Alexander Aleksandrov

Denis Efimov

On stability of mechanical systems with homogeneous and delayed forces

Keywords: Mechanical systems, Homogeneous systems, Time-delay systems

The problem of delay-independent stability is studied for a class of mechanical systems under dissipative, non-conservative and potential forces, which are described by homogeneous terms. The obtained conditions are extended to the case of switched force model under arbitrary and restricted commutation laws. The proof is based on analysis of a complete Lyapunov-Krasovskii functional (common and multiple ones are considered for switched scenario). The proposed stability conditions are illustrated by simulation examples.

Introduction

The analysis and control design problems for time-delay mechanical systems are challenging and practically important [START_REF] Arajo | Special issue on control of second-order vibrating systems with time delay[END_REF][START_REF] Hu | Dynamics of controlled mechanical systems with delayed feedback[END_REF]. Appearance of delay may be related with distributed nature of the process, or coming from distant location of actuators and sensors, as it is usually the case in haptic devices controlled through a communication network [START_REF] Gil | Stability Boundary for Haptic Rendering: Influence of Damping and Delay[END_REF]. Frequently, linear time-delay models of mechanical systems are used [START_REF] Keller | Time-delay systems with application to mechanical engineering process dynamics and control[END_REF] due to a relative simplicity of their analysis, but in many situations the nonlinearity of forces has to be taken into account, e.g., in delayed Duffing equation [START_REF] Rusinek | Dynamics of a time delayed Duffing oscillator[END_REF], where delay appears to represent a regenerative force effect, or in magnetic bearing systems [START_REF] Post | Stability issues in ambient temperature passive magnetic bearing systems[END_REF].

In general, stability investigation of time-delay systems is a rather sophisticated issue [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF][START_REF] Hale | Theory of functional differential equations[END_REF][START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF], and for nonlinear systems there is no structured way to select a Lyapunov-Krasovskii functional for analysis. Exceptions exist for particular classes of systems, as it is demonstrated in a recent work for homogeneous time-delay dynamics in [START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF].

For this class systems the independent on delay kind of stability is recurrent [START_REF] Efimov | Weighted homogeneity for time-delay systems: Finite-time and independent of delay stability[END_REF].

Considering switching systems is another popular extension for modeling of complex mechanical processes, where commutation may come from dependence of the force shape on the coordinates, from time variation of parameters, from impacts or reconfiguration of multi-component plants. Utility of this modeling frameworks is originated by a well-developed theory of stability analysis for switched systems [START_REF] Liberzon | Switching in systems and control[END_REF]. Usually, a common Lyapunov-Krasovskii functional allows the stability conditions to be obtained under arbitrary switching, while the multiple Lyapunov-Krasovskii functional approach is coupled with (dwell-time) restrictions on the commutation sequence [START_REF] Aleksandrov | Stability of hybrid mechanical systems with switching linear force fields[END_REF]Aleksandrov, Kosov, & Chen, 2011;[START_REF] Aleksandrov | On the asymptotic stability of switched homogeneous systems[END_REF][START_REF] Liberzon | Switching in systems and control[END_REF].

The topic of this work lies on the intersection of all above mentioned issues: delayindependent stability analysis for a class of (switched) mechanical systems with nonlinear homogeneous forces. The results are based on investigation of a complete Lyapunov-Krasovskii functional. The derived applicability conditions are mild and easily verifiable, they are only related with the form of the forces. Contrarily the conventional results obtained for linear models, where delay may destroy the system stability [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF][START_REF] Keller | Time-delay systems with application to mechanical engineering process dynamics and control[END_REF], it is shown in the present work that for a strongly nonlinear systems the delay impact is only quantitative (i.e., appearance of a lag cannot lead to instability, but it may reduce the domain of convergence and attenuate the decay rates).

The outline of the paper is as follows. Preliminaries are surveyed in Section 2. The detailed problem statement and basic hypotheses are presented in Section 3. Sections 4 and 5 are devoted to the construction of Lyapunov-Krasovskii functional for the cases without and with commutation, respectively. Illustrative examples with the simulation results are shown in Section 6. Final conclusions are summarized in Section 7.

Preliminaries

The set of real numbers is denoted by R, R + = {s ∈ R : s ≥ 0}, R n and R n×n are used to denote the space of vectors and matrices of dimensions n and n × n, respectively. The Euclidean norm of a vector x ∈ R n is denoted by x , and |s| is the absolute value for a scalar s ∈ R. Denote by C 1 ([-τ, 0], R n ) for τ > 0 the space of continuously differentiable functions ϕ(ξ) : [-τ, 0] → R n with the uniform norm

ϕ τ = max ξ∈[-τ,0] ( ϕ(ξ) + φ(ξ) ) .
The Young's inequality claims that for any a, b ∈ R + and γ > 0, δ > 0 [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]:

a γ b δ ≤ 1 p a γp + p -1 p b δp p-1
for any p > 1, while Hölder's inequality for any f, g : I → R with I ⊂ R ensures that 2 [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]:

I |f (s)g(s)|ds ≤ I |f (s)| p ds 1 p I |g(s)| p p-1 ds p-1 p
for any p > 1.

Using the properties of homogeneous functions the following result can be obtained [START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF]:

Lemma 1. Let a, b ∈ R + and > 0, α > 0, β > 0, γ > 0, δ > 0 be given, then

a α + b β -a γ b δ ≥ 0 provided that 1) max{a α , b β } ≥ 1 1- γ α -δ β and γ α + δ β < 1, 2) max{a α , b β } ≤ 1 1- γ α -δ β and γ α + δ β > 1.
For definitions of various conventional stability concepts used in the paper the reader is referenced to [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF][START_REF] Hale | Theory of functional differential equations[END_REF][START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF].

Statement of the Problem

Consider a mechanical system whose dynamics are defined for t ≥ t 0 ≥ 0 by the equation

Aẍ(t) + B ẋ(t) + ∂Π 1 (x(t)) ∂x + ∂Π 2 (x(t -τ )) ∂x + F (t, x) = 0, (1)
where x(t), ẋ(t) ∈ R n are vectors of generalized coordinates and velocities, respectively; A, B ∈ R n×n are constant positive definite matrices; Π 1 , Π 2 : R n → R are continuously differentiable and homogeneous (with respect to the standard dilation) functions of the order µ + 1, µ > 0, i.e., Π i (λx) = λ µ+1 Π i (x), ∀x ∈ R n , ∀λ > 0, i = 1, 2;

the vector function F (t, x) is continuous for t ≥ 0 and x ∈ R n ; τ is a constant positive delay. Assume that initial functions for solutions of (1) belong to the space C 1 ([-τ, 0], R n ). Let x t be the restriction of a solution x(t) of (1) to the segment [t-τ, t], i.e., x t : ξ → x(t + ξ), ξ ∈ [-τ, 0]. We will impose the following constraints on the force fields in (1):

Assumption 1. The function Π(x) = Π 1 (x) + Π 2 (x) is positive definite.
Note that Π can be interpreted as a potential energy of (1).

Assumption 2. The force -F (t, x) is non-conservative, i.e., x F (t, x) = 0 for all t ≥ 0, x ∈ R n .

Hence, the considered mechanical system is under the action of dissipative, potential and non-conservative forces, and the term ∂Π 2 (x(t -τ ))/∂x can be interpreted as a control with a constant delay in the feedback loop. The positional forces in (1) have potential and non-conservative components, which corresponds to the general scenario [START_REF] Zubov | Problems of mechanics of a solid deformable body[END_REF]. Note also that Assumption 1 includes the cases when in the absence of the delayed part (without control) the system (1) is unstable.

