Singular Perturbation Method Applied for BVP, IVP and Optimal
Control to One Parameter Armature Controlled DC Servo Motor
KISHOR BABU GUNTI *
Department of EEE,
Gudlavalleru Engineering College,
Gudlavalleru, A. P. 521356,
INDIA.
SREE KRISHNARAYALU MOVVA
Department of EEE,
VR Siddhartha Engineering College
Vijayawada, Andhra Pradesh 520007,
INDIA.
Abstract: DC servo motor discrete model of one-parameter Singular Perturbation Method (SPM) is enlarged
showing zero-order, first-order and second-order approximations. In this paper, a one parameter SPM based
two time scale model is considered for evaluation. Now by applying this SPM a real time Boundary Value
Problems (BVP), Initial Value Problem (IVP) and Optimal Control Problem (OCP) are premeditated. Such
evaluated SPM have a boundary layer correction (BLC) solution and an outer series solution. To improve
degenerate solution and to recover initial and boundary conditions a BLC solution of SPM is used. The DC
servo-motor control model of second order approximation is carried out for BVP, IVP and OCP. The results
thus obtained are presented in comparison to the precise solution. The efficacy of the present model is
evaluated in MATLAB environment.
Keywords: Discrete one-parameter servo system; optimal control problem; singular perturbation method;
boundary layer correction; boundary value problems; initial value problem.
Received: May 13, 2021. Revised: January 17, 2022. Accepted: February 7, 2022. Published: March 5, 2022.
1 Introduction
Conventionally the continuous and discrete control
systems are modeled based on high order
differential equations and difference equations. The
order of the system is described by the energy
storage elements like time constants, moments of
inertia, masses, capacitances and inductances
present in the system. Such obtained higher order
BVP and optimal solution leads to the usage of
shooting method [26]. SPM eases this crisis by
repressing the small- parameters. Thus developed
SPM is meant to eliminate systems firmness,
decreases control system order and results in exact
solution fulfilling the set boundary conditions.
Time-scale and singularly perturbed systems are
same [1-14]. The researchers are aspiring towards
discrete control systems [3-6, 9-20] rather than
matured continuous control systems [1, 2, 7, 18, 21-
24]. To study that gap, a DC servomotor with two
time scales is considered in this paper. Later the
studied model is developed as one parameter
Singularly Perturbed System (SPS) for evaluating
the discrete control systems [5-20, 25, 26]. Further,
SPM based discrete control systems developed as
BVP, IVP and OCP with performance index are
approximated and extended upto second order.
2. Discrete One-Parameter Problem
2.1 Discrete One-Parameter Problem for BVP
and IVP:
Here we are considering the two time scale discrete
control system described in Fig. 1. One and multi
parameter problems in discrete systems are
premeditated broadly [15, 16, 17]. We consider slow
and fast state variables are and
respectively, u(k) is input variable of step signal, ε is
the small parameter, A and B are state and input
matrices and have written state equation from Fig.2
and initial conditions are
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Fig. 1. A linear time invariant two-time-scale
system
=A
ε+Bu(k) (1)
Where A=
, B=
In the process of degeneration discard the state
variable of for ε is zero in (1) and initial
conditions
(0) and
(0).Consequentially the reduced system is
specified by
=
+Bu(k) (2)
The SPM which formulates the process of recovery
of the lost initial conditions is discussed in the next
secession.
2.2 Discrete One-Parameter Problem for OCP:
Here we are consider the singularly perturbed
one-parameter discrete control system (1). The
performance index to be minimized is
J=1/2
(3)
where w(k) = [ ε. indicates transpose.
D is a real positive-semidefinite symmetric matrix
of order (n1+n2) x (n1+n2). R is real positive-definite
symmetric matrix of order (1 x 1and N is a fixed
integer indicating the terminal (final) time. Here
note that the states are incorporated in an
appropriate manner to bring the resulting Two Point
Bundary value Problem(TPBVP) into SPS[26]. The
Hamiltonian of the problem is
where the co-state vector p(k) =
Using the results of digital optimal control theory
[18, 20-26]
(4)
Form (1) and (4), the co-states and states are
obtained as
=C
(5a)
Where C =
.
