A constructive test for exponential stability of linear time-varying discrete-time systems

We complete the stability results of the paper Bourlès et al. (SIAM J Control Optim 53:2725–2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators.


Introduction
We complete the stability results of [4] and use the notions, in particular the linear timevarying (LTV) discrete-time behaviors and the exponential stability (e.s.), of this paper, but extend the theory from the coefficient field C(z) of rational functions to the larger field C << z >> of locally convergent Laurent series with at most a pole at 0. In the main Theorem 1.1 together with Corollary 3.11 we characterize e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. Due to, for instance, [6,14] This test can and has to be applied in connection with the construction of stabilizing compensators; cf. [12] in the case of differential LTV systems. The proof of Theorem 1.1 on state space behaviors is contained in Sect. 2. Section 3 presents the theory of discrete-time LTV behaviors for the coefficient field C << z >>, but we expose the necessary modifications of [4] only. In particular, behaviors and their morphisms, autonomous behaviors and their e.s. are recalled from [4] and adapted. Essential properties of these are stated in Corollary 3.10 on modulebehavior duality, in Corollary 3.12 on closure properties of the class of e.s. autonomous behaviors and in Corollary 3.11. The latter enables the application of Theorem 1.1 to arbitrary autonomous behaviors instead of state space behaviors only. In Sect. 4 we shortly extend the results to the still larger field of locally convergent Puiseux series (cf. [16], [5, §3.1]). The latter field seems to be the largest coefficient field for which a reasonable stability theory for general LTV systems can be developed. We refer to the books [15, pp. 423-461] and [8, pp. 193-368] for comprehensive surveys of exponential stability of state space systems. Part II of the book [3] contains a detailed theory of general LTV behaviors and their stability that was modified in the papers [4,5]. We also refer to the recent papers [1,2,7,10,13]. Theorem 1.1 requires some preliminary explanations: let C < z > denote the local principal ideal domain of locally convergent power series in the variable z and K =: C << z >> its quotient field. A formal power series a = ∞ i=0 a i z i is called locally convergent if The power series a is a unit (invertible) if and only if a(0) = a 0 = 0, and z is the unique prime of C < z >, up to units. Each nonzero a ∈ C < z > has a unique representation This implies that every nonzero a ∈ K has the analogous unique representation The element a is a power series if and only if k = ord(a) ≥ 0. The representation a = ∞ j=k b j−k z j shows that a is a locally convergent Laurent series with at most a pole at zero, and that indeed K = C << z >> consists of all these Laurent series. If k < 0 the function a(z) is a holomorphic function in the pointed open disc D(ρ(a)) \ {0}. In particular, the function The sequences a(t −1 ), t ∈ N, t > σ(a), are the time-varying coefficients of the difference equations of the present paper that in [4] were used for a ∈ C(z) = C(z −1 ). The coefficient functions a(t −1 ), a ∈ K, are of at most polynomial growth on each closed interval [σ 1 , ∞), σ 1 > σ(a), i.e., there are c > 0 and m ∈ N such that |a(t −1 )| ≤ ct m for t ≥ σ 1 [5, (29)]. For nonzero a there is σ 2 > σ(a) such that a(t −1 ) = 0 for t ≥ σ 2 . These properties of the coefficient functions are essential for the module-behavior duality and for the definition and properties of exponential stability of autonomous behaviors. Let, more generally, A = (A μν ) 1≤μ,ν≤n ∈ K n×n be any square matrix and define Then the function t → A(t −1 ) is a smooth matrix function on the open real interval (σ (A), ∞). For t 0 ∈ N we consider the signal space of complex sequences or discrete signals starting at the initial time t 0 . For n ∈ N we use the column spaces C n and W (t 0 ) n and identify If t 0 > σ(A), t 0 ∈ N, the matrix A gives rise to the state space equation resp. the behavior or solution space The transition matrix [15, p. 392] associated to (9) is There is the obvious isomorphism For ξ = (ξ 1 , . . . , ξ n ) ∈ C n and M ∈ C n×n we use the maximum norms ξ := max i |ξ i | and M := max Mξ ; ξ ∈ C n , ξ = 1 .
The system, i.e., the matrix A and the equation and behavior from (9) Here a sequence ϕ ∈ C t 0 +N is called of at most polynomial growth (p.g.) if |ϕ(t)| ≤ ct m , t ≥ t 0 , for some c > 0 and m ∈ N. In [4] we used the notation ρ := e −α < 1 and ρ t−t 1 = e −α(t−t 1 ) . In particular, e.s. implies asymptotic stability, i.e., lim t→∞ A (t, t 1 ) = 0 for t 1 ≥ t 0 . The system is called uniformly e.s. (u.e.s.) [15,Def. 22.5] if ϕ in (13) can be chosen constant. Notice that (13) is a property of the behavior family (K(A, t 1 )) t 1 ≥t 0 and of the trajectories for sufficiently large t 0 . This is appropriate for stability questions where the behavior of x(t) for t → ∞ is investigated. In [15] the author considers LTV state space equations All stability results in [15, require additional properties of F. Our choice in [4] and in the present paper is A nonzero A admits a unique representation where the exponent −k is chosen for notational convenience in Theorem 1.1.

