On local ISS of nonlinear second‐order time‐delay systems without damping

For a second‐order system with vector position, either constant or time‐varying delays, and a power nonlinearity of the degree higher than one, which does not contain a velocity‐proportional damping term, the conditions of local input‐to‐state stability are proposed. The result is based on application of Lyapunov‐Razumikhin approach, for which new time estimates on decay of solutions are obtained. The approach is extended to attitude stabilization of a rigid body, and it is illustrated by simulations.


INTRODUCTION
2][3][4] Indeed, any kind of networked communications usually leads to emergence of various lags, samplings or packet losses, then the time delays represent an efficient and common modeling framework. 5,6requently, the role of delays is considered to be negative since their parasitic inclusion in the loop implies the quality of transients degradation in dynamical systems. 7][10][11][12][13][14][15][16][17][18][19] A usual price is an accurate and sophisticated stability analysis of functional differential equations, which is more complex than in the delay-free counterparts.
A benchmark scenario, where introduction of a small delay ensures asymptotic stability for a neutrally stable planar system, was studied in Reference 1: ẍ(t) = −a 1 x(t) + a 2 x(t − ) with a 1 , a 2 ,  > 0 and x(t) ∈ R, and an extension of that result was obtained in the works, 20,21 where a second-order linear time-varying system is considered including a delayed position term with a negative gain, and it is proven that it can be stabilized by a position feedback with positive gain and with a sufficiently small delay.
In general, different kinds of delayed planar dynamics are omnipresent in modeling the mechanical or power systems controlled by or connected to a network providing communication or resources [22][23][24][25] (in this case the delays appear through the lags in control or measurement channels).The contribution of this paper consists in extension of the results of References 20,21 to a nonlinear case with external bounded perturbations.Using Lyapunov-Razumikhin approach we will demonstrate that in the nonlinear setting the local input-to-state stability (ISS) can be guaranteed for any value of the delay (under some mild restrictions) with the attraction domain dependent on the magnitude of delays and other parameters (only qualitative estimates will be given).The method will be extended to attitude regulation of rigid body.The Lyapunov-Razumikhin approach will be complemented by a rate of convergence estimation result.
The outline of this work is as follows.Preliminaries are given in Section 2. The considered analysis problem is described in Section 3. The main stability results are formulated in Section 4. Two illustrative examples are shown in Section 5. A conference version of this paper is restricted to disturbance-free analysis of the scalar systems. 26

Notation
The real numbers are denoted by R, R + = {s ∈ R ∶ s ≥ 0}, and |s| is an absolute value for s ∈ R. A norm for a vector x ∈ R n is defined as for any p ∈ [1, +∞), then ||x|| = ||x|| 2 is the usual Euclidean norm.For a matrix A ∈ R n×n its induced norm is denoted by ||A||; if it is symmetric, then  min (A) and  max (A) correspond to minimal and maximal eigenvalues, respectively, and A > 0 means that it is positive definite.For v ∈ R n , diag [v] denotes a diagonal matrix with the vector v on the main diagonal. We

