A New Methodology for Frequency Domain Design of Robust Decentralized Controllers

IFAC

Copyright © IFAC Control Systems Design, Bratislava, Slovak Republic, 2003

~

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A NEW METHODOLOGY FOR FREQUENCY DOMAIN DESIGN OF ROBUST DECENTRALIZED CONTROLLERS

Alena Kozakova " Vojtech Vesely'

• Department ofAutomatic Control Systems, Faculty ofElectrical Engineering and Information Technology Slovak University of Technology, I1kovicova 3, 81219 Bratislava, Slovak Republic [email protected]

Abstract: The paper proposes an original frequency domain approach to decentralized controller design for performance applied within a robust design strategy. The novelty of the proposed approach consists in assessing the influence of interactions by means of characteristic functions/loci of the plant interaction matrix, and further using them to modify mathematical models of subsystems thus defining the so called "equivalent subsystems". The independent design carried out for equivalent subsystems provides local controllers that guarantee a prespecified performance of the full system. The developed design procedure applied to the nominal model yields local controllers that provide robust properties within a specified range of uncertainty. Theoretical conclusions are supported by an example. Copyright © 2003 1FAC Keywords: decentralized control, Nyquist's criterion, spectral Nyquist plot, characteristic loci. robustness

techniques remain probably the most popular among control engineers. in particular the frequency domain ones which provide insightful solutions and link to the classical control theory.

INTRODUCTION Industrial plants are complex systems typical by multiple inputs and multiple outputs (MIMO systems). Usually they arise as interconnection of a finite number of subsystems. If strong interactions within the plant are to be compensated for then multi variable controllers are used. However. there may be practical reasons that make restrictions on controller structure necessary or reasonable. In an extreme case. the controller is split into several local feed backs and becomes a decentralized controller. Compared with centralized full-controller systems such controller structure constraints bring about certain performance deterioration. However. this drawback is weighted against important benefits such as hardware simplicity, operation simplicity and reliability improvement as well as design simplicity [2.7]. Due to them. decentralized control (DC) design

Major multivariable frequency-response Nyquisttype design techniques were developed in the late 60' and throughout the 70's: almost simultaneously with the non-interacting Characteristic locus (CL) technique [3.4]. Development of decentralized control (DC) techniques dates back to the 70': the research, though not so excessive. is still going on. The DC design has two main steps: first. a suitable control structure (pairing inputs with outputs) has to be selected: then. local controllers are designed for individual subsystems. There are three general approaches to the local controller design: simultaneous. sequential and independent designs [2. 7].

393

In the independent design. effect of interactions in the full system is assessed. and transformed into bounds for individual subsystems which are to be considered in the local controller design in order to proyide stability and a required performance of the full system.

H (s) E R m • m is the closed-loop transfer function matrix. H(s)=Q(s)[I+Q(s)r

J

R denotes the lield of rational functions in s. The Nyquis( D-contour is a large contour In the complex plane consisting of the imaginary axis s = )w and an infinite semi-circle into the right-

In the sequel the control structure selection is supposed to be already completed. To account for plant interactions. an original approach is proposed based upon representing interactions by means of their characteristic loci (eigenyalue loci) [1.3.4.5) and further using them to modilY mathematical models of indiyidual subsystems. For such modified subsystems is applied then. Designed local controllers guarantee fulfilment of performance requirements imposed on the full system: moreover. for a specified range of operating conditions the proposed design procedure applied to the nominal model guarantees closed-loop robustness. Theoretical results are supported by an example.

half plane. It has to ayoid locations where Q(S) has jw -axis poles (e.g. if R(s) includes integrators) by making small indentations around these points such as to include them to the left-half plane. Thus. unstable poles of Q(S) will be considered those in the open right-half plane. l\/yquist plot of a complex function g( s) is the image of the Nyquist under g(s). whereby

the

D-contour is traversed clockwise: number of anticlockwise

:\" [ k. g( s)J denotes

encirclements of the point (k. )0) by the Nyquist plot ofg(s) The characteristic polynomial of the system in Fig.1 IS

The paper is organized as It)lIows: theoretical preliminaries of the proposed technique are surveyed in Section 2. problem lormulation is in Section 3. main results along with the proposed design procedure and robustness study are presented in Section 4 and verified on an example in Section 5. Conclusions are given at the end of the paper.

det FrS) = detll +Q(S)J = detll +G(s)R( s)j

(2)

