Delay-Independent Stability Test for Systems With Multiple Time-Delays Ismail Ilker Delice and Rifat Sipahi, Associate Member, IEEE

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 4, APRIL Delay-Independent Stability Test for Systems With Multiple Time-Delays Ismail Ilker Delice and Rifat Sipahi, Associate Member, IEEE Abstract Delay-independent stability (DIS) of a general class of linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in the entire delay-parameter space. Stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays. We build an analytical approach that requires the inspection of the roots of finite number of single-variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays, excluding infinite delays. The approach enables to test the necessary and sufficient conditions of DIS of LTI- MTDS, technically known as weak DIS, as well as the robust stability of single-delay systems against all variations in delay ratios. The proposed approach, which does not require any parameter sweeping and graphical display, becomes possible by establishing a link between the infinite spectrum and algebraic geometry. Case studies are provided to show the effectiveness of the approach. Index Terms Algebraic geometry, delay-independent stability (DIS), iterated discriminants, multiple time-delay systems. I. INTRODUCTION T IME-DELAY systems (TDS) arise in many applications from diverse areas such as economy, biology, machine tool chatter, population dynamics, communication systems; see [1] [4] and the references therein for other applications. Investigation of asymptotic stability of TDS with respect to delays is essential; however, asymptotic stability analysis is a nontrivial task even for linear time-invariant (LTI) systems, primarily due to the infinite-dimensional nature of TDS [3] [6]. The following state-space representation, which is the general class of LTI multiple time-delay systems (MTDS), covers all such systems considered earlier: where and are constant system matrices, is the state vector, and are nonnegative delay parameters. The system in (1) is of retarded class since the delays do not pervade through the highest order derivative of the states. Stability analysis of this system requires investigating the eigenvalues of (1) that are on the imaginary axis of Manuscript received April 20, 2010; revised November 24, 2010; accepted August 30, Date of publication September 22, 2011; date of current version March 28, This work was supported in part by an award from the National Science Foundation ECCS A preliminary version of this work was presented in part at the IFAC Time Delay Systems Workshop 2009 in Sinaia, Romania. Recommended for publication by Associate Editor H. Zhang. The authors are with the Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA USA ( delicei@coe.neu.edu; rifat@coe.neu.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TAC (1) the complex plane for some critical delay values [7]. It is these eigenvalues which may cross the imaginary axis at and may cause stability reversals/switches as is perturbed [5]. The frequency parameter indicates the pathways of the eigenvalues across the imaginary axis. In this sense, the set of all nonnegative values, called the crossing frequency set, carries key information about the stability and spectral properties of (1). In the delay-dependent stability case (see the stability maps found by [5], [6], [8]), we have, that is, a stable system will become unstable for some delay values. Since the finite upper-bound of is known to exist [9], one can sweep in a range starting from zero up to a conservative upper-bound in order to solve all the eigenvalues of a TDS [10]. Although this may need graphical tools, frequency sweeping methodology [11] is applicable to robustness analysis [12] and to extracting the stability features of MTDS in 2-D ( ) [8], [35], [47] and 3-D ( ) delay space [13], [14]. When ; however, the system s stability/instability becomes delay-independent. Many papers are published along these lines, where delay-independent stability (DIS) sufficient [11], [15], and necessary and sufficient conditions are proposed [12]. The starting point in many studies is that TDS cannot possess imaginary eigenvalues with respect to the entire delay-parameter space. When, graphical display in these analyses is instrumental in order to easily verify, by sweeping, whether or not larger values reveal any eigenvalue solutions. There are other techniques to test DIS of TDS as well. DIS conditions are studied in [4] for subclasses of (1). In another study, one of the most complicated MTDS is studied for robustness via frequency sweeping [12], but the characteristic function treated in the cited work does not cover the general problem in (1). Furthermore, the studies in [16] [23] are applicable for only single-delay cases ( ), and [24] [26] are feasible only for two-delay cases ( ). When ; [16] studies the DIS problem by means of two-variable zero criterion, which is limited to single-delay