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Lyapunov Spectrum Local Assignability of Linear Discrete Time-Varying Systems by Static Output Feedback

We consider a linear discrete time-varying input-output system. Our goal is to study the problem of local assignability of the Lyapunov spectrum by static output feedback control. To this end we introduce the notion of uniform consistency for discrete-time linear systems which is the extension of the notion of uniform complete controllability to input-output systems. The property of uniform consistency is investigated, some necessary and sufficient conditions for this property are obtained. The notion of uniform local attainability is introduced for the closed-loop system. We prove that uniform consistency implies uniform local attainability of the closed-loop system. The property of local Lyapunov reducibility is introduced for the closed-loop system. We prove that uniform local attainability implies local Lyapunov reducibility. We prove that, for a locally Lyapunov reducible system, the Lyapunov spectrum is locally assignable, if the free system is diagonalizable or regular (in the Lyapunov sence) or has the stable Lyapunov spectrum.


I. INTRODUCTION
It is well known that a well-designed feedback controller is expected not only to produce the required output, but also to ensure the satisfactory quality for the transition process, e.g., provide the required overall decay rate of the solutions or some appropriate oscillatory properties. In many cases, these properties are determined by the asymptotic behavior of some linear system, which usually arises as a system in variations for the original system and most often turns out to be nonstationary. Mathematical problems arising here are diverse and often difficult.
The simplest case of such a problem and, at the same time, its classic example is the stabilization problem, where the The associate editor coordinating the review of this manuscript and approving it for publication was Donatella Darsena . deviations of the process parameters from the target values are to be suppressed in the shortest possible time. The state-ofthe-art for the stabilization problem of continuous-time linear systems is described in [1] (see also [2]), and for discrete time systems in [3]. Another classic examples are the pole assignment problem for a stationary system and the problem of assigning the multiplier spectrum for a periodic system. Here the ultimate goal is not only to influence the decay rate of the solutions, but also on other characteristics of the transient process. It is well known that a necessary and, in the stationary case, a sufficient condition for the solvability of these problems is the complete controllability of the open system [4]- [7].
The spectrum of eigenvalues of a stationary system determines almost all features of this system. That is why for stationary systems it is possible to fine-tune their asymptotic properties even with the help of stationary linear feedback. In the non-stationary case, it is difficult to provide something like this, and we are to seek for some alternative approaches.
One of the useful way to handle the asymptotic properties of time-varying linear systems is the use of the Lyapunov spectrum and some related characteristics of these systems such as the Bohl exponents, the dichotomy spectrum, the properties of stability, reducibility, regularity in the Lyapunov sense, etc. For example, the stabilization problem as a rule can be reduced to the problem of assigning the higher Lyapunov exponent to be negative. In turn, to ensure uniform stability, the upper Bohl exponent is to be assigned. It should be stressed that by assigning some asymptotic characteristics of a linear system, we can influence various properties of this system that are not reduced only to the overall decay rate of solutions. In particular, by the simultaneous assignment of all Lyapunov exponents of a system (i.e., the assignment of the Lyapunov spectrum), it is possible to influence the conditional stability of its solutions. By assigning a zero value to the irregularity coefficient, it is possible to ensure a more reliable preservation of stability under the action of nonlinear perturbations.
All the above characteristics and properties are studied within the framework of the theory of Lyapunov exponents, the foundations of which were laid by A.M. Lyapunov in his doctoral thesis of 1892 [8]. Since then, the exponents theory for both differential and discrete case has been intensively developed in many directions and is now a well-established mathematical theory having many applications. The current state-of-the-art and basic definitions can be found in [9]- [14]. Some necessary definitions are also given below.
Thus, we may assert that the problem of ensuring the required quality of the transition process leads to the problem of assigning some prescribed asymptotic properties of a given linear control system by introducing some appropriate linear feedback into it. These problems have been intensively investigated for the continuous-time case, and the monograph [15] contains a summary and history of this research before 2012. Recently, some substantially new results have been obtained in this direction. In particular, necessary and sufficient conditions for assignability of the dichotomy spectrum for continuous time-varying linear systems are obtained in [16].
There exist few alternative approaches to the problem of assigning asymptotic properties of a linear system. An approach based on reducing of a periodic system to a stationary form using special feedback were considered, for example, in [17]- [19]. Starting from [20], a number of authors have tried to solve the problem of assigning asymptotic properties for a system with smooth coefficients by reducing it to the second canonical Luenberger form with subsequent transformation into stationary one by means of suitable feedback, see e.g. [21]- [23]. These results are quite advanced and provide a well-developed computational technique for practical applications. However, they have significant limitations of the scope due to the requirements for the original control system.
