Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution.The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.


Introduction
This study is motivated by the fact that many frequencydomain stability criteria are developed and exploited as parametric and graphical tools for stability analysis and stabilization in classes of time-delayed systems, subject to various structural, analytical or technical assumptions, and constraints.Due to intuitive conception, simple geometry, experimental availability, and frequency-domain methods are extended to stability analysis and stabilization in distributed [1,2], periodically time-varying [3], and timedelayed ones with or without uncertainties [4][5][6][7][8][9].Some stability criteria are given in Nyquist-like form, which involve open-loop unstable poles and frequency-domain plots in general and thus are not useful for time-delayed systems with time-varying factors [10][11][12].However, when time-delayed systems are dealt with [13][14][15][16][17], poles distribution is not available numerically and frequency responses need to be examined with prior sweeping.Stability criteria for timedelayed systems are created also via LMI [18,19].Complexvalued neural networks with time delays are attached in [20][21][22].As a matter of fact, when time delay occurs in dynamics and/or their derivatives, the characteristic polynomials are quasi-polynomials in one variable and its exponential powers [4,23] that are of infinite degree.Although finitely many poles can be computed by truncating infinite series of exponentials or other numerical algorithms, approximate poles are not meaningful for analytically exploiting Nyquist-type criteria, let alone unavoidable algorithm convergence [24] and numerical pathology.
Lately, a series of studies [25][26][27][28] are developed in various systems based on complex scaling, which enables us to cope with structure and stability issues by means of frequencydomain properties of the concerned systems without invoking pole distribution and specific frequency responses.More precisely, the study [25] talks about stability of sampleddata systems; the work [26] deals with that of continuoustime fractal-order systems; the study [27] considers the Lur' etype feedback nonlinear systems.The paper [28] generalizes the complex scaling tool for investigating structural features in LTI systems, say controllability, observability, and so on.Tentative discussion via complex scaling for stability in openloop time-delayed systems was reported in [29].In this study, we carefully work through the technical details and provide rigorous proofs for internal stability criteria for LTI feedback systems with multiple time delays.

Mathematical Problems in Engineering
More specifically, to surmount multiple time-delay dependence, infinite-dimensionality, polynomial cancellation, and other complex analysis issues in quasi characteristic polynomials of time-delayed continuous systems in feedback configuration, the complex scaling technique is coined by using auxiliary Hurwitz polynomial to modify the return difference relationships such that the modified ones reflect exactly and completely asymptotical/exponential stability while analytical properties are retained for applying the argument principle [30].The suggested stability criteria produce necessary and sufficient conditions for asymptotical and exponential stability (or simply internal stability), which involve neither open-loop unstable poles that are hard to know, nor prespecified contour and locus encirclement orientations.The stability criteria are implementable graphically with locus plotting or merely by numerical complex argument integration.Also using the complex scaling approach, we further contrive a new technique for stabilizing timedelayed systems with static state feedback.The stabilization results in asymptotical/exponential stability internally as well as  2 -stability externally that are robust with respect to time delay.
Outlines.Section 2 collects preliminaries to LTI time-delayed continuous systems of retarded type, their feedback configuration, and return difference relationships.Section 3 states the complex scaling return difference relationships and proves the generalized criteria.Stabilization is explained in Section 4. Illustrations are sketched in Section 5, and Section 6 gives our conclusions.Notations 1. R and C are the sets of real and complex numbers, respectively, while C + = { ∈ C : Re() > 0}.  is the  ×  identity matrix.(⋅) * means the complex conjugate transpose of a matrix (⋅).det(⋅) means the determinant of a square matrix (⋅).A polynomial is Hurwitz polynomial if all its roots have negative real parts, and deg(⋅) means its degree. 2 [0, ∞) denotes the functions space such that where ‖ ⋅ ‖ is the Euclidean norm of a vector (⋅) or the induced matrix norm as appropriately.