problems. Since some trigonometric identities are utilized in [16] and [17], these methods remain restricted to scalar TDS ( ), as recognized in [17], see also [19]. Moreover, the resultant theory is applied to the DIS problem in [18], [19], followed by [20] and [21] which use a similar logic, but a different set of two polynomial equations for the resultant computation. Procedures in [18] [21] are applicable to TDS with only single time-delay, with no restriction on system order. Furthermore, [22] transforms frequency sweeping conditions in [27] to easily testable algebraic conditions by utilizing the resultant theory. These conditions are, however, valid for single-delay cases. Finally, [23] concludes DIS property of TDS, but with a single-delay, /$ IEEE

2 964 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 4, APRIL 2012 based on the roots of a polynomial constructed by utilizing a bilinear transformation. When ; [24] considers a specific second-order damped vibration problem, which has only two time delays. The techniques in [25] and [26] are also limited to a specific dynamic system only with two delays. In all the cited papers, extensions to cases is restrictive due to two main reasons: 1) the DIS test for is an NP hard problem since each delay needs to be treated as an independent parameter [28]; and 2) the number of available equations to be solved for DIS analysis is less than the number of unknowns in the respective analysis. While many linear matrix inequality (LMI)-based approaches, which can be conservative, exist for DIS test [4], [10], [29], recent results show that non-conservative LMI techniques can also be constructed to test DIS of LTI systems with a single delay [30], [31]; see also [32] for results using LMIs in the context of robust control. To the best of our knowledge, existing DIS tests using LMIs for systems with more than one delay are, however, based on sufficient conditions [33]. In this paper, we construct a DIS test that is based on necessary and sufficient conditions, and that is not limited by the number of delays. The results are valid for the general class of systems represented by (1), but studies on time-varying delays, see for instance [34], are kept outside the scope of this paper. The main objectives of this paper are to compute the upper bound and lower bound of, and to test, based on necessary and sufficient conditions, whether or not the system in (1) is DIS. Technically speaking, the DIS analysis here refers to the weak case, but not the strong case, that is, delays at infinity are not considered here. This assumption, however, does not lose the essence of practical control problems, where delays remain finite. Weak DIS, hereafter called DIS, means that given and, system in (1) is asymptotically stable in the entire -dimensional delay parameter space, excluding the infinite delays [3]. DIS holds when the following two conditions are satisfied, 1) all the eigenvalues of have negative real parts, and 2) with respect to the entire delay parameter space is empty,. We start by considering the DIS test of (1) as a problem of existence/absence of the upper bound and the lower bound of. This problem formulation leads to our iterated discriminants development, which eventually yields a finite number of single-variable polynomials. The roots of these polynomials are directly related to the necessary and sufficient conditions of DIS of (1), or equivalently, to the absence of the bounds and. Our approach is practical, efficient, and inclusive, and it does not impose any limitations on the number of delays, system order, and on the ranks and the entries of system matrices in (1). To the best of the authors knowledge, the problems addressed in this paper have not been resolved in the open literature due to the assumptions and simplifications involved in the existing approaches. The proposed DIS test can verify whether or not the stability of (1) is insensitive to multiple delays. This is especially useful in control problems, where delays are unknown and stability is required regardless of the amount of delays, as a worst case scenario. Furthermore, if the bounds and exist, the test precisely detects them. This serves for two purposes; one reveals that the system is actually delay-dependent stable/unstable, and one can use these bounds in order to analyze stability with respect to delays, see the frequency-sweeping-based techniques cited in the introduction. This suggests improvement over the existing delay-dependent stability techniques that numerically sweep in conservatively large ranges. It is crucial to note here that the proposed DIS test is for analysis purposes, leaving its utilization for control synthesis to future work. In this regard, analyzing whether or not (1) is DIS does not bring any conservatism. Notations used in the text are standard. We have,,, and for the set of real numbers, the set of nonnegative real numbers, the set of positive real numbers, and the set of natural numbers (including zero), respectively. Open