The final conditions of the system (5a) are
P0(N)= 0 and P1(N)= 0 (5b)
The optimal control is obtained as
uq(k) = -R-1
(k+1)], j=0,1. (5c)
Where q is the order of approximation.
The set of equations (5) constitutes the open loop
optimal control problem. The 2(n1+n2)th order
discrete TPBVP represented by (5) is in the
singularly perturbed form in the sense that the
degenerate TPBVP
(6)
Obtained by suppressing the small parameters in
(5a) is of order 2n1 and can satisfy the boundary
conditions
and
(7)
In general, the other boundary conditions are
and
(8)
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That is the boundary conditions
are lost during the time spent degeneration and the
deficiency of these boundary conditions adds to the
presence of boundary layers at initial and terminal
points. The 2n2 boundary conditions lost during the
time spent degeneration are recuperated by the
accompanying particular strategy which gives a
approximate answer for the stiff/hardened TPBVP
addressed by (5c). Subsequently it results in a
problematic control.
3. Singular Perturbation Method
SPM consists of two solutions, the outer and
boundary layer solution. The initial and boundary
conditions of SPM, recovers the fast variable in the
discrete control systems. The stiffed small
parameter ‘ε’ results in outer solution. Then
boundary layers are produced due to loss of fast
variable and to improve the solution [15- 17, 20-
26]. First start the flow chart with degenerate
solution and then to zero-order solution to improve
it as show in Fig. 3 and 4. The outer and BLC
solutions are obtained from the respective equations
and conditions. The zero-order solution is obtained
with total series solution (TSS) addition
approximation of outer and BLC solution. This is
further enhanced and approximated from first to
higher order.
3.1 Singular perturbation method for IVP
The outer series solution with asymptotic power
series expansions, we gain a set of equations for the
zero-order, first-order and second-order
approximations. Then formulate boundary layer
corrections to enhance the degenerate solution
ensuring unique boundary layer solution[15]. Then
add them according to total series solution to get the
order of solutions which satisfy the given initial
conditions as show in Fig. 3.
3.2 Singular perturbation method for BVP
We are taking into consideration of two modes of
stable system. Here consider first mode is slow and
second mode is fast mode, the boundary layers are
occurred to stable fast mode at initial point[16, 17].
Now the problem has become two point BVP
(TPBVP) which is as show in Fig. 2. Then add them
according to total series solution to get the order of
solutions which satisfy the given initial and
boundary conditions as show in Fig. 3.
3.3 Singular perturbation method for OCP
Algorithm is similar to the SPM of a TPBVP of
discrete control systems as the formulation of this
optimal control problem results in TPBVP [20, 25,
26]. For a specific order of approximate solution,
first track down the external answer for states and
co-states. Then, at that point add the BLC
comparing to the most singular transformation.
Proceed with this cycle lastly add the BLC
comparing to the most singular transformation.
When a specific order of solution is gotten for states
and co-states utilizing total series solution, then
obtain the corresponding suboptimal control using
control law and asymptotic correctness [28-30].
4. Application of Discrete One
Parameter Singular Perturbation
Method(DOPSPM)
DC Servo Motor Analysis
The armature controlled DC servo motor may be
modeled as
=
ea(t) (9a)
where
x01 = Angular position of rotor
x02 = Angular velocity of rotor
x1 = Armature current
ea = Armature voltage
Ra= Armature resistance
La= Armature inductance
Kb = Back electromagnetic field constant
Kt =Torque constant
J = Moment of inertia of motor-load set
B = Viscous coefficient of motor-load set
The 125V, 1500rpm and 3hp DC servomotor
specifications are [27]:
Ra= 0.6Ω, La= 6mH, Kt= 0.7274, Kb= 0.6,
J = 0.093kgm, B = 0.008N-mrad-1s-1
Obviously armature inductance La is very small
leading to a two-time-scale singularly perturbed
system. This continuous system is discretized with a
sampling interval of 0.012 sec for first forward
difference. The control signal is taken as unit step
function. The subsequent system is shown as below:
=
+
u(k)
(9b)
Here = {,}= set of slow state variables,
represents fast state variables and u(k) is unit step
control function. The eigen spectrum of this system
1.0000; 0.8962; -0.0973)
is of the two time scale nature one with fast mode
and two slow modes. The one-parameter system
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with above eigen spectrum is represented as shown
in equation (9c).