Estimation of rates of convergence through Lyapunov-Razumikhin approach
Consider an autonomous functional differential equation of retarded type: 2 where , and is such that solutions in forward time for the system (1) exist and are unique. 2Denote such a unique solution satisfying the initial condition x 0 ∈ C([−, 0], R n ) by x(t, x 0 ), which is defined on time interval [−, T) with 0 < T ≤ +∞ (we will use the notation x(t) to reference x(t, x 0 ) if the origin of x 0 is clear from the context).For a locally Lipschitz continuous function V ∶ R n → R + the upper directional Dini derivative is defined as follows: ) is decreasing to zero for any fixed r > 0. For any  ∈  we can define   ∈ , verifying the properties   (s, 0) = s and   (  (s, t 1 ), t 2 ) =   (s, t 1 + t 2 ) for any s, t 1 , t 2 ∈ R + , such that y(t) ≤   (y(0), t) for all t ≥ 0 and y(0) ∈ R + , where y(t) satisfies ̇y(t) ≤ −(y(t)) for t ≥ 0.
Proof.Since all conditions of the Lyapunov-Razumikhin theorem 2 are satisfied, then the system (1) is asymptotically stable at the origin.Take any x 0 ∈ Ω C , the solution x(t, x 0 ) is well defined for all t ≥ 0, and in particular for all t ≥ 0. 31 First, assume that the implication given in the formulation of the theorem is not satisfied and max for an interval of time t ∈ [0, t 1 ], where t 1 ≥ 0 is (possibly infinite) instant of time.This inequality implies that for any t ∈ [0, t 1 ] there exists where we introduce the minimum over  ∈ [−, 0] to resolve the issue of existence of several time instants, where V(x t ()) reaches the maximum on the interval [−, 0].Note that the inequality  t ≤ − x 0 is satisfied for some  x 0 ∈ (0, ] dependent on initial conditions x 0 , since the maximum is calculated under the restriction that max ∈[−,0] V(x t ()) > (V(x(t))) with (s) > s > 0 and the solution x(t, x 0 ) is bounded for t ≥ 0. Note that in such a case there exists  ∈ (0, 1) such that  −1 (s) ≤ s for all s ∈ [0,  2 ()) ⊃ [0, max ∈[−,0] V(x 0 ())], thus, Recursively applying this estimate, that is, we obtain and by induction, for all t ∈ [0, t 1 ] (i.e., for t ≥ 0 sufficiently small it could be t +  t < 0 and the sum −( t +  t+ t + • • • ) above belongs to the interval [t, t + ]).Now, suppose that for t ∈ [t 1 , t 2 ) the implication max ∈[−,0] V(x t ()) ≤ (V(x(t))) holds, where t 2 ≥ t 1 is (possibly again infinite) the time instant that the relation stated in the theorem fails for the first time higher than t 1 (by their definitions, t 1 + t 2 > 0).By construction, in such a case for all t ∈ [t 1 , t 2 ) and, consequently, using standard comparison theorem result we obtain that Hence, due to properties of   , we substantiated that the estimate (2) is satisfied for all t ∈ [0, t 2 ).Next, the analysis above can be iterated for all t ≥ 0, and the estimate for x(t) can be derived using the bounds of V. Indeed, let us check by contradiction that (2) is valid for all t ≥ 0. Recall that V(x(t)) ≤ max ∈[−,0] V(x 0 ()) for all t ≥ 0 and let t 3 ≥ 0 be a time instant such that and ( 2) is valid for all t ∈ [0, t 3 ).These properties imply that max ∈[−,0] V ( x t 3 () ) ≤ (V(x(t 3 ))) and, consequently, D + V(x(t 3 ))f (x t 3 ) ≤ −(V(x(t 3 ))), which means that (2) cannot be violated since the argument functions of max{⋅} above are not intersecting, and the inequality (2) has also to be preserved at any such instant t 3 .▪ The result is formulated for the case of local asymptotic stability, and its modification for a global/practical/ISS analysis is straightforward.It can be interpreted as follows: the Razumikhin condition implicitly defines two rates of convergence presented in the system: first, the inverse of max ∈[−,0] V(()) ≤ (V((0))) guarantees at least exponential decreasing for V with the time constant − ln   , second, the dynamics D + V((0))f () ≤ −(V((0))) ensures another velocity of decay for V; finally, the established by the Lyapunov-Razumikhin approach convergence rate is the minimum of these two.

STATEMENT OF THE PROBLEM
In References 20,21, the problem of asymptotic stability for the scalar equation was studied, where x(t) ∈ R, a and b are constant positive coefficients, h and g are constant positive delays (all these parameters can also be time-varying).The system (3) has no damping proportional to the velocity ̇x ∈ R, and in the delay-free case (when h = g = 0) under the restriction a > b it has purely oscillating trajectories.It was proven that if the inequalities a > b, gb > ah are satisfied and the values of a − b and gb − ah are sufficiently small, then the system (3) is asymptotically stable.
In the present paper, we consider a nonlinear vector counterpart of (3): where are twice continuously differentiable and homogeneous of the order  + 1 for  > 1: g > 0 and h ≥ 0 are constant positive delays; d ∈  n ∞ is the exogenous disturbance.The instantaneous value of the state vector of ( 4) is , where  = max{h, g}.Assume that initial functions, x 0 and ̇x0 , for solutions of (4) belong to the space Under introduced restrictions, for  > 1 being a rational number with an odd numerator and denominator, taking ) we obtain a nonlinear counterpart of (3) (coinciding with (3) for n = 1 and  = 1).In the system (4), the term G(x(t−h)) x can be interpreted as a part of its own dynamics, while