right half and imaginary axis of complex plane are represented by and, respectively, and. We use for the Laplace variable; for the real and for the imaginary part of. stands for the resultant of multivariate polynomials and with eliminating, where, and. The delay vector is denoted by ; is the pseudo-delay vector [4], [6], [10]; is the commensurate degree of, and indicates a fixed value of a variable. Arguments are omitted for easier reading when no confusion occurs. II. PRELIMINARIES A. On Delay Systems and Their Imaginary Spectra Characteristic function of the system in (1) is given by where are polynomials in terms of with real coefficients, and. In order to account for system (1) being of retarded class, we have that the largest power of is in, and hence does not multiply any exponential functions, [5]. The characteristic function (2) possesses infinitely many zeros due to the presence of transcendental terms. For a given, MTDS (1) is asymptotically stable if and only if all these zeros have negative real parts, that is,. Examination of whether or not (1) is DIS requires to verify the asymptotic-stability condition for all. Obviously, this verification is impossible by numerically sweeping and.in order reduce the complications, one can exploit the continuity property of the zeros of (2) on, which holds since (1) is of retarded class [7]. With the continuity property at hand, it is known that stability of (1) may change only when,, is a zero of (2). All nonnegative values, where is a zero of (2) for some nonnegative delays, define the crossing frequency set [1] [6] (2) for some (3) Remark 1: Weak and Strong Delay-Independent Stability [12]. For the system in (1) to be delay-independent stable, it is necessary that the system is asymptotically stable for zero delays [3]. Under this condition, can satisfy (2) only when some delays approach infinity. Such a possibility is, however, excluded (or included) in the analysis of weak (or

3 DELICE AND SIPAHI: DELAY-INDEPENDENT STABILITY TEST FOR SYSTEMS WITH MULTIPLE TIME-DELAYS 965 strong) delay-independent stability. Since we ignore here the infinite delays for practical purposes, we restrict ourselves to in the remaining of the text without loss of generality [12], [36]. Some stability analysis results inspire the study here, see [6], [17], [18], [23], [35], [37], which convert the infinite-dimensional characteristic function (2) to a finite-dimensional characteristic function with continuous coefficients. This is achieved with the Rekasius transformation [52] which is exact for with the existence of a back-transformation from to, see the references cited above. Upon substitution of (4) into (2) and with a manipulation to remove the fractions, the transformed characteristic function is obtained as which is a complex function in terms of and. At this point, define, as was done in [6], the crossing frequency set of (5) as (4) (5) for some (6) Corollary 1 ([6] [52]): The identity holds. Remark 2: Since holds, we seek for by studying from the algebraic function (5) instead of searching for directly from the transcendental function (2). This was also done in similar forms in [17], [18], [23], and [37]. Lemma 1 ([38]): Delay-independent stability of the singledelay TDS is robust (well-posed) against all perturbations in the delay coefficients if and only if the MTDS in (1) is delay-independent stable. Lemma 1 states that testing the robustness of DIS property of (7) against all delay ratios determined by, with, is equivalent to testing the DIS property of MTDS in (1). Excluding cases, the contributions of this paper also address the robust stability of (7) against all delay perturbations. B. On Resultant Theory and Discriminant of Polynomials Consider the two multivariate polynomials in terms of with real coefficients (7) (8) (9) where and have positive degrees in terms of, and. The resultant of and with respect to is defined by (10) which is the determinant of the well-known Sylvester matrix [39]. Theorem 1 ([40]): If is a common root of (8)-(9), then for. Conversely, if for some, then at least one of the following four conditions holds: 1) there exists that is a common root of both (8) and (9); 2) leading coefficients of both and vanish, ; 3) all the coefficients in vanish, ; 4) all the coefficients in vanish,. Definition 1: a) Let,, then the discriminant of the polynomial with respect to is defined as (11) b) Let,, then the discriminant of the polynomial with respect to and is defined as (12) Polynomial is treated as a univariate polynomial in (11) and a bivariate polynomial in (12) [39], [41]. Readers are referred to the Appendix for a geometric interpretation of discriminant. III. MAIN RESULT We first develop a theoretical framework that enables practical and direct computation of the finite lower and upper bounds of. These bounds are crucial as they determine the range of the frequency that delineates the stability transitions of (1) with respect to the delay parameters. Obviously, absence of such bounds will help assess the DIS property of (1). Recall that studying requires studying the zeros of (5), which can be rewritten as (13)

