The document describes a novel mixed method for order reduction of discrete linear systems. The method uses particle swarm optimization (PSO) to determine the denominator polynomials of the reduced order model. It then uses a polynomial technique to derive the numerator coefficients by equating the original and reduced order transfer functions. This leads to a set of equations that can be solved for the numerator coefficients. The proposed method is illustrated on an 8th order example system from literature. It is found to provide a stable 2nd order reduced model. A lead compensator is then designed and connected to improve the steady state response of the original and reduced order systems.

1. N.Nagendra, M.Sudheer Kumar, T.Madhubabu / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2331-2334
Design Of Discrete Controller Via Novel Model Order Reduction
Method
N.NAGENDRA* (M.E), M.SUDHEER KUMAR*, M.E,
T.MADHUBABU *,(M.E)
Asst. professor
ANITS ENGG.COLLEGE,
SANGIVALAS, BHEEMUNIPATNAM MANDAL
VISAKHAPATNAM.
ABSTRACT
In this paper, a novel mixed method for Evolutionary techniques particles swarm
discrete linear system is used for reducing the optimization appears as a promising algorithm. PSO
higher order system to lower order systems. The algorithm shares many similarities with the genetic
denominator polynomials are obtained by the algorithm. One of the most promising advantages of
PSO Algorithm and the numerator coefficients PSO over the GA is its algorithmic simplicity. In the
are derived by the polynomial method. This present paper to overcome above problem a new
method is simple and computer oriented. If the mixed method is proposed. The denominator
original system is stable then reduced order polynomials of the reduced order model are obtained
system is also stable. Finally the lead compensator by PSO technique and the numerator Coefficients are
is designed and connected to original and reduced determined by using polynomial technique. And lead
order systems to improve steady state responses. compensator is designed for the original and reduced
The proposed method is illustrated with the help order systems. The proposed method is compared
of typical numerical examples considered from the with the other well known order reduction techniques
literature. available in the literature.
Keywords: PSO optimization, polynomial REDUCTION PROCEDURAL STEPS FOR
method, order reduction, transfer function, THE PROPOSED MIXED
discrete system, lead compensator, stability. METHOD:
STEP:1 Let the transfer function of high order
INTRODUCTION original system of the order „n‟ be
The order reduction of a system plays an 𝑁(𝑧) 𝑎 0 +𝑎 1 𝑧+𝑎 2 𝑧 2 +⋯…….𝑎 𝑚 −1 𝑧 𝑛 −1
Gn(z) = =
important role in many engineering applications 𝐷(𝑧) 𝑏0 +𝑏1 𝑧+𝑏 2 𝑧 2 +⋯…….𝑏 𝑛 𝑧 𝑛
especially in control systems. The use of reduced (1)
1+𝑤
order model is to implement analysis, simulation and STEP:2 Applying bilinear transformation Z=
1−𝑤
control system designs. The use of original system is to Gn(z ), to obtain Gn(w )
tedious and costly. So to avoid the above problems 𝑁(𝑤 ) 𝐴0 +𝐴1 𝑤 +𝑎 2 𝑤 2 +⋯…….𝐴 𝑛 𝑤 𝑛
order reduction implementation is necessary. Gn(w) = =
𝐷(𝑤 ) 𝐵0 +𝐵1 𝑤 +𝐵2 𝑤 2 +⋯…….𝐵 𝑛 𝑤 𝑛
Many approaches have been proposed to (2)
reducing order from higher to lower order system for STEP:3 Let the transfer function of the reduced
linear discrete systems. S.Mukherjee, V.Kumar and model of the order „k‟ for w-domain is
R.Mitra [1] proposed a reduction method for discrete 𝑁 𝑘 (𝑤 ) 𝐷0 +𝐷1 𝑤 +𝐷2 𝑤 2 +⋯…….𝐷 𝑘−1 𝑤 𝑘−1
Rk(w )= =
time systems using Error minimization technique. 𝐷 𝑘 (𝑤) 𝐸0 +𝐸1 𝑤+𝐸2 𝑤 2 +⋯…….𝐸 𝑘𝑘
This method becomes complex when the input (3) STEP:4 Determination of denominator by PSO:
polynomial is of high order. Y. Shamash and The PSO method is a member of wide
Feinmesser [2] proposed a method of reduction using category of swarm intelligence methods for solving
Routh array . The disadvantage of this method is, it the optimization problems. Particle swarm
doesn‟t guarantees stable reduced models even optimization technique is computationally effective
though original system is stable. Stability Equation and easier. PSO is started with randomly generated
Method of R.Prasad[3] requires separation of even solution as an initial population called particles. Each
and odd parts of denominator polynomial of original particle is treated as a point in D dimensional space.
