STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS IN THE ...

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INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Number 4, Pages 303–322

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS IN THE SENSE OF LYAPUNOV D.LJ. DEBELJKOVIC1 , I.M. BUZUROVIC2 , AND G.V. SIMEUNOVIC3 Abstract. In this article a comprehensive overview of the stability of linear discrete descriptor systems in the sense of Lyapunov was presented. The complex character of descriptor systems introduces more challenges in the numerical and analytical investigation, which is not a case for the system in its normal form. A discrete Lyapunov equation for discrete implicit systems has been analyzed in details. Stability conditions for both regular and irregular linear discrete descriptor systems were presented. Furthermore, attention was focused on the asymptotic stability of equilibrium points. The bounds on perturbation matrix have been determined in order to preserve the attraction property of origin for both structured and unstructured independent perturbations. The stability robustness for linear discrete descriptor systems was discussed. Key Words. Discrete descriptor systems, Lyapunov stability, Asymptotic stability

1. Introduction In some control systems the character of both dynamic and static states should be mutually analyzed. Descriptor systems (also referred to as degenerate, singular, generalized, differential-algebraic or semi-state systems) are the ones in which the system dynamics was described using algebraic and differential equations. The investigation of the descriptor system concept started in the beginning of the 80s and reached its peak during the 90s. However, the descriptor systems are still of the major research field in control theory. Descriptor systems arise naturally as a linear approximation of the system models in many physical systems such as electrical networks, aircraft dynamics, time delay systems, chemical, thermal and diffusion processes, large-scale systems, interconnected systems, economics, optimization problems, feedback systems, robotics, biology, economy, etc. The applications of descritor systems are various. Using some results on nonnegative matrices, and properties which relate both monotonicity and nonnegative rank factorization, sufficient conditions for the existence of nonnegative solutions for the Leontief price model were presented in [29]. The applications in electrical engineering were presented in [5]. In [26], an effort to characterise the state of the art of modelling, analysing and designing dynamical processes, particularly mechatronic systems by the descriptor system approach has been made. Without any doubts, the modelling of dynamical systems by differential-algebraic equations has many advantages and is superior to the state space modelling,[26]. To investigate descriptor systems, it was necessary to develop various mathematical techniques, such as a Drazin inverse for infinite matrices [7], or developing a set of necessary and sufficient conditions for a set of dynamic equations to contain an Received by the editors March 16, 2012 and, in revised form, March 20, 2012. 303

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embedded state-space representation, [24]. Since that time a considerable progress has been made investigating such systems. A brief historical review of linear singular systems, followed by a survey of results on their solution and properties was presented in [21] and [9, 10].The results for nonlinear singular systems were described in [1, 4]. The algebraic conditions for stability and boundedness of singular systems were derived in terms of the conditions on system matrices. Investigating stability of singular systems, many results on stability in the sense of Lyapunov stability were derived, [1] and [38, 39, 40]. The practical examples of the discrete descriptor systems are not numerous comparing to the linear descriptor systems. However, they appear in some optimal problems and in demography, so their investigation deserves a careful attention. This article presents the results on stability of linear discrete descriptor systems (LDDS) in the sense of Lyapunov. 2. Basic Notations R Real vector space C Complex vector space C Complex plane I Unit matrix F = (fij ) ∈ Rn×n Real matrix F T Transpose of matrix F F > 0 Positive definite matrix F ≥ 0 Positive semi definite matrix ℜ (F ) Range of matrix F N Nilpotent matrix N (F ) Null space (kernel) of matrix F λ (F ) Eigenvalue of matrix F σ() (F ) Singular values of matrix F σ {F } Spectrum of matrix F kF k Euclidean matrix norm kF k= (λmax (AT A))1/2 F D Drazin inverse of matrix F ⇒ Follows 7→ Such that 3. Stability of regular linear discrete descriptor systems Generally, the time invariant discrete descriptor control systems can be written as: (1)

Ex (k + 1) = Ax (k)

(x(k0 ) = x0 )

where is a generalized state space (co-state, semi-state) vector, is a possibly singular matrix, with rank E = r < n. Matrices E and A are of the appropriate dimensions and they are defined over the field of real numbers. In the general case, the initial conditions for an autonomous system and one operating in the forced regime does not need to be identical. The system models of this form have advantages in comparison with models in the normal form, e.g. when E =I. An appropriate discussion can be found in

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS

305

[1] and [14, 17, 18]. However, the complex nature of the descriptor systems causes many difficulties in analytical and numerical analysis which is not a case for the systems in normal form. Due to that fact, the questions of existence, solvability, and uniqueness have to be solved in satisfactory manner. In the discrete case, the concept of smoothness does not play a role but the idea of consistent initial conditions x 0 , that generates solution (x(k) : k ≥ 0) has a physical meaning. The fundamental geometric tool in the characterization of the subspace of consistent initial conditions is the subspace sequence (2)

W0 = Rn , Wj+1 = A−1 (EWj ) , (j ≥ 0)

Here A−1 ( · ) denotes inverse image of ( · ) under the operator A and we will denote by N (F ) and ℜ (F ) the kernel and range of any operator F respectively. Lemma 3.1. The subspace sequence {W0 , W1 , W2 , ...} has the following property (3)

