LMI approach to normalised HN loop-shaping
design of power system damping controllers
R. Majumder, B. Chaudhuri, H. El-Zobaidi, B.C. Pal and I.M. Jaimoukha
Abstract: Application of the normalised HN loop-shaping technique for design and simplification
of damping controllers in the LMI framework is illustrated. A linearised model of the power system
is pre- and post-compensated using the loop-shaping approach. The problem of robust stabilisation
of a normalised coprime factor plant description is translated to a generalised HN problem. The
solution is sought numerically using LMIs with additional pole-placement constraints. This ensures
that the time-domain specifications are met besides robust stabilisation.
1
Introduction
Damping inter-area oscillations is one of the major concerns
for electric power system operators [1]. With ever-increasing
power exchange between utilities over the existing transmission network, the problem has become even more
challenging. Secure operation of power systems thus
requires application of robust controllers to damp these
inter-area oscillations. Power system stabilisers (PSSs) are
the most commonly used devices for this purpose [2].
Nowadays, flexible AC transmission systems (FACTS) [3]
devices are receiving growing importance as an alternative
to transmission system reinforcement, which is otherwise
restricted for economic and environmental reasons. Besides
power flow and voltage control, supplementary control is
being incorporated into these FACTS devices to damp
inter-area oscillations at not much additional cost.
The objective of the control design exercise is to ensure
adequate damping under all credible operating conditions.
Recently, many researchers have investigated the use of HN
optimisation [4–6] and m-synthesis [7, 8] for power system
damping control design. The resulting controller has the
ability to maintain stability and achieve the desired
performance while being insensitive to the perturbations.
A mixed-sensitivity design formulation with linear matrix
inequality (LMI)-based solution is illustrated in [9–11]. In
this approach, the designer specifies the performance
requirements in terms of the weighted closed-loop transfer
functions and a stabilising controller is obtained that
satisfies these criteria. One of the difficulties with this
approach is that the appropriate selection of the mixedsensitivity weights is not straightforward. Moreover, it is
possible for the closed-loop specifications to be made
without considering the properties of the nominal plant,
which can often be undesirable. The selection of weights for
the relevant closed-loop transfer functions, such as the
sensitivity and the complementary sensitivity functions, is
r IEE, 2005
IEE Proceedings online no. 20045175
doi:10.1049/ip-gtd:20045175
Paper first received 14th September 2004 and in final revised form 23rd June
2005
The authors are with the Department of Electrical and Electronic Engineering,
Imperial College London, Exhibition Road, London SW7 2BT, UK
E-mail: b.chaudhuri@imperial.ac.uk
952
done without much regard to the actual limitations of the
closed loop. This may lead to unrealistic designs.
A loop-shaping design methodology, which does not
suffer from the above drawbacks, was proposed by
McFarlane and Glover [12–14]. It combines the characteristics of both classical open-loop-shaping and HN optimisation. Zhu et al. [15] and Farsangi et al. [16] have applied
this technique for power system damping control design.
However, the problem was solved analytically using the
standard normalised coprime factorisation approach,
wherein time-domain specifications in terms of minimum
damping ratios (pole-placement) could not be considered
explicitly in the design stage. Although the analytic
procedure has a non-iterative solution, the design requirements can only be captured through proper selection of
weights, which is not always straightforward.
In this paper, we have converted the problem of robust
stabilisation of a normalised coprime factor plant description into a generalised HN problem. The problem is solved
using LMIs [17–19] with additional pole-placement constraints. In addition to robust stabilisation of the shaped
plant, a minimum damping ratio can thus be ensured for
the critical inter-area modes.
2
Design approach
The normalised coprime factorisation approach for loopshaping design was proposed by McFarlane and Glover
[12–14]. The two-stage design procedure is based on HN
robust stabilisation combined with classical loop-shaping.
