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Matrix convex directions for time delay systems1
IFAC
Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001
~
Publications www.elsevier.comllocate/ifac
MATRIX CONVEX DIRECTIONS FOR TIME DELAY SYSTEMS l J. Santos 2
,
S. Mondie and V. Kharitonov
Departamento de Control Automatico CINVESTAV-IPN, Av. IPN 2508, A.P. 14-740, 07300 Mexico, D.F., Mexico, [khar, smondie,jsantos}@ctrl.cinvestav. mx
Abstract: The characterization of convex direction for real Hurwitz matrices is extended to the case of multivariable time delay linear systems. The above is achieved using the results of convex directions for quasipolynomials. Copyright © 2001 IFAC Keywords: time delay systems, convex directions. 1. INTRODUCTION
delay system corresponding to each candidate, which is a quasipolynomial of the retarded type (Bellman and Cooke, 1963), is then analyzed with the help of results on convex directions for quasipolynomials (Kharitonov and Zhabko, 1993).
Many actual systems have the property of aftereffect, i.e., the future states depend not only on the present but also on the past history. Delay is believed to occur in such areas as mechanics, chemistry, biology, medicine. economics, atomic energy, information theory, etc.
New proofs of the basic results on matrix convex directions obtained in Kokame et all. (1994) are also given. They are essentially based on the ideas of the proofs developed for delayed systems and on results about convex directions for polynomials.
For an accurate modelling of such systems, the effect that the delays may have on the behavior of the system response must indeed be taken into account.
2. PRELIMINARIES
Stability is one of the important characteristics of dynamical systems. The concept of convex directions for polynomials and quasipolynomials proves to be a useful tool in stability and robust stability study, (see Rantzer (1992), Kharitonov and Zhabko (1993)). Recently some interesting extention were made to the case of matrices, see Kokame et all. (1994). There, a complete characterization of convex directions for real Hurwitz matrices has been given. Similar results for discrete time systems were reported in Xie (1997).
We introduce first some background on convex directions.
Convex Directions of Polynomials A notion of convex direction has been proposed in Rantzer (1992) and is defined as follows: Definition 1. A polynomial g( s) is a convex direction if for any stable polynomial p(s), the stability of p( s) + g( s) implies that of p(s)
The aim of this paper is to characterize matrix convex directions for time delay systems. Since this characterization must be valid for all values of delay, it must also be valid when the delay is equal to zero, therefore, matrix convex directions of Kokame et all. (1994) are the only matrix candidates for our characterization. It worths to be mentioned the fact that the caracterization does not depend of delay does not imply that we are talking about stability independent of delay. The characteristic function of the time 1 2
+ J.lg(s) ,
J.l E [0,1]
(1)
where deg(p(s)) > deg(g(s)). In the following we will use the following two results. Remark 1. Any first or zero degree polynomial g( s) with real coefficients is a convex direction for real Hurwitz stable polynomials. Remark 2. Any constant polynomial g( s) is a convex direction for complex Hurwitz stable polynomials.
Supported by Conacyt, Mexico 31951-A. Financed by Conacyt 130255.
147
Matrix convex directions
where D ij ERn; xn j , for i, j = 1,2. If D is a matrix convex direction, so are both D 11 and D 22 .
Consider a linear time-invariant system of the form
x(t) = Ax(t)
+ ILDx(t)
The next theorems describe main results from Kokame et all., (1994). We supply them with new proofs, because we will use later on the same arguments for the case of time delay systems.
(2)
x(t) ERn, A,D E R nxn , IL E R. where matrix A is Hurwitz and IL is a scalar parameter.
Theorem 5. (Kokame et all., 1994) For n = 2 a matrix D E R 2x2 is a matrix convex direction if and only if D, up to similarity transformations, has one of the following three forms:
Definition 2. (Kokame et all., 1994) Matrix D E R nxn is a convex direction if for any Hurwitz matrix A E Rnxn , the stability of x(t) = Ax(t)
Dl =
+ Dx(t)
0 0 [Od]
,D2 =
[d0 0] d ,D3 =
[Xl 0 0X2 ] ' (3)
where d, Xl, x2 are real numbers and Xl X2 :::; O. implies that of (2) for alllL E [0,1]. Geometric interpretation: the previous definition means that system (2) has only one open interval of stability on the real axis IL. The size and location of the interval depends on matrices A and D, but the fact that (2) has only one stability interval depends on matrix D. When system (2) is stable for IL = 0 and IL = 1 the stability interval contains segment [0. 1]. We can say even more: if D is a convex direction then for any (not necessarily Hurwitz) real square matrix A there exist no more than one stability interval on the IL-axis. In other words, the definition implies that if matrix D is a convex direction, then for any Hurwitz matrix A the system (2) has only one stability interval and vice versa, if for every Hurwitz matrix A the system (2) has only one stability interval, then matrix D is a convex direction.
Proof. 1. For D l the characteristic polynomial of the system (2) is
p(s) = S2
+ s( -0'22 -
0'11)
+ (0'110'22 -
0'210'12) -lLd0'21,
and can be written as p(s) = Po(s) + ILg(8), where g( s) = -d0'2l' Since g( s) is a constant polynomial it is a convex direction, then it follows from remark 1 that D l is a matrix convex direction. 2. For D 2 , the statement follows directly from lemma 1 and lemma 2.