W0 ⊃ W1 ⊃ W2 ⊃ W3 ⊃ · · ·

Moreover (4)

N (A) ⊂ Wj

,

(j ≥ 0)

and there exists an integer k ≥ 0 such that (5)

Wk+1 = Wk

and hence Wk+1 = Wk for j ≥ 1. If k ∗ is the smallest integer with the property that (6)

Wk ∩ N (E) = {0}

(k ≥ k ∗ )

provided that (λE − A) is invertible for some λ ∈ R, [27]. Theorem 3.1. Under conditions in Lemma 1, x 0 is a consistent initial condition for (1) iff x0 ∈ Wk∗ . Moreover x 0 generates a unique solution x (t )∈Wk∗ (k ≥0) that is real-analytic on { k :k ≥0}, [27]. Theorem 3.1 is the geometric counterpart of the algebraic results in [5]. Further explanations may be found in [11, 12, 13]. The survey of latest results for singular (descriptor) systems and a broad bibliography can be found in [1], [5, 6], [21, 22], [12, 13, 14] and in two special issues of the journal Circuits, Systems and Signal Processing (1986, 1989). 3.1. Necessary definitions. Definition 3.1. Linear discrete descriptor system (1) is said to be regular if det (sE − A)is not identically zero,[9]. Remark 3.1. Note that the regularity of matrix pair (E, A) guarantees the existence and uniqueness of solution x (.) for any specified initial condition, and the impulse immunity avoids the impulsive behavior at the initial time for inconsistent initial conditions. For nontrivial case E 6= 0, impulse immunity implies regularity. Definition 3.2. The linear discrete descriptor system (1) is assumed to be nondegenerate (or regular), i.e.det (zE − A) 6= 0, [30].

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Otherwise, it will be called degenerate. If (zE − A)is non-degenerate, we define the spectrum of(zE − A), denoted as σ{E,A} as those isolated values of where det (zE − A) 6= 0 fails. The usual spectrum of (zI − A) will be denoted as σ{A}. Note that due to (possible) singularity of E, σ{E, A} may contain finite and infinite values of z. The finite spectrum will be denoted as σ f {E, A}. Since the system is regular, then there exist orthogonal transformations U and V such that,[31]:     Ef E1 Af A1 T T (7) U EV = , U AV = 0 E∞ 0 A∞

where matrices Ef , A∞ are nonsingular and E ∞ is nilpotent. The pencil (zEf Af ) contains the finite elementary divisors and the pencil (zE ∞ - A∞ ) contains the infinite elementary divisors of (zE - A). Furthermore, the spectrum of (zEf - Af ), σ {Ef , Af } coincides withσ f {E, A}. The structure of Jordan blocks of E ∞ determines the chains at infinity of pencil (zE - A). The discrete-time implicit system admits one-sided solutions when u(k )=0, for every k ≥ 0, i.e. for every admissible initial condition x (k ) there exists a solution that does not depend on states x (k ) for k > 0, if and only if it has trivial chains at infinity. This is known as the anticipation phenomenon. For any regular pair (E, A) there exist two real nonsingular matrices M and N such that ˆ M AN = Aˆ M EN = E,

(8) where (9)

ˆ= E



Ir 0

0 J



,

Aˆ =



A1 0

0 In−r



The pair (E, A) is called Weierstrass form of (E, A). In (9) matrix J ∈ R(n−r)×(n−r) is a nilpotent matrix, and r is the number of finite eigenvalues of (E, A). Since M and N are nonsingular and det (sJ − In−r ) = (−1)n−r , we have o n ˆ Aˆ = σ {Ir , A1 } (10) σ {E, A} = σ E,

i.e.the stability of any regular descriptor system (E, A) can be completely determined analyzing stability of state-space system (Ir , A1 ) of smaller dimension. Note that (E, A) is impulse–free if and only if J =0. If the discrete descriptor system is regular, the transfer function matrix from u(k ) to y(k ) can be uniquely defined as

(11)

Wzu (z) = C (sE − A)

−1

B+D

Any nonzero vector ν 1 satisfing Eν 1 = 0 is called a grade 1 (infinite generalized ) eigenvector of the pair (E, A) and any nonzero vector ν k , (k ≥ 2) satisfing Eν k =Aν k−1 is called a grade k (infinite generalized ) eigenvector of pair (E, A). Definition 3.3. Linear discrete descriptor system (1) is said to be causal if (1) is regular and degree det (sE − A) = rank E,[9]. Definition 3.4. Pair (E, A) is said to be admissible if it is regular, impulse-free and stable, [20]. Lemma 3.2. Linear discrete-time descriptor system (1) is regular, causal and stable if and only if there exists invertible symmetric matrix H ∈ Rn×n such that the following two inequalities hold, [34] (12)

E T HE ≥ 0

(13)

AT HA − E T HE < 0.