First, the open-loop plant is augmented by pre- and postcompensators to give the desired shape to the open-loop
frequency response. Then the resulting shaped plant is
robustly stabilised with respect to coprime factor uncertainties by solving the HN optimisation problem. In this paper,
the standard normalised coprime factorisation-based problem is converted into a generalised HN problem in the
LMI framework with additional pole-placement constraints
[18, 19].
2.1
Loop-shaping design
The basic principle of HN loop-shaping design is to preand post-compensate the plant for shaping the open-loop
frequency response. The idea is to specify the performance
requirements prior to robust stabilisation [13]. If W1 and
W2 are the pre- and post-compensators, respectively, the
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
shaped plant Gs is given by Gs ¼ W2 GW1 , as shown in
Fig. 1. The controller K is designed by solving the robust
stabilisation problem for the shaped plant Gs, as described
in Section 2.2. The equivalent feedback controller for the
original plant G is obtained by augmenting the designed
controller K with the compensators, i.e. Keq ¼ W1 KW2 , as
shown in Fig. 1. The primary task in loop-shaping design is
to choose appropriate pre- and post-compensators. Based
on the recommendations in [20], the following guidelines are
normally used for shaping the open-loop plant [21]:
such that
MM þ NN ¼ I
where M ðsÞ ¼ M ðsÞ. The block diagram for the
normalised coprime factorisation robust stabilisation problem is shown in Fig. 2.
G∆
+
N
Gs
G
W2
b
G
K
W2
c
Fig. 1
+
M −1
Normalised coprime factor robust stabilisation problem
K eq
Loop-shaping design procedure
a Shaped plant
b Compensated plant
c Equivalent controller
It should, however, be noted that the procedure is specific to
the particular application and some trial and error is
involved. The maximum stability margin emax (see (3))
provides an indication as to whether the choice of the
compensators is appropriate or not. If the margin is too
small, emax o0:2, then the compensators need to be
modified following the above guidelines. When emax 40:2,
the choice is considered to be acceptable.
2.2 Robust stabilisation with
pole-placement
The robust stabilisation of a plant described in terms of its
normalised coprime factors is discussed in detail in [12, 14].
A normalised left coprime factorisation of a plant GðsÞ is
GðsÞ ¼ M 1 ðsÞN ðsÞ
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
The largest positive number, eð¼ emax Þ, such that
GD ¼ ðM þ DM Þ1 ðN þ DN Þ can be stabilised by a
single controller, K, for all D ¼ ½DN ; DM ; kDk1 oe is
given by
1
I
1 1 emax ¼ inf ðI
GKÞ
M
ð3Þ
K
K
1
where K is chosen over all stabilising controllers [12].
The objective of the robust stabilisation problem is to
ensure stability under uncertainties in the plant model. The
larger is the uncertainty against which the controller is
able to ensure stability, the better is the design. In other
words, obtaining a robust stabilising controller is equivalent
to maximising the uncertainty measure e. Therefore, the
control design problem boils down to minimising the cost
function
I
1 1 ðI
GKÞ
M
ð4Þ
min
K
K2S
1
K
W1
+
W2
a
W1
−
K
Fig. 2
G
∆M
∆N
The plant inputs and outputs are properly scaled to
improve conditioning of the design problem.
The compensators are chosen in such a way that singular
values of the shaped plant are desirable. This would
normally correspond to high gain at low frequencies, rolloff rates of approximately 20 dB/decade at the desired
bandwidth(s), with higher rates at high frequencies [21].
Integral action is added at low frequencies.
W1
ð2Þ
T
ð1Þ
where S is the set of all stabilising controllers.
The above formulation can be used for robust stabilisation of systems. However, it can not be extended to robust
stabilisation with pole-placement. Therefore, the problem
needs to be translated from the standard coprime factor
robust stabilisation formulation (4) to a generalised HN
problem [21, 22] format, which is described next.