𝜕Q(x(t−g)) 𝜕x
is the stabilizing term introduced by a control, which uses only delayed position measurements. 32In such a case it is necessary to select the shape of Q and g providing ISS for (4) (see References 4,33 for the basic definitions and results on ISS of time-delay systems).
In this work, we will formulate the conditions of local ISS for (4), that is, that there exist  ∈ ,  ∈  and  > 0 such that It is worth highlighting that our conditions are less conservative in the nonlinear case than those for (3).In addition, we will extend our result obtained for constant delays to time-varying ones, and next apply it to the rotation stabilization of rigid body using delayed feedback.Note that since  > 1, the local asymptotic stability of (4) cannot be derived from References 20,21 using the linearization techniques.

MAIN RESULTS
For the proof of the results below we will use the approaches proposed in References 32,34.

General case
Theorem 2. Let the function Π(x) = G(x) − Q(x) be positive definite, and the matrix be positive definite for every fixed x ≠ 0, then the system (4) is locally input-to-state stable.
Remark 1.Compared with the result of References 20,21, in Theorem 2 it is not assumed that the values of x 2 are sufficiently small for y ∈ R n with ||y|| = 1 (the analogues of smallness of a − b and gb − ah), while the delay h can be zero.
Proof.By adding and subtracting the delay-free terms G(x(t)) x and Q(x(t)) x , using the Mean Value Theorem, for  1 (t),  2 (t) ∈ (0, 1) n (for brevity of presentation we use the notation , and again by adding and subtracting the delay-free terms h  2 G(x(t)) x 2 ̇x(t), the system (4) can be represented in the form where ) .
Hence, in (5) the nominal part takes a form of the Liénard equation, which is asymptotically stable at the origin, [35][36][37] and the remaining terms d(t) + Δ(x t , ̇xt ) are considered as perturbations.For the rest of the proof, all computations are done for the case h > 0, and if h = 0, then the arguments stay unchanged by imposing the respective terms to be zero.Let us choose a Lyapunov function for (5) as follows: where  and  are positive coefficients,  ≥ 1,  ≥ 1 are powers to be properly selected.Note that the full energy E(x, ̇x) = 1 2 ̇x⊤ ̇x + Π(x) can be used to establish global asymptotic stability of the Liénard Equation ( 6), however, E is not a strict Lyapunov function in this case, then a variant of such a Lyapunov function V was proposed in Reference 37 being strict and guaranteeing asymptotic stability of the origin locally.Using properties of homogeneous functions (i.e., a 1 ||x|| +1 ≤ Π(x) ≤ a 2 ||x|| +1 for all x ∈ R n for some real a 1 ≤ a 2 , and Π(x) x is also a homogeneous vector function of degree ) we obtain where all a i , i = 1, … , 9 are positive constants (a 1 > 0 due to positive definiteness of Π).In what follows, the main derivations are done for the case d ≡ 0, since local input-to-state stability can be concluded from local asymptotic stability for functional differential equations. 