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

We complete the stability results of [4] and use the notions, in particular the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.), of this paper, but extend the theory from the coefficient field \({\mathbb {C}}(z)\) of rational functions to the larger field \({\mathbb {C}}<<z>>\) 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], Theorem 1.1 furnishes a constructive test for e.s. 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 \({\mathbb {C}}<<z>>\), 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 module-behavior 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 \({\mathbb {C}}<z>\) denote the local principal ideal domain of locally convergent power series in the variable z and \(\mathbf{K}=:{\mathbb {C}}<<z>>\) its quotient field. A formal power series \(a=\sum _{i=0}^\infty a_i z^i \) is called locally convergent ifis its convergence radius and a(z) is holomorphic in the open discThe power series a is a unit (invertible) if and only if \(a(0)=a_0\ne 0\), and z is the unique prime of \({\mathbb {C}}<z>\), up to units. Each nonzero \(a \in {\mathbb {C}}<z>\) has a unique representationThis implies that every nonzero \(a\in \mathbf{K}\) has the analogous unique representationThe element a is a power series if and only if \(k={\text {ord}}(a)\ge 0\). The representation \(a=\sum _{j=k}^\infty b_{j-k} z^j\) shows that a is a locally convergent Laurent series with at most a pole at zero, and that indeed \(\mathbf{K}={\mathbb {C}}<<z>>\) consists of all these Laurent series. If \(k<0\) the function a(z) is a holomorphic function in the pointed open disc \(\mathbf{D}({\rho }(a))\setminus \left\{ 0\right\} \). In particular, the function \(a(t^{-1})=t^{-k}b(t^{-1})\) is a smooth function on the real open interval \(({\sigma }(a),\infty ):=\left\{ t\in {\mathbb {R}};\; t>{\sigma }(a)\right\} \), henceThe sequences \(a(t^{-1}),\; t\in {\mathbb {N}} ,\; t>{\sigma }(a),\) are the time-varying coefficients of the difference equations of the present paper that in [4] were used for \(a \in {\mathbb {C}}(z)={\mathbb {C}}(z^{-1})\). The coefficient functions \(a(t^{-1}),\; a\in \mathbf{K},\) are of at most polynomial growth on each closed interval \([{\sigma }_1,\infty ),\; {\sigma }_1 >{\sigma }(a),\) i.e., there are \(c>0\) and \(m\in {\mathbb {N}} \) such that \(|a(t^{-1})|\le ct^m \) for \(t\ge {\sigma }_1\) [5, (29)]. For nonzero a there is \({\sigma }_2>{\sigma }(a)\) such that \(a(t^{-1})\ne 0\) for \(t\ge {\sigma }_2\). 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, \(A=(A_{\mu \nu })_{1\le \mu ,\nu \le n}\in \mathbf{K}^{n\times n}\) be any square matrix and defineThen the function \(t\mapsto A(t^{-1})\) is a smooth matrix function on the open real interval \(({\sigma }(A),\infty )\). For \(t_0\in {\mathbb {N}} \) we consider the signal spaceof complex sequences or discrete signals starting at the initial time \(t_0\). For \(n\in {\mathbb {N}} \) we use the column spaces \({\mathbb {C}}^n\) and \(W(t_0)^n\) and identifyIf \(t_0 > {\sigma }(A),\; t_0\in {\mathbb {N}} ,\) 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) isThere is the obvious isomorphismFor \( \xi =(\xi _1,\ldots ,\xi _n)^\top \in {\mathbb {C}}^n\) and \( M\in {\mathbb {C}}^{n\times n}\) we use the maximum normsThe system, i.e., the matrix A and the equation and behavior from (9), are called exponentially stable (e.s.) [4, Def. 1.7, Cor. 3.3] ifHere a sequence \({\varphi }\in {\mathbb {C}}^{t_0+{\mathbb {N}} }\) is called of at most polynomial growth (p.g.) if \(|{\varphi }(t)|\le ct^m,\; t\ge t_0,\) for some \(c>0\) and \(m\in {\mathbb {N}} \). In [4] we used the notation \({\rho }:=e^{-{\alpha }}<1\) and \({\rho }^{t-t_1}=e^{-{\alpha }(t-t_1)}\). In particular, e.s. implies asymptotic stability, i.e., \(\lim _{t\rightarrow \infty }{\Phi }_A(t,t_1)=0\) for \(t_1\ge t_0\). The system is called uniformly e.s. (u.e.s.) [15, Def. 22.5] if \({\varphi }\) in (13) can be chosen constant. Notice that (13) is a property of the behavior family \(({\mathcal {K}}(A,t_1))_{t_1\ge t_0}\) and of the trajectories \(x(t)={\Phi }_{A}(t,t_1)x(t_1),\; t\ge t_1\ge t_0,\) for sufficiently large \(t_0\). This is appropriate for stability questions where the behavior of x(t) for \(t\rightarrow \infty \) is investigated. In [15] the author considers LTV state space equations \(x(t+1)=F(t)x(t),\; t\ge 0,\) with an arbitrary sequence of complex matrices \(F=(F(0),F(1),\ldots )\in \left( {\mathbb {C}}^{n\times n}\right) ^{\mathbb {N}} = \left( {\mathbb {C}}^{\mathbb {N}} \right) ^{n\times n} \). All stability results in [15, Chs. 22–24] require additional properties of F. Our choice in [4] and in the present paper isA nonzero A admits a unique representationwhere the exponent \(-k\) is chosen for notational convenience in Theorem 1.1.