Remark 1. Assumption 2 implies that F (t, 0) = 0, see [START_REF] Rouche | Stability Theory by Liapunov's Direct Method[END_REF]. Hence, the system (1) admits the steady-state solution

x(t) = ẋ(t) = 0, t ≥ 0.
(2)

We will look for conditions ensuring the delay-independent asymptotic stability of the equilibrium position (2) or delay-independent ultimate boundedness for solutions of (1).

It is worth noting that asymptotic stability conditions for the corresponding delayfree system (τ = 0) were obtained in (Aleksandrov, Chen, et al., 2011;Aleksandrov, Kosov, & Chen, 2011) with the aid of a special construction of a strong Lyapunov function. On the other hand, in [START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous diffrential-difference systems[END_REF][START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF], an approach to the construction of a complete-type Lyapunov-Krasovskii functional for some classes of nonlinear time-delay systems was developed. In the present contribution, we will apply the results of (Aleksandrov, Chen, et al., 2011;Aleksandrov, Kosov, & Chen, 2011;[START_REF] Aleksandrov | Complete type functionals for some classes of homogeneous diffrential-difference systems[END_REF][START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF] to the analysis of dynamics of the system (1).

Constructing a Complete-Type Lyapunov-Krasovskii Functional

We will consider the cases where the vector of non-conservative forces satisfies one of the following conditions: Assumption 3. There exist numbers ∆ > 0, c > 0 and λ > (µ + 1)/2 such that F (t, x) ≤ c x λ for all t ≥ 0, x < ∆.

Assumption 4. There exist numbers ∆ > 0, c > 0, d > 0 and λ < (µ+1)/2 such that F (t, x) ≤ c x λ for all t ≥ 0, x > ∆ and F (t, x) ≤ d for all t ≥ 0, x ≤ ∆.

The difference between assumptions 3 and 4 consists in the domain, where the potential forces are related with the non-conservative one: close to the origin or far enough, respectively. Examples of such nonlinear forces satisfying assumptions 1-4 can be found in [START_REF] Blekhman | Vibrational mechanics -a general approach to solving nonlinear problems[END_REF][START_REF] Burton | On the multi-scale analysis of strongly non-linear forced oscillators[END_REF][START_REF] Gendelman | Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium[END_REF][START_REF] Rakaric | Approximations for motion of the oscillators with a nonnegative real-power restoring force[END_REF].

Theorem 1. Let assumptions 1 and 2 be fulfilled and consider (1) with any τ > 0:

• if µ > 1 and Assumption 3 is valid, then the equilibrium position (2) is asymptotically stable;

• if µ < 1 and Assumption 4 is valid, then solutions of (1) are uniformly ultimately bounded.

Proof. First, using results of (Aleksandrov, Chen, et al., 2011;Aleksandrov, Kosov, & Chen, 2011), choose a Lyapunov function candidate as follows:

V (x, ẋ) = α 2 ẋ A ẋ + 1 2 x Bx + x A ẋ,
where α is a positive parameter. There exist positive numbers α 0 , a 1 , a 2 , a 3 , a 4 such that

αa 1 ẋ 2 + a 2 x 2 ≤ V (x, ẋ) ≤ αa 3 ẋ 2 + a 4 x 2
for α ≥ α 0 and all x, ẋ ∈ R n .

Considering the derivative of V along the solutions of the system (1), we obtain:

V = -α ẋ (t)B ẋ(t) + ẋ (t)A ẋ(t) -α ẋ (t) ∂Π 1 (x(t)) ∂x + ∂Π 2 (x(t -τ )) ∂x -α ẋ (t)F (t, x(t)) -x (t) ∂Π(x(t)) ∂x + x (t) ∂Π 2 (x(t)) ∂x - ∂Π 2 (x(t -τ )) ∂x ≤ -(αa 5 -a 6 ) ẋ(t) 2 + αa 7 ẋ(t) ( x(t) µ + x(t -τ ) µ ) +α ẋ(t) F (t, x(t)) -a 8 x(t) µ+1 + x (t) ∂Π 2 (x(t)) ∂x - ∂Π 2 (x(t -τ )) ∂x ,
where a 5 , a 6 , a 7 , a 8 are positive constants, and the Euler theorem on homogeneous functions was used: x ∂Π(x) ∂x = (µ + 1)Π(x). Next, according to the approach developed in [START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF], construct a Lyapunov-Krasovskii functional by the formula

V (x t ) = V (x(t), ẋ(t)) -x (t) t t-τ ∂Π 2 (x(ξ)) ∂x dξ (3) + t t-τ (β + γ(ξ -t + τ )) x(ξ) µ+1 dξ,
where β and γ are positive parameters. If α ≥ α 0 , then

αa 1 ẋ(t) 2 + a 2 x(t) 2 + β t t-τ x(ξ) µ+1 dξ -a 9 x(t) t t-τ x(ξ) µ dξ ≤ V (x t ) ≤ αa 3 ẋ(t) 2 + a 4 x(t) 2 + (β + γτ ) t t-τ x(ξ) µ+1 dξ + a 9 x(t) t t-τ x(ξ) µ dξ,
where a 9 > 0. Differentiating the functional (3) along the solutions of (1), we obtain

˙ V ≤ -(αa 5 -a 6 ) ẋ(t) 2 + αa 7 ẋ(t) ( x(t) µ + x(t -τ ) µ ) +α ẋ(t) F (t, x(t)) -(a 8 -β -γτ ) x(t) µ+1 -β x(t -τ ) µ+1 +a 9 ẋ(t) t t-τ x(ξ) µ dξ -γ t t-τ
x(ξ) µ+1 dξ.

Let α 0 ≥ 2a 6 /a 5 and 2(β + γτ ) < a 8 , with the aid of Hlder's inequality, we arrive at the estimates:

αa 1 ẋ 2 + a 2 x 2 + β t t-τ x(ξ) µ+1 dξ -a 10 x(t) t t-τ x(ξ) µ+1 dξ µ µ+1 ≤ V (x t ) ≤ αa 3 ẋ 2 + a 4 x 2 +(β + γτ ) t t-τ x(ξ) µ+1 dξ + a 10 x(t) t t-τ x(ξ) µ+1 dξ µ µ+1 and ˙ V ≤ - 1 2 αa 5 ẋ(t) 2 - 1 2 a 8 x(t) µ+1 + αa 7 ẋ(t) ( x(t) µ + x(t -τ ) µ ) +α ẋ(t) F (t, x(t)) -β x(t -τ ) µ+1 +a 10 ẋ(t) t t-τ x(ξ) µ+1 dξ µ µ+1 -γ t t-τ
x(ξ) µ+1 dξ for α ≥ α 0 , where a 10 is another positive constant.