Much less is currently known for discrete systems. Results related to the canonical Luenberger form are presented in [24]- [26]. An approach similar to the approach of [17] for discrete systems is developed in [27]. Sufficient conditions for assignability of the dichotomy spectra for discrete time-varying linear control systems were obtained in [28]. Necessary and sufficient conditions for assignability of the dichotomy spectrum for one-sided discrete time-varying linear systems are obtained in [29].
A series of papers [30]- [33] discussed investigations of the problem of Lyapunov exponents placement for discrete-time systems. In these works, sufficient conditions are obtained for the solvability of the problem of assigning the Lyapunov spectrum of discrete non-stationary systems in various formulations. The main one among these conditions is, as in the continuous case, the uniform complete controllability of the original (open-loop) control system.
More precisely, in [32] it was established that if a linear discrete-time system with time-varying coefficients is uniformly completely controllable and the free system is diagonalizable or regular (in the sense of Lyapunov) or has the stable Lyapunov spectrum, then the Lyapunov spectrum of the closed-system (by linear state feedback) is proportionally locally assignable. Here proportional local assignability means that for an arbitrary set of numbers lying in a small neighborhood of the Lyapunov spectrum of the free system (2), we can construct a small-norm control U (t) t∈Z such that the Lyapunov spectrum of the closed-loop system (3) coincides with the given set. Moreover, we can choose the control U (t) t∈Z so that the value of U (t) satisfies some Lipshitz-type estimate with respect to the required exponents shift. An essential feature of the above result is the use of static state feedback. Such a restriction significantly narrows the scope of the result, but makes it easier to obtain. Our main goal in this paper is to overcome this deficiency. Here we consider the problem of assignment of the Lyapunov spectrum for a linear input-output discrete-time system with time-varying coefficients where K = R or K = C, by means of linear static output feedback that is for the closed-loop system of the form The problem is considered in a local setting, i.e., for an arbitrary set of numbers lying in a small neighborhood of the Lyapunov spectrum of the free system (2) one needs to construct a small-norm control U (t) t∈Z such that the Lyapunov spectrum of the closed-loop system (7) coincides with the given set. Note that we do not assume the control U (t) t∈Z to have any Lipshitz-type estimate with respect to the required exponents shift.
To extend the results obtained in [32] to system (7), we use the concept of uniform consistency of system (4), (5), which is a generalization of the concept of uniform complete controllability of system (4). The definition of uniform consistency was given in [34] for continuous-time systems, and in [35] for discrete-time systems. This new notion allows us to obtain sufficient condition for the above formulated problem. However, unlike the case of state feedback, we failed to obtain proportional local controllability. In [32] we construct the control explicitly and due to that we easily obtain the desired estimate. To construct a control in the case of output feedback, we have to use some sophisticated technique that severely restricts our options.
The paper is organized as follows. In Section II, we introduce the definition of uniform consistency for a linear discrete-time input-output system. The properties of uniformly consistent systems are investigated. Necessary conditions and sufficient conditions for uniform consistency are established, in terms of the coefficients of the original system, as well as in terms of the coefficients of the big system. In Section III, we recall the concept of dynamical equivalence and its sufficient condition. In Section IV, criteria for uniform consistency are obtained for time-invariant systems. In Section V, we introduce and investigate the definition of uniform local attainability for a closed-loop system by static output feedback. In Section VI, we establish an interrelation between the properties of the uniform consistency and uniform local attainability. It is proved that uniform consistency of the open-loop system implies uniform local attainability of the closed-loop system but the converse is not true. In Section VII, we introduce the definition of the property of local Lyapunov reducibility for the closed-loop system and prove that the uniform local attainability is a sufficient condition for local Lyapunov reducibility. In Section VIII, we introduce the definition of local assignability of the Lyapunov spectrum for a closed-loop system by static output feedback. We prove that, under some additional assumptions on the matrix of the free system, uniform local attainability implies local assignability of the Lyapunov spectrum. Corollaries are obtained on local assignability of the Lyapunov spectrum for uniformly consistent systems. In Section IX, an example is presented to illustrate the results obtained. In Section X, conclusion comments are given.
Notation. Relations α := β and β =: α mean that α is assumed, by definition, equal to β. Let K = C or K = R; K n = {x = col (x 1 , . . . , x n ) : x i ∈ K} is the linear space of vectors over K; M m,n (K) is the space of m × n-matrices over K; M n (K) := M n,n (K); I ∈ M n (K) is the identity matrix; [e 1 , . . . , e n ] := I ; set A 0 := I for any A ∈ M n (K); A is the complex conjugation of a matrix A; T is the transposition and * is the Hermitian conjugation of a matrix or a vector treated as a matrix; |x| = √ x * x is the norm in K n ; A = max |x|=1 |Ax| is the norm in M m,n (K); B ε (H ) := {G ∈ M m,n (K) : G − H ≤ ε}; N and Z are the sets of natural numbers and integers, respectively; an interval [t 0 , t 1 ), where t 0 , t 1 ∈ Z, t 0 < t 1 , is understood as the set of integer points t 0 , t 0 + 1, . . . , A quadratic form V P (y) := y * Py is identified with its Hermitian matrix P = P * ; the inequalities P > Q and P ≥ Q for Hermitian matrices P, Q are understood in the sense of quadratic forms, i.e., P > Q iff V P (y) > V Q (y) for all y ∈ K n \ {0}, and P ≥ Q iff V P (y) ≥ V Q (y) for all y ∈ K n . Let For every matrix H ∈ N n (α) the inequality holds; therefore, the set N n (α) is nonempty only for α ≥ 1. For this reason, below we consider the set N n (α) only for α ≥ 1. We say that a sequence L(·) = L(t) t∈Z ⊂ M n forms a Lyapunov sequence if there exists α ≥ 1 such that For any sequence F(·) = F(t) t∈Z ⊂ M m,n we define

