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Invariance results for linear, time-invariant, discrete-time systems
Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Invariance results for linear, time-invariant, discrete-time systems Anthony N. Michel a,∗ , Ling Hou b a
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
b
Department of Electrical and Computer Engineering, St. Cloud State University, St. Cloud, MN 56301, USA
article
info
Article history: Received 25 June 2010 Accepted 19 November 2010 Keywords: Discrete-time LTI systems Asymptotic stability Invariance stability results Observability
abstract We establish a general invariance result for linear, time-invariant, discrete-time systems. This result, which constitutes necessary and sufficient conditions for asymptotic stability in the large is very general and yields several important variants. The results presented herein are in the same spirit as the authors’ earlier results for linear, time-invariant continuoustime systems. We demonstrate the applicability of the present results by means of a specific example. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction We consider finite-dimensional discrete-time dynamical systems determined by a system of linear, time-invariant difference equations given by x(k + 1) = Ax(k)
(1)
where x ∈ R , A ∈ R and k ∈ N = {0, 1, 2, . . .}. Recall that (1) possesses the trivial solution x(k, x0 , k0 ) = x(k) ≡ 0 for all k ≥ k0 ≥ 0, k0 ∈ N, so x = 0 ∈ Rn is an equilibrium (A is a real n × n matrix and x is a real n vector). It is well known that the equilibrium x = 0 of (1) is asymptotically stable in the large if and only if for every negative definite matrix C ∈ Rn×n (i.e., C = C T < 0) there exists a positive definite matrix P ∈ Rn×n (i.e., P = P T > 0), where n
n×n
AT PA − P = C
(2)
(refer to, e.g., [1, p. 293]). In the present paper we establish necessary and sufficient conditions under which the equilibrium x = 0 of system (1) is asymptotically stable in the large when P > 0 and the matrix C in (2) is negative semi-definite (i.e., C ≤ 0). Our result constitutes an invariance stability result (in the sense of LaSalle [2] and Barbashin and Krasovskii [3]) for linear, discrete-time, time-invariant dynamical systems determined by (1). In the next section of this paper we present essential background material. The main results of this paper are established in Section 3. In Section 4 we address a specific example. 2. Preliminaries The original invariance stability results due to LaSalle [2] and Barbashin and Krasovskii [3] were established for continuous-time autonomous dynamical systems determined by ordinary differential equations. Subsequently, these results were generalized to continuous-time and discrete-time dynamical systems determined by semigroups defined on locally
∗
Corresponding author. E-mail addresses: [email protected] (A.N. Michel), [email protected] (L. Hou).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.11.001
536
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
compact metric spaces (see, e.g., [1,4, Section 4.2]). In the proof of our main result, we will require an invariance stability result for discrete-time autonomous dynamical systems determined by systems of ordinary differential equations given by x(k + 1) = f (x(k)),
(3)
where x ∈ R , f : R → R and k ∈ N. We will assume that x = 0 is an equilibrium for system (3) which requires that f (0) = 0. Before stating the invariance result, we recall the definition of asymptotic stability in the large (in the sense of Lyapunov) of the equilibrium x = 0 of system (3). n
n
n
Definition 1 ([1,5]). The equilibrium x = 0 of system (3) is (a) stable if for every ε > 0 and any k0 ∈ N, there exists a δ = δ(ε) > 0 (δ depends only on ε ) such that for all solutions of (3),
|x(k, x0 , k0 )| < ε for all k ≥ k0 ≥ 0 whenever |x(k0 , x0 , k0 )| = |x0 | < δ; (b) asymptotically stable in the large if (i) it is stable, and (ii) for all x0 ∈ Rn and k0 ∈ N , limk→∞ |x(k, x0 , k0 )| = 0.
In the following result, we employ Lyapunov functions v ∈ C [Rn , R+ ] (i.e. v is a continuous mapping of Rn into R+ = [0, ∞)). We will employ the following characterizations. Definition 2. A function v ∈ C [Rn , R+ ] is said to be positive definite if (i) v(0) = 0, and (ii) v(x) > 0 for all x ̸= 0. A positive definite function v ∈ C [Rn , R+ ] is said to be radially unbounded if v(x) → ∞ as |x| → ∞.
We are now in a position to state the following result for system (3). Theorem 1 ([1, p. 260], [6, p. 9]). Assume that there exists a positive definite and radially unbounded function v ∈ C [Rn , R+ ] such that
v(f (x)) ≤ v(x) for all x ∈ Rn . Suppose that the set {0} ⊂ Rn is the only invariant subset (with respect to system (3)) of the set Z = {x ∈ Rn : v(f (x)) = v(x)}. Then the equilibrium x = 0 of system (3) is asymptotically stable in the large.