3. For D 3 , the characteristic polynomial of system (2) has the form
p( 8) = 82 + 8( -0'22 - 0'11) + (0'11 0'22 - 0'120'2I)
+ 0'22Xl - (Xl + X2)8) + 1L2XlX2 = PO(8) + ILgl(8) + 1L 292(8), IL E [0,1]. (4) +1L(0'11X2
Initial Problem: Characterize the set of matrix convex directions for system (2).
It is not possible to apply here directly arguments based on polynomial convex directions because (4) depends quadratically on IL. Therefore we imbed first the family into the bigger one
The characterization has been given in Kokame et all., (1994), where the following auxiliary results were also stated.
Lemma 1. (Kokame et all., 1994) The identity matrix is a matrix convex direction.
where new parameters ILl' 1L2' vary in the triangle ABC (see Figure 1). This new family contains the original one for ILl = IL, 1L2 = 1L 2, IL E [0,1] (see curve TJ on Figure 2). On the other hand (5) is the convex hull of three polynomials: PO(8), PO(8 )+gl (8), Po(8 )+gl (8 )+g2(8), and therefore we may apply the edge theorem to check stability of the new family. Indeed, this theorem says that the stability of the three edges ll, l2, l3 implies that of (5).
Lemma 2. (Kokame et all., 1994) If DE Rnxn is a matrix convex direction. so is kD for an arbitrary real scalar k. Lemma 3. (Kokame et all., 1994) If DE R nxn is a matrix convex direction, so is 8- 1D8, where 8 is a real nonsingular matrix. Lemma 4. (Kokame et all., 1994) Let D be a block triangular matrix
Polynomials, corresponding to extreme points A and B are stable because (4) is stable for IL = 0 and IL = 1. We need first to verify stability of polynomial Po (8) + gl (8), which corresponds to the
D _ [D11 D12] 0 D 22 '
148
third extreme point, C. Comparing coefficients of PO(8) + 91(8) + 92(8) and PO(8) + 91(8) we see that the only difference is in the free coefficient. This coefficient of PO(8) + 91 (8) is bigger on -X1X2 (remember that X1X2 :::; 0) than the corresponding coefficient of Po (8) + 91 (8) + 92 (8). Both polynomials are of degree two, therefore stability of PO(8) + 91(8) + 92(8) implies that of PO(8) + 91(8).
Since 9( 8) is a polynomial of the first degree, then it is a convex direction, therefore D 4 is a matrix convex direction.
Now the stability of the three edges is verified with the help of the concept of convex direction.
Theorem 7. (Kokame et all., 1994) A matrix DE Rn X n, n 2: 4, is a matrix convex direction if a only if D is the identity matrix multiplied a real scalar
2. For matrix D 5 , we can observe that D 5 = dI3x3 is the identity matrix multiplied by a real scalar d. By lemma 1 and lemma 2 one can conclude that matrix D 5 is a matrix convex direction. •
For the edge h we have to verify the stability of PO(8) + /1[91(8) + 92(8)] for all /1 E [0,1]. Since 91 (8) + 92 (8) is of the first degree, then it is a convex direction, and stability of polynomials corresponding to extreme points A, B, imply that of the edge 11 .
d.
Proof. It follows directly from lemma 1 and lemma 2. • 3. MATRlX CONVEX DIRECTIONS FOR TIME DELAY SYSTEMS
For the edge 12 , we have to verify the stability of PO(8) + /1191(8) for all /11 E [0,1]. Again 91(8) is of the first degree, so it is a convex direction, and edge 12 is stable.
Given a linear delay system of the form
x(t)
For the edge 13 , we have to verify the stability of the segment Po (8) + 91 (8) + /1292 (8) for all /12 E [0, 1]. Polynomial 92 (8) is constant and therefore it is a convex direction, so edge 13 is stable. Since all three edges, h, 12 , h, are stables, this implies the stability of the family (5) and, of course, that of (4). Therefore matrix D 3 is a matrix convex direction. •
= Ax(t) + /1Dx(t - h)
(7)
where x(t) E Rn, A, DE R nxn , h 2: 0, /1 E R is the system parameter. The following definition of matrix convex directions for time delay systems, is motivated by our geometric interpretation of matrix convex directions for systems without time delay.
Definition 3. A matrix D E Rnxn is a convex direction for time delay systems if for any Hurwitz matrix A E R nxn , and for any h 2: 0 the system (7) has only one interval of stability on the real axis of /1.
B
I,
---
It is important to emphasize that this definition introduces matrix convex directions which do not depend on the value of the delay. But as it has been mentioned already in the introduction it does not imply that stability should be independent of delay.
Fig. 1
Theorem 6. (Kokame et all., 1994) A matrix D E R 3x3 is a matrix convex direction if and only if D coincides up to similarity transformation with one of the following two matrices:
Problem Statement: Characterize the set of matrix convex directions for (7). The following remark is a direct consequence of definition 3.
Remark 3. If matrix D E Rnxn is a convex direction, so is kD for a real scalar k.
where d is a real number.
Proof. 1. For matrix D 4 , the characteristic polynomial of the system (2) is p(8)
= PO(8) + /19(8),
3X3 where Po () 8 = d et (8I
9(8)
Machinery for the characterization Since we look for a characterization of matrix convex direction independent of the value of h, they must also be matrix convex directions for h = O. Therefore, the only candidates to be matrix convex direction in this sense are· those matrices
(6)
-A) ,
= d( -cx218 + CX21CX33 - CX31CX23).