=
ε+
u(k)
whereε=0.1. (9c)
IVP
The initial conditions are specified at initial point
(k=0)
x01(0) =1; x02(0) = 1; x1(0) = 1.
BVP
The boundary conditions are listed as
x01(10) =2; x02(10) = 2; x1(0) = 1.
The TPBVP illustrated as x01 and x02 is specified at
k =10 and x1 is specified at initial point (k=0).
The SPM discussed in the present paper solves the
BVP and IVP. The exact solution is compared and
illustrated with zeroth, first and second-order
approximation solutions. When we are drown
graphs between state variable x(k) vs k. By
observing the graph it clear including second order
solution was reached to exact solution as shown in
the Fig. 4 and Fig. 5 for IVP and BVP respectively.
The correct values of all solutions were clearly
tabulated as shown in the table 1 and table 2 for IVP
and BVP respectively.
OCP
The initial conditions are given as
x01(0) =1; x02(0) =1; x1(0) = 1. (10a)
and the final conditions are
p01(10) =0; p02(10) =0; p1(10) = 0. (10b)
and the performance index
J = ½ ′′
(11)
where R = 1, D =
, w(k) =
The singularly perturbed TPBVP of fourth-order
corresponding to (9) is
(12)
Utilizing the singularly perturbed strategy created in
the past segment, the degenerate, zero, first and
second-order solutions are assessed and contrasted
and the optimal siolution in table 3 and table 4.
Perceptions from this table are
The degenerate solution, acquired by making ε
equivalent to focus in (6), can't fulfill the limit states
of quick state and co-state determined as x1(0) = 1
and p1(10) = 0.
Fig. 2. Flow Chart for TPBVP
End
Star
t
Kronecker delta
initial conditions [26]
Hence computing correct Missing Initial Conditions (MIC) is the real
problem. xh and xf indicate homogeneous and forced solutions of x.
Obtain the forced solution from xf(0) to xf(N) by shooting process.
Force initialization as xf(0) = [x0(0); 0]
We compute the Missing Initial Conditions (MIC) [26].
Now solve the TPBVP as IVP using the given IC and MIC. The final boundary
and initial conditions are satisfied if MIC and the solution of x(k) is accurate.
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Fig. 3. Flow Chart for Singular Perturbation Methods
High-order solution
Zero-order solution
First-order solution
& q
Outer solution:
,v=0,1.
where q is desired approximation
order.
Boundary layer correction solutions:
& q
Initial / Boundary conditions [15, 17]
Total series solution [15, 17]
=
; =
(0) and (0).
get degenerate singularly perturbed form [15].
Obtained by suppressing the small parameterin the linear, DOPSPM
and can satisfy the boundary conditions of slow modes only.
... Aij is matrices of suitable dimensionality and
state vector, j=0, 1.
Start
Select matrices Aij and Bij suitable for
operation dimensionality. The control vector
is u(k)[15] is free from small parameters.
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The zero-request arrangement, gotten from (9), joins
BLCs and recuperates these limit conditions x1(0)
and p1(10).
The first-order solution further develops the zero-
order solution. The second-order solution further
develops the main request solution and is a lot
nearer to the optimal solution. The optimal solution
of the fourth-order singularly perturbed discrete
TPBVP given by (5a) is acquired by the technique
for integral capacities recommended for consistent
'hardened' issues. This requires a mathematical
calculation to be carried out on a computerized PC
[26]. Then again, by utilizing the present SPM, the
different series solutions are effectively acquired as
the firmness is taken out and simultaneously are
exceptionally near the optimal solution.
Subsequently it is seen that the singularly perturbed
strategy decreases the request as well as eliminates
the 'hardness' of the issue. This can be proven from
the eigen values of full and ruffian optimal control
system.
