LMI approach to normalised H∞ loop-shaping design of power system damping controllers

LMI approach to normalised HN loop-shapingdesign 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 simplificationof damping controllers in the LMI framework is illustrated. A linearised model of the power systemis pre- and post-compensated using the loop-shaping approach. The problem of robust stabilisationof a normalised coprime factor plant description is translated to a generalised HN problem. Thesolution is sought numerically using LMIs with additional pole-placement constraints. This ensuresthat the time-domain specifications are met besides robust stabilisation.

1 Introduction

Damping inter-area oscillations is one of the major concernsfor electric power system operators [1]. With ever-increasingpower exchange between utilities over the existing trans-mission network, the problem has become even morechallenging. Secure operation of power systems thusrequires application of robust controllers to damp theseinter-area oscillations. Power system stabilisers (PSSs) arethe most commonly used devices for this purpose [2].Nowadays, flexible AC transmission systems (FACTS) [3]devices are receiving growing importance as an alternativeto transmission system reinforcement, which is otherwiserestricted for economic and environmental reasons. Besidespower flow and voltage control, supplementary control isbeing incorporated into these FACTS devices to dampinter-area oscillations at not much additional cost.

The objective of the control design exercise is to ensureadequate damping under all credible operating conditions.Recently, many researchers have investigated the use ofHN

optimisation [4–6] and m-synthesis [7, 8] for power systemdamping control design. The resulting controller has theability to maintain stability and achieve the desiredperformance while being insensitive to the perturbations.A mixed-sensitivity design formulation with linear matrixinequality (LMI)-based solution is illustrated in [9–11]. Inthis approach, the designer specifies the performancerequirements in terms of the weighted closed-loop transferfunctions and a stabilising controller is obtained thatsatisfies these criteria. One of the difficulties with thisapproach is that the appropriate selection of the mixed-sensitivity weights is not straightforward. Moreover, it ispossible for the closed-loop specifications to be madewithout considering the properties of the nominal plant,which can often be undesirable. The selection of weights forthe relevant closed-loop transfer functions, such as thesensitivity and the complementary sensitivity functions, is

done without much regard to the actual limitations of theclosed loop. This may lead to unrealistic designs.

A loop-shaping design methodology, which does notsuffer from the above drawbacks, was proposed byMcFarlane and Glover [12–14]. It combines the character-istics of both classical open-loop-shaping andHN optimisa-tion. Zhu et al. [15] and Farsangi et al. [16] have appliedthis technique for power system damping control design.However, the problem was solved analytically using thestandard normalised coprime factorisation approach,wherein time-domain specifications in terms of minimumdamping ratios (pole-placement) could not be consideredexplicitly in the design stage. Although the analyticprocedure has a non-iterative solution, the design require-ments can only be captured through proper selection ofweights, which is not always straightforward.

In this paper, we have converted the problem of robuststabilisation of a normalised coprime factor plant descrip-tion into a generalised HN problem. The problem is solvedusing LMIs [17–19] with additional pole-placement con-straints. In addition to robust stabilisation of the shapedplant, a minimum damping ratio can thus be ensured forthe critical inter-area modes.

2 Design approach

The normalised coprime factorisation approach for loop-shaping 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 post-compensators to give the desired shape to the open-loopfrequency response. Then the resulting shaped plant isrobustly stabilised with respect to coprime factor uncertain-ties by solving theHN optimisation problem. In this paper,the standard normalised coprime factorisation-based pro-blem is converted into a generalised HN problem in theLMI framework with additional pole-placement constraints[18, 19].

2.1 Loop-shaping designThe basic principle of HN loop-shaping design is to pre-and post-compensate the plant for shaping the open-loopfrequency response. The idea is to specify the performancerequirements prior to robust stabilisation [13]. If W1 andW2 are the pre- and post-compensators, respectively, the

The authors are with the Department of Electrical and Electronic Engineering,Imperial College London, Exhibition Road, London SW7 2BT, UK

E-mail: [email protected]

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 June2005

952 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005

shaped plant Gs is given by Gs ¼ W2GW1, as shown inFig. 1. The controller K is designed by solving the robuststabilisation problem for the shaped plant Gs, as describedin Section 2.2. The equivalent feedback controller for theoriginal plant G is obtained by augmenting the designedcontroller K with the compensators, i.e. Keq ¼ W1KW2, asshown in Fig. 1. The primary task in loop-shaping design isto choose appropriate pre- and post-compensators. Basedon the recommendations in [20], the following guidelines arenormally used for shaping the open-loop plant [21]:

� The plant inputs and outputs are properly scaled toimprove conditioning of the design problem.