149
4. BASIC LEMMAS
which are convex directions for systems without delay. This observation reduces drastically the set of possible candidates.
We present now some basic statements which will be used later on for deriving matrix convex directions.
For each candidate, our approach is based on the fact that it is possible to factorize the characteristic function of (7) in order to apply to each factor stability conditions based on the concept of convex direction for quasipolynomials.
Lemma 10. The identity matrix I E Rnxn , is a matrix convex direction. Proof. When D = I E Rnxn system (7) has the following characteristic function
Convex directions for quasipolynomials It follows from the above arguments that the char-
f(s) = det(sI - A - p1e- sh ).
acterization of matrix convex directions for linear systems with delays and for quasipolynomials are intimately related. For this reason. we recall now important results on convex directions of quasipolynomials (Kharitonov and Zhabko, 1993).
There exists a nonsingular matrix S such that
S-lAS =
= {P(s) = sn + a1s n- 1 + ... + an-IS + an} p(s) =
a
=}
Re(s) <
where Ai, i = 1,2, ... , n, are eigenvalues of A and f (s) can be factorized as
f(s) = (s - Al - /-Le-sh)(s - A2 - /-Le- sh ) ... (s - An - /-Le- sh ).
g(s) = q(s)e TS
Each factor is of the form fi (s) = Pi (s) + /-L9i (s), where Pi(S) = s - Ai, and gi(S) = qi(s)e- sh , qi (s) = -1. It follows from lemma 8 and lemma 9 that 9i(S) is a convex direction (real or complex, depending of Ai)' Therefore stability of Pi(S) and Pi(S) + gi(S) implies that of Pi(S) + /-L9i(S) for all /-L E [a,l]. This means that Inxn is a matrix convex direction. •
is a convex direction for 'H if (8) and the inequality 8 arg( q(jw)) < Isin(2 arg( q(jw))) I 8w 2w
(9)
holds for all positive frequencies w where q(jw)
a.
i
Lemma 11. If DE Rnxn is a convex direction, so is S-l DS, where S is a real nonsingular matrix.
Lemma 9. Let
= {P(S) = sn + C1 Sn - 1 + ... + Cn-1 S + Cn } p(s) = a =} Re(s) < a
be the set of Hurwitz stable polynomials with complex coefficients, and q(s) be a complex polynomial of degree less or equal n. Then
g(s) = q(s)e
a,
= Ax(t)
and the inequality
holds for all real w where q(jw)
+ /-LS-1 DSx(t -
h)
(13)
which is stable if and only if the original one is stable. It is clear that the new system has only one stability interval on the axis /-L when it is the case for system (7). In other words S-l DS is a matrix convex direction if and only if D is a matrix convex direction. •
(10)
8arg(q(jw)) < a 8w -
Proof. Since D E Rn X n is a convex direction, then system (7) has one interval of stability on the real axis /-L for all Hurwitz matrix A, so that, applying the similarity transformation to the system we obtain the new one
x(t) = S-l (Ax(t) + /-LDx(t - h)) S
TS
is a convex direction for 'H if T ~
a
a
be the set of Hurwitz stable polynomials with real coefficients, and q( s) be a real polynomial of degree less or equal to n. Then
'H
a
*
Lemma 8. Let 'H
(12)
(11)
Lemma 12. Given a complex number z, then cp( s) = s + A - ze- sh is stable if A > IzI.
i a. 150
Proof. Suppose, by contradiction that cp(s) has a root So such that Re( so) ~ O. Then So + A soh ze= 0 and Iso + AI = Izlle-sohl· Since le-sohl :::; 1, we can write that Iso + AI :::; Izl. By our assumption Re(so) ~ 0, A :::; Iso + AI. Therefore A :::; Izl. This inequality leads indeed to a contradiction. •
Then system (14) has the form x(t)
= [AJ1
-~I] x(t) + f-L [D~l g~~]
x(t - h).(18)
The characteristic equation of (18) is 1(s)
Lemma 13. Let D be a block triangular matrix:
= det(sh -
Ail - f-LDlle- sh )
. det(sI2 + >.12
-
f-LD22e-sh).
(19)
We know that 1(s) is stable for f-L = 0 and for = 0:' but is unstable for some f-La E (0,0:'). However D is a matrix convex direction, therefore the system (18) should be stable for all f-L E [0,0:']. The contradiction proves the lemma. • f-L
where D ij ERn; xn j , for i, j = 1,2. If D is a matrix convex direction, so are both D ll and D 22 . Proof. If D is a matrix convex direction then the system x(t) = Ax(t)
+ f-LDx(t -
h)
5. CHARACTERIZATION OF MATRIX CONVEX DIRECTIONS
(14)
We are now able to verify if the candidates, namely, the matrix convex directions for linear systems, are also matrix convex directions for time delay systems.
has only one stability interval for every Hurwitz stable matrix A. Let us assume by contradiction that D ll is not a matrix convex direction, i.e., there exists matrix Ail such that the system
Matrix convex directions in R 2x2 For 2 x 2 matrices we have to analyze the following three cases:
(15)
Case 1: Consider the following matrix candidate
is stable for f-L = 0 and f-L = 0:' > 0, but it is not stable for some f-Lo E (0,0:'). Now, let us choose a matrix A 22 such that the system
Do=[~~],
The characteristic equation of system (7) for this matrix is 1(s) = p(s) + f-Lg(s) where 1(s) = S2 + s( -0:'22 - O:'ll) + (0:'1l0:'22 - 0:'210:'12), g(s) = d0:'21e-sh = q(s)e TS with q(s) = d0:'21, and T = -h. It is clear that T = -h < 0, and
is stable for all f-L E [0,0:']. Let A 22 = -AI where A > 0, then the characteristic equation of system (16) is given by 1(s) = det(sI
+ >.1 -
f-LDne- sh ).