II. UNIFORM CONSISTENCY FOR LINEAR DISCRETE-TIME SYSTEMS
A. DEFINITIONS Consider a linear discrete time-varying control system (4), (5) with a Lyapunov sequence A(t) t∈Z ⊂ M n (K) and bounded sequences B(t) t∈Z ⊂ M n,m (K), C(t) t∈Z ⊂ M n,k (K). We assume that the inclusion and the inequalities hold for some finite a ≥ 1, b > 0, c > 0. By X A (t, τ ), t, τ ∈ Z, denote the transition matrix of the corresponding free system (2), that is From (9) it follows that, for any t ∈ Z, hence, by definition of X A (t, τ ), for every t, τ ∈ Z the estimation holds.
Recall that system (4) is said to be completely reachable on (completely controllable on) [t 0 , t 1 ) if for any x ∈ K n there exists a control function u(t), t ∈ [t 0 , t 1 ), steering the solution of system (4) from the state Let us construct the reachability gramian and the controllability gramian . It is known the following proposition.
Proposition 4: System (4) is ϑ-uniformly completely controllable if and only if there exists a number 1 > 0 such that for any τ ∈ Z and for any x 1 ∈ K n there exists a control function u(t), t ∈ [τ, τ + ϑ), steering the solution of system (4) from the state Let us introduce the following definition by analogy with Definition 2.
Theorem 1: System (4), (5) is ϑ-uniformly consistent if and only if there exists 2 > 0 such that for any τ ∈ Z and for any G ∈ M n (K) there exists a control function , steering the solution of system (12) from the state Z (τ ) = 0 into the state Z (τ + ϑ) = G, and the inequality U (t) ≤ 2 G holds for all t ∈ [τ, τ + ϑ).
The aim of this section is studying the property of uniform consistency.

