Arbitrary Coefﬁcient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays

: We consider a linear control system deﬁned by a scalar stationary linear differential equation in the real or complex space with multiple non-commensurate lumped and distributed delays in the state. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, we study the problem of arbitrary coefﬁcient assignment for the characteristic function by linear static output feedback with lumped and distributed delays. We obtain necessary and sufﬁcient conditions for the solvability of the arbitrary coefﬁcient assignment problem by the static output feedback controller. Corollaries on arbitrary ﬁnite spectrum assignment and on stabilization of the system are obtained. We provide an example illustrating our results.


Introduction
A large number of works have been devoted to the problem of stability of time-delay systems and problems of stabilization for control systems with delays (see reviews [1][2][3][4]). A number of methods have been developed to solve this problem. One of the methods, known as the Lyapunov-Krasovskii functional approach [5], allows one to obtain sufficient conditions for asymptotic and exponential stabilization of delayed systems [6][7][8]. This method historically traces back to the second Lyapunov method. Another approach to studying problems of stability and stabilization of time-delay systems is an eigenvaluebased approach [9]. This approach traces back to the first Lyapunov method. Here, the goal is to find conditions providing the desired assignment of the spectrum of the system, that is, the set of zeros of the characteristic function of the system.
The classical problem of spectrum assignment (for systems without delays) is usually studied as the problem of coefficient assignment for characteristic function (in another terminology as the problem of modal control) and is as follows: consider a linear timeinvariant control systemẋ = Ax + Bu, x ∈ K n , u ∈ K m (here K = C or K = R). Let the controller have the form of linear static state feedback u = Qx.
The closed-loop system (1), (2) takes the forṁ (1) is called arbitrary spectrum assignable by means of static state feedback (2) if, for any γ i ∈ K, i = 1, n, there exists a gain matrix Q such that the characteristic polynomial of the matrix A + BQ of the system (3) coincides with the polynomial λ n + γ 1 λ n−1 + . . . + γ n .

