Interval Stability and Stabilization of Linear Stochastic Systems _x = Ax ...

810

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009

are provided to demonstrate the effectiveness of the novel Filtering algorithms. Indeed, this paper has provided a new optimal approach for FI problems of stochastic systems with non-Gaussian variables. The results in this paper can also be generalized to multivariate systems where (x), ! (x) and (x) are joint PDFs under the assumption of vector-value Borel measurable functions for f (xk ; k ; !k ) and h(xk ; k ). See [9] for the filtering problem.

REFERENCES [1] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation With Applications to Tracking and Navigation. London, U.K.: Wiley, 2001. [2] M. Basseville and I. Nikiforov, “Fault isolation for diagnosis: Nuisance rejection and multiple hypothesis testing,” Annu. Rev. Control, vol. 26, pp. 189–202, 2002. [3] R. H. Chen, D. L. Mingori, and J. L. Speyer, “Optimal stochastic fault detection filter,” Automatica, vol. 39, pp. 377–390, 2003. [4] W. Chen and M. Saif, “Fault detection and accommodation in nonlinear time-delay systems,” in Proc. ACC, Denver, CO, Jun. 2003, pp. 4255–4260. [5] X. B. Feng and K. A. Loparo, “Active probing for information in control system with quantized state measurements: A minimum entopy approach,” IEEE Trans. Automat. Control, vol. 42, no. 2, pp. 216–238, Feb. 1997. [6] P. M. Frank and S. X. Ding, “Survey of robust residual generation and evaluation methods in observer-based fault detection systems,” J. Process Control, vol. 7, pp. 403–424, 1997. [7] L. Guo and H. Wang, “Fault detection and diagnosis for general stochastic systems using B-spline expansions and nonlinear filters,” IEEE Trans. Circuits Syst. I, vol. 52, no. 8, pp. 1644–1652, Aug. 2005. [8] L. Guo, H. Wang, and T. Chai, “Fault detection for nonlinear non-Gaussian stochastic systems using entropy optimization principle,” Trans. Inst. Meas. Control, vol. 28, pp. 145–161, 2006. [9] L. Guo and H. Wang, “Minimum entropy filtering for multivariate stochastic systems with non-gaussian noises,” IEEE Trans. Automat. Control, vol. 51, no. 4, pp. 695–700, Apr. 2006. [10] R. Isermann and P. Balle, “Trends in the application of model-based fault detection and diagnosis of technical process,” Control Eng. Practice, vol. 7, pp. 709–719, 1997. [11] P. Kabore and H. Wang, “Design of fault diagnosis filters and faulttolerant control for a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. 46, no. 11, pp. 1805–1810, Nov. 2001. [12] A. Papoulis, Probablity, Random Variables and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [13] A. Renyi, A Diary on Information Theory. New York: Wiley, 1987. [14] A. A. Stoorvogel, H. H. Niemann, and A. Saberi et al., “Optimal fault signal estimation,” Int. J. Robust Nonlin. Control, vol. 12, pp. 697–727, 2002. [15] H. Wang and W. Lin, “Applying observer based FDI techniques to detect faults in dynamic and bounded stochastic distributions,” Int. J. Control, vol. 73, pp. 1424–1436, 2000. [16] H. Yue and H. Wang, “Recent developments in stochastic distribution control: A review,” J. Meas. Control, vol. 36, pp. 209–216, 2003.

Interval Stability and Stabilization of Linear Stochastic Systems Weihai Zhang and Lihua Xie

Abstract—This technical note applies the spectrum technique to deal with the interval stability and stabilization of linear stochastic time-in)-stability of stochastic variant systems. First, the notion of ( )-stability, systems is defined and some relationships among the ( the decay rate of system state response, and the second-order moment )-staLyapunov exponent are revealed. We then address the ( bilization problem and give a sufficient condition via a linear matrix inequality approach. )-stability, Index Terms—Asymptotic mean square stability, ( )-stabilization, second-order moment Lyapunov exponent, ( spectrum assignment.

