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Stability analysis by means of modern linear systems theory
Economics Letters North-Holland
STABILITY
53
28 (1988) 53-56
ANALYSIS
BY MEANS OF MODERN
LINEAR SYSTEMS
THEORY
J.W. NIEUWENHUIS Unicersity
Recieved Accepted
of
Groningen,
16 December 12 February
9700 AV Gronmgen,
The Netherlands
1987 1988
By means of modem tools from linear systems theory we give a new procedure to calculate all stable solutions of a set of homogeneous difference equations. We apply our procedure to a small model analyzed by Anderson and Moore (1985).
1. Description of tools Notation Z + := {all non-negative
integers}, C := complex plane, R y := q-dimensional Euclidean space, (Ry)z+ := { w:z+ + lR4}, 0 : QR*y+ - (lRqZ+ is defined by (GE)(~) := Qt + l), Vr E Z +, (a is called the shift operator) uk := u.uk-‘, V’k 2 1, p:’ :;‘s~:=, {matrices R such that R,,, is a polynomial in the indeterminate, j=l,2 ,. -,n,, ,-.., n2). 2. Introduction of 9,
representation
results
(Ry+ is a linear vector space with componentwise addition and scalar multiplication. We endow this space with the topology of pointwise convergence as follows: W, wiE(Wq)‘+, iE Z+, WI-’ WI aPV,(t): + w(t), VtE H+, where the last convergence is the i-m-
usual one in lR4.
gq := {B G (rWq)‘+ (1, 2, 3 hold}, where (1) B is a linear subspace, (2) u B c B (II is shift invariant), (3) B is closed.
Definition.
The following statements are equivalent: Theorem I (see Willems (19864 b, 1987)]. (1) B EJp (2) 3 integer g I q, 3R(s) E IWRXq[s] with (a) rank R(c) = g for all, but a finite number, c E @, (b) B= {M,E(IW~)~+ IR(u)y=O}, whereR(u)~:=(~,Rl,(u)w,...~,R,,(u)w). 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
54
J. W. Nieuwenhuis
/ Stability analysis by modern linear systems themy
Definition. When g = q in Theorem 1 then B is called autonomous. Theorem 1, then lE8is called controllable.
When rank R(c)
= g, Vc E C, in
Theorem 2 [see Willems (1986a, b, 1987) and Nieuwenhuis and Willems (1986, 1987)]. (1) B E gq autonomous iff it is finite dimensional; (2) B c&Fq is controllable iff 34(s) E Rqxp[s] such that BE{w)~~E([W~)~+ withy=M(a)cz}, (3) VB E_Y~~B, ~gq, 38, •2~ with B = B, + B,. and with B, autonomous and B,. controllable. _ nxn s IS unimodular if V’(s) Definition. when det U(~(~)d=“W, c! 2 0.
E tRflx” [s]. Hence U(S) is unimodular precisely
Theorem 3 [see Willems (19864 b, 1987) and Anderson and Moore (I 98511. (1) Let R, E IWRIXq[s], R, E (WR2xq[s] be such that, with B, := {E 1R,(a)y = 0}, i = 1, 2., B, c US,, then there is a matrix K(s) E oxR2Xq[s] such that R*(S) = K(s)R,(s). (2) When, in addition, B, c B,, then one can take g, = g, and K(s) to be unimodular.
3. An application, stability analysis Anderson and Moore (1985) study the following homogeneous linear difference equation: e
c
H,X,+i=O,VtEZ+,
,= --7 where the matrices H, are given, where r and 8 are given positive integers. X, E Iwq, Qt E Z +. The authors give a procedure to check whether the following holds: (a) There is for every set of initial conditions X, = x,, i = -7, . . . , - 1, a unique solution sequence { X,, t E Z + } (b) Every solution sequence { X,, t E Z + } is stable, i.e. lim X, = 0 f+OO By means of the theorems stated above we immediately conclude that (a) holds only if the set of solutions to the model is an autonomous B E ea-yLq. Proof. Every B E gq is the sum of an autonomous and a controllable part, i.e., ll3 = B, + 5,. From Theorem 2(2) we deduce that for (a) to hold B, has to be equal to (0). In the rest of this note we give a new procedure to calculate the set of stable solutions of a given autonomous B egg. In order to do so we need some lemmata. Lemma
1.