B. AUXILIARY STATEMENTS
Let us give some auxiliary statements.
Lemma 1: Let W ∈ M p (K) be a Hermitian, positive definite matrix, and 0 < µ 1 I ≤ W ≤ µ 2 I . Then the matrix W −1 is also Hermitian, positive definite and Lemmas 1 and 2 are clear. The proofs are given, e.g., in [35, Lemma 8 and Lemma 9].
Proposition 5 follows from inequality (14) of Definition 3 and Proposition 2. The converse is not true, in general, by the following example.

2) THE BIG SYSTEMS
Let us construct the control system (so-called the big system) By Ω(t, τ ) denote the transition matrix of the free system Then, by properties of the Kronecker product, we get Theorem 3: System (4), (5) is ϑ-uniformly consistent iff system (24), (25) is ϑ-uniformly completely controllable.

3) INTERCONNECTION BETWEEN UNIFORM COMPLETE CONTROLLABILITY AND UNIFORM CONSISTENCY
The following statements establish an interconnection between the properties of uniform complete controllability of system (4) and uniform consistency of system (4), (5).

III. DYNAMIC EQUIVALENCE
Definition 4 (see [42, p. 15]): Let L : Z → M n (K) be a Lyapunov sequence. A linear transformation of the space K n is called a Lyapunov transformation. Definition 5 (see [42, p. 15]): We say that system (2) is dynamically equivalent to the system if there exists a Lyapunov transformation (42) which connects these systems, i.e., for every solution x(t) of system (2) the function θ(t) = L(t)x(t) is a solution of system (43) and for every solution θ(t) of system (43) the function x(t) = L −1 (t)θ(t) is a solution of system (2). Let us note that if a Lyapunov transformation (42) establishes the dynamic equivalence between systems (2) and (43), then hence, Thus, systems (2) and (43) are dynamically equivalent if and only if there exists a Lyapunov sequence L(·) = L(t) t∈Z , such that the equality (44) is satisfied.
Denote by Θ(t, τ ) the transition matrix of the free system (43). There is the following criterion of dynamic equivalence [30,Lemma 4.5].
Lemma 4: Suppose that A(t) t∈Z and A(t) t∈Z are Lyapunov sequences. Assume that Then systems (2) and (43) are dynamically equivalent.
Remark 5: Lemma 4 was proved in [30, Lemma 4.5] for positive semiaxis of integers but the proof remains the same for the whole axes of integers.

V. UNIFORM LOCAL ATTAINABILITY
Consider system (4), (5). Suppose that the control in this system is constructed as static output feedback (6). The closed-loop system has the form (7). By Φ U (t, τ ), t ≥ τ , denote the transition matrix of system (7). In particular, Definition 6: System (7) is said to be: (a) ϑ-uniformly locally attainable if there exists γ ≥ 1 such that, for any ε > 0, there exists δ > 0 such that, for any matrix H ∈ B δ (I ) ⊂ M n (K) and any t 0 ∈ Z, there exists a control function U : [t 0 , t 0 +ϑ) → B ε (0) ⊂ M m,k (K) ensuring the equality and the following inclusion holds: uniformly locally attainable if there exists ϑ > 0 such that the system is ϑ-uniformly locally attainable.
Let us prove some properties of uniformly locally attainable systems.
Let (b) hold. From (56) and (57), it follows that From (58), it follows that Multiplying (64) from the left by X A (t 0 , t 0 + ϑ), we obtain Applying vec to (65), we obtain Solving (66) with respect to vec V (s) in the left-hand side, we obtain vec V (t) Next, from (63), it follows that Applying vec to (69), we get Substituting (67) for vec V (s) in the first summand of (70), we get vec Y (t) Dividing in (71) the summing over s from t 0 to t 0 + ϑ − 1 into two parts -from t 0 to t − 1 and from t to t 0 + ϑ − 1, and using the equality Thus, from (68) and (72), it follows (a). All above arguments are reversible, hence, (a) ⇒ (b). So, (A) is true.
Let us prove the existence of a solution to system (61), (62) by a fixed-point theorem.
Remark 6: By Corollary 2, uniform consistency of system (4), (5) is a sufficient condition for uniform local attainability of system (7). But it is not a necessary condition. The following example 2 confirms this.
Example 2: Consider system (4), (5) with n = 2, m = 1, Let us show that the system is not uniformly consistent. We have From (98) and (99), it follows that, for any τ ∈ Z and ϑ ∈ N, the first row of the matrix T 1 (τ, ϑ) is equal to zero. Hence, rank T 1 (τ, ϑ) < n 2 . Thus, the system is not consistent on [τ, τ +ϑ), hence, is not ϑ-uniformly consistent. Nevertheless, the system is ϑ-uniformly locally attainable for ϑ = 12. The proof of this assertion is the same as the proof of the similar assertion for the corresponding continuous-time system. This proof was given in [43, Theorem 3] on the base of Lemmas 2 and 3 [43], and there was constructed the control U (·) ensuring (51). The proof of Lemma 2 [43] remains the same with a simpler estimation for |u 2 |. The proofs of Lemma 3 and Theorem 3 [43] remains the same. The constructed control U (·) is ϑ-periodic. This will imply estimations (52) for some γ ≥ 1.