4 966 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 4, APRIL 2012 where and are the real and imaginary parts of (5), respectively. When (13) holds, and are concurrently zero. It can be shown that these equations are in the following form: (14) is necessary that points of. For the minima/maxima of to exist, it. From [42], for the regular,wehave (17) and (15) Notice the difference between versus, and versus by comparing (8)-(9) with (14) (15). The commensurate degree denoted by here is also known to be the largest power of in both (14) and (15) [6]. Moreover, since the functions in (2) and (5) are identical, respectively, when and [6], and since it is necessary that (1) with is asymptotically stable for DIS (see Remark 1), we can conclude that and do not simultaneously vanish for all when. We also assume that, without loss of generality, and do not have common factors, which can be separately treated. Furthermore, in (14) (15), and terms can either vanish (become identical to zero) or become zero for some ( ) values. With this understanding, we maintain the highest power of as in the summations. Next, we utilize the resultant theory to eliminate from the two multivariate polynomials and [39]. A -order Sylvester matrix is constructed via (10), and its determinant is a function of and. Remark 3: The singularity of Sylvester s matrix, i.e., the resultant being zero,, is a necessary condition for and to have common roots. Hence, studying the solutions of is adequate for studying the solutions of. This is exactly what we do in order to benefit the advantages of the resultant theory. Based on the implicit function theorem [42], for the regular points of the resultant and discriminant expressions calculated below, is differentiable with respect to, because the partial derivatives of these expressions are multivariate polynomials, and hence continuous with respect to [43]. The remaining few singular points, if any, should also be considered [44], as explained in the Appendix. With this knowledge, we now provide the theorem that reveals the exact positive lower and upper bounds of. Theorem 2: Minimum and maximum positive real zeros of the iterated discriminants (16) that correspond to are the exact positive lower bound and the exact upper bound of the crossing frequency set, respectively. Proof: As per Remark 3, all that give rise to solution in (13) also satisfy for some, where a mapping to exists through (14) (15). It is therefore adequate to seek and by studying Since it is necessary that, for (17) to hold, the new equation should also hold as for regular points 1. One can now search for the common solutions between and. Among these solutions lie and for some. For this search, one can eliminate by constructing via (10). With this, and solutions are embedded into the solutions of in domain, with mappings to and domains via, and. If and exist, then it is also necessary that, which can be analyzed with the same logic used above in (17). The repetition of the same procedure until only the parameter remains and all are eliminated leads to the following univariate polynomial in which is (16) as per Definition 1(a). The minima/maxima, and, if they exist, are among the roots of. For each real root of, there exists found via sequential back-substitutions into single-variable polynomials, ;, ; ;. The minimum and maximum positive real zeros of the polynomial that correspond to are the exact positive lower bound and the exact positive upper bound of, respectively. Note that (14) (15) are interrelated and can be expressed as. This new equation can be used to start the elimination procedure in Theorem 2, instead of starting with. Nevertheless, this choice leads to higher powers of in (16) and is therefore not preferable from computational efficiency point-of-view. Furthermore, in case, this can be detected by developing arguments along the lines of [36]. We state that Theorem 2 treats both the regular and singular points of the resultants (see also the Appendix), except when the singular points arise from repeated factors of the arguments of the resultants. That is, so long the arguments of the discriminants do not have repeated factors, Theorem 2 is applicable since the parametric discriminant operation does not exclude the singularity points [45]. Furthermore, the objective here is to detect and regardless of identifying whether or not the points are singular. For this objective, one only needs to check if the roots of have a mapping in. We next study how Theorem 2 needs to be modified in the case when the arguments of the discriminants have repeated factors. 1 Notice that R = =@T = 0are also necessary conditions for singular points to exist. Hence, proceeding with the common solutions of these two equations does not exclude the singular points from the theorem, permitting us to capture also the singular points as candidate extrema points.