� The compensators are chosen in such a way that singularvalues of the shaped plant are desirable. This wouldnormally correspond to high gain at low frequencies, roll-off rates of approximately 20dB/decade at the desiredbandwidth(s), with higher rates at high frequencies [21].

� Integral action is added at low frequencies.

It should, however, be noted that the procedure is specific tothe particular application and some trial and error isinvolved. The maximum stability margin emax (see (3))provides an indication as to whether the choice of thecompensators is appropriate or not. If the margin is toosmall, emaxo0:2, then the compensators need to bemodified following the above guidelines. When emax40:2,the choice is considered to be acceptable.

2.2 Robust stabilisation withpole-placementThe robust stabilisation of a plant described in terms of itsnormalised coprime factors is discussed in detail in [12, 14].A normalised left coprime factorisation of a plant GðsÞ is

GðsÞ ¼ M�1ðsÞNðsÞ ð1Þ

such that

MM� þ NN � ¼ I ð2Þwhere M�ðsÞ ¼ MT ð�sÞ. The block diagram for thenormalised coprime factorisation robust stabilisation pro-blem is shown in Fig. 2.

The largest positive number, eð¼ emaxÞ, such that

GD ¼ ðM þ DM Þ�1ðN þ DN Þ can be stabilised by asingle controller, K, for all D ¼ ½DN ;DM �; Dk k1oe isgiven by

emax ¼ infK

IK

� �ðI � GKÞ�1M�1

��1

� ��1ð3Þ

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. Thelarger is the uncertainty against which the controller isable to ensure stability, the better is the design. In otherwords, obtaining a robust stabilising controller is equivalentto maximising the uncertainty measure e. Therefore, thecontrol design problem boils down to minimising the costfunction

minK2S

IK

� �ðI � GKÞ�1M�1

��1

ð4Þ

where S is the set of all stabilising controllers.The above formulation can be used for robust stabilisa-

tion of systems. However, it can not be extended to robuststabilisation with pole-placement. Therefore, the problemneeds to be translated from the standard coprime factorrobust 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 notaffect the overall infinity norm. Therefore,

I

K

� �ðI � GKÞ�1M�1

��1

¼I

K

� �ðI � GKÞ�1M�1½M N �

��1

¼I

K

� �ðI � GKÞ�1½ I G �

��1

¼S SG

K S K SG

� ��1

where S ¼ ðI � GKÞ�1 is the sensitivity. The problemof robust stabilisation of the standard normalisedcoprime factor plant description is thus translated into a

GW1 W2

a

GW1 W2

K

Gs

b

W1W2K

G

Keqc

Fig. 1 Loop-shaping design procedurea Shaped plantb Compensated plantc Equivalent controller

N M −1

K

∆N ∆M

+

+

−

G∆

+

Fig. 2 Normalised coprime factor robust stabilisation problem

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005 953

generalised HN problem, which can be equivalently statedas follows:

minK2S

S SGK S K SG

� ��1

ð5Þ

It can be noted that the closed-loop transfer functions in (5)corresponds to robustness against the following specificplant/controller perturbations:

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.

Therefore, minimising (5) maximises the amount ofallowable perturbations with guaranteed stability.

The generalised plant [21, 22] P for minimising theinfinity norm of the closed-loop quantities in (5) is given by:

P ¼

A B 0 BC 0 I 00 0 0 IC 0 I 0

2664

3775 ¼

A B1 B2

C1 D11 D12

C2 D21 D22

24

35 ð6Þ

where B1 ¼ ½B 0 �, B2 ¼ ½B�, C1 ¼ ½C 0 �T , D11 ¼0 I0 0

� �, D12 ¼ ½ 0 I �T and D21 ¼ ½ 0 I �.