Using Lemma 1 and Young's inequality and taking into account assumptions 3 and 4, it is easy to show the existence of positive numbers δ 1 , δ 2 such that

• if µ > 1, then αa 1 ẋ(t) 2 + 1 2 a 2 x(t) 2 + 1 2 β t t-τ x(ξ) µ+1 dξ ≤ V (x t ) (4) ≤ αa 3 ẋ(t) 2 + 2a 4 x(t) 2 + 2(β + γτ ) t t-τ x(ξ) µ+1 dξ, ˙ V ≤ - 1 4 αa 5 ẋ(t) 2 - 1 4 a 8 x(t) µ+1 - 1 2 γ t t-τ x(ξ) µ+1 dξ (5) for ẋ(t) 2 + x(t) 2 + t t-τ x(ξ) µ+1 dξ < δ 1 ; • if µ < 1, then the inequalities (4), (5) hold for ẋ(t) 2 + x(t) 2 + t t-τ x(ξ) µ+1 dξ > δ 2 .
Hence (see [START_REF] Efimov | Analysis of robustness of homogeneous systems with time delays using Lyapunov-Krasovskii functionals[END_REF]), in such a case

˙ V ≤ -h 1 V µ+1 2 (6)
for µ > 1 in the domain where V < δ 3 , and

˙ V ≤ -h 2 V µ+1 2 (7)
for µ < 1 in the domain where V > δ 4 for some positive numbers h 1 , h 2 , δ 3 , δ 4 . This completes the proof.

Remark 2. There are dissipative and potential stabilizing forces in (1). Theorem 1 guarantees the asymptotic stability of the equilibrium position (2) or ultimate boundedness of solutions for any positive delay and any non-conservative force satisfying Assumption 3 or Assumption 4, respectively. It is important to highlight that, contrarily the existing results on the stability and ultimate boundedness by homogeneous approximations (see [START_REF] Zubov | Mathematical methods for the study of automatical control systems[END_REF]), if µ > 1 (µ < 1), then the order of non-conservative forces may be less (greater) than that of the potential forces. Thus, Theorem 1 complements the analysis based on the first-order approximation.

With the aid of the differential inequalities ( 6) and ( 7), estimates on the convergence rates for solutions of the system (1) can be derived.

Corollary 1. Let assumptions 1 and 2 be fulfilled,

• if µ > 1, Assumption 3 is valid and for a solution x(t) of (1) the inequalities t 0 ≥ 0, V (x t0 ) < δ 3 hold, then

αa 1 ẋ(t) 2 + 1 2 a 2 x(t) 2 ≤ V (x t0 )(1 + 1 2 h 1 (µ -1) V µ-1 2 (x t0 )(t -t 0 )) -2 µ-1
for t ≥ t 0 ;

• if µ < 1, Assumption 4 is valid and for a solution x(t) of (1) the inequalities t 0 ≥ 0, V (x t0 ) > δ 4 hold, then

αa 1 ẋ(t) 2 + 1 2 a 2 x(t) 2 ≤ δ 4
for t ≥ t 0 + T , where

T = 2 h 2 (1 -µ) V 1-µ 2 (x t0 ) -δ 1-µ 2 4 .
Proof. The claim can be obtained as a direct consequence of ( 6) and ( 7) by calculating their solutions analytically.

Mechanical Systems with Switched Force Fields

In the present section, we will apply the proposed approach to the analysis of dynamics of mechanical systems with switched forces, where commutation may represent instantaneous modification of the force shape or parameter values.

Consider the system

Aẍ(t) + B ẋ(t) + ∂Π (σ) 1 (x(t)) ∂x + ∂Π 2 (x(t -τ )) ∂x + F (σ) (t, x) = 0, (8)
where σ = σ(t) : [0, +∞) → {1, . . . , N } is a piecewise constant right-continuous function defining switching signal, integer N > 0 determines the number of subsystems in (8), Π

1 : R n → R are continuously differentiable and homogeneous (with respect to the standard dilation) functions of the order µ + 1, vector functions F (s) (t, x) are continuous for t ≥ 0, x ∈ R n , s = 1, . . . , N ; and the remaining notation is the same as for (1).

Let σ(t) be a non-Zeno switching law. This means (see [START_REF] Liberzon | Switching in systems and control[END_REF]) that on every bounded time interval the set of discontinuity instants for σ(t) is finite. Such a type of switching signals will be called admissible ones. Denote the respective switching instants by θ j , j = 1, 2, . . ., which satisfy the property 0 < θ 1 < θ 2 < . . . under the absence of Zeno behavior. For brevity, the characteristics of the orchestrating signal σ(t) are formulated as for a function of time, but a similar consideration can be done if the switching is governed by a function of state.

Assumption 5. Functions Π (s) (x) = Π (s) 1 (x) + Π 2 (x) are positive definite, s = 1, . . . , N . Assumption 6. Forces -F (s) (t, x) are non-conservative, i.e., x F (s) (t, x) = 0 for all t ≥ 0, x ∈ R n , s = 1, . . . , N .
Thus, in (8), non-conservative forces and the delay-free part of potential forces are switched, whereas the delayed and dissipative components of forces are non-switched. In addition, consider the following assumptions: Assumption 7. There exist numbers ∆ > 0, c > 0 and λ s > (µ + 1)/2 such that F (s) (t, x) ≤ c x λs for all t ≥ 0, x < ∆ and s = 1, . . . , N .

Assumption 8. There exist numbers ∆ > 0, c > 0, d > 0 and λ s < (µ + 1)/2 such that F (s) (t, x) ≤ c x λs for all t ≥ 0, x > ∆ and F (s) (t, x) ≤ d for all t ≥ 0, x ≤ ∆, s = 1, . . . , N .

Assumptions 5-8 are direct projections to the case of (8) of assumptions 1-4, respectively.

Theorem 2. Let assumptions 5 and 6 be fulfilled and consider (8) with any τ > 0 and any admissible switching law σ:

• if µ > 1 and Assumption 7 is valid, then the equilibrium position (2) is asymptotically stable;

• if µ < 1 and Assumption 8 is valid, then solutions of (8) are uniformly ultimately bounded.

Proof. Similarly to the proof of Theorem 1, it is easy to verify that the functional (3) is a common Lyapunov-Krasovskii functional for the family of subsystems associated with the switched system (8). By standard argumentation [START_REF] Liberzon | Switching in systems and control[END_REF], existence of such a functional guarantees the asymptotic stability of the equilibrium position (2) for µ > 1 and uniform ultimate boundedness of solutions for µ < 1 for any admissible switching signal σ.

Next, consider the case where all the forces in the considered mechanical system are switched:

Aẍ(t) + B (σ) ẋ(t) + ∂Π (σ) 1 (x(t)) ∂x + ∂Π (σ) 2 (x(t -τ )) ∂x + F (σ) (t, x) = 0, (9)
where B (s) is a constant positive definite matrix of dissipative forces; Π

2 : R n → R is a continuously differentiable and homogeneous (with respect to the standard dilation) function of the order µ + 1, s = 1, . . . , N ; and the remaining notation is the same as for ( 1) and (8).

Assumption 9. Functions Π (s) (x) = Π (s) 1 (x) + Π (s) 2 (x) are positive definite, s = 1, . . . , N .
Assumption 10. The switching instants θ 1 , θ 2 , . . . are given, but we have no information on the order of commutation of the subsystems in (9).

In this case, to derive delay-independent conditions of asymptotic stability and ultimate boundedness, we will use multiple Lyapunov-Krasovskii functionals, whose application needs additional constraints to be imposed on switching signal (frequently, dwell-time restrictions are used [START_REF] Liberzon | Switching in systems and control[END_REF], which however cannot be applied in highly nonlinear cases as here [START_REF] Efimov | On input-to-output stability of switched nonlinear systems[END_REF], and have to be revisited).