We conclude this section by recalling the notion of observability. Definition 3 ([7]). Let U ∈ Rm×n and let W ∈ Rn×n . We say that the pair (U , W ) is observable if and only if the matrix U UW
··· = O UW n−1 has full rank.
3. Main results We are now in a position to state and prove the main result of this paper. Theorem 2. For system (1), let
v(x) = xT Px
(4)
and let
∆(1) v(x(k)) = v(x(k + 1)) − v(x(k))
(5)
or
∆(1) v(x) = xT Cx C = AT PA − P where P is positive definite (P = P T > 0). Assume that C is negative semi-definite (C ≤ 0). Define M = {x ∈ Rn : ∆(1) v(x) = 0}.
(6)
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
537
Let M0 be the largest A-invariant subset of M. Suppose that there exists a matrix P1 ∈ Rn×n such that M01 = M0 where M01 denotes the largest A-invariant subset of the set M1 = {x ∈ Rn : P1 x = 0}. Suppose there exists a matrix P2 ∈ Rn×n such that M02 ⊃ M01 where M02 denotes the largest A-invariant subset of the set M2 = {x ∈ Rn : P2 x = 0} ⊃ M1 . Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (P1 , A − P2 ) is observable. Proof. Consider a trajectory x(k) ∈ M , k ∈ N for system (1). If x(k1 ) ∈ M0 for some k1 ∈ N, then x(k) ∈ M0 for all k ∈ N. Then P1 x(k) ≡ 0, so P2 x(k) ≡ 0 and 0 = P1 x(k + 1) = P1 Ax(k) = P1 (A − P2 )x(k) 0 = P1 (A − P2 )x(k + 1) = P1 (A − P2 )Ax(k) = P1 (A − P2 )2 x(k)
··· 0 = P1 (A − P2 )n−1 x(k). Since, by assumption, (P1 , A − P2 ) is observable, it follows that x(k) ≡ 0, k ∈ N. Therefore, M0 = {0} ⊂ Rn . Therefore, by the invariance theorem, Theorem 1, it follows that the equilibrium x = 0 of system (1) is asymptotically stable in the large. Conversely, assume that the equilibrium x = 0 of system (1) is asymptotically stable in the large and suppose that (P1 , A − P2 ) is not observable. Then there exists x0 ̸= 0 such that P1 (A − P2 )k x0 ≡ 0. Let x(k) = (A − P2 )k x0 . Since by assumption, M01 = M0 , it follows that x(k) ∈ M0 , k ∈ N, and since by assumption M02 ⊃ M01 , we have that x(k + 1) = (A − P2 )x(k) = Ax(k) − 0 = Ax(k), i.e. x(k) solves Eq. (1). Therefore,
∆(1) v(x(k)) = xT (k)Cx(k) ≡ 0, and
v(x(k)) ≡ v(x0 ) > 0. Thus, x(k) cannot tend to the origin as k → ∞. Therefore, the equilibrium x = 0 of (1) is not asymptotically stable. We have arrived at a contradiction. Therefore, (P1 , A − P2 ) must be observable. We may restate Theorem 2 in the following more compact form. Corollary 1. For system (1), assume that (4) and (6) are true with P = P T > 0 and C ≤ 0. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (E , A − H ) is observable, where E = Q1 C and H = Q2 C , where Q1 ∈ Rn×n is a nonsingular matrix and Q2 ∈ Rn×n is an arbitrary matrix. Proof. In Theorem 2, choose P1 = C , Q1 = I and P2 = Q2 C where I denotes the n × n identity matrix.
For Theorem 2, there are many possible choices for the matrices P1 and P2 , including: (1) P1 = C and either P2 = 0 or P2 = ±C . (2) Since C is symmetric and negative semi-definite, there exists a matrix P1 such that P1T P1 = −C . For this choice of P1 we may choose P2 = 0 or P2 = ±P1 or P2 = ±C . In view of these observations, we wish to single out the following special cases. Proposition 1. For system (1), assume that (4) and (6) are true, such that P = P T > 0 and C ≤ 0. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (C , A) is observable. Proof. In Theorem 2, choose P1 = C and P2 = 0. Then M1 = M and M01 = M0 .