(17)
oarg(q(jw)) ow
Matrix D 22 can be written as D 22 = T- 1JT, where Zl
J
=
*
= sin(2arg(q(jw)) = 0 2w
So, g(s) satisfies lemma 8 and convex direction.
o
Do
.
is a matrix
Case 2: Consider the matrix candidate
0 [
o
o
o We notice that D{3 = dI 2x2 is the identity matrix multiplied by a real scalar d. Due to lemma 10 and remark 3 we infer that matrix D{3 is a matrix convex direction.
is the canonical form of matrix D 22 . We can apply the last transformation to (17) in order to obtain 1(s)
dER.
=
II~~l (s + A - f-LZke-sh)
Case 3: Consider the matrix candidate
It follows from lemma 12 that system (16) is stable for all f-L E [0,0:'] if A> O:'max1:S;k:S;n2{lzkl} = Aa·
Now let A > Aa. Consider matrix We are able to analyze only the subcase X1X2 = O. The analysis of the general case, X1X2 < 0, seems
A* = [
1<;1
Theorem 14. A matrix D is a convex direction for time delay systems in R 2X2 , if D coincides, up to similarity transformations, with one of the following three matrices:
to be rather involved. Let us assume that Xl = X2 i- O. The characteristic equation of system (7) is then
o while
f(8) =
8
2
-
(022
+ 011)8 + (011 0 n
+{t(011 X2 - x2 8)e-
sh
-
01202d
Do:
,
where g(8) = q(8)e TS , q(8) = X2(011 - 8), and T = -h. Since f(8) has real coefficients it follows from 8 that it is a convex direction, and therefore in this particular case (Xl = 0, X2 i- 0) D"'( is matrix convex direction. The analysis of the subcase X2 = 0, similar to the previous one.
Xl
i-
= [ Od] 0 0 ' D{3 = [d0 0] d ' D"'( = [Xl 0 0X2 ] '
where d,Xl,X2 are real numbers and X1X2 =
o.
Theorem 15. A matrix D is a convex direction for time delay systems in R 3X3 , if and only if D coincides, up to similarity transformations, with one of the following two matrices:
0 is
o
Matrix convex directions in R 3x3
d
o
For 3 x 3 matrices, we have the following possibilities.
where d is a real number.
Case 4: The matrix candidate
Db =
d
0
00] dO,
[ 00 d
Theorem 16. A matrix D is a convex direction for time delay systems in Rnxn, n 2: 4, if and only if D is the identity multiplied by a real scalar d.
dE R,
The analysis of Case 3 when X1X2 < 0 is open due to the complexity of the corresponding quasipolynomial. Further developments of the above characterization are, for instance: the delay dependent matrix convex directions characterization, the analysis of matrix convex directions for two or more delays, and, given the system x(t) = Ax(t)+ {t[Dlx(t - h l ) + D 2x(t - h2)], where A is Hurwitz and h l , h 2 are delays, the characterization of matrices D l , D 2 such that the segment Dlx(t - h l ) + D 2 x(t - h 2 ) is a convex direction.
is a matrix convex direction. This is a direct consequence of lemma 10 and remark 3. Case 5: Consider the candidate
DE =
0d0] 0 0 0 , [000
d E R.
The characteristic equation of system (7) for this matrix is
f(8)
= det(8I 3X3 - A) + +{td( -0218
+ (021033 -
REFERENCES
03l 0 23))e-
sh
H. Kokame, H. Ito and T. Mori (1994). Entire Family of Convex Directions for Real Hurwitz Matrices, Proc. 33rd Conference on Decision and Control, Lake Buena Vista, pp. 3835-3836. L. Xie (1997). Issues on Robustness Analysis of Linear Control Systems with Parametric Uncertainties, Ph. D. Thesis, Department of Electrical and Computer Engineering, The University of Newcastle. R. Bellman and K. L. Cooke (1963). DifferentialDifference Equations, Academic Press, New York. V. L. Kharitonov and A. P. Zhabko (1994). Robust Stability of Time Delay Systems, IEEE Tran8. Automat. Contr., vol 39, pp. 2388-2397. A. Rantzer (1992). Stability Conditions for Polytopes of Polynomials, IEEE Tran8. Automatic Control, vo1.37, pp. 79-89.
.
It can be written as f(8) = p(8) + {tg(8), where g(8) = q(8)eTS • Using lemma 8 it is easy to verify that quasipolynomial f (8) is a convex direction. Therefore DE is a matrix convex direction. Matrix convex directions in Rn X n, n 2: 4 Consider the candidate
where Inxn is the matrix identity and d is a real scalar. From lemma 10 and remark 3 we conclude that matrix Dq, is a matrix convex direction. 6. SUMMARY AND FURTHER DEVELOPMENTS Finally, we are able to state the theorems resulting from the analysis of convex directions for time delay systems performed in this paper.