VII. LOCAL LYAPUNOV REDUCIBILITY
Along with the free system (2) we consider the perturbed system The perturbation R(·) = R(t) t∈Z ⊂ M n (K) will be called a multiplicative perturbation of system (2).
Definition 7 (see [44]): A multiplicative perturbation R(·) is called admissible if the sequence R(t) t∈Z is a Lyapunov sequence.
The set of all admissible multiplicative perturbations will be denoted by R and the subset of R consisting of perturbations satisfying the condition R − I ∞ < δ will be denoted by R δ .
Let X AR (t, s) be the transition matrix of system (100). Below we will need the following lemma.
Theorem 11: If system (7) is uniformly locally attainable, then this system has the property of local Lyapunov reducibility.

Note that
By Definition 6, for this matrix H s there exists a control U s : J s → B ε (0) ⊂ M m,k (K) ensuring the equality , and for every integer s the following equalities hold: From this, it follows, due to Lemma 4, that systems (100) and (7) with the constructed control U (·) are dynamically equivalent.
Let the control in the system (4), (5) have the form of static output feedback (6). We identify (6) with the sequence U (·) = U (t) t∈Z .
Definition 9: A bounded function U : Z → M m,k (K) is said to be an admissible feedback control for system (4), (5) if A(t) + B(t)U (t)C * (t) t∈Z is a Lyapunov sequence.
Definition 10: The Lyapunov spectrum of system (7) is called: 1) locally assignable if for any ε > 0 there exists δ > 0 such that for any µ ∈ O δ λ(A) there exists an admissible feedback control U : Z → B ε (0) ⊂ M m,k (K) for system (4), (5), ensuring the equality 2) proportionally locally assignable if there exist > 0 and δ > 0 such that for any sequence µ = µ 1 , . . . , µ n ) ∈ O δ λ(A) there exists an admissible feedback control U (·) for system (4), (5), satisfying the estimate It is clear that the property of proportional local assignability implies the property of local assignability, and the reverse implication is generally not true.
It turns out that the concepts of local and proportional local assignability of the Lyapunov spectrum of system (7) are closely related to the concept of proportional global assignability of the Lyapunov spectrum of the system (100), in which the multiplicative perturbation R(·) is understood as a control.
Definition 11: The Lyapunov spectrum of system (100) is called proportionally globally assignable if for any > 0 there exists = ( ) > 0 such that for any sequence µ = µ 1 , . . . , µ n ∈ O λ(A) there exists a perturbation R(·) ∈ R satisfying the estimation and providing the validity of the relation Theorem 12: Suppose that system (7) has the property of local Lyapunov reducibility. If the Lyapunov spectrum of (100) is proportionally globally assignable, then the Lyapunov spectrum of (7) is locally assignable.
This corollary follows from Corollary 3 and Theorem 12.
Consider a linear control system (4). Let the control in this system have the form of static state feedback We get the closed-loop system (3). This system has the form (7) with k = n and C(t) ≡ I , t ∈ Z, therefore, for this system, one can introduce the concepts of local assignability and proportional local assignability of the Lyapunov spectrum.
Corollary 5: Suppose that system (4) is uniformly completely controllable. If the Lyapunov spectrum of (100) is globally proportionally assignable, then the Lyapunov spectrum of (3) is locally assignable.
This corollary follows from Theorem 7 and Corollary 4. Note that in the paper [32] the more strong assertion was proved.
The method of proving this theorem used in [32] is not applicable to the input-output system (4), (5).
Now we present results about local assignability of the Lyapunov spectrum of system (7). They are expressed in the forms of certain concepts from the asymptotic theory of linear systems, which are defined below.
Definition 12: System (2) is called diagonalizable if it is dynamically equivalent to system (43) with a diagonal matrix A(t), t ∈ Z.
Definition 13 (see [24, p. 63]): System (2) is called regular (in the Lyapunov sense) if the following equality holds: The notion of regularity of linear differential systems was introduced in the famous paper of Lyapunov [8]. Some facts about regularity of discrete equations may be found in the works [24], [45], [46]. Let us notice that all time-invariant or all periodic systems are regular.
Definition 14 (see [44]): The Lyapunov spectrum of system (2) is called stable if for any ε > 0 there exists The effect of instability of the Lyapunov spectrum under the influence of small coefficient perturbations for linear continuous-time systems was discovered by O. Perron [47]. Later the stability property of the Lyapunov spectrum for these systems was investigated in [48], [49]. The study of this property for discrete-time systems was started in [44].
In [32], sufficient conditions were obtained for proportional global assignability of the Lyapunov spectrum of system (100).
Theorem 14 (see [32]): Assume that at least one of the following conditions holds: (i) system (2) is regular; (ii) system (2) is diagonalizable; (iii) the Lyapunov spectrum of system (2) is stable. Then the Lyapunov spectrum of system (100) is proportionally globally assignable.
Theorem 15: Suppose that system (7) has the property of local Lyapunov reducibility and at least one of the conditions (i), (ii), (iii) of Theorem 14 holds. Then the Lyapunov spectrum of system (7) is locally assignable.
This Theorem follows from Theorems 12 and 14.
Corollary 6 follows from Corollary 4 and Theorem 15. Corollary 7: Suppose that system (4) is uniformly completely controllable. If at least one of the conditions (i), (ii), (iii) of Theorem 14 holds, then the Lyapunov spectrum of (3) is locally assignable.
Corollary 7 follows from Theorem 7 and Corollary 6. From Theorem 13, it follows a more strong assertion, which have been proved in [32].
Theorem 16 (see [32]): If system (4) is uniformly completely controllable and at least one of the conditions (i), (ii), (iii) of Theorem 14 holds, then the Lyapunov spectrum of (3) is proportionally locally assignable.