The controller, K ¼ Ak Bk

Ck Dk

� �can be obtained by

solving HN optimisation problem given in (5).For analytical solution, additional constraints (e.g. pole-

placement) cannot be imposed in the synthesis stage.Therefore, in this work, the solution is obtained using anLMI formulation [18, 19] as it offers the flexibility to imposeadditional pole-placement constraints that directly addressesthe damping improvement issue.

The transfer matrix between the exogenous inputs andoutputs of P in (6) is given by:

T ðsÞ ¼ CclðsI � AclÞ�1Bcl þ Dcl ð7Þwhere

Acl ¼Aþ B2DkC2 B2Ck

BkC2 Ak

� �ð8Þ

Bcl ¼B1 þ B2DkD21

BkD21

� �ð9Þ

Ccl ¼ ½C1 þ D12DkC2 D12Ck � ð10Þ

Dcl ¼ D11 þ D12DkD21 ð11ÞWith the help of the bounded real lemma [23], it is possibleto show that the HN norm of Tzw is less than g and theclosed-loop system stable if there exists a symmetric X suchthat

ATclX þ XAcl Bcl XCT

clBT

cl �gI DTcl

CclX Dcl �gI

24

35o0 ð12Þ

The pole-placement objective is formulated in terms of LMIregions of the complex plane. There exists a general class ofLMI 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 anappropriate LMI region for power system damping controlapplication as it defines a minimum damping for thedominant closed-loop poles (see Fig. 3). The closed-loopsystem is guaranteed to have all its pole in the conic sector

with apex at the origin and internal angle y if and only ifthere exists X40 such that [19]

sin yðAclX þ XATclÞ cos yðAclX � XAT

clÞcos yðXAT

cl � AclX Þ sin yðAclX þ XATclÞ

!o0 ð13Þ

The damping ratio of the placed poles within the conic

sector is at least equal to cos y2

[17]. The value isappropriately chosen to achieve the required specifications.Therefore, the controller design exercise boils down tosolving the matrix inequalities (12) and (13). However, both(12) and (13) contains AclX, BclX and CclX. Acl and Ccl arefunctions of the controller parameters Ak, Bk, Ck and Dk

and the controller parameters themselves are functions of Xmaking the products AclX, BclX, CclX nonlinear in X. Toconvert the problem into a linear one i.e. to obtain the set ofLMIs, a change of variable is required. The expression forthe new controller variables and the LMIs in terms of thetransformed variables are given in the Appendix. Interestedreaders can refer to [18, 19] for further details.

3 Case study

The control design and simplification exercise was carriedout on a 16-machine, five-area study system, as shown inFig. 4. The details of the study system with all theparameters can be found in [10, 24]. A thyristor-controlledseries capacitor (TCSC) is installed in the system forstrengthening the transmission corridor between NYPS andarea 5. An eigenvalue analysis on the linearised model of thesystem revealed that the system has three critical inter-areamodes, as shown in Table 1. The objective is to damp thesemodes by designing a supplementary damping controllerfor the TCSC. Appropriate feedback stabilising signals werechosen for each mode using the modal observabilityanalysis, see [10] for details. The open-loop plant wasconstructed using the linearised system matrix A, the inputmatrix B corresponding to the output of the TCSC and theoutput matrix C corresponding to the measured signals.

3.1 Loop-shapingThe original plant was of 132th order, which could besimplified to a 9th order equivalent using balanced

——————————————

− infinity

innerangle

imag

real

S-plane

all poles should be

placed within conic sector

θ

0

Fig. 3 Conic sector region for LMI pole placements


truncation [21] technique. The frequency response of thefull-order plant and reduced-order plant is shown in Fig. 5.

Prior to solving the HN problem, the open-loop planthad to be shaped following the recommendations in Section2.1. A pre-compensator was used to introduce an integralaction in the low frequency region and also to reduce theoverall gain of the plant in order to suit the desired

performance requirements. The transfer function and thefrequency response of the pre-compensator is (see Fig. 6):

W1ðsÞ ¼0:106sþ 0:1096

sþ 0:001ð14Þ

The three output channels were scaled with static weightingproperly to improve the conditioning of the open-loopplant. Scale factors of 1.0, 2.0, and 0.6 were used for the1st, 2nd and 3rd outputs, respectively, resulting in a post-compensator W2 of the form:

W2 ¼1:0 0 00 2:0 00 0 0:6

24

35 ð15Þ

The frequency response of the resulting shaped plant andthe original reduced plant is shown in Fig. 7.