152
Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001
~
Publications www.elsevier.comllocate/ifac
MATRIX CONVEX DIRECTIONS FOR TIME DELAY SYSTEMS l J. Santos 2
,
S. Mondie and V. Kharitonov
Departamento de Control Automatico CINVESTAV-IPN, Av. IPN 2508, A.P. 14-740, 07300 Mexico, D.F., Mexico, [khar, smondie,jsantos}@ctrl.cinvestav. mx
Abstract: The characterization of convex direction for real Hurwitz matrices is extended to the case of multivariable time delay linear systems. The above is achieved using the results of convex directions for quasipolynomials. Copyright © 2001 IFAC Keywords: time delay systems, convex directions. 1. INTRODUCTION
delay system corresponding to each candidate, which is a quasipolynomial of the retarded type (Bellman and Cooke, 1963), is then analyzed with the help of results on convex directions for quasipolynomials (Kharitonov and Zhabko, 1993).
Many actual systems have the property of aftereffect, i.e., the future states depend not only on the present but also on the past history. Delay is believed to occur in such areas as mechanics, chemistry, biology, medicine. economics, atomic energy, information theory, etc.
New proofs of the basic results on matrix convex directions obtained in Kokame et all. (1994) are also given. They are essentially based on the ideas of the proofs developed for delayed systems and on results about convex directions for polynomials.
For an accurate modelling of such systems, the effect that the delays may have on the behavior of the system response must indeed be taken into account.
2. PRELIMINARIES
Stability is one of the important characteristics of dynamical systems. The concept of convex directions for polynomials and quasipolynomials proves to be a useful tool in stability and robust stability study, (see Rantzer (1992), Kharitonov and Zhabko (1993)). Recently some interesting extention were made to the case of matrices, see Kokame et all. (1994). There, a complete characterization of convex directions for real Hurwitz matrices has been given. Similar results for discrete time systems were reported in Xie (1997).
We introduce first some background on convex directions.
Convex Directions of Polynomials A notion of convex direction has been proposed in Rantzer (1992) and is defined as follows: Definition 1. A polynomial g( s) is a convex direction if for any stable polynomial p(s), the stability of p( s) + g( s) implies that of p(s)
The aim of this paper is to characterize matrix convex directions for time delay systems. Since this characterization must be valid for all values of delay, it must also be valid when the delay is equal to zero, therefore, matrix convex directions of Kokame et all. (1994) are the only matrix candidates for our characterization. It worths to be mentioned the fact that the caracterization does not depend of delay does not imply that we are talking about stability independent of delay. The characteristic function of the time 1 2
+ J.lg(s) ,
J.l E [0,1]
(1)
where deg(p(s)) > deg(g(s)). In the following we will use the following two results. Remark 1. Any first or zero degree polynomial g( s) with real coefficients is a convex direction for real Hurwitz stable polynomials. Remark 2. Any constant polynomial g( s) is a convex direction for complex Hurwitz stable polynomials.
Supported by Conacyt, Mexico 31951-A. Financed by Conacyt 130255.
147
Matrix convex directions
where D ij ERn; xn j , for i, j = 1,2. If D is a matrix convex direction, so are both D 11 and D 22 .
Consider a linear time-invariant system of the form
x(t) = Ax(t)
+ ILDx(t)
The next theorems describe main results from Kokame et all., (1994). We supply them with new proofs, because we will use later on the same arguments for the case of time delay systems.
(2)
x(t) ERn, A,D E R nxn , IL E R. where matrix A is Hurwitz and IL is a scalar parameter.
Theorem 5. (Kokame et all., 1994) For n = 2 a matrix D E R 2x2 is a matrix convex direction if and only if D, up to similarity transformations, has one of the following three forms:
Definition 2. (Kokame et all., 1994) Matrix D E R nxn is a convex direction if for any Hurwitz matrix A E Rnxn , the stability of x(t) = Ax(t)
Dl =
+ Dx(t)
0 0 [Od]
,D2 =
[d0 0] d ,D3 =
[Xl 0 0X2 ] ' (3)
where d, Xl, x2 are real numbers and Xl X2 :::; O. implies that of (2) for alllL E [0,1]. Geometric interpretation: the previous definition means that system (2) has only one open interval of stability on the real axis IL. The size and location of the interval depends on matrices A and D, but the fact that (2) has only one stability interval depends on matrix D. When system (2) is stable for IL = 0 and IL = 1 the stability interval contains segment [0. 1]. We can say even more: if D is a convex direction then for any (not necessarily Hurwitz) real square matrix A there exist no more than one stability interval on the IL-axis. In other words, the definition implies that if matrix D is a convex direction, then for any Hurwitz matrix A the system (2) has only one stability interval and vice versa, if for every Hurwitz matrix A the system (2) has only one stability interval, then matrix D is a convex direction.
Proof. 1. For D l the characteristic polynomial of the system (2) is
p(s) = S2
+ s( -0'22 -
0'11)
+ (0'110'22 -
0'210'12) -lLd0'21,
and can be written as p(s) = Po(s) + ILg(8), where g( s) = -d0'2l' Since g( s) is a constant polynomial it is a convex direction, then it follows from remark 1 that D l is a matrix convex direction. 2. For D 2 , the statement follows directly from lemma 1 and lemma 2.