5 DELICE AND SIPAHI: DELAY-INDEPENDENT STABILITY TEST FOR SYSTEMS WITH MULTIPLE TIME-DELAYS 967 A. Discriminant of With Repeated Factors When the arguments of the discriminants have repeated factors, the iterated discriminants treatment in Theorem 2 needs a modification as explained next. Lemma 2 ([46]): Let,, then the discriminant in (12) is identically zero if and only if has a repeated factor. Lemma 2 states that partial derivatives and will make the discriminant defined in Definition 1(b) vanish if and only if has repeated factors. Let us investigate how this information affects in Definition 1(a), which is used iteratively in Theorem 2. In general, we have, where, the polynomials and carry the variable, and the polynomial has no variable. It then follows that both and have a common factor of. Therefore, all the roots of the repeated factor make the partial derivatives vanish. These roots are also some of the singular points of [42]. It is now easy to see that the discriminant in Theorem 2 also becomes identically zero (always vanishes) due to the repeated factor ([45. p. 142]). When this discriminant becomes identically zero, the subsequent discriminant in Theorem 2 cannot be calculated. This issue can be resolved with the following modification. The repeated factor is eliminated and a modified resultant (18) is to be found first. One should proceed with in order to execute the remaining steps of Theorem 2. Notice that this manipulation does not lose the insight of the problem, but it carefully separates the multiplicity of the roots arising particularly from repeated factors, incorporating them with multiplicity one into the discriminant calculations in Theorem 2. Since is square-free, that is, it does not have repeated factors, the discriminant in Theorem 2 can be easily calculated. Once the analysis provided in the proof of this theorem is complete, one can revisit to separately identify the multiplicity of the roots. This procedure is demonstrated over two explanatory examples next. 1) Explanatory Example 1: Consider the characteristic function of a MTDS given by Using (19), the equation corresponding to (13) becomes (19) (20) We first eliminate by calculating the resultant, which is (21) Notice that has a repeated factor in terms of variable, thus the discriminant of by eliminating,, is identically zero not permitting us to solve for. Therefore, we need a modification as discussed above. By using a symbolic manipulator, we modify the resultant as explained in (18). It is then possible to compute, which becomes (22) where roots are neglected, see Remark 1 and Remark 4. It is found that the univariate polynomial has three positive real zeros, , , and By back substitution of the minimum and the maximum of these roots into, and the characteristic (20), we compute the corresponding,, which are found to be real numbers. Therefore, it is concluded that and. The multiplicity of the roots are revealed by inspecting the roots of, which really show multiple roots in for both and. The results are as follows 2 : (1.7304, , ), (1.7304, , ), (1.9179, , ), (1.9179, , ). In this example, there exists an easier procedure that does not need a resultant modification. Since the order of elimination in (16) is immaterial, it is possible to eliminate before eliminating by calculating. This way the repeated factor does not cause a problem in eliminating in the discriminant calculation, since multiple roots in the repeated factor do not arise in domain. Without the need for identifying the repeated factors, we find directly as [1.7304, ]. One can now use this range of and any technique compatible with frequency sweeping in order to extract the stability maps in domain by sweeping the frequency from to Notice that we reveal only and, i.e., the precise lower and upper bounds of the crossing frequency set, but this does not claim that one will find feasible solutions in (2) for all. 2) Explanatory Example 2: Although the factor of is not repeated, it may also be eliminated. This elimination can be done only if is a univariate polynomial in terms of. This is because the derivatives of resultants with respect to are never calculated in Theorem 2, and therefore the resultant can always be modified without affecting the results in the subsequent discriminants. In such cases, the roots of should be separately studied. Moreover, the roots of may satisfy 2 Four-digit precision is used for numerical values in order to conserve space.

6 968 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 4, APRIL 2012 either one of the arguments of, i.e., either or polynomials (corresponding to Conditions (III)-(IV) of Theorem 1), or satisfy both and (corresponding to Condition (I) of Theorem 1). Consider the characteristic function of a MTDS given by Equation (13) becomes (23) (24) We first eliminate by calculating the resultant, which is and next find (25) (26) Notice that factor is arising from the non-repeated factor of ; hence, the subsequent discriminant is not identically zero, and the modified resultant is not needed. It is preferred, however, to proceed by eliminating in order to ease the numerical computations. Proceeding further, we calculate as (27) where roots are neglected, see Remark 1 and Remark 4. The smallest and the largest real zero of (27) are and , which can be confirmed to correspond to admissible satisfying,, and (24). Moreover, positive