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307

Remark 3.2. In the case of E =I, i.e. discrete descriptor system (1) reduces to a state-space one. Lemma 2 coincides with the stability results of discrete-time state-space systems based on Lyapunov inequalities. Furthermore Lemma 2 naturally extends existing results on discrete-time state-space systems to singular ones. Having in mind that Lemma 2 was obtained without assuming the regularity of system (1), the usage of Lemma 2 is convenient due to fact that it was represented in terms of the coefficient matrices of the whole system. Lemma 3.3. If the pair (E, A) is not impulse-free, then it has at least one grade 2 eigenvector ν 2 such that Eν 2 = Aν 1 and Eν 1 = 0, where ν 1 is the corresponding grade 1 eigenvector. ˆ A) ˆ share the same The following Lemma clarifies that the pairs (E, A) and (E, properties of regularity and impulse immunity, with A being certain linear combination of E and A. However, the two sets of finite eigenvalues are related in affine linear manner. ˆ αE + βA, then the following Lemma 3.4. Letα, β∈R with β 6= 0. Let A= properties can be written: ˆ A) ˆ is regular and (1) (E, A)is regular and impulse-free if and only if (E, impulse-free (2) Each  pair ofcorresponding finite eigenvalues of the two systems is related ˆ Aˆ = α + βλi (E, A), [20]. by λi E, 3.2. Stability definitions. Stability plays a central role in the theory of systems and control engineering. There are different types of stability that arise in the dynamic systems, such as Lyapunov stability, finite time stability, practical stability, technical stability, and BIBO stability. The first part of this section deals with the asymptotic stability of equlibrium points of linear discrete descriptor systems. For the linear systems, this the asymptotic stability is equivalent the stability itsef. The Lyapunov direct method (LDM) was investigated rigorously in literature. Here we present different approaches to this problem, including the previous contributions of the authors of this article. Definition 3.5. Linear discrete descriptor system (1) is said to be stable if (1) is regular and all of its finite poles are within Ω(0,1), [9]. Definition 3.6. The system in (1) is asymptotically stable if all finite eigenvalues of pencil (zE - A) are inside the unit circle, and anticipation free if every admissible x (0) in (1) admits one-sided solutions when u (k ) = 0, [30]. Definition 3.7. Linear discrete descriptor system (1) is said to be asymptotically stable iff, for all consistent initial conditions x 0 , it is x (t) →0 as t → +∞, [27]. 3.3. Stability theorems. In the first part the system stability in the sense of Lyapunov, applied to the linear discrete descriptor systems, was presented [27]. The attention was restricted to the case of singular (i.e. noninvertible) E. Furthermore, the deriveation of the geometric conditions for causal solutions x 0 existence for (1) was presented. The solution was expressed in terms of the relative subspace structure of matrices E and A. The results are a geometric counterpart of the algebraic theory in [7]. The required form of x 0 was established in terms of the Drazin inverse where E and A were replaced by commuting operators. This idea is valid with E and A directly, and the commutative property was not assumed. The geometric theory of consistency

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D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

describes a class of positive-definite quadratic forms on the subspace containing all solutions. This fact makes possible the derivation of Lyapunov stability theory for linear discrete descriptor systems in the sense that asymptotic stability is equivalent to the existence of symmetric, positive-definite solutions to a weak form of Lyapunov equation. Throughout the article it is assumed that (λE − A) is invertible at all but finite number of points λ ∈ C and if solution x (k) , (k ≥ 0) of (x (k)):k=0,1,.., exists for a given choice of x 0 , than it is unique [5]. The linear discrete descriptor systems is said to be stable if (1) is regular and all of its finite poles are within Ω(0,1),[9]. A careful analysis shows there is no need for matrix A to be invertible, comparing to the continuous case, [12]. Thus, the matrix A could be noninvertible. Theorem 3.2. Linear discrete descriptor system (1) is asymptotically stable if, and only if there exists a real number λ∗ >0 such that, for all λ in the range 0< |λ| 0,

(15)

∀x (t) ∈ Wk∗ \ {0}

Theorem 3.3. Suppose that matrix A is invertible. Then linear discrete descriptor system (1) is asymptotically stable if, and only if, there exists a self-adjoint, positivedefinite solution H in Rn satisfying AT HA − E T HE = −Q

(16)

where Q is self-adjoint and positive in the sense that, [27] xT (t) Qx (t) >0,

(17)

∀x (t) ∈ Wk∗ \ {0}

Theorem 3.4. Linear discrete descriptor system (1) is asymptotically stable if and only if there exists real number λ∗ >0 such that, for all λ in the range0< |λ| 0 there exist unique positive semidefinite solution H to Lyapunov equation, [36] (41)

AT HA − E T HE = −E T QE

satisfying (42)


rank E T HE = rank E = r

Theorem 3.12. Suppose that EA=AE and there exists z such that (zE - I )=I, then linear discrete descriptor system (1) is regular and asymptotically stable if and only if for any given positive definite matrix H there exist positive semi-solution to the Lyapunov equation,[36]
T
T
T (43) E h AT Ω A E h A − E h+1 Ω E h+1 = − E h+1 H E h+1

satisfying (44)

rank Ω = rank E h+1


T

ΩE h+1 = r

Theorem 3.13. Linear discrete descriptor system (1) is regular, causal and asymptotically stable if and only if the Lyapunov function, (45)

V (Ex (k)) = xT (k) E T H E x (k)

satisfying (46)

∆V (Ex (k)) = V (Ex (k + 1)) − V (Ex (k)) < 0

whenEx (k) 6= 0, where H ≥ 0 is unique and fulfills rank (E T HE ), [37]. Theorem 3.14. Linear discrete descriptor system (1) is regular, causal and asymptotically stable if and only if Lyapunov function (45) satisfies Eq.(46), where matrix H satisfies, [37]

(47) rank E T H E = r, rank E T AT = n.