Theorem 1.1 Consider a matrix A(z)
∈ C << z >> n×n and the state space system defined by the data from (9) to (10).
i.e., the system is unstable. (iii) Assume k > 0 and B 0 nilpotent in (15) and If kn > then (16) holds and the system is not e.s. The significance of Theorem 1.1 for arbitrary autonomous behaviors instead of state space behaviors follows from Corollary 3.11. Example 1.2 That B 0 in item (ii) of Theorem 1.1 is not nilpotent cannot be omitted. To see this consider the nilpotent matrix B 0 := 0 1 0 0 ∈ C 2×2 . For 0 = λ ∈ C and ρ := |λ| = e −α > 0, α ∈ R, define For ρ ≥ 1 the sequence A (t, t 0 ) does not converge to zero and therefore the system is not e.s. The sum t−1 i=t 0 i grows polynomially. If ρ = e −α < 1 or α > 0 the transition matrix A (t, t 0 ) decreases exponentially with a decay factor e −α (t−t 0 ) for every α with 0 < α < α. So the system defined by A is e.s. for |λ| = e −α < 1.

Remark 1.3
The ring C < z > is defined by analytic conditions on the coefficients of the power series that imply its good algebraic properties. These are inherited by K. Also the e.s. of an autonomous behavior is defined by analytic conditions on its trajectories [4,Def. 1.7]. In contrast, the construction of the category of behaviors and the derivation of the module-behavior duality proceed algebraically. This explains the necessity for both analytic and algebraic arguments in [4] and the present paper.

The proof of Theorem 1.1 (i) Since A is a power series we can write
The function C(t −1 ) is bounded for t ≥ t 0 and therefore t −1 C(t −1 ) is a disturbance term that is small for large t.
(a) If |λ| < 1 for all eigenvalues of A 0 the system is uniformly exponentially stable (u.e.s.). According to [15,Thm. 24.7], [4,Cor. 3.17] the equation is also u.e.s. and therefore e.s. (b) Assume that A 0 has an eigenvalue λ with |λ| > 1. According to [4,Thm. 3.21] the system is exponentially unstable and, in particular, This implies that the system is not e.s. (c) Assume that A 0 has an eigenvalue λ with |λ| = 1 and that the system Now consider the modified system The matrix ρ −1 A 0 has the eigenvalue ρ −1 λ with |ρ −1 λ| = ρ −1 = e α > 1. From (b) we infer The first and the last line of (23) are in contradiction and therefore This completes the proof of part (i) of the theorem. (ii) Recall that a square complex matrix is nilpotent if and only if 0 is its only eigenvalue. Under the conditions of (ii) the matrix B 0 has a nonzero eigenvalue λ. Choose ρ > |λ| −1 so that |ρλ| > 1 for the eigenvalue ρλ of the matrix ρ B 0 .
According to (i)(b) ρ B is exponentially unstable and indeed where the last implication follows as in (24) due to kn − > 0. If the sequence was bounded so would be the sequence of determinants | det( A (t, t 1 ))|.