real zeros of (26) should be separately studied. There is only one such root,, which makes all the coefficients of in (24) zero. It can be confirmed that for, also becomes zero whenever. That is, is a member of, and multiple roots of (24) occur in domain where. Since, we conclude that the lower and upper bounds of are respectively and Remark 4: For the system in (1) to be DIS, it is necessary that the delay-free system is asymptotically stable [3]; hence, can be a characteristic root only when some, recall Remark 1. Moreover, because is not a part of the weak DIS analysis, roots of can be ignored. Furthermore, and has a mapping in domain only when [36], but in the converse, can correspond to either a finite value of or. The case with finite is detectable from the roots of, and the case with can be ignored since it corresponds to, which is not considered in the weak DIS analysis. Notice that Condition (II) of Theorem 1 requires that a leading coefficient to vanish, that is, the parameter multiplying this coefficient becomes unbounded,. In light of the above discussions, such cases are taken care of by the iterated discriminants if is finite, and solutions can be disregarded in the context of weak DIS. Finally, we note that all the conditions of Theorem 1 are covered in the DIS analysis since the calculations are performed by studying the zeros of the resultants. B. Delay-Independent Stability Analysis for MTDS The following theorem is the main result of this paper. Theorem 3: The MTDS in (1) is delay-independent stable in the entire -D delay domain if and only if the following two conditions are satisfied simultaneously: The polynomial in Theorem 2, with modified resultants when necessary, has no positive real zeros corresponding to. The matrix is Hurwitz stable. Proof: Excluding, Theorem 2 and the procedure of modified resultants together form the first condition of the theorem guaranteeing that is empty set. Since generates the stability switches/reversals, the condition does not yield such switches, and vice versa, in the entire delay parameter space. As a result, if there exist no stability switches, then the entire -D delay domain exhibits the delay free system s stability behavior, which is stable by construction. In order to make the DIS approach computationally more tractable, developments in computer algebra on the computation of resultants are extremely important. Hence, improvements in this field favor the feasibility and applicability of our approach. Interested readers are referred to [48] and [49] for details on computer algebra. Lemma 3: The polynomial is an even and real polynomial. Proof: is a polynomial in terms of the coefficients of and. Ignoring its factor, is an even polynomial as is. By inspection of the product formula of the resultant [39] where are zeros of, one sees that the resultant yields either an even or an odd polynomial with respect to.if is an odd polynomial, it can be converted to an even polynomial by eliminating the factors of. In this way, we again obtain an even polynomial with respect to, excluding roots as per Remark 4. Moreover, the power of remains even throughout the discriminant steps since the multiplication of two even polynomials is also an even polynomial. Hence, is an even polynomial with respect to. This polynomial is also a real polynomial, since the coefficients of the resultants are all real. Remark 5: Since is an even polynomial with respect to, one can alternatively study the existence of the positive zeros of this polynomial. The number of positive real zeros of even polynomials can be found via a procedure proposed in [50], and thus without numerically solving, one can detect whether or not has positive zeros. The count of the number of solutions can be used in place of the first condition of Theorem 3. If this count is zero and the delay-free system is asymptotically stable, then the system in (1) is guaranteed to be delay-independent stable. If this count is not zero, one should use solutions to check whether or not the corresponding

7 DELICE AND SIPAHI: DELAY-INDEPENDENT STABILITY TEST FOR SYSTEMS WITH MULTIPLE TIME-DELAYS 969 solutions are real. If no real solutions exist, one can still claim DIS of the system. Remark 6: It is noted that are kept finite here, yet as large as possible, with the pre-assumption that in a practical control system, is unrealistic. On the other hand, the framework proposed in this paper could be extended to studying cases. This would require a separate treatment of the problem, which is nontrivial to solve. The main reason is because all the infinitely many delay ratios,,,, should be considered when. In this sense, when all or some of the delays converge infinity, one cannot simply assume that these delays become equal to each other and that the problem has fewer number of delays. Indeed, the problem is still a multiple-delay problem, as we explain next. The extensions to studying DIS for infinite delays could be possible by noticing that when,wehave and, yet can be finite or infinite, see [36] and Remark 4. For instance, if all converge infinity, we then let in (4), as was done in [36] only for,to obtain (28) which one should analyze for solutions in the limit. This analysis is obviously cumbersome in -dimensional parametric space of,, although converge infinity. We leave this part of the study to future work, as we