X. CONCLUSION
In the paper, linear discrete time-varying input-output systems have been studied. The problem of local assignability of the Lyapunov spectrum by static output feedback control have been investigated. The notion of uniform consistency for discrete-time systems have been introduced. This property is, in some sense, the extension of the notion of uniform complete controllability for input-output systems. The property of uniform consistency have been developed in detail, the necessary and sufficient conditions for this property have been obtained. The notions of uniform local attainability and local Lyapunov reducibility have been introduced, which were previously introduced for continuous-time systems. We have proved that uniform consistency implies uniform local attainability of the closed-loop system. In turn, uniform local attainability implies local Lyapunov reducibility. We have proved that, for a locally Lyapunov reducible system, the Lyapunov spectrum is locally assignable, if the free system is diagonalizable, or regular (in the Lyapunov sence), or has the stable Lyapunov spectrum. This is an extension of the corresponding results proved earlier for continuous-time systems and for discrete-time systems with static state feedback.
Further development of the results of the paper could be as follows. We plan to study in more detail the properties of uniform consistency and uniform local attainability, including for the case when conditions (9) and (10) are not satisfied. In addition, the invariance of these properties under the Lyapunov transformations will be proved. We also plan to prove that, in Theorem 15, the conditions (i)-(iii) could be weakened. Further, we plan to extend the theory of uniformly consistent systems to more general systems, namely, to bilinear systems of the form x(t +1) = A(t)+u 1 (t)A 1 (t) + . . . + u r (t)A r (t) x(t) (105) and to obtain the corresponding results on uniform local attainability, local Lyapunov reducibility and Lyapunov spectrum assignability for systems (105). Some results concerning global assignability of Lyapunov spectrum for timeinvariant consistent systems of the form (105) were obtained in [26], [37].