G703

G9

G6

G4

G5

G3

67

22

24

21

19 66

23

06

37

05

07

68

27

64

65

62

26

2809

29

2004

G2

58

56

55

52

02

57

59

60

G8 G1

01

5425

08

63

36

1131

53

51

49

38

30

39

35

33

32

34

43

13

17 4512

46

61

50

48 4047

G12

G13

G10

G11

18

14

15

16

41

42

G14

G15

G16NYPS

area 5

area 3

area 4

10

TCSC

remote signal links

NETS

u

y

K

K controller to be designed

Fig. 4 Sixteen-machine five-area study system with TCSC

Table 1: Inter-area modes of study system

z f (Hz)

0.0626 0.3913

0.0435 0.5080

0.0554 0.6232

10−1 100 101 102−5

0

5

10

15

20

25

gain

, dB

frequency, rad/s

full plantreduced plant

Fig. 5 Frequency response of full-order plant and reduced-orderplant

10−2 10−1 100 101 102 103 104−20

−15

−10

−5

0

5

10

15

20

25

gain

, dB

frequency, rad/s

Fig. 6 Frequency response of pre-compensator


3.2 Control DesignThe matrices A, B, C and D of the shaped plant are used toformulate the generalised plant P following (6). The hinfmixfunction available in the LMI Control Toolbox [17] wasused to perform the necessary computations. The pole-placement constraint was specified in terms of a conic sectoras shown in Fig. 3 with apex at the origin and an innerangle of 2 cos�1ð0:15Þ, which ensures a minimum dampingof 0.15 for all the three inter-area modes. The designconverged to an optimum HN performance index gopt of

4.873. As stated earlier, if the problem had been solved

analytically using the standard normalised coprime factor-isation approach, time-domain specifications in terms ofminimum damping ratios (pole-placement) could not beconsidered explicitly in the design stage. Table 2 gives theopen-loop damping of the critical inter-area modes and theclosed-loop damping with both approaches. It is seen fromthis Table that two control modes with 0.1042 damping at0.3941Hz and 0.0812 damping at 0.5399Hz are introducedand the inter-area mode damping remains poor at 0.0288 at0.3968Hz and 0.0855 damping at 0.4989Hz when thecontroller is synthesised analytically.

The order of the designed controller by solving the LMIswas 11, which was subsequently simplified to a 10th orderone using balanced truncation. The state–space representa-tion of the reduced controller is given in the Appendix.

3.3 Simulation resultsOne of the most severe disturbances stimulating poorlydamped inter-area oscillations is a three-phase fault in one

10−2 10−1 100 101 102 103 104−70

−60

−50

−40

−30

−20

−10

0

10

20

30

gain

, dB

frequency, rad/s

reduced plantshaped plant

Fig. 7 Frequency response of reduced-order original and shapedplant

Table 2: Open- and closed-loop system dampings

Open loop Analytical solution LMI coprime

Damping Frequency(Hz)