As M, N are the normalised left coprime factors of G,
satisfying (1) and (2), introduction of ½M N in (4) does not
affect the overall infinity norm. Therefore,
I
1 1 K ðI GKÞ M 1
I
1
1
¼
ðI
GKÞ
M
½
M
N
K
1
I
1
¼
K ðI GKÞ ½ I G 1
S
SG
¼
K S K SG 1
where S ¼ ðI GKÞ1 is the sensitivity. The problem
of robust stabilisation of the standard normalised
coprime factor plant description is thus translated into a
953
generalised HN problem, which can be equivalently stated
as follows:
S
SG min
ð5Þ
K S K SG 1
K2S
S-plane
It can be noted that the closed-loop transfer functions in (5)
corresponds to robustness against the following specific
plant/controller perturbations:
inner
angle θ
S: parametric perturbation of the plant
SG: additive perturbation of the controller
K S: additive perturbation of the plant
K SG: input multiplicative perturbation of the plant.
real
0
−infinity
Therefore, minimising (5) maximises the amount of
allowable perturbations with guaranteed stability.
The generalised plant [21, 22] P for minimising the
infinity norm of the closed-loop quantities in (5) is given by:
2
3
2
3
A B 0 B
———————
A B1 B2
6C 0 I 0 7
5
7 4
ð6Þ
P ¼6
4 0 0 0 I 5 ¼ C1 D11 D12
C2 D21 D22
C 0 I 0
where B1 ¼ ½ B 0 , B2 ¼ ½B, C1 ¼ ½ C 0 T , D11 ¼
0 I
, D12 ¼ ½ 0 I T and D21 ¼ ½ 0 I .
0 0
Ak Bk
The controller, K ¼
can be obtained by
Ck Dk
solving HN optimisation problem given in (5).
For analytical solution, additional constraints (e.g. poleplacement) cannot be imposed in the synthesis stage.
Therefore, in this work, the solution is obtained using an
LMI formulation [18, 19] as it offers the flexibility to impose
additional pole-placement constraints that directly addresses
the damping improvement issue.
The transfer matrix between the exogenous inputs and
outputs of P in (6) is given by:
T ðsÞ ¼ Ccl ðsI Acl Þ1 Bcl þ Dcl
where
A þ B2 Dk C2 B2 Ck
Acl ¼
Bk C2
Ak
B1 þ B2 Dk D21
Bcl ¼
Bk D21
Ccl ¼ ½ C1 þ D12 Dk C2
Dcl ¼ D11 þ D12 Dk D21
ð7Þ
D12 Ck ð8Þ
all poles should be
placed within conic sector
Fig. 3
Conic sector region for LMI pole placements
with apex at the origin and internal angle y if and only if
there exists X 40 such that [19]
!
sin yðAcl X þ XATcl Þ cos yðAcl X XATcl Þ
o0 ð13Þ
cos yðXATcl Acl X Þ sin yðAcl X þ XATcl Þ
The damping ratio of the placed poles within the conic
sector is at least equal to cos y2 [17]. The value is
appropriately chosen to achieve the required specifications.
Therefore, the controller design exercise boils down to
solving the matrix inequalities (12) and (13). However, both
(12) and (13) contains AclX, BclX and CclX. Acl and Ccl are
functions of the controller parameters Ak, Bk, Ck and Dk
and the controller parameters themselves are functions of X
making the products AclX, BclX, CclX nonlinear in X. To
convert the problem into a linear one i.e. to obtain the set of
LMIs, a change of variable is required. The expression for
the new controller variables and the LMIs in terms of the
transformed variables are given in the Appendix. Interested
readers can refer to [18, 19] for further details.