3. For D 3 , the characteristic polynomial of system (2) has the form
p( 8) = 82 + 8( -0'22 - 0'11) + (0'11 0'22 - 0'120'2I)
+ 0'22Xl - (Xl + X2)8) + 1L2XlX2 = PO(8) + ILgl(8) + 1L 292(8), IL E [0,1]. (4) +1L(0'11X2
Initial Problem: Characterize the set of matrix convex directions for system (2).
It is not possible to apply here directly arguments based on polynomial convex directions because (4) depends quadratically on IL. Therefore we imbed first the family into the bigger one
The characterization has been given in Kokame et all., (1994), where the following auxiliary results were also stated.
Lemma 1. (Kokame et all., 1994) The identity matrix is a matrix convex direction.
where new parameters ILl' 1L2' vary in the triangle ABC (see Figure 1). This new family contains the original one for ILl = IL, 1L2 = 1L 2, IL E [0,1] (see curve TJ on Figure 2). On the other hand (5) is the convex hull of three polynomials: PO(8), PO(8 )+gl (8), Po(8 )+gl (8 )+g2(8), and therefore we may apply the edge theorem to check stability of the new family. Indeed, this theorem says that the stability of the three edges ll, l2, l3 implies that of (5).
Lemma 2. (Kokame et all., 1994) If DE Rnxn is a matrix convex direction. so is kD for an arbitrary real scalar k. Lemma 3. (Kokame et all., 1994) If DE R nxn is a matrix convex direction, so is 8- 1D8, where 8 is a real nonsingular matrix. Lemma 4. (Kokame et all., 1994) Let D be a block triangular matrix
Polynomials, corresponding to extreme points A and B are stable because (4) is stable for IL = 0 and IL = 1. We need first to verify stability of polynomial Po (8) + gl (8), which corresponds to the
D _ [D11 D12] 0 D 22 '
148
third extreme point, C. Comparing coefficients of PO(8) + 91(8) + 92(8) and PO(8) + 91(8) we see that the only difference is in the free coefficient. This coefficient of PO(8) + 91 (8) is bigger on -X1X2 (remember that X1X2 :::; 0) than the corresponding coefficient of Po (8) + 91 (8) + 92 (8). Both polynomials are of degree two, therefore stability of PO(8) + 91(8) + 92(8) implies that of PO(8) + 91(8).
Since 9( 8) is a polynomial of the first degree, then it is a convex direction, therefore D 4 is a matrix convex direction.
Now the stability of the three edges is verified with the help of the concept of convex direction.
Theorem 7. (Kokame et all., 1994) A matrix DE Rn X n, n 2: 4, is a matrix convex direction if a only if D is the identity matrix multiplied a real scalar
2. For matrix D 5 , we can observe that D 5 = dI3x3 is the identity matrix multiplied by a real scalar d. By lemma 1 and lemma 2 one can conclude that matrix D 5 is a matrix convex direction. •
For the edge h we have to verify the stability of PO(8) + /1[91(8) + 92(8)] for all /1 E [0,1]. Since 91 (8) + 92 (8) is of the first degree, then it is a convex direction, and stability of polynomials corresponding to extreme points A, B, imply that of the edge 11 .
d.
Proof. It follows directly from lemma 1 and lemma 2. • 3. MATRlX CONVEX DIRECTIONS FOR TIME DELAY SYSTEMS
For the edge 12 , we have to verify the stability of PO(8) + /1191(8) for all /11 E [0,1]. Again 91(8) is of the first degree, so it is a convex direction, and edge 12 is stable.
Given a linear delay system of the form
x(t)
For the edge 13 , we have to verify the stability of the segment Po (8) + 91 (8) + /1292 (8) for all /12 E [0, 1]. Polynomial 92 (8) is constant and therefore it is a convex direction, so edge 13 is stable. Since all three edges, h, 12 , h, are stables, this implies the stability of the family (5) and, of course, that of (4). Therefore matrix D 3 is a matrix convex direction. •
= Ax(t) + /1Dx(t - h)
(7)
where x(t) E Rn, A, DE R nxn , h 2: 0, /1 E R is the system parameter. The following definition of matrix convex directions for time delay systems, is motivated by our geometric interpretation of matrix convex directions for systems without time delay.
Definition 3. A matrix D E Rnxn is a convex direction for time delay systems if for any Hurwitz matrix A E R nxn , and for any h 2: 0 the system (7) has only one interval of stability on the real axis of /1.
B
I,
---
It is important to emphasize that this definition introduces matrix convex directions which do not depend on the value of the delay. But as it has been mentioned already in the introduction it does not imply that stability should be independent of delay.
Fig. 1
Theorem 6. (Kokame et all., 1994) A matrix D E R 3x3 is a matrix convex direction if and only if D coincides up to similarity transformation with one of the following two matrices:
Problem Statement: Characterize the set of matrix convex directions for (7). The following remark is a direct consequence of definition 3.
Remark 3. If matrix D E Rnxn is a convex direction, so is kD for a real scalar k.
where d is a real number.
Proof. 1. For matrix D 4 , the characteristic polynomial of the system (2) is p(8)
= PO(8) + /19(8),
3X3 where Po () 8 = d et (8I
9(8)
Machinery for the characterization Since we look for a characterization of matrix convex direction independent of the value of h, they must also be matrix convex directions for h = O. Therefore, the only candidates to be matrix convex direction in this sense are· those matrices
(6)
-A) ,
= d( -cx218 + CX21CX33 - CX31CX23).