Construct a family of Lyapunov-Krasovskii functionals

V s (x t ) = 1 2 α s ẋ (t)A ẋ(t) + 1 2 x (t)B (s) x(t) +x (t)A ẋ(t) -x (t) t t-τ ∂Π (s) 2 (x(ξ)) ∂x dξ + t t-τ (β s + γ s (ξ -t + τ )) x(ξ) µ+1 dξ, s = 1, . . . , N,
where α s , β s , γ s are positive parameters. Denote by ˙ V s (x t ) the derivative of V s (x t ) along the solutions of the sth subsystem from the family associated with (9), s = 1, . . . , N , and introduce a norm-like variable

ω(x t ) = ẋ(t) 2 + x(t) 2 + t t-τ x(ξ) µ+1 dξ.
With the aid of the same arguments as in the proof of Theorem 1, it can be shown that, for an appropriate choice of α s , β s , γ s (see the proof of Theorem 1 for the method of their selection), there exist positive numbers δ1 , δ2 , b l , d l , l = 1, . . . , 4, such that

• if µ > 1, then b 1 ω(x t ) ≤ V s (x t ) ≤ b 2 ω(x t ), (10) ˙ V s (x t ) ≤ -b 3 V µ+1 2 s (x t ), V s (x t ) ≤ b 4 V r (x t )
for ω(x t ) < δ1 , s, r = 1, . . . , N ;

• if µ < 1, then

d 1 ω(x t ) ≤ V s (x t ) ≤ d 2 ω(x t ), (11) ˙ V s (x t ) ≤ -d 3 V µ+1 2 s (x t ), V s (x t ) ≤ d 4 V r (x t )
for ω(x t ) > δ2 , s, r = 1, . . . , N .

Theorem 3. Let assumptions 6, 9, 10 be fulfilled and consider (9) with any τ > 0 and an admissible σ:

• if µ > 1, Assumption 7 is valid and k-1 j=1 b (µ-1)(j-k) 2 4 (θ j+1 -θ j ) → ∞ as k → ∞, (12)
then the equilibrium position (2) of ( 9) is asymptotically stable;

• if µ < 1, Assumption 8 is valid and k-1 j=1 d (µ-1)j 2 4 (θ m+j+1 -θ m+j ) → ∞ as k → ∞ (13)
uniformly with respect to m ∈ {0, 1, . . .}, then solutions of (9) are uniformly ultimately bounded.

Proof. Let µ > 1. Consider the system (9) with a switching law satisfying the condition ( 12). Define a piecewise continuously differentiable Lyapunov-Krasovskii functional by the formula V (t, x t ) = V σ(t) (x t ). Let x(t, ϕ, t 0 ) be a solution of (9) such that x(t 0 + ξ, ϕ, t 0 ) = ϕ(ξ) for ξ ∈ [-τ, 0], ω(x t (ϕ, t 0 )) < δ1 for t ∈ [t 0 , t], where t 0 ≥ 0 with some t > t 0 . Find m, k ∈ {0, 1, 2, . . .} satisfying the conditions t 0 ∈ [θ m , θ m+1 ), t ∈ [θ k+m , θ k+m+1 ). Using (10) we arrive at the following estimates:

b -µ-1 2 1 ω -µ-1 2 (xt(ϕ, t 0 )) ≥ ṽ-µ-1 2 ( t) ≥ ṽ-µ-1 2 (t 0 ) + µ -1 2 b 3 ( t -t 0 ) ≥ b -µ-1 2 2 ω -µ-1 2 (ϕ) + µ -1 2 b 3 ( t -t 0 ) for k = 0; b -µ-1 2 1 ω -µ-1 2 (xt(ϕ, t 0 )) ≥ ṽ-µ-1 2 ( t) ≥ b -(µ-1)k 2 4 ṽ-µ-1 2 (t 0 ) + µ -1 2 b 3 b -(µ-1)k 2 4 (θ m+1 -t 0 ) + b -(µ-1)(k-1) 2 4 (θ m+2 -θ m+1 ) + . . . +b -(µ-1) 2 4
(θ m+k -θ m+k-1 ) + t -θ m+k for k ≥ k and all m ∈ {0, 1, . . .}. Therefore, if ω(x t(ϕ, t 0 )) < δ2 for some t ≥ t 0 , then ω(x t (ϕ, t 0 )) ≤ d 2 d k 4 δ2 /d 1 for t ≥ t. This implies the delay-independent uniform ultimately boundedness of solutions of the system (9) with the following estimate of the ultimate boundedness domain:

x 2 + ẋ 2 ≤ d 2 d 1 d k 4 δ2 ,
which is necessary to prove.

Note that the conditions ( 12) and ( 13) do not necessarily imply a dwell-time property [START_REF] Liberzon | Switching in systems and control[END_REF] of σ(t), but they can be used to determine how long the current subsystem may become active.

Examples

Consider a point of mass with a linear friction force proportional to velocity, a potential force shifting the point from the origin, a non-conservative rotating force, and a stabilizing potential delayed force, then the system (1) can be presented in the form

ẍ(t) + b ẋ(t) -x(t) 2 x(t) + ρ   x 3 1 (t -τ ) x 3 2 (t -τ ) x 3 3 (t -τ )   (14) + x(t) λ-1 ψ(t) × x(t) = 0,
where we assume that x(t), ẋ(t) ∈ R 3 (coordinates of the point and respective velocities), b and ρ are positive parameters, ψ : [0, +∞) → R 3 is a bounded continuous function of time, λ ≥ 1, τ is a constant positive delay, and × denotes the vector product. Hence, Π 1 (x) = -x 4 /4, Π 2 (x) = ρ(x 4 1 + x 4 2 + x 4 3 )/4 and assumptions 1 and 2 are satisfied for ρ > 3. For F (t, x) = x(t) λ-1 ψ(t) × x(t) Assumption 3 is also verified with c = sup t≥0 ψ(t) . Following the result of Theorem 1, we obtain that if λ > 2, then the equilibrium position (2) of ( 14) is asymptotically stable for any τ > 0. Note that for ρ = 0 or when there is no delayed part, the system ( 14) is unstable. For simulations we take:

b = 2, λ = 3, ρ = 4, ψ(t) = [1 0 -1] sin(t),
then the trajectories of x(t) and ẋ(t) for four different initial conditions are presented in Fig. 1 for two values of the delay (τ = 0.1 and τ = 1), demonstrating an asymptotic convergence to zero.

Next, consider the system (8) derived from ( 14):

ẍ(t) + b ẋ(t) - x(t) 2 x(t) σ(t) + ρ   x 3 1 (t -τ ) x 3 2 (t -τ ) x 3 3 (t -τ )   (15)
+ x(t) λσ(t)-1 ψ σ(t) (t) × x(t) = 0, where σ : [0, +∞) → {1, 2, 3, 4} is an admissible switching signal, and the used notation is similar to (14) except that now Π (s) 1 (x) = -x 4 4s and F (s) (t, x) = x(t) λs-1 ψ s (t) × x(t) for λ s > 2 and a bounded continuous function ψ s : [0, +∞) → R 3 , s = 1, . . . , 4. It is straightforward to check that assumptions 5-7 are satisfied, then by Theorem 2 the equilibrium position (2) of ( 15) is asymptotically stable for any τ > 0. For simulations we setup: b = 2, λ s = 2 + s, ρ = 4, ψ s (t) = [-1 1 1] sin(st), σ(t) = 1 + ceil (1 + sin(t) + cos(πt)) , then the trajectories of x(t) and ẋ(t) for four different initial conditions are presented in Fig. 2 for two values of the delay (τ = 0.1 and τ = 1). The given results of numeric experiments confirm the theoretical findings of this work.