Notations and abbreviations \(\mathbf{D}({\rho }):=\left\{ z\in {\mathbb {C}};\; |z|<{\rho }\right\} ,\; {\rho }>0\), e.s. = exponentially stable, exponential stability, f.g.= finitely generated, LTV = linear time-varying, resp. = respectively, u.e.s. = uniformly e.s., w.e.s. = weakly e.s., \(X^{p\times q}\) = set of \(p\times q\)-matrices with entries in X, \(X^{1\times q}\)=rows, \(X^q:=X^{q\times 1}\) = columns, \(X^{\bullet \times \bullet }:=\bigcup _{p,q\ge 0} X^{p\times q}\).

We explain the basic notions of a variant of the theory from [4] since we use the difference field \(\mathbf{K}={\mathbb {C}}<<z>>\) instead of the field \({\mathbb {C}}(z) \subset \mathbf{K}\) of rational functions in [4]. Recall \(W(t_0)= {\mathbb {C}}^{t_0+{\mathbb {N}} }\) for \(t_0\in {\mathbb {N}} \) from (7). The space \({\mathbb {C}}^{t_0+{\mathbb {N}} }=W(t_0)\) is also a difference \({\mathbb {C}}\)-algebra with the componentwise multiplication and the shift algebra homomorphismIt gives rise to the noncommutative skew-polynomial algebra of difference operators [9, Section 1.2.3], [4, (20)]The space \(W(t_0)\) is a left \(\mathbf{B}(t_0)\)-module with the action \(f\circ w\) for \(f=\sum _{j=0}^\infty f_j s^j\in \mathbf{B}(t_0)\) and \(w \in W(t_0)\) [4, (21)], defined byOf course, almost all, i.e., up to finitely many, \(f_j\) are zero so that the sums \(\sum _j\) are actually finite, here and in later occurrences. As usual the action is extended to the action \(R\circ w\) [4, (22)] of a matrixThe behavior or solution space defined by R isFor \({\sigma }>0\) the algebra \({\text {C}}^\infty ({\sigma },\infty )\) is also a difference algebra with the algebra endomorphismIt gives rise to the skew-polynomial algebraFor \(t_0 > {\sigma }\) the mapis a difference algebra homomorphism since \({\Phi }_d \Psi _s=\Psi _s{\Phi }_s\) and therefore its extension(denoted with the same letter) is an algebra homomorphism. The algebras \({\text {C}}^\infty ({\sigma },\infty )\) and \({\mathbb {C}}^{t_0+{\mathbb {N}} }\) 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 sequenceswhere t is chosen sufficiently large as explained in Lemma 3.1, for instance \(t> {\sigma }(a)={\rho }(a)^{-1}\). In the latter case we have \((t+1)^{-1}=\frac{t^{-1}}{1+t^{-1}} <{\rho }(a)\) andThis suggests to make \({\mathbb {C}}<<z>>\) a difference field [16, Ex. 1.2], [4, §4.7] via the field automorphismIfThe corresponding skew-polynomial algebra of difference operators isThe \({\mathbb {C}}\)-algebra \(\mathbf{A}\) is a noncommutative euclidean domain [9, §1.2], especially principal, and the f.g. left \(\mathbf{A}\)-modules are precisely known [9, Thm. 1.2.9, § 5.7., Cor. 5.7.19]. The operators in \(\mathbf{A}\) have the form \(f:=\sum _j f_j s^j\in \mathbf{A}, \; f_j \in \mathbf{K}, \) where almost all \(f_j\) are zero. We define \({\rho }(f):={\text {min}}_j\left\{ {\rho }(f_j);\; f_j \ne 0\right\} ,\; {\sigma }(f):={\rho }(f)^{-1}.\)