0.0626 0.3013 0.1042 0.3941 0.1681 0.3913

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

0 5 10 15 20 25−35

−30

−25

−20

−15

−10

−5

angl

e(G

1−G

15),

deg

time, s

fault at bus 60 with autoreclosing

0 5 10 15 20 25−30

−25

−20

−15

−10

angl

e(G

1−G

15),

deg

time, s

fault at bus 53 with line 53−54 out

0 5 10 15 20 25−28

−26

−24

−22

−20

−18

−16

−14

angl

e(G

1−G

15),

deg

time, s


0 5 10 15 20 25−35

−30

−25

−20

−15

−10

−5

angl

e(G

1−G

15),

deg

time, s


without controlwith control




Fig. 8 Dynamic response of system


0 5 10 15 20 2520

30

40

50

60

70

angl

e(G

14−G

13),

deg

time, s


0 5 10 15 20 2530

35

40

45

50

55

angl

e(G

14−G

13),

deg

time, s


0 5 10 15 20 2535

40

45

50

55

angl

e(G

14−G

13),

deg

time, s


0 5 10 15 20 2520

30

40

50

60

70

angl

e(G

14−G

13),

deg

time, s


without control

with control

without control

with control

without control

with control

without control

with control


0 5 10 15 20 2510

20

30

40

50

angl

e(G

15−

G13

), d

eg

time, s


0 5 10 15 20 2515

20

25

30

35

40

45

angl

e(G

15−

G13

), d

eg

time, s


0 5 10 15 20 2515

20

25

30

35

40

45

angl

e(G

15−G

13),

deg

time, s


0 5 10 15 20 2510

20

30

40

50

angl

e(G

15−G

13),

deg

time, s


without control

with control

without control

with control

without control

with control

without control

with control



of the key transmission corridors. For temporary faults, thecircuit breaker ‘auto-recloses’ and normal operation isrestored: otherwise one or two lines might have to be takenout. There might be other types of disturbances in thesystem like change of load characteristics, sudden change inpower flow etc., which are less severe compared to faultsand are not considered here.

To evaluate the performance and robustness of thedesigned controller, simulations were carried out corre-sponding to some of the probable fault scenarios inthe NETS and NYPS interconnection. There are threetransmission corridors between NETS and NYPS connect-ing buses 60–61, 53–54 and 27–53, respectively. Each ofthese corridors consists of two tie-lines. Outage of one ofthese lines weakens the corridor considerably. The followingdisturbances were considered for simulation. A three-phasesolid fault for 80ms (five cycles)

� at bus 60 followed by auto-reclosing of the circuitbreaker

� at bus 53 followed by outage of one of the tie-linesconnecting buses 53 and 54

� at bus 53 followed by outage of one of the tie-linesconnecting buses 27 and 53

� at bus 60 followed by outage of one of the tie-linesconnecting buses 60 and 61.

The designed controller is supposed to settle the inter-area oscillations within 12–15 s following any of thedisturbances. Moreover, it should be able to achieve thisfollowing any of the above disturbances (robustness)although the design is based on a nominal operatingcondition (no outage).

The simulations were carried out in Matlab Simulinkfor 25 s employing the trapezoidal method with a variablestep size. The disturbance was created 1 s after thestart of the simulation. The dynamic response of thesystem following the disturbance is shown in Figs. 8–10.These Figures exhibit the relative angular separationbetween the generators located in separate geographicalregions. Inter-area oscillations are mostly manifested inthese angular differences and are therefore chosenfor displaying. It can be seen that inter-area oscillationsettles within 12–15 s for a range of post-fault operatingconditions and thus abides by the robustness requirement aswell. A hard limit of 0.1–0.8 was imposed on the variationof the percentage compensation of the TCSC, which isshown in Fig. 11.

4 Conclusions

This paper has demonstrated the applicationof the normalised HN loop-shaping technique for designof damping controllers in the LMI framework. Thefirst step in this design approach was to pre- and post-compensate the linearised model of the power systemusing the McFarlane and Glover loop-shaping technique.The problem of robust stabilisation of a normalisedcoprime factor plant description was translated to ageneralised HN problem. The solution was soughtnumerically using LMIs with additional pole-placementconstraints. By imposing the constraints, a minimumdamping ratio could be ensured for the critical inter-areamodes, which resulted in settling of oscillations within thespecified time.

0 5 10 15 20 2510

20

30

40

50

60

70

80

perc

enta

ge c

ompe

nsat

ion

of T

CS

C

time, s


0 5 10 15 20 2510

20

30

40

50

60

70

80

perc

enta

ge c

ompe

nsat

ion

of T

CS

C

time, s


0 5 10 15 20 2510

20

30

40

50

60

70

80

perc

enta

ge c

ompe

nsat

ion

of T

CS

C

time, s


0 5 10 15 20 2520

30

40

50

60

70

80

perc

enta

ge c

ompe

nsat

ion

of T

CS

C

time, s


without control

with control

without control

with control

without control

with control

without control

with control



5 Acknowledgment

The authors would like to acknowledge EPSRC, UK, forfunding this research.