high order system which will become Each particle in PSO flies through the search space
computationally tedious when applied to very high- with an adaptable velocity that is dynamically
order original systems. modified according to its own flying experience and
One of the most proposing research fields is also flying experience of the other particles. Each
“Evolutionary techniques” [9]. Among all the particle has a memory and hence it is capable of
2331 | P a g e

2. N.Nagendra, M.Sudheer Kumar, T.Madhubabu / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2331-2334
remembering the best position in the search space 1+𝑤
Step2: Applying the bilinear transformation Z=1−𝑤
ever visited by it.
to the reduced order model.
The position corresponding to the best
Step3: Draw the Bode plot for the reduced order
fitness is known as pbest and the overall best of all
system and find the phase margin αm and consider the
the particles in the population is called gbest .
required value of 𝛼1𝑚 .
By using iteration the values of pbest and gbest
Step4: If αm > 𝛼1𝑚 then no compensation is required,
are calculated. Velocity and particle positions are
Otherwise lead compensator is to be design.
updated by using below formulae.
Step5: Determine the phase lead required using
𝑣 𝑖𝑑+1 = 𝑣 𝑖𝑑 + 𝑐1 ∗ 𝑟1 ∗ 𝑝 𝑖𝑑 − 𝑥 𝑖𝑑 + 𝑐2 ∗ 𝑟2 ∗
𝑘 𝑘 𝑘 𝑘
αl = 𝛼1𝑚 - αm+ €
𝑘 𝑘
(𝑝 𝑔𝑑 − 𝑥 𝑖𝑑 ) (14)
(4) Where € = 50 or 60
𝑥 𝑖𝑑+1 = 𝑥 𝑖𝑑 + 𝑣 𝑖𝑑
𝑘 𝑘 𝑘
Step6: Determine the 𝛽 by using 𝛽 =
1−sin 𝛼 𝑙
and
(5) 1+sin 𝛼 𝑙
1
Where „v‟ is the velocity, „x ‟is the determine the ωm by using -20log(1/√𝛽), T =
ωm √β
position, 𝑝 𝑖𝑑 and 𝑔 𝑖𝑑 are the pbest and gbest, „k‟ is Step7: After finding the T value, design the lead
iteration and c1,c2 are the cognition and social compensator, Then
parameter. These parameters are variable or constant. 1 𝑤+
1
𝑇
Generally these values are „2‟and r1, r2 are the random Gc(w) = 𝛽 𝑤+
1
𝛽𝑇
numbers in the range (0, 1). The parameters c1 and c2
determine the relative pull of pbset and gbset and the (15)
parameters r 1 and r2 help in stochastically varying Step8: Applying inverse bilinear transformation w
𝑧−1
these pulls.. = 𝑧+1 , and cascade the compensator with the original
𝑤 = 𝑤 𝑓 + 𝑤 𝑓 − 𝑤 𝑖 (max 𝑖𝑡 − 𝑖𝑡) 𝑖𝑡 and reduced order system. Obtain the closed loop
(6) responses of original and reduced order systems.