In the sequel we make use of the equation \({\Phi }(a)(t^{-1})=a((t+1)^{-1})\) for \(t>{\sigma }(a)\). Since \({\rho }({\Phi }(a))\) may be smaller than \({\rho }(a)\), cf. Example 3.3, the left side of this equation is not defined a priori. To solve this problem we introduce difference subalgebras \(\mathbf{K}({\rho })\subset \mathbf{K},\; {\rho }>0,\) such that the map \(\Psi _{\rho }:\; \mathbf{K}({\rho }) \rightarrow {\text {C}}^\infty ({\rho }^{-1},\infty ),\; a \mapsto a(t^{-1}),\) is a well-defined difference algebra homomorphism. We need the following detour: For an open set \(U\subseteq {\mathbb {C}}\) let \({\mathcal {O}}(U)\) denote the \({\mathbb {C}}\)-algebra of holomorphic functions in U. So any \(a\in {\mathbb {C}}<<z>>\) defines the holomorphic function \(a(z)\in {\mathcal {O}}\left( \mathbf{D}({\rho }(a))\setminus \left\{ 0\right\} \right) \). In general, this can be extended to larger connected open sets. For \({\rho }>0\) we define the subset \(\mathbf{K}({\rho }) \subset \mathbf{K}\) as follows: An element \(a\in \mathbf{K}\) belongs to \(\mathbf{K}({\rho })\) if there is an open connected neighborhood U(a) of 0 and a holomorphic function \(f\in {\mathcal {O}}\left( U(a)\setminus \left\{ 0\right\} \right) \) such that \([0,{\rho })\subset U(a) \) and \(f(z)=a(z)\) for \(z\ne 0\) near 0. In other terms, the germ of f at 0 is a. Since \(U(a)\setminus \left\{ 0\right\} \) is connected the function f is unique with these properties, due to the identity theorem. The \(\mathbf{K}({\rho })\) obviously satisfyAll entire functions \(a\in {\mathcal {O}}({\mathbb {C}})\subset {\mathbb {C}}<<z>>\) and \(z^{-1}\) belong to all \(\mathbf{K}({\rho })\).

The Lemmas 3.1 and 3.2 imply thatis a subalgebra of \(\mathbf{A}\) and that the extended map, denoted by the same letter,is an algebra monomorphism. Since any \(f=\sum _j f_j s^j\) belongs to \(\mathbf{K}({\rho }(f))\), \({\rho }(f)={\text {min}}\left\{ {\rho }(f_j);\; f_j\ne 0\right\} \), we again have \(\mathbf{A}=\bigcup _{{\rho }>0} \mathbf{A}({\rho })\).

For \(t_0> {\rho }^{-1}\) we compose \(\Psi _{\rho }\) and \(\Psi _s\) and obtain the algebra homomorphism

The homomorphisms \({\Phi },{\Phi }_d, \Psi \) are extended to matrices componentwise. For a matrix \(R=\sum _j R_j s^j \in \mathbf{A}^{p\times q} ,\; R_j \in \mathbf{K}^{p\times q}, \) we define, cf. (6),A matrix \(R \in \mathbf{A}({\rho })^{p\times q}\) and \(t_0>{\rho }^{-1}\) give rise toand the solution space or behaviorAs in Definition and Corollary 3.6 the equationholds and is decisive for the duality theory, recalled below.

For \(A\in \mathbf{K}({\rho })^{q\times q}\) the state space behavior from (9) now obtains the form

Two behavior familiesare called equivalent, cf. [4, (7)], ifSince \(t_1\) can be chosen large one may always assume that \({\rho }_i={\rho }(R_i)\) for \(i=1,2\). The equivalence class of \({\mathcal {B}}_1\) is denoted by \({\text {cl}}({\mathcal {B}}_1)\). The study of this class means to study the behaviors \({\mathcal {B}}(R_1,t_2)\) for large \(t_2\). This is appropriate for stability questions where the trajectories w(t) of a behavior are studied for \(t\rightarrow \infty \).

If M is a f.g. \(\mathbf{A}\)-module with a given list \(\mathbf{w}:= (\mathbf{w}_1,\ldots ,\mathbf{w}_q)^\top \) of generators there is the canonical isomorphismSince \(\mathbf{A}\) is noetherian the submodule U is f.g. and thus generated by the rows of some matrix \(R\in \mathbf{A}^{p\times q}\), i.e., \(U=\mathbf{A}^{1\times p}R.\) Since \(\mathbf{A}\) is even a principal ideal domain U is free and one may assume that \(\dim _\mathbf{A}(U)={\text {rank}}(R)=p\). The matrix R gives rise to behaviorsEquation (51) is used to show that this class depends on U only and not on the special choice of R [4, Lemma 2.5]. The class is denoted byand called the behavior defined by U. These behaviors \({\mathcal {B}}(U)\) were introduced in [4, (9)] for the coefficient field \({\mathbb {C}}(z)={\mathbb {C}}(z^{-1})\) of rational functions instead of \(\mathbf{K}\supset {\mathbb {C}}(z)\). In particular, a matrix \(A\in \mathbf{K}^{n\times n}\) gives rise to the solution spaces \({\mathcal {B}}(s{\text {id}}_n-A,t_0)={\mathcal {K}}(A,t_0)\) from (9) to (52) and the Kalman state space behavior