Eigen values of full optimal control system
= {10.7812; 1.2292; 1.0101; 0.9891; 0.8130; -
0.0921}
Eigenvalues of degenerate optimal control system
= {1.0010; 1.0000; 1.0000; 0.9990}
Fig. 4(a). Comparisons of various series solution of
x01/x01with exact solution for IVP
Fig. 4(b). Comparisons of various series solution of
x02/x02 with exact solution for IVP
Fig. 4(c). Comparisons of various series solution of
x1/x1with exact solution for IVP
Fig. 5(a). Comparisons of various series solution of
x01/x01with exact solution for BVP
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Fig. 5(b). Comparisons of various series solution of
x02/x02 with exact solution for BVP
Fig. 5(c). Comparisons of various series solution of
x1/x1with exact solution for BVP
Table 1. Comparison illustration of IVP exact
solution with various prior series solutions.
x(k)
Degener
ate
Solution
Zero
Order
Solutio
n
First
Order
Solutio
n
Second
Order
Solutio
n
Exact
Solutio
n
x01(0)
x02(0)
x1(0)
1
1
-1.2012
1
1
1
1
1
1
1
1
1
1
1
1
x01(1)
x02(1)
x1(1)
1.0120
0.9990
0.8000
1.0120
0.9990
0.8000
1.0120
1.0929
0.6000
1.0120
1.0929
0.6000
1.0120
1.0929
0.6000
x01(2)
x02(2)
x1(2)
1.0240
0.9980
0.8012
1.0240
0.9980
0.8012
1.0251
1.1669
0.5285
1.0251
1.1499
0.5530
1.0251
1.1481
0.5685
x01(3)
x02(3)
x1(3)
1.0360
0.9970
0.8024
1.0360
0.9970
0.8024
1.0392
1.2410
0.4394
1.0389
1.2097
0.5009
1.0389
1.2004
0.5085
x01(4)
x02(4)
x1(4)
1.0479
0.9960
0.8036
1.0479
0.9960
0.8036
1.0540
1.3151
0.3503
1.0535
1.2314
0.4425
1.0533
1.2469
0.4578
x01(5)
x02(5)
x1(5)
1.0599
0.9950
0.8048
1.0599
0.9950
0.8048
1.0698
1.3892
0.2612
1.0689
1.2931
0.4022
1.0683
1.2887
0.4121
x01(6)
x02(6)
x1(6)
1.0718
0.9940
0.8060
1.0718
0.9940
0.8060
1.0864
1.4634
0.1720
1.0847
1.3563
0. 3521
1.0837
1.3261
0.3712
x01(7)
x02(7)
x1(7)
1.0837
0.9930
0.8072
1.0837
0.9930
0.8072
1.1040
1.5376
0.0827
1.0999
1.3812
0.3220
1.0996
1.3596
0.3345
x01(8)
x02(8)
x1(8)
1.0957
0.9920
0.8084
1.0957
0.9920
0.8084
1.1225
1.6119
-0.0066
1.1171
1.3976
0.3110
1.1160
1.3897
0.3016
x01(9)
x02(9)
x1(9)
1.1076
0.9910
0.8096
1.1076
0.9910
0.8096
1.1418
1.6862
-0.0959
1.1398
1.4357
0.2315
1.1326
1.4166
0.2721
x01(10)
x02(10)
x1(10)
1.1195
0.9900
0.8108
1.1195
0.9900
0.8108
1.1621
1.7605
-0.1853
1.1505
1.4554
0.2253
1.1496
1.4407
0.2457
Table 2. Comparison illustration of BVP exact
solution with various prior series solutions
x(k)
Degene
rate
Solutio
n
Zero
Order