I I' Q(s) has n q unstable poles. the closed-loop stability can be determined using the generalized Nyquist stability criterion [I. 3. 4. 5.

n

Theorem I (Generali=ed Nyquist Stability Theorem) Thefeedback system in Fig. I is stable if and only if

2 PRELIMINARIES

a. detF(s)7'-O

b. /1(O.detF(s)j=n'l

Consider a MIMO system G(s) and the controller R(~) in a standard feedback conliguration (Fig. I)

(3)

where nq is the number (il its open-loop unstable poks. D As rational functions in s form a lield. standard arithmetical manipulations with transfer functions can be carried out. Thus. eigenvalues of a square matrix

E

It"·,

and

function matrices and

R( s) E R'·m

It". U.)".

are transfer

are giyen by

e. dare rcspecti\Cly

yectors of reference. eontrol. output. control error and disturbance of compatible dimensions. Hercafter. only square matrices \\ill be considered. i.e. m= I.

detlq,(sil m -Q(sll=O Then

i

n

= I . ... fI1

(4)

m

det F( s i = dell I + Q( s ) J =

The necessary and sufticient condition for the closedloop stability prmides the generalized Nyquist stability theorem based upon the concept of the system return difference 15. 71 detined as

Fr s ) = II + Q( s ) I

R mxm (whose elements are rational

functions in s) are themselyes rational functions in s. They are called characteristic functions [3. 4. 51. The m characteristic functions If, ( SI. i = I ..... fI1 of Q(.I")

Fig. I Standard feedback configuration whcre C( s)

Q(S)E

II + If, ( S

)

1

(5)

/ ..; ;J

Characteristic loci (CL! q,rjW). i = 1.2..... m

are

the set of loci in the complex plane traced out by the characteristic functions of Q(S) as s tra\erses the Nyquist D-contour. This set is called the spectral \vquist plot Ill. The deg.ree of the spectral Nyquist plot is the sum of anticlockwise encirclements with respect to a spccilied point in the complex plane. contributed by the characteristic loci of Q( si. A

(I )

where

F( s ) E R m . m is the system return-difference matrix Q( s) E R m . m is the (open) loop transfer function matrix. !t)r the system in Fig. I Q( s) = (J( s IR( s)

stability test analogous to Theorem I derived in terms of the CL's 11. 3. 4. 5. 71

394

has been

Theorem 2

where R, (s) is transfer function of the i-th subsystem local controller.

The closed-loop system with the open-loop transfer function Q(s) is stable if and only if del! J + Q( s)J 0 and the degree of the spectral Nyquist plot of Q(s) with respect to (-I . Dj) fulfils the following condition (with nq being the number of unstable poles of Q(s))

In compliance with the independent design framework, the point at issue is to appropriately assess the effect of interactions and to translate it into constraints for local controller designs to guarantee stability of the full closed-loop system and its required performance.

'*

m

(6a)

LN[-I. q,(s)]=n q 1=1

or alternatively. plot of[1+Q(s)]

if considering

4

MAIN RESULTS

the spectral Nyquist

4. I Theoretical Background

m

TI¥'i proposed decentralized control design technique is based upon factorising the closed-loop characteristic polynomial of the full system (2) under the decentralized controller (9) in the following way

(6b)

LN{D.[I+ q,(s)]}=n q ,=1

o

Remarks: I. In the sequel, whenever possible. matrices and their corresponding characteristic transfer functions will be denoted by like capital and small letters, respectively. 2. The CL's are not uniquely defined since whenever two particular loci cross, the identity of the correct continued path is lost. Furthermore. each locus does not in general form a closed contour. Fortunately, as the underlying characterristic functions are piecewise analytic, it is possible to concatenate the m CL's of [I + Q( s)] to form one or more close contours [I]. 3. Theorems I and 2 are equivalent, therefore

det F(s)

= det{1 + R(sj((Gd(s)+Gn,(s)]} =

= del R(s)det[ R- 1(s)+Gd(s)+Gm(s)]

(10)

Existence of R- 1 (s) is implied by the assumption that det R( s) '# D. Denote Fds) = R- 1(s)+GAs)+Gm(s) Then. with respect to (8)-(10), corollary holds.

the

(11)

following

Corollary 1 A closed-loop system comprising the system (9) and Ihe decentrali=ed controller (10) is stable if and only

if

m

N{D.det[ 1 + Q(s)]} = L N{D.[ 1+ q,(s)]} = n q

a. del

;=1

F1 ( s) '# 0

b. N[O.detFJ(s)]+N[O.deIR(s)]=n q

(7)

(12)

If

R(s) is stable. N{O. del[R(s)]}=O and the encirclement condition (l3b) reduces to

In general, for points other than (O,Oj), degrees of the respective plots are different [I].

