Robust model reference adaptive control for a second order system 2

INTERNATIONAL Electrical EngineeringELECTRICAL ENGINEERING
International Journal of
JOURNAL OF and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 1, January- February (2013), © IAEME
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ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 4, Issue 1, January- February (2013), pp.09-18 IJEET
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ROBUST MODEL REFERENCE ADAPTIVE CONTROL FOR A
SECOND ORDER SYSTEM

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 has been
proposed in presence of first order and second order noise. The noise has been applied to the
second order system. Simulation is done in MATLAB-Simulink and the results are compared
for varying adaptation mechanisms for different value of adaption gain in presence of noise
and without noise, which show that system is stable.

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

1. INTRODUCTION

Robustness in Model reference adaptive Scheme is established for bounded
disturbance and unmodeled dynamics. Adaptive controller without having robustness
property may go unstable in the presence of bounded disturbance and unmodeled dynamics.
Model reference adaptive controller has been developed to control the nonlinear system.
MRAC forces the plant to follow the reference model irrespective of plant parameter
variations. i.e decrease the error between reference model and plant to zero[2]. Effect of
adaption gain on system performance for MRAC using MIT rule for first order system[3] and
for second order system[4] has been discussed. Comparison of performance using MIT rule
& Lyapunov rule for second order system for different value of adaptation gain is discussed
[1].

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

In this paper adaptive controller for second order system using MIT rule in the presence of
first order and second order bounded disturbance and unmodeled dynamics has been
discussed first and then simulated for different value of adaptation gain in MATLAB and
accordingly performance analysis is discussed for MIT rule for second order system in the
presence of bounded disturbance and unmodeled dynamics.

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 γ 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 IN PRESENCE OF BOUNDED AND
UNMODELED DYNAMICS

Model Reference Adaptive Control Scheme is applied to a second order system using
MIT rule has been discussed [3] [4]. 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).
In this section mathematical modeling of model reference adaptive control (MRAC) scheme
for MIT rule in presence of first order and second order noise has been discussed

Considering a Plant: = -a - by + bu (6)

Consider the first order disturbance is = - +

Where 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:

= - - y+ r (7)

Where 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:

11


u = θ1 r − θ 2 y p (8)
e = y p − y m = G p Gd u − Gm r (9)

 b  k 
y p = G p Gd u =  2  (θ 1 r − θ 2 y p )
 s + as + b  s + k 
bkθ 1
yp = 2 r
( s + as + b)( s + k ) + bkθ 2
Where Gd = Noise or disturbances.

bkθ 1
e= 2
r − Gm r
( s + as + b)( s + k ) + bkθ 2
∂e bk
= 2 r
∂θ 1 ( s + as + b)( s + k ) + bkθ 2
∂e b 2 k 2θ 1
=− 2 r
∂θ 2 [( s + as + b)( s + k ) + bkθ 2 ] 2
bk
=− 2
yp
( s + as + b)( s + k ) + bkθ 2
If reference model is close to plant, can approximate:

( s 2 + as + b)(s + k ) + bkθ 2 ≈ s 2 + a m s + bm
bk ≈ b (10)
∂e bm
= b / bm 2 uc
∂θ 1 s + a m s + bm
(11)
∂e bm
= −b / b m 2 y plant
∂θ 2 s + a m s + bm
Controller parameter are chosen as θ 1 = bm /b and θ 2 = ( b − bm )/b

Using MIT

dθ 1 ∂e  bm  (12)
= −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
(13)
dt ∂θ 2  m m 

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

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5. SIMULATION RESULTS FOR MIT RULE WITHOUT BOUNDED
DISTURBANCE AND UNMODELED DYNAMICS

To analyze the behavior of the adaptive control the designed model has been simulated in
Matlab-Simulink.

The simulated result for different value of adaptation gain for MIT rule is given below:

Figure 2

Figure 3 Figure- 4

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Figure 5 Figure 6

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.

Results with different value of adaptation gain for MIT rule is summarized below:
Without any With MRAC
controller
γ =0.1 γ =2 γ =4 γ =5
Maximum 61% 0 0 0 0
Overshoot
(%)
Undershoot 40% 0 0 0 0
(%)
Settling 1.5 22 2.4 2.25 2.1
Time
(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. 3, fig. 4, fig. 5 & fig. 6) in chosen range (0.1< γ <5).

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6. MIT RULE IN PRESENT OF BOUNDED DISTRUBANCE AND UNMODELED
DYNAMICS

Consider the first order disturbance:

1
Gd =
s +1
Time response for different value of adaption gain for MIT rule in presence of first order
disturbance is given below:

Figure 7 Figure 8

Figure 9 Figure 10

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Simulation results with different value of adaptation gain for MIT rule in presence of first
order bounded and unmodeled dynamics is summarized below:

Without any In presence of first order bounded and unmodeled
controller dynamics
γ =0.1 γ =2 γ =4 γ =5
Maximum 61% 0 27% 30% 35%
Overshoot
(%)
Undershoot 40% 0 15% 18% 21%
(%)
Settling Time 1.5 26 8 6 4
(second)

In the presence of first order disturbance, if the adaptation gain increases the overshoot and
undershoot increases, but the settling time decreases. This overshoot and undershoot are due
to the first order bounded and unmodeled dynamics. It shows that even in the presence of first
order bounded and unmodeled dynamics, system is stable.

Consider the second order disturbance:

25
Gd = 2
s + 30s + 25

Time response for different value of adaption gain for MIT rule in presence of first order
disturbance is given below:

Figure 11 Figure 12

16


Figure 13 Figure 14

Simulation results with different value of adaptation gain for MIT rule in presence of second
order bounded and unmodeled dynamics is summarized below:

Without any In presence of second order bounded and unmodeled
controller dynamics
γ =0.1 γ =2 γ =4 γ =5
Maximum 61% 0 35% 40% 45%
Overshoot
(%)
Undershoot 40% 0 10% 15% 20%
(%)
Settling Time 1.5 30 12 10 7
(second)

In the presence of second order disturbance, if the adaptation gain increases the overshoot
and undershoot increases, but the settling time decreases. This overshoot and undershoot are
due to the second order bounded and unmodeled dynamics. It shows that even in the presence
of second order bounded and unmodeled dynamics, system is stable.

7. 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 in the presence of first order & second order bounded and unmodeled dynamics.
Time response is studied in the presence of first order & second order bounded and
unmodeled dynamics using MIT rule with varying adaptation gain. It has been observed
that, disturbance added in the conventional MRAC has some oscillations at the peak of
signal, hence these disturbances can be considered as a random noise. These oscillations
reduce with the increase in adaptation time. These overshoots and undershoots are due to the
presence of bounded and unmodeled dynamics or noise. It can be concluded that even in the
presence of bounded and unmodeled dynamics, system performance is stable.

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