Where „it‟ is the number of iteration
𝑣 𝑖𝑑+1 = 𝑤 ∗ 𝑣 𝑖𝑑 + 𝑐1 ∗ 𝑟1 ∗ 𝑝 𝑖𝑑 − 𝑥 𝑖𝑑 + 𝑐2 ∗ 𝑟2 ∗
𝑘 𝑘 𝑘 𝑘 NUMERICAL EXAMPLE:
𝑘 𝑘
(𝑝 𝑔𝑑 − 𝑥 𝑖𝑑 ) Example1: Consider the 8 𝑡𝑕 order discrete
(7) system[15]
STEP:5 Determination of numerator by polynomial G(z)
0.165 𝑧 7 +0.125 𝑧 6 −0.0025 𝑧 5 +0.00525 𝑧 4
method −0.02263 𝑧 3 −0.00088 𝑧 2 +0.003 𝑧−0.000413
The numerator polynomial is obtained by equating = 𝑧 8 −0.62075 𝑧 7 −0.415987 𝑧 6 +0.076134 𝑧 5 −0.05915 𝑧 4
+0.190593 𝑧 3 +0.097365 𝑧 2 −0.016349 𝑧+0.002226
the original system with its reduced order model. 1+𝑤
𝐴+𝐴1 𝑤 +𝐴2 𝑤 2 +⋯…….𝐴 𝑛 𝑤 𝑛 Applying bilinear transformation Z= 1−𝑤 to
𝐵0 +𝐵1 𝑤 +𝐵2𝑤 2 +⋯…….𝐵 𝑛 =
𝑛 𝑤 Gn(z ), to obtain Gn(w )
𝐷0 +𝐷1 𝑤+𝐷2 𝑤 2 +⋯…….𝐷 𝑘−1 𝑤 𝑘−1
(8) Gn(w )
𝐸0 +𝐸1 𝑤 +𝐸2 𝑤 2 +⋯…….𝐸𝑤 𝑘 −0.0151 𝑤 8 −0.4607 𝑤 7 −2.0074 𝑤 6 −2.3844 𝑤 5 −
Equate the same power‟s of „s‟ on both sides, we = 1.4988 𝑤 4 +1.8399𝑤 3 +2.7271 𝑤 2 +1.5338 𝑤+0.2656
get 1.0007 𝑤 8 +9.8311 𝑤 7 +36.613 𝑤 6 +65.846 𝑤 5 +
73.0179 𝑤 4 +50.0351 𝑤 3 +17.4061 𝑤 2 +1.9988𝑤+0.2493
𝐴0 𝐸0 = 𝐵0 𝐷 𝑜
(9) For implementing PSO algorithm, to obtain
𝐴0 𝐸1 + 𝐴1 𝐸0 = 𝐵0 𝐷1 + 𝐵1 𝐷0 the reduced denominator several parameters are to be
(10) considered.
𝐴0 𝐸2 + 𝐴1 𝐸1 + 𝐴2 𝐸0 = 𝐵0 𝐷2 + 𝐵1 𝐷 +1 𝐵2 𝐷0 The values of c1 and c2 are „2‟
(11) The range of random numbers r 1 and r2 are
𝐴 𝑛−1 𝐸 𝑘 = 𝐵 𝑛 𝐷 𝑘−1 (0,1).
(12) Swarm size = 20 (Number of reduced order
From the above equations we can get the values of models)
D0, D1….Dq. Unknown coefficients = 2
𝑧−1
After finding D0, D1….Dq. apply w = 𝑧+1 for Number of iterations = 200
obtaining reduced order in Z-domain. The denominator polynomial obtained using PSO
𝑁 𝑘 (𝑧 ) 𝑑 0 +𝑑 1 𝑧+𝑑 2 𝑧 2 +⋯…….𝑑 𝑘 𝑧 𝑘 algorithm is
Rk(z ) = 𝐷 𝑘 (𝑧)
= 𝑒0 +𝑒1 𝑧+𝑒2 𝑧 2 +⋯…….𝑒 𝑘 𝑧 𝑘 D2(w) = 0.027957 + 0.107706𝑤 + 𝑤 2
(13) For finding the numerator values, use the
polynomial technique. Equate original transfer
STEPS FOR DESIGNING A LEAD function and reduced order transfer function with
COMPENSATOR: obtained denominator.
Step1: Consider the transfer function of the reduced
order model.