N[O,detF,(s)]= = N{D. del[ R- J(s)+Gd(s)+Gm(s)]} = n q

3 PROBLEM FORMULATION

( 13) o

Consider a complex system with m subsystems (m=/), described by a square transfer matrix

As the term [WJ(s)+Gd(s)] in (11) is diagonal and includes all necessary information on subsystem dynamics. it is possible to specify a required performance of individual subsystems using an appropriately chosen diagonal matrix P(s): , WJ(s)+Gd(s)=P(s) (14)

mxm

G( s) E R that can be split into the diagonal and off-diagonal parts. The transfer matrix collecting diagonal entries of G(s) is the model of decoupled (isolated) subsystems; interactions between subsystems are represented by the off-diagonal entries. i.e. G(s)=Gd(s)+Gn,(s) (8) where

From ( 15) results 1 + R(s)[G,JS)- P(s)]=O or. on the subsystem level

G d (s) = diag{ G, (s )},=I.m whereby det Gd(s),*D

and

where P(s)=diag{ p,(s)},=J .m

'<:Is

1+ R,(s)G,eq(s) = 0

i=I.2, .... m

(16)

G,eq(s)=G,(s)- p,(s)

i=I.2..... m

(17)

Gn,(s) = G(s) - Gd(s)

where

For the system (8). a decentralized controller is to be designed R(s) =diag{R,(s)}'"1 .m.

detR(s)'#D

'<:Is

(15)

q

G,e (s) denotes the i-th subsystem transfer function modified by p,(s) and will be called the transfer

(9)

395

function of the i-th equivalent subsystem or simply the equivalent transfer function. Similarly, (17) will be denoted the i-th equivalent characteristic equation.

P(s) = p(s)1 motivated by (20) and (21) have been adopted: I. If p(s)= p,(s) = -g,(s) for fixed then

ModifYing (11) using (14) we obtain det Fls) = det[P(s) + Gm (s)}

2.1f P(s)=pds(a)}=-gds(a)} , (23) fixed fE{I, .. ,m} where we consider the generalized frequency s = -a + jW. a ~ 0 ; W E (-00,00) and s(a) indicates the particular case when a > 0, then m

detF,[s(a)} =

(l9b)

/;;/

m,(s), i = I, ... , m

are

1=1

Thus, as det FJ [s(a)} = 0, the modified closed-loop system is exactly on the stability boundary "shifted to ( -a )", hence it is stable with the decay rate a. The equivalent subsystems transfer functions are G,~q [s(a)}

n",is the 0

'I 'I 1=/

=

4.2 Development ofthe Decentrali=ed Controller Design Technique

where m~q [s(a)}

N{O, m,eUs(a)}} = nm

= (pt! s(a)} + g,(s)} = =(-gds(a)} + g,(s)},

i = I, ... ,m will be called equivalent characteristic functions of M(s)=[P(s)+Gm(s)};n m is the number of

unstable poles of M ( s) . If for any s=-a+jw,

a~O, WE(-OO,oo)

m

detF;(s) =TI{pds(a)}+g,(s)} =

(20)

l=!

For identical entries in the diagonal of P(s), substituting (14) in (18) and equating to zero actually defines the m characteristic functions of [- Gm(s)} i=I.... m

(26)

I=}

For the independent design the p,( s). i = I ..... m actually represent bounds for local controller designs. To guarantee closed-loop stability of the full system they should be chosen such as to appropriately account for the interaction term G", ( s) . According to (4). characteristic functions of Gm(s) are defined

det[p,(s)I+Gm}=O

= 1,2, ... , m

N{O, [pds(a)} + g,(s)}} =

the choice of p;(s) is discussed in detail.

i=I, ... ,m

i

According to (19b) and considering (22) and (23), the encirclement condition for the closed-loop stability in terms of the spectral Nyquist plot of Fj ( s) is then

Thus far, only stability in the DC design has been considered. Any additional performance requirements need to be included in the design by a suitable choice of P(s) = diag{ p, (s )},=Im' Next,

det[g,(s)I-Gm(s)}=0

= G,[s(a)}- Pt [s(a)}

(25) It is noteworthy that a closed-loop system stable with a decay rate a ~ 0 it is necessarily stable also fora = O.

