Performance analysis of a second order system using mrac

INTERNATIONAL JOURNAL OF ELECTRICALISSN 0976 – 6545(Print),
International Journal of Electrical Engineering and Technology (IJEET), ENGINEERING
ISSN 0976 – 6553(Online) Volume 3, Issue 3, October – December (2012), © IAEME
& TECHNOLOGY (IJEET)
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 3, Issue 3, October - December (2012), pp. 110-120 IJEET
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PERFORMANCE ANALYSIS OF A SECOND ORDER SYSTEM USING
MRAC

Rajiv Ranjan1, Dr. Pankaj Rai2
1
(Assistant Manager (Projects)/Modernization & Monitoring, SAIL, Bokaro Steel Plant, India,
rajiv_er@yahoo.com)
2
(Head of Deptt., Deptt. Of Electrical Engineering, BIT Sindri, Dhanbad ,India,
pr_bit2001@yahoo.com)

ABSTRACT

In this paper model reference adaptive control (MRAC) scheme for MIT rule and Lyapunov rule
has been discussed. These rules have been applied to the second order system. Simulation is
done in MATLAB- Simulink for different value of adaptation gain and the results are compared
for varying adaptation mechanisms due to variation in adaptation gain.

Keywords: adaptive control, MRAC (Model Reference Adaptive Controller ), adaptation gain,
MIT rule, Lyapunov rule

1. INTRODUCTION

Traditional non-adaptive controllers are good for industrial applications, PID controllers are
cheap and easy for implementation [1]. Nonlinear process is difficult to control with fixed
parameter controller. Adaptive controller is best tool to improve the control performance of
parameter varying system.
Adaptive controller is a technique of applying some system identification to obtain a model and
hence to design a controller. Parameter of controller is adjusted to obtained desired output
[2].Model reference adaptive controller has been developed to control the nonlinear system.
MRAC forcing the plant to follow the reference model irrespective of plant parameter variations.
i.e decrease the error between reference model and plant to zero[5]. MRAC implemented in
feedback loop to improve the performance of the system [3].There are many adaptive control
schemes [4] but in this paper mainly MRAC control approach with MIT rule and Lyapunov rule
has been discussed. Effect of adaption gain on system performance for MRAC using MIT rule

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),

for first order system[11] and for second order system[12] has been discussed. Compression of
performance using MIT rule & and Lyapunov rule for first order system for different value of
adaptation gain is discussed [13],[14]. In this paper adaptive controller for second order system
using MIT rule and Lyapunov rule has been discussed first and then simulated for different value
of adaptation gain in MATLAB and accordingly performance analysis is discussed for MIT rule
and Lyapunov rule for second order system .

2. MODEL REFERENCE ADAPTIVE CONTROL

Model reference adaptive controller is shown in Fig. 1. The basic principle of this adaptive
controller is to build a reference model that specifies the desired output of the controller, and
then the adaptation law adjusts the unknown parameters of the plant so that the tracking error
converges to zero [6]

Figure 1

3. MIT RULE

There are different methods for designing such controller. While designing an MRAC using the
MIT rule, the designer selects the reference model, the controller structure and the tuning gains
for the adjustment mechanism. MRAC begins by defining the tracking error, e. This is simply the
difference between the plant output and the reference model output:

system model e=y(p) −y(m) (1)

The cost function or loss function is defined as
F (θ) = e2 / 2 (2)

The parameter θ is adjusted in such a way that the loss function is minimized. Therefore, it is
reasonable to change the parameter in the direction of the negative gradient of F, i.e
1 2 (3)
J (θ ) = e (θ )
2

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dθ δJ δe
= −γ = −γe (4)
dt δθ δθ

– Change in is proportional to negative gradient of J

J (θ ) = e(θ )
dθ δe (5)
= −γ sign(e)
dt δθ
 1, e > 0

where sign(e) =  0, e = 0
− 1, e < 0


From cost function and MIT rule, control law can be designed.