mainly focus here on establishing the ground rules of utilizing algebraic geometry and the iterated discriminants concepts. Remark 7: The DIS test presented in this paper applies to the most general LTI problem in (1) with necessary and sufficient conditions. Extensions of this test to slowly time-varying delays could be possible following the author s previous work [51], where connections between frequency domain and time domain are established. This connection allows reflecting stability analysis in the time domain to an equivalent analysis in frequency domain, but only with sufficient stability conditions. C. DIS Test With Respect to the Number of Delays As reviewed in the Introduction section, DIS test using frequency domain and time domain techniques can be achieved with necessary and sufficient conditions for systems with a single delay and two delays. The DIS test developed in this article handles these cases as follows. When, wehave only two unknowns and in and (29) (30) In this case, upper and lower bounds of are found as follows. One eliminates from (29) and (30) via a resultant operation, which then leads to defined in Theorem 2. The claim in the theorem follows similarly; one should investigate the zeros of and the corresponding common solutions in order to identify whether or not (1) is DIS. As the nature of the problem, discriminant operation is not necessary when, see Case 1 in Section IV. When, we have three unknowns in (14) and (15); they are,, and. The elimination of using the resultant leads to a polynomial with two unknowns and.at this point, studying the extrema points of is beneficial. For to exhibit an extremum, it is necessary that the partial derivative of with respect to vanishes. This condition produces a new polynomial as explained in Theorem 2. One now has two equations ( and ) and two unknowns ( and ), and thus can use another resultant operation to eliminate. The elimination of a variable from a polynomial and its derivative with respect to that variable is called discriminant operation. The discriminant operation creates the polynomial, which is used to test DIS as explained above, see also the explanatory examples in this section. The DIS test presented in this paper similarly extends to the cases with, with the difference that new polynomials are generated by introducing additional conditions for partial derivatives of with respect to to vanish, and all these polynomials are used in sequential resultant operations that generate the iterated discriminants concept. This is exactly where the novelty of the proposed DIS test lies; it is able to generate sufficiently many new polynomials and process them using the salient features of algebraic geometry so that all the variables can be eliminated using resultants, and only remains in the polynomial. Once this final polynomial is obtained, testing DIS becomes as easy as inspecting the roots of univariate polynomials. IV. CASE STUDY Two case studies are presented to test the DIS property of TDS. In Case 1, we investigate the robustness of the DIS property against perturbations in delay ratios. In achieving this, both single-delay dynamics and multiple-delay dynamics are tested for DIS property. In Case 2, we consider three-delay dynamics with a commensurate delay and a delay cross-talk term. A. Case 1 1) DIS Test: Consider the single-delay TDS ( )governed by (31) The eigenvalues of the delay-free system, which are and, have negative real parts and thus the delay-free system is asymptotically stable. The characteristic function of the system in (31) is (32)

8 970 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 4, APRIL 2012 The procedure to test the DIS property of the characteristic function (32) is as follows. First, the Rekasius substitution in (4) is used in order to convert (32) to (5) for. Then, the parameter in (5) is eliminated using the resultant theory as shown in Theorem 2. The elimination leads to a univariate polynomial which is given by excluding solutions. In the above equation, all s are real constants and omitted for brevity. We solve for the zeros of and find that none of these zeros are positive real, i.e.,. As per Theorem 3, it is concluded that the TDS in (31) is delay-independent stable. 2) Robustness of DIS Property Against Perturbations in Delay Ratios: We next test the robustness of the DIS property of (32) against perturbations in delay ratios in (7) of Lemma 1. To do this, as instructed in [38], the terms,, and in (32) are replaced by,, and, respectively, where. We now have a characteristic function of a MTDS with three delays: Fig. 1. (a) 3-D figure of F (!; T ;T )=0. (b) 2-D figure of F (!; T ;T )= 0 on the! 0 T plane. Following the same procedure as in Case 1, we find that has no positive real zeros, that is,. Since the delay-free system is asymptotically stable, it is concluded from Theorem 3 that the MTDS represented by (34) is delay-independent stable. 2) Computational Efficiency: The computation time to test DIS of the MTDS represented by (34) is approximately 0.6 seconds. The DIS approach follows from Theorem 2 and leads to (33) excluding roots. In the above equation, all and they are omitted for conciseness. It is verified that has no positive real zeros, indicating that. Since the delay-free system is asymptotically stable, it is concluded as per Theorem 3 that the MTDS in (33) is delay-independent stable. As per Lemma 1, it is concluded that the DIS property of (32) is robust (well-posed) against all perturbations in delay ratios. 