On the Stability of Linear Discrete Systems and Related Problems

and Related Problems

M. Mansour^l and E. I. Jury2

1 Institute of Automatie Control, Swiss Federal Institute of Teehnology, ETH-Zürieh CH-8092 Zürich, Switzerland

2 Department of Eleetrieal and Computer Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA

In this report some developments in the area of stability of discrete systems originally motivated by a paper by Kaiman and Bertram are overviewed. It is shown that an analog to the Schwarz-form was developed for diserete systems. This form was applied in determining the margin of stability, Identification, signal proeessing using lattiee filters and model reduetion of one-dimensional and multi-dimensional systems

1 Introduction

Wall [1] has shown that a polynomial

(1) has all its roots with negative real parts if and only if Cl' C2 , ••• , Cn in the following continued fraction are aB positive

where

Let

g(s) = ________ 1 ________ _ f(s) Cls+ 1 +---1---¹

C2S

+---

C3S

+

i = 1, .. . ,n and ^Co = 1

(2)

(3)

A. C. Antoulas (ed.), Mathematical System Theory

© Springer-Verlag Berlin Heidelberg 1991

372 M. Mansour and E. I. Jury

Then g(s) f(s)

f(s) is Hurwitz-stable if and only if b1 , ... ,bn are all positive.

(4)

It was proved also in [1] that if f(s) is Hurwitz-stable then there exists uniquely determined polynomials fO,f1"'" fn-1 of degrees 0,1, ... , n - 1 having the same property as f(s) and which are connected with f(s) by the recurrence formula

fm+ ¹ = sfm

+

bm+ 1fm-1 m = 0, 1, ... ,n-1

where fo = 1,f -1 = 1 and fn = f(s).

(5)

Here f1 (S),f2(S), ... ,fis) are the successive denominators of the continued fraction (4).

It was also shown in [1] that

f(s) = det

i.e. f(s) is the characteristic polynomial of the matrix -b1 b2

-1

°

(6)

(7)

Other variations of the matrix (7) which have the same characteristic equation are

[=:: ~

^-h.

J[ ^~1

(8a, b,c)

It is noted here that this matrix or one of its variations has been called in the literature the "Schwarz-matrix."

Schwarz [2J developed a numerical method of elementary transformations to transform a given matrix A to the above nurmal-form (8a) so that the Hurwitz- stability ofthe matrix A is determined from b1,b2 , ••. ,bn i.e. A is Hurwitz-stable if and only if b1 , b2 , • •• , bn are all positive.

KaIman and Bertram [3J used the second method of Lyapunov to prove the Hurwitz-stability of a system matrix in Schwarz-form, e.g. if we consider the linear system

where Al is in Schwarz-form (8a) then the Lyapunov function

v=

:s.TP1:s.

where

b./b,b, ... b.

l

can be used to prove Hurwitz-stability

V=

-:s.TQ1X 1 where

(9)

(10)

(11 )

(12)

V

is negative semidefinite and vanishes identically only at the origin. Therefore b1 , b2 , •. • , bn > 0 are necessary and sufficient conditions for the stability of the system (9).

Parks [4J showed that the first column of the Routh array consists of the elements

(13) As the connection between Routh and Hurwitz criteria is known (see Gantmacher [5J) i.e. the elements of the first column of the Routh array are

(14)

where D, . .. , Dn are the Hurwitz determinants.

374 M. Mansour and E. I. Jury

From (13) and (14) we get

D2 D3 DID4

bl =D I,b2 =-,b3 =--,b4=--,

DI DID2 D2D3

(15) Thus the link between Lyapunov-Routh-Hurwitz was obtained.

In [4] the inverse problem of stability was solved, i.e. the coefficients of the characteristic equation as a function of the stability conditions are determined.

In [6] a recursion formula is derived which expresses the coefficients of the characteristic polynomial (1) as a function of bl , b2,.", bn. Here we have

ar,n=ar,n-l +ar-2,n-2 bn

al,l = bl ao,o = 1 aj,k = 0 for j < 0 and k <j (16) For n = 1 all = bl

For n = 2 a12 = b l a22 = b2

For n = 3 a13 = bl a23 = b2

+

b3 a33 = bl b3 The relation between al " .an and bl " .bn is given by

al 1 0 0 0 0 bl

a2 1 1 1 0 b2

a3 al,l a l ,2 al ,n-2 b3

a4 a2,2 a2,n-2 b4 (17)

an an-2,n-2 bn

Also in [6] a transformation matrix T, which transforms a matrix A in companion form to a matrix A, in Schwarz-form is derived. Here Al = Tl A T~ I

where

A=

[0

-an 1

o

(18)

al,l 1

T

I = a2,2 a l ,2

a3,3 a2,3 a l ,3

an-l,n-l an- 2,n-1 an-3,n-1 1

A similar transformation was also derived in [8].