Laurent coefficients
We explain the basic notions of a variant of the theory from [4] since we use the difference field K = C << z >> instead of the field C(z) ⊂ K of rational functions in [4]. (7). The space C t 0 +N = W (t 0 ) is also a difference C-algebra with the componentwise multiplication and the shift algebra homomorphism It gives rise to the noncommutative skew-polynomial algebra of difference operators [9, Section 1.2.3], [4, (20)] The space Of course, almost all, i.e., up to finitely many, f j are zero so that the sums j are actually finite, here and in later occurrences. As usual the action is extended to the The behavior or solution space defined by R is For σ > 0 the algebra C ∞ (σ, ∞) is also a difference algebra with the algebra endomorphism It gives rise to the skew-polynomial algebra For t 0 > σ the map is a difference algebra homomorphism since d s = s s and therefore its extension (denoted with the same letter) is an algebra homomorphism. The algebras C ∞ (σ, ∞) and C t 0 +N are not noetherian and have many zero-divisors and therefore very little is known about the rings of difference operators from (27) to (32) and their modules.
Therefore we restrict the time-varying coefficients of discrete difference equations to sequences where t is chosen sufficiently large as explained in Lemma 3.1, for instance t > σ (a) = ρ(a) −1 . In the latter case we have (t The corresponding skew-polynomial algebra of difference operators is The In the sequel we make use of the equation (a)(t −1 ) = a((t + 1) −1 ) for t > σ(a).
Since ρ( (a)) may be smaller than ρ(a), cf. Example 3.3, the left side of this equation is not defined a priori. To solve this problem we introduce difference subalgebras K(ρ) ⊂ K, ρ > 0, such that the map ρ : is an algebra monomorphism . The function f 1 +/ * f 2 is holomorphic on (U (a 1 ) U (a 2 ))\ {0} and hence on U 3 \ {0} and obviously coincides with a 1 + / * a 2 near zero, hence a 1 + / * a 2 ∈ K(ρ). For t > ρ −1 this implies Hence K(ρ) is a subalgebra of K and ρ is an algebra homomorphism.

Example 3.3 Let
For the germ a ∈ C < z > of f at 0 we get

Remark 3.4
The preceding example can be generalized. Assume that a ∈ C << z >> can be extended to f ∈ O(C \ S) where S is a discrete closed subset of C and S is the set of singularities of f . So a is the germ of f at zero, U := C \ S is connected and both S and f are uniquely determined by a. Therefore a(z) := f (z), z ∈ U, is well-defined. In particular, a(t −1 ) is well-defined for t ∈ N and t −1 ∈ (0, ρ) ∩ U . The inclusion (0, ρ) ⊂ U is not required in this case, but was essential for the definition and properties of K(ρ).

The Lemmas 3.1 and 3.2 imply that
is a subalgebra of A and that the extended map, denoted by the same letter, is an algebra monomorphism. Since any f = j f j s j belongs to K(ρ( f )), ρ( f ) = min ρ( f j ); f j = 0 , we again have A = ρ>0 A(ρ).
For t 0 > ρ −1 we compose ρ and s and obtain the algebra homomorphism Proof Since is a homomorphism and z is a unit in K(ρ) it suffices to show that for a power series a ∈ K(ρ) the equation (a) = (a(t −1 )) t≥t 0 = 0 implies a = 0. This holds by the identity theorem since 0 is an accumulation point of the sequence t −1 .