Preliminaries to Time-Delayed Systems
2.1.Linear Time-Delayed Continuous-Time Systems of Retarded Types.By linear time-delayed continuous-time system of retarded type, we mean a system in form of where () ∈ R  , () ∈ R  , and () ∈ R  are the state, control input, and measured output, respectively. > 0 is an integer, and   ∈ R × ,   ∈ R × ,   ∈ R × , and  ∈ R × , respectively, are constant matrices.In (1), the time delays are assumed to satisfy (2) That is, , , and ] are short notations for the corresponding multiple time delays.
The transfer function matrix of system (1) is Conventionally, det(Δ  (, )) is called the characteristic function, and det(Δ  (, )) = 0 is the characteristic equation, whose zeros are the characteristic roots.Since det(Δ  (, )) is quasi when  ̸ = 0, there are infinitely many characteristic roots.Due to infinite-dimensionality, it is considerably difficult to deal with stability investigation directly in terms of numerically computing the characteristic roots.
We collect properties about the characteristic roots in the following remark, which pave the way for stability analysis and stabilization.
Remark 2. (i) det(Δ  (, )) is holomorphic; therefore, all characteristic roots are isolated; (ii) there are only finitely many characteristic roots in any vertical strip { ∈ C :  < R() < } of the complex plane, where  <  ∈ R; (iii) for any  ∈ R with || < ∞, only finitely many characteristic roots exist to the right-hand side of the vertical line R() = , while infinitely many ones are located to the left-hand side of R() =  [24, p. 9, Corollary 1.9]; (iv) the characteristic roots are continuous with respect to delays and coefficients [24, p. 10, Theorem 1.14]; (v) in any bounded rectangular area { ∈ C :  < R() < ,  < I() < } š D, where  <  ∈ R and  <  ∈ R, there are no infinite sequences of characteristic roots that converge to points in D.
Asymptotical and exponential stabilities in (1) are summarized by Definition 3. Proposition 4 gives necessary and sufficient conditions for exponential stability, in terms of the characteristic roots and in light of [24].Definition 3. Let ()(⋅) be the null solution of (1) with respect to the initial condition () ∈ ([− max , 0], R  ), where  max is defined in (2).
(i) The null solution of (1) is asymptotically stable, or simply Σ  is asymptotically stable, if, for any  > 0, there exist  > 0 and (ii) The null solution of ( 1) is exponentially stable, or simply Σ  is exponentially stable, if there exist constants  > 0 and  > 0 such that ‖( Together with Remark 2, asymptotical and exponential stabilities are equivalent whenever LTI time-delayed systems of retarded type are concerned.Hence, only exponential stability conditions are claimed in Proposition 4 explicitly by restating Proposition 1.6 [24].Proposition 4. The null solution of ( 1) is exponentially stable if and only if there is  > 0 such that R( , ) < − for each , where  , means a root of the characteristic equation det(Δ  (, )) = 0.

Output Feedback Configuration and Return Difference
Relationship.By the notation Σ  : (Σ  , Σ  ), we denote an output feedback connection with Σ  being the time-delayed system (1) and Σ  being an LTI controller, which has the state-space equation where Λ ∈ R × , Γ ∈ R × , Θ ∈ R × , and Π ∈ R × are constant matrices.Apparently, the transfer function matrix of Σ  is On the one hand, the open-loop characteristic function is denoted by   (, ) and satisfies On the other hand, the state-space equations for the closedloop system Σ  are given by where Ξ fl (  + Π) −1 .The closed-loop characteristic function   (, , , ]) is given by By ( 6) and ( 8), if Σ  is well-posed (i.e.,   + Π is nonsingular), both the open-loop system, denoted by Σ  , and the closed-loop system Σ  are retarded type.Hence, Definition 3 and Proposition 4 apply to Σ  as well.
To attack stability in Σ  , we need to address how to compute   (, , , ]).This relates us to the return difference relationship given below.Since the derivation algebras are standard and trivial, we omit the details: Although the return difference relationship ( 9) is also in form as what we know in LTI systems, we must be cautious in connecting (9) with the argument principle so as to create stability criteria for the concerned time-delayed systems.Note the following: (i) Exponentials in delays are involved.Open-loop unstable poles in Σ  (i.e., the zeros of det(Δ  (, )) = 0), if any, cannot be calculated simply.det(Δ  (, )) = 0 has infinitely many zeros; it is almost meaningless to numerically compute unstable poles, if any.