4. MATHEMATICAL MODELLING

Model Reference Adaptive Control Scheme is applied to a second order system using MIT rule.
It is a well known fact that an under damped second order system is oscillatory in nature. If
oscillations are not decaying in a limited time period, they may cause system instability. So, for
stable operation, maximum overshoot must be as low as possible (ideally zero).
This can automatically reduce the transient period of the system and improve the system
performance. A critically damped second order system gives a characteristic without any
oscillations and this characteristic is similar to the first order system. But it is not feasible to
achieve such system practically. In this paper a second order under damped system with large
settling time, very high maximum overshoot and with intolerable dynamic error is taken as a
plant. The object is to improve the performance of this system by using adaptive control scheme.
For this purpose, a critically damped system is taken as the reference model. Let the second order
system be described by:
Y ( s)
Where system = kG ( s ) where K is constant.
U ( s)

and Y ( s)
= koG ( s)
U c (s)

using plant Gm ( s ) = ko G ( s )
1 2 dθ δe
Let cost function as J (θ ) = e (θ ) 
→ = −γ ' e (6)
2 dt δθ
γ ' usually kept small

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Tuning γ ' is crucial to adaptation rate and stability.

Considering a Plant: ‫ݕ‬௣ = -a‫ݕ‬ሶ௣ - by + bu
ሷ (7)

Where ‫ݕ‬௣ is the output of plant (second order under damped system) and u is the controller
output or manipulated variable.

Similarly the reference model is described by:

‫ݕ‬௠ = -ܽ௠ ‫ݕ‬ሶ௠ - ܾ௠ y + ܾ௠ r
ሷ (8)

Where ‫ݕ‬௠ is the output of reference model (second order critically damped system) and r is the
reference input (unit step input).

Let the controller be described by the law:

u = θ1r − θ 2 y p (9)
e = y p − ym = G p u − Gm r (10)

 b 
y p = G pu =  2 (θ 1 r − θ 2 y p )
 s + as + b 
bθ 1
yp = 2 r
s + as + b + bθ 2

bθ1
e= 2
uc − Gm r
s + as + b + bθ 2
∂e b
= 2 r
∂θ1 s + as + b + bθ 2
∂e b 2θ1
=− r
∂θ 2 (
s 2 + as + b + bθ 2 ) 2

b
=− 2
yp
s + as + b + bθ 2
If reference model is close to plant, can approximate:

s 2 + as + b + bθ 2 ≈ s 2 + a m s + bm
∂e bm (11)
= b / bm 2 uc
∂θ 1 s + a m s + bm
∂e bm
= −b / b m 2 y plant (12)
∂θ 2 s + a m s + bm

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Controller parameter are chosen as θ1 = bm /b and θ 2 = ( b − bm )/b

Using MIT

dθ 1 ∂e  bm  (13)
= −γ e = −γ  2 
 s + a s + b u c e
dt ∂θ 1  m m 
dθ 2 ∂e  bm 
= −γ e =γ 2 
 s + a s + b y plant e
(14)
dt ∂θ 2  m m 

Where γ = γ ' x b / bm = Adaption gain
Considering a =10, b = 25 and am =10 , bm = 1250

5. SIMULATION RESULTS FOR MIT RULE

To analyze the behavior of the adaptive control the following model has designed in Matlab/Simulink

Time response for different value of adaption gain for MIT rule is given below:

Figure 2

Figure 3 Figure 4

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Figure 5 Figure 6

Figure 7 Figure 8

The time response characteristics for the plant and the reference model are studied. It is observed that the
characteristic of the plant is oscillatory with overshoot and undershoot whereas the characteristic of the
reference model having no oscillation. Dynamic error between these reduced to zero by using model
reference adaptive control technique.

115


Results with different value of adaptation gain for MIT rule is summarized below:
Without any With MRAC
controller
γ =0.1 γ =2 γ =3 γ =5
Maximum 61% 0 0 0 0
Overshoot (%)
Undershoot (%) 40% 0 0 0 0
Settling Time 1.5 22 2.4 2.25 2.1
(second)

Without controller the performance of the system is very poor and also having high value of undershoot
and overshoot(fig. 2). MIT rule reduces the overshoot and undershoot to zero and also improves the
system performance by changing the adaptation gain. System performance is good and stable (fig. 4, fig.
5 & fig. 6) in chosen range (0.1< γ >5) . Beyond the chosen range of adaption gain (0.1< γ >5)
performance of system is very poor and become unstable (fig.7 & fig. 8). So for only suitable value
of adaptation gain MIT rule can be used to track the plant output closer to the reference output[12].