Let y E (rWq)‘+ be stable and K(s)
Proof.
Trivial.
E IWRxq[s]. Then K(u)M,
Lemma 2 [see Kalaith (1980)]. Let R(s) E Iwgxq[s], V(s) such that A(s) := U(s)R(s)V(s) is diagonal.
is stable.
then there are unimodular
matrices
U(s)
and
J. W. Nieuwenhuis
/ Stability
55
analysis by modern linear systems theory
For B ~~~ we define B,, := {z E B 1y_ is stable}. For every polynomial p(s) E R[s] Definition. we write p(s) =p,(s).pp(s) where p,(s) contains all the roots of p(s) in magnitude smaller than one. As a mnemonic device y stands for good and p for bad. Lemma 3. (1) Let B E gp” be autonomous, then B, E g and is automous. (2) Let B ELI be given by B = {w 1p(a)y = 0) f or some po@nomial p(s). = O}. Proof. As B is autonomous and B, is linear it follows that B, autonomous. (2) follows by applying Theorem 3.
4. Procedure
to calculate
Then US, = {E 1p,(a)~
is finite dimensional
and hence
B,
Let B E&Z’~ be given by B = {w ] R(u)M, = 0} for some R(s) E Rqxq[s] where rank R(c) = q for almost all c E C. (i). Calculate U(s) and V(s) such that A(s):= U(s)R(s)V( s ) is diagonal and such that U(s) and V(s) are unimodular. Let A(s):= diag(p,(s),..., P,(s>) and A(s),:= dWp,.,(s),..., P,.,(s)). (ii). B, = {w ( A(c~),V~‘(a)w = 0). From Theorem 3 it follows that B = {M/] URN = 0} Proof of the correctness of the procedure. = {w ] A(u)V-‘(a)~ = O}. Now notice that {w ] A(a)V-‘(u)! = 0} = {w = V(u)@ with A(u)@ = 0}, hence the procedure is justified because of Lemma 1 and Lemma 3.
5. An example
We take the following example from Anderson and Moore (1985).
l.E0:=
w=(E,,w2)
t
Iiu-t-r
iw=o 1
U u-l+8
-
’
where r > 0 and 0 < S < 1 are given numbers. We take (Y and /3 such that (Y- p = - 1 and (~(6 - 1) = - /3(1-t r). then as U(s) is unimodular,
p&..
(li w
u-1-r 0
Define V(s) := (t
P(u-1-r) a-1+8
,“)
w=.
i-
1
thenU(s)R(s)V(s)=(“-~-r
’ o-1+6
).
56
J. W. Nieuwenhuis
/ Stability analysis by modern linear systems theor?;
Because of the conditions on r and 6 it now follows that A(.s)y=(i
’ a-1+6
),hence
and this agrees with the result derived by Anderson and Moore.
References Anderson, B. and B. Moore, 1985, A linear algebraic procedure for solving linear perfect foresight models, Economics Letters 17, 247-252. Kailath, T., 1980, Linear systems (Prentice-Hall, Englewood Cliffs, NJ). Nieuwenhuis, J.W. and J.C. Willems, 1986, Deterministic ARMA models, Lecture Notes in Control and Informational Sciences 83, 429-439. Nieuwenhuis, J.W. and J.C. Willems, 1987. About reachable and autonomous systems. reprint. Willems, J.C., 1986a. From time series to linear system, Part I, Automatica 22, no. 5, 561-580. Willems, J.C., 1986b, From time series to linear system, Part II. Automatica 22, no. 6, 6755694. Willems, J.C., 1987, From time series to linear system, Part III, Automatica 23, no. 1, 87-113.