149
4. BASIC LEMMAS
which are convex directions for systems without delay. This observation reduces drastically the set of possible candidates.
We present now some basic statements which will be used later on for deriving matrix convex directions.
For each candidate, our approach is based on the fact that it is possible to factorize the characteristic function of (7) in order to apply to each factor stability conditions based on the concept of convex direction for quasipolynomials.
Lemma 10. The identity matrix I E Rnxn , is a matrix convex direction. Proof. When D = I E Rnxn system (7) has the following characteristic function
Convex directions for quasipolynomials It follows from the above arguments that the char-
f(s) = det(sI - A - p1e- sh ).
acterization of matrix convex directions for linear systems with delays and for quasipolynomials are intimately related. For this reason. we recall now important results on convex directions of quasipolynomials (Kharitonov and Zhabko, 1993).
There exists a nonsingular matrix S such that
S-lAS =
= {P(s) = sn + a1s n- 1 + ... + an-IS + an} p(s) =
a
=}
Re(s) <
where Ai, i = 1,2, ... , n, are eigenvalues of A and f (s) can be factorized as
f(s) = (s - Al - /-Le-sh)(s - A2 - /-Le- sh ) ... (s - An - /-Le- sh ).
g(s) = q(s)e TS
Each factor is of the form fi (s) = Pi (s) + /-L9i (s), where Pi(S) = s - Ai, and gi(S) = qi(s)e- sh , qi (s) = -1. It follows from lemma 8 and lemma 9 that 9i(S) is a convex direction (real or complex, depending of Ai)' Therefore stability of Pi(S) and Pi(S) + gi(S) implies that of Pi(S) + /-L9i(S) for all /-L E [a,l]. This means that Inxn is a matrix convex direction. •
is a convex direction for 'H if (8) and the inequality 8 arg( q(jw)) < Isin(2 arg( q(jw))) I 8w 2w
(9)
holds for all positive frequencies w where q(jw)
a.
i
Lemma 11. If DE Rnxn is a convex direction, so is S-l DS, where S is a real nonsingular matrix.
Lemma 9. Let
= {P(S) = sn + C1 Sn - 1 + ... + Cn-1 S + Cn } p(s) = a =} Re(s) < a
be the set of Hurwitz stable polynomials with complex coefficients, and q(s) be a complex polynomial of degree less or equal n. Then
g(s) = q(s)e
a,
= Ax(t)
and the inequality
holds for all real w where q(jw)
+ /-LS-1 DSx(t -
h)
(13)
which is stable if and only if the original one is stable. It is clear that the new system has only one stability interval on the axis /-L when it is the case for system (7). In other words S-l DS is a matrix convex direction if and only if D is a matrix convex direction. •
(10)
8arg(q(jw)) < a 8w -
Proof. Since D E Rn X n is a convex direction, then system (7) has one interval of stability on the real axis /-L for all Hurwitz matrix A, so that, applying the similarity transformation to the system we obtain the new one
x(t) = S-l (Ax(t) + /-LDx(t - h)) S
TS
is a convex direction for 'H if T ~
a
a
be the set of Hurwitz stable polynomials with real coefficients, and q( s) be a real polynomial of degree less or equal to n. Then
'H
a
*
Lemma 8. Let 'H
(12)
(11)
Lemma 12. Given a complex number z, then cp( s) = s + A - ze- sh is stable if A > IzI.
i a. 150
Proof. Suppose, by contradiction that cp(s) has a root So such that Re( so) ~ O. Then So + A soh ze= 0 and Iso + AI = Izlle-sohl· Since le-sohl :::; 1, we can write that Iso + AI :::; Izl. By our assumption Re(so) ~ 0, A :::; Iso + AI. Therefore A :::; Izl. This inequality leads indeed to a contradiction. •
Then system (14) has the form x(t)
= [AJ1
-~I] x(t) + f-L [D~l g~~]
x(t - h).(18)
The characteristic equation of (18) is 1(s)
Lemma 13. Let D be a block triangular matrix:
= det(sh -
Ail - f-LDlle- sh )
. det(sI2 + >.12
-
f-LD22e-sh).
(19)
We know that 1(s) is stable for f-L = 0 and for = 0:' but is unstable for some f-La E (0,0:'). However D is a matrix convex direction, therefore the system (18) should be stable for all f-L E [0,0:']. The contradiction proves the lemma. • f-L
where D ij ERn; xn j , for i, j = 1,2. If D is a matrix convex direction, so are both D ll and D 22 . Proof. If D is a matrix convex direction then the system x(t) = Ax(t)
+ f-LDx(t -
h)
5. CHARACTERIZATION OF MATRIX CONVEX DIRECTIONS
(14)
We are now able to verify if the candidates, namely, the matrix convex directions for linear systems, are also matrix convex directions for time delay systems.
has only one stability interval for every Hurwitz stable matrix A. Let us assume by contradiction that D ll is not a matrix convex direction, i.e., there exists matrix Ail such that the system
Matrix convex directions in R 2x2 For 2 x 2 matrices we have to analyze the following three cases:
(15)
Case 1: Consider the following matrix candidate
is stable for f-L = 0 and f-L = 0:' > 0, but it is not stable for some f-Lo E (0,0:'). Now, let us choose a matrix A 22 such that the system
Do=[~~],
The characteristic equation of system (7) for this matrix is 1(s) = p(s) + f-Lg(s) where 1(s) = S2 + s( -0:'22 - O:'ll) + (0:'1l0:'22 - 0:'210:'12), g(s) = d0:'21e-sh = q(s)e TS with q(s) = d0:'21, and T = -h. It is clear that T = -h < 0, and
is stable for all f-L E [0,0:']. Let A 22 = -AI where A > 0, then the characteristic equation of system (16) is given by 1(s) = det(sI
+ >.1 -
f-LDne- sh ).