6. LYAPUNOV RULE

In order to derive an update law using Lyapunov theory, the following Lyapunov function is
defined [15],[16].
ଵ ଵ ଵ
V = ߛ݁ ଶ + (ܾߠଵ − ܾ௠ )ଶ + ଶ௕ (ܾߠଶ + ܽ − ܾܽ௠ )ଶ (15)
ଶ ଶ௕

The time derivative of V can be found as

ሶ ሶ
ܸሶ = ߛ݁݁ሶ + ߠଵ (ܾߠଵ − ܾ௠ ) + ߠଶ (ܾߠଶ + ܽ − ܾܽ௠ ) (16)

And its negative definiteness would guarantee that the tracking error converge to zero along the
system trajectories.

For first order system:

The process dynamics ‫ݕ‬ሶ௣ + ܽ‫ݕ‬௣ = bu (17)

Reference dynamics ‫ݕ‬ሶ௠ + ܽ௠ ‫ݕ‬௠ = ܾ௠ r (18)

Inserting the dynamic equations of the plant and reference model in above equation

ሶ ሶ
ܸሶ = ߛ݁൫‫ݕ‬ሶ௣ − ‫ݕ‬ሶ௠ ൯+ ߠଵ (ܾߠଵ − ܾ௠ ) + ߠଶ (ܾߠଶ + ܽ − ܾܽ௠ )

ሶ
= -ߛܽ௠ ݁ ଶ + (ߛ݁‫ߠ + ݎ‬ଵ )(ܾߠଵ − ܾ௠ ) + (ߠଶ − ߛ݁‫ݕ‬௣ )(ܾߠଶ + ܽ − ܾܽ௠ ) (19)

116


Eq.(19) give following condition for negative definiteness and thus the update laws:

(20)
(21)
ௗఏమ
= ߛ݁‫ݕ‬௣
ௗ௧

For MIT rule , above equation can be derived as :

ௗఏభ ೘ ௔
= −ߛ݁ (௦ା ௔ ‫ݎ‬
ௗ௧ ೘)

ௗఏమ ௔೘
= ߛ݁ (௦ା ௔ ‫ݕ‬௣
ௗ௧ ೘)

From the above equation it is observed that MIT rule and Lyapunov theory is similar only
difference is that MIT rule comprises an additional filter operation with reference model.

Lyapunov updates laws for second order system can we written as:

ௗ ௔೘
ߠ
ௗ௧ ଵ
= − ߛ݁ (௦మ ା௔ ‫ݎ‬ (22)
೘ ௦ା ௕೘ )

ௗ ௔೘
ߠ = − ߛ݁ (௦మ ‫ݕ‬ (23)
ௗ௧ ଶ ା௔೘ ௦ା ௕೘ ) ௣

7. SIMULATION RESULTS FOR LYAPUNOV THEORY

To analyze the behavior of the adaptive control the following model was designed in Matlab/Simulink

Time response for different value of adaption gain for Lyapunov rule is as:

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Figure. 9 Figure. 10



In case of MIT rule, if adaptation gain is increased to chosen range (0.1< γ >5) the performance
of the system become very poor and system also become unstable (fig.7 & fig. 8). In this section
Lyapunov rule is used for designing of system in chosen range as well as beyond the chosen
range and then performance of the system is analyzed for different value of adaptation gain.

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Without any With Lyapunov
controller
γ =0.1 γ =2 γ =3 γ =5
Maximum 61% 0 0 0 40%
Overshoot (%)
Undershoot 40% 0 0 0 30%
(%)
Settling Time 1.5 18 1.9 1.6 1.4
(second)

In chosen range of adaption gain (0.1< γ >5) performance of system by using Lyapunov rule is
better as compared to the MIT rule response (fig. 10, fig.11 & fig. 12).
Beyond the chosen range performance of the system has also been analyzed for γ =7 and γ =10
and it is observed that system having little oscillation but system is stable (fig. 13 & fig. 14).
System is even also stable for the higher value of adaption gain γ =50.

8. CONCLUSION

Adaptive controllers are very effective where parameters are varying. The controller parameters
are adjusted to give desired result. This paper describes the MRAC by using MIT rule and
Lyapunov rule for second order system.
Time response is studied for second order system using MIT rule and Lyapunov rule with
varying the adaptation gain. It is observed that in case of MIT rule, if adaptation gain increases
the time response of the system also is improved for the chosen range of adaption gain and
further system is unstable in the upper range. In Lyapnove rule, system is stable beyond the
chosen range of adaptation gain. So with suitable value of adaptation gain in MIT rule and
Lyapunov rule plant output can be made close to reference model. It can be concluded that
performance using Lyapunov rule is better than the MIT rule.

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Performance analysis of a second order system using mrac

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