STABILITY
53
28 (1988) 53-56
ANALYSIS
BY MEANS OF MODERN
LINEAR SYSTEMS
THEORY
J.W. NIEUWENHUIS Unicersity
Recieved Accepted
of
Groningen,
16 December 12 February
9700 AV Gronmgen,
The Netherlands
1987 1988
By means of modem tools from linear systems theory we give a new procedure to calculate all stable solutions of a set of homogeneous difference equations. We apply our procedure to a small model analyzed by Anderson and Moore (1985).
1. Description of tools Notation Z + := {all non-negative
integers}, C := complex plane, R y := q-dimensional Euclidean space, (Ry)z+ := { w:z+ + lR4}, 0 : QR*y+ - (lRqZ+ is defined by (GE)(~) := Qt + l), Vr E Z +, (a is called the shift operator) uk := u.uk-‘, V’k 2 1, p:’ :;‘s~:=, {matrices R such that R,,, is a polynomial in the indeterminate, j=l,2 ,. -,n,, ,-.., n2). 2. Introduction of 9,
representation
results
(Ry+ is a linear vector space with componentwise addition and scalar multiplication. We endow this space with the topology of pointwise convergence as follows: W, wiE(Wq)‘+, iE Z+, WI-’ WI aPV,(t): + w(t), VtE H+, where the last convergence is the i-m-
usual one in lR4.
gq := {B G (rWq)‘+ (1, 2, 3 hold}, where (1) B is a linear subspace, (2) u B c B (II is shift invariant), (3) B is closed.
Definition.
The following statements are equivalent: Theorem I (see Willems (19864 b, 1987)]. (1) B EJp (2) 3 integer g I q, 3R(s) E IWRXq[s] with (a) rank R(c) = g for all, but a finite number, c E @, (b) B= {M,E(IW~)~+ IR(u)y=O}, whereR(u)~:=(~,Rl,(u)w,...~,R,,(u)w). 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
54
J. W. Nieuwenhuis
/ Stability analysis by modern linear systems themy
Definition. When g = q in Theorem 1 then B is called autonomous. Theorem 1, then lE8is called controllable.
When rank R(c)
= g, Vc E C, in
Theorem 2 [see Willems (1986a, b, 1987) and Nieuwenhuis and Willems (1986, 1987)]. (1) B E gq autonomous iff it is finite dimensional; (2) B c&Fq is controllable iff 34(s) E Rqxp[s] such that BE{w)~~E([W~)~+ withy=M(a)cz}, (3) VB E_Y~~B, ~gq, 38, •2~ with B = B, + B,. and with B, autonomous and B,. controllable. _ nxn s IS unimodular if V’(s) Definition. when det U(~(~)d=“W, c! 2 0.
E tRflx” [s]. Hence U(S) is unimodular precisely
Theorem 3 [see Willems (19864 b, 1987) and Anderson and Moore (I 98511. (1) Let R, E IWRIXq[s], R, E (WR2xq[s] be such that, with B, := {E 1R,(a)y = 0}, i = 1, 2., B, c US,, then there is a matrix K(s) E oxR2Xq[s] such that R*(S) = K(s)R,(s). (2) When, in addition, B, c B,, then one can take g, = g, and K(s) to be unimodular.
3. An application, stability analysis Anderson and Moore (1985) study the following homogeneous linear difference equation: e
c
H,X,+i=O,VtEZ+,
,= --7 where the matrices H, are given, where r and 8 are given positive integers. X, E Iwq, Qt E Z +. The authors give a procedure to check whether the following holds: (a) There is for every set of initial conditions X, = x,, i = -7, . . . , - 1, a unique solution sequence { X,, t E Z + } (b) Every solution sequence { X,, t E Z + } is stable, i.e. lim X, = 0 f+OO By means of the theorems stated above we immediately conclude that (a) holds only if the set of solutions to the model is an autonomous B E ea-yLq. Proof. Every B E gq is the sum of an autonomous and a controllable part, i.e., ll3 = B, + 5,. From Theorem 2(2) we deduce that for (a) to hold B, has to be equal to (0). In the rest of this note we give a new procedure to calculate the set of stable solutions of a given autonomous B egg. In order to do so we need some lemmata. Lemma
1.