Definition 1. System
The problem of arbitrary spectrum assignment was solved in [10] (for K = C) and in [11] (for K = R)-namely, it has been proven that complete controllability of (1) is a necessary and sufficient condition for arbitrary spectrum assignability (i.e., for modal controllability) of system (3). This property is a sufficient condition for exponential stabilization of the system (with any given decay of rate).
For systems with delays, the spectrum is in general infinite. The spectrum depends on coefficients of the characteristic function. The problem of arbitrary spectrum assignment of linear time-delay systems (in contrast to systems without delays) is not equivalent to the problem of arbitrary coefficient assignment (ACA) for the characteristic function of the closed-loop system. A number of early works dealt with coefficient assignment, spectrum assignment, and stabilizability problems for linear time-delay systems by means of static state feedback with delays: sufficient conditions for ACA were obtained in [12] for systems with multiple lumped delays; decoupling and canonical forms were used for coefficient assignment in [13,14]; stabilizability and spectrum assignment for linear autonomous systems with general time delays were studied in [15]; problems of stabilizability independent of delay were developed for the class of delay differential systems of the retarded type with commensurate time delays in [16]; spectrum placement problem was studied in [17] by using a ring of delay operators.
In a study [18], an approach was developed for assigning an arbitrary finite spectrum for linear systems with delays. Later, the finite spectrum assignment problem for time-delay systems by linear state feedback was studied in [19] for systems with one lumped delay in the states with the scalar controller, in [20] for systems with multiple commensurate lumped delays in the states with the scalar controller, in [21] for systems with multiple commensurate lumped delays in the states and control with the scalar controller, in [22] for systems with multiple commensurate lumped delays in the states and control with the multidimensional controller, in [23] for systems with multiple commensurate lumped and distributed delays in the states with the scalar controller, and in [24] for systems of neutral type with multiple commensurate lumped delays in the states with the scalar controller. In [25], the ACA problem was studied for single-input single-output (SISO) systems with commensurate lumped delays in the states by the dynamic output feedback controller. In [26], the problem of stabilization of linear systems with both input and state delays by observer-predictors was studied. In the paper [27], the assignment of the poles of a second-order vibrating system through state feedback with one lumped delay was studied by means of the linear matrix inequality (LMI) approach. A partial pole assignment approach was presented in [28] for second-order systems with time delay.
Some recent important works on stochastic time-delay systems have been reported in [29][30][31]. In [29], the event-triggered control problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control was studied. In [30], the stability problem for a class of stochastic delay nonlinear systems driven by G-Brownian motion was studied. The global stabilization of stochastic nonlinear systems with time-varying delay, unknown powers, and SISS stochastic inverse dynamics was studied in [31].
The problem of spectrum assignment by static output feedback (for systems without delays) is as follows: consider a linear time-invariant control systeṁ x ∈ K n , u ∈ K m , y ∈ K k . Let the controller have the form of linear static output feedback where u = Ky.
The closed-loop system (5), (6) takes the forṁ Definition 2. System (5) is called arbitrary spectrum assignable by means of static output feedback (6) if, for any γ i ∈ K, i = 1, n, there exists a gain matrix K such that the characteristic polynomial of the matrix F + GKH of the system (7) coincides with the polynomial (4).
The static output feedback problem of eigenvalue assignment (in particular, of stabilization) is one of the most important open questions in control theory; see reviews [32,33]. This problem has been studied for over 40 years by various authors. The most significant results have been obtained in [34] for K = C, in [35][36][37] for K = R.
The problems of stabilization and spectrum assignment by static output feedback for time-delay systems are more difficult to study. A study [38] considered the problem of stabilization of system (5) by static output feedback with a delay u(t) = Ky(t − τ). For SISO systems with delays, necessary conditions for the existence of static output feedback stabilizing controllers were derived in [39]. Another study [40] considered the output feedback stabilization problem for a class of linear SISO systems with I/O network delays. The problem of stabilization of linear time-varying systems with input delays via delayed static output feedback is studied in [41]. Consider a control system defined by a linear differential equation of nth order where the input is a linear combination of m variables and its derivatives of order ≤ n − p, and the output is a k-dimensional vector of linear combinations of the state x and its derivatives of order ≤ p − 1 (1 ≤ p ≤ n): x (n) + a 1 x (n−1) + . . . + a n x = = b p1 u (n−p) 1 Here, x ∈ K is a state variable, u α ∈ K are control variables, y β ∈ K are output variables, a i , b lα , c νβ ∈ K, i = 1, n, l = p, n, ν = 1, p, α = 1, m, β = 1, k. Construct the vectors u = col(u 1 , . . . , u m ) ∈ K m , y = col(y 1 , . . . , y k ) ∈ K k . Let the control in system (8), (9) have the form of linear static output feedback: System (8), (9), (10) can be rewritten in the form (5), (6). We say that system (8), (9) is arbitrary coefficient assignable by linear static output feedback (10) if for any γ i ∈ K, i = 1, n, there exists a linear static output feedback control (10) such that the characteristic polynomial of the closed-loop system (8), (9), (10) has the form (4). The conditions imposed on the orders of derivatives in (8) and (9) are natural because one needs the orders of the derivatives on the right-hand side of the closed-loop system to be less than n. For the scalar system (8), (9), (10), the arbitrary coefficient assignment problem by static output feedback has been solved in [42]. Construct the matrices B = {b lα }, l = 1, n, α = 1, m, and C = {c νβ }, ν = 1, n, β = 1, k, where b lα := 0 for l < p and c νβ := 0 for ν > p. Let J := {ϑ ij } ∈ M n (R) where ϑ ij = 1 for j = i + 1 and ϑ ij = 0 for j = i + 1. Let T denote the transposition of a matrix. The following theorem holds [42]: (9) is arbitrary coefficient assignable by linear static output feedback (10) if and only if the matrices C T B, C T JB, . . . , C T J n−1 B are linearly independent.
Extension of Theorem 1 to the case when the state, input, and output are multidimensional (i.e., x ∈ K s , u α ∈ K s , y β ∈ K s , s ≥ 1) was obtained in [43].
In the present paper, we extend Theorem 1 on arbitrary coefficient assignment by static output feedback to systems with non-commensurate lumped and distributed delays in the state variable. Notation.
a is the complex conjugation of a; T is the transposition of a vector or a matrix; * is the Hermitian conjugation, i.e., A * = A T ; Sp H is the trace of a matrix H ∈ M n (K); for a matrix H ∈ M n (K), we use the denotation denote by vec : M p,q (K) → K pq the mapping, which "unrolls" a matrix Z = {z ij }, i = 1, p, j = 1, q, by rows into the column vector vec Z = col (z 11 , . . . , z 1q , . . . , z p1 , . . . , z pq ) ∈ K pq .