Theorem 3.15. Linear discrete-time descriptor system (1) or matrix pair (E, A) is admissible if and only if there exists a matrix H = H T ∈ Rn×n satisfying following generalized discrete Lyapunov inequality, [20] (48)

AT H A < E T H E

(49)

E T H E ≥ 0.

Corollary 3.6. Linear discrete-time descriptor system (1) or matrix pair (E, A) is regular, impulse-free, and D (c, r )-stable, i.e. σ {E, A} ⊂ D (c, r), if and only if there exists matrix H = H T ∈ Rn×n such that, [20]

(50) c2 − r2 E T H E + AT H A < c E T H A + AT H E (51)

E T H E ≥ 0.

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D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

Theorem 3.16. Regular linear discrete-time descriptor system (1) is asymptotically stable if and only if for any given Q > 0 there exists solution to Lyapunov equation, [38, 39]
T (52) Eh H = H T Eh ≥ 0

(53)

satisfying (54)

Eh


T

AT HA − E T AT E h+1 = − E h+1

rank E h


T


T

Q E h+1

H = degree det (zE − A) = rank E h .

Theorem 3.17. The zero solution of linear discrete descriptor system (1) is asymptotically stable if and only if there exist inverse matrices P and Q such that     I1 0 A1 0 (55) QEP = , QAP = 0 N 0 I2 and for any λ ∈ σ {A 1 } , λ < 1, where A1 is n1 × n1 matrix, and Ij is ni × ni unit matrix, N is a nilpotent matrix, i = 1, 2, n 2 = n − n 1 , [23].

Theorem 3.18. The zero solution of linear discrete descriptor system (1) is asymptotically stable if and only if there exist inverse matrices  
Q1 (56) P = P1 P2 , Q = Q2

such that: (1) Q 2 EP 2 is a nilpotent matrix; (2) σ{Q1 AP1 } ⊂ Υ, where Υ = {λ, |λ| < 1}, Qi , Pi are ni × n and n × ni matrices, respectively,i=1, 2, n2 = n – n1 , [23].

Theorem 3.19. Linear discrete-time descriptor system (1) is regular, causal and asymptotically stable if and only if there exists an invertible symmetric matrix H fulfilling the following inequalities, [35]: (57) (58)

ET H E ≥ 0 AT H A − E T H E < 0

system (59)

ˆ xˆ (k + 1) = Aˆ ˆx (k) , E

x ˆ (0) = x ˆ0

is called the system restriction equivalent to system (1) if there exist two nonsingular matrices P and Q such that ˆ (60) x (k) = P x ˆ (k) , QEP = E, QAP = Aˆ Theorem 3.20. Assume that A is invertible. Then, the following statements are equivalent: (1) System (1) is stable. (2) For any positive definite matrix Q >0, generalized Lyapunov equation (61)

ˆ Aˆ − Eˆ T H ˆ Eˆ = −Q ˆ AˆT H

has unique symmetric solution H satisfying       ˆ = n 1 , In+ H ˆ + In− H ˆ =n (62) In+ H ˆ ∈ Rn×n and Q ˆ ∈ Rn×n are symmetric matrices, [41]. where H

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS

There exist invertible matrices U, V such that    Iq 0 A11 (63) U EV = , U AV = 0 0 A21

then the following theorem can be written.

A12 A22



313

,

Theorem 3.21. Linear discrete-time descriptor system (1) is regular, causal and asymptotically stable if and only if there exists positive definite matrix   X1 0 −T (64) X =V V −1 > 0 0 X2

fulfill the following conditions (65)

˜ T X EA ˜ −X 0

where, [32]. ˜=V E

(67)



Iq 0

0 0



U

Theorem 3.22. Regular linear discrete-time descriptor system (1) is asymptotically stable if and only if for any given Q > 0 there exists unique positive definite solution H to Lyapunov equation ˆT H E ˆ A − H = −Q (68) AT E where, [32] ˆ=Q E

(69)



Iq1 0

0 0



P

Theorem 3.23. Suppose that Ax e (t )=0 and the norm of the eigenvalue of (zE A) is less than 1, then linear discrete-time descriptor system (1) is globally asymptotically stable if balance state x eq = 0, [34]. Theorem 3.24. Suppose that for any given real symmetric matrix Q, there exists unique, positive definite symmetric solution H of the Riccati equation AT HA − E T H E = −2Q

(70)

then linear discrete descriptor system (1) is globally asymptotically stable if the balance state of system (1) x eq = 0, [34]. 4. Lyapunov stability of irregular linear discrete descriptor systems 4.1. Introduction. In this section we analyzed LDDS described by following equation (71)

Ey (k + 1) = Ay (k) n×n

(k = k0 , k0 + 1, ...) , n

y (k0 ) = y0

with E, A ∈ R , where y (k) ∈ R is the phase vector (i.e. the generalized state space vector) and where matrix E can be singular. The notion of discrete descriptor system was introduced in [24]. The qualitative and stability analysis of linear discrete descriptor systems was investigated by several authors. [8] investigated bounds of response of discrete nonlinear descriptor systems. With the use of LDM some particular classes of nonlinear non-stationary discrete descriptor systems were analyzed in [25] and [3]. Large-scale discrete descriptor systems were also investigated in [25]. [27] derived some necessary and sufficient conditions for asymptotic stability of LDDS using the a geometric approach. [27] investigated