Conclusion

For a class of mechanical systems having dissipative, non-conservative and potential forces, the problems of delay-independent asymptotic stability of the origin and ultimate boundedness of the trajectories were investigated. It was assumed that the potential forces are formed by gradients of homogeneous functions, and delay is constant. The given stability conditions are direct for verification and do not require a sophisticated analysis, they are also weaker than existing in the literature [START_REF] Zubov | Mathematical methods for the study of automatical control systems[END_REF]. To substantiate these results, a complete Lyapunov-Krasovskii functional was utilized. Further, these stability conditions were extended to a switched model of mechanical system, and the cases of arbitrary and restricted commutation laws were explored by analyzing common and multiple Lyapunov-Krasovskii functionals, respectively. The efficiency of proposed stability conditions were demonstrated in numeric experiments.

Figure 1 .Figure 2 .

 12 Figure 1. The results of simulation for (14)

Section 4 is supported by the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-573), other chapters are partially supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

where ṽ(t) = V (t, x t (ϕ, t 0 )). Similarly to the proof of Theorem 2 in [START_REF] Aleksandrov | On the asymptotic stability of switched homogeneous systems[END_REF], it can be shown that for (12) these inequalities imply delay-independent asymptotic stability of the steady-state solution (2).

Next, consider the system (9) with µ < 1 and a switching signal satisfying the condition (13). Let x(t, ϕ, t 0 ) be a solution of ( 9) such that

) and applying the inequalities (11), we obtain the following estimates:

Using these estimates and taking into account the constraint (13) imposed on the switching signal, it can be verified that, for any ∆ > 0, there exists T > 0 such that if t 0 ≥ 0, ϕ τ < ∆, then ω(x t(ϕ, t 0 )) ≤ δ2 for some t ∈ [t 0 , t 0 + T ]. In addition, there exists k ≥ 1 such that


Maoyu Mao

Zhongcheng Xiao

Xiangui Kang

Xiang Li

Liang Xiao

ELECTRIC NETWORK FREQUENCY BASED AUDIO FORENSICS USING CONVOLUTIONAL NEURAL NETWORKS

Keywords: network frequency based audio forensics using convolutional neural networks Audio forensics, electric network frequency, neural networks

Introduction

Audio editing software is often used by malicious actors to reduce the reliability of judicial evidence and defeat intellectual property protection. Audio tampering detection methods mostly rely on fingerprint information embedded in audio signals. Since fragile watermarks cannot assist in detecting private audio signal tampering [START_REF] Arnold | Audio watermarking: Features, applications and algorithms[END_REF], passive forensic schemes based on extracted audio features can provide lightweight solutions. Researchers have developed detection methods based on local noise levels of audio signals [START_REF] Pan | Detecting splicing in digital audios using local noise level estimation[END_REF] and voice activity detection [START_REF] Imran | Blind detection of copymove forgery in digital audio forensics[END_REF]. The electric network frequency (ENF) of audio signals demonstrates that power grid features are applicable to digital media forensics [START_REF] Garg | Seeing" ENF: Power-signature-based timestamps for digital multimedia via optical sensing and signal processing[END_REF]. Specifically, electric network frequency signals can be used to verify recording features such as time and location [START_REF] Hajj-Ahmad | ENF-based region-ofrecording identification for media signals[END_REF][START_REF] Vatansever | Factors affecting ENF-based time-of-recording estimation for video[END_REF], detect synchronization between audio and video data [START_REF] Su | Exploring the use of ENF for multimedia synchronization[END_REF] and verify the authenticity of multimedia [START_REF] Liu | Application of power system frequency for digital audio authentication[END_REF].

Electric network frequency based audio tampering detection techniques can verify if audio recordings have been edited at low computational cost. Ideally, the grid signal is a real sinusoid that fluctuates around its nominal value of 50 Hz or 60 Hz. Given that control mechanisms and power supply parameters are different in different parts of the world, electric network frequency signals display different fluctuations and peak frequency transformations. When signal-to-noise ratio (SNR) conditions are poor, disturbances near the electric network frequency component may be confused with the peak corresponding to the true electric network frequency [START_REF] Lin | Robust electric network frequency estimation with rank reduction and linear prediction[END_REF]. Furthermore, due to legal restrictions, it is difficult to obtain concurrent reference datasets of power systems [START_REF] Hajj-Ahmad | ENF-based region-ofrecording identification for media signals[END_REF]. Additionally, many edit detection schemes based on electric network frequency variations adjust the classification thresholds manually. Although some automated tampering detection schemes do not rely on concurrent power reference datasets, new techniques are required to improve detection accuracy and reduce computational costs.

This chapter proposes an electric network frequency based audio forensic scheme that detects tampering. The scheme assumes a signal model containing the electric network frequency component, where the background noise is low enough to ensure that the electric network frequency signal is the energy-dominant signal around the nominal frequency. Audio tampering is detected without using concurrent reference electric network frequency signals from power networks. The scheme applies two-stage -low-rank and bandpass -filtering to purify electric network frequency signals in a narrow spectral vicinity and compensate for time delays in order to obtain accurate estimates of the real-time edit locations. Based on the sensitivity of electric network frequency features to phase discontinuity changes, variations in the electric network frequency based features extracted from the multiple signal classification, Hilbert linear prediction and Welch algorithms are combined as eigenvectors and input to an automatic classifier. A convolutional neural network (CNN) is employed in the audio tampering detection scheme to improve the generalization ability in practical situations. Simulation results demon-strate that the proposed audio tampering detection scheme has good accuracy and an expanded application scope.

Related Work

Hua et al. [START_REF] Hua | On practical issues of electric network frequency based audio forensics[END_REF] have discussed the limitations of electric network frequency based tampering detection systems and the challenges posed by noise and interference. Several electric network frequency extraction algorithms such as the short-time Fourier transform and time recursive iterative adaptive algorithms are incorporated in instantaneous frequency estimation techniques to achieve high-precision extraction by measuring the maximum energy or weighted energy recorded from the average frequencies of spectrograms [START_REF] Hajj-Ahmad | Instantaneous frequency estimation and localization for ENF signals[END_REF].

A systematic assessment of parametric and non-parametric extraction techniques for electric network frequency signals has demonstrated that time-domain-based extraction algorithms are susceptible to frequency anomalies caused by sudden changes in noise or speech activity [START_REF] Karantaidis | Assessing spectral estimation methods for electric network frequency extraction[END_REF]. In addition, parametric algorithms such as the multiple signal classification and Welch algorithms can improve resolution frequency estimation of sinusoidal signals by using fewer data series than spectrogram-based extraction algorithms.

An electric network frequency extraction scheme proposed by Lin and Kang [START_REF] Lin | Robust electric network frequency estimation with rank reduction and linear prediction[END_REF] applies robust principle component analysis to remove noise interference and purify the electric network frequency when signal-to-noise conditions are poor. It adopts the Hilbert linear prediction algorithm to capture the electric network frequency from fewer audio recordings in an efficient manner.