3) Computational Efficiency: It is remarked that the computation times for testing the DIS property of (31) and the MTDS represented by (33) are approximately seconds and 0.35 seconds, respectively. The test is satisfactorily fast possibly because it does not require any hand calculations, parameter sweeping, and graphical displays. B. Case 2 1) DIS Test: The DIS property for system (33) can also be checked via existing frequency-domain tools. We now present an example, which is slightly different than (33), yet extremely difficult to treat with the existing DIS test tools cited in Section I. We multiply and terms in (33) by and, respectively, and obtain the following characteristic function: V. CONCLUDING REMARKS An approach for revealing the exact positive lower and upper bounds of the crossing frequency set of the most general linear time-invariant multiple time-delay system (MTDS) is developed. The approach also captures the weak delay-independent stability (DIS) properties of such systems and tests the robust stability (well-posed) of single-delay systems directly, without sweeping any parameter and using graphical display. To achieve this with necessary and sufficient conditions, a connection between polynomials, DIS, and transcendental functions is established for the first time via the iterated discriminants in algebraic geometry. We demonstrate the effectiveness of the approach in several case studies, which are difficult, if not impossible, to analyze through the existing practice. APPENDIX A. Geometric Interpretation of Discriminant in a 3-D Topology Let there be a polynomial that implicitly depends on three variables. We wish to find the maximum/minimum of satisfying for some. Since is implicit, it is not possible to solve from ; however, we can visualize this polynomial in domain as shown in Fig. 1(a). In order to assist the reader, Fig. 1(b) is provided to show the view of from plane. If exhibits an extremum in domain, then it is necessary that and. Let us focus on condition. This condition can be formulated using, paying attention to singularities. The regular points of satisfy (34)

9 DELICE AND SIPAHI: DELAY-INDEPENDENT STABILITY TEST FOR SYSTEMS WITH MULTIPLE TIME-DELAYS 971 Fig. 2. Discriminant of F (!; T ;T )=0with respect to T, D (F )=0. Hence, can be alternatively studied with. For the singular points of,wehave and. Such points can also be eligible to be one of the extrema points [44]. However, regardless of being regular or singular, for the extrema points to exist, it is necessary that and. At this point, one can eliminate from the last two equations using the resultant, which is called the discriminant of by eliminating,. Geometrically speaking, is a curve in domain, and all the critical points of the surface are among those points on these curves. These critical points are the projections of the tangent points and singular points of. Here tangent points are all those points at which lines drawn parallel to axis become tangent to the surface. In other words, among all points satisfying are those that are candidates for to exhibit an extremum, compare Figs. 1(b) and 2. With a similar logic, one can next eliminate by computing, which becomes a polynomial in terms of only. The zeros of are, which are candidate values, where could make an extremum. The existence of the extremum can be checked by confirming if maps to numerical values via back substitutions into the polynomial pairs forming the resultants. Finally, it is noted that singularity points of and can be identified, although this is not necessary in the process of detecting the extrema points. The singular points of satisfy both and, while the singular points of satisfy both and. The example presented above explains visually the concepts of discriminant using a 3-D topology. 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His research interests include delay-dependent and delay-independent stability analysis of multiple time-delay systems, synthesizing structural controllers for delay-independent stability, model predictive control, fuzzy logic control, and neural networks. Dr. Delice is the recipient of the 2010 Ferretti Academic Excellence Award of the Department of Mechanical and Industrial Engineering, Northeastern University. Rifat Sipahi (A 07) received the B.Sc. degree in mechanical engineering from Istanbul Technical University, Istanbul, Turkey, in 2000, and the M.Sc. and Ph.D. degrees in mechanical engineering from the University of Connecticut, Storrs, in 2003 and 2005, respectively. He joined the Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, as an Assistant Professor in 2006, and since then he has been directing the Complex Dynamical Systems and Control Laboratory (CDSCL). Dr. Sipahi was awarded the Chateaubriand Postdoctoral Scholarship of the French Government to conduct research at Universite de Technologie de Compiegne, France, during He is the recipient of a 2011 DARPA Young Faculty Award. He is a member of ASME Dynamic Systems and Control Division and IEEE. His research interests include system level approach to understanding the interconnections among delayed dynamical systems, stability, and network/topology of coupled systems arising in engineering, and physics problems.