Let P 1 = P 11 2 where Plis given by (10), i.e.

[(bd/2 ^{(b1/b2 )1 / 2}

1

P11 = .

(b db2b3 . ... bn)1 /2

(19)

Using the transformation matrix T2 = P 11 Tl we get the system matrix

o

⁽²⁰⁾

b;12 - b!/2 0 In this case V = ~T ~ Le. P 2 = I = Identity matrix and

(21)

It is noted that the matrix (20) was first used in [9J. In [10J it was shown that the Schwarz-matrix of a complex polynomial can be related to that of areal polynomial using the Lienard-Chipart-type simplification of the stability pro- perties of areal polynomial.

Motivated by the work of KaIman and Bertram [3J and by their statement:

"The analog of the canonic matrix (Schwarz-matrix) is so far not yet available for descrete systems" and solving the inverse problem of stability similar to the work of Parks [4J, Mansour [7J obtained the analog of Schwarz-matrix for discrete systems which is called in the literature the discrete Schwarz-form or Mansour-form. In the next section the development leading to this form is presented, as weIl as some extensions. In section III the application of this form in different areas, like determining the stability margin of discrete systems, Identification, Lattice filters, Model reduction of one-dimensional systems and model reduction of multi-dimensional systems.

2 The Discrete Schwarz-Form or Mansour-Form

The Schur-Cohn stability criterion of a linear discrete system gives the condi- tions under which all the roots of the characteristic polynomial lie inside the unit circle.

Given the system

(22) A is in companion form.

376 M. Mansour and E. 1. Jury

The characteristic equation of the system (22) is given by

F(z) = Fn(z) = zn + a, zn-l + ... + an = 0 (23) The necessary and sufficient condition for the roots of (22) to lie inside the unit circle is that the zero order terms in Fn(z) and the n - 1 polynomials obtained successively through the transformation

(24)

are of magnitude smaller than unity [l1J, [12].

L1r is the zero order term of the polynomial of degree r

lL1

^r

l<1

r=1,2, ... ,n (25)

The L1r can be obtained from Jury table [13].

In [7J the inverse problem of stability for discrete systems was solved giving the recursion formula

ar,n = ar,n-l

+

an-r,n-l L1n

aj,k = 0 for k <j, aj,k = 1 for j = 0 i.e.

a 1,l=L1l , a12=L1l +L1lL1 z, a22=L1 z,···

This can be expressed in matrix form as follows 1 al,l az,z

a1,z 1

a3,3 aZ,3 al ,3

an-l,n-l

an-Z,n-l an - 3,n-1

If we use the transformation matrix

an-l,n-I]

an-Z,n-Z 1

we can transform the system (22) to the Mansour-form

l-L1~_1

- L1n -2L1n -1 ^{1 - .12} n-2

(26)

(27)

(28)

(29)

Choose a Lyapunov-function V = ~T P 3~' where

o ... 0J

o ...

⁰

·

.

·

.

· .

o ...

⁰

(31)

AVis negative semidefinite and does not vanish identically for any general sequence of vectors.

Thus the Schur-Cohn stability criterion is proved using the second method of Lyapunov and a diagonal matrix is used here for the proof similar to [3].

Let 1-

A/

⁼

b/

^{and P3 =} p3 / . then

(32)

Using the transformation matrix T4

=

P 33 T3 then we get - An-lAn

- An-2Anbn-l

- AlAnb2 .. ·bn- l - A l An- l b2 ···bn - 2

-Anblb2"·bn-l -An-lblb2···bn-2

(33) In this case V

=

~ ^T ~ i.e. P 4 = land

J

⁽³⁴⁾

Another form can be obtained using the transformation matrix T [14], [15]

378 M. Mansour and E. 1. Jury

whose elements are given by

i-j+l

Tij= T i - 1,j-l

+ L

Lin-i+1Lin-i+kTi-k,j-l k=2

T ii = 1, Tij = 0 for j > 0 T iQ = 0 for' i > 0

-.1.- 2.1.(1-.1;_1) -.1.- 2.1.- 1 [

-.1._1.1. 1

As = TAr- 1 = -L1.-3L1.(I-f;-2)(I-L1;-1) -.1.-3.1.-:1(1-.1;_2) -L1.(1-L1~) ... (1-L1;_1) -L1._1(1-L1~) ... (1-L1;_1)

For the proof of Schur-Cohn criterion we use here

and get

Q,~ [0 J

(35)

-J

⁽³⁶⁾

(38)

In [10] the results were extended to the complex case and simplifications analogous to those provided by the Lienard-Chipart criterion are obtained for the real case.