38ith the aid of Lemma 1 and Young's inequality, we obtain that if and in the case where  = 2 +1 or  = +1 2 the values of parameters  or  are sufficiently small, respectively, there exist positive numbers a 10 , a 11 , D 1 such that for all ||x|| 2 + || ̇x|| 2 < D 1 .Using Lemma 2, it is straightforward to show that if then there exists a positive number D 2 such that for any  ∈ [t − 2, t], for some positive constants c 1 and c 2 .Therefore, we obtain and considering the first term, an upper bound can be derived: where as before  3ij (t) ∈ (0, 1), c 3 , c 4 > 0, and the Razumikhin condition was used on the last step.A similar estimate can be derived for another term: Applying Lemma 2, Young's and Jensen's inequalities, it can be shown that if the inequalities ( 9) are fulfilled and the value of D 3 is sufficiently small, then Finally, consider the terms . For the former one we obtain: where  4i (t) ∈ (0, 1) n , and applying the Razumikhin condition as above we get an upper bound: ) .
2 in the sense of Reference 39 where it has been proven that if a homogeneous system of positive degree is locally asymptotically stable in a given vicinity of the origin for any  > 0, then it is globally asymptotically stable independently of the delay value.In Theorem 2, due to (12) the estimate on the domain, where V(x(t), ̇x(t)) ≥ 0 and V(t) ≤ 0, determined by D 3 is inversely proportional to  (depending as well on other parameters), hence, the global result from Reference 39 cannot be applied.In addition, the delay free model (4), has purely oscillating trajectories, therefore, it does not confirm the asymptotically stable behavior of the system for vanishing delays, whose appearance changes qualitatively the kind of stability in (4).Remark 2. In the proof of Theorem 2 many positive constants are introduced (a i for i = 1 … 11, c j for j = 1 … 9 and D k for k = 1, 2, 3) without providing estimations of their values.The reason is that often, in nonlinear setting, such estimations are very conservative, and only qualitative bounds can be established indicating the kind of behavior presented in the system.To illustrate this claim, let us consider how the constants a 10 and a 11 can be evaluated.Therefore, we need to show, for example, that 7) and a proper choice of D 1 , then a 10 = min{ 1 4 , a 1 2 }.The inequalities above are implied by whose validity follows from Lemma 1 provided that 1 +1 +  2 > 1 and  +1 + 1 2 > 1, respectively, which are verified if (7) holds with strict signs of inequalities.If (7) takes the form of equalities, then using Young's inequality we can show that the desired relations hold for sufficiently small  and .To derive the upper bounds in (8), the Young's inequality can be applied with posterior utilization of ( 7) and smallness of D 1 : Characterizing the input-to-state stability property, the shape of the asymptotic gain of the system, , has been found in the proof of Theorem 2, while evaluation of a qualitative rate of convergence follows from the result of Theorem 1: Corollary 1.Let the all conditions of Theorem 2 be satisfied, then Proof.Since  > 1 and  > 1 due to (9), then q > 1, and the dynamics V ≤ −c 9 V q has a slower convergence than any exponential close to the origin, which gives the estimate of the corollary.▪ The obtained result can be extended to the time-varying case: with continuous and bounded for t ≥ 0 delays h(t) ≥ 0 and g(t) > 0 with  = sup t≥0 {h(t), g(t)}.
Corollary 2. Let the function Π(x) = G(x) − Q(x) be positive definite and the there exist R > 0 such that for all x ∈ R n and t ≥ 0, then the system (13) is locally input-to-state stable.
Proof.The proof repeats all steps of Theorem 2 (application of the Lyapunov-Razumikhin approach is not much influenced by dependence of the delays on time).▪