Nicolalde Rodriguez and Apolinario [START_REF] Rodriguez | Evaluating digital audio authenticity with spectral distances and ENF phase change[END_REF] have developed a digital audio authenticity evaluation scheme that detects electric network frequency phase transitions and leverages the spectral distance using an adaptive filter as a linear indicator. An electric network frequency based edit detection scheme for speech recordings designed by Esquef et al. [START_REF] Esquef | Edit detection in speech recordings via instantaneous electric network frequency variations[END_REF] yields low equal error rate (EER) values by comparing electric network frequency variations around the nominal frequency with the upper limit of the normal variations observed in an unedited signal. Hua et al. [START_REF] Hua | Audio authentication by exploring the absolute error map of ENF signals[END_REF] have analyzed the absolute error map between an electric network frequency database and test electric network frequency signals to perform timestamp verification and detect tampering via insertion, deletion and splicing attacks with image erosion.

Nicolaide Rodriguez et al. [START_REF] Rodriguez | Audio authenticity: Detecting ENF discontinuity with high precision phase analysis[END_REF] have also developed an automated authenticity detection scheme for audio recordings via phase analysis of high-order electric network frequency harmonics. Reis et al. [START_REF] Reis | ESPRIT-Hilbert-based audio tampering detection with SVM classifier for forensic via electrical network frequency[END_REF] have designed an adulteration detection scheme for audio recordings that integrates the kurtosis features of electric network frequency signals in rotational invariance techniques and Hilbert linear prediction in poor signal-to-noise conditions to autonomously classify audio recordings using a support vector machine. Although kurtosis extraction speeds up the classification, some characteristic information is lost.

Wang et al. [START_REF] Wang | Digital audio tampering detection based on ENF consistency[END_REF] have developed a detection scheme that applies discrete Fourier transforms of audio signals to achieve instantaneous phase estimation using a support vector machine classifier. However, the accuracy of the scheme is unsatisfactory and the cost of using a support vector machine to evaluate the decision function is linearly related to the number of training samples. This results in high computational costs for large datasets.

Researchers have also applied convolutional neural networks to analyze audio recapture [START_REF] Lin | Audio recapture detection with convolutional neural networks[END_REF] and perform median filtering [START_REF] Chen | Median filtering forensics based on convolutional neural networks[END_REF]. However, no research has applied convolutional neural networks to electric network frequency based audio tampering forensics.

System Model

Figure 1 shows a schematic diagram of the proposed audio forensic scheme, which uses a convolutional neural network in conjunction with the multiple signal classification (MUSIC), Hilbert linear prediction and Welch algorithms.

The system initially reduces the sampling rate of an audio signal under test x(m). Let ω 0 be the nominal electric network frequency. According to convention, the new sampling frequency f s is adjusted to 20 times the nominal frequency ω 0 . Therefore, the sampled signal x ds (n) where 0 < n ≤ m is obtained using a 1,000 Hz or 1,200 Hz sampling frequency.

The low-rank structure of the electric network frequency signal in the short-time Fourier transform (STFT) domain is leveraged to separate grid signals from interference by robust principal component analysis (RPCA). Let X ds be the amplitude spectrum of the sampled signal x ds (n). Then, the robust principal component analysis objective is given by: min

XC ,X E rank XC + λ X E 0 s.t. XC + X E = X ds (1)
where • 0 is the L0-norm, λ > 0 is a parameter that trades off the low-rank part with the electric network frequency component XC , and X E is the sparsity part containing the impulse noise and speech activity signal. Due to the non-convex optimization objective, a relaxation is applied to Equation (1) according to [START_REF] Zhao | Robust principal component analysis with complex noise[END_REF]. Thus, the principal component analysis objective becomes:

ˆc x (n) c x (n) M x (n) L x (n) W x (n) M L W F(X X X )
min XC ,X E XC * + λ X E 1 s.t. XC + X E = X ds (2)
where

• * is a nuclear norm and • 1 is the L1-norm.
Next, XC is derived by the augmented Lagrange multiplier method [START_REF] Lin | Robust electric network frequency estimation with rank reduction and linear prediction[END_REF] and the inverse short-time Fourier transform (ISTFT) is employed to determine the low-rank filtered signal sequence denoted by xC (n).

Grid signals with electric network frequency components x C (n) are insulated from interference falling into the low-order space by filtering xC (n). Instead of a finite impulse response filter, a fourth-order elliptic filter is adopted with a phase that is approximately linear and adjacent to the bandpass region. This reduces the computational complexity and computational costs.

The grid signal x k C (n) obtained from the audio recording k after twostage filtering is similar to a narrow-band pseudo-sinusoidal signal. To simplify the presentation, the audio recording index k in the superscript is omitted. Accordingly, the time-domain representation of the electric network frequency signal in the interval is modeled as:

x C (n) = a cos 2π x F (n) f s n + φ 1 ≤ n ≤ L ( 3
)
where L is the length of time-domain signal, x F (n) is the electrical network frequency to be estimated, f s is the sampling frequency and a and φ are related to the magnitude and phase, respectively.

ENF-Based Forensics with CNN

During electrical network frequency extraction, the captured grid signal x C (n) is typically divided into Y time frames of fixed-length l containing overlapping portions where Y ∈ {L/l} 0≤l≤L . A frequency estimation algorithm is then used to obtain the components of the electrical network frequency characteristics in frame i where

1 ≤ i ≤ Y .
The MUSIC algorithm is a subspace spectrum estimation algorithm based on feature structure decomposition. The algorithm decomposes the covariance matrix of a signal sequence into a singular value. By constructing the orthogonal signal and noise subspaces, the algorithm provides spatial spectral functions that can be used to estimate electrical network frequency features. Since electrical network frequency signals contain one real sinusoid, two complex frequency sinusoids are embedded in the white noise P for electrical network frequency signals.

The algorithm first computes the M × N sample data matrix A based on the power grid signal X (i) C of audio frame i:

A = a C (1) a C (2) . . . a C (N -1) T (4) where a C (n) = [x C (n), x C (n + 1), . . . , x C (n + M -1)
] T , M is the order of the covariance matrix that is chosen to be larger than P and M ∈ N 3 , 2N 3 [START_REF] Hajj-Ahmad | Instantaneous frequency estimation and localization for ENF signals[END_REF]. Next, the eigenvalue decomposition of the auto-covariance matrix R = 1 N A H A is computed. Since the signal and noise are independent, the covariance can be decomposed and the space comprising the eigenvectors corresponding to the large eigenvalues (q 1 q 2 . . . q P ) is the signal subspace S. Also, the space comprising the eigenvectors corresponding to the small eigenvalues q P +1 q P +2 . . . q P +M is the noise subspace G n .

The following assumptions are made for the complex form of the observation model described above: Different x (i) C (n) signals are linearly independent of each other. The additive noise u(n) is the complex noise with zero mean additive, uncorrelated and the same variance σ 2 u . Given the orthogonality property of the white noise eigenvectors and signal steering vectors v X (i) C , the signal steering vectors v X (i) C can be written in complex form as:

v X (i) C = 1, e j2πX (i) C , e j4πX (i) C , . . . , e j2(M -1)πX (i) C H ( 5
)
where j = √ -1 and [ • ] H denotes the conjugate transposition. The pseudo-spectral function P MU is computed as:

P MU = 1 v * X (i) C G i n G i * n v X (i) C ( 6
)
where * is the element conjugate. Ultimately, the estimated electrical network frequency X MU is obtained by searching for the spectral peak of the spatial-spectral function P MU . The MUSIC algorithm estimates a fixed parameter for each frame, which is the best electrical network frequency value in the least mean square sense for a given signal sequence.