ð9Þ
3
ð10Þ
ð11Þ
With the help of the bounded real lemma [23], it is possible
to show that the HN norm of Tzw is less than g and the
closed-loop system stable if there exists a symmetric X such
that
2 T
3
T
Acl X þ XAcl Bcl XCcl
4
ð12Þ
BTcl
gI DTcl 5o0
Ccl X
Dcl gI
The pole-placement objective is formulated in terms of LMI
regions of the complex plane. There exists a general class of
LMI regions for the above purpose, i.e. discs, conic sectors,
vertical/horizontal strips etc. or intersections of the above. A
‘conic sector’ of inner angle y and apex at the origin is an
appropriate LMI region for power system damping control
application as it defines a minimum damping for the
dominant closed-loop poles (see Fig. 3). The closed-loop
system is guaranteed to have all its pole in the conic sector
954
imag
Case study
The control design and simplification exercise was carried
out on a 16-machine, five-area study system, as shown in
Fig. 4. The details of the study system with all the
parameters can be found in [10, 24]. A thyristor-controlled
series capacitor (TCSC) is installed in the system for
strengthening the transmission corridor between NYPS and
area 5. An eigenvalue analysis on the linearised model of the
system revealed that the system has three critical inter-area
modes, as shown in Table 1. The objective is to damp these
modes by designing a supplementary damping controller
for the TCSC. Appropriate feedback stabilising signals were
chosen for each mode using the modal observability
analysis, see [10] for details. The open-loop plant was
constructed using the linearised system matrix A, the input
matrix B corresponding to the output of the TCSC and the
output matrix C corresponding to the measured signals.
3.1
Loop-shaping
The original plant was of 132th order, which could be
simplified to a 9th order equivalent using balanced
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
area 5
K controller to be designed
G16
NETS
NYPS
G3
G7
16
G2
03
G5
07
23
02
05
G4
59
19
G12
12
36
17
21
37
35
49
32 38
30
24
46
11
28
29
08
G10
47
54
41
40
48
01
G8
14
G1
G14
remote signal links
Fig. 4
42
10
53
25
26
15
31
G11
09
G15
33
55
27
G9
45
34
61
52
22
TCSC
50
51
area 4
56
68
39
43
57
66
67
18
K
u
60
58
64
G6
y
13
63
20 65
04
06
62
G13
area 3
Sixteen-machine five-area study system with TCSC
25
Table 1: Inter-area modes of study system
f (Hz)
0.0626
0.3913
0.0435
0.5080
0.0554
0.6232
20
15
10
gain, dB
z
25
5
0
−5
full plant
reduced plant
−10
20
−15
gain, dB
15
− 20
10 −2
10 −1
100
101
102
frequency, rad/s
103
104
10
Fig. 6
5
Frequency response of pre-compensator
performance requirements. The transfer function and the
frequency response of the pre-compensator is (see Fig. 6):
0
W1 ðsÞ ¼
−5
10 −1
100
101
102
frequency, rad/s
Fig. 5
plant
Frequency response of full-order plant and reduced-order
truncation [21] technique. The frequency response of the
full-order plant and reduced-order plant is shown in Fig. 5.
Prior to solving the HN problem, the open-loop plant
had to be shaped following the recommendations in Section
2.1. A pre-compensator was used to introduce an integral
action in the low frequency region and also to reduce the
overall gain of the plant in order to suit the desired
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
0:106s þ 0:1096
s þ 0:001
ð14Þ
The three output channels were scaled with static weighting
properly to improve the conditioning of the open-loop
plant. Scale factors of 1.0, 2.0, and 0.6 were used for the
1st, 2nd and 3rd outputs, respectively, resulting in a postcompensator W2 of the form:
2
3
1:0 0
0
W2 ¼ 4 0 2:0 0 5
ð15Þ
0
0 0:6
The frequency response of the resulting shaped plant and
the original reduced plant is shown in Fig. 7.