Proposition 2. For system (1), assume that there exists a positive definite matrix P ∈ Rn×n and a matrix G ∈ Rn×n such that the matrix AT PA − P = −GT G is negative semi-definite. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (G, A) is observable. Proof. In Theorem 2, choose G = P1 and P2 = 0.
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A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
We conclude by noting that for linear, time-invariant, continuous-time dynamical systems determined by equations of the form x˙ = Ax
(7)
where x ∈ Rn and A ∈ Rn×n , results which are in the same spirit as Theorem 2 and Proposition 1 have been reported in [8]. Also, a result which is in the same spirit as Proposition 2 has been reported for system (7) in [9]. In the next section, we present another variant of Theorem 2 (or Corollary 1). 4. An example We consider a plant described by the difference equation
w(k + 1) = Aw(k)
(8)
where w ∈ R , A ∈ R and k ∈ N. We assume that the equilibrium x = 0 of (8) is not asymptotically stable. Stabilizing system (8) using linear feedback results in a system of the form n×n
n
x(k + 1) = Ax(k) + Bu(x(k))
(9)
u(x(k)) = Fx(k) where B ∈ Rn×m , F ∈ Rm×n and k ∈ N. We rewrite (9) as x(k + 1) = (A + BF )x(k)
(10)
x(k + 1) = (A + AF )x(k) and we assume that BF = AF is of the form AF = Q2 C
(11) n×n
where C is defined as before (see (6)) and Q2 ∈ R is arbitrary. We now address the following problem: Given A (or system (8)), determine conditions on BF = AF such that the eigenvalues of A + BF are in the left half of the complex plane. This is accomplished through the result given below, where the following notation is employed:
v(x) = xT Px ∆(10) v(x) = xT CF x
(12)
CF = (A + AF ) P (A + AF ) − P T
where P = P T ∈ Rn×n . Proposition 3. For system (10), assume that there exists a positive definite matrix P = P T ∈ Rn×n (P > 0) such that the matrix CF in (12) is negative semi-definite (CF ≤ 0). Then the equilibrium x = 0 of system (10) is asymptotically stable in the large if and only if the pair (CF , A) is observable. Proof. We apply Corollary 1 which requires that P > 0, CF ≤ 0 and that (CF , (A + AF ) − H ) is observable, where E = ICF = CF , I ∈ Rn×n is the identity matrix and H = Q2 C , where Q2 is arbitrary. Choosing H = AF , we have that (CF , (A+AF )−AF ) = (CF , A) must be observable. 5. Concluding remarks We established an invariance result for discrete-time, linear, time-invariant dynamical systems determined by difference equations (Theorem 2). This result, which constitutes necessary and sufficient conditions for asymptotic stability in the large, is very general and flexible and yields several variants (Corollary 1 and Propositions 1 and 2). In the proof of the main result, we made use of an invariance result for general discrete-time dynamical systems determined by autonomous difference equations (Theorem 1). We applied the present results to a specific example (to establish conditions under which all eigenvalues of a linear stabilizing feedback system are in the left-hand side of the complex plane (Proposition 3)). Finally, Theorem 2 and Proposition 1 are in the same spirit as some of the authors’ earlier results for systems determined by linear, time-invariant differential equations [8], while Proposition 2 is in the same spirit as a result reported in [9], also applicable to linear time-invariant differential equations. References [1] A.N. Michel, L. Hou, D. Liu, Stability of Dynamical Systems, Birkhauser, Boston, 2008. [2] J.P. LaSalle, Asymptotic Stability Criterion, in: Symposia in Applied Mathematics, vol. 13, American Math. Society, Providence, RI, 1962, pp. 229–307. [3] E.A. Barbashin, N.N. Krasovskii, Stability of motion in the large, Dokl. Akad. Nauk SSSR 86 (1952) 453–456.
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539 [4] [5] [6] [7] [8] [9]
J.K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969) 39–59. W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967. J.P. LaSalle, The Stability and Control of Discrete Processes, in: Applied Mathematical Sciences, vol. 63, Springer-Verlag, New York, 1986. P.J. Antsaklis, A.N. Michel, Linear Systems, Birkhauser, Boston, 2005. R.K. Miller, A.N. Michel, Asymptotic stability of systems: results involving the system topology, SIAM J. Control Optim. 18 (2) (1980) 181–190. D.L. Russell, Mathematics of Finite Dimensional Control Systems, Marcel Dekker, New York, 1979.