Stability Analysis of Time-Delayed Systems
3.1.Contour, Assumptions, and Problem Formulation.Now we describe a Cauchy contour for utilizing the argument principle of complex analysis in light of [30,33].In Figure 1, the contour is a simply closed curve marked by the dashed line, whose vertical portion is overlapped actually with the imaginary axis.Let us denote by Int(N) the interior circumscribed by N to its right.Thus, Int(N) is an open set and Int(N) = C + .In the paper, the entry-belonging  ∈ N is equivalent to saying  ∈ N or the point  is on the imaginary axis; namely,  ∈ (−∞, ∞).In the cases involving -exponential stability, we need to use the -shifted Cauchy contour that is denoted by N +  with  > 0 being a constant.The exact meaning of the relation To deal with the issues explained in Section 2.2 about the return difference equations, we introduce a polynomial (), namely, the scaling free-term, into (9), and obtain which is a generalized return difference relation.By means of (),   (, ) and   (, , , ]) can be handled separately so that characteristic roots distribution of   (, ) and coprimeness between them are surmounted.About the scaling freeterm, we assume the following.
(A1) () is a Hurwitz polynomial with all the roots being far enough from the imaginary axis; thus it has no zeros in N ∪ Int(N) and |(0)| is as large as desired.
Obviously, (A1) guarantees that ( 10) is meromorphic, and () vanishes nowhere along N; even if there exists any cancellation between () and   (, , , ]), those cancelled closed-loop characteristic roots must be outside N ∪ Int(N).By (A1) and (A2), the locus can be plotted within a prespecified bounded area around the origin without prior frequency sweeping.This brings us numerical and graphical convenience.One will understand this by the remarks about Theorem 5.
Our first problem is to establish frequency-domain criteria for testing stability of Σ  .