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: If \(M_i=\mathbf{A}^{1 \times q_i}/U_i,\; U_i=\mathbf{A}^{1\times p_i}R_i,\; R_i\in \mathbf{A}^{p_i\times q_i},\; i=1,2,\) are two modules any \(\mathbf{A}\)- linear map \( {\varphi }: M_1\rightarrow M_2 \) has the form [4, (47), (48)]If \(R_1,R_2,P,X\in \mathbf{A}({\rho })^{\bullet \times \bullet }\), for instance if \({\rho }< {\text {min}}({\rho }(R_1),{\rho }(R_2),{\rho }(P),{\rho }(X))\), and if \(t_0>{\rho }^{-1}\) Eq. 51 impliesThe equivalence class \({\mathcal {B}}({\varphi }):={\text {cl}}\left( (P\circ :{\mathcal {B}}(R_2,t_0) \rightarrow {\mathcal {B}}(R_1,t_0))_{t_0 >{\rho }^{-1}}\right) \) is defined in analogy to (54), cf. [4, (33), (50)], and defines the behavior morphismThe map \({\varphi }\) is an isomorphism if and only if \({\mathcal {B}}({\varphi })\) is one, i.e., if \(P\circ :{\mathcal {B}}(R_2,t_1)\rightarrow {\mathcal {B}}(R_1, t_1)\) is an isomorphism for sufficiently large \(t_1\).

A f.g. module \(M=\mathbf{A}^{1 \times q_1}/U_1\) is a torsion module, cf. [4, §2.5], if and only if \(n:= \dim _\mathbf{K}(M)< \infty \) or if and only if it is isomorphic to a module in state space form, i.e.,The behavioris called autonomous if and only if there is \(t_1>{\sigma }(R)\) such that all trajectories \(w\in {\mathcal {B}}(R,t_2),\; t_2\ge t_1,\) are determined by the initial vector \((w(t_2),\ldots , w(t_2+d))\) of some fixed length d. This is the case if and only if the module \(M=\mathbf{A}^{1\times q}/U\) is a torsion module [4, Thm. 3.18].

Exponential stability (e.s.) of an autonomous behavior \({\mathcal {B}}(U)\) and of its torsion module \(M=\mathbf{A}^{1\times q}/U\) [4, Def. 1.7] is defined by a more general version of (13) and preserved by behavior isomorphisms [4, § 3.2]. The state space behavior \({\mathcal {B}}(U_2)\) in (64) is e.s. if and only if the matrix A is e.s. in the sense of (13) [4, Cor. 3.3].

To a large extent Theorem 1.1 can be extended to the difference field \(\mathbf{P}\) of locally convergent Puiseux series, cf. [5, §3.1], whereand its \({\Phi }\)-invariant Bézout and valuation subdomainThe field \(\mathbf{P}\) is the algebraic closure of \(\mathbf{K}={\mathbb {C}}<<z>>\) [11]. If \(m_1\) divides \(m_2\) then \(z^{1/m_1}=(z^{1/m_2})^{m_2/m_1}\) and hence \({\mathbb {C}}<z^{1/m_1}>\subseteq {\mathbb {C}}<z^{1/m_2}>\). The nonzero elements of \({\mathbb {C}}<<z^{1/m}>>\) have the unique formSuch an \( a(z^{1/m})\) induces the smooth function \(a(t^{-1/m})= t^{-k/m}\sum _i b_it^{-i/m}\) on the real interval \(({\sigma }(a)^m,\infty )\). The coefficient rings give rise to the operator domainsAgain \(\mathbf{B}\) is a left and right euclidean domain. A matrixacts on \(w\in W(t_0)^q=\left( {\mathbb {C}}^q\right) ^{t_0+{\mathbb {N}} }\) with \(t_0 >{\sigma }(R)^m\) viaIn particular, a matrix \(A(z^{1/m})\in \mathbf{P}^{n\times n}\) induces the state space equation and behaviors

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 ifThe difference to e.s. is the exponent \(\mu >0\). If \(\mu <1 \) the factor \(\exp (-{\alpha }t^\mu )\) decreases more slowly than \(\exp (-{\alpha }t)\). Corollary 4.2 obtains a slightly weaker form.