Solutio
n
First
Order
Solutio
n
Second
Order
Solutio
n
Exact
Solution
x01(0)
x02(0)
x1(0)
1.7586
2.0201
-2.4265
1.7586
2.0201
1.0000
1.7346
2.2785
1.0000
1.7281
2.4785
1.0000
1.7233
2.5162
1.0000
x01(1)
x02(1)
x1(1)
1.7829
2.0180
-0.4241
1.7829
2.0180
-0.4241
1.7620
2.3701
-0.9342
1.7604
2.5338
-1.1242
1.7535
2.6076
-1.2195
x01(2)
x02(2)
x1(2)
1.8071
2.0160
-0.4217
1.8071
2.0160
-0.4217
1.7904
2.3279
-0.7593
1.7856
2.4581
-0.8348
1.7847
2.4905
-0.8852
x01(3)
x02(3)
x1(3)
1.8313
2.0140
-0.4192
1.8313
2.0140
-0.4192
1.8184
2.2860
-0.7091
1.8151
2.3843
-0.7978
1.8146
2.4049
-0.8115
x01(4)
x02(4)
x1(4)
1.8554
2.0120
-0.4168
1.8554
2.0120
-0.4168
1.8458
2.2443
-0.6593
1.8437
2.3154
-0.7194
1.8435
2.3262
-0.7235
x01(5)
x02(5)
x1(5)
1.8796
2.0100
-0.4144
1.8796
2.0100
-0.4144
1.8727
2.2029
-0.6098
1.8715
2.2511
-0.6466
1.8714
2.2560
-0.6468
x01(6)
x02(6)
x1(6)
1.9037
2.0080
-0.4120
1.9037
2.0080
-0.4120
1.8992
2.1618
-0.5606
1.8985
2.1916
-0.5774
1.8985
2.1930
-0.5778
x01(7)
x02(7)
x1(7)
1.9278
2.0060
-0.4096
1.9278
2.0060
-0.4096
1.9251
2.1210
-0.5118
1.9248
2.1364
-0.5158
1.9248
2.1365
-0.5160
x01(8)
x02(8)
x1(8)
1.9519
2.0040
-0.4072
1.9519
2.0040
-0.4072
1.9506
2.0804
-0.4632
1.9505
2.0856
-0.4618
1.9504
2.0859
-0.4606
x01(9)
x02(9)
x1(9)
1.9759
2.0020
-0.4048
1.9759
2.0020
-0.4048
1.9755
2.0401
-0.4150
1.9755
2.0404
-0.4113
1.9755
2.0406
-0.4110
x01(10)
x02(10)
x1(10)
2.0000
2.0000
-0.4024
2.0000
2.0000
-0.4024
2.0000
2.0000
-0.3681
2.0000
2.0000
-0.3666
2.0000
2.0000
-0.3665
Table 3. Comparison of various series sub-optimal
solutions with the optimal solution
x(k)
Degener
ate
Solution
Zero
Order
Solution
First
Order
Solutio
n
Second
Order
Solutio
n
Optimal
Solution
x01(0)
x02(0)
x1(0)
p01(0)
p02(0)
1.0000
1.0000
-1.2022.
10.5375
10.4888
1.0000
1.0000
1.0000
10.5385
10.4898
1.0000
1.0000
1.0000
10.4532
6.3825
1.0000
1.0000
1.0000
10.4295
5.9598
1.0000
1.0000
1.0000
10.4177
5.2907
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p1(0)
u(0)
8.8133
0.0000
8.8123
0.0000
4.1386
-1.5571
4.0074
-0.6312
3.7934
-0.5352
x01(1)
x02(1)
x1(1)
p01(1)
p02(1)
p1(1)
u(1)
1.0130
0.9990
-1.2000
9.5385
9.3847
7.7859
0.0000
1.0120
0.9990
-1.2000
9.5385
9.3847
7.7859
0.0000
1.0120
1.0929
-1.9586
9.4657
4.9993
3.1562
-1.3542
1.0120
1.0929
-2.2596
9.4251
4.5943
2.9502
-0.6125
1.0120
1.0929
-2.4704
9.4177
4.5034
2.6762
-0.4302
x01(2)
x02(2)
x1(2)
p01(2)
p02(2)
p1(2)
u(2)
1.0239
0.9980