6 References

1 Paserba, J.: ‘Analysis and control of power system oscillation’, CIGRESpecial Publication 38.01.07, Technical Brochure 111, 1996

2 Kundur, P.: ‘Power system stability and control’ (McGraw Hill, USA,1994)

3 Hingorani, N., and Gyugyi, L.: ‘Understanding FACTS’ (IEEE Press,USA, 2000)

4 Klein, M., Le, L., Rogers, G., Farrokpay, S., and Balu, N.: ‘HN

damping controller design in large power system’, IEEE Trans. PowerSyst., 1995, 10, (1), pp. 158–166

5 Taranto, G., and Chow, J.: ‘A robust frequency domain optimizationtechnique for tuning series compensation damping controllers’, IEEETrans. Power Syst., 1995, 10, (3), pp. 1219–1225

6 Zhao, Q., and Jiang, J.: ‘Robust SVC controller design for improvingpower system damping’, IEEE Trans. Power Syst., 1995, 10, (4),pp. 1927–1932

7 Djukanovic, M., Khammash, M., and Vittal, V.: ‘Sequentialsynthesis of structured singular value based decentralizedcontrollers in power systems’, IEEE Trans. Power Syst., 1999, 14,(2), pp. 635–641

8 Chen, S., and Malik, O.: ‘Power system stabilizer designusing m synthesis’, IEEE Trans. Energy Convers., 1995, 10, (1),pp. 175–181

9 Chaudhuri, B., Pal, B., Zolotas, A.C., Jaimoukha, I.M., and Green,T.C.: ‘Mixed-sensitivity approach to HN control of power systemoscillations employing multiple facts devices’, IEEE Trans. PowerSyst., 2003, 18, (3), pp. 1149–1156

10 Chaudhuri, B., and Pal, B.: ‘Robust damping of multiple swing modesemploying global stabilizing signals with a TCSC’, IEEE Trans. PowerSyst., 2004, 19, (1), pp. 499–506

11 Pal, B., Coonick, A., Jaimoukha, I., and Zobaidi, H.: ‘A linear matrixinequality approach to robust damping control design in powersystems with superconducting magnetic energy storage device’, IEEETrans. Power Syst., 2000, 15, (1), pp. 356–362

12 McFarlane, D., and Glover, K.: ‘Robust controller design usingnormalized coprime factor plant descriptions’ (Lecture Notes inControl and Information Sciences, Springer-Verlag, Berlin, Germany,1990)

13 McFarlane, D., and Glover, K.: ‘A loop shaping design procedureusing HN synthesis’, IEEE Trans. Autom. Control, 1992, 37, (6),pp. 759–769

14 Glover, K., and McFarlane, D.: ‘Robust stabilization of normalizedcoprime factor plant descriptions with HN-bounded uncertainty’,IEEE Trans. Autom. Control, 1989, 34, (8), pp. 821–830

15 Zhu, C., Khammash, M., Vittal, V., and Qiu, W.: ‘Robust powersystem stabilizer design using HN loop shaping approach’, IEEETrans. Power Syst., 2003, 18, (2), pp. 810–818

16 Farsangi, M., Song, Y., Fang, W., and Wang, X.: ‘Robust factscontrol design using HN loop-shaping method’, IEE Proc. Gener.Trans. Distrib., 2002, 149, (3), pp. 352–357

17 Gahinet, P., Nemirovski, A., Laub, A., and Chilali, M.: ‘LMIcontrol toolbox for use with matlab’ (The Math Works Inc., USA,1995)

18 Chilali, M., and Gahinet, P.: ‘Multi-objective output feedback controlvia LMI optimization’, IEEE Trans. Autom. Control, 1997, 42, (7),pp. 896–911

19 Scherer, C., Gahinet, P., and Chilali, M.: ‘HN design with poleplacement constraints: an LMI approach’, IEEE Trans. Autom.Control, 1996, 41, (3), pp. 358–367

20 Hyde, R., and Glover, K.: ‘The application of scheduled HN

controllers to a VSTOL aircraft’, IEEE Trans. Autom. Control,1993, 38, (7), pp. 1021–1039

21 Skogestad, S., and Postlethwaite, I.: ‘Multivariable feedback control’(John Wiley and Sons, UK, 2001)

22 Zhou, K., Doyle, J., and Glover, K.: ‘Robust and optimal control’(Prentice Hall, USA, 1995)

23 Gahinet, P., and Apkarian, P.: ‘A linear matrix inequality approachto HN control’, Int. J. Robust Nonlinear Control, 1994, 4, (4),pp. 421–448