Complex Scaling Stability Criteria
Theorem 5. Suppose that the closed-loop time-delayed system Σ  is well-posed.Then, Σ  is asymptotically stable, if and only if, for some arbitrarily chosen finite-degree polynomial () satisfying (A1) and (A2), the generalized locus vanishes nowhere, and its number of clockwise encirclements around the origin (0, 0) is equal to that of counterclockwise ones.The contour N is defined in Figure 1.
Proof.As is illustrated in Figure 1, let  ∈ N move from a point  0 clockwise along N; or we start from  =  − 0 on N. When we return to  0 from the other side along N, we arrive at  =  + 0 .We assert readily by applying the argument principle [30] to (10) that where (⋅) and (⋅) are the numbers for zeros and poles of (⋅) in Int(N) (up to multiplicity), respectively, and   (⋅) and   (⋅) are the numbers of clockwise and counterclockwise encirclements of the locus around the origin, respectively.∠(⋅) is the complex argument of (⋅).The last equation of ( 12 as long as (, ) vanishes nowhere over  ∈ N. In (14),   means how many net times that the locus (, )| ∈N encircles the origin (0, 0).
Equipped with (14), we are ready to show the main assertion.
To see sufficiency, we see that, under the assumptions on () and the locus (, )| ∈N , ( 14) can be equivalently interpreted as which says by Proposition 4 that Σ  is asymptotically stable.
To see necessity, assume that Σ  is asymptotically stable.Hence, no closed-loop characteristic root belongs to N ∪ Int(N).To complete the proof, it remains to show that some Hurwitz polynomial () can be constructed such that (A1) and (A2) are satisfied, and the corresponding locus vanishes nowhere over  ∈ N, and   (⋅) =   (⋅).
(i) The conditions are necessary and sufficient, dependent on time delays, independent of any openloop structural features, and stated via generalized frequency-domain features.In addition, the locus encirclement and orientation can be self-defined.
(ii) Note that the scaling () is not unique.Actually, () can be arbitrary as long as it is a Hurwitz polynomial.However, as graphical implementation is concerned, it is prudent to have deg(()) =  + .If deg(()) >  + , the locus may approach the origin numerically as  → ∞, and this makes it difficult to count encirclements, whereas if deg(()) <  + , the corresponding locus may contain portions at infinity as  → ∞.
Together with (A1), deg(()) =  +  guarantees that the whole locus can be plotted within a bounded region around the origin as compactly as possible by choosing the roots of () to be located far away sufficiently to the left of the imaginary axis.In view of this, no frequency sweeping is prerequisite for correctly testing the conditions.
(iii) When using the -shifted Cauchy contour N + , possible -exponential stability of the concerned time-delayed system can be reflected, that is, if all the closed-loop poles are at the left-hand side of the vertical line Re() = .Also, if Σ  and/or Σ  have any poles on the imaginary axis, we need the contour N+ to detour such poles and guarantee numerical welldefinedness of the right-hand side of (10).It should be stressed that analytically there is no need to use the shifted contour even if there are imaginary poles.
(iv) Though the conditions are stated in a graphical manner for stability testing and robustness evaluation as known in LTI continuous-time systems, they can be interpreted and implemented purely in a numerical way.Indeed, the criterion conditions can be examined via computing the argument incremental, namely, Let () =   (, ) and note that det(  + Π) ̸ = 0 is a constant.The desired assertion follows from Theorem 5.
Corollary 6 implies that the conventional Nyquist loci are merely special cases of the generalized ones.The former generally reveal partially the frequency-domain features of a concern system, which may or may not be adequate or complete in reflecting stability and robustness.

Problem Formulation with Time-Delayed State Feedback.
Let us consider the time-delayed system where the matrices and vectors are the same as those in (1).To differential-difference state-space equation (19), we introduce the state feedback where V() ∈ R  is a new reference input and   ∈ R × ,  = 0, 1, . . ., , are static feedback gains.Then, the closed-loop system can be described by and the associated closed-loop characteristic polynomial is The characteristic polynomial   (, ) for ( 21) can be rewritten as where  0 () = det(Δ(,  0 )) and Δ(,  0 ) =   −  0 +  0  0 .Clearly,  0 () is the closed-loop characteristic polynomial of the LTI system ẋ =  0  +  0  subject to the state feedback  =  0 .Hence, the stabilization problem is formulated as follows: design the static feedback gains   ∈ R × ,  = 0, 1, . . .,  such that all the roots of   (, ) = 0 possess negative real parts.

Stabilization via Time-Delayed Dynamics Cancellation.
In the control theory, stabilization in LTI systems by static state feedback has been well addressed.In view of the specific expression of the closed-loop time-delayed system (21), we see that if the time-delayed states can be cancelled by the state feedback, then the closed-loop system reduces to an LTI one, which can be stabilized by means of some pole assignment as usual.
Proof.Under the solvability assumption about   =  0   , there are   with  = 1, . . .,  such that   −  0   = 0 and thus the closed-loop characteristic polynomial   (, ) turns out to be   (, ) = det(  −  0 +  0  0 ).Note that ( 0 ,  0 ) is stabilizable.It follows that all the roots of   (, ) can be assigned to left half-plane by choosing  0 as desired.Proposition 7 is significant merely in the theoretical sense and applies to extremely special time-delayed systems.However, the time-delayed states cancellation idea is inspiring; that is, if the impact of the time-delayed states can be reduced in some sense, then stabilization of the original time-delayed system is achievable by working with a corresponding LTI system in the sense of perturbation analysis.This is what we do in the subsequent subsections.