I. INTRODUCTION This technical note applies the spectrum technique developed in [17] to study the stability and stabilization of linear stochastic time-invariant systems. For a long time, the study on stability analysis and stabilization for stochastic systems is mainly dominated by the Lyapunov function-based approach and the Lyapunov exponents; see monographs [1], [10], [11] and a recent book [14]. In particular, for linear stochastic time-invariant systems, many criteria have been presented for testing the asymptotic mean square stability in terms of linear matrix inequalities (LMIs) and algebraic Riccati equations, which are derived from the application of the Lyapunov function approach. As is well known, the time-invariant deterministic system

x_ = Ax;

x(0) = x0

2 Rn

(1)

is asymptotically stable if and only if (iff) all eigenvalues of A belong to the open left-half complex plane, which is the so-called “eigenvalue criterion” or “spectrum criterion” for asymptotic stability. One of the advantages of spectrum characterization is its direct linkage with the system dynamic behavior. Intuitively speaking, the further left the spectrum set of (A) is located to, the faster the system decay rate is. Secondly, the spectrum characterization leads to an important control design technique named as pole assignment, which plays an important role in linear systems theory, and has been studied extensively by many researchers; see [9], [13]. Likewise, for linear time-invariant systems governed by Itô-type stochastic differential equations, the spectrum technique has been respectively introduced in [7], [8] and [17] recently, and three different spectrum criteria for mean square stabilization [8], [17] and stability [7] were also given. In particular, by means of the spectrum technique, [17] presented a necessary and sufficient condition called “PBH Criterion” for stochastic observability,

Manuscript received September 01, 2008; revised October 02, 2008. Current version published April 08, 2009. This work was supported by the Academic Research Fund, Ministry of Education, Singapore, and the National Natural Science Foundation of China (60874032 and 60828006). Recommended by Associate Editor J. Lygeros. W. Zhang is with the College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266510, China (e-mail: [email protected]). L. Xie is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2008.2009613 0018-9286/$25.00 © 2009 IEEE

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which is shown to be useful in the study of generalized algebraic Riccati equation (GARE), generalized Lyapunov-type equation (GLE) and control; see [6] and [17]. stochastic 2 This technical note is a further development of [17]. First, similar to a closed-loop operator LK in [17], for open-loop stochastic systems, a linear symmetric operator LA;C ;C ;...;C is defined. Based on the spectrum of LA;C ;C ;...;C , the notion of interval stability, named as (0 0 )-stability, is defined. Moreover, some spectrum properties of LA;C ;C ;...;C , HA;C ;C ;...;C [7], and TA;C ;C ;...;C [8] are also discussed. Second, some relationships among the interval stability, the convergence rate of system state response, and the second-order moment Lyapunov exponent [1] are revealed. Third, the problem of interval stabilization called (0 0 )-stabilization is introduced via the spectrum of the closed-loop operator LK , and a sufficient condition for (0 0 )-stabilization is presented by a LMI-based technique [15]. For convenience, we adopt the following standard notations: R(C ) is the set of all real (complex) numbers and Rm n the set of all 2 real matrices. Sn denotes the set of all 2 symmetric matrices whose entries may be complex whereas Hn (Cn ) the set of all 2 ) is the ( Hermitian matrices (all 2 complex matrices). is transpose (conjugate transpose) of a matrix . 0 means a positive definite symmetric matrix and the identity matrix. ( ) denotes the spectrum set of the operator or matrix and C stands for the open left-half complex plane. k k (j j) denotes the norm of a matrix or vector (absolute value of the real number ).