(17)
oarg(q(jw)) ow
Matrix D 22 can be written as D 22 = T- 1JT, where Zl
J
=
*
= sin(2arg(q(jw)) = 0 2w
So, g(s) satisfies lemma 8 and convex direction.
o
Do
.
is a matrix
Case 2: Consider the matrix candidate
0 [
o
o
o We notice that D{3 = dI 2x2 is the identity matrix multiplied by a real scalar d. Due to lemma 10 and remark 3 we infer that matrix D{3 is a matrix convex direction.
is the canonical form of matrix D 22 . We can apply the last transformation to (17) in order to obtain 1(s)
dER.
=
II~~l (s + A - f-LZke-sh)
Case 3: Consider the matrix candidate
It follows from lemma 12 that system (16) is stable for all f-L E [0,0:'] if A> O:'max1:S;k:S;n2{lzkl} = Aa·
Now let A > Aa. Consider matrix We are able to analyze only the subcase X1X2 = O. The analysis of the general case, X1X2 < 0, seems
A* = [
1<;1
Theorem 14. A matrix D is a convex direction for time delay systems in R 2X2 , if D coincides, up to similarity transformations, with one of the following three matrices:
to be rather involved. Let us assume that Xl = X2 i- O. The characteristic equation of system (7) is then
o while
f(8) =
8
2
-
(022
+ 011)8 + (011 0 n
+{t(011 X2 - x2 8)e-
sh
-
01202d
Do:
,
where g(8) = q(8)e TS , q(8) = X2(011 - 8), and T = -h. Since f(8) has real coefficients it follows from 8 that it is a convex direction, and therefore in this particular case (Xl = 0, X2 i- 0) D"'( is matrix convex direction. The analysis of the subcase X2 = 0, similar to the previous one.
Xl
i-
= [ Od] 0 0 ' D{3 = [d0 0] d ' D"'( = [Xl 0 0X2 ] '
where d,Xl,X2 are real numbers and X1X2 =
o.
Theorem 15. A matrix D is a convex direction for time delay systems in R 3X3 , if and only if D coincides, up to similarity transformations, with one of the following two matrices:
0 is
o
Matrix convex directions in R 3x3
d
o
For 3 x 3 matrices, we have the following possibilities.
where d is a real number.
Case 4: The matrix candidate
Db =
d
0
00] dO,
[ 00 d
Theorem 16. A matrix D is a convex direction for time delay systems in Rnxn, n 2: 4, if and only if D is the identity multiplied by a real scalar d.
dE R,
The analysis of Case 3 when X1X2 < 0 is open due to the complexity of the corresponding quasipolynomial. Further developments of the above characterization are, for instance: the delay dependent matrix convex directions characterization, the analysis of matrix convex directions for two or more delays, and, given the system x(t) = Ax(t)+ {t[Dlx(t - h l ) + D 2x(t - h2)], where A is Hurwitz and h l , h 2 are delays, the characterization of matrices D l , D 2 such that the segment Dlx(t - h l ) + D 2 x(t - h 2 ) is a convex direction.
is a matrix convex direction. This is a direct consequence of lemma 10 and remark 3. Case 5: Consider the candidate
DE =
0d0] 0 0 0 , [000
d E R.
The characteristic equation of system (7) for this matrix is
f(8)
= det(8I 3X3 - A) + +{td( -0218
+ (021033 -
REFERENCES
03l 0 23))e-
sh
H. Kokame, H. Ito and T. Mori (1994). Entire Family of Convex Directions for Real Hurwitz Matrices, Proc. 33rd Conference on Decision and Control, Lake Buena Vista, pp. 3835-3836. L. Xie (1997). Issues on Robustness Analysis of Linear Control Systems with Parametric Uncertainties, Ph. D. Thesis, Department of Electrical and Computer Engineering, The University of Newcastle. R. Bellman and K. L. Cooke (1963). DifferentialDifference Equations, Academic Press, New York. V. L. Kharitonov and A. P. Zhabko (1994). Robust Stability of Time Delay Systems, IEEE Tran8. Automat. Contr., vol 39, pp. 2388-2397. A. Rantzer (1992). Stability Conditions for Polytopes of Polynomials, IEEE Tran8. Automatic Control, vo1.37, pp. 79-89.
.
It can be written as f(8) = p(8) + {tg(8), where g(8) = q(8)eTS • Using lemma 8 it is easy to verify that quasipolynomial f (8) is a convex direction. Therefore DE is a matrix convex direction. Matrix convex directions in Rn X n, n 2: 4 Consider the candidate
where Inxn is the matrix identity and d is a real scalar. From lemma 10 and remark 3 we conclude that matrix Dq, is a matrix convex direction. 6. SUMMARY AND FURTHER DEVELOPMENTS Finally, we are able to state the theorems resulting from the analysis of convex directions for time delay systems performed in this paper.
152