characteristic

functions of M(s)=P(s)+Gm(s); number of its unstable poles,

(24)

= TI{-gds(a)j+ g,[s(a)}} = 0

1=1

where

TI{ pds(a)} + g,[s(a)}} =

m

any ofthe following conditions is satisfied

'IN[O, m,(s)} = nm

1=/

(22)

can be stabili=ed by its related local controller R,(s), i.e. each equivalent characteristic polynomial q CLCp,e =1+R,(s)G,eq(s) i=I,2, ... ,m has stable roots; (I9a)

I=J

I, ... , m}

and the closed-loop system is not stable (it is on the stability boundary);

Corollary 2 A closed-loop system comprising the system (8) and a stable decentralized controller (9) is stable if 1. there exists such a diagonal matrix P(s) = diag{ pJs )},;j,.,,,, that each equivalent q subsystem G,e (s) = G,( s) - p,( s). i = I, .... m

N[O, det[ P(s)+Gm(s)} = nm

m

E {

detF/s) = f1[ AS)+g,(S)} = f1[-gls)+g,(s)} =0

(18)

Considering (18) we can formulate the encirclement stability conditions for the closed-loop system under a decentralized controller in terms of the spectral Nyquist plot of Fj ( s) .

2.

m

f

(27)

m

= TI{-gds(a)}+g,(s)}=0 1=1

i.e. if at any frequency, say

(21)

pd

To be able to include the performance issue in the design. the following considerations about

-a + jW j )

g,( w,), i

396

E

and

any

wj

,

the particular

characteristic

locus

{I, ... ,m} happen to cross, the closed

4.4 Robust Controller Design Procedure

loop system is not stable. The main theoretical results from the preceding section are summarized next.

The developed DC design procedure is applicable within the decentralized robust controller design strategy developed in [2]. In the robust approach the DC is to be designed for the uncertain system

Definition 1 (Set ofstable characteristic functions) P l [ s(a)] will be called a stable characteristic function if \I g,(s), i = 1, ... ,m, s

G(s) = Go(s) + G",(s) + L1(s)

= -a+ jW, a? 0, WE (--00,00)

where the diagonal matrix Go ( s) is considered as the nominal model, while interconnections along with the uncertainty are included in the generalized multiplicative perturbation matrix

l. ( pd s(a)] + g,( s)} *- 0;

2. IN{O,

m,~q [sea)]} = n",

1=/

L R( s) = [G( s) - G o ( s)]G~J (s)

where m~q[s(a)]={Pl[s(a)]+g,(s)} and nmis of unstable poles of the number M(s) = [P(s)+G",(s)]. The set of all stable characteristic functions for a 0 system will be denoted Ps.

pds( a)] s

=- gJs( a)]

= -a + jW,

a? 0,

, fIXed

H o(s)= Go (s)R(s)[ 1 + Go(s)R(s)]-J

belongs

The robust DC design strategy evolves from the sufficient condition for robust stability of systems under DC (SCRS-DC) [2]

toPs and the closed-loop characteristic polynomials of all equivalent subsystems (equivalent characteristic polynomials CLCP/q =l+R,,(s)G,'q(s), i=I,2, ... ,m are stable. 0 Proof of Lemma results from previous considerations.

2.

IILRH 011 < 1

3.

GIl(jw)R,(jw)

m,',q [sea)]

i=l, ... ,m:

\lw

I

Mo(w)= 11

[G( jW) - Go( jW)] G~J (jw)

(34) I1

Mo (w) does not depend on the controller. it actually assesses uncertainty and interactions within the system. The expression on the I..h,s. in (33) is the local closed loop magnitude IH,( jW)I. hence (33) becomes

loci

IH,(jW~
= (p,[s(a)] + g,(s)}, i = I, ... ,m

If no such p, [ s( a)] E p'~ can be found, no stabilizing decentralized controller can be designed using this approach. and the procedure stops; else 4.


(33)

b. specify a > Find the characteristic functions of Gm(s): g,(s), i= I, ... ,m Choose p,[s(a)]=-gtfs(a)]: p,[s(a)]E Ps by examining equivalent characteristic

(32)

I1+ GIl (jw)R, (jw)

Partition the controlled system into subsystems (diagonal part) and interactions (off-diagonal part) a. G(s)=Gd(s)+G",(s)

°;

2.