H =H1

;

;

;

2 n n

n n

m n A 0 A3 A AA> I 0 L L A x x

n n

A

811

By solving the following characteristic equation:

X = AX + XA0 + CXC 0 = X; X 6= 0 we obtain (LA;C ) = f03 + i; 03 0 i; 02g. LA;C

By means of the spectrum of LA;C ;C ;...;C , we may give a necessary and sufficient condition for the mean square stability for system (2). Definition 2.1 [10]: System (2) or ( 1 ... k ) is called asymptotically mean square stable if lim k ( )k2 = 0 for any initial state t n (0) 2 R . Proposition 2.1: System (2) is asymptotically mean square stable iff (LA;C ;C ;...;C ) C . Proof: This is a corollary of Theorem 1 in [17] with = 0. Proposition 2.1 can be called a spectrum criterion for stochastic mean square stability for real stochastic systems. It should, however, be noted that the result does not hold for complex stochastic systems. Proposition 2.2: The following complex system:

;

A; C ; ; C E !1 x t

x

where

A~ = BA 0AB ; C~i = DCii 0CDi i ; i = 1; 2; 1 1 1 ; k: 0 0 00 Proof: Let x = x1 + jx2 and x ~ = [x1 ; x2 ] . Then (4) is equiv-

alent to

dx1 (t) = (Ax1 (t) 0 Bx2 (t)) dt k + i=1 (Ci x1 (t) 0 Di x2 (t)) dwi (t) dx2 (t) = (Ax2 (t) + Bx1 (t)) dt k + i=1 (Ci x2 (t) + Di x1 (t)) dwi (t)

i=1

Ci x(t)dwi (t); x(0) = x0

A; C1 ; 1 1 1 ; Ck are real constant matrices of suitable dimensions, x 2 Rn is the system state, wi (t), i = 1; 1 1 1 ; k are independent, standard 1-D Wiener processes defined on the filtered probability space ( ; F ; P ; Ft ). Similar to a linear symmetric operator LK in [17], we LA;C ;C ;...;C

:

;C ;...;C

associated with (2) as fol-

X 2 Sn 7! AX + XA0 +

k i=1

Ci XCi0 2 Sn :

X X (LA;C ;C ;...;C ) = fi : LA;C ;C ;...;C Xi = i Xi ;Xi 6= 0 2 Sn ; i = 1; 2; 1 1 1 ; n(n + 1)=2g. It can be seen that the number of elements in (LA;C ;C ;...;C ) is n(n + 1)=2. The following example tells us how to compute all spectra of LA;C ;C ;...;C . Example 2.1: In (2), take k = 1 A = 0013 10=12 ; C1 = 20 00 ; X = xx1112 xx1222 : X

k

C~ix~(t)dwi (t):

i=1

(6)

Obviously,

tlim !1 E kx(t)k

=0

tlim !1 E kx~(t)k

=0

2

is equivalent to 2

By the operator theory, we know that is a spectrum or eigen2 Sn , value of LA;C ;C ;...;C , if there exists a nonzero = . In this case, is called an such that LA;C ;C ;...;C eigenvector of LA;C ;C ;...;C corresponding to . We denote by (LA;C ;C ;...;C ) the spectral set of LA;C ;C ;...;C , i.e.,

X

dx~(t) = A~x~(t)dt +

(2)

where

introduce a linear operator LA;C lows:

(5)

or

Consider the following Itô-type stochastic differential system:

dx(t) = Ax(t)dt +

K

dx(t) = (A + jB)x(t)dt + ki=1 (Ci + jDi )x(t)dwi (t) (4) j 2 = 01; x(0) = x0 2 Rn 0 is asymptotically mean square stable iff (LA; ~ C ~ ;C ~ ;111;C ~ ) C ,

A. Spectral Criteria for Stochastic Stability

k

0

II. INTERVAL STABILITY In this section, we shall apply the spectrum technique to study stochastic stability, a new concept called (0 0 )-stability is introduced and investigated. Some other existing spectral characterizations for stochastic stability are also discussed and compared.