24 Rogers, G.: ‘Power system oscillations’ (Kluwer Academic Publishers,USA, 2000)

7 Appendix

7.1 Obtaining the controller through LMIsTo linearise the matrix inequalities described in Section 2, achange of variable is required. The new controller variablesare defined as:

A ¼VAkU T þ VBkC2Rþ SB2CkU T

þ SðAþ B2DkC2ÞR ð16Þ

B ¼ VBk þ SB2Dk ð17Þ

C ¼ CkU T þ DkC2R ð18Þ

D ¼ Dk ð19Þ

where

UV T ¼ I � RS ð20Þ

If U and V have full row rank, then the controllermatrices Ak, Bk, Ck and Dk can always be computed

from A, B, C, D, R, S, U and V. Moreover, the controllermatrices can be determined uniquely if the controllerorder is chosen to be equal to that of the generalisedplant [19].

From the matrix inequalities described in Section 2, thefollowing LMIs are obtained in terms of the new controllervariables:

R II S

� �40 ð21Þ

C11 CT21

C21 C22

� �o0 ð22Þ

sin yð�þ �T Þ cos yð�� T Þcos yð�T � �Þ sin yð�T þ �Þ

� �o0 ð23Þ

where

C11 ¼ARþ RAT þ B2C þCT BT

2 B1 þ B2DD21

ðB1 þ B2DD21ÞT �gI

� �ð24Þ

C21 ¼ Aþ ðAþ B2DC2ÞT SB1 þBD21

C1Rþ D12C D11 þ D12DD21

� �ð25Þ

C22 ¼ AT S þ SAþBC2 þ CT2B

T ðC1 þ D12DC2ÞTC1 þ D12DC2 �gI

� �

ð26Þ

� ¼ ARþ B2C Aþ B2DD21

A SAþBC2

� �ð27Þ

The system of LMIs in (21), (22) and (23) can be solved

for R, S, A, B, C and D. A full-rank factorisationUV T ¼ I � RS of the matrix I � RS is computedvia the SVD approach such that U and V are square

and invertible. With known values R, S, A, B, C, D, Uand 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 � AkÞ�1Bk.

7.2 Controller state-space dataThe state–space representation of the designed controlleris given by the A, B, C and D matrices in (28), (29),


(30) and (31):

A ¼

�0:001 0:0018 �0:092 0:0118 �0:0530 �2:445 23:12 �0:541 30:33

0 4:822 44:25 �3:551 66:69

0 22:94 �25:38 �15:37 �58:960 �49:2 �175 38:68 �201:10 �70:74 �558:7 84:35 �667:50 72:95 416:8 �70:38 484:3

0 13:46 76:54 �10:17 96:49

0 2:284 38:9 �1:919 67:49

0 �6:7 �130:5 16:22 �141:6

2666666666666666666664

�0:132 0:059 0:001 �0:016 0:084

46:4 �0:014 8:171 �0:896 �8:40892:96 4:268 3:092 �2:185 �16:05�63:49 �29:81 �52:44 11:39 �28:04�330:3 17:36 59:18 �4:388 125:1

�1102 11:37 21:82 17:68 320:7

797:2 �43:73 �55:83 3:497 �262:9148:4 10:01 �9:31 �2:473 �36:9792:71 20:7 9:99 �9:978 �2:857�247 22:86 �10:97 �0:433 76:76

3777777777777777777775

ð28Þ

B ¼ 1� 104�0 0:014 �0:003 0:001 0:046

0 0:151 0:131 �0:382 �0:1890 �0:083 �0:167 0:167 0:416

264

0:398 �0:216 �0:063 �0:039 0:118

�0:495 0:275 0:208 0:085 0:095

1:531 �0:1057 �0:202 �0:208 0:318

375

ð29Þ

C ¼½ 0:109 0:002 �0:0098 0:0012 �0:0056�0:014 0:006 0:0001 �0:0017 0:0089 �

ð30Þ

D ¼ ½ 0 0 0 � ð31Þ


Documents

LMI approach to normalised H∞ loop-shaping design of power system damping controllers