Complex Scaling Stabilization via Pole Assignment. To facilitate our following statements, let us define
where () is a prescribed Hurwitz polynomial.Clearly, (, ) =  −1 ()  (, ).Now let us examine what happens to (, ) as the variable  travels along the Cauchy contour N. We observe the following facts: (i) If the pair ( 0 ,  0 ) is completely controllable, the eigenvalues of  0 −  0  0 can always be assigned arbitrarily on the left half-plane such that  0 () is a Hurwitz polynomial.
(ii) With respect to some given feedback gains {  } =1,..., and eigenvalues {  } =1,..., that are sufficiently far left to the imaginary axis, it is always possible to determine  0 by some poles assignment algorithm such that  0 () is a Hurwitz polynomial and Based on the above observations, we claim the following results for stabilization.Theorem 8. Consider time-delayed system (19), to which the static state feedback (20) is introduced.Let the pair ( 0 ,  0 ) be controllable.If there exists a feedback gain sequence {  } =0,1,..., satisfying (25) and  0 −  0  0 is a Hurwitz matrix (all its eigenvalues have negative real parts), then closed-loop system ( 21) is asymptotically stabilized.
Proof.Under the given assumptions, there exists a feedback gain sequence {  } =0,1,..., such that both of the above two observations are satisfied.Bearing this in mind, let () =  0 (), which is a Hurwitz polynomial according to our poles assignment with  0 .Then, we can use the following generalized locus with respect to the contour N for stability analysis Then, the stabilization is attainable if we can assert that the corresponding locus does not encircle the origin under the given {  } =0,1,..., .Or equivalently, a static state feedback (20) with {  } =0,1,..., exists such that all the roots of   (, ) = 0 are assigned to the left half-plane.
To this end, let us simply denote the th eigenvalue by with  = 1, . . ., .Next, it is also clear that, for any specific Using {  ()} =1,..., , we notice by the determinant theory that from which the following points follow immediately.
Secondly, lim →±∞ (, ) = 1 and lim ‖ 0 ‖→∞ (, ) = 1 for each fixed  ∈ N hold.This implies that, for any  ∈ N, each complex phaser (1−  ()) must be a complex vector from the origin (0, 0) to a point in some closed -related neighborhood around (1, 0), whose complex argument ∠(1−   ()), which is the angle subtended by the phaser itself and the positive real axis, satisfies This says exactly nothing but that the entire locus (, )| ∈N must be within a bounded range around the point (1, 0) that excludes the origin (0, 0), when ‖ 0 ‖ is sufficiently large and all eigenvalues of  0 −  0  0 have negative real parts.It follows that the entire locus (, )| ∈N has no encirclements around the origin.Based on the above observations and the argument principle, it follows that the closed-loop characteristic polynomial   (, ) cannot have zeros in the closed right half-plane.This completes the proof arguments eventually.Some remarks about Theorem 8 are as follows: (i) Stabilization in Theorem 5 is meant in asymptotical stability.If the state decreasing ratio is expected, the stabilization design should be constructed in exponential stability.It is simple to see that such stabilization design can be achieved by replacing the contour N with some -shifted one, that is, N+, with  > 0 being an expected decreasing ratio.
(ii) Compared to Proposition 7, the controllability assumption about the pair ( 0 ,  0 ) cannot be relaxed to a stabilizability one in general.Otherwise, some eigenvalues of  0 −  0  0 may not be able to be positioned as appropriately to satisfy the robustness inequality (25).
(iv) A stabilization design procedure is suggested as follows. Step It is simple to see that inequality ( 25) is satisfied as long as ( 31) is true.As a matter of fact, the following holds for each  ∈ (−∞, ∞): In addition, it is well known that sup ∈N {‖Δ −1 ( 0 , )‖} can be manipulated arbitrarily by using the  ∞ norm performance optimization technique [35].
The so-designed static feedback gains sequence {  } =0,1,..., has nothing to do with the time delays in the state and control input.In view of this, we say that the suggested stabilization is robust with respect to the time delays in the state and control input.(v) With some rough words and abuse of terminologies, the stabilization design consists of two design procedures: firstly, the embedding LTI differential equation ẋ =  0  +  0  is stabilized as a continuoustime subsystem; secondly, the embedding difference relations in terms of the time-delayed states are stabilized as a discrete-time subsystem.In this sense, the suggested stabilization technique is hybrid and quite intuitive.