Let y E (rWq)‘+ be stable and K(s)
Proof.
Trivial.
E IWRxq[s]. Then K(u)M,
Lemma 2 [see Kalaith (1980)]. Let R(s) E Iwgxq[s], V(s) such that A(s) := U(s)R(s)V(s) is diagonal.
is stable.
then there are unimodular
matrices
U(s)
and
J. W. Nieuwenhuis
/ Stability
55
analysis by modern linear systems theory
For B ~~~ we define B,, := {z E B 1y_ is stable}. For every polynomial p(s) E R[s] Definition. we write p(s) =p,(s).pp(s) where p,(s) contains all the roots of p(s) in magnitude smaller than one. As a mnemonic device y stands for good and p for bad. Lemma 3. (1) Let B E gp” be autonomous, then B, E g and is automous. (2) Let B ELI be given by B = {w 1p(a)y = 0) f or some po@nomial p(s). = O}. Proof. As B is autonomous and B, is linear it follows that B, autonomous. (2) follows by applying Theorem 3.
4. Procedure
to calculate
Then US, = {E 1p,(a)~
is finite dimensional
and hence
B,
Let B E&Z’~ be given by B = {w ] R(u)M, = 0} for some R(s) E Rqxq[s] where rank R(c) = q for almost all c E C. (i). Calculate U(s) and V(s) such that A(s):= U(s)R(s)V( s ) is diagonal and such that U(s) and V(s) are unimodular. Let A(s):= diag(p,(s),..., P,(s>) and A(s),:= dWp,.,(s),..., P,.,(s)). (ii). B, = {w ( A(c~),V~‘(a)w = 0). From Theorem 3 it follows that B = {M/] URN = 0} Proof of the correctness of the procedure. = {w ] A(u)V-‘(a)~ = O}. Now notice that {w ] A(a)V-‘(u)! = 0} = {w = V(u)@ with A(u)@ = 0}, hence the procedure is justified because of Lemma 1 and Lemma 3.
5. An example
We take the following example from Anderson and Moore (1985).
l.E0:=
w=(E,,w2)
t
Iiu-t-r
iw=o 1
U u-l+8
-
’
where r > 0 and 0 < S < 1 are given numbers. We take (Y and /3 such that (Y- p = - 1 and (~(6 - 1) = - /3(1-t r). then as U(s) is unimodular,
p&..
(li w
u-1-r 0
Define V(s) := (t
P(u-1-r) a-1+8
,“)
w=.
i-
1
thenU(s)R(s)V(s)=(“-~-r
’ o-1+6
).
56
J. W. Nieuwenhuis
/ Stability analysis by modern linear systems theor?;
Because of the conditions on r and 6 it now follows that A(.s)y=(i
’ a-1+6
),hence
and this agrees with the result derived by Anderson and Moore.
References Anderson, B. and B. Moore, 1985, A linear algebraic procedure for solving linear perfect foresight models, Economics Letters 17, 247-252. Kailath, T., 1980, Linear systems (Prentice-Hall, Englewood Cliffs, NJ). Nieuwenhuis, J.W. and J.C. Willems, 1986, Deterministic ARMA models, Lecture Notes in Control and Informational Sciences 83, 429-439. Nieuwenhuis, J.W. and J.C. Willems, 1987. About reachable and autonomous systems. reprint. Willems, J.C., 1986a. From time series to linear system, Part I, Automatica 22, no. 5, 561-580. Willems, J.C., 1986b, From time series to linear system, Part II. Automatica 22, no. 6, 6755694. Willems, J.C., 1987, From time series to linear system, Part III, Automatica 23, no. 1, 87-113.