-1.19880
8.5265
8.2917
6.7710
0.0000
1.0239
0.9980
-1.19880
8.5265
8.2917
6.7710
0.0000
1.0250
0.9791
-1.4448
8.4537
4.6202
3.0628
-1.1535
1.0251
0.8701
-1.5948
8.4130
3.6142
2.7486
-0.5529
1.0251
0.8598
-1.6779
8.4057
3.5713
2.1513
-0.3355
x01(3)
x02(3)
x1(3)
p01(3)
p02(3)
p1(3)
u(3)
1.0359
0.9970
-1.19760
7.5025
7.2109
5.7677
0.0000
1.0359
0.9970
-1.19760
7.5025
7.2109
5.7677
0.0000
1.0358
0.8655
-1.3089
7.4286
3.2463
2.7648
-0.9552
1.0354
0.7455
-1.3487
7.4097
3.0946
2.0644
-0.4317
1.0354
0.7014
-1.3673
7.3806
2.8271
1.6777
-0.2569
x01(4)
x02(4)
x1(4)
p01(4)
p02(4)
p1(4)
u(4)
1.0479
0.9960
-1.1964
6.4666
6.1424
4.7762
0.0000
1.0479
0.9960
-1.1964
6.4666
6.1424
4.7762
0.0000
1.0472
0.7522
-1.1158
6.3918
2.8790
2.1586
-0.7593
1.0440
0.6227
-1.1005
6.3719
2.5074
1.8599
-0.3513
1.0438
0.5723
-1.0820
6.3452
2.2059
1.2845
-0.1903
x01(5)
x02(5)
x1(5)
p01(5)
p02(5)
p1(5)
u(5)
1.0598
0.9950
-1.1952
5.4186
5.0865
3.7966
0.0000
1.0598
0.9950
-1.1952
5.4186
5.0865
3.7966
0.0000
1.0565
0.6391
-1.1020
5.3444
1.9867
1.7568
-0.5658
1.0515
0.5713
-1.9725
5.3149
1. 8065
1.2087
-0.2096
1.0507
0.4701
-0.8510
5.3014
1.6858
0.9515
-0.1332
x01(6)
x02(6)
x1(6)
p01(6)
p02(6)
p1(6)
u(6)
1.0718
0.9940
-1.1940
4.3588
4.0432
2.8290
0.0000
1.0718
0.9940
-1.1940
4.3588
4.0432
2.8290
0.0000
1.0639
0.5262
-0.9958
4.2882
1.4679
1.0481
-0.3747
1.0598
0.4759
-0.7950
4.2698
1.3677
0.6489
-0.1789
1.0563
0.3897
-0.6604
4.2506
1.2459
0.6662
-0.0834
x01(7)
x02(7)
x1(7)
p01(7)
p02(7)
p1(7)
u(7)
1.0837
0.9930
-1.1928
3.2869
3.0128
1.8736
0.0000
1.0837
0.9930
-1.1928
3.2869
3.0128
1.8736
0.0000
1.0702
0.4136
-0.7058
3.2242
1.0305
0.8947
-0.1861
1.0653
0.4003
-0.6749
3.2200
0.9746
0.4084
-0.1318
1.0610
0.3273
-0.5025
3.1943
0.8688
0.4173
-0.0395
x01(8)
x02(8)
x1(8)
p01(8)
p02(8)
p1(8)
u(8)
1.0956
0.9920
-1.1916
2.2032
1.9953
0.9305
0.0000
1.0956
0.9920
-1.1916
2.2032
1.9953
0.9305
0.0000
1.0752
0.3012
-0.5689
2.1540
0.9031
0.6591
0.0000
1.0717
0.2926
-0.5007
2.1453
0.6958
0.1494
-0.0358
1.0649
0.2798
-0.3713
2.1332
0.5401
0.1976
0.0050
x01(9)
x02(9)
x1(9)
p01(9)
p02(9)
p1(9)
u(9)
1.1075
0.9910
-1.1904
1.1075
0.9910
0.0000
0.0000
1.1075
0.9910
-1.1904
1.1075
0.9910
0.0000
0.0000
1.0787
0.2990
-0.4085
1.0787
0.1891
0.1790
0.0000
1.0702
0.2607
-0.2325
1.0716
0.2259
0.0075
0.0000
1.0683
0.2446
-0.2514
1.0683
0.2446
-0.0251
0.0000
x01(10)
1.1194
1.1194
1.0910
1.0910
1.0712
x02(10)
x1(10)
p01(10)
p02(10)
p1(10)