Stabilization Interpreted with Small-Gain Theorem.
In this subsection, we show that some stabilization results similar to those of Theorem 8 follow readily by the small-gain theorem [36,37] under the inequality condition max To this end, let us note that closed-loop time-delayed system (21) can be equivalently interpreted as a feedback configuration in form of where () = ∑  =1 [  −  0   ]( −   ) is viewed as a static state feedback.Similar feedback connection remodeling in time-delayed systems can be found in [10], for dealing with absolute stability in some Lur' e problem settings.
Theorem 9. Consider time-delayed system (19), to which the static state feedback (20) is introduced.Let the pair ( 0 ,  0 ) be controllable.If there exists a feedback gain sequence {  } =0,1,..., satisfying (33) and  0 −  0  0 is a Hurwitz matrix, then the closed-loop system ( 21) is  2 -stabilized.Some remarks about Theorem 9 are as follows: (i) At first glance, the same assumptions are seemingly supposed for guaranteeing stabilization.In Theorem 8, stability of  0 −  0  0 is for the closed-loop characteristic polynomial   (, ) to be reexpressed such that generalized locus formula ( 26) is welldefined for applying the argument principle, while in Theorem 9 stability of  0 −  0  0 is for welldefinedness of the  ∞ -norm of Δ −1 (,  0 ).
(ii) Note that ( 31) yields (33) to us.Interpreting this in light of Theorem 9, one can easily see that the stabilization procedure suggested in the remarks for Theorem 8 can also be adopted for stabilization design in the  2 -stability sense.
(iii) Inequality ( 31) implies (33) readily; however, the reverse may not be true.This tells us that the smallgain condition may be less conservative than that of (31).It is stressed that the relationships between ( 25) and ( 33) are not definite.

Numerical Illustrations
where  > 0 is the time delay in the states, and  > 0 is that in the control input.

Stability Analysis When 𝜏
Varies.Now we investigate stability with  ∈ [0.1, 0.5], while  = 4 and  = 0.3 are fixed.The generalized loci are plotted in Figure 5 with  ∈ {0.1, 0.2, 0.3, 0.4, 0.5}, respectively.By Figure 5, when  ∈ {0.1, 0.2}, the loci have zero net encirclement around the origin; thus the closed-loop time-delayed system is asymptotically stable by Theorem 5; however, when  ∈ {0.3, 0.4, 0.5}, each of the corresponding loci has two net encirclements around the origin; thus the closed-loop time-delayed system is unstable.In short, the longer the time delay  in the states is, the more likely the closed-loop system tends to be unstable.

Stability Analysis When 𝜎
Varies.Now we investigate stability with  ∈ [0.2, 1.0], while  = 4 and  = 0.22 are fixed.The generalized loci are plotted in Figure 6 with  ∈ {0.2, 0.4, 0.6, 0.8, 1.0}, respectively.By Figure 6, when  ∈ {0.2, 1.0}, the loci have zero net encirclement around the origin; thus the closed-loop time-delayed system is asymptotically stable by Theorem 5; however, when  ∈ {0.4,0.6, 0.8}, each of the corresponding loci has two net encirclements around the origin; thus the closed-loop timedelayed system is unstable.In short, the time delay  of the control input may scatter in magnitude as far as stability is concerned.