\lw

Basically, the DC design for robust stability consists in breaking down (32) into local bounds for subsystems, which if fulfilled, guarantee closed-loop stability of the overall uncertain system (3 I). For the standard unity feedback configuration applied in each local loop the subsystem constraints are as follows

4.3 Decentrali=ed Controller Design Procedure 1.

(30)

as well as of the overall uncertain system (robust stability) J R( s) = G(s)R(s)[ 1 + G(s)R(s)r (31)

eE {l, ... ,m},

WE (-00,00)

(29)

The global uncertain system G( s) is to be controlled using the controller (9), which has to guarantee closed-loop stability of the nominal (diagonal) system (nominal stability)

Lemma I The closed-loop system comprlslng the system (8) and a stable decentralized controller (9) is stable if l.

(28)

i=I, ... ,m:

(35)

Put simply, a system under a decentralized controller is robustly stable if there are such local controllers. which provide stability of related isolated subsystems and simultaneously comply with (33). Graphical interpretation of (33) provides useful robust stability criteria. the simplest-to-use being the following,

Design local controllers R,(s) for all m equivalent subsystems q G,e [s(a)] = G,[ sea)] - p,[sea)] i = l, ... ,m: using any suitable design technique. e.g. the Neymark D-partition method [6].

Magnitude robust stability criterion [2]: Let Go (s) and G( s) have the same number of unstable poles. The overall uncertain closed-loop

397

a model of mean parameter values (7], The chosen decay rate is a = O. I

system (31) is guaranteed to be stable in face of specified uncertainty (both structured and/or unstructured) as long as there are such local controllers R, (.I). i = I..... m which stabilize related

Equivalent characteristic loci for (= I . are in Fig. 2, Obviously. p,( s.a) E P,- as nnt = O.

subsystems and simultaneously ensure that over the given frequency range. the rH, ( jW)< -versus- CV plot is upper-bounded by the

;\f () (W)

0.6

-versus- CV plot.

This robust stability criterion enables an at-a-glance testing of stability of the global uncertain system as well as of individual subsystems. In addition. information on subsystem performance in terms of bandwidth and resonance peak is provided. Besides testing robust stability. the Magnitude criterion can be used as a tool for heuristic tuning of local controllers. whereby stability of the global system is improved by increasing relative stability measures of subsystems.

- r----

-

'",

......

~04

,

-E.

~O.3

~

::: 0.2

E \

~Ii

0

------- --

\

The proposed robust controller design procedure is illustrated on an example. 5

L

0.5

'().2

o

0.05

0.\

0.\5

Re{

EXAMPLE

.-

,

~- ~

0.2

-,

-

0.25

03

035

0.4

0.45

n1;,eq [s(a1p-n)ll

Fig. 2 Equivalent characteristic loci ( (= I and a = O. I )

The proposed design technique has been applied for the design of a robust decentralized controller for a TITO system G(sj

=[ GJI{ .I)

G:,f s)

For the related equivalent subsystems. local PI controllers were designed in fi.)rm

Glef .I)]

.I

The process is given by two transfer function

applying the Neymark D-partition method to both equivalent characteristic equations (Fig. 3).

matrices eland e: considered as perturbed models. defining the operating range of the system.

1'1 operating point -

e

I(

R," (s) + {G,! s(a)] - PI! s(a)]/= 0

ejS)

i = 1.2

.I ) :

0.0172.1" - O. 10-10.1 + 04487 .I"

1'1

+-

R( s) = r{/

G:,,f s)

,

+2.232.1: +2.098.1+0.6122 0012.1: -00346s-0//97

I SI equivalent subsystem

,

s" + 2. 74/s: + 1.88Is+.5701 () 0203s" - O. 0535s + 1.1 14

-

/'

----"_.-

+4.129.1-' +6.22&+3.795 () 0174.1: - O. 1067 s + 0.3996

S3

~

./

8

-

/

4

-

-

/

--------

"-.

,

\

-

\

-

/

s" + 3.257.1: + 1.809.1 + O.34J4

0

\

2'''' operating poim - G:f .I):

o

05

,

, 5

5

5

35

.0

00163\" - 00995s + 04272 --_._------------+ 2. 193r' + 2.049s + 06091

2

.I"

00191.1"- - 0040-1s - 0 IOJ4

06

+2.367.1: +1.686s+05165 00062.1: - OOJ4& + 092 I 3

.I"

05

equivalent subsystem I

1

~

1 /'

I 04

I

+ 3. 724.1: + 5.402.1 + 3.3 0.041.1: -O.I.J37s+048.J3

.I"

Go': {.I j

nd

07

"

03

0'

o,

.\--' + 3 ~72s: + 22 Is + 0.J33

1

L

/

/

, I (

I

I -1

-'

. .