(3)

0

111

while the latter holds iff (LA; ~ C~ ;C~ ;...;C~ ) C by Proposition 2.1. Similar to LA;C ;C ; ;C , the operators HA;C ;C ;...;C and TA;C ;C ;...;C were, respectively, introduced in [7] and [8]

HA;C ;C ;...;C TA;C ;C ;...;C

:

:

k X 2 Hn 7! AX + XA0 + Ci XCi0 2 Hn ; i=1 k

X 2 Cn 7! AX + XA0 + Ci XCi0 2 Cn : i=1

111

Note that the only difference among LA;C ;C ; ;C , HA;C ;C ;...;C and TA;C ;C ;...;C is that they are defined in different domains. However, they are equivalent in characterizing asymptotic mean square stability of system (2).


812


Lemma 2.1: System (2) is asymptotically mean square stable iff 0 [7] or ( A;C ;C ;...;C ) 0 [8]. ( A;C ;C ;...;C ) = i : Theorem 2.1: Let ( A;C ;C ;...;C )

H

C

AX2 + X2 A0 +

C

T

f H HA;C ;C ; ;C Ui = i Ui ;Ui 6= 0 2 Hn ; i = 1; 2; 1 1 1 ; ng and (TA;C ;C ; ;C ) = fi : TA;C ;C ; ;C Vi = i Vi ;Vi 6= 0 2 Cn ; i = 1; 2; 1 1 1 ; n g. Then, we have := , where 0 i) maxi Re(i ) = maxi Re(i ) = 0 maxi Re(i ); ii) (TA;C ;C ; ;C ) = (A I + I A + ki Ci Ci ), where A B stands for the Kronecker product of A and B . Proof: By contradiction and without loss of generality, we 2 R assume that 0 < maxi Re(i ). Then take any such that 0 < 0 < maxi Re(i ). It is easily tested that (LA = I;C ;C ; ;C )= fi + : i 2 (LA;C ;C ; ;C );i = 0 1; 2; 1 1 1 ; n(n + 1)=2g C . So by Proposition 2.1, the following ...

...

0

0

+(

2)

stochastic system

0

dx(t) = A +

2

...

0

...

I x(t)dt +

k i

Ci x(t)dwi (t); x(0) = x0 (7)

=1

is asymptotically mean square stable. In addition, we also know that

HA

+

I;C ;C ; ;C

=

...

fi +

0

C 0

:

i 2 (HA;C ;C ;...;C

g

)

does not belong to 0 , which, in view of Lemma 2.1, results in a contradiction. Hence, one must have = maxi Re(i ). Similarly, it = maxi Re(i ). can be shown that To show (ii), we only need to note that

TA;C ;C ;

0

;C Vi = AVi + Vi A0 +

...

k

j

Cj Vi Cj0

=

i Vi

1 X1 0 2 X2 = 1 X1 + 2 X2 ) 2 X2 = 0; 1 2 + 2 X1 = 0 1 X2 0 2 X1 ) 2 X1 = 0:

is equivalent to

A I +I A+

k j

=1

in terms of Kronecker matrix product theory, where V~i is a column vector formed by Vi ; see [3]. Remark 2.1: It should be noted that we generally do not have mini Re(i ) = mini Re(i ) = mini Re(i ). For instance, in Example 2.1, it is easy to compute ( A;C ) = 3 + i; 3 i; 2; 4 , ( A;C ) = 2; 4 . Therefore, mini Re(i ) = 3, mini Re(i ) = mini Re(i ) = 4. Obviously, ( A;C ;C ;...;C ), ( A;C ;C ;...;C ) ( A;C ;C ;...;C ) ( A;C ;C ;...;C ), but there is no inclusive relationship between ( A;C ;C ;...;C ) and ( A;C ;C ;...;C ). I+I A+ Remark 2.2: It is conjectured by [12] that (A k Cj Cj ) must have repeated eigenvalues, but Example 2.1 and j =1 Remark 2.1 show that the assertion is wrong. Lemma 2.1 still holds for complex stochastic systems and in this case in the definitions of A;C ;C ;...;C and A;C ;C ;...;C , A0 ; Ci0 are respectively replaced ; k. by A3 and Ci3 , i = 1; 2; for the real Furthermore, we can show ( A;C ;C ;...;C ) system (2). . Proposition 2.3: For the real system (2), ( A;C ;C ;...;C ) Proof: Assume = 1 + i2 is any eigenvalue of A;C ;C ;111;C associated with the eigenvector X = X1 + iX2 , , X1 where 1 ; 2 n and X2 0n with n and 0n being the sets of real symmetric and anti-symmetric matrices, respectively. Moreover, X1 and X2 cannot be zero simultaneously. From A;C ;C ;111;C X = X , it follows that:

f0 L T L

T f0 0 g 0 H

0 0 0 0g H 0 T H

T

H

111

R

H

R

H

H

2R

2R

H

AX1 + X1 A0 +

k

i

=1

2R

R

Ci X1 Ci0 = 1 X1 0 2 X2 ;

R

(8)

(11)

Since X1 and X2 cannot be simultaneously zero, (10) and (11) imply that 2 = 0. The proof is complete. B. An Interval Stability Theorem By means of the spectrum technique, we define a class of internal stability called ( ; )-stability. Definition 2.2: System (2) is said to be ( ; )-stable with 0 0 < , if ( A;C ;C ;...;C ) 0 := : < Re() < . Definition 2.3 [10], [14]: For system (2), the second-order := moment Lyapunov exponent Le2 is defined as Le2 2 limt!1 sup(1=t) log(E x(t) ). The following theorem shows that the ( ; )-stability is closely related to the decay rate of the system state response and the Lyapunov exponent Le2 [1]. Theorem 2.2: Assume that system (2) is ( ; )-stable for 0 < . Then, the following hold: i) System (2) is exponentially mean square stable and converges faster than O(e0(+")t ) but slower than O(e(0 +")t ) for any sufficiently small " > 0. That is, E x(t) 2 C2 x0 2 e(0 +")t for some C1 x0 2 e0(+")t , E x(t) 2 constants C1 ; C2 > 0. ii) < Le2 < . iii) Furthermore, Le2 = maxi Re(i ). Proof: Without loss of generality, we assume that maxi Re(i ) = Re(1 ) < , mini Re(i ) = Re(n(n+1)=2 ) > , where i ( A;C ;C ;...;C ), i = 1; 2; ; n(n + 1)=2. Let X (t) = E [x(t)x0 (t)], then by Itô’s formula, we have

0 0

0 0 f 0

C

L

k

k

0g

0 0

0 0

k k

k

k

k k

k

k

0

0

Cj Cj V~i = i V~i

(10)

0 X

0

=1

(9)

By symmetry and anti-symmetry properties, (8) and (9) give

...

=1

Ci X2 Ci0 = 1 X2 + 2 X1 :

i

=1

2

...

k

0

2 L

111

X_ (t) = LA;C ;C ;111;C X (t) 0 0 = AX (t) + X (t)A + CX (t)C ; 0 X (0) = x(0)x (0):

(12)

2

Equation (12) is an n n symmetric matrix equation, and since X is symmetric, (12) is in fact equivalent to an n(n + 1)=2-order vector equation as seen below. As in [17], for any Y = (Yij )n2n n , if we let Y~ = ~(Y ) = 0 (Y11 ; Y12 ; . . . ; Y1n ; Y22 ; Y23 ; . . . ; Y2n ; . . . ;Yn01;n01 ; Yn01;n ; Ynn ) , then there exists a unique matrix n(n+1)=22n(n+1)=2 , such that (12) is LA;C ;C ;111;C equivalent to

2S

L

2 R

X~_ (t) = L~ (LA;C ;C ;111;C X (t)) = LA;C ;C ;111;C X~ (t); (13) X~ (0) = X~ 0 :