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3. N.Nagendra, M.Sudheer Kumar, T.Madhubabu / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2331-2334
−0.0151 𝑤 8 −0.4607 𝑤 7 −2.0074 𝑤 6 −2.3844 𝑤 5 − Design procedure for lead compensator:
1.4988 𝑤 4 +1.8399𝑤 3 +2.7271 𝑤 2 +1.5338 𝑤+0.2656
1.0007 𝑤 8 +9.8311 𝑤 7 +36.613𝑤 6 +65.846𝑤 5 +
= The transfer function of reduced order system
73.0179 𝑤 4 +50.0351 𝑤 3 +17.4061 𝑤 2 +1.9988𝑤+0.2493 R2(z)
𝐷0 +𝐷1 𝑤
0.0238708 z 2 +0.057609 z+0.033739
0.027957 +0.117706 𝑤+𝑤 2 = (proposed)
1.1445 𝑧 2 −1.94564 𝑧+0.90984
On cross multiplying the above equations Applying the Bilinear transformation, we get
and comparing the same power of „w‟ on the both 0.028847 −0.004861 w
sides, we get numerator value and multiply the R2(w) = 0.027957 +0.117706 𝑤+𝑤 2
numerator with „k.‟ From the Bode plot phase margin αm = 55.40 and
Therefore, the numerator by polynomial method required phase margin 𝛼1𝑚 =700 , Then αl = 19.60
is and β = 0.497. Then the lead compensator is
2.012 (𝑤 +0.169)
N2(w) = 0.028847-0.004861w Gc(w) =
𝑤 +0.341
The proposed second order reduced model using Lead compensator in Z-domain is
mixed method is obtained as follows: 2.012 (1.169𝑧−0.831 )
0.028847 −0.004861 w Gc(z) = 1.341 𝑧+0.341
R2(w) = 0.027957 +0.117706 𝑤+𝑤 2
𝑧−1 The reduced order system with lead compensator is
Apply w = for obtaining reduced order in Z- 0.05614 zz 3 +0.09558 z 2 −0.01696 z−0.05634
𝑧+1 R(z) = 1.591𝑧 3 −3.2677 𝑧 2 +2.4850 𝑧−0.5996
domain
0.0238708 z 2 +0.057609 z+0.033739
R2(z) = 1.1445 𝑧 2 −1.94564 𝑧+0.90984
(proposed model)
The comparison is made by computing the Step Response
3.5
error index known as integral square error ISE in
between the transient parts of the original and 3
reduced order model is calculated by
𝑡
ISE= 0 ∞[ 𝑦 𝑡 − 𝑦 𝑟 𝑡 ]2 (16) 2.5
Where y(t) and yr(t) are the unit step
responses of original and reduced order systems for a 2
Amplitude
second order reduced respectively. The error is
calculated for various reduced order models and 1.5
proposed method is shown below.
1
Method Reduced order models ISE
0.5 w ith compensator
Proposed R2(z) w ithout compensator
mixed = 0
method 0.0238708 z 2 +0.057609 z+0.033739 0.0260 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
1.1445 𝑧 2 −1.94564 𝑧+0.90984 42
Time (sec)
Improved R2(z)
bilinear 0.409429 z−0.29467 Fig2: Step response of reduced order system with and
= 1.235 𝑧 2 −1.9485 𝑧+0.8158
Routh without compensator
approximati 0.0902
on 25 Step Response
0.9
Routh R2(z)
Approximati = 0.0699 0.8
on Method 0.15208 z 2 +0.049726 z−0.102354 72
0.7
1.203709 𝑧 2 −1.955526 𝑧+0.840765
0.6
Step Response
0.5
Amplitude
1.5
0.4
1 0.3
Amplitude
0.2
w ith compensator
0.5
0.1 w ithout compensator
ORIGINAL
PRPPOSED
IBRAM
RAM 0
0 1 2 3 4 5 6
0
0 1 2 3 4 5 6
Time (sec)
Time (sec)
Fig3:Step response of original system with and
Fig1: Step responses of original and reduced order
models without compensator
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4. N.Nagendra, M.Sudheer Kumar, T.Madhubabu / International Journal of Engineering Research
and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2331-2334
CONCLUSION: international journal of computational and
In this paper, design of discrete lead mathematical sciences 3:0 2009.
compensator via a novel mixed method is used for [10]. Shamash‟s “linear system reduction using
discrete linear time system to reduce the high order pade approximation to allow retention of
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denominator polynomial is obtained by PSO pp.257-272, 1975.
algorithm technique and numerator coefficients are [11]. Shailendra K.MITTAL DINESH
derived by polynomial method.PSO method is based CHANDRA and BHARATHI DURIDEVI
on the minimization of the integral squared error A computer -aided approach for routh pade
(ISE) between the transient responses of original approximation of SISO systems based on
higher order model and the reduced order model multi objective optimization IAGSIT int.J.
pertaining to a unit step input. Of engineering and technology vol2, no2,
This method guarantees stability of the aprl-2010 ISSN: 1993-8236.
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stable and which exactly matches the steady state particle swarm optimization and genetic
value of the original system. From these algorithm for FACTS- based controller
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[14]. T.C. CHEN, G.Y.CHANG and K.W HAN
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