I I

I

I

I

,

I

I I

I

11

I ·0 ,

The designed controller has to guarantee a required performance of the nominal model and robust stability of the global system mer the operating range. The nominal model G(sj has been obtained as

-'

i

·0 , ·05

, 5

05

, 5

Fig 3 Neymark D-contours for hoth equi\alent suhs\stems.

398

a

=

{O.O. I,'

35

For both equivalent subsystems. parameters of local PI controllers have been chosen from the boundary of the closed areas in Fig. 3. which correspond to the closed-loop decay rate of the full system a = O. I . The upper contours correspond to a = O. the inside of the closed areas to a > O. I. Design results are summarized in Tab I.

Subsystem

whole operating range specified by transfer matrices in two operating points. CONCLUSION In this paper a novel frequency-domain approach to the decentralized controller design lor perl{)rmance has been proposed. It is based on including plant interactions in the design of local controllers through their characteristic functions. modi lied so as to achie\'e a required closed-loop perlorrnance in terms of the full system decay rate. The independent design is carried out tor the so called ..equivalent subsystems" - models of individual decoupled subsystems modified using characteristic functions of the plant interaction matrix. Local controllers designed tor equivalent subsystems guarantee fulfilment of perlorrnance requirements imposed on the full system. When applied within the robust control strategy [2] the developed design procedure applied to the nominal model yields local controllers that provide robust properties over a specilied range of uncertainty. Theoretical results are supported with results obtained by solving several examples one of which is included.

Achieved decay rate a

Controller

R1-- I.-77-I 0 + 0.21-19 s

0.1007

R. =0.9221+ 0.1-129

2

s

Tab. 1 Results of the local controller design The whole closed-loop eigem'alue set is

;\ = {-0, lom ± 0 009 I j,

- 0 2 -199 ± 0, -1113 j.

- 0367 ± 07166 j: - 0,-1163 ± 0,3871 j. - /.0801 HJ, 9296/. - 13172. - 1,70-13: -/.81-15.-2,9918 :

-

-

50-

~

ACKNOWLEDGEMENT

-

0-

f-

:::::

·5(-

-

-

10< -

-

-

1\

This work has been supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences under Grant No. I/O 158/03.

f--

f-

REFERENCES 15<

-+

-

I

11 10'

'"

-

10'

,

I 10'

I

'"

I I

10"

[I] DeCarlo. R.A.. Saeks. R. .. Interconnected dynamical systems". Marcel Dekker. Inc.. 1981. [2J Kozakova. A.: Robust Decentralized Control of Complex Systems in the Frequency Domain. 2nd IFA(' Workshop on New Trend~ in Design 4 Control Systems, 7-10 Sepl. 1997. Smolenice. Slomkia. Else\ier. Kidlington. UK. 1998.

f-

10'

10'

w (rad/secl

50,-

-

e-

.

5C

[31 MacFarIane. A.G,J.. Bclletrutti. Characteristic Locus Design A wommica. 9, pp. 575-588 ( 1(73).

=

Of---

It

,

"

-

1IIII

1

't I: I : " d

I'i .

1

I

-'",",

,

:11 10'

14 J MacFarlane. .G.J.. KoU\aritakis. B.: A design

,

, ,

I

I

I1II1 10'

I I:

I:

I

I11 10"

III

W

I 10'

11

LI. "Thc Method".

technique lor linear multi\ariable feedback systems. Int, Journal of Control. 2S. No.] (1977).837-874.

I" d:

II(i' '"

i'

,

151 MacFarlane. A.G.J.. Postlethwaite. I.: The Generalized yquist Stability Criterion and Multi\'ariable Root Loci. 1nl. Journal (if ('ontrol. 2S. No. I (1977).81-127.

~I

~l

10'

161 Neymark. J..I. "Dynamical systems and controlled processes". Nauka. MosCtm. 1978 (in Russian J.

(fad/sec)

Fig.4 Magnitude robust stability test I{)r both subsystems

P]

The robust stability test using the Magnitude criterion prm'es robust stability of the full system oyer the

399

Skogestad. S.. Postlethwaite. I. ",\/ultimriahle Fedhack Control: Ana~\'sis and Design". John Wiley & Sons. 1996.