The Hilbert linear prediction extraction algorithm is more sensitive to sharp phase changes than the MUSIC algorithm. However, the MU-SIC algorithm is more robust to noise interference. According to Equation (3), the electrical network frequency value ĥC (n) can be estimated by the transient phase change of the Hilbert transform from the realvalued estimate x C (n) as follows:

ĥC (n) = x C (n) + jH x C (n) ( 7
)
where j = √ -1 and H is the Hilbert operator. Since the analytical version of a pseudo-sinusoidal signal is equivalent to the real-valued signal with respect to x F (n), the linearly predictable property can be applied to the complex model. This yields:

ĥC (n) = ae " j2π x F (n) fs n+φ " = β 1 ĥC (n -1) (8)
where

β 1 = e " j2π x F (n) fs "
is the first-order prediction coefficient. The signal entry s(n) = x F (n) + u(n) is then obtained by adding the additive complex noise u(n).

Given the assumption that additive complex noise is always equivalent, the approximation s(n) ≈ β 1 s(n -1) is obtained according to Equation [START_REF] Hua | On practical issues of electric network frequency based audio forensics[END_REF]. Extending this equation to the entire audio recording yields: S 1 ≈ β 1 S 2 where S 1 = [s(n -1), s(n -2), . . . , s(n)] 0<n≤m and S 2 is the sequence with one sample shift from S 1 .

Therefore, the crux of electrical network frequency estimation is to minimize the weighted linear prediction error in the minimum squared sense as follows:

min J(β 1 ) = e T We = (S 2 -β 1 S 1 ) H W (S 2 -β 1 S 1 ) ( 9
)
where W is a symmetric weighting matrix, H is the conjugate transposition operator and J(β 1 ) is the total cost function denoted by the weighted squared error e.

The symmetric weighted matrix W, which is obtained by Markov estimation, is given by:

W = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1+ β 1 2 -β 1 0 0 . . . 0 -β * 1 1+ β 1 2 -β 1 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 . . . 0 -β * 1 1+ β 1 2 -β 1 0 0 . . . 0 -β * 1 1+ β 1 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -1
(10) where * and [ • ] -1 are the element conjugate and matrix inverse, respectively.

Upon setting the differential in Equation ( 9) to zero, the prediction coefficient β 1 is given by:

β 1 = S 1 H WS 2 S 1 H WS 1 (11)
Equations ( 9) through [START_REF] Karantaidis | Assessing spectral estimation methods for electric network frequency extraction[END_REF] reveal that the computation of β 1 is an iterative process. Having obtained β 1 , the Hilbert linear prediction of the electrical network frequency X L is computed as:

X L = f s 1 2π ∠(β 1 ) (12)
For consistency with other characteristics, X L is divided into fragments and the maximum value of the absolute values is taken as the i th segment electrical network frequency estimate X The Welch algorithm is an improved periodogram method. The algorithm reduces noise in the estimated power spectrum by enhancing the frequency resolution, yielding the largest maximum correlation coefficient around the nominal frequency compared with the MUSIC algorithm and other methods.

The Welch estimate is obtained from the power spectral density. The algorithm divides each audio recording into overlapped segments multiplied by a Hamming window. The frequency sample w, which corresponds to the maximum periodogram value, is extracted as the Welchbased electrical network frequency estimate denoted by X (i) W . Next, a quadratic interpolation is employed to fit the quadratic model of w. The Welch algorithm with a Hamming window improves the spectral distortion caused by the large-side lobe of the rectangular window, yielding an accurate electrical network frequency estimate X W that is not affected by interference.

Figure 2 clearly shows the electrical network frequency based fingerprints used to verify the effectiveness of the proposed audio forensic scheme. The fingerprints are located in three recording fragments named 1p, 2p and 3p at 60 Hz with slight offsetting for easy viewing. The three recording fragments, all with the same number of samples, are derived from the MUSIC, Hilbert linear prediction and Welch (MHPW) feature estimation algorithms. The forensic fingerprints simultaneously display high stability and sensitivity to tampering operations. For example, the fingerprints in recording fragments 1p and 2p show stable pseudo-sinusoidal fluctuations whereas the fingerprints in the tampered fragment 3p show sensitive mutations.

Instead of manually determining the threshold, a novel deep learning approach is applied to identify tampered audio recordings. The electrical network frequency signals extracted by the three algorithms are directly modeled as features to avoid information loss when extracting the representative values of features.

Three-dimensional feature vectors F = X k L , X k MU , X k W 0<k≤N are obtained from the N audio recordings. When the lengths of the recordings in the audio dataset are different, the nominal frequency of 50 Hz or 60 Hz is applied to fill the feature vectors to the same length. The three feature channels with the same length constitute the input layer of the neural network structure.

Figure 3 shows the network architecture of the audio tampering detection scheme. The convolutional neural network model has four convolution layers, two pooling layers, one full connection layer and an input layer and output layer. Before processing the features, min-max normalization is used to amplify the differences and variation rules of the features. Next, given the overlaps of the adjacent electrical network frequency components, convolution is used to refine the energy changes in the electrical network frequency signals, which improves the detection accuracy. Finally, the tag distribution obtained by the convolutional neural network model is used to compute the detection performance metrics.

Experiments and Results

This section describes the simulation experiments and the results obtained.

Experimental Setup

The experiments were performed on Matlab and Python 3.6 platforms with the scikit-learn package.

The electrical network frequency based features were extracted from two classical audio databases. The first was the Carioca 1 database [START_REF] Rodriguez | Audio authenticity: Detecting ENF discontinuity with high precision phase analysis[END_REF], a telephone recording database of the public switched telephone network containing 16-bit mono waves at a 44.1 kHz sampling rate and coded by pulse code modulation with an electrical network frequency component around 60 Hz, The second was the Spanish Speech database [START_REF] Esquef | Improved edit detection in speech via ENF patterns[END_REF] sampled at 16 kHz with a nominal electrical network frequency component around 50 Hz. The databases each contain 100 original voice audio recordings and 100 edited versions of the original voice audio recordings.

Simulations were performed to evaluate the performance of the audio forensic scheme with N = 400 audio recordings. Each recording was divided into time frames of length l = 1 second with an overlap of 0.5 seconds. In the simulations, 70% of the original audio recordings and tampered audio recordings were randomly chosen to train the convolutional neural network. The remaining 30% of all the recordings were randomly-chosen for the testing set. Distributing the data into training and testing datasets in this manner ensured that every portion of the data would be more representative. The data randomness had to be high due to the large number of parameters and strong learning ability of the convolutional neural network, and so that the random gradient descent optimization function did not get stuck in a local minimum.

Multiple evaluations were performed to achieve fair comparisons with the benchmark strategy proposed by Reis et al. [START_REF] Reis | ESPRIT-Hilbert-based audio tampering detection with SVM classifier for forensic via electrical network frequency[END_REF]. The detection error tradeoff (DET) curves were obtained by plotting the false negative rate (FNR) versus false positive rate (FPR) curves for various thresholds [START_REF] Martin | The DET curve in assessments of detection task performance[END_REF]. In general, as the false positive rate increases to 100%, the false negative rate decreases, and vice versa. The equal error rate is the point at which the false negative rate and false positive rate are equal. The overall error rate (OER) is computed as the average of the false negative rate and false positive rate.