0.9900
-1.1892
0.0000
0.0000
-0.0015
0.9900
-1.1892
0.0000
0.0000
0.0000
0.2770
-0.3715
0.0000
0.0000
0.0000
0.2395
-0.2007
0.0000
0.0000
0.0000
0.2208
-0.2433
0.0000
0.0000
0.0000
Table 4. Comparison of various series sub-optimal
solutions with the optimal solution
S. No.
Performance Index(PI)
1
Degenerate Solution
17.3087
2
Zero Order Solution
17.3087
3
First Order Solution
14.5304
4
Second Order Solution
13.8785
5
Optimal Solution
13.7855
5. Conclusion
Singular perturbation methodology in discrete
control systems is being developed for one
parameter discrete systems with two time scales. An
armature controlled DC servo system is taken as one
parameter system as armature inductance is very
small giving rise to two time scales. A first forward
difference discrete model is developed for the
continuous system considered. SPM is applied to
this DC servo system for BVP, IVP and optimal
control. The simulation results that obtained justify
the proposed method. The same system may be
considered for closed loop optimal control with one
parameter as future work.
References
[1] Saksena V. R., O’Reiley J. and Kokotovic P. V.,
Singular perturbations and time-scale methods in
control theory: Survey 1976-1983,
Automatica(1984), 20: 273-293.
[2] Yong Chen and Yongqiang Liu, Summary of
Singular perturbation modeling of multi-time scale
power systems, In 2005 IEEE/PES Transmission &
Distribution Conference & Exposition: Asia and
Pacific. Dalian, China: IEEE. pp. 1-4.
[3] Koichi F. and Kunihiko K, Bifurcation cascade as
chaotic itinerancy with multiple time scales, Chaos:
An Interdisciplinary J Nonlinear Science (2003),
13: 1041-1056.
[4] F. Hoppensteadt and W. Miranker, Multitime
methods for systems of difference equations, Stud.
Appl. Math., vol. 56, pp. 273–289(1977).
[5] D. S. Naidu, Singular Perturbations and Time Scales
in Control Theory and Applications: An Overview,
Dynamics of Continuous, Discrete and Impulsive
Systems Series B: Application and
Algorithm(2002), 9, 2, 233-278. ISSN: 1492-8760.
[6] Adel Tellili, Nouceyba Abdelkrim. Diagnosis of
discrete-time singularly perturbed systems based on
slow subsystem. ACTA Mechanical
Automatica(2014); 8: 175-180.
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.5
Kishor Babu Gunti, Sree Krishnarayalu Movva
E-ISSN: 2224-266X
47
Volume 21, 2022
[7] Dongdong Zheng, Wen-Fang Xie, Identification and
control for singularly perturbed systems using
multitime-scale neural networks, IEEE T on Neural
Networks and Learning Systems(2017); 28: 321-
333.
[8] Rajasekhar Ananthoju, A. P. Tiwari, and Madhu N.
Belur, A Two-Time-Scale Approach for Discrete-
Time Kalman Filter Design and Application to
AHWR Flux Mapping. IEEE T on Nuclear Sci.
(2016); 63: 359-370.
[9] Magdi S. Mahmoud , Yuliu Chen & Madan G.
Singh, Discrete two-time-scale systems,
International Journal of Systems Science, 17:8,
1187-1207(1986), doi:
10.1080/00207728608926881.
[10] Hardev Singh, Ronald H. Brown, D. Subbaram
Naidu., Discrete-time timescale analysis via a new
separation ratio and unified approach, International
Journal of Systems Science 34:6(2003), pages 403-
412.
[11] Patre, B. M., & Bandyopadhyay, B., Robust Control
for Two-Time-Scale Discrete Interval Systems,
Reliable Computing, 12(1), 45–
58(2006), doi:10.1007/s11155-006-2971-x
[12] E. C. Bekir, Singular perturbation methods for
discrete time systems, in Proc. IEEE Conf. Decis.
Control(1985), vol. 24, pp. 86–94.