3 Applications

3.1 Estimation of the Margin of Stability

In [14] the margin of stability of a discrete system given by the nth order difference equation

(39) is determined using the weighted square error.

The state variable representation is given by

~(k

+

1)

=

A~(k) (40)

where A is in the companion form. A can be transformed to the form (36) g(k

+

¹⁾

=

Asg(k)

Let the Lyapunov function be V=gTPsg, L1V= _gTQsg

(41)

(42) Psis given by (37) and Qs by (38). The margin of stability is determined using the weighted sum ofsquare error J = 'L.:'=Ok'y2(k). Due to the fact that T defined in (35) is a lower triangular matrix with (1,1) entry

=

1, then

<Xl <Xl <Xl

J =

L

^{k'y2(k) =}

L

^{k'xi(k) =}

L

^k'xi(k)

k=O k=O k=O

For r=O,

Jo

= gT(O{k~O k'(A~)TQsA~

^]g(O)

which can be computed directly for different initial conditions. For example if y(O) = y(1) = ... = y(n - 2) = 0 and y(n - 1) = 1

then the margin of stability [14]

1

(J

=

1 - max

I

Ai

I

> 22n - 1 J 0

(43)

(44)

(45) where Ai are the roots of (39). For weighted sum of square error (r ~ 1) J, can be obtained in terms of solutions of the Lyapunov equation for linear discrete systems [14].

3.2 Identification and Approximation of Systems

Dourdoumas [16] transformed the system

~(k

+

¹⁾

=

^A~(k)

+

lzu(k) y(k) = fT ~(k)

where A is in companion form, lzT = [0 ... 1] to the form g(k

+

¹⁾

=

Mg(k)

+

Qu(k)

y(k) =

fT

g(k)

(46)

(47) where M is in Mansour-form (29). The transformation matrix T is determined from AT= TM and b = Tb.

380 M. Mansour and E. 1. Jury

In this case there are 2n parameters to be identified in the system (47). The advantage of the transformation to the form (47) is that M is in Hessenberg form which has some numerical advantages. The performance criterion is chosen as

N

J =

L

Iy*(k) - y(k)1 (48)

k;1

where y*(k) is the measured output.

Stability is assured if ,1;1 < 1. To use optimization without constraints on the parameters ,1; is transformed as follows

,1; = - (1 - e) + 2(1 - e)sin² ,1;

e«l (49)

3.3 Representation of Lattice Filters

Takizawa et al. [17] uses the Mansour-form to represent lattice filters. The system (46) can be transformed to

where

g(k

+

¹⁾ = Mg(k)

+

fu(k) y(k) =

fT

g(k) M is in Mansour-form

~ T

!z. = [L1n -1L1n - 2 ·•· ,111]

fT

=

[flfz··· fn]

(50)

The system (50) can be represented by the following cascade structure Fig. 1 with a diagonal matrix transformation

T=diagona{1 +enL1n^,(1 +enL1n)(1 +en-1L1n-l), ...

lII

(1 +e;L1J]

where e; =

±

1.

~

I

+

Fig.l (51 )

u

-

'---..z ^)---~ - - - -- - - - 1 0 - ( )----~ Y

The above representation (50) can be transformed to

r -

^{.1.- 1.1.} 1 - 6.- 1.1.- 1 - .1.-2.1.(1 + 6.- 1.1.-d - .1.-2.1.- 1

M'= : :

.11.1.U7~2(1 + 6j.1;) .. . .1.U;:II(I+eA) .. .

~'I

^{= [} .1.- 1,(1 + 6 • .1.)···

U7~

1(1 + 6j.1 j)

T

(:y ⁼

[<.

^··e'l]

-.1 1.12

- .12(1 + 61 .1 1)

Fig.2

(52) The system (52) can be represented by the lattice digital filter Fig.2. This realization is with the least number of multipliers. Normally, there is a direct coupling between u and y. The signs in Fig. 2 must be chosen in the same sequence and are determined so that the coefficient sensitivity and the output noise become smalI. The structure corresponds to a lossless cascade transmission line.