955
30
Table 2: Open- and closed-loop system dampings
reduced plant
shaped plant
20
Open loop
10
0
gain, dB
Analytical solution
LMI coprime
Damping Frequency Damping Frequency Damping Frequency
(Hz)
(Hz)
(Hz)
− 10
0.0626
0.3013
0.1042
0.3941
0.1681
0.3913
− 20
0.0435
0.5080
0.0288
0.3968
0.1410
0.4926
0.0554
0.6232
0.0855
0.4989
0.1154
0.6344
0.0812
0.5399
0.0560
0.6267
− 30
−40
− 50
− 60
− 70
10 −2
Fig. 7
plant
3.2
10 −1
100
101
102
frequency, rad/s
103
104
Frequency response of reduced-order original and shaped
Control Design
The matrices A, B, C and D of the shaped plant are used to
formulate the generalised plant P following (6). The hinfmix
function available in the LMI Control Toolbox [17] was
used to perform the necessary computations. The poleplacement constraint was specified in terms of a conic sector
as shown in Fig. 3 with apex at the origin and an inner
angle of 2 cos1 ð0:15Þ, which ensures a minimum damping
of 0.15 for all the three inter-area modes. The design
converged to an optimum HN performance index gopt of
4.873. As stated earlier, if the problem had been solved
3.3
Simulation results
One of the most severe disturbances stimulating poorly
damped inter-area oscillations is a three-phase fault in one
fault at bus 53 with line 53−54 out
fault at bus 60 with autoreclosing
−5
−10
without control
with control
without control
with control
angle(G1−G15), deg
− 10
angle(G1−G15), deg
analytically using the standard normalised coprime factorisation approach, time-domain specifications in terms of
minimum damping ratios (pole-placement) could not be
considered explicitly in the design stage. Table 2 gives the
open-loop damping of the critical inter-area modes and the
closed-loop damping with both approaches. It is seen from
this Table that two control modes with 0.1042 damping at
0.3941 Hz and 0.0812 damping at 0.5399 Hz are introduced
and the inter-area mode damping remains poor at 0.0288 at
0.3968 Hz and 0.0855 damping at 0.4989 Hz when the
controller is synthesised analytically.
The order of the designed controller by solving the LMIs
was 11, which was subsequently simplified to a 10th order
one using balanced truncation. The state–space representation of the reduced controller is given in the Appendix.
− 15
− 20
− 25
−15
−20
−25
− 30
− 35
0
5
10
15
20
−30
25
0
5
10
time, s
fault at bus 53 with line 27− 53 out
angle(G1− G15), deg
angle(G1− G15), deg
without control
with control
−10
− 18
− 20
− 22
− 24
−15
−20
−25
−30
− 26
0
5
10
15
time, s
956
25
fault at bus 60 with line 60−61 out
without control
with control
− 16
Fig. 8
20
−5
− 14
− 28
15
time, s
20
25
−35
0
5
10
15
20
25
time, s
Dynamic response of system
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
fault at bus 53 with line 53−54 out
fault at bus 60 with autoreclosing
55
70
without control
with control
50
angle(G14−G13), deg
angle(G14− G13), deg
60
50
40
30
20
without control
with control
45
40
35
0
5
10
15
20
30
25
0
5
10
time, s
fault at bus 53 with line 27− 53 out
20
25
fault at bus 60 with line 60−61 out
70
55
without control
with control
without control
with control
60
50
angle(G14−G13), deg
angle(G14−G13), deg
15
time, s
45
40
50
40
30
35
0
5
10
15
20
20
25
0
5
10
time, s
Fig. 9
15
20
25
time, s
Dynamic response of system
fault at bus 53 with line 53−54 out
fault at bus 60 with autoreclosing
50
45
without control
with control
40
40
angle(G15 − G13), deg
angle(G15− G13), deg
without control
with control
30
20
35
30
25
20
10
0
5
10
15
20
15
25
0
5
10
time, s
20
25
fault at bus 60 with line 60 − 61 out
fault at bus 53 with line 27− 53 out
50
45
without control
with control
without control
with control
angle(G15− G13), deg
40
angle(G15 − G13), deg
15
time, s
35
30
25
40
30
20
20
15
0
5
10
15
20
time, s
Fig. 10
25
10
0
5
10
15
20
25
time, s
Dynamic response of system
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
957
fault at bus 53 with line 53−54 out
fault at bus 60 with autoreclosing
80
without control
with control
70
percentage compensation of TCSC
percentage compensation of TCSC
80
60
50
40
30
20
10
0
5
10
15
20
60
50
40
30
20
10
25
without control
with control
70
0
5
10
time, s
fault at bus 53 with line 27 − 53 out
60
50
40
30
20
0
5
10
15
20
60
50
40
30
20
25
without control
with control
70
0
5
time, s
Fig. 11
25
fault at bus 60 with line 60−61 out
without control
with control
70
10
20
80
percentage compensation of TCSC
percentage compensation of TCSC
80
15
time, s
10
15
20
25
time, s
Dynamic response of system
of the key transmission corridors. For temporary faults, the
circuit breaker ‘auto-recloses’ and normal operation is
restored: otherwise one or two lines might have to be taken
out. There might be other types of disturbances in the
system like change of load characteristics, sudden change in
power flow etc., which are less severe compared to faults
and are not considered here.