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D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

the geometric description of initial conditions that generate solution sequences {y (k) : k = 0, 1, ...}. The results were expressed directly in terms of matrices E and A. In this approach, there is no need to introduce the algebraic transformations as a prerequisite for stability analysis. The geometric approach provided possibility for a basis-free analysis of the dynamic properties for this class of systems. The initial problem was in the fact that the properties of certain matrices were required to hold only on a linear submanifold of the system’s phase space and not in the whole phase space. Later on this problem was overcome by results of [26]. The results presented in the preceding section were established only for regular LDDS. 4.2. Preliminaries. Now we investigate the irregular systems for the same class the systems. Model (71) of a LDDS can be transformed into a more convenient form for the intended analysis by a suitable transformation. For that purpose, consider LDDS given by (71). It is assumed that matrix E is of form E ={In1 , On2 }, where Ip and Op stand for p × p identity matrix and p × p is null matrix. If matrix E is not in this form, then transformation E → T EQ, where T and Q are suitable nonsingular matrices [10], can convert it to the desired form. Using the transformation, broad class of systems (71) can be brought to form (72)

x1 (k + 1) = A1 x1 (k) + A2 x2 (k) ,

(73)

0 = A3 x1 (k) + A4 x2 (k) ,

where column vector x = (x 1 , x 2 )∈ Rn is composed of subcolumns x 1 (k )∈ Rn1 and x 2 (k )∈ Rn2 , with n = n 1 + n 2 . Let k ∈ K denote the current discrete moment. Here K = {k0 , k0 + 1, ...} is a discrete time interval, where k 0 is an integer. Matrices Ai , (i = 1,...,4) are of appropriate dimensions. The form (72-73) for LDDS is also known as the second equivalent form, [9]. For system (72-73), det E = 0. Since the system considered is time-invariant, it is sufficient to consider its current solution value x (k ) as depending only on current discrete moment k and initial value x 0 at initial moment. k 0 . Hence let us x = (k, x0 ) denote solution x of (72-73), at instant k ∈ K, which emanated from x 0 at k = k 0 . In abbreviated form value of solution x at instant k = k 0 was denoted by x (k ). When matrix pencil {(sE - A):s ∈ C} is regular, i.e. when there exists s ∈ C such that (74)

det (sE − A) 6= 0,

then solutions of (71) exist and are unique for the so-called consistent initial values y 0 . Moreover a closed form of the solutions exists, [5]. If A4 is nonsingular, then regularity conditions (74) for system (72-73) is considerably simplified and reduces to, [19]   −1 det (c In 1 − A1 ) det −A4 − A3 (cIn 1 − A1 ) A2 (75)
n = (−1) 2 det A4 det (cIn 1 − A1 ) + A2 A−1 4 A3 6= 0

It was proved by [27] that, under the conditions of appropriate Lemma, y 0 is a consistent initial condition for (71) if y 0 belongs to certain subspace Wk of consistent initial conditions. Moreover, y 0 generates discrete solution sequence {y (k) : k = 0, 1, ...} (in this case  k 0 = 0)such that y(k ) × Wk for all k = 0,1. . . The ˆE ˆ D , where ED is the so-called Drazin inverse subspace is given by Wk = N 1 − E of matrix E. Note that Wk is independent of the particular choice of s, which can be any complex number such that (sE − A) is nonsingular.

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS

315

Remark 4.1. Following discussion of the consistent initial values was investigated in details in [2]. Let us denote the set of the consistent initial values of (72-73) by M 1 . Consider manifold M ∈ Rn determined by (72) asM = {x ∈ Rn : 0 = A 3 x1 + A 4 x2 }. For systems (72-73) in general case, M 1 ⊆ M. Thus consistent value x 0 =[x 10 x 20 ]T has to satisfy 0 = A3 x10 + A4 x20 or equivalently (76)

x0 ∈ M 1 ⊆ M ≡ N ((A 3 , A 4 )) .

However, if (77)

rank (A 3 , A 4 ) = rank A 4 ,

then M 1 = M = N((A3 , A4 )), and the determination of M 1 requires no additional computation, except to convert (71) into form (72-73). In this case, under the assumption rank A4 = r ≤ n 2 , when x 0 ∈ N((A3 , A4 )) (i.e. when x 0 satisfies (73), (n 1 +n 2 -r ) components of vector x 0 can be chosen arbitrarily in order to achieve consistent initial conditions of system governed by (72-73). More precisely, all components of x 10 and some (the case rank A4 = r ≤ n 2 ,) or none (the case rank A4 = n 2 ,) of the components of x 20 are free for choice. In both these cases, (72-73) is reducible to a lower-order normal form system with x 1 (k ) as the state variable and for the case rank A4 = r < n 2 with some of the components of x 2 (k ) as free. In the later case, the reduced-order model will be of the form x 1 (k +1)= F 1 x 1 (k )+ F 2 x 2 ∗ (k ), where F 1 and F 2 are matrices of appropriate dimensions, and where x 2 ∗ (k ) represents a vector composed of those components of x 2 that can be chosen as free. Thus the existence of solutions is guaranteed for such selected x 0 , and M 1 = M = N((A3 , A4 )). Note that the uniqueness of solutions is not guaranteed when rank A4 = r ≤ n 2 . Since the transformation from (71) to (72-73) is nonsingular, the convergence of solutions y(k ) of (71) toward the origin of the phase space of (71) and that of x (k ) toward the origin of (72-73) are equivalent problems. Let k( · )k denote the Euclidean vector norm or induced matrix norm. The potential (weak) domain of attraction of null solution x (k, 0) = 0 (k ∈ K) of (72-73) is defined by (78) A = {x0 ∈ M 1 : ∃ {x (k) : k = 0, 1, ...} ⇒ x (0) = x0 and lim |x (k)| = 0} k→∞