L

Moreover, ( A;C ;C ;...;C ) = (LA;C ;C ;...;C ). According to the ordinary differential equation theory, if 1 and n(n+1)=2 are respectively eigenvalues of A;C ;C ;...;C with multiplicity r and r , then

L

X~ (t)

C

0

r

X~ (0)

i r

ti01 eRe( )t ;

=1

X~ (t)

C

0

X~ (0)

i

ti01 eRe(

)t

(14)

=1

for some scalars C0 ; C 0 > 0. Next, we show that the following relationships hold:

p

E kxk2 nkX~ k;


(15)


n + 1 E kxk2 :

X

k ~k

(16)

2

E kxk

X

n

=

i=1

2

k ~k = =

E jxi j

1ij n

n

E xx

n 2

n

E jxij2 2 ;

i=1

2

+

LA;C

1i 0:

0 +

2

i01 1 sup t2[0;1) i=1 = 1 k 0 k2 0(+")t

e[Re( )++"]t e0(+")t

t

C x e ; E kx(t)k2 C 0 kx0 k2 n(n2+ 1) r

1

ti01 e[Re(

i=1

)+ 0"]t

e(0 +")t

C 0 kx0 k2 n(n2+ 1) e(0 +")t = C2 kx0 k2 e(0 +")t t

(20)

Example 2.2: For the system considered in Example 2.1, because (LA;C ) = f03 + 03 0 02g, it yields m = 03, 2 2 M = 02, and e = limt!1 sup(1 ) log( k ( )k )= 2 limt!1 inf(1 ) log( k ( )k ) = 02. By Corollary 2.1, the fastest exponential convergence rate is ( 03t ), and the slowest exponential convergence rate is ( 02t ). Remark 2.3: By Remark 2.1, we generally have mini ( i ) mini ( i ), so if we use the spectrum of TA;C ;C ;...;C to define (0 0 )-stability, we will obtain a conservative estimate of state decay rate. Remark 2.4: Theorem 2.2(iii) reveals a relationship between the second-order moment Lyapunov exponent e2 which has been well studied for linear stochastic systems [1] and the spectrum technique studied in the present technical note. Remark 2.5: The problem of matrix root-clustering in subregions of the complex plane is an important research topic; see [5], [9], [13] for detail. We believe that a similar problem for the spectrum of LA;C ;111;C deserves further study.

=t

L E xt

i;

i;

=t

E xt

Oe

Re

Re ;

L

where

r i01 sup t2[0;1) i=1

; ;

Oe

C1 = C0 n(n2+ 1)

1 2 Le2 = tlim !1 sup t log E kx(t)k 1 = lim inf log E kx(t)k2 t!1 t ~ (t) log X = lim = M : t!1 t

n(n + 1) r

L

; ;

Substituting (15) and (16) into (14), we have 2

:

Re < Re < ; ; 0. (ii) of subject to (32). Next, we present an interesting result for the following 1-D stochastic system: k

We observe that (0 ; 0)-stabilization of system (21) is in fact a special spectral assignment problem, which requires us to find a K such that (LK ) f : 0 < Re < 0; ; 0g. A natural extension was presented in [17]: For any given 1 ; 2 ; 1 1 1 ; n(n+1)=2 2 C , can we find a feedback gain matrix K , such that (LK ) = f1 ; 2 ; 1 1 1 ; n(n+1)=2 g? If such K exists, we say that the spectrum of LK associated with system (21) can be assigned arbitrarily. Proposition 3.1 reveals that the spectrum of LK of system (33) can be assigned arbitrarily iff d = 0, b 6= 0. In this case, for any 2 R, (LK ) = with K = ( 0 ( ki=1 ci )2 0 2a)=2b.

REFERENCES

2 Rm2n ), where 111

dx = (ax + bu)dt +

815

ci2 + 2a + > 0

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