(49)
A matrix R ∈ A(ρ) p×q and t 0 > ρ −1 give rise to As in Definition and Corollary 3.6 the equation holds and is decisive for the duality theory, recalled below. For A ∈ K(ρ) q×q the state space behavior from (9) now obtains the form Remark 3.7 Notice that W (t 0 ) is not an A-, but only an A(ρ)-left module for t 0 > ρ −1 .
In contrast to the well-known algebraic structure of A that of A(ρ) and its f.g. left modules is unknown. To enable a module-behavior duality between f.g. A-left modules and behaviors those from (50) have to be modified as in [4, (7), (9) This signal space W (∞) was already defined in [16, p. 5]. In [3, Thm. 839] it was shown for the coefficient field C(z) instead of C << z >> here that it is a large injective Acogenerator and thus enables a module-behavior duality. This signal module W (∞) is unsuitable for the stability theory of LTV systems since the signals w+C (N) do not have well-defined values w(t) ∈ C and in particular no initial value w(t 0 ). So (uniform) exponential stability of state space behaviors as in (13) or of general behaviors [4, Def. 1.7] cannot be defined. However, W (∞) is even a commutative ring since C (N) is an ideal of C N with the componentwise multiplication. The shift on C N induces that on W (∞) and makes it a difference ring. The ring homomorphism is well-defined, injective and indeed a monomorphism of difference rings. The algebra W (∞) and the preceding homomorphism can be used instead of : K(ρ) → C t 0 +N from (47) to derive the duality theory, recalled below, with different, but similar arguments.
Two behavior families are called equivalent, cf. [4, (7)], if Since t 1 can be chosen large one may always assume that ρ i = ρ(R i ) for i = 1, 2. The equivalence class of B 1 is denoted by cl(B 1 ). The study of this class means to study the behaviors B(R 1 , t 2 ) for large t 2 . This is appropriate for stability questions where the trajectories w(t) of a behavior are studied for t → ∞.
If M is a f.g. A-module with a given list w : = (w 1 , . . . , w q ) of generators there is the canonical isomorphism Since A is noetherian the submodule U is f.g. and thus generated by the rows of some matrix R ∈ A p×q , i.e., U = A 1× p R. Since A is even a principal ideal domain U is free and one may assume that dim A (U ) = rank(R) = p. The matrix R gives rise to behaviors and called the behavior defined by U . These behaviors B(U ) were introduced in [4, (9)] for the coefficient field C(z) = C(z −1 ) of rational functions instead of K ⊃ C(z).
In particular, a matrix A ∈ K n×n gives rise to the solution spaces B(s id n −A, t 0 ) = K(A, t 0 ) from (9) to (52) and the Kalman state space behavior Remark 3.8 The following three properties of the coefficient sequences a(t −1 ) t>σ (a) , a ∈ K, are decisive: 1. Any nonzero a ∈ C < z > can be written as a = a 0 + zc(z) where c(z) is bounded for |z| ≤ ρ < ρ(a) and hence a( 3. The sequences have no zeros for large t, i.e., ∃t 1 > σ (a)∀t ≥ t 1 : a(t −1 ) = 0. [4] hold for the coefficient field K and the behaviors B(U ) defined in (54).

Result 3.9 (Meta-theorem) With the obvious necessary modifications all essential notions and results from
For the preceding result one checks that the proofs of [4] use the properties of Remark 3.8 only. In particular, there is a canonical definition of behavior morphisms: The equivalence class B(ϕ) The behavior and its -invariant Bézout and valuation subdomain The field P is the algebraic closure of K = C << z >> [11]. If m 1 divides m 2 then z 1/m 1 = (z 1/m 2 ) m 2 /m 1 and hence C < z 1/m 1 >⊆ C < z 1/m 2 >. The nonzero elements of C << z 1/m >> have the unique form Again B is a left and right euclidean domain. A matrix In particular, a matrix A(z 1/m ) ∈ P n×n induces the state space equation and behaviors Result 4.1 Since P satisfies the conditions of Remark 3.8 the notions and theorems of [4] also hold for P like for C(z) in [4] and for K = C << z >> in Sect. 3. One can weaken the definition of exponential stability to weak exponential stability (w.e.s.) as in [5,Def. 2.4]. This is also preserved by behavior isomorphisms. For the state space system (72) w.e.s. holds if and only if ∃t 0 > σ (A) m ∃ρ = e −α < 1 (α > 0)∃μ > 0∃ p.g. ϕ ∈ C t 0 +N , ϕ > 0, ∀t 1 ≥ t 0 : A(t −1/m ) (t, t 1 ) ≤ ϕ(t 1 )ρ t μ −t μ 1 = ϕ(t 1 )e −α(t μ −t μ 0 ) .