3.4 Model Reduction of Discrete Time Systems

In [18] the Badreddin-Mansour method of model reduction of SISO discrete systems was presented. The idea of model reduction is explained as follows. Let

~(k

+

¹⁾ = M~(k)

+

Qu(k) y(k) =

!/

~(k) where M is given by

-.11 1-L1i 0 0 0

-.12 -.11.12

M=

Q=[1 O,,·O]T

!/=[C1 C2 "'Cn ]

(53)

382 M. Mansour and E.1. Jury

It is assumed that (53) is stable, Le. ILiil < 1. If I(Lin/Lin - 1)1 is sufficiently small then X n will reach its quasi steady state much faster than X n -1' From last system equation in (53) we get after putting xn(k) instead of xn(k

+

1) on the left side of the equation

- Li^{n •} (k Li^{n -} 2Li^{n .I>} (k)

~n(k)= Xl )_ ••• - An-1

1

+

Lin _ 1 Lin 1

+

Lin - 1 Lin

substituting in equation (53) we get the new system matrix

where

A _ Lin

+

Lin - 1 L.ln-1 -

1

+

Lin - 1Liⁿ

o

0

o

1 - Li;_2 - Lin - 2L1n - 1

(54)

(55)

The matrix

Nt

is in the same form. One can easily prove that stability and steady state response are preserved after reduction. Further reduction can be continued in the same manner.

In [19J and [20J two multivariable Mansour forms for model reduction are derived. The first one is obtained from the Luenberger first form using similarity transformation.

x : x·

•• 0 • • • • • • • • • • • • : • • • • • • • • • • • • ~

x : x:

: : 1 · _{· · .}

. .

A= ... ~.~ ... ~ .. ~ ^{B= (56)}

x : x:

. . . ..

. .

... ; ... : ... .

x x; x

:

x x: 1 x

~

X X

IM^{22 \}

M= ^X x (57)

x x x x

The diagonal blocks are in Mansour form. The extra degree of freedom in the coupling elements in (57) can be used to get a nice structure of M. For example, for 2 inputs (m = 2)

A [ All

I

^{0 ]}

= O···OPl A22

(58) The reduction of the first or the second subsystems can be done independently by the same method used in the single input case.

The second multivariable normal form [20J is obtained by first transforming the Luenberger first form to the block-controllability form by means of elemen- tary similarity transformations and then applying a generalized matrix Schur- Cohn-Jury table. This second form is then constructed in the same manner as the single input case.

Let the controllability of the nth order system with m inputs be described by the block controllability form

[0 -AqJ

A= 1^m . . , :

.. Im -:A₁

The second multivariable Mansour form will be

- (}l 1 - ()~

- (}i - (}1(}2 M=

(59)

(60)

The elements (}l' (}2' ... ' (}q are m x m matrices obtained from the matrix Schur- Cohn-Jury table.

The reduction procedure is similar to the single input case. (For details of transformation and model reduction see [20J).

In [21J further justification of the Badreddin-Mansour method is presented where the connection between root location properties, coefficient properties and Schur-Cohn coefficient properties are established. It is also shown that the lattice realization of the reduced system differs from the lattice realization of the original system by replacing the last delay by unity gain element.

384 M. Mansour and E. I. Jury

In [22] it was shown that the elimination of row 1 and column 1 or elimination of row i-I and column i for i = 2, 3" ... ,n of the Mansour matrix (29) will result in a Mansour matrix of dimension n - 1. The above corresponds to elemination of the first subsystem or the ith subsytem i = 2, 3, ... ,n in the cascade realization.

3.5 Model Reduction of Two Dimensional Discrete Systems

In [23] the one dimensional reduction method of Badreddin-Mansour is extenned to two-dimensional (2-D) discrete systems. The Roesser model of 2-D system utilizes two kinds of state variables; one which propagates horizontally and the other vertically. The model for a 2-D linear shift invariant SISO discrete system is given by

[ ^~:(~

^{-: 1,j)] =}

[~I~] [~:(~'~)] +

[Bl]U(i,j)

~ (z,J

+

1) A3 A4 ~ (z,J) B2

(61)

y(i,j) = [Cl

I

^C

2] ^[~h(~,~)]

_~V(z,J)

⁺

^Du(i,j)

A similarity transformation transforms the companion forms Aland A4 to Mansour matrices M ¹ and M ^4' The matrices M ¹ and M ⁴ are the horizontally propagating and the vertically propagating sections of the 2-D systems. The off-diagonal matrices represent the interconnection between the two sections.

The model reduction of the horizontal or vertical section is done similarly to the I-D case. For the special ca se of separable systems (M 2 = 0 or M 3 = 0) the reduced system will preserve the stability of the original system and also the steady state unit response. The same will be the ca se for 2-D 1 h-l v system. In the other cases stability is not, generally, guaranteed.

In [24] the above results are extended to 2-D multivariable discrete systems.