To evaluate the performance and robustness of the
designed controller, simulations were carried out corresponding to some of the probable fault scenarios in
the NETS and NYPS interconnection. There are three
transmission corridors between NETS and NYPS connecting buses 60–61, 53–54 and 27–53, respectively. Each of
these corridors consists of two tie-lines. Outage of one of
these lines weakens the corridor considerably. The following
disturbances were considered for simulation. A three-phase
solid fault for 80 ms (five cycles)
The simulations were carried out in Matlab Simulink
for 25 s employing the trapezoidal method with a variable
step size. The disturbance was created 1 s after the
start of the simulation. The dynamic response of the
system following the disturbance is shown in Figs. 8–10.
These Figures exhibit the relative angular separation
between the generators located in separate geographical
regions. Inter-area oscillations are mostly manifested in
these angular differences and are therefore chosen
for displaying. It can be seen that inter-area oscillation
settles within 12–15 s for a range of post-fault operating
conditions and thus abides by the robustness requirement as
well. A hard limit of 0.1–0.8 was imposed on the variation
of the percentage compensation of the TCSC, which is
shown in Fig. 11.
at bus 60 followed by auto-reclosing
breaker
at bus 53 followed by outage of one
connecting buses 53 and 54
at bus 53 followed by outage of one
connecting buses 27 and 53
at bus 60 followed by outage of one
connecting buses 60 and 61.
4
of the circuit
of the tie-lines
of the tie-lines
of the tie-lines
The designed controller is supposed to settle the interarea oscillations within 12–15 s following any of the
disturbances. Moreover, it should be able to achieve this
following any of the above disturbances (robustness)
although the design is based on a nominal operating
condition (no outage).
958
Conclusions
This paper has demonstrated the application
of the normalised HN loop-shaping technique for design
of damping controllers in the LMI framework. The
first step in this design approach was to pre- and postcompensate the linearised model of the power system
using the McFarlane and Glover loop-shaping technique.
The problem of robust stabilisation of a normalised
coprime factor plant description was translated to a
generalised HN problem. The solution was sought
numerically using LMIs with additional pole-placement
constraints. By imposing the constraints, a minimum
damping ratio could be ensured for the critical inter-area
modes, which resulted in settling of oscillations within the
specified time.
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
5
Acknowledgment
The authors would like to acknowledge EPSRC, UK, for
funding this research.
6
B^ ¼ VBk þ SB2 Dk
ð17Þ
C^ ¼ Ck U T þ Dk C2 R
ð18Þ
^ ¼ Dk
D
ð19Þ
UV T ¼ I RS
ð20Þ
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7 Appendix
7.1 Obtaining the controller through LMIs
To linearise the matrix inequalities described in Section 2, a
change of variable is required. The new controller variables
are defined as:
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
If U and V have full row rank, then the controller
matrices Ak, Bk, Ck and Dk can always be computed
^ B,
^ D,
^ C,
^ R, S, U and V. Moreover, the controller
from A,
matrices can be determined uniquely if the controller
order is chosen to be equal to that of the generalised
plant [19].