Application to the problem of the attitude stabilization of a rigid body
In this subsection, we will extend the developed approach to the design of a control torque ensuring the triaxial stabilization of a rigid body.Consider a rigid body that rotates around its mass center O with an angular velocity  ∈ R 3 .Let Oxyz be the principal central axes of inertia of the body.The attitude motion of the body under the action of a control torque M is modeled by the Euler equations (for vectors u, w ∈ R 3 , u × v denotes its vector product) where diagonal matrix 0 < J ∈ R 3×3 is inertia tensor of the body in the axes Oxyz, 40 d(t) ∈ R 3 is the perturbation.Let two right triples of mutually orthogonal unit vectors s 1 , s 2 , s 3 and r 1 , r 2 , r 3 be given.Vectors s 1 , s 2 , s 3 are fixed in the inertial frame, whereas vectors r 1 , r 2 , r 3 are fixed in the frame connected with the body.Hence, vectors s 1 , s 2 , s 3 rotate with respect to the system Oxyz with the angular velocity −, and we arrive at the Poisson kinematic equations Our objective is the robust triaxial stabilization of the body.This means that we should design a control torque M for which the system ( 14), (15) admits the asymptotically stable equilibrium position for d(t) = 0: It is known (see References 41,42) that, to solve the stated problem, we can use the torque of the form M = D() + F(s 1 , s 2 ), where the dissipative component D ∶ R 3 → R 3 is a continuous vector function such that the product  ⊤ D() is negative definite, while the restoring component F(s 1 , s 2 ) is defined as follows: a 1 , a 2 are positive constants, and  ≥ 0. However, it is worth noticing that creating damping forces and constructing special damping devices for practical mechanical systems may be difficult problems, especially for the attitude stabilization of satellites due to the limited resources of reactive control systems. 40,43,44To overcome this difficulty, in Reference 42, it was proposed to use attitude control systems in which dissipative torques tend to zero as time increases.On the over hand, the approach developed in the present paper permits us to provide the triaxial stabilization with the aid of a control that does not contain a dissipative component at all.
Denote F(t) = F(s 1 (t), s 2 (t)) and select M(t) = c 1 F(t) − c 2 F(t − ), where c 1 , c 2 are positive constants,  is a positive delay.Then the Euler equations ( 14) take a form corresponding to (4) with h = 0 and g = , but containing an additional rotation torque in the left-hand side with a more complex shape of the potential terms and having two equilibria (( 16) and its inverse): Theorem 3. Let c 1 > c 2 > 0 and  > 0. Then the equilibrium position (16) of the system (15), ( 17) is locally input-to-state stable.
Let us note that where  5 ,  6 are positive constants.
The remaining part of the proof is similar to that of Theorem 2. ▪ For brevity, only one delay is considered in ( 17), but the result can be easily extended to the case of two delays in the restoring term.

EXAMPLES
In this section we restrict ourselves by n = 2 and  = 3.For an illustration of the result of Theorem 2, consider the case with h = 0, g = 3, and In the case g = 0 (or b = 0) the system (4) takes the form of a nonlinear mechanical oscillator.The appearance of delayed part for g ≠ 0 can be interpreted as a stabilizing control that uses delayed position measurements.By Theorem 2, the system is locally ISS.The results of simulation of x(t) for the initial conditions x(s) = [−0.In order to evaluate the dependence of the domain of convergence on the value of the delay , the simulations are performed for h = 0 and g ∈ {1, 3, 9, 27} with initial conditions x(s) = [1 1] ⊤ and ̇x(s) = [0 0] ⊤ for all s ∈ [−, 0], where the parameter  > 0 was tuned to obtain the maximal value providing the convergent trajectories in the disturbance-free case.The behavior of the norm of the state for different values of delays and the respective values of  is presented in Figure 2. As we can conclude, in these simulations there is an approximately linear inverse dependence of  in , which is aligned with the theoretical observations given in the previous section.Selecting h(t) = 0.2(1 + cos(0.5t)),g(t) = 1 + sin 2 (10t) and keeping the values of all other parameters unchanged, the conditions of Corollary 2 are verified.The respective results of simulations for the same initial conditions and disturbances as in Figure 1 are given in Figure 3.As we can remark, the time variation of delays does not destroy stability under the proposed restrictions.These results confirm the conclusions of the theorems that the systems are locally input-to-state stable under introduced mild restrictions (for bigger initial conditions or amplitude of the input, the trajectories become unbounded).

CONCLUSION
For a second-order vector system with either constant or time-varying delays, smooth power nonlinearity and a bounded external disturbance, without a velocity damping term, the conditions of local input-to-state stability were formulated.These conditions are simple for checking.It is seen from the proof and the examples that the domain of attraction depends on the value of delays and parameters, but there is no restriction on admissible maximal value of delays.The proposed method was developed for attitude stabilization of a rigid body.New estimates on the decreasing of solutions were derived for the Lyapunov-Razumikhin approach.Development of these results to the case with the power  ∈ (0, 1) can be considered as a direction for future research.

F I G U R E 1
The illustration for Theorem 2. (A) Noise-free case.(B) The disturbed case.

F I G U R E 2
The illustration of dependence of the domain of convergence for different values of delay.F I G U R E 3 The illustration for Corollary 2. (A) Noise-free case.(B) The disturbed case.