Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefﬁcients by Linear Stationary Feedback

: We consider a control system deﬁned by a linear time-varying differential equation of n -th order with uncertain bounded coefﬁcients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefﬁcients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufﬁcient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufﬁcient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefﬁcients by linear time-invariant static output feedback. Illustrative examples are considered.


Introduction
Consider a control system defined by an ordinary differential equation with time-varying coefficients of n-th order x (n) + p 1 (t)x (n−1) + . . . + p n (t)x = u, where x ∈ R is the state variable, u ∈ R is the control input, t ∈ R + := [0, +∞). We suppose that the functions p i (t) are measurable but exact values of these functions at time moments t are unknown, we know only that the functions are bounded on R + and lower and upper bounds (α i and β i ) are known: Functions p i (t) can be arbitrary, in particular, they can vary fast or slowly. Denote x = (x,ẋ, . . . , x (n−1) ). We consider a problem of feedback stabilization for system (1). One needs to construct a function u(t, x), u(t, 0) = 0, such that, for system (1) closed-loop by u = u(t, x), the zero solution is exponentially stable and has a given decay rate. The stated problem essentially relates to the problems of robust stabilization.
Let us assume that p i (t) are time-invariant (and hence, are known), i.e., p i (t) ≡ p i (= α i = β i ). In that case, the stabilization problem is trivial. In fact, we construct where φ i ∈ R, i = 1, n, are chosen such that the polynomial is stable (i.e., Re λ j < −θ < 0 for all roots λ j , j = 1, n, of (4)). Then system (1) closed-loop by the control has the form x (n) + φ 1 x (n−1) + . . . + φ n x = 0, (6) and the zero (and hence, every) solution of (6) is exponentially stable. Now, assume that p i (t) are time-varying. Then we can not construct the control by using (3) because p i (t) are unknown. Let the feedback control law have the form (5), where v i are constant. The closed-loop system has the form We study the following problem: construct constants v 1 , . . . , v n ∈ R such that all solutions of (7) are exponentially stable with a given decay of rate. This problem is non-trivial due to the following reasons. For studying this problem, we need use some sufficient conditions for exponential stability of linear time-varying systems. The problem of obtaining some sufficient conditions for (asymptotic, exponential) stability of linear time-varying systemṡ is one of the important and difficult problems in the theory of differential equations and control theory [1]. In contrast to systems with constant coefficients (A(t) ≡ A), the condition Re λ j < 0, j = 1, n, fulfilled for the eigenvalues of the matrix of the system (8) is neither a sufficient nor a necessary condition for the asymptotic stability of the system (8) (see, e.g., [2], ( [3], § 9)). Some sufficient conditions for asymptotic and exponential stability of linear time-varying systems (8) and linear time-varying differential equations were obtained in [1][2][3][4][5][6][7][8][9][10][11]. The following theorem take place.
Theorem 1. Suppose the functions q i (t) are measurable and bounded on R + and the following inequalities hold: Let the polynomial have only real roots. Then all solutions of (9) are exponentially tends to 0 as t → +∞.
Uncertain systems (13), (14) were studied in [34][35][36][37] and in other works of A.H. Gelig and I.E. Zuber. In particular, it follows from results of [34] that system (13) is exponentially stabilizable by feedback control (14). This result is supplemented and developed in this paper. The difference between this result and the results obtained in the work is as follows. Firstly, we achieve exponential stabilization of (7) not only with some decay rate as it follows from [34] but with an arbitrary pregiven decay rate. Secondly, in contrast to [34], which uses the Second Lyapunov Method (Method of Lyapunov Function), we apply, in some sense, the First Lyapunov Method (which uses the roots of characteristic polynomial) and non-oscillation theory. Thirdly, we extend these stabilization results to systems with static output feedback control.
In this work, using Theorem 1, we prove results on exponential stabilization with any pregiven decay rate by linear stationary static state or output feedback for a control system defined by a linear time-varying differential equation of the n-th order with uncertain coefficients.
Proof. At first, suppose that the theorem is proved for any η ≥ 1. Let us construct, for η = 1, the polynomials (15), (16) providing properties (i), (ii), (iii), and denote them by f 1 (λ), g 1 (λ). Now, let η ∈ (0, 1). Then, let us set f (λ) := f 1 (λ), g(λ) := g 1 (λ). Hence, conditions (i), (ii) are satisfied. Since −η > −1, condition (iii) holds as well. Thus, without loss of generality, one can assume that η ≥ 1. Let us give the proof by induction on n. The statements that we have to prove are different for odd and even numbers n: for even n, we need to ensure inequalities (17), in addition to (i) and (ii), and for odd n, we need to ensure inequalities (18). Therefore, the induction base as well as the induction hypothesis and the induction step should depend on whether the number n is even or odd. That is why we should check the induction base for n = 1 and n = 2.
Let us put forward the induction hypothesis. Suppose that the assertion of the theorem is true for n = k. Then, let us prove that the assertion of the theorem is true for n = k + 1. We will carry out the induction step for even and odd k separately.
By the induction hypothesis, there exist polynomials such that Let us prove that there exist polynomials such that We assume that δ 0 := 1, γ 0 := 1. Set for the case if k = 2 , and for the case if k = 2 + 1. Then They intersect at the point M 0 (x 0 , y 0 ) with the coordinates The set Ω 0 is a cone, with a vertex at the point M 0 , located in the first quadrant of the xOy-plane and bounded by half-lines (36) where The solution of system (37) is the set Then condition (31) is satisfied. Next, since x > a k , it follows that Next, since ( x, y) is a solution of (37), we have Thus, it follows from inequalities (41), (42), equalities (38) and induction hypothesis (26), (27) that inequalities (32) are satisfied if k = 2 , and inequalities (33) are satisfied if k = 2 + 1.
Let us prove inequalities (30). From the definition (40) of the polynomials F(λ), G(λ) and equalities (38), (25) we obtain that Substituting (22), (23) and (28), (29) into (43) and opening the brackets, we obtain equalities for the case if k = 2 , and equalities for the case if k = 2 + 1. The inequalities Γ i > 0, i = 1, k + 1, are satisfied due to inequalities (24) and are equivalent to the inequality system for the case if k = 2 , and are equivalent to the inequality system for the case if k = 2 + 1. System (45) is equivalent to the inequality system System (46) is equivalent to the inequality system For the case if k = 2 , the following inequalities hold: For the case if k = 2 + 1, the following inequalities hold: Thus, it follows from definitions (34), (35) that to satisfy inequalities (47) (for the case if k = 2 ) and inequalities (48) (for the case if k = 2 + 1) it is sufficient to satisfy inequalities By (39), inequalities (49) hold because ( x, y) ∈ Ω 0 . Therefore, inequalities (44) are satisfied. Hence, (30) are satisfied. Thus, the induction step is proved. The theorem is proved.