[13] R. G. Phillips, Reduced order modelling and control
of two-timescale discrete systems, Int. J. Control,
vol. 31(1980), pp. 765–780.
[14] D.S Naidu, and D.B Price, Singular perturbations
and time scales in the design of digital flight control
systems, NASA Technical paper 2844. 1988,
doi:10.1007/BFb0015123.
[15] Naidu, D. S., & Kailasa Rao, A., Singular
perturbation method for initial-value problems with
inputs in discrete control systems. International
Journal of Control, 33(5), 953–
965(1981). doi:10.1080/00207178108922967
[16] K.W. Chang, Singular perturbations of a general
boundary value problem, SIAM J. Math.
Anal. 3 (1972), 520–526.
[17] Krishnarayalu, M. S. and Naidu, D. S., Singular
perturbation method for boundary value problems in
two-parameter discrete control systems, Int. J.
Systems Sci. 19 (1988) 2131–2143.
[18] Vasil'eva, A.B., Dmitriev, M.G. Singular
perturbations in optimal control problems. J Math
Sci 34, 1579–1629 (1986).
doi:10.1007/BF01262406
[19] Dmitriev M. and Kurina G. Singular perturbations
in control problems. Automation and Remote
Control Volume 67,Issue 1 (2006), pp 1–43,
doi:10.1134/S0005117906010012.
[20] M. Bidani, N. E. Radhy, B. Bensassi, Optimal
control of discrete-time singularly perturbed
systems, International Journal of Control 75:13,
pages 955-966(2002).
[21] Robert R. Wilde and Petar V. Kokotović, Optimal
open-and closed-loop control of singularly
perturbed linear systems, IEEE Trans. Automatic
Control AC-18 (1973), 616–626.
[22] P.J. Moylan and J.B. Moore, Generalizations of
singular optimal control theory, Automatica J.
IFAC 7 (1971), 591–598.
[23] Marvin I. Freedman and James L. Kaplan,
Perturbation analysis of an optimal control problem
involving Bang-Bang controls, J. Differential
Equations 25 (1977), 11–29.
[24] Sannuti, P. and Reddy, P. Optimal control of a
coupled-core nuclear reactor by a singular
perturbation method, IEEE Transactions on
Automatic Control, 20(6), 766– 769(1975),
doi:10.1109/tac.1975.1101096.
[25] G. K. Babu and M. S. Krishnarayalu, Suboptimal
control of singularly perturbed multiparameter
discrete control system, 2015 International
Conference on Power, Instrumentation, Control and
Computing (PICC), pp. 1-8(2015), doi:
10.1109/PICC.2015.7455794.
[26] Krishnarayalu M. S. and Kishore Babu G., Shooting
Methods for Two-Point Boundary Value Problems
of Discrete Control Systems. International Journal
of Computer Applications, Volume 111 – No
6(2015), pp 16-20, doi:10.5120/19543-1394.
[27] Jide Julius Popoola1, Modelling and Simulation of
Armature-Controlled Direct Current Motor Using
MATLAB. SSRG International Journal of Electrical
and Electronics Engineering (SSRG-IJEEE) –
volume 2 Issue 3. pp 1-18(2015), ISSN: 2348 –
8379.
[28] Kurina G., Nguyen Thi Hoai, Zero-Order
Asymptotic Solution of a Class of Singularly
Perturbed Linear-Quadratic Problems With Weak
Controls in a Critical Case, Optim. Control Appl.
Methods, 40:5 (2019), 859–879.
[29] Kurina G., Nguyen Thi Hoai, Higher Order
Asymptotic Approximation to a Solution of
Singularly Perturbed Optimal Control Problem With
Intersecting Solutions of the Degenerate Problem,
23rd International Conference on System Theory,
Control and Computing (ICSTCC), International
Conference on System Theory Control and
Computing, ed. Precup R., IEEE(2019), 727–732.
[30] G. A. Kurina, Asymptotic solution of singularly
perturbed linear-quadratic optimal control problems
with discontinuous coefficients, Comput. Math.
Math. Phys., 52:4 (2012), 524–547
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