Two new 2-D multivariable canonical forms are derived on the same line as the two 1-D multivariable canonical forms obtained in [19] and [20].

Conclusion

The Schwarz canonical form of continuous systems and the discrete Schwarz canonical form or Mansour-form for discrete systems are discussed. An overview is given on the characteristics and applications of the Mansour-form in the areas of determination of the margin of stability, identification, signal processing using Lattice filters and model reduction of one-dimensional and two- dimensional systems. The work of KaIman and Bertram in 1960 on using Lyapunov theory to prove the stability of Schwarz form has inspired the sub se quent work for the discrete case.

References

[1] Wall, H.S.: Polynomials whose zeros have negative real parts. Amer Math Monthly, 52 (1945), pp 308-322

[2] Schwarz, H.: Ein Verfahren zur Stabilitätsfrage bei M atrizen-Eigenwertproblemen, Zeit f angew Math u Physik, 1956, pp 473-500

[3] Kaiman, R., Bertram, J.: Control System analysis and design via the second method of Lyapunov.

J of Basic Engineering, June 1960, p 371

[4] Parks, P.: A new proof of the Hurwitz-stability criterion by the second method of Lyapunov with applications to optimum transfer functions. Fourth Joint Automatie Control Conference, June 1963

[5] Gantmacher, F.R.: Applications ofthe theory ofmatrices. Interscience, New York, 1959 [6] Mansour, M.: Stability criteria of linear systems and the second method of Lyapunov. Scientia

Electrica, Vol XI, 1965, pp 65-104

[7] Mansour, M.: Die Stabilität linearer Abtastsysteme und die zweite Methode von Lyapunov.

Regelungstechnik, Heft 12, 1965, pp 592-596

[8] Chen, C.F., Chu, H.: A matrixfor evaluating Schwarz-Jorm. IEEE Trans on Aut Control, 1966, pp 303-305

[9] Puri, N., Weygandt, C.: Second method of Lyapunov and Routh canonicalform. J. ofthe Franklin Institute, Nov 1963, p 365

[10] Anderson, B.D.O., Jury, E.I., Mansour, M.: Schwarz-Matrix Properties for Continous and Discrete Time Systems. Int J Control, Vol 23, 1976, pp 1-16

[11] Schur, 1.: Ueber Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine u.

Angewandte Math. 147 (1917), pp 205-232

[12] Cohn, A.: Ueber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreis, Math Zeit 14 (1922), pp 110-148

[13] Jury, E.I.: Theory and Applications ofthe z-Transform Method. Krieger, 1982

[14] Mansour, M., Jury, E.I., Chaparro, L.F.: Estimation of the margin of stability for linear continuous and discrete systems. Int J Control, Vol 30, 1979, pp 49-69

[15] Mansour, M.: A note on the stability of Linear Discrete Systems and Lyapunov Method. IEEE Trans on Aut Control, Vol AC-27, No 3, June 1982, pp 707-708

[16] Dourdoumas, N.: Ein Beitrag zur Identifikation und Approximation von Systemen mit Hilfe linearer diskreter mathematischer Modelle. Archiv für Elektrotechnik 62 (1980), pp 1-4 [17] Takizawa, M., Kishi, H., Hamada, N.: Synthesis of Lattice digital filters by the state variable

method. Electron Commun Japan, 65-A, pp 27-36

[18] Badreddin, E., Mansour, M.: Model reduction of discrete time systems using the Schwartz canonicalform. Electron. Lett., Vol 16, No 20, Sep 25,1980, pp 782-783

[19] Badreddin, E., Mansour, M.: A multivariable normal-formfor model reduction of discrete-time sytems. Syst. Control. Lett. 4 (1983), pp 271-285

[20] Badreddin, E., Mansour, M.: A second multivariable normal-Jorm for model reduction of discrete-time systems. Syst Control Lett 4 (1984), pp 109-117

[21] Anderson, B.D.O., Jury, E.I., Mansour, M.: On Model Reduction of Discrete Time Systems.

Automatica, Vol 22, No 6, 1986, pp 717-721

[22] Antoulas, A.C., Mansour, M.: On Stability and the Cascade Structure. Technical Report 1990 [23] Jury, E.I., Premaratne, K.: Model Reduction ofTwo-Dimensional Discrete Systems. IEEE Trans

on Circuit and Systems, Vol CAS-33, No 5, May 1986, pp 558-562

[24] Premaratne, K., Jury, E.I., Mansour, M.: Multivariable Canonical Formsfor Model Reduction of 2D Discrete Time Systems. Vol CAS-37, No 4, IEEE Trans on Circuits and Systems, April 1990, pp 488-501