Stabilization Illustrations
where  > 0 is a single time delay in the state vector.Theorem 5 shows that the concerned system itself is unstable for any  > 0. And, the pair ( 0 ,  0 ) is controllable.
Our problem here is to fix the parameters  01 ,  02 ,  11 , and  12 such that the closed-loop time-delayed system is asymptotically stable.
In the closed-loop state-space equation, let us take  11 = 0.125 and  12 = 0.Then, it follows that ‖ 1 − 0  1 ‖ = 1.4151.Furthermore, the characteristic polynomial for from which the eigenvalue formula is with  > 0 and  ≥ 0. Based on this formula, let us take (44) Then, the eigenvalues of  0 − 0  0 will be assigned at −± as expected.By numerical computation, when − ±  = −50 ± 1 are assigned by means of the feedback gain  0 = [104.50,2917.00],we can see in Figure 7 that the generalized loci for  ∈ {0.01, 0.1, 1} under the same  0 neither pass through nor encircle the origin; namely,  = 0. Hence, the closedloop time-delayed system is stabilized under the designed  0 and  1 according to Theorem 5, robustly with respect to  ∈ [0.01, 1].
To complete our understanding about the suggested stabilization approach, let us observe the generalized loci in Figure 8 where the eigenvalues of  0 −  0  0 are specified in four different cases, when  = 0.2 is fixed.Clearly, in each of the last three cases the closed-loop system is stabilized, while stabilization is not attained in the first case.By the loci of Figure 8, we also see that the more left to the imaginary axis the expected eigenvalues of  0 −  0  0 are specified (i.e., ‖ 0 ‖ is taken bigger), the more right to the imaginary axis the corresponding generalized loci are situated.This verifies not only the proof arguments for Proposition 7 about loci positioning, but also stability robustness with respect to ‖ 0 ‖.
In Figure 9, time responses under the static state feedback gains  0 adopted in Figure 8 are plotted.The convergence tendency of the time response curves coincides with the stability assertions that we conclude from Figure 8.

Conclusions
In the study, we claim several complex scaling stability criteria for internal stability in linear time-delayed continuous-time systems, as summarized in Theorem 5 and its corollary.Possible advantages of the criteria include the following: (a) stability conditions are necessary and sufficient, delay-dependent, both graphically, and numerically implementable; (b) the criteria involve no open-loop characteristic roots distribution and interconnection structural facts, which are hard to know generally in time-delayed systems; (c) the locus encirclement and orientation can be self-defined after the locus is ready; (d) the criteria hold true for internal stability.Another contribution of the paper is the stabilization design in timedelayed systems, namely, Theorems 8 and 9, again based on the complex scaling technique.
Although the results are stated only for linear timedelayed systems of retarded type, it is trivial to extend the methodology for time-delayed systems of neutral type.Due to page limitation, all details are omitted.Furthermore, how to extend the complex scaling approach to other complicated and hybrid systems, for example, linear time-varying systems and sampled-data systems with time-delayed plants, wherever some return difference relationships can be retrieved, is one of our subsequent topics.Indeed, latest results were reported in [26][27][28] in various types of complicated systems as reviewed in the Introduction.
(b), the time responses for  ∈ {−4, −2} diverge as time grows.Stability/instability of the closed-loop system under  ∈ [−4, 4] can also be examined by Corollary 6.The simulation results are plotted in Figure 4. Clearly, the locus encirclements in terms of Corollary 6 completely coincide with those in terms of Theorem 5.
A self-evident corollary of Theorem 5 is as follows.Corollary 6. Suppose that the closed-loop time-delayed system Σ  is well-posed.Also assume that the open-loop system is asymptotically stable.Then, Σ  is asymptotically stable, if and only if the locus det(  + ()(, , , ]))| ∈N vanishes nowhere, and its number of clockwise encirclements around the origin (0, 0) is equal to that of counterclockwise ones.Proof.Since the open-loop system is asymptotically stable, it follows that the open-loop characteristic polynomial det(Δ  (, ))det(Δ  ()) =   (, ) is a Hurwitz polynomial.