Main Results
Consider a control system defined by a linear time-invariant differential equation of nth order with multiple non-commensurate lumped and distributed delays in the state variable x ∈ K; the input is a linear combination of m variables and its derivatives of order ≤ n − p; the output is a k-dimensional vector of linear combinations of the state x and its derivatives of order ≤ p − 1, . . , u m ) ∈ K m is a control vector and y = col (y 1 , . . . , y k ) ∈ K k is an output vector; p ∈ {1, n}; the complex conjugation to c νβ is used for convenience of notation (for consistency with previous works). Let the controller in system (11), (12) have the form of linear static output feedback with lumped and distributed delays: Hence, for any α = 1, m, The closed-loop system (11), (12), (13) takes the form Denote by ϕ(λ) the characteristic function of the closed-loop system (14). Then The set Λ = {λ ∈ C : ϕ(λ) = 0} is the spectrum of system (14). If Λ is contained in the open left half-plane, then system (14) is exponentially stable. The spectrum of system (14) is uniquely determined by the coefficients of system (14). We study the problem of assigning an arbitrary coefficients to the characteristic function (15) of the closed-loop system. Definition 3. System (11), (12) is said to be arbitrary coefficient assignable by static output feedback (13) if, for any integer ≥ 0, for any given 0 = ω 0 < ω 1 < . . . < ω , for any numbers γ iµ ∈ K, i = 1, n, µ = 0, , and for any integrable functions δ iξ : such that the characteristic function (15) of the closed-loop system (14) satisfies the equality Remark 1. The problem of arbitrary coefficient assignment was studied and solved in [44] for system (11), (12), (13) with only lumped delays (g iη (τ) ≡ 0, i = 1, n, η = 1, s; R κ (τ) ≡ 0 ∈ M m,k (K), κ = 1, θ); in [45] for systems with only one lumped and one distributed delays (s = 1, θ = 1, h 1 = σ 1 = h > 0); in [46] for systems with multiple commensurate delays (h j = jh, j = 0, s; σ ρ = ρh, ρ = 0, θ; h > 0). Here, we consider a more general case.
are linearly independent.
Suppose that the delays in system (11), (12) and in feedback (13) are commensurate, i.e., for some h > 0, In this case, the closed-loop system (14) contains only commensurate delays. Under conditions (34) and (35), the statements of the problem is as follows: system (11), (12) is said to be arbitrary coefficient assignable by static output feedback (13) if, for any integer ≥ 0, for any numbers γ iµ ∈ K, i = 1, n, µ = 0, , and for any integrable functions such that the characteristic function (15) of the closed-loop system (14) satisfies the equality (34) and (35), system (11), (12) is arbitrary coefficient assignable by linear static output feedback (13) if and only if the matrices (16) are linearly independent.

Remark 2.
Let us indicate to differences between systems with commensurate and non-commensurate delays. In systems with commensurate delays, one needs, for a given triplet to construct a triplet (36), while in systems with non-commensurate delays, one needs, for a given quadruple to construct a quadruple ensuring the equality from Definition 3. Thus, the problem statements are different. Corollary 4, for systems with commensurate delays, was proved in [46] (Theorem 1). Here we prove a more general result. Difference and difficulty here, with respect to [46], is in choosing the required numbers σ ρ . The proof given in [46] does not provide an algorithm for constructing the indicated numbers σ ρ and the corresponding gain coefficients Q ρ and R κ (·). Here, we overcome these difficulties. It provides the novelty of the results obtained.

Conclusions
Necessary and sufficient conditions were obtained for the problem of arbitrary coefficient assignment for the characteristic function of the closed-loop system by static output feedback for a linear differential equation with non-commensurate lumped and distributed delays. The obtained results extend the earlier corresponding results for systems with commensurate delays and for systems with only lumped delays. Corollaries on arbitrary finite spectrum assignment and on stabilization were stated. We provided an example illustrating our results. In future works, we expect to extend these results to control systems defined by non-scalar systems of differential equations. Moreover, this approach could be applied to problems of stabilization by static output feedback for linear quasi-differential equations and for nonlinear differential equations with delays based on linear approximation.