539
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Invariance results for linear, time-invariant, discrete-time systems Anthony N. Michel a,∗ , Ling Hou b a
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
b
Department of Electrical and Computer Engineering, St. Cloud State University, St. Cloud, MN 56301, USA
article
info
Article history: Received 25 June 2010 Accepted 19 November 2010 Keywords: Discrete-time LTI systems Asymptotic stability Invariance stability results Observability
abstract We establish a general invariance result for linear, time-invariant, discrete-time systems. This result, which constitutes necessary and sufficient conditions for asymptotic stability in the large is very general and yields several important variants. The results presented herein are in the same spirit as the authors’ earlier results for linear, time-invariant continuoustime systems. We demonstrate the applicability of the present results by means of a specific example. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction We consider finite-dimensional discrete-time dynamical systems determined by a system of linear, time-invariant difference equations given by x(k + 1) = Ax(k)
(1)
where x ∈ R , A ∈ R and k ∈ N = {0, 1, 2, . . .}. Recall that (1) possesses the trivial solution x(k, x0 , k0 ) = x(k) ≡ 0 for all k ≥ k0 ≥ 0, k0 ∈ N, so x = 0 ∈ Rn is an equilibrium (A is a real n × n matrix and x is a real n vector). It is well known that the equilibrium x = 0 of (1) is asymptotically stable in the large if and only if for every negative definite matrix C ∈ Rn×n (i.e., C = C T < 0) there exists a positive definite matrix P ∈ Rn×n (i.e., P = P T > 0), where n
n×n
AT PA − P = C
(2)
(refer to, e.g., [1, p. 293]). In the present paper we establish necessary and sufficient conditions under which the equilibrium x = 0 of system (1) is asymptotically stable in the large when P > 0 and the matrix C in (2) is negative semi-definite (i.e., C ≤ 0). Our result constitutes an invariance stability result (in the sense of LaSalle [2] and Barbashin and Krasovskii [3]) for linear, discrete-time, time-invariant dynamical systems determined by (1). In the next section of this paper we present essential background material. The main results of this paper are established in Section 3. In Section 4 we address a specific example. 2. Preliminaries The original invariance stability results due to LaSalle [2] and Barbashin and Krasovskii [3] were established for continuous-time autonomous dynamical systems determined by ordinary differential equations. Subsequently, these results were generalized to continuous-time and discrete-time dynamical systems determined by semigroups defined on locally
∗
Corresponding author. E-mail addresses: [email protected] (A.N. Michel), [email protected] (L. Hou).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.11.001
536
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
compact metric spaces (see, e.g., [1,4, Section 4.2]). In the proof of our main result, we will require an invariance stability result for discrete-time autonomous dynamical systems determined by systems of ordinary differential equations given by x(k + 1) = f (x(k)),
(3)
where x ∈ R , f : R → R and k ∈ N. We will assume that x = 0 is an equilibrium for system (3) which requires that f (0) = 0. Before stating the invariance result, we recall the definition of asymptotic stability in the large (in the sense of Lyapunov) of the equilibrium x = 0 of system (3). n
n
n
Definition 1 ([1,5]). The equilibrium x = 0 of system (3) is (a) stable if for every ε > 0 and any k0 ∈ N, there exists a δ = δ(ε) > 0 (δ depends only on ε ) such that for all solutions of (3),
|x(k, x0 , k0 )| < ε for all k ≥ k0 ≥ 0 whenever |x(k0 , x0 , k0 )| = |x0 | < δ; (b) asymptotically stable in the large if (i) it is stable, and (ii) for all x0 ∈ Rn and k0 ∈ N , limk→∞ |x(k, x0 , k0 )| = 0.
In the following result, we employ Lyapunov functions v ∈ C [Rn , R+ ] (i.e. v is a continuous mapping of Rn into R+ = [0, ∞)). We will employ the following characterizations. Definition 2. A function v ∈ C [Rn , R+ ] is said to be positive definite if (i) v(0) = 0, and (ii) v(x) > 0 for all x ̸= 0. A positive definite function v ∈ C [Rn , R+ ] is said to be radially unbounded if v(x) → ∞ as |x| → ∞.
We are now in a position to state the following result for system (3). Theorem 1 ([1, p. 260], [6, p. 9]). Assume that there exists a positive definite and radially unbounded function v ∈ C [Rn , R+ ] such that
v(f (x)) ≤ v(x) for all x ∈ Rn . Suppose that the set {0} ⊂ Rn is the only invariant subset (with respect to system (3)) of the set Z = {x ∈ Rn : v(f (x)) = v(x)}. Then the equilibrium x = 0 of system (3) is asymptotically stable in the large.