Detection Performance

Table 1 shows the overall error rates for feature vectors obtained by combining fusion features. Combining all three features (MUSIC, Hilbert linear prediction and Welch (MHPW)) yields the lowest overall error rate of 3.2% compared with using any one feature or any two features. When all three features are used together, the overall error rates fall by 4.3%, 1.9% and 3.1%, respectively, from the overall error rates when the MUSIC, Hilbert linear prediction and Welch algorithms are used alone.

However, the overall error rates obtained for the mixed features extracted by two of the three algorithms may be suboptimal to those extracted by a single algorithm; this is due to the cancellation of the sharp peak features of the two algorithms. For example, the overall error rate for the fusion features extracted by the MUSIC and Welch algorithms is reduced by 1.0% compared with the overall error rate of the features extracted by the MUSIC algorithm alone. The proper choice of features plays an important role in the accurate detection of audio tampering. Table 2 shows that the proposed scheme using the convolutional neural network with MUSIC, Hilbert linear prediction and Welch features has the lowest overall error rate compared with the other classifiers. For example, the overall error rate is 5.9% less than that obtained by the neural network scheme and is 1.0% less than that obtained by the support vector machine scheme. Additionally, the detection performance of the proposed scheme using the convolutional neural network with the MUSIC, Hilbert linear prediction and Welch features has an overall error rate that is 1.3% less than that obtained by the benchmark strategy with a support vector machine described in [START_REF] Reis | ESPRIT-Hilbert-based audio tampering detection with SVM classifier for forensic via electrical network frequency[END_REF]. Table 3 shows the results of cross-domain evaluations when the Carioca 1 and Spanish Speech databases were used for training and testing, respectively, and vice versa. Using the combination of MUSIC, Hilbert linear prediction and Welch features trained with the Carioca 1 database yields slightly better prediction results (4.3%) compared with when the Spanish Speech database was used for training (4.5%). This could be because the extracted features of Caricoa 1 are more obvious, which renders the trained model more representative and the testing results more accurate.

However, the difference between the two overall error rates is small (0.2%), which may be due to the number of training sessions, number of iterations, final convergence and small differences in only one set of random values in the convolutional neural network. As observed above, a mixed training dataset yields better detection performance than using a single dataset for training. For example, the detection with the mixed (Carioca 1 and Spanish Speech) training dataset decreases the overall error rate by 1.3% compared with the single Carioca 1 training dataset and 1.1% compared with the single Spanish Speech dataset. The key insight is that the increased data diversity provided by mixed training increases the generality of the learned model, which improves the detection performance and reduces the overall error rate.

Figure 4 compares the detection performance of the proposed scheme using the convolutional neural network with MUSIC, Hilbert linear prediction and Welch features (CNN with MHPW) versus the detection performance of the benchmark strategy by Reis et al. [START_REF] Reis | ESPRIT-Hilbert-based audio tampering detection with SVM classifier for forensic via electrical network frequency[END_REF]. The same Carioca 1 and Spanish Speech databases with N = 400 audio record- ings and l = 1 second were used in the comparison. The DET curve of the proposed scheme (CNN with MHPW) is much closer to the y-axis (i.e., lower false negative rates) compared with the benchmark strategy. Moreover, the proposed scheme has an equal error rate of 3.3%, which is less than the 4% equal error rate of the benchmark strategy.

Results for Different SNR Conditions

This section evaluates the performance of the proposed audio forensic scheme under signal-to-noise ratios ranging from 5 dB to 30 dB. The speech activity detector of Esquef et al. [START_REF] Esquef | Edit detection in speech recordings via instantaneous electric network frequency variations[END_REF] was used to separate the noise from speech signals in the Carioca 1 and Spanish Speech databases, following which various levels of additional background Gaussian noise were introduced. It is important to note that the results of this evaluation can be generalized to any audio recording.

Figure 5 shows the performance of the proposed audio forensic scheme (CNN with MHPW) under various signal-to-noise ratios. The equal error rates obtained for the datasets corrupted by Gaussian noise decrease with increasing signal-to-noise ratio because the classifier acquires more accurate electric network frequency information with less noise. The performance gap is much wider at lower signal-to-noise ratios, which validates the effectiveness of low-rank filtering for noise correction. For example, the proposed scheme decreases the equal error rate from 20.8% to 17.9% for the lowest signal-to-noise ratio of 5 dB. In fact, the proposed scheme achieves optimal performance faster than the benchmark strategy with less signal information. Specifically, the equal error rate of the proposed scheme decreases from 17.9% at 5 dB to 4.0% at 25 dB whereas the benchmark strategy achieves the same equal error rate only at 30 dB. Moreover, despite showing a consistent performance trend, the proposed scheme is more effective than the benchmark strategy even for low signal-to-noise ratios. This demonstrates that the low-rank filtering incorporated in the proposed scheme improves the accuracy of detecting audio tampering, especially in poor signal-to-noise conditions.

Conclusions

The audio forensic scheme described in this chapter leverages a convolutional neural network classifier to evaluate electric network frequency features in audio signals to detect tampering without manual regulation or information about the concurrent reference frequency from the power grid. The experimental results demonstrate that the audio forensic scheme increases the accuracy of tamper detection and is better adapted to noisy environments than the benchmark strategy of Reis et al. [START_REF] Reis | ESPRIT-Hilbert-based audio tampering detection with SVM classifier for forensic via electrical network frequency[END_REF]. For example, the proposed scheme reduces the overall error rate by 1.3% and increases the equal error rate by 0.7% compared with the benchmark strategy. Additionally, it increases the equal error rate up to 2.9% compared with the benchmark strategy under different signalto-noise conditions. The tamper detection performance and robustness in noisy environments help ensure the reliability of audio evidence and protect intellectual property.

Future research will attempt to enhance detection and efficiency in more aggressive scenarios, and develop an online detection system that identifies the specific locations of audio tampering. Additionally, future research will explore the application of electric network frequency signals in video forensics.

Figure 1 .

 1 Figure 1. Proposed audio forensic scheme.

Figure 2 .

 2 Figure 2. Electrical network frequency fingerprints in three recording fragments.

Figure 4 .

 4 Figure 4. Comparison of detection performance.

Figure 5 .

 5 Figure 5. Performance of the audio forensic scheme under different SNR conditions.

  Network architecture of the proposed audio forensic scheme.

	MHPW								Tamper
	Features								Detection
									Tags
	Input Layer Convolution	Convolution	Pooling	Convolution	Convolution	Pooling	Fully	Output
		Layer	Layer	Layer	Layer	Layer	Layer	Connected	Layer
								Layer
	50×60	50×60×32	50×60×32	25×30 25×30×64	25×30×64	12×15	256	2
	Figure 3.

Table 1 .

 1 Overall error rates for combinations of fusion features.

	Feature

Table 2 .

 2 Overall error rates for various classifiers.

	Classifier	Overall Error Rate
	Neural Network	9.1%
	Random Forest	9.1%
	Decision Tree	7.3%
	Logistic Regression	6.7%
	Support Vector Machine	4.2%
	Convolutional Neural Network	3.2%

Table 3 .

 3 Cross-domain evaluations of the audio databases.

	Training Database Testing Database Overall Error Rate
	Carioca 1	Spanish Speech	4.3%
	Spanish Speech	Carioca 1	4.5%

Acknowledgement

This research was supported by the Natural Science Foundation of China under Grant nos. 61772571, U1536204, 61971366 and 61671396.