1 , Q 2 = 

P 2 -1 , U = V -1 , L 1 = F 1 Q 1 and L 2 = F 2 Q 2 . Hence the robust 

AWC for the cascade control system can be designed by 

minimizing γ for LMI (14). The weighted value W p1 , W p2 

and W r are positive definite weighting matrices that reflect 

the relative importance of performance and robustness, 

respectively, and chosen by designer. 

VI. SIMULATION RESULTS OF CONSTRAINT 

SYSTEM WITH ANTI-WINDUP CONTROLLER 

As we have seen in Fig. 8 that incorporating the constraint 

to the system produces the instability in the system. To 

produce the stability in the constraint system the Anti- 

Windup controller will be incorporated. 

The value of F 1 and F 2 that we get after solving LMI (14) 

is as follow: 

F1 

= [0.0001 0.0038] 

F = × 

2 

−14 

1 10 *[0 0.3301] 

(15) 

The response of the constraint system with the full order 

robust anti-windup controller is shown in Fig. 11 in the 

left column: 

From the simulation result shown in Fig.11 that by 

incorporating the anti-windup controller the overall system 

has become stable. Although there is steady state error, but 

it is always present because we have limit the output pf the 

actuator. 

VII. 

CONCLUSION 

In this paper we design the PID controller for the ball on 

the beam system using the ITAE equations. The 

incorporation of the actuator limitation produces the 

instability in the system. The full order anti-windup 

controller is designed for the cascaded ball on the beam 

control system. The LMI based anti-windup controller 

gives the optimal results with dynamic gains and it only 

dependent on plant parameter. By incorporating the full 

order anti-windup controller the constraint system has 

become stable and there is no more instability in the system. 

ACKNOWLEDGEMENT 

I gratefully acknowledge the support of Sir Ashab Mirza 

(Associate Professor, IIEE), Sir Yousuf Kakakhail 

(Professor, PIEAS) for their guidance as my teacher and 

course in-charge. I would also like to thanks Mr. Ufaq Shah 

who helped us a lot in implementing the Ball & Beam 

Control System. 



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