We conclude this section by recalling the notion of observability. Definition 3 ([7]). Let U ∈ Rm×n and let W ∈ Rn×n . We say that the pair (U , W ) is observable if and only if the matrix U UW
··· = O UW n−1 has full rank.
3. Main results We are now in a position to state and prove the main result of this paper. Theorem 2. For system (1), let
v(x) = xT Px
(4)
and let
∆(1) v(x(k)) = v(x(k + 1)) − v(x(k))
(5)
or
∆(1) v(x) = xT Cx C = AT PA − P where P is positive definite (P = P T > 0). Assume that C is negative semi-definite (C ≤ 0). Define M = {x ∈ Rn : ∆(1) v(x) = 0}.
(6)
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
537
Let M0 be the largest A-invariant subset of M. Suppose that there exists a matrix P1 ∈ Rn×n such that M01 = M0 where M01 denotes the largest A-invariant subset of the set M1 = {x ∈ Rn : P1 x = 0}. Suppose there exists a matrix P2 ∈ Rn×n such that M02 ⊃ M01 where M02 denotes the largest A-invariant subset of the set M2 = {x ∈ Rn : P2 x = 0} ⊃ M1 . Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (P1 , A − P2 ) is observable. Proof. Consider a trajectory x(k) ∈ M , k ∈ N for system (1). If x(k1 ) ∈ M0 for some k1 ∈ N, then x(k) ∈ M0 for all k ∈ N. Then P1 x(k) ≡ 0, so P2 x(k) ≡ 0 and 0 = P1 x(k + 1) = P1 Ax(k) = P1 (A − P2 )x(k) 0 = P1 (A − P2 )x(k + 1) = P1 (A − P2 )Ax(k) = P1 (A − P2 )2 x(k)
··· 0 = P1 (A − P2 )n−1 x(k). Since, by assumption, (P1 , A − P2 ) is observable, it follows that x(k) ≡ 0, k ∈ N. Therefore, M0 = {0} ⊂ Rn . Therefore, by the invariance theorem, Theorem 1, it follows that the equilibrium x = 0 of system (1) is asymptotically stable in the large. Conversely, assume that the equilibrium x = 0 of system (1) is asymptotically stable in the large and suppose that (P1 , A − P2 ) is not observable. Then there exists x0 ̸= 0 such that P1 (A − P2 )k x0 ≡ 0. Let x(k) = (A − P2 )k x0 . Since by assumption, M01 = M0 , it follows that x(k) ∈ M0 , k ∈ N, and since by assumption M02 ⊃ M01 , we have that x(k + 1) = (A − P2 )x(k) = Ax(k) − 0 = Ax(k), i.e. x(k) solves Eq. (1). Therefore,
∆(1) v(x(k)) = xT (k)Cx(k) ≡ 0, and
v(x(k)) ≡ v(x0 ) > 0. Thus, x(k) cannot tend to the origin as k → ∞. Therefore, the equilibrium x = 0 of (1) is not asymptotically stable. We have arrived at a contradiction. Therefore, (P1 , A − P2 ) must be observable. We may restate Theorem 2 in the following more compact form. Corollary 1. For system (1), assume that (4) and (6) are true with P = P T > 0 and C ≤ 0. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (E , A − H ) is observable, where E = Q1 C and H = Q2 C , where Q1 ∈ Rn×n is a nonsingular matrix and Q2 ∈ Rn×n is an arbitrary matrix. Proof. In Theorem 2, choose P1 = C , Q1 = I and P2 = Q2 C where I denotes the n × n identity matrix.
For Theorem 2, there are many possible choices for the matrices P1 and P2 , including: (1) P1 = C and either P2 = 0 or P2 = ±C . (2) Since C is symmetric and negative semi-definite, there exists a matrix P1 such that P1T P1 = −C . For this choice of P1 we may choose P2 = 0 or P2 = ±P1 or P2 = ±C . In view of these observations, we wish to single out the following special cases. Proposition 1. For system (1), assume that (4) and (6) are true, such that P = P T > 0 and C ≤ 0. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (C , A) is observable. Proof. In Theorem 2, choose P1 = C and P2 = 0. Then M1 = M and M01 = M0 .