Time-Invariant Stabilization by Static Output Feedback
Consider a linear control system defined by a linear differential equation of n-th order with time-varying uncertain coefficients satisfying (2); the input is a stationary linear combination of m variables and their derivatives of order ≤ n − p; the output is a k-dimensional vector of stationary linear combinations of the state x and its derivatives of order ≤ p − 1: w = col(w 1 , . . . , w m ) ∈ R m is an input vector; y = col(y 1 , . . . , y k ) ∈ R k is an output vector. Let the control in (64), (65) have the form of linear static output feedback We suppose that the gain matrix U is time-invariant. The closed-loop system has the form where the coefficients q i (t) of (67) depends on p i (t), b lτ , c νj , U. On the basis of system (64), (65), we construct the n × m-matrix B = {b lτ }, l = 1, n, τ = 1, m, and the n × k-matrix C = {c νj }, ν = 1, n, j = 1, k, where b lτ = 0 for l < p and c νj = 0 for ν > p. Denote by J the matrix whose entries of the first superdiagonal are equal to unity and whose remaining entries are zero; we set J 0 := I. By Sp Q denote the trace of a matrix Q.
Definition 2. We say that system (64), (65) is exponentially stabilizable with the decay rate θ > 0 by linear stationary static output feedback (66) if there exists a constant m × k-matrix U such that every solution x(t) of the closed-loop system (67) is exponentially stable with the decay rate θ.
Theorem 4. Suppose that linear stationary output feedback (66) bring system (64), (65) to the closed system (67). Then the coefficients q i (t), i = 1, n, of (67) satisfy the equalities The proof of Theorem 4 is identical to the proof of Theorem 1 [38]. Let us introduce the mapping vec that unwraps an n × m-matrix H = {h ij } row-by-row into the column vector vec H = col (h 11 , h 12 , . . . , h 1m , . . . , h n1 , . . . , h nm ). For any k × m-matrices X, Y, the obvious equality holds: Denote r = col (r 1 , . . . , r n ) ∈ R n , ψ = vec (U T ). Equalities (68) represent a linear system of n equations with respect to the coefficients of the matrix U. Taking into account (69), one can rewrite system (68) in the form P T ψ = r.
Suppose that matrices (70) are linearly independent. Then rank P = n. Hence, the system of linear equations (71) is solvable for any vector r ∈ R n . In particular, system (71) has the solution ψ = P(P T P) −1 r.
Next, let us construct matrices (70) and P. We obtain P = . Obviously, rank P = 3 and matrices (70) are linearly independent. Resolving system (71) where r i has the form (75), we obtain Thus, the gain matrix has the form We obtain that feedback (66) with the matrix (81) exponentially stabilizes the system (72), (73) with the decay rate θ.
System (84) is exponentially stable with the decay rate θ = 1. Some graphs of the solutions to system (84) are shown in Figure 2.

Conclusions
We examined the problem of exponential stabilization with any pregiven decay rate for a linear time-varying differential equations with uncertain bounded coefficients by means of stationary linear static feedback. We have received sufficient conditions for the solvability of this problem by state and output feedback. For this purpose, the first Lyapunov method and the Levin theorem on non-oscillatory absolute stability were used. We plan to extend these results to systems of differential equation including systems with delays. A further development of these results may be their extension to systems