From the matrix inequalities described in Section 2, the
following LMIs are obtained in terms of the new controller
variables:
R I
40
ð21Þ
I S
C11
C21
sin yð þ T Þ
cos yðT Þ
CT21
o0
C22
ð22Þ
cos yð T Þ
o0
sin yðT þ Þ
ð23Þ
where
C11 ¼
AR þ RAT þ B2C^ þC^T BT2
^ 21 ÞT
ðB1 þ B2DD
C21 ¼
C22 ¼
^ 2 ÞT
A^ þ ðA þ B2DC
^
C1 R þ D12 C
^ 21
B1 þ B2 DD
gI
^ 21
SB1 þBD
^ 21
D11 þ D12 DD
^ 2 þ C TB^T
AT S þ SA þBC
2
^ 2
C1 þ D12DC
ð24Þ
^ 2 ÞT
ðC1 þ D12 DC
gI
ð25Þ
ð26Þ
¼
AR þ B2C^
A^
^ 21
A þ B2DD
^
SA þBC2
ð27Þ
The system of LMIs in (21), (22) and (23) can be solved
^ B,
^ C^ and D.
^ A full-rank factorisation
for R, S, A,
T
UV ¼ I RS of the matrix I RS is computed
via the SVD approach such that U and V are square
^ B,
^ D,
^ C,
^ U
and invertible. With known values R, S, A,
and V the system of linear equations (16), (17),
(18) and (19) is solved for Dk, Bk, Ck and Ak in that order.
The controller is obtained as KðsÞ ¼ Dk þ Ck
ðsI Ak Þ1 Bk .
7.2 Controller state-space data
A^ ¼VAk U T þ VBk C2 R þ SB2 Ck U T
þ SðA þ B2 Dk C2 ÞR
where
ð16Þ
The state–space representation of the designed controller
is given by the A, B, C and D matrices in (28), (29),
959
(30) and (31):
2
0:001
6
6 0
6
6 0
6
6
6 0
6
6
6 0
A ¼6
6 0
6
6
6 0
6
6
6 0
6
6 0
4
0
0:0018
0:092
0:0118
2:445
23:12
0:541
30:33
4:822
44:25
3:551
66:69
22:94
25:38 15:37
175
38:68
201:1
70:74
558:7
84:35
667:5
72:95
416:8
70:38
484:3
13:46
76:54
10:17
96:49
2:284
38:9
1:919
67:49
6:7
130:5
16:22
141:6
0:059
0:001
46:4
0:014
8:171
92:96
4:268
3:092
63:49
29:81
52:44
330:3
17:36
59:18
1102
11:37
21:82
797:2
43:73
55:83
148:4
10:01
9:31
92:71
20:7
9:99
247
22:86
10:97
0:016
0
6
B ¼ 1 10 4 0
0
4
0:084
0:003 0:001
0:046
0:131 0:382 0:189
0:083 0:167
0:216
0:167
0:063
0:495
0:275
0:208
1:531 0:1057 0:202
0:416
0:039 0:118
3
7
0:085 0:095 5
0:208 0:318
ð29Þ
C ¼½ 0:109 0:002 0:0098 0:0012 0:0056
ð30Þ
0:014 0:006 0:0001 0:0017 0:0089 3
7
0:896 8:408 7
7
2:185 16:05 7
7
7
11:39 28:04 7
7
7
4:388 125:1 7
7
17:68
320:7 7
7
7
3:497 262:9 7
7
7
2:473 36:97 7
7
9:978 2:857 7
5
0:433
0:014
0:151
0:398
58:96
49:2
0:132
2
0:053
D ¼ ½0
0
0
ð31Þ
76:76
ð28Þ
960
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005