Proposition 2. For system (1), assume that there exists a positive definite matrix P ∈ Rn×n and a matrix G ∈ Rn×n such that the matrix AT PA − P = −GT G is negative semi-definite. Then the equilibrium x = 0 of system (1) is asymptotically stable in the large if and only if the pair (G, A) is observable. Proof. In Theorem 2, choose G = P1 and P2 = 0.
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A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539
We conclude by noting that for linear, time-invariant, continuous-time dynamical systems determined by equations of the form x˙ = Ax
(7)
where x ∈ Rn and A ∈ Rn×n , results which are in the same spirit as Theorem 2 and Proposition 1 have been reported in [8]. Also, a result which is in the same spirit as Proposition 2 has been reported for system (7) in [9]. In the next section, we present another variant of Theorem 2 (or Corollary 1). 4. An example We consider a plant described by the difference equation
w(k + 1) = Aw(k)
(8)
where w ∈ R , A ∈ R and k ∈ N. We assume that the equilibrium x = 0 of (8) is not asymptotically stable. Stabilizing system (8) using linear feedback results in a system of the form n×n
n
x(k + 1) = Ax(k) + Bu(x(k))
(9)
u(x(k)) = Fx(k) where B ∈ Rn×m , F ∈ Rm×n and k ∈ N. We rewrite (9) as x(k + 1) = (A + BF )x(k)
(10)
x(k + 1) = (A + AF )x(k) and we assume that BF = AF is of the form AF = Q2 C
(11) n×n
where C is defined as before (see (6)) and Q2 ∈ R is arbitrary. We now address the following problem: Given A (or system (8)), determine conditions on BF = AF such that the eigenvalues of A + BF are in the left half of the complex plane. This is accomplished through the result given below, where the following notation is employed:
v(x) = xT Px ∆(10) v(x) = xT CF x
(12)
CF = (A + AF ) P (A + AF ) − P T
where P = P T ∈ Rn×n . Proposition 3. For system (10), assume that there exists a positive definite matrix P = P T ∈ Rn×n (P > 0) such that the matrix CF in (12) is negative semi-definite (CF ≤ 0). Then the equilibrium x = 0 of system (10) is asymptotically stable in the large if and only if the pair (CF , A) is observable. Proof. We apply Corollary 1 which requires that P > 0, CF ≤ 0 and that (CF , (A + AF ) − H ) is observable, where E = ICF = CF , I ∈ Rn×n is the identity matrix and H = Q2 C , where Q2 is arbitrary. Choosing H = AF , we have that (CF , (A+AF )−AF ) = (CF , A) must be observable. 5. Concluding remarks We established an invariance result for discrete-time, linear, time-invariant dynamical systems determined by difference equations (Theorem 2). This result, which constitutes necessary and sufficient conditions for asymptotic stability in the large, is very general and flexible and yields several variants (Corollary 1 and Propositions 1 and 2). In the proof of the main result, we made use of an invariance result for general discrete-time dynamical systems determined by autonomous difference equations (Theorem 1). We applied the present results to a specific example (to establish conditions under which all eigenvalues of a linear stabilizing feedback system are in the left-hand side of the complex plane (Proposition 3)). Finally, Theorem 2 and Proposition 1 are in the same spirit as some of the authors’ earlier results for systems determined by linear, time-invariant differential equations [8], while Proposition 2 is in the same spirit as a result reported in [9], also applicable to linear time-invariant differential equations. References [1] A.N. Michel, L. Hou, D. Liu, Stability of Dynamical Systems, Birkhauser, Boston, 2008. [2] J.P. LaSalle, Asymptotic Stability Criterion, in: Symposia in Applied Mathematics, vol. 13, American Math. Society, Providence, RI, 1962, pp. 229–307. [3] E.A. Barbashin, N.N. Krasovskii, Stability of motion in the large, Dokl. Akad. Nauk SSSR 86 (1952) 453–456.
A.N. Michel, L. Hou / Nonlinear Analysis: Hybrid Systems 5 (2011) 535–539 [4] [5] [6] [7] [8] [9]
J.K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969) 39–59. W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967. J.P. LaSalle, The Stability and Control of Discrete Processes, in: Applied Mathematical Sciences, vol. 63, Springer-Verlag, New York, 1986. P.J. Antsaklis, A.N. Michel, Linear Systems, Birkhauser, Boston, 2005. R.K. Miller, A.N. Michel, Asymptotic stability of systems: results involving the system topology, SIAM J. Control Optim. 18 (2) (1980) 181–190. D.L. Russell, Mathematics of Finite Dimensional Control Systems, Marcel Dekker, New York, 1979.
539