satisfying (72-73). The term potential or weak was used because solutions of (7273) does not need to be unique. Thus, for every x 0 ∈ A, there also may exist the solutions which do not converge toward the origin. Our goal was to estimate the set A. Lyapunov’s direct method was used to obtain an underestimate AU of the set A (i.e. AU ⊆ A). In the derivation regularity conditions (74) of the matrix pencil {(sE - A):s ∈ C, were not required, [13]. Some other aspects of irregular singular systems were considered in [10]. 4.3. Attraction of the origin and its potential domain. This section introduces a stability result which was employed for the robustness analysis of the attraction of origin. We assume that rank condition (77) holds, which implies M 1 = N((A3 , A4 )) for systems (72-73). Consequently, there exists matrix L which satisfies matrix equation (79)

0 = A 3 + A 4 L,

where 0 is null matrix of the same dimensions as A3 . From (77) and (79), it follows that, if solutions of (72-73) exist, then there exist solutions x (k ) whose components

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D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

satisfy (80)

x2 (k) = Lx1 (k)

(k ∈ K) .

Fulfilling rank condition (77), it follows that, [1] N((L, -In2 ))⊆N((A3 , A4 )). To show this, consider an arbitrary x ∗ (k )∈ N((L, -In2 )), i.e. x 2 ∗ (k )=L x1 ∗ (k ), where L is any matrix that satisfies (79). Then, multiplying (79) from the right by x 1 (k ) and using (80), one gets 0= A3 x 1 ∗ (k ) + A4 Lx 2 ∗ (k ) = A3 x 1 ∗ (k ) + A4 x 2 ∗ (k ), which shows that x ∗ (k )∈ N((L, -In2 )). Hence N((L, -In2 ))⊆N((A3 , A4 )). Consequently those solutions of (72-73) that satisfy (80) also have to satisfy constraints (73). For all solutions of (72-73) for which (80) holds, the following conclusions are important, [19]. (1) The solutions of (72-73) belong to set N((L, -In2 )). (2) If, under rank conditions (77), the existence of a solution x (k ) of (72-73) which satisfies (80) and converges to the origin is proved, then the potential domain of attraction of the origin for (72-73) can be underestimated by (81)

AU = N ([L, −In2 ]) ⊆ A.

Let pd and nd stand for positive definite and negative definite, respectively. For the system (72-73), the Lyapunov function can be selected as (82)

V (x (k)) = xT 1 (k) Hx1 (k) ,

where H is pd symmetric. The total time difference of V (x (k)) is defined by expression (83)

∆V (x (k)) = V (x (k + 1)) − V (x (k)) ,

and its value, calculated along the solutions of (72-73), is then (84)

T ∆V (x (k)) = xT 1 (k + 1) Hx1 (k + 1) − x1 (k) Hx1 (k) = T T = α1 + α2 + α2 + α3 − x1 (k) Hx1 (k)

where (85)

T α1 = xT 1 (k) A1 HA1 x1 (k) T α2 = x2 (k) AT 2 HA1 x1 (k) T α3 = xT 2 (k) A2 HA2 x2 (k)

Employing (80) and (84), one obtains   T ∆V (x (k)) = xT 1 (k) (A1 + A2 L) H (A1 + A2 L) − H x1 (k) (86) = −xT 1 (k) Zx1 (k) , where (87)

Z = − (A1 + A2 )T H (A1 + A2 ) + H

note that Z is a real symmetric matrix. Since H is pd matrix, V (x (k)) is a pd function with respect to x 1 (k ).Thus, if Z is pd, then V (x (k)) would tend to zero as k → ∞, provided that solutions x (k ) exist when k → ∞. This implies k x1 (k) k → 0 when k → ∞. To find pair of matrices H and Z to comply with these requirements can be achieved by means of a discrete Lyapunov matrix equation (88)

AT L HAL − H = −Z,

with (89)

AL = A1 + A2 L,

where Z can be an arbitrary real symmetric pd matrix, and the corresponding H, which is a symmetric pd matrix, can be found as a unique solution of (88), if and

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS

317

only if AL is a discrete stable matrix, i.e. a matrix whose eigenvalues lie in the open unit circle of the complex plane. Note that the order of H is n1 × n1 , so that (88) can be considered as reduced-order discrete Lyapunov matrix equation when compared to the number of the components of x (k ) for system given (71). Theorem 4.1. Let (77) holds. Then underestimate AU of potential domain A of attraction of the null solution LDDS (72-73) is determined by (81), provided that L is any solution of (79) and AL = A1 + A2 L is a discrete stable matrix. Moreover, AU contains more than one element, [18]. 5. Robustness and attraction property Physical systems are often modeled by idealized and simplified models, so information obtained from such models are not always sufficiently accurate. This motivates investigating the robustness properties of the systems with respect to model inaccuracies. The quantitative measures of robustness for multivariable systems in the presence of time-dependent nonlinear perturbations were first investigated in [28]. The bounds were obtained for the perturbation vector such that the nominal system remains stable. The stability robustness of linear discrete-time system in the time domain using Lyapunov’s approach was analyzed in [19]. The bounds on linear time-dependent perturbations that maintain the stability of the asymptotically stable nominal systems were obtained for both structured and unstructured independent perturbations. A general overview of results concerning problems of stability robustness in the area of nonlinear time-dependent descriptor systems was published in[1], while some other robustness results for linear discrete descriptor systems were presented in [9]. In the sequel we present some results on the attraction of origin for LLDS. The results presented here are similar to those given for the continuous linear singular systems, as in [2]. In this section the Lyapunov stability robustness of both regular and irregular LLDS were considered. The bounds on perturbation matrix were determined so that the attraction property of the origin of nominal system was preserved for all perturbation matrices of a specific class. The results presented here rely to some extent on those given in [1] and [15]. This section, also, presents the results on the robustness of attraction of the origin under unstructured and structured perturbation in the model of LLDS. Here, we consider a perturbed version of (71) which is in the form (90)

Ey (k + 1) = Ay (k) + AP y (k) ,

y (k0 ) = y0 ,

where matrix AP represents the perturbations in the model. To analyze the robustness of attraction of the origin of (90), we consider (90) transformation to (91)

x1 (k + 1) = (A1 + B1 ) x1 (k) + (A2 + B2 ) x2 (k) ,

(92)

0 = A3 x1 (k) + A4 x2 (k) + B34 x (k) ,

where x (k ) = [x 1 (k ), x 2 (k )]T does not need to represent the original variables y(k ) of system (90), [13]. To simplify the formulation of results on stability robustness, we introduce the following assumption. Assumption 1 : Matrix B 34 in (92) is null matix. To perform analysis of robustness for system (91-92), we employ Lyapunov function V (x (k )) denoted by (82). Let rank condition (77) holds. Then, taking into

318

D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

account (80) and (85), expression ∆V (x (k)) given by (83) along the solutions of (91-92) is obtained as   T ∆V (x (k)) = xT (k) (Θ + Θ L) H (Θ + Θ L) x1 (k) 1 2 1 2 1 (93) T T − x1 (k) Hx1 (k) = x1 (k) ZP x1 (k) , (94)

T

ZP = (Θ1 + Θ2 L) H (Θ1 + Θ2 L) − H

Θi = Ai + Bi ,

i = 1, 2

note that ZP is a real symmetric matrix. Define the singular values of real matrix X to be the square roots of the eigenvalues of X T X. Let σ M {X } denote the maximal singular value of matrix X , while λM (S ) and λm (S ) stand for maximal eigenvalue and minimal eigenvalue of symmetric matrix S respectively. Note that σ M {S } is also the spectral norm of S. Now it is possible to present the following result which concerns the unstructured perturbations in model (91-92). Theorem 5.1. Let rank condition (77), Assumption 1 and all conditions of Theorem 25 hold. Let Z and H be two real, symmetric, and pd matrices satisfying discrete Lyapunov matrix equation (88), and let AU be the matrix from (88). Then underestimate AU of the potential domain of attraction of system (91-92) was determined by (81) if  1 λM (Z) 2 2 (95) σM {BL } < − σM {AL } + σM {AL } + , λM (H)

where BL = B 1 + B 2 L.

Moreover, AU contains more than one element, [16]. In order to invetigate the structured perturbations in model (91-92), we introduce the following assumption. Assumption 2 : LetBL = B1 + B2 L = [bij : i, j = 1, ..., n1 ] , where L is any solution of (79). Let constraints |bij | ≤ πij hold. Theorem 5.2. Let rank condition (77), Assumption 1 and Assumption 2 and all conditions of Theorem 25 hold. Let Z and H be two real symmetric pd matrices satisfying discrete Lyapunov matrix equation (88), and let AL be the matrix from (88). Let℘ = max 1≤i,j≤n1 ℘ij . Then underestimate AU of the potential domain of attraction of system (91-92) is determined by (81) if  1 ! 1 λM (Z) 2 2 (96) ℘< −σM {AL } + σM {AL } + . n1 λM (H) Moreover AU contains more than one element, [16]. Consider the linear descriptive discrete system represented by its state space model in the following form (97)

Ey (k + 1) = (A) y (k) + fp (y (k)) , y (k0 ) = y0

usually with k 0 =0. Function f p (y) represents the vector of general system perturbations. Introducing a suitable nonsingular linear transformation (98)

T x (k) = y (k) ,

det T 6= 0

the class of LDDS, (97), can be transformed in to the following form (99)

x1 (k + 1) = A1 x1 (k) + A2 x2 (k) + f1p (T x)

(100)

0 = A3 x1 (k) + A4 x2 (k) + f2p (T x)

STABILITY OF LINEAR DISCRETE DESCRIPTOR SYSTEMS

(101)

ET =



I 0

0 0



,

AT =



A1 A3

A2 A4

319



where x (t ) = [x 1 (k )T x 2 (k )T ]T ∈ Rn is the state decomposed vector, with x 1 (k )T ∈ Rn1 and x 1 (k )T ∈ Rn2 with n=n 1 +n 2 . Matrices Ai , i=1,2,. . . ,n, have appropriate dimensions. Moreover, it can be concluded that (102)

det (ET ) det E det T = 0

det T 6= 0. Under the applied transformation, the perturbation vector can be expressed as:     f1 (x (k)) f1p (T x (k)) (103) fp (y (k)) = fp (T x (k)) = f (x (k)) = = f2 (x (k)) f2p (T x (k)) The vector f (x (k)) being decomposed on two subvectors f 1 (x (k )) and f 2 (x (k )). 5.1. Stability robustness of linear descriptor systems with unstructured perturbations. System described by (99-100) has been analyzed under the following assumption. Assumption 3 : The perturbation vector can be adopted in the following form  T (104) f (x (k)) = f1T (x (k)) 0T Quasi-Lyapunov function is adopted as in the form given in (82). Having in mind that H is a symmetric matrix, implying that (105)

T xT 1 (k) A1 H f1 (x (k)) = f1 (x (k)) HA1 x1 (k)

(106)

T T T xT 2 (k) A2 HA1 x1 (k) = x1 (k) A1 HA2 x2 (k)

the forward difference is given with (107)


∆V (x (k)) = xT1 (k) ATL HAL − H x1 (k) + 2xT1 (k) ATL H f1 (x (k)) + f1T (x (k)) H f1 (x (k))

Theorem 5.3. Let the rank condition (77), be satisfied and let L be any solution of matrix equation (79), so that (80) is valid. The perturbation subvector may be adopted in the following manner (108)

f1 (x (k)) = P x1 (k)

with P being the perturbation matrix. Let Assumption 3 is satisfied. Then the system (99-100) possesses a subset of solutions convergent to the origin of phase space if the following condition is satisfied: 1  σmin (Z) 2 , (109) σmax (P) < − σmax (AL ) + (σmax (AL ))2 + σmax (H) AL = A1 + A2 L being a discrete stable matrix (89), with real symmetric positive definite matrix H T = H >0 being the solution of the discrete Lyapunov matrix equation (88) for any symmetric matrix Z T = Z >0, [16]. Remark 5.1. As well as in the case of time-invariant time continuous systems, [28] the constraint given with (109) has its maximum when Z =I, i.e. when σ min (Z )=1, [16]. 5.2. Stability robustness of linear descriptor systems with structured perturbations.

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D.LJ. DEBELJKOVIC, I.M. BUZUROVIC, AND G.V. SIMEUNOVIC

5.2.1. Structural independent perturbations. Let the rank condition (77) be satisfied, matrix L is any solution of the matrix equation (79), so (80) is also fulfilled. The perturbation subvector, having in mind Assumption 3, may be adopted in a convenient form such as (108). Constants π ij and π are defined in the following manner. The elements ℘ij (k ) of the matrix P (k) fulfill (110)

℘ij (k) ≤ | ℘ij |max = πij ,

π = max πij

Theorem 5.4. System (99-100) is stable if the following condition is satisfied  1! 1 σmin (Z) 2 2 (111) π< −σmax (AL ) + (σmax (AL )) + n σmax (H) It can be noted that σ max (P1 )=nπ for any matrix all elements of which are π, and it is obvious that σ max (P) ≤ σ max (P1 ) since π ij ≤ π, so the result follows directly from Theorem 28, [16]. Theorem 5.5. System (99-100) is stable if the following condition is satisfied !1/2  2 σmax (AT |HAL |) σmax (AT |HAL |) σmin (Z) S S π< − σmax (AT |H|A) + + σmax (AT |H|A) (AT |H|A) (112) the matrix A having all positive elements, so that (113)

|P (k)| ≤ π A

Remark 5.2. If matrix P (k) is known or we can estimate the maximum values of all elements in (110), then matrix A may be formed as: A = [uij ] , uij = πij /π,. In this case it can be concluded that 0 ≤ uij ≤ 1. If perturbations ℘ij were not explicitly known, uij can be used simply as a positive real number. If the perturbation ℘ij of aLij elements of the matrix AL is equal to zero, then directly follows uij =0,[16]. 5.2.2. Structural dependent perturbations. In some systems a relatively small number of unknown parameters exist. In such cases, uncertain time-dependent matrix P (k) may be formed in the following way (114)

P (k) = ki i

where i , are constant matrices and ki , are uncertain parameters which can vary independently. Let us define (mn × mn) and (n × n) symmetric matrices


  T T T ··· 1 H 1
S 1 H 2
S 1 H m
S T T   T2 H 1 ··· 2 H2 S 2 Hm S  S  (115) Hpp =   .. .. .. ..  .
 .
. .
T T · · · Tm H m S 1 Hm S 2 Hm S and

Hapi = ATL H i

(116)




S

Theorem 5.6. System (99-100) with the structural perturbation term given by (114) possesses the subset of solutions converge to the origin of phase space if the following condition is satisfied, [16] m m X X 2 (117) |ki | σmax (Hpp ) + 2 |ki | σmax (Hapi )