Graphical Stability Criteria for Large-Scale Nonlinear Multiloop Systems

Graphical Stability Criteria for Large-Scale Nonlinear Multiloop Systems

GRAPHICAL STABILITY CRITERIA FOR LARGE -SCALE NONLINEAR MULTILOOP SYSTEMS J .D. Blight Electrical Engineer Tektronix, Inc. Beaverton, Oregon U.S . A...
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GRAPHICAL STABILITY CRITERIA FOR LARGE -SCALE NONLINEAR MULTILOOP SYSTEMS

J .D. Blight Electrical Engineer Tektronix, Inc. Beaverton, Oregon U.S . A.

N.H. McClamroch Associate Professor Computer, Information and Control Engineering The University of Michigan Ann Arbor, Michigan U.S.A .

ABSTRACT

11.

This paper presents a derivation of stability criteria for nonlinear multiloop systems . A new system description, the Standard Multiloop Form, is used to express system dynamics in terms of scalar subsystems and can be shown to include many of the common vector descriptions of multiloop systems. The stability criteria presented in the paper involve the individual Nyquist plots of the linear scalar subsystems and a certain positivity condi tion on the nonlinear subsystems. The method allows relatively convenient computations even where the number of subsystems is large.

Stabi lit y criteria will be derived for systems having a particular structure called the Standard Multiloop Form.

1.

SYSTEM DESCRIPTION

2.1 Definition ~. (t)

Aixi(t) + bifi(t)

1

c 'Xi (t) i n L l/J .. [u.(t) j =1 1J J

INTRODUCTION

Nonlinear stability theory has received the attention of researchers for many years . The most widely known method of analyzing the stability of nonlinear systems is Lyapunov's direct method which requires the discovery of a scalar function of the system state vector such that certain properties are satisfied in the time domain. The difficulty of discovering such a "Lyapunov function" is the primar y disadvantage of the method. In 1961 V.M. Popov introduced a method of stability analysis based on the use of the frequency domain [8], which greatly simplified the analysis for systems having a particular structure, namely, systems having a linear time-invariant "plant", one nonlinear element, and a single feedback loop. Since the introduction of Popov ' s method and the associated Circle Condition, many researchers have generalized the Popov method to include systems having multiple nonlinearities and multiple feedback loops, e.g . [1, 3, 4, 6, 7]. Unfortunately, these multiloop stability criteria do not have a convenient graphical interpretation which is expressed in terms of the frequency response of the several linear subsystems. It is the purpose of this paper to introduce a method of analysis which does lead to such a convenient graphical interpretation in the frequency domain. There have been a few recent results [5, 9, 10, 11] where an attempt has been made to preserve a graphica l interpretation; however in these works various system representations were considered but the difficulty of handling largescale mu lt i l oop systems remains.

A system of equations of the form

for i

n

L 'k(Yk,t),t ] k=l J

1, ... , n

=

(2 .1)

is said to be in the Standard Multiloop Form. The function xi(t) is an n - vector, and fi(t) and i Yi(t) are scalar functions . The function ui(t) is a scalar input function to the ith subsystem . time varying and nonlinear functions l/J

The

and ij

ij are assumed to be continuous functions of their arguments with ij(O)

=

assume throughout that n

0 and l/J ij(O) >

=

O.

We shall

1.

The transfer function for the ith linear subsystem is given by G. (s) 1

Yi (s)

F.(s) 1

-1 =

ci '( sI - Ai)

bi ,

(2.2)

and a block diagram of the system is shown in Figure 1.

It may be shown that a great many multi-

loop feedback systems may be placed in this form through suitable selection of the nonlinearities l/J

ij

and ij [2]. The reason for considering our system model in the above "decomposed" Standard Multiloop Form is that our objective is to obtain conditions for stability which involve individual conditions on the scalar linear subsystems Gi(s) and the scalar functions l/J ij and ij'

600

In this work we

co~sider

the stability of the multi-

loop system only for zero inputs. 2.2 Definition

3.2 Theorem

The system (2.1) is globally stable

= 0,

with degree y if, for ui(t)

i = 1,

n, and

and for any xi(O), i = 1, ... , n, there exist nu~bers

qi are parameters. The system (2.1) with (irreducible)

transfer functions (2.2) is globally stable with degree y if the pair [Ai + yl , bil is controllable i and the pair [A. + yI., c.l is observable for i = 1, ~

~

~

... , n, where Ii is the n

K. >O such that ~

and:

x n identity matrix, i i If G.(s) has all poles in Res <-y: One of the ~

following holds: i

Ill.

=

1, ... , n.

(a)

STABILITY CRITERIA

The stability criteria are based upon a variation of a lemma developed by V.M. Popov for single-loop systems [81.

(b)

The Nyquist locus of Gi(s-y ) is inside the

closed disk D , where Pi
3.1 Basic Lemma

closed half plane where

If the linear systems

Aixi(t) + bifi(t)

(d)

-1

with transfer functions Gi(s) = ci'(sI - Ai)

If Gi(s) has Ni b

Res >-q-' where O=p.
i

~

~l

Res -<--p"~

where p.~ - y : The Nyquist

times in the counterclockwise direction and does

i observable), and the functions Gi(s) have all poles

not enter the disk; where O
in Res
For the nonlinear functions:

Res ~O,

condition

~i>O

and vi >O, i = i, ... , n

The positivity

n

such that n < L

i=l

f~

villx.(O) ~

II

Q(y) ~ - L p . (l + Pi )y.2 ;=1 ~ ----- ~ • qi -p i

2

-~

t

+

o

(1+

i=l

f.( T)y . ( T)d T

i=l ~

~

2Pi)Y.~~·-l w. . [-~k-l ~ 'k(Yk)J

qi-Pi

~J

-

J

n _ b

1 i=l qi-Pi

The lemma will be useful in the development of

holds for all y = (Yl' Y2' .•. , Yn)'

conditions for the multiloop system in

the following way:

J-

(3.1)

holds for all t > O.

stability

~

lo cus of Gi(s-y ) encircles the disk Di exactly Ni

i

are each irreducible (i.e., controllable and

then there exist

~

The Nyquist locus of G.(s-y) lies in the

closed half plane where

ci'xi(t) + d/i(t), i = 1, ... , n

+ d

The Nyquist locus of Gi(s-y) does not encircle

nor enter the closed disk Di' where O
Proof:

suppose the conditions of the

Basic Lemma are satisfied for each linear sub-system G (s) (or some modification of G (s)), i = 1, ... , i i n; then if the nonlinear functions W. . and ~ .. are ~J

E:

Rn.

'V

Let Y.(s) and Y. (s) denote the Laplace ~

~'V

tr a nsform of y.(t) and

y.(t), etc., and note that ~ t Yi(s-y) is the Laplace transform of e Y Yi(t). ~

Define the following variables for i=l, ... ,n:

~J

such that the integral in (3.1), according to the system interconnections, is nonpositive for all t

~

0 it follows that xi(t) are bounded, for i

1 'V Y; (s) ~ Y (s-y) + - - F . (s) . i qi -Pi ~ •

'V

1,

... , n and hence the system is globally stable with

Then 'V

y = O.

In the basic result of this paper to follow

_:~_.(_s_) = _G....:i,,-(_s_-_y_)_ _ + _1_

we use this approach to develop conditions for global stability.

Fi(s)

1 + PiGi(s-y)

qi-Pi

It is convenient to state the results

in terms of the Nyquist locus of Gi(s) and the closed disk Di in the complex plane with center at - (qi+Pi) + jO and with radius Iqi-Pi l , where Pi and 2Piqi 2Piqi

'V

Now define Xi(s) ~ Xi(s-y ), where Xi(s) is the Laplace transform of xi(t).

601

Now

for all t > O.

(sI-.A i ) - \ i Fi (s) and

(sI -Ai-yI)

-1

biFi(s-y). '\,

'\,

'\,

Thus

(sI-Ai-yI)X (s) i

f.

L

Pibici'Xi(s)

Yi 1

E f .y .. i=l L L

We have

yt e yt f . + Pie Yi L

'\,

and '\,

n '\, '\,

Now consider the integrand

e

yt

Yi

yt +_1_ [eytf. + Pie Yi 1 L qi-Pi

'\,

c.'X.(s) +---F.(s), L L qi-Pi L

and

which has a realization '\,

x.

(Ai-Pibici

L

,

'\,

'\,

'\,

'\,

+ yI)x

i

+ b . f. L

L

'\,

(3.3)

c. 'x. +_1_ f , L L qi-Pi i

Yi

~~

+ _1_ { .. [_ kn_E _ l


i=l, ... ,n which is minimal by hypothesis. imposed on Gi(s) by the theorem

ensur~,

by the

Thus, from (3.2), E f.y . i=l L L n

After some simplification we

(!+q.G.(S-y»(!+P.G.(S-y»} LL LL

ReGi(s) = Re {

2

O.

<

Therefore,

n

'\,

2.. E v· llx.(O)11 i=l

L

2

L

n 2 -2yt n 2 E ~.llx.(t)11 2.. e E v. llx.(O)11 i=l L L i=l L L

or

which proves the theorem.

(qi-Pi)ll+PiGi(S-y)12

where the bar denotes complex conjugate.

'\,

E ~.llx.(t)11 i=l L L

have '\,

2

n '\, '\,

The conditions

Nyquist condition, that the poles of Gi(s) are in Res
(Yk~} J

The conditions of Theorem 3.2 require that the

In cases

Nyquist locus for each Gi(s-y) remain outside of a

(a) and (b):

"forbidden" region in the complex plane.

This for-

bidden region is either the inside or outside of a critical disk.

{IGi(s-y) +

~~::~12 - [~~::~r}.

The requirements zn the Nyquist locus of Gi(s)

This condition is in the familiar form of the Circle Criterion used in the analysis of single-loop nonlinear systems. Application of Theorem 3.1 is straightforward,

guarantee that ReGi(jw»O for all w (making the standard allowances for poles So such that Re so=Y).

The main difficulty is in the verAlthough it is

possible to examine the inequality for each system

= ReGi(s-y) + :i'

encountered, we shall consider, as one typical

'\,

so that Re Gi(jw) > 0 for all w; and for case (d): '\,

-1

ReGi(s)

I!+p. G. (s -y) I L

2

Re lG. (s-y) +.1..-1 L Pi

so that again ReGi(j w)

~

0 for all w by hypothesis. '\,

Therefore, the functions Gi(s), the conditions of Lemma 3.1. ~i>O

illustration, a special case where inequality (3.2) can be more directly verified. We consider the special case of our multiloop

L

'\,

exist numbers

in principle.

ification of inequality (3.2).

For case (c):

Re~i(s)

In the special situation where

Pi=O or qi=O, the disk degenerates into a half plane.

i~l,

... ,n, satisfy

It follows that there

and vi >O such that

system (2.1) where the feedforward interconnection nonlinearities

~

.. (v.) are of simple form and the

LJ

J

feedback interconnection nonlinearities
LJ

Let

~ ii(vi,t)=vi

(v.,t)=O for i#j. J

is satisfied if n>l and

602

for i=l, ... ,n and

The positivity condition (3.1)

1 '\, +- $ n-l jj

(3.3)

and also f o r i#j

__1_ _ qi-Pi

~. .2

_ _ 1_ qj-Pj

~J

~ .. 2_2 ~

_1_ k=l qk-Pk

J~

~k. ~k.1 ~

J\

Cons ider the term

~ii

[ Yi -

qi~Pi ~iiJ.

If y i=O, then any terms invo l v ing Y va ni sh and i canno t lessen the value of the exp ress i on ; th e ref ore, l e t Yi #O for all i.

By hypothesis,

$ii

y. ::.

Bii + Pi ::. qi

~

thus There are n(n-l) /2 pairs of inequa lities in (3.4). Proof:

The exp re ss i on for Q(y) from (3.2) in the

case where ~ ii(vi,t)

I t follows tha t

v i a nd ~ ij( Vj ,t) = 0, i#j

is gi ve n by p.

n L

Q(y)

Pi

i=l

(1

+ -~-)

2

and

q. -p. Yi ~

'\,

~

1

'\,

$ ii [Yi - q.-p. $ii 1 ~ ~

I f we def ine

>

'\,

~ .. (Y .) ~J

0 '\,

Now consider the t erm $ki (Yi ) $kj (Y j )'

J

then

then

If YiYj
Now

'\,

'\,

$ki $kj < - YiY j Bki Bkj

Q(y) so in all cases

'\,

'\,

$ki $kj ::. Bki Bkjl Yi Yj l . No te finally '\,

Yi $ij 'V

'V

'V

'V

'V

'V }

+ 2$·1 $ ~·2 + ... +2 $ ~·1$·~n + ... +2 $i ,n- 1$ ~n · ~

and

603

~ - Bij lYi Yjl

If y . y. >0, ~

J

It follows that Q(y) >

~

connections.

~

'!_l_ a ~~. . [1

smaller as the number of loops increases.

i=l j=l+l !n-l

+ --.-Ll a.. nJJ

IV.

[1 -~

(3" ]

J

B. .

Yj

As one might expect, the stable

sectors for the interconnections necessarily become

EXAMPLE

2

Consider the system shown in Figure 3.

J

This

system has two third-order linear subsystems, two

2

- B . . ly.y . 1 - ~ y. J~ J ~ qi-Pi J

2

feedback loops, and four nonlinearities.

By

plotting the Nyquist diagrams of the two transfer functions, we find that we may choose PI = 0.227, ql = 0.5, P2 = -1.0, q2 = 2.2 [2].

Applying

Theorem 3.2 in conjunction with Lemma 3.3, we find that the system is stable if n 1:

0.327 < ~ll(Yl) ~ 0.427, -0.900 < ~ 22(Y2) < -0.600

n 1:

i=l j=l+l Y2 V.

The ± signs are taken due to the ly . y.1 terms. ~

J

The

inequalities (3.4) guarantee that each matrix is non-negative definite, and there are n(n-l)/2 matrices. In Lemma 3.3 the inequalities (3.3) require that the graph of each of the nonlinearities

~

. . lie in

~J

CONCLUSION

We have focused on a multiloop system as an interconnection of scalar subs ystems. This viewpoint has allowed US to obtain conditions for global stability whi ch involve the Nyquist diagram for each of the scalar linear subsystems separately; a certain inequality involving the nonlinearities must also be satisfied. We have also considered, as a special case, conditions where the constraints on the nonlinearities can be expressed in terms of sector conditions. The main advantage of the method is that it is relatively easy to use, even in the cases where most other methods [1, 3, 4, 6, 7, 10] are intractable, such as the case where the number of nonlinearities is large.

certain sectors which are illustrated in Fig . 2. The inequalities require that 0

~

a

< Bii for ii i=l, ..• ,n but the number Pi' taken from the Nyquist

plot of Gi(s-y), may be pdsitive or negative and has the effe c t of "rotating" the sector for Consequently

~ ii(Yi)

~ ii.

may lie in any of the four

quadrants. A close examination of Theorem 3.2 in this case reveals that we have required that the uncoupled subsystems (i.e. if

~

.. (v.) = 0, i I j) are nec-

~J

essarily globally stable.

J

It can be shown [2] that

Several 'possible extensions of the work described here should be mentioned. The special conditions on the nonlinearities given in Lemma 3.3 to guarantee that Q(y) > 0 are only typical; other conditions can also be developed to guarantee satisfaction of the inequality [2]. Second, although we have considered global stability for the case of zero inputs, the same criteria also guarantee bounded input-bounded state stability for (2.1). Finally, we mention that the basic result presented in this paper, Theorem 3.2, can be improved by introduction of certain multipliers. Weighting multipliers among the subsystems [1,5] and Popov multipliers for each subsystem [5,8] can be used. VI.

REFERENCES

there always exist nontrivial gain sectors for the interconnection nonlinearities ~ ij(Yj)' i I j, such

(1)

B.D.O. Anderson, "Stability of Control Systems with Multiple Nonlinearities," J. of Franklin Institute, 282, 1966, 155-160.

(2)

J .D. Blight, "Scalar Stability Criteria for Nonlinear Multivariable Feedback Systems," Ph.D. Dissertation, The University of Michigan, 1973.

(3)

R.F. Estrada, "On the Stability of Multiloop Feedback Systems," IEEE Trans. Automatic

that the interconnected system is globally stable. In particular we can state the following:

under

the stated assumptions, if n globally stable subs y stems are interconnected the multiloop system will be globally stable for sufficiently small inter-

604

Control, AC-17, 1972, 781-791. (4)

R.P. Iwens, "Input-Output Stability of Continuous and Discrete-Time Nonlinear Control Systems," Ph.D. Dissertation, University of California, Berkeley, 1967.

(5)

N.H. McClamroch and G.D. Iancelescu, "Conditions for Global Stability of Two Linearly Interconnected Nonlinear Systems," 1974 JACC Proceedings, Austin, Texas.

(6)

K.S. Narendra and C.P. Neuman, "Stability of Continuous Time Systems with n-Feedback Nonlinearities," AIAA Journal, 11, 1967, 2021-2027.

(7)

S. Partovi and N.E. Nahi, "Absolute Stability of Dynamic Systems Containing Nonlinear Functions of Several State Variables," Automatica, 5, 1969, 465-473.

(8)

(9)

+ Pi ----------------~~~~-----------------yi

Figure 2a

V.M. Popov, Hyperstability of Control Systems (in Rumanian), Bucharest; Publishing House of the Academy, 1966.

Sector Conditions for Non1inearities


D.W. Porter and A.N. Michel, "Input-Output Stability of Time Varying Nonlinear Multiloop Feedback Systems," 1974 JACC Proceedings, Austin, Texas.

Y. J

(10) H.H. Rosenbrock, "Multivariable Circle Theorems", Recent Mathematical Developments in Control, ed. D. J. Bell, Academic Press, 1973, 345-365. (11) D. Siljak, "Stability of Large-Scale Systems under Structural Pertubations," IEEE Trans. Systems, Man, Cybernetics, 2, 1972, 657-663.

Figure 2b

Sector Conditions for Non1inearities

1 s(0.5s + 1) (0.2s + 1) U. 1

i

1, ... ,n

Block Diagram of the i th Subsystem of the Standard Multi